id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
dict |
|---|---|---|---|---|---|
1007.1417
|
CALT 68-2797
Unitary Evolution and Cosmological Fine-Tuning
Sean M. Carroll and Heywood Tam
California Institute of Technology
seancarroll@gmail.com, heywood.tam@gmail.com
###### Abstract
Inflationary cosmology attempts to provide a natural explanation for the
flatness and homogeneity of the observable universe. In the context of
reversible (unitary) evolution, this goal is difficult to satisfy, as
Liouville’s theorem implies that no dynamical process can evolve a large
number of initial states into a small number of final states. We use the
invariant measure on solutions to Einstein’s equation to quantify the problems
of cosmological fine-tuning. The most natural interpretation of the measure is
the flatness problem does not exist; almost all Robertson-Walker cosmologies
are spatially flat. The homogeneity of the early universe, however, does
represent a substantial fine-tuning; the horizon problem is real. When
perturbations are taken into account, inflation only occurs in a negligibly
small fraction of cosmological histories, less than $10^{-6.6\times 10^{7}}$.
We argue that while inflation does not affect the number of initial conditions
that evolve into a late universe like our own, it nevertheless provides an
appealing target for true theories of initial conditions, by allowing for
small patches of space with sub-Planckian curvature to grow into reasonable
universes.
###### Contents
1. 1 Introduction
2. 2 The Evolution of our Comoving Patch
1. 2.1 Autonomy
2. 2.2 Unitarity
3. 3 The Canonical Measure
4. 4 Minisuperspace
1. 4.1 Canonical scalar field
2. 4.2 Scalar perfect fluid
3. 4.3 The flatness problem
4. 4.4 Likelihood of inflation
5. 5 Perturbations
1. 5.1 Description of perturbations
2. 5.2 Computation of the measure
3. 5.3 Likelihood of inflation
6. 6 Discussion
7. 7 Appendix: Eternal Inflation
## 1 Introduction
Inflationary cosmology [1, 2, 3] has come to play a central role in our modern
understanding of the universe. Long understood as a solution to the horizon
and flatness problems, the success of inflation-like perturbations (adiabatic,
Gaussian, approximately scale-invariant) at explaining a multitude of
observations has led most cosmologists to believe that some implementation of
inflation is likely to be responsible for determining the initial conditions
of our observable universe.
Nevertheless, our understanding of the fundamental workings of inflation lags
behind our progress in observational cosmology. Although there are many
models, we do not have a single standout candidate for a specific particle-
physics realization of the inflaton and its dynamics. The fact that the scale
of inflation is likely to be near the Planck scale opens the door to a number
of unanticipated physical phenomena. Less often emphasized is our tenuous grip
on the deep question of whether inflation actually delivers on its promise:
providing a dynamical mechanism that turns a wide variety of plausible initial
states into the apparently finely-tuned conditions characteristic of our
observable universe.
The point of inflation is to make the evolution of our observable universe
seem natural. One can take the attitude that initial conditions are simply to
be accepted, rather than explained – we only have one universe, and should
learn to deal with it, rather than seek explanations for the particular state
in which we find it. In that case, there would never be any reason to
contemplate inflation. The reason why inflation seems compelling is because we
are more ambitious: we would like to understand why the universe seems to be
one way, rather than some other way. By its own standards, the inflationary
paradigm bears the burden of establishing that inflation is itself natural (or
at least more natural than the alternatives).
It has been recognized for some time that there is tension between this goal
and the underlying structure of classical mechanics (or quantum mechanics, for
that matter). A key feature of classical mechanics is conservation of
information: the time-evolution map from states at one time to states at some
later time is invertible and volume-preserving, so that the earlier states can
be unambiguously recovered from the later states. This property is
encapsulated by Liouville’s theorem, which states that a distribution function
in the space of states remains constant along trajectories; roughly speaking,
a certain number of states at one time always evolves into precisely the same
number of states at any other time. In quantum mechanics, an analogous
property is guaranteed by unitarity of the time-evolution operator; most of
our analysis here will be purely classical, but we will refer to the
conservation of the number of states as “unitarity” for convenience.
The conflict with the philosophy of inflation is clear. Inflation attempts to
account for the apparent fine-tuning of our early universe by offering a
mechanism by which a relatively natural early condition will robustly evolve
into an apparently finely-tuned later condition. But if that evolution is
unitary, it is impossible for any mechanism to evolve a large number of states
into a small number, so the number of initial conditions corresponding to
inflation must be correspondingly small, calling into question their status as
“relatively natural.” This point has been emphasized by Penrose [4], and has
been subsequently discussed elsewhere [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. As
long as it operates within the framework of unitary evolution, the best
inflation can do is to move the set of initial conditions that creates a
smooth, flat universe at late times from one part of phase space to another
part; it cannot increase the size of that set.
As a logical possibility, the true evolution of the universe may be non-
unitary. Indeed, discussions of cosmology often proceed as if this were the
case, as we discuss below. The justification for this perspective is that a
comoving patch of space is smaller at earlier times, and therefore can
accommodate fewer modes of quantum fields. But there is nothing in quantum
field theory, or anything we know about gravity, to indicate that evolution is
fundamentally non-unitary. The simplest resolution is to imagine that there
are a large number of states that are not described by quantum fields in a
smooth background (e.g., with Planckian spacetime curvature or the quantum-
mechanical version thereof). Even if we don’t have a straightforward
description of the complete set of such states, the underlying principle of
unitarity is sufficient to imply that they must exist.
If unitary evolution is respected, there is nothing special about “initial”
states; the state at any one moment of time specifies the evolution just as
well as the state at any other time. In that light, the issue of cosmological
fine-tuning is a question about _histories_ , not simply about initial
conditions. Our goal should not be to show that generic initial conditions
give rise to the early universe we observe; Liouville’s theorem forbids it.
Given the degrees of freedom constituting our observable universe, and the
macroscopic features of their current state, the vast majority of possible
evolutions do not arise from a smooth Big Bang beginning. Therefore, a
legitimate explanation for cosmological fine-tuning would show that not all
histories are equally likely – that the history we observe is very natural
within the actual evolution of the universe, even though it belongs to a tiny
fraction of all conceivable trajectories. In particular, a convincing scenario
would possess the property that when the degrees of freedom associated with
our observable universe are in the kind of state we currently find them in, it
is most often in the aftermath of a smooth Big Bang.
We can imagine two routes to this goal: either our present condition only
occurs once, and the particular history of our universe is simply highly non-
generic (perhaps due to an underlying principle that determines the wave
function of the universe); or conditions like those of our observable universe
occur many times within a much larger multiverse, and the dynamics has the
property that most appearances of our local conditions (in some appropriate
measure) are associated with smooth Big-Bang-like beginnings. In either case,
inflation might very well play a crucial role in the evolution of the
universe, but it does not by itself constitute an answer to the puzzle of
cosmological fine-tuning.
In this paper we try to quantify the issues of cosmological fine-tuning in the
context of unitary evolution, using the canonical measure on the space of
solutions to Einstein’s equations developed by Gibbons, Hawking, and Stewart
[5]. Considering first the measure on purely Robertson-Walker cosmologies
(without perturbations) as a function of spatial curvature, there is a
divergence at zero curvature. In other words, curved RW cosmologies are a set
of measure zero – the flatness problem, as conventionally understood, does not
exist. This divergence has no immediate physical relevance, as the real world
is not described by a perfectly Robertson-Walker metric. Nevertheless, it
serves as a cautionary example for the importance of considering the space of
initial conditions in a mathematically rigorous way, rather than relying on
our intuition. We therefore perform a similar analysis for the case of
perturbed universes, to verify that there is not any hidden divergence at
perfect homogeneity. We find that there is not; any individual perturbation
can be written as an oscillator with a time-dependent mass, and the measure is
flat in the usual space of coordinate and momentum. The homogeneity of the
universe represents a true fine tuning; there is no reason for the universe to
be smooth.
We also use the canonical measure to investigate the likelihood of inflation.
In the minisuperspace approximation, we find that inflation can be very
probable, depending on the inflaton potential considered. However, this
approximation is wildly inappropriate for this problem; it is essential to
consider perturbations. If we restrict ourselves to universes that look
realistic at the epoch of matter-radiation equality, we find that only a
negligible fraction were sufficiently smooth at early times to allow for
inflation. This simply reflects the aforementioned fact that there are many
more inhomogeneous states at early times than smooth ones.
We are not suggesting that inflation plays no role in cosmological dynamics;
only that it is not sufficient to explain how our observed early universe
arose from generic initial conditions. Inflation requires very specific
conditions to occur – a patch of space dominated by potential energy over a
region larger than the corresponding Hubble length [15] – and these conditions
are an extremely small fraction of all possible states. However, while
inflationary conditions are very few, there is something simple and compelling
about them. Without inflation, when the Hubble parameter was of order the
Planck scale our universe needed to be smooth over a length scale many orders
of magnitude larger than the Planck length. With inflation, by contrast, a
smooth volume of order the Planck length can evolve into our entire observable
universe. There are fewer such states than those required by conventional Big
Bang cosmology, but it is not hard to imagine that they are somehow easier to
create. In other words, given that the history of our observable universe
seems non-generic by any conceivable measure, it seems very plausible that
some hypothetical theory of initial conditions (or multiverse dynamics)
creates the necessary initial conditions through the mechanism of inflation,
rather than by creating a radiation-dominated Big Bang universe directly. We
argue that this is the best way to understand the role of inflation, rather
than as a solution to the horizon and flatness problems.
The lesson of our investigation is that the state of the universe does appear
unnatural from the point of view of the canonical measure on the space of
trajectories, and that no choice of unitary evolution can alleviate that fine-
tuning, whether it be inflation or any other mechanism. Inflation can alter
the set of initial conditions that leads to a universe like ours, but it
cannot make it any larger. Inflation does not remove the need for a theory of
initial conditions; it brings that need into sharper focus.
## 2 The Evolution of our Comoving Patch
For many years, the paradigm for fundamental physics has been information-
conserving dynamical laws applied to initial data. A consequence of
information conservation is reversibility: the state of the system at any one
time is sufficient to recover its initial state, or indeed any state in the
past or future. The goal of this section is to lay out the motivations for
treating the degrees of freedom of our observable universe as a system obeying
reversible dynamics, and to establish the limitations of that approach.
Both quantum mechanics and classical mechanics feature this kind of unitary
evolution.111The collapse of the wave function in quantum mechanics is an
apparent exception. We will not address this phenomenon, implicitly assuming
something like the many-worlds interpretation, in which wave function collapse
is only apparent and the true evolution is perfectly unitary. In the
Hamiltonian formulation of classical mechanics, a state is an element of phase
space, specified by coordinates $q^{i}(t)$ and momenta $p_{i}(t)$. Time
evolution is governed by Hamilton’s equations,
$\dot{q}^{i}=\frac{\partial{\mathcal{H}}}{\partial
p_{i}}\,,\quad\dot{p}_{i}=-\frac{\partial{\mathcal{H}}}{\partial q^{i}}\,,$
(1)
where ${\mathcal{H}}$ is the Hamiltonian. In quantum mechanics, a state is
given by a wave function $|\psi(t)\rangle$ which defines a ray in Hilbert
space. Time evolution is governed by the Schrödinger equation,
$\hat{{\mathcal{H}}}|\Psi\rangle=i\partial_{t}|\Psi\rangle,$ (2)
where $\hat{{\mathcal{H}}}$ is the Hamiltonian operator, or equivalently by
the von Neumann equation,
$\partial_{t}\hat{\rho}=-i[\hat{{\mathcal{H}}},\hat{\rho}]\,,$ (3)
where $\hat{\rho}(t)=|\psi(t)\rangle\langle\psi(t)|$ is the density operator.
In either formalism, knowledge of the state at any one moment of time is
sufficient (given the Hamiltonian) to determine the state at all other times.
While we don’t yet know the complete laws of fundamental physics, the most
conservative assumption we could make would be to preserve the concept of
unitarity. Even without knowing the Hamiltonian or the space of states, we
will see that the principle of unitarity alone offers important insights into
cosmological fine-tuning problems.
Although the assumption of unitary evolution seems like a mild one, there are
challenges to applying the idea directly to an expanding universe. We can only
observe a finite part of the universe, and the physical size of that part
changes with time. The former feature implies that the region we observe is
not a truly closed system, and the latter implies that the set of field modes
within this region is not fixed. Both aspects could be taken to imply that,
even if the underlying laws of fundamental physics are perfectly unitary, it
would nevertheless be inappropriate to apply the principle of unitarity to the
the part of the universe we can observe.
We will take the stance that it is nevertheless sensible to proceed under the
assumption that the degrees of freedom describing our observable universe
evolve according to unitary dynamical laws, even if that assumption is an
approximation. In this section we offer the justification for this assumption.
In particular we discuss two separate parts to this claim: that the observable
universe evolves autonomously (as a closed system), and that this autonomous
evolution is governed by unitary laws.
### 2.1 Autonomy
We live in an expanding universe that is approximately homogeneous and
isotropic on large scales. We can therefore consider our universe as a
perturbation of an exactly homogenous and isotropic (Robertson-Walker)
background spacetime. Defining a particular map from the background to our
physical spacetime involves a choice of gauge. Nothing that we are going to do
depends on how that gauge is chosen, as long as it is defined consistently
throughout the history of the universe. Henceforth we assume that we’ve chosen
a gauge.
The map from the RW background spacetime to our universe provides two crucial
elements: a foliation into time slices, and a congruence of comoving geodesic
worldlines. The time slicing allows us to think of the universe as a fixed set
of degrees of freedom evolving through time, obeying Hamilton’s equations. At
each moment in time there exists an exact value of the (background) Hubble
parameter and all other cosmological parameters.
Figure 1: The physical system corresponding to our observable universe. Our
comoving patch is defined by the interior of the intersection of our past
light cone with a cutoff surface, for example the surface of last scattering.
This illustration is not geometrically faithful, as the expansion is not
linear in time. Despite the change in physical size, we assume that the space
of states is of equal size at every moment.
The notion of comoving worldlines, orthogonal to spacelike hypersurfaces of
constant Hubble parameter, allows us to define what we mean by our comoving
patch. If there is a Big Bang singularity in our past, there is a
corresponding particle horizon, defined by the intersection of our past light
cone with the singularity. However, independent of the precise nature of the
Big Bang, there is an effective limit to our ability to observe the past; in
practice this is provided by the surface of last scattering, although in
principle observations of gravitational waves or other particles could extend
the surface backwards. The precise details of where we draw the surface aren’t
important to our arguments. What matters is that there exists a well-defined
region of three-space interior to the intersection of our past light cone with
the observability surface past which we can’t see. Our comoving patch,
$\Sigma$, is simply the physical system defined by the extension of that
region forward in time via comoving worldlines, as shown in Figure 1.
Our assumption is that this comoving patch can be considered as a set of
degrees of freedom evolving autonomously through time, free of influence from
the rest of the universe. This is clearly an approximation, as an observer
stationed close to the boundary of our patch would see particles pass both
into and out of that region; our comoving patch isn’t truly a closed system.
However, the fact that the observable universe is homogenous implies that the
net effect of that exchange of particles is very small. In particular, we
generally don’t believe that what happens inside our observable universe
depends in any significant way on what happens outside.
Note that we are not necessarily assuming that our observable universe is in a
pure quantum state, free of entanglement with external degrees of freedom;
such entanglements don’t affect the local dynamics of the internal degrees of
freedom, and therefore are complete compatible with the von Neumann equation
(3). We are, however, assuming that the appropriate Hamiltonian is local in
space. Holography implies that this is not likely to be strictly true, but it
seems like an effective approximation for the universe we observe.
### 2.2 Unitarity
Autonomy implies that we can consider our comoving patch as a fixed set of
degrees of freedom, evolving through time. Our other crucial assumption is
that this evolution is unitary (reversible). Even if the underlying
fundamental laws of physics are unitary, it is not completely obvious that the
effective evolution of our comoving patch evolves this way. Indeed, this issue
is at the heart of the disagreement between those who have emphasized the
amount of fine-tuning required by inflationary initial conditions [4, 8, 9,
11, 12] and those who have argued that they are natural [10, 14].
The issue revolves around the time-dependent nature of the cutoff on modes of
a quantum field in an expanding universe. Since we are working in a comoving
patch, there is a natural infrared cutoff given by the size of the patch, a
length scale of order $\lambda_{\rm IR}\sim aH_{0}^{-1}$, where $a$ is the
scale factor (normalized to unity today) and $H_{0}$ is the current Hubble
parameter. But there is also a fixed ultraviolet cutoff at the Planck length,
$\lambda_{\rm UV}\sim L_{\rm pl}=\sqrt{8\pi G}$. Clearly the total number of
modes that fit in between these two cutoffs increases with time as the
universe expands. It is therefore tempting to conclude that the space of
states is getting larger.
We can’t definitively address this question in the absence of a theory of
quantum gravity, but for purposes of this paper we will assume that the space
of states is _not_ getting larger – which would violate the assumption of
unitarity – but the nature of the states is changing. In particular, the
subset of states that can usefully be described in terms of quantum fields on
a smooth spacetime background is changing, but those are only a (very small)
minority of all possible states.
The justification for this view comes from the assumed reversibility of the
underlying laws. Consider the macrostate of our universe today – the set of
all microstates compatible with the macroscopic configuration we observe. For
any given amount of energy density, there are two solutions to the Friedmann
equation, one with positive expansion rate and one with negative expansion
rate (unless the expansion rate is precisely zero, when the solution is
unique). So there are an equal number of microstates that are similar to our
current configuration, except that the universe is contracting rather than
expanding. As the universe contracts, each of those states must evolve into
some unique state a fixed time later; therefore, the number of states
accessible to the universe for different values of the Hubble parameter (or
different moments in time) is constant.
Most of the states available when the universe is smaller, however, are not
described by quantum fields on a smooth background. This is reflected in the
fact that spatial inhomogeneities would be generically expected to grow,
rather than shrink, as the universe contracted. The effect of gravity on the
state counting becomes significant, and in particular we would expect copious
production of black holes. These would appear as white holes in the time-
reversed expanding description. Therefore, the overwhelming majority of states
at early times that could evolve into something like our current observable
universe are not relatively smooth spacetimes with gently fluctuating quantum
fields; they are expected to be wildly inhomogeneous, filled with white holes
or at least Planck-scale curvatures.
We do not know enough about quantum gravity to explicitly enumerate these
states, although some attempts to describe them have been made (see e.g.
[16]). But we don’t need to know how to describe them; the underlying
assumption of unitarity implies that they are there, whether we can describe
them or not. (Similarly, the Bekenstein-Hawking entropy formula is
conventionally taken to imply a large number of states for macroscopic black
holes, even if there is no general description for what those individual
states are.)
This argument is not new, and it is often stated in terms of the entropy of
our comoving patch [4, 12]. In the current universe, this entropy is dominated
by black holes, and has a value of order $S_{\Sigma}(t_{0})\sim 10^{104}$
[17]. If all the matter were part of a single black hole it would be as large
as $S_{\Sigma}({\rm BH})\sim 10^{122}$. At early times, when inhomogeneities
were small and local gravitational effects were negligible, the entropy was of
order $S_{\Sigma}({\rm RD})\sim 10^{88}$. If we assume that the entropy is the
logarithm of the number of macroscopically indistinguishable microstates, and
that every microstate within the current macrostate corresponds to a unique
predecessor at earlier times, it is clear that the vast majority of states
from which our present universe might have evolved don’t look anything like
the smooth radiation-dominated configuration we actually believe existed
(since $\exp[10^{104}]\gg\exp[10^{88}]$).
This distinction between the number of states implied by the assumption of
unitarity and the number of states that could reasonably be described by
quantum fields on a smooth background is absolutely crucial for the question
of how finely-tuned are the conditions necessary to begin inflation. If we
were to start with a configuration of small size and very high density, and
consider only those states described by field theory, we would dramatically
undercount the total number of states. Unitarity could possibly be violated in
an ultimate theory, but we will accept it for the remainder of this paper.
## 3 The Canonical Measure
In order to quantify the issue of fine-tuning in the context of unitary
evolution, we review the canonical measure on the space of trajectories, as
examined by Gibbons, Hawking, and Stewart [5]. Despite subtleties associated
with coordinate invariance, GR can be cast as a conventional Hamiltonian
system, with an infinite-dimensional phase space and a set of constraints. The
state of a classical system is described by a point $\gamma$ in a phase space
$\Gamma$, with canonical coordinates $q^{i}$ and momenta $p_{i}$. The index
$i$ goes from $1$ to $n$, so that phase space is $2n$-dimensional. The
classical equations of motion are Hamilton’s equations (1). Equivalently,
evolution is generated by a Hamiltonian phase flow with tangent vector
$V=\frac{\partial{\mathcal{H}}}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial{\mathcal{H}}}{\partial q^{i}}\frac{\partial}{\partial
p_{i}}.$ (4)
Phase space is a symplectic manifold, which means that it naturally comes
equipped with a symplectic form, which is a closed 2-form on $\Gamma$:
$\omega=\sum_{i=1}^{n}{\rm d}p_{i}\wedge{\rm d}q^{i}\,,\qquad{\rm
d}\omega=0\,.$ (5)
The existence of the symplectic form provides us with a naturally-defined
measure on phase space,
$\Omega=\frac{(-1)^{n(n-1)/2}}{n!}\omega^{n}\,.$ (6)
This is the Liouville measure, a $2n$-form on $\Gamma$. It corresponds to the
usual way of integrating distributions over regions of phase space,
$\int f(\gamma)\Omega=\int f(q^{i},p_{i})d^{n}qd^{n}p\,.$ (7)
The Liouville measure is conserved under Hamiltonian evolution. If we begin
with a region $A\subset\Gamma$, and it evolves into a region $A^{\prime}$,
Liouville’s theorem states that
$\int_{A}\Omega=\int_{A^{\prime}}\Omega\,.$ (8)
The infinitesimal version of this result is that the Lie derivative of
$\Omega$ with respect to the vector field $V$ vanishes,
${\cal L}_{V}\Omega=0\,.$ (9)
These results can be traced back to the fact that the original symplectic form
$\omega$ is also invariant under the flow:
${\cal L}_{V}\omega=0\,,$ (10)
so any form constructed from powers of $\omega$ will be invariant.
In classical statistical mechanics, the Liouville measure can be used to
assign weights to different distributions on phase space. That is not
equivalent to assigning _probabilities_ to different sets of states, which
requires some additional assumption. However, since the Liouville measure is
the only naturally-defined measure on phase space, we often assume that it is
proportional to the probability in the absence of further information; this is
essentially Laplace’s “Principle of Indifference.” Indeed, in statistical
mechanics we typically assume that microstates are distributed with equal
probability with respect to the Liouville measure, consistent with known
macroscopic constraints.
In cosmology, we don’t typically imagine choosing a random state of the
universe, subject to some constraints. When we consider questions of fine-
tuning, however, we are comparing the real world to what we think a randomly-
chosen history of the universe would be like. The assumption of some sort of
measure is absolutely necessary for making sense of cosmological fine-tuning
arguments; otherwise all we can say is that we live in the universe we see,
and no further explanation is needed. (Note that this measure on the space of
solutions to Einstein’s equation is conceptually distinct from a measure on
observers in a multiverse, which is sometimes used to calculate expectation
values for cosmological parameters based on the anthropic principle.)
GHS [5] showed how the Liouville measure on phase space could be used to
define a unique measure on the space of solutions (see also [6, 7, 13]). In
general relativity we impose the Hamiltonian constraint, so we can consider
the $(2n-1)$-dimensional constraint hypersurface of fixed Hamiltonian,
$C=\Gamma/\\{{\mathcal{H}}={\mathcal{H}}_{*}\\}\,.$ (11)
For Robertson-Walker cosmology, the Hamiltonian precisely vanishes for either
open or closed universes, so we can take ${\mathcal{H}}_{*}=0$. Then we
consider the space of classical trajectories within this constraint
hypersurface:
$M=C/V\,,$ (12)
where the quotient by the evolution vector field $V$ means that two points are
equivalent if they are connected by a classical trajectory. Note that this is
well-defined, in the sense that points in $C$ always stay within $C$, because
the Hamiltonian is conserved.
As $M$ is a submanifold of $\Gamma$, the measure is constructed by pulling
back the symplectic form from $\Gamma$ to $M$ and raising it to the $(n-1)$th
power. GHS constructed a useful explicit form by choosing the $n$th coordinate
on phase space to be the time, $q^{n}=t$, so that the conjugate momentum
becomes the Hamiltonian itself, $p_{n}={\mathcal{H}}$. The symplectic form is
then
$\omega={\widetilde{\omega}}+{\rm d}{\mathcal{H}}\wedge{\rm d}t\,,$ (13)
where
${\widetilde{\omega}}=\sum_{i=1}^{n-1}{\rm d}p_{i}\wedge{\rm d}q^{i}\,.$ (14)
The pullback of $\omega$ onto $C$ then has precisely the same coordinate
expression as (14), and we will simply refer to this pullback as
${\widetilde{\omega}}$ from now on. It is automatically transverse to the
Hamiltonian flow (${\widetilde{\omega}}(V)=0$), and therefore defines a
symplectic form on the space of trajectories $M$. The associated measure is a
$(2n-2)$-form,
${\Theta}=\frac{(-1)^{(n-1)(n-2)/2}}{(n-1)!}{\widetilde{\omega}}^{n-1}\,.$
(15)
We will refer to this as the GHS measure; it is the unique measure on the
space of trajectories that is positive, independent of arbitrary choices, and
respects the appropriate symmetries [5].
To evaluate the measure we need to define coordinates on the space of
trajectories. We can choose a hypersurface $\Sigma$ in phase space that is
transverse to the evolution trajectories, and use the coordinates on phase
space restricted to that hypersurface. An important property of the GHS
measure is that the integral over a region within a hypersurface is
independent of which hypersurface we chose, so long as it intersects the same
set of trajectories; if $S_{1}$ and $S_{2}$ are subsets, respectively, of two
transverse hypersurfaces $\Sigma_{1}$ and $\Sigma_{2}$ in $C$, with the
property that the set of trajectories passing through $S_{1}$ is the same as
that passing through $S_{2}$, then
$\int_{S_{1}}{\Theta}=\int_{S_{2}}{\Theta}\,.$ (16)
The property that the measure on trajectories is local in phase space has a
crucial implication for studies of cosmological fine-tuning. Imagine that we
specify a certain set of trajectories by their macroscopic properties today –
cosmological solutions that are approximately homogeneous, isotropic, and
spatially flat, suitably specified in terms of canonical coordinates and
momenta. It is immediately clear that the measure on this set is independent
of the behavior in very different regions of phase space, e.g. for high-
density states corresponding to early times. Therefore, no choice of early-
universe Hamiltonian can make the current universe more or less finely tuned.
No new early-universe phenomena can change the measure on a set of universes
specified at late times, because we can always evaluate the measure on a late-
time hypersurface without reference to the behavior of the universe at any
earlier time.222On the other hand, if the effective Hamiltonian is time-
dependent, what looks like a generic state at early times can evolve into a
non-generic state at later times, as energy can be injected into the system.
This is related to the recent proposal of weak gravity in the early universe
[18]. At heart, this is a direct consequence of Liouville’s theorem.
## 4 Minisuperspace
In this section, we evaluate the measure on the space of solutions to
Einstein’s equation in minisuperspace (Robertson-Walker) cosmology with a
scalar field, applying the results to the flatness problem and the likelihood
of inflation. We will look at two specific models: a scalar with a canonical
kinetic term and a potential, and a scalar with a non-canonical kinetic term
chosen to mimic a perfect-fluid equation of state.
A scalar field coupled to general relativity is governed by an action
$S=\int d^{4}x\,\sqrt{-g}\left[\frac{1}{2}R+P(X,\phi)\right],$ (17)
where $R$ is the curvature scalar and $P$ is the Lagrange density of the
scalar field $\phi$. We have set $m_{\rm pl}^{-2}=8\pi G=1$ for convenience.
The scalar Lagrangian is taken to be a function of the field value and and the
kinetic scalar $X$, defined by
$X\equiv-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi.$ (18)
We will consider homogeneous scalar fields $\phi(t)$ defined in a Robertson-
Walker metric,
$ds^{2}=-N^{2}{\rm d}t^{2}+a^{2}(t)\left[\frac{{\rm
d}r^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right]\,,$ (19)
where the spatial curvature parameter $k$ can be normalized to $-1$, $0$, or
$+1$ (so that $a(t_{0})$ is not normalized to unity). $N$ is the lapse
function, which acts as a Lagrange multiplier. We then have
$X=\frac{1}{2}N^{-2}\dot{\phi}^{2}.$ (20)
### 4.1 Canonical scalar field
We start with the canonical case,
$P(X,\phi)=X-V(\phi).$ (21)
The Lagrangian for the combined gravity-scalar system in minisuperspace is
$L=-3N^{-1}a\dot{a}^{2}+3Nak+\frac{1}{2}N^{-1}a^{3}\dot{\phi}^{2}-Na^{3}V(\phi)\,.$
(22)
The canonical coordinates can be taken to be the lapse function $N$, the scale
factor $a$, and the scalar field $\phi$. The conjugate momenta are given by
$p_{i}=\partial L/\partial q^{i}$, implying
$p_{N}=0\,,\quad p_{a}=-6N^{-1}a\dot{a}\,,\quad
p_{\phi}=N^{-1}a^{3}\dot{\phi}\,.$ (23)
The vanishing of $p_{N}$ reflects the fact that the lapse function is a non-
dynamical Lagrange multiplier. We can do a Legendre transformation to
calculate the Hamiltonian, obtaining
$\displaystyle{\mathcal{H}}$ $\displaystyle=$ $\displaystyle\sum
p_{i}\dot{q}^{i}-L(p_{i},q^{i})$ (24) $\displaystyle=$ $\displaystyle
N\left(-\frac{p_{a}^{2}}{12a}+\frac{p_{\phi}^{2}}{2a^{3}}+a^{3}V(\phi)-3ak\right).$
(25)
Varying with respect to $N$ gives the Hamiltonian constraint,
${\mathcal{H}}=0$, which is just the Friedmann equation,
$H^{2}=\frac{1}{3}\left(\frac{1}{2}{\dot{\phi}}^{2}+V(\phi)-\frac{3k}{a^{2}}\right).$
(26)
Henceforth we will set $N=1$ (consistent with the equations of motion),
leaving us with a four-dimensional phase space,
$\Gamma=\\{\phi,p_{\phi},a,p_{a}\\}\,.$ (27)
The GHS measure on the space of trajectories is just the the Liouville measure
subject to the constraint that $\mathcal{H}=0$,
${\Theta}=(dp_{a}\wedge da+dp_{\phi}\wedge d\phi)|_{\mathcal{H}=0}.$ (28)
Note that the measure in this example is a two-form; the full phase space is
four-dimensional, the Hamiltonian constraint surface is three dimensional, so
the space of trajectories is two-dimensional.
To express the measure in a convenient form, we use the Friedmann equation to
eliminate one of the phase-space variables. Solving for $p_{\phi}$ gives us
$p_{\phi}=\left[\frac{1}{6}a^{2}p_{a}^{2}-2a^{6}V(\phi)+6a^{4}k\right]^{1/2}.$
(29)
We can change variables from $p_{a}$ to $H$ using $p_{a}=-6a^{2}H$, so that
$p_{\phi}=\left(6a^{6}H^{2}-2a^{6}V+6a^{4}k\right)^{1/2}.$ (30)
Our coordinates on the constraint hypersurface $C$ are therefore
$\\{\phi,a,H\\}$. The basis one-forms appearing in (28) are
$dp_{a}=-12aHda-6a^{2}dH$ (31)
and
$dp_{\phi}=\frac{6a^{4}HdH-a^{4}V^{\prime}d\phi+6a(3a^{2}H^{2}-a^{2}V+2k)da}{(6a^{2}H^{2}-2a^{2}V+6k)^{1/2}},$
(32)
where $V^{\prime}(\phi)=dV/d\phi$. Plug into the expression (28) for the
measure, whose components become
$\displaystyle{\Theta}_{\phi H}$ $\displaystyle=$
$\displaystyle-\frac{6a^{4}}{\left(6a^{2}H^{2}-2a^{2}V+6k\right)^{1/2}}$ (33)
$\displaystyle{\Theta}_{Ha}$ $\displaystyle=$ $\displaystyle-6a^{2}$ (34)
$\displaystyle{\Theta}_{a\phi}$ $\displaystyle=$ $\displaystyle
6\frac{3a^{3}H^{2}-a^{3}V+2ak}{\left(6a^{2}H^{2}-2a^{2}V+6k\right)^{1/2}}.$
(35)
The measure is calculated by choosing some transverse surface $\Sigma$ in
phase space, and integrating ${\Theta}$ over a subset of that surface. If we
choose coordinates such that one coordinate is constant over $\Sigma$, we
simply integrate the orthogonal component of ${\Theta}$ with respect to the
other coordinates. One possible choice of the surface $\Sigma$ is to fix the
Hubble parameter,
$\Sigma:\\{H=H_{*}\\}\,.$ (36)
Any consistent definition is equally legitimate; however, this choice
corresponds to our informal idea that initial conditions are set in the early
universe when the Hubble parameter is near the Planck scale. The measure
evaluated on a surface of constant $H$ is then the integral of
${\Theta}_{a\phi}$,
$\mu=-6\int_{H=H_{*}}\frac{3a^{3}H_{*}^{2}-a^{3}V+2ak}{\left(6a^{2}H_{*}^{2}-2a^{2}V+6k\right)^{1/2}}dad\phi,$
(37)
where the minus sign indicates that we have chosen an orientation that will
give us a positive final answer. We can make this expression look more
physically transparent by introducing variables
$\Omega_{\dot{\phi}}\equiv\frac{\dot{\phi}^{2}}{6H_{*}^{2}},\quad\Omega_{V}\equiv\frac{V(\phi)}{3H_{*}^{2}},\quad\Omega_{k}\equiv-\frac{k}{a^{2}H_{*}^{2}},$
(38)
so that the Friedmann equation is equivalent to
$\Omega_{\dot{\phi}}+\Omega_{V}+\Omega_{k}=1.$ (39)
The scale factor is strictly positive, so that integrating over all values of
$\Omega_{k}$ is equivalent to integrating over all values of $a$. Note that
$-k/\Omega_{k}=1/|\Omega_{k}|$. We therefore have
$da=-\frac{1}{2H_{*}|\Omega_{k}|^{3/2}}d\Omega_{k},$ (40)
and the measure becomes
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
3\sqrt{\frac{3}{2}}H_{*}^{-2}\int_{H=H_{*}}\frac{1-\Omega_{V}-\frac{2}{3}\Omega_{k}}{|\Omega_{k}|^{5/2}\left(1-\Omega_{V}-\Omega_{k}\right)^{1/2}}\,d\Omega_{k}d\phi$
(41) $\displaystyle=$ $\displaystyle
3\sqrt{\frac{3}{2}}H_{*}^{-2}\int_{H=H_{*}}\frac{\Omega_{\dot{\phi}}-\frac{1}{3}\Omega_{k}}{|\Omega_{k}|^{5/2}\Omega_{\dot{\phi}}^{1/2}}\,d\Omega_{k}d\phi,$
(42)
where $\Omega_{\dot{\phi}}(\phi,\Omega_{k})$ is defined by (39).
This integral is divergent. One divergence clearly occurs for small values of
the curvature parameter, $\Omega_{k}\rightarrow 0$, as the denominator
includes a factor of $|\Omega_{k}|^{5/2}$. The integrand also blows up at
$\Omega_{\dot{\phi}}=0$ (or equivalently at $\Omega_{V}+\Omega_{k}=1$), but
the integral in that region remains finite. The integral would also diverge if
$\Omega_{k}$ or $\Omega_{\dot{\phi}}$ were allowed to become arbitrarily
large, but that could be controlled by only integrating over a finite range
for those quantities, e.g. under the theory that Planckian energy densities or
curvatures should not be included in this classical description.
The important divergence, therefore, is the one at $\Omega_{k}\rightarrow 0$,
i.e. for flat universes. We discuss the implications of this divergence in
Section 4.3.
### 4.2 Scalar perfect fluid
In conventional Big Bang cosmology, we generally consider perfect-fluid
sources of energy such as matter or radiation, rather than using a single
scalar field. This situation is slightly more difficult to analyze as a
problem in phase space, as homogeneity and isotropy are only recovered after
averaging over many individual particles. However, we can model a perfect
fluid with an (almost) arbitrary equation of state by a scalar field with a
non-canonical kinetic term [19].
Consider the action (17), where the scalar Lagrangian takes the form
$P(X,\phi)$, where $X=-(\nabla_{\mu}\phi)^{2}/2$. In a Robertson-Walker
background, the energy-momentum tensor takes the form of a perfect fluid,
$T_{\mu\nu}=(\rho+P)U_{\mu}U_{\nu}-Pg_{\mu\nu},$ (43)
where the pressure is equal to the scalar Lagrange density itself (thereby
accounting for the choice of notation). The fluid has four-velocity
$U_{\mu}=(2X)^{-1/2}\nabla_{\mu}\phi$ (44)
and energy density
$\rho=2X\partial_{X}P-P.$ (45)
We will be interested in a vanishing potential but a non-canonical kinetic
term,
$P(X,\phi)=\frac{2^{n-1}}{n}X^{n}=\frac{1}{2n}N^{-2n}\dot{\phi}^{2n}.$ (46)
This gives a fluid with a density
$\rho=\frac{2n-1}{2n}N^{-2n}\dot{\phi}^{2n},$ (47)
corresponding to a constant equation-of-state parameter
$w=P/\rho=\frac{1}{2n-1},$ (48)
as can easily be checked. Therefore we can model the behavior of radiation
($w=1/3$) by choosing $n=2$, and approximate matter ($w=0$) by choosing $n$
very large.
The scalar-Einstein Lagrangian in a Robertson-Walker background takes the form
$L=-3N^{-1}a\dot{a}^{2}+3Nak+\frac{1}{2n}N^{-(2n-1)}a^{3}\dot{\phi}^{2n},$
(49)
and the Friedmann equation is
$H^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{2n-1}{6n}\right)\dot{\phi}^{2n}-\frac{k}{a^{2}},$
(50)
where we have set $N=1$. We can duplicate the steps taken in the previous
section, to evaluate the GHS measure in terms of coordinates $\\{\phi,a,H\\}$.
We end up with
$\displaystyle{\Theta}_{\phi H}$ $\displaystyle=$
$\displaystyle-\left(\frac{2n-1}{2n}\right)^{1/2n}\frac{6a^{6n/(2n-1)}}{\left[a^{6n/(2n-1)}3H^{2}+3a^{2(n+1)/(2n-1)}k\right]^{1/2n}}$
(51) $\displaystyle{\Theta}_{Ha}$ $\displaystyle=$ $\displaystyle-6a^{2}$ (52)
$\displaystyle{\Theta}_{a\phi}$ $\displaystyle=$ $\displaystyle
6(2n-1)^{(1-2n)/2n}\frac{na^{(4n+1)/(2n-1)}3H^{2}+(n+1)a^{3/(2n-1)}k}{\left[2na^{6n/(2n-1)}3H^{2}+6na^{2(n+1)/(2n-1)}k\right]^{1/2n}}.$
(53)
To calculate the measure of a set of trajectories over a surface of constant
$H=H_{*}$, we integrate ${\Theta}_{a\phi}$ over $a$ and $\phi$. This yields
$\mu=-3\left(\frac{2n-1}{6n}\right)^{(1-2n)/2n}\int_{H=H_{*}}a^{2}\frac{H_{*}^{2}+\frac{(n+1)}{3n}a^{-2}k}{\left(H_{*}^{2}+a^{-2}k\right)^{1/2n}}dad\phi.$
(54)
Note that the integrand has no dependence on $\phi$, since there was no
potential in the original action. We therefore define
$x\equiv 3\left(\frac{2n-1}{6n}\right)^{(1-2n)/2n}\int d\phi,$ (55)
which contributes an overall multiplicative constant to the measure. As
before, it is convenient to change variables from $a$ to
$\Omega_{k}=-k/a^{2}H_{*}^{2}$. This leaves us with
$\mu=\frac{x}{2H_{*}^{(n+1)/n}}\int\frac{1-\frac{(n+1)}{3n}\Omega_{k}}{|\Omega_{k}|^{5/2}\left(1-\Omega_{k}\right)^{1/2n}}d\Omega_{k}.$
(56)
This will diverge for small $\Omega_{k}$ for any value of $n$; all of the
measure is at spatially flat universes. This of course includes the case of
radiation, $n=2$. Therefore, the divergence we found in the previous
subsection for flat universes does not seem to depend on the details of the
matter action.
### 4.3 The flatness problem
Let’s return to the expression for the measure (42) we derived for Robertson-
Walker universes with a scalar field featuring a canonical kinetic term and a
potential,
$\mu\propto\int_{H=H_{*}}\frac{1-\Omega_{V}-\frac{2}{3}\Omega_{k}}{|\Omega_{k}|^{5/2}\left(1-\Omega_{V}-\Omega_{k}\right)^{1/2}}\,d\Omega_{k}d\phi.$
(57)
We have left out the numerical constants in front, as the overall
normalization is irrelevant. It is clear that this is non-normalizable as it
stands; the integral diverges near $\Omega_{k}=0$, which is certainly a
physically allowed region of parameter space. This non-normalizability is
problematic if we would like to interpret the measure as determining the
relative fraction of universes with different physical properties.
We propose that the proper way of handling such a divergence is to regularize
it. That is, we define a series of integrals that are individually finite, and
which approach the original expression as the regulator parameter $\epsilon$
is taken to zero. We can then isolate an appropriate power of $\epsilon$ by
which we can divide the regulated expression, so that we isolate the finite
part of the result as $\epsilon$ goes to zero.
The divergence in (57) can be regulated by “smoothing” the factor
$|\Omega_{k}|^{-5/2}$ in an $\epsilon$-neighborhood around $\Omega_{k}=0$ to
get a finite integral. Consider the function
$f_{\epsilon}(x)=\begin{cases}|x|^{-5/2}&\text{if $|x|\geq\epsilon$,}\\\
\epsilon^{-5/2}&\text{if $|x|<\epsilon$.}\end{cases}$ (58)
Clearly $\lim_{\epsilon\rightarrow 0}f_{\epsilon}(x)=|x|^{-5/2}$, our original
function. The integral of $f_{\epsilon}(x)$ over all values of $x$ is
$\frac{10}{3}\epsilon^{-3/2}$. So we obtain a normalized integral by
introducing the function
$F_{\epsilon}(x)=\begin{cases}\displaystyle{\frac{3\epsilon^{3/2}}{10|x|^{5/2}}}&\text{if
$|x|\geq\epsilon$,}\\\ \displaystyle{\frac{3}{10\epsilon}}&\text{if
$|x|<\epsilon$,}\end{cases}$ (59)
which satisfies $\int F_{\epsilon}(x)dx=1$. We can therefore regularize the
integral in (57) by replacing $|\Omega_{k}|^{-5/2}$ by
$F_{\epsilon}(\Omega_{k})$, and take the limit as $\epsilon\rightarrow 0$:
$\mu\propto\lim_{\epsilon\rightarrow
0}\epsilon^{-3/2}\int_{H=H_{*}}F_{\epsilon}(\Omega_{k})\frac{1-\Omega_{V}-\frac{2}{3}\Omega_{k}}{\left(1-\Omega_{V}-\Omega_{k}\right)^{1/2}}\,d\Omega_{k}d\phi.$
(60)
The multiplicative factor of $\epsilon^{-3/2}$ goes to infinity in the limit,
but only the finite integral is physically relevant. We interpret this
integral as defining the normalized measure on the space of cosmological
spacetimes.
However, it is clear that the limit of $F_{\epsilon}(x)$ is simply a delta
function,
$\lim_{\epsilon\rightarrow 0}F_{\epsilon}(x)=\delta(x),$ (61)
in the sense that the integral over a test function $\psi(x)$ gives
$\lim_{\epsilon\rightarrow
0}\int_{-\infty}^{\infty}F_{\epsilon}(x)\psi(x)\,dx=\psi(0).$ (62)
Consequently, the measure is entirely concentrated on exactly flat universes;
universes with nonvanishing spatial curvature are a set of measure zero. The
integrated measure (60) is equivalent to
$\mu\propto\int_{H=H_{*}}\sqrt{1-\Omega_{V}}\,d\phi,$ (63)
with $\Omega_{k}$ fixed to be $0$.
Therefore, our interpretation is clear: almost all universes are spatially
flat. In terms of the measure defined by the classical theory itself, a
“randomly chosen” cosmology will be flat with probability one. The flatness
problem, as conventionally understood, does not exist; it is an artifact of
informally assuming a flat measure on the space of initial cosmological
parameters. Of course, any particular specific theory of initial conditions
might actually have a flatness problem, if it predicts spatially-curved
universes with high probability; but that problem is not intrinsic to the
standard Big Bang model by itself.
Classical general relativity is not a complete theory of gravity, and our
notions of what constitutes a “natural” set of initial conditions are
inevitably informed by our guesses as to how it will ultimately be completed
by quantum gravity. At the level of the classical equations of motion, initial
data for a solution may be specified at any time; Hamilton’s equations then
define a unique solution for the complete past and future. However, we
generally impose a cutoff on the validity of a classical solution when some
quantity – the energy density, Hubble parameter, or spatial curvature –
reaches the Planck scale. It therefore makes sense to us to imagine that some
unknown physical process sets the initial conditions near the Planck regime.
In Robertson-Walker cosmology, we might imagine that the space of allowed
initial conditions consists of all values of the phase-space variables such
that the energy density and curvatures are all sub-Planckian; in terms of the
density parameters $\Omega_{i}$, this corresponds to $|\Omega_{i,{\rm
pl}}|<1$, where the subscript “pl” denotes that the quantity is evaluated when
$H\sim m_{\rm pl}$.
What this means in practice is that we tend to assign equal probability – a
flat prior – to all the allowed $\Omega_{i,{\rm pl}}$’s when contemplating
cosmological initial conditions. As a matter of principle, it is necessary to
invoke some kind of prior in order to sensibly discuss fine-tuning problems; a
quantity is finely-tuned if it is drawn from a small (as defined by some
measure) region of parameter space. The lesson of the GHS measure is that a
flat prior on $\Omega_{i,{\rm pl}}$ ignores the structure of the classical
theory itself, which comes equipped with a unique well-defined measure. In
Figure 2 we plot two different measures on the value of $\Omega_{k}$ at the
Planck scale; the informal flat prior assumed in typical discussions of the
flatness problem, and the GHS measure (evaluated at $\Omega_{V}=0$ for
convenience). We see that using the measure defined by the classical equations
of motion leads to a dramatic difference in the probability density.
Figure 2: Two measures as a function of the curvature parameter $\Omega_{k}$.
The GHS measure is highly peaked near the origin, indicating a divergence at
spatially flat universes. (We’ve drawn the unnormalized measure; a normalized
version would simply be a $\delta$-function.) This is in stark contrast with
the flat distribution generally assumed in the discussion of the flatness
problem, which we’ve plotted for $\Omega_{k}$ between $\pm 1$.
Note that the model of a canonical scalar field with a potential will allow
for the possibility of inflation if the potential is chosen appropriately;
however, the divergence at flat universes is _not_ because inflation is
secretly occurring. For one thing, the divergence appears for any choice of
potential, and also in the perfect-fluid model where there is no potential at
all. For another, we could always choose to evaluate the measure at late times
– i.e., we could pick the fixed Hubble parameter $H_{*}$ to be very small. The
measure on trajectories is independent of this choice, so the divergence for
flat universes cannot depend on whether inflation occurs.
This divergence was noted in the original GHS paper [5], where it was
attributed to “universes with very large scale factors” due to a different
choice of variables. This is not the most physically transparent
characterization, as any open universe will eventually have a large scale
factor. It is also discussed by Gibbons and Turok [13], who correctly
attribute it to nearly-flat universes. However, they advocate discarding all
such universes as physically indistinguishable, and concentrating on the non-
flat universes. To us, this seems to be throwing away almost all the
solutions, and keeping a set of measure zero. It is true that universes with
almost identical values of the curvature parameter will be physically
indistinguishable, but that doesn’t affect the fact that almost all universes
have this property. In Hawking and Page [6] and Coule [7] the divergence is
directly attributed to flat universes, but they do not seem to argue that the
flatness problem is therefore an illusion.
The real world is not precisely Robertson-Walker, so in some sense the
flatness problem is not rigorously defined; a super-Hubble-radius perturbation
could lead to a deviation from $\Omega=1$ in our observed universe, even if
the background cosmology were spatially flat. Nevertheless, the unanticipated
structure of the canonical measure in minisuperspace serves as a cautionary
example for the importance of considering the space of initial conditions in a
rigorous way. More directly, it raises an obvious question: if the canonical
measure is concentrated on spatially flat universes, might it also be
concentrated on smooth universes, thereby calling into question the status of
the horizon problem as well as the flatness problem? (We will find that it is
not.)
### 4.4 Likelihood of inflation
A common use of the canonical measure has been to calculate the likelihood of
inflation [5, 6, 7]. Most recently, Gibbons and Turok [13] have argued that
the fraction of universes that inflate is extremely small. However, they threw
away all but a set of measure zero of trajectories, on the grounds that they
all had negligibly small spatial curvature and therefore physically
indistinguishable. Inflation, of course, tends to make the universe spatially
flat, so this procedure is potentially unfair to the likelihood of inflation.
We therefore re-examine this question, following the philosophy suggested by
the above analysis, which implies that almost all universes are spatially
flat. We choose to look only at flat universes, and calculate the fraction
that experience more than sixty $e$-folds of inflation. We will look at two
choices of potential: a massive scalar, and a pseudo-Goldstone boson. (We will
argue in the next section that these results are physically irrelevant, as
perturbations play a crucial role.)
We start with a massive scalar field with $m_{\phi}=3\times 10^{-3}m_{\rm
pl}$, which yields an amplitude of perturbations that agrees with
observations. We choose to evaluate the measure on the hypersurface
$H=1/\sqrt{3}$, so that the Friedmann equation becomes
$1=\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}m^{2}\phi^{2}$. After replacing the
divergence at zero curvature in (56) by a delta function, the normalized
measure becomes
$\mu=\frac{\sqrt{2}m}{\pi}\int_{H=1/\sqrt{3}}\sqrt{1-\frac{1}{2}m^{2}\phi^{2}}\,d\phi.$
(64)
(Recall that we have set $m_{\rm pl}=1/\sqrt{8\pi G}=1$.) The range of
integration is $|\phi|\leq\sqrt{2/m_{\phi}^{2}}$ (or $\rho_{\dot{\phi}}\leq
1$), corresponding to $V\leq 1$.
We used the Euler method with a time step $\Delta t=10^{-3}$ to numerically
follow the evolution of the scale factor and the scalar field. We find that
the universe undergoes more than sixty $e$-folds of inflation for all initial
values of $\phi$ except for the range $-24$ to $6$ if $\dot{\phi}>0$ (for
$\dot{\phi}<0$, the range would be $-6$ to $24$ due to the symmetry of the
potential). For simplicity, we disregard in our calculation the expansion that
occurs after the first period of slow-roll inflation. (We verified numerically
that subsequent periods of slow-roll expansion are relatively brief and lead
to very little further expansion.) Excluding the region $-24\leq\phi\leq 6$,
the measure (64) integrates to 0.99996. It seems highly likely to have more
than sixty $e$-folds of inflation by this standard.
As another example we consider inflation driven by a pseudo-Goldstone boson
[20] with potential
$V(\phi)=\Lambda^{4}(1+\cos(\phi/f)).$ (65)
In our calculation, we use $f=\sqrt{8\pi}$ and $\Lambda=10^{-3}$, so that the
model is consistent with WMAP3 data [21]. We evaluate the measure on the
hypersurface $H=H_{*}=\sqrt{4/3}\Lambda^{2}$, so that
$3H_{*}^{2}=2V_{max}=4\Lambda^{4}$. In this case, the normalized measure
becomes
$\mu=\frac{1}{8\sqrt{\pi}E[-1]}\int_{H_{*}=\sqrt{4/3}\Lambda^{2}}\sqrt{1-\frac{1}{4}\left(1+\cos{\frac{\phi}{\sqrt{8\pi}}}\right)}d\phi,$
(66)
where $E[m]$ is the elliptic integral
$\int_{0}^{2\pi}\sqrt{1-m\sin^{2}{t}}dt$. Numerically we find that the
universe expands by more than $60$ efolds for $-4.0<\phi<2.4$ if
$\dot{\phi}>0$ at $H=H_{*}=\sqrt{4/3}\Lambda^{2}$. (The evenness of the
potential allows us to consider only this branch of solutions.) Evaluating the
measure gives a probability of $0.171$. Notice that this is not too different
from the calculation in [20], which gives $0.2$ by assuming that $\phi$ is
randomly distributed between $0$ and $\sqrt{8\pi}$. We also note that the
probability is rather sensitive to the value of $f$; numerical evidence
suggest that it increases with $f$ (a flatter potential).
Both of these examples lead to the conclusion that inflation has a very
reasonable chance of occurring. Indeed, it is sometimes claimed that inflation
is an “attractor” (see e.g. [24]), but that is a misleading abuse of
nomenclature. It is a basic feature of dynamical systems theory that there are
no attractors in true Hamiltonian mechanics; Liouville’s theorem implies that
the total volume of a region of phase space remains constant under time
evolution. Attractors, in the rigorous sense of the word, only occur for
systems with dissipation. Inflation appears to be an attractor only because it
is often convenient to portray “phase portraits” in terms of the inflaton
$\phi$ and its time derivative, $\dot{\phi}$. But $\dot{\phi}$ is not the
momentum conjugate to $\phi$; as seen in (23), with the lapse function set to
$N=1$, it is $p_{\phi}=a^{3}\dot{\phi}$. Trajectories drawn on a
$(\phi,\dot{\phi})$ plot tend to approach a fixed point, but only because the
scale factor $a$ is dramatically increasing, not because of any true attractor
behavior.
These calculations of the likelihood of inflation are of dubious physical
relevance. Examining a single scalar field in minisuperspace is an extremely
unrealistic scenario. At a very simple level, if there are other massless
fields in the problem, any of them may share some of the energy density,
reducing the probability that the inflaton potential dominates. More
importantly, the role of perturbations is crucial. The real reason why
inflation is unlikely from the point of view of the canonical measure is not
because it is unlikely in minisuperspace, but because perturbations can easily
be sufficiently large to prevent inflation from ever occurring. We examine
this issue in detail in the next section.
## 5 Perturbations
The horizon problem is usually formulated in terms of the absence of causal
contact between widely-separated points in the early universe. Operationally,
however, it comes down to the fact that the universe is smooth over large
scales. We can investigate the measure associated with such universes by
looking at perturbed Robertson-Walker cosmologies. While the set of all
perturbations defines a large-dimensional phase space, in linear perturbation
theory we can keep things simple by looking at a single mode at a time. We
will find that, in contrast with the surprising result of the last section,
the measure on perturbations is just what we would expect – there is no
divergence at nearly-smooth universes. However, this implies that only an
imperceptibly small fraction of spacetimes were sufficiently smooth at early
times to allow for inflation to occur.
To calculate the measure for scalar perturbations, we need to first compute
the corresponding action. We are interested in universes dominated by
hydrodynamical matter such as dust or radiation. For linear scalar
perturbations, the coupled gravity-matter system can be described by a single
independent degree of freedom, as discussed by Mukhanov, Feldman and
Brandenberger [22]; we will follow closely the discussion in [23]. After
obtaining the action, we can isolate the dynamical variables and construct the
symplectic two-form on phase space, which can then be used to compute the
measure on the set of solutions to Einstein’s equations. A slight subtlety
arises because the corresponding Hamiltonian is time-dependent, but this is
easily dealt with.
### 5.1 Description of perturbations
In this section it will be convenient to switch to conformal time,
$\eta=\int a^{-1}dt.$ (67)
Derivatives with respect to $\eta$ are denoted by the superscript ′, and
$\widetilde{H}\equiv a^{\prime}/a$ is related to the Hubble parameter
$H=\dot{a}/a$ by $\widetilde{H}=aH$. The Friedmann equations become
$\displaystyle\widetilde{H}^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}a^{2}\bar{\rho}-k,$ (68) $\displaystyle\widetilde{H}^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{4\pi G}{3}a^{2}(\bar{\rho}+3\bar{p}),$
(69)
where $\bar{\rho}$ and $\bar{p}$ are the background density and pressure. In a
flat universe with only matter and radiation, in the radiation-dominated era
we have
$\eta({\rm RD})=\frac{a}{H_{0}\sqrt{a_{\rm eq}}},\qquad\widetilde{H}({\rm
RD})=\eta^{-1},$ (70)
where $a_{\rm eq}$ is the scale factor at matter-radiation equality, and now
we set the current scale factor to unity, $a_{0}=1$. Numerically, the
conformal time in the radiation-dominated era is approximately
$\eta(T)\approx\frac{5\times 10^{30}}{T({\rm eV})}\,{\rm eV}^{-1}.$ (71)
The metric for a flat RW universe in conformal time with scalar perturbations
is
$ds^{2}=a^{2}(\eta)\left[-(1+2\Phi)d\eta^{2}+2B_{,i}d\eta
dx^{2}+((1-2\Psi)\delta_{ij}+2E_{,ij})dx^{i}dx^{j}\right],$ (72)
where $\Phi$, $\Psi$, $E$, and $B$ are scalar functions characterizing metric
perturbations, and commas denote partial derivatives. It is useful to define
the gauge-invariant Newtonian potential,
$\Phi=\phi-\frac{1}{a}\left[a(B-E)^{\prime}\right]^{\prime},$ (73)
and the gauge-invariant energy-density perturbation,
$\widetilde{\delta\rho}=\delta\rho-\bar{\rho}^{\prime}(B-E^{\prime}).$ (74)
For scalar perturbations in the absence of anisotropic stress, these are
related by
$\widetilde{\delta\rho}=\frac{1}{4\pi
Ga^{2}}\left[\nabla^{2}\Phi-3\widetilde{H}(\Phi^{\prime}+\widetilde{H}\Phi)\right].$
(75)
For adiabatic perturbations ($\delta S=0$), the potential obeys an autonomous
equation,
$\Phi^{\prime\prime}+3(1+c_{s}^{2})\widetilde{H}\Phi^{\prime}-c_{s}^{2}\nabla^{2}\Phi+[2\widetilde{H}^{\prime}+(1+3c_{s}^{2})\widetilde{H}^{2}]\Phi=0,$
(76)
where $c_{s}^{2}=\partial p/\partial\rho$ is the speed of sound squared in the
fluid. This equation simplifies if we introduce the rescaled perturbation
variable
$u\equiv\frac{\Phi}{\sqrt{\bar{\rho}+\bar{p}}},$ (77)
and the time-dependent parameter
$\theta=\exp\left[\frac{3}{2}\int(1+c_{s}^{2})\widetilde{H}d\eta\right]\Phi=\frac{1}{a}\left[\frac{2}{3}\left(1-\frac{\widetilde{H}^{\prime}}{\widetilde{H}^{2}}\right)\right]^{-1/2}.$
(78)
In terms of these (76) becomes
$u^{\prime\prime}-c_{s}^{2}\nabla^{2}u-\frac{\theta^{\prime\prime}}{\theta}u=0.$
(79)
The variable $u$ is a single degree of freedom that encodes both the
gravitational potential [through (77)] and the density perturbation [through
(75)]. The equation of motion (79) corresponds to an action
$S_{u}=\frac{1}{2}\int d^{4}x\left(u^{\prime
2}-c_{s}^{2}u_{,i}u_{,i}+\frac{\theta^{\prime\prime}}{\theta}u^{2}\right).$
(80)
Defining the conjugate momentum $p_{u}=\partial{\mathcal{L}}/\partial
u^{\prime}=u^{\prime}$, we can describe the dynamics in terms of a Hamiltonian
density for an individual mode with wavenumber $k$,
$\mathcal{H}=\frac{1}{2}p_{u}^{2}+\frac{1}{2}\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)u^{2}.$
(81)
This is simply the Hamiltonian for a single degree of freedom with a time-
dependent effective mass $m^{2}=c_{s}^{2}k^{2}-\theta^{\prime\prime}/\theta$.
### 5.2 Computation of the measure
Given the Hamiltonian (81), we can straightforwardly compute the invariant
measure on phase space. One caveat is that now the Hamiltonian is time-
dependent, because the effective mass evolves. The carrier manifold of the
Hamiltonian therefore has an odd number of dimensions. We can retain the
symplecticity of a time-dependent Hamiltonian system (which requires an even
number of dimensions) by promoting time to be an addition canonical
coordinate, $q^{n+1}=t$. The conjugate momentum is minus the Hamiltonian,
$p_{n+1}=-\mathcal{H}$. We can then define an extended Hamiltonian by
$\mathcal{H}_{+}=\mathcal{H}(p,q,t)+p_{n+1}.$ (82)
This is formally time-independent, and recovers the original Hamiltonian
equations via
$\dot{q}_{i}=\frac{\partial\mathcal{H}_{+}}{\partial
p^{i}}\,,\quad\dot{p}^{i}=-\frac{\partial\mathcal{H}_{+}}{\partial q^{i}},$
(83)
along with two additional trivial equations $\dot{t}=1$ and
$\dot{\mathcal{H}}=\partial\mathcal{H}/\partial t$.
With $t$ promoted to a coordinate, the time-dependent Hamiltonian system also
comes equipped naturally with a closed symplectic two-form, now with an
additional term:
$\omega=\sum_{i=1}^{n}dp_{i}\wedge dq^{i}-d\mathcal{H}\wedge dt.$ (84)
The invariance of the form of Hamilton’s equations ensures that the Lie
derivative of $\omega$ with respect to the vector field generated by
$\mathcal{H}_{+}$ vanishes. The top exterior power of $\omega$ is then
guaranteed to be conserved under the extended Hamiltonian flow, and can thus
play the role of the Liouville measure for the augmented system. The GHS
measure can then be obtained by pulling back the Liouville measure onto a
hypersurface intersecting the trajectories and satisfying the constraint
$\mathcal{H}_{+}=0$.
In our case, the original system, with coordinate $u$ and conjugate momentum
$p_{u}$, is augmented to one with two coordinates $u$ and $\eta$ and their
conjugate momenta $p_{u}$ and $-\mathcal{H}$. The extended Hamiltonian,
$\mathcal{H}_{+}=\frac{1}{2}p_{u}^{2}+\frac{1}{2}\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)u^{2}-\mathcal{H},$
(85)
is time-independent and set to zero by the equations of motion. Its
conservation is analogous to the Friedmann equation constraint in the analysis
of the flatness problem. Using (84), the GHS measure $\Theta$ for the
perturbations is the two-form
$\displaystyle\Theta$ $\displaystyle=$ $\displaystyle dp_{u}\wedge
du-(d\mathcal{H}\wedge d\eta)|_{\mathcal{H}_{+}=0}$ (86) $\displaystyle=$
$\displaystyle dp_{u}\wedge
du-\frac{1}{2}d\left[p_{u}^{2}+\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)u^{2}\right]\wedge
d\eta$ $\displaystyle=$ $\displaystyle dp_{u}\wedge du-p_{u}(dp_{u}\wedge
d\eta)-u\left(c_{s}^{2}k^{2}-\frac{\theta^{\prime\prime}}{\theta}\right)du\wedge
d\eta\,.$
One convenient hypersurface in which we can evaluate the flux of trajectories
is $\eta=\eta_{*}=\mbox{constant}$. (This is equivalent to a surface of
$H=\mbox{constant}$ or $a=\mbox{constant}$, although those are not coordinates
in the phase space of the perturbation.) As $\eta$ is always positive in a
matter- and radiation-dominated universe, this surface intersects all
trajectories exactly once. We then have
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\int_{\eta=\eta_{*}}\Theta_{p_{u}u}dudp_{u}$ (87)
$\displaystyle=$ $\displaystyle\int_{\eta=\eta_{*}}dudp_{u}.$
The flux of trajectories crossing this surface is unity, implying that all
values for $u$ and $p_{u}$ are equally likely. There is nothing in the measure
that would explain the small observed values of perturbations at early times.
Hence, the observed homogeneity of our universe does imply considerable fine-
tuning; unlike the flatness problem, the horizon problem is real.
### 5.3 Likelihood of inflation
We can use the canonical measure on perturbations to estimate the likelihood
of inflation. Our strategy will be to consider universes dominated by matter
and radiation – i.e., the supposed post-inflationary era in the universe’s
history – and ask what fraction of them could have begun with inflation. This
is somewhat contrary to the conventional approach, which might start with an
assumed early state of the universe and ask whether inflation will begin; but
it is fully consistent with the philosophy of unitary and autonomous
evolution, and takes advantage of the feature of the canonical measure that it
can be evaluated at any time.
If inflation does occur, perturbations will be very small when it ends.
Indeed, perturbations must be sub-dominant if inflation is to begin in the
first place [15], and by the end of inflation only small quantum fluctuations
in the energy density remain. It is therefore a necessary (although not
sufficient) condition for inflation to occur that perturbations be small at
early times. For convenience, we will take inflation to end near the GUT
scale, $T_{\rm G}=10^{16}$ GeV ($\eta_{\rm G}\approx 6\times 10^{5}\,{\rm
eV}^{-1}$), although this choice is not crucial.
We therefore want to calculate what fraction of perturbed Robertson-Walker
universes are relatively smooth near the GUT scale. We take “smooth” to mean
that both the density contrast $\delta=\widetilde{\delta\rho}/\bar{\rho}$ and
the Newtonian potential $\Phi$ are less than one. Because the phase space is
unbounded and the measure (87) is flat, it is necessary to cut off the space
of perturbations in some way. We might define “realistic” cosmologies as those
that match the homogeneity of our observed universe at the redshift of
recombination $z\sim 1200$, when CMB temperature anisotropies are observed.
Our expressions will be much less cumbersome, however, if we demand smoothness
at matter-radiation equality, $z_{\rm eq}\sim 3000$ ($\eta_{{\rm eq}}\approx
10^{31}\,{\rm eV}^{-1}$), within an order of magnitude of recombination. Since
the observed temperature anisotropies are of order one part in $10^{5}$, we
therefore define a realistic universe as one with $\delta_{\rm eq}\leq
10^{-5}$ and $\Phi_{\rm eq}\leq 10^{-5}$.
There is a long-distance cutoff on the modes we consider given by the size of
our comoving observable universe, extrapolated back to matter-radiation
equality. The size $L_{0}$ of our observable universe today is a few times
$H_{0}^{-1}=10^{33}~{}{\rm eV}^{-1}$, and the size of our comoving patch at
equality is $a_{\rm eq}=1/3000$ times that, so
$L_{\rm eq}\approx 10^{30}\,{\rm eV}^{-1}.$ (88)
We will also impose a short-distance cutoff at the Hubble radius at equality,
$H_{\rm eq}^{-1}\approx m_{\rm pl}\left(\frac{a_{\rm
eq}^{2}}{T_{0}^{2}}\right)\approx 10^{28}\,{\rm eV}^{-1}.$ (89)
The total number of modes we consider is therefore
$n=\left(\frac{L_{\rm eq}}{H_{\rm eq}^{-1}}\right)^{3}\approx 10^{6}.$ (90)
Our short-distance cutoff is chosen primarily for convenience; there is a
natural ultraviolet cutoff set by the scale below which the hydrodynamical
approximation becomes invalid, but that is much shorter than $H_{\rm
eq}^{-1}$. It is clear that we are neglecting a large number of modes that
could plausibly have large amplitudes at early times; our result will
therefore represent a generous overestimate of the fraction of inflationary
spacetimes. The final numerical answer will be small enough that this shortcut
won’t matter.
With this setup in place, we would like to compare the measure on trajectories
that are smooth near the GUT scale to the measure on those that are realistic
near matter-radiation equality. We therefore only need to consider a single
kind of evolution – long-wavelength modes (super-Hubble-radius,
$c_{s}k\eta<1$) in a radiation-dominated universe. In that case the general
solution to our evolution equation (79) is
$u=c_{1}\theta+c_{2}\theta\int_{\eta_{0}}\theta^{-2}\,d\eta,$ (91)
where $c_{1}$ and $c_{2}$ are constants. During radiation domination we have
$\theta=\frac{\sqrt{3}}{2\sqrt{a_{\rm eq}}H_{0}}\eta^{-1}.$ (92)
The solution for $u$ is therefore
$u=\alpha\eta^{-1}+\beta\eta^{2},$ (93)
where $\alpha$ and $\beta$ are constants. The conjugate momentum is
$p_{u}=-\alpha\eta^{-2}+2\beta\eta.$ (94)
The potential is related to $u$ by (77). In the radiation era we have
$(\bar{\rho}+\bar{p})^{1/2}=\gamma\eta^{-2},$ (95)
where we have defined
$\gamma=\frac{2m_{\rm pl}}{\sqrt{a_{\rm eq}}H_{0}}.$ (96)
Our general solution is therefore
$\Phi=\gamma(\alpha\eta^{-3}+\beta).$ (97)
Finally we turn to the density perturbation, which is given by (75). The
$\nabla^{2}\Phi=-k^{2}\Phi$ term is negligible for long wavelengths, so we’re
left with
$\widetilde{\delta\rho}=\frac{12m_{\rm pl}^{3}}{a_{\rm
eq}^{3/2}H_{0}^{3}}(2\alpha\eta^{-7}-\beta\eta^{-4}),$ (98)
which in turn implies
$\delta\equiv\frac{\widetilde{\delta\rho}}{\bar{\rho}}=2\gamma(2\alpha\eta^{-3}-\beta).$
(99)
To calculate the measure, it is convenient to use $\alpha$ and $\beta$ as the
independent variables that specify a mode. The measure is simply
$\mu=\int dudp_{u}=3\int d\alpha d\beta.$ (100)
This comes from taking the derivative of (93) and (94), treating $\alpha$ and
$\beta$ as the independent variables, and computing $du\wedge dp_{u}$. No
$\eta$-dependent factors appear when we write the measure in terms of $\alpha$
and $\beta$. We can also express it in terms of the density contrast and
Newtonian potential,
$d\Phi d\delta=6\gamma^{2}\eta^{-3}d\alpha d\beta.$ (101)
Therefore, a region in the $\Phi$-$\delta$ plane at time $\eta_{\rm G}$ has a
measure that is larger than the same coordinate region at time $\eta_{\rm eq}$
by a factor of
$\left(\frac{\eta_{\rm eq}}{\eta_{\rm G}}\right)^{3}\approx 10^{76}.$ (102)
The coordinate area of our initial region at the GUT scale is $\Delta\Phi_{\rm
G}\Delta\delta_{\rm G}\approx 1$, while the coordinate area of our region at
equality is $\Delta\Phi_{{\rm eq}}\Delta\delta_{{\rm
eq}}\approx(10^{-5})^{2}=10^{-10}$. For each mode, we therefore have
$\frac{\mu(\mathrm{inflationary})}{\mu(\mathrm{realistic})}=\left(\frac{\eta_{\rm
G}}{\eta_{\rm eq}}\right)^{3}\frac{\Delta\Phi_{\rm G}\Delta\delta_{\rm
G}}{\Delta\Phi_{{\rm eq}}\Delta\delta_{{\rm eq}}}\approx 10^{-66}.$ (103)
This is saying that, for a given wave vector, only $10^{-66}$ of the allowed
amplitudes that are realistic at matter-radiation equality are small at the
GUT scale. To allow for inflation, we require that modes of every fixed
comoving wavelength and direction be less than unity at the GUT scale; the
fraction of realistic cosmologies that are eligible for inflation is therefore
$P({\rm inflation})\approx(10^{-66})^{n}\approx 10^{-6.6\times 10^{7}}.$ (104)
This is a small number, indicating that a negligible fraction of universes
that are realistic at late times experienced a period of inflation at very
early times. We derived this particular value by assuming the universe was
realistic at matter-radiation equality, but similarly tiny fractions would
apply had we started with any other time in the late universe. We also looked
at only a fraction of possible modes, so a more careful estimate would yield a
much smaller number. Indeed, using entropy as a proxy for the number of states
yields estimates of order $10^{-10^{122}}$ [4]. Clearly, the precise numerical
answer is not of the essence; the conclusion is that inflationary trajectories
are a negligible fraction of all possible evolutions of the universe.
A crucial feature of this analysis is that we allowed for the possibility of
_decaying_ cosmological perturbations; if all we know about the perturbations
is that they are small at matter-radiation equality, the generic case is that
many have been decaying since earlier times. Such decaying modes are often
neglected in cosmology, but for our purposes that would be begging the
question. A successful theory of cosmological initial conditions will account
for the absence of such modes, not presume it.
## 6 Discussion
We have investigated the issue of cosmological fine-tuning under the
assumption that our observable universe evolves unitarily through time. Using
the invariant measure on cosmological solutions to Einstein’s equation, we
find that the flatness problem is an illusion; in the context of purely
Robertson-Walker cosmologies, the measure diverges on flat universes. In the
case of deviations from homogeneity, however, we recover something closer to
the conventional result; in appropriate variables, the measure on the phase
space of any particular mode of perturbation is flat, so that a generic
universe would be expected to be highly inhomogeneous.
Inflation by itself cannot solve the horizon problem, in the sense of making
the smooth early universe a natural outcome of a wide variety of initial
conditions. The assumptions of unitarity and autonomy applied to our comoving
patch imply that any set of states at late times necessarily corresponds to an
equal number of states at early times. Different choices for the Hamiltonian
relevant in the early universe cannot serve to focus or spread the
trajectories, which would violate Liouville’s theorem; they can only deflect
the trajectories in some overall way. Therefore, whether or not a theory
allows for inflation has no impact on the total fraction of initial conditions
that lead to a universe that looks like ours at late times.
This basic argument has been appreciated for some time; indeed, its essential
features were outlined by Penrose [4] even before inflation was invented.
Nevertheless, it has failed to make an important impact on most discussions of
inflationary cosmology. Attitudes toward this line of inquiry fall roughly
into three camps: a small camp who believe that the implications of
Liouville’s theorem represent a significant challenge to inflation’s purported
ability to address fine-tuning problems [8, 9, 11, 12, 13]; an even smaller
camp who explicitly argue that the allowed space of initial conditions is much
smaller than the space of later conditions, in apparent conflict with the
principles of unitary evolution [10, 14]; and a very large camp who choose to
ignore the issue or keep their opinions to themselves.
But this issue is crucial to understanding the role of inflation (or any
alternative mechanism) in accounting for the apparent fine-tuning of our
universe. The part of the universe we observe consists of a certain set of
degrees of freedom, arranged in a certain way – a few hundred billion
galaxies, distributed approximately uniformly through an expanding space – and
apparently evolving from a very finely-tuned smooth Big-Bang-like beginning.
Understanding why things are this way could have crucial consequences for our
view of other features of the universe, much as the inflationary scenario
revolutionized our ideas about the origin of cosmological perturbations.
There seem to be two possible ways we might hope to account for the apparent
fine-tuning of the history of the observable universe:
1. 1.
The present configuration of the universe only occurs once. In this case, the
evolution from the Big Bang to today is highly non-generic, and the question
becomes why this evolution, rather than some other one. The answer might be
found in properties of the wave function of the universe (e.g. [25]).
2. 2.
Degrees of freedom arrange themselves in configurations like the observable
universe many times in the history of a much larger multiverse. In this case,
there is still hope that the overall evolution may be generic, if it can be
shown that configurations like ours most often occur in the aftermath of a
smooth Big Bang. The apparent restrictions of Liouville’s theorem may be
circumvented by imagining that the degrees of freedom of our current universe
do not describe a closed system for all time, but interact strongly with other
degrees of freedom at some times (e.g. by arising as baby universes [12]).
In either case, inflation could play an important role as part of a more
comprehensive picture. While inflation does not make universes like ours more
numerous in the space of all possible universes, it might provide a more
reasonable target for a true theory of initial conditions, from quantum
cosmology or elsewhere. (This is a possible reading of [10, 14], although
those authors seem to exclude non-smooth initial conditions a priori, rather
than relying on some well-defined theory of initial conditions.)
As we have shown in this paper, most universes that are smooth at matter-
radiation equality were wildly inhomogeneous at very early times. But the
converse is not true; most universes that were wildly inhomogeneous at early
times simply stay that way. The process of smoothing out represents a
violation of the Second Law of Thermodynamics, as entropy decreases along the
way. Even though the vast majority of trajectories that are smooth at matter-
radiation equality were inhomogeneous at early times, it seems intuitively
unlikely that the real universe behaves this way; much more plausible is the
conventional supposition that the universe was smoother (and entropy was
lower) all the way back to the Big Bang.
One way of expressing why this seems more natural to us is that the
corresponding initial states are very simple to characterize: they are smooth
within an appropriate comoving volume. In contrast, the much more numerous
histories that begin inhomogeneously and proceed to smooth out are impossible
to characterize in terms of macroscopically observable quantities at early
times; the fact that they will ultimately smooth out is hidden in extremely
subtle correlations between a multitude of degrees of freedom.333The situation
resembles the time-reversal of a glass of water with an ice cube that melts
over the course of an hour. At the end of the melting process, if we reverse
the momentum of every molecule in the glass, we will describe an initial
condition that evolves into an ice cube. But there’s no way of knowing that,
just from the macroscopically available information; the surprising future
evolution is hidden in subtle correlations between different molecules. It
seems much easier to imagine that an ultimate theory of initial conditions
will produce states that are simple to describe rather than ones that feature
an enormous number of mysterious and inaccessible correlations. It may be true
that a randomly-chosen universe like ours would have begun in a wildly
inhomogeneous state; but it’s clear that the history of our observable
universe is not a randomly-chosen evolution of the corresponding degrees of
freedom.
Given that we need some theory of initial conditions to explain why our
universe was not chosen at random, the question becomes whether inflation
provides any help to this unknown theory. There are two ways in which it does.
First, inflation allows the initial patch of spacetime with a Planck-scale
Hubble parameter to be physically small, while conventional cosmology does
not. If we extrapoloate a matter- and radiation-dominated universe from today
backwards in time, a comoving patch of size $H_{0}^{-1}$ today corresponds to
a physical size $\sim 10^{-26}H_{0}\sim 10^{34}L_{\rm pl}\sim 1$ cm when
$H=m_{\rm pl}$. In contrast, with inflation, the same patch needs to be no
larger than $L_{\rm pl}$ when $H=m_{\rm pl}$, as emphasized by Kofman, Linde,
and Mukhanov [10, 14]. If our purported theory of initial conditions, whether
quantum cosmology or baby-universe nucleation or some other scheme, has an
easier time making small patches of space than large ones, inflation would be
an enormous help.
The other advantage is in the degree of smoothness required. Without
inflation, a perfect-fluid universe with Planckian Hubble parameter would have
to be extremely homogeneous to be compatible with the current universe, while
an analogous inflationary patch could accommodate any amount of sub-Planckian
perturbations. While the actual number of trajectories may be smaller in the
case of inflation, there is a sense in which the requirements seem more
natural. Within the set of initial conditions that experience sufficient
inflation, all such states give us reasonable universes at late times; in a
more conventional Big Bang cosmology, the perturbations require an additional
substantial fine tuning. Again, we have a relatively plausible target for a
future comprehensive theory of initial conditions: as long as inflation
occurs, and the perturbations are not initially super-Planckian, we will get a
reasonable universe.
These features of inflation are certainly not novel; it is well-known that
inflation allows for the creation of a universe such as our own out of a small
and relatively small bubble of false vacuum energy. We are nevertheless
presenting the point in such detail because we believe that the usual sales
pitch for inflation is misleading; inflation does offer important advantages
over conventional Friedmann cosmologies, but not necessarily the ones that are
often advertised. In particular, inflation does not by itself make our current
universe more likely; the number of trajectories that end up looking like our
present universe is unaffected by the possibility of inflation, and even when
it is allowed only a tiny minority of solutions feature it. Rather, inflation
provides a specific kind of set-up for a true theory of initial conditions –
one that is yet to be definitively developed.
## Acknowledgments
This work was supported in part by the U.S. Dept. of Energy and the Gordon and
Betty Moore Foundation. We thank Andy Albrecht, Adrienne Erickcek, Don Page,
and Paul Steinhardt for helpful conversations.
## 7 Appendix: Eternal Inflation
Eternal inflation [26, 27, 28, 29] is sometimes held up as a solution to the
puzzle of the unlikeliness of inflation occurring. In many models of
inflation, the process is eternal – while some regions reheat and become
radiation-dominated, other regions (increasing in physical volume) continue to
inflate. This may be driven by the back-reaction of large quantum fluctuations
in the inflaton during slow-roll inflation, or simply by the failure of
percolation in a false vacuum with a sufficiently small decay rate.
Through eternal inflation, a small initially inflating volume grows without
bound, creating an ever-increasing number of pocket universes that expand and
cool in accordance with conventional cosmology. Therefore, the reasoning goes,
it doesn’t matter how unlikely it is that inflation ever begins; as long as
there is some nonzero chance that it starts, it creates an infinite number of
universes within the larger multiverse, and questions of probability become
moot.
If unitary evolution is truly respected, this reasoning fails. Consider the
state of the universe at some late time $t_{*}$ (long after inflation has
begun), in some particular slicing. Let us imagine that the basic idea of
eternal inflation is correct, and the multiverse consists of more and more
localized universes of ever-increasing volume as time passes. According to the
reasoning developed in this paper, the macroscopic state of the multiverse
(that is, the set of microstates with macroscopic features identical to the
multiverse at time $t_{*}$) will be compatible with a very large number of
past histories, only a very small fraction of which will begin in a single
inflating patch. The more the volume grows and the more universes that are
created, the _less_ likely it is that this particular configuration began with
such a patch. It requires more and more fine-tuning to take all of the degrees
of freedom and evolve them all backward into their vacuum states in a Planck-
sized region. Therefore, while eternal inflation can create an ever-larger
volume, it does so at the expense of starting in an ever-smaller fraction of
the relevant phase space.
To say the same thing in a different way, if a multiverse mechanism is going
to claim to solve the cosmological fine-tuning problems, it will have to be
the case that the mechanism applies to generic (or at least relatively common)
initial data. We should be able to start from a non-finely-tuned state, evolve
it into the future (and the past), and see universes such as our own arise. As
conventionally presented, models of eternal inflation usually presume a
starting point that is a smooth patch with a Planckian energy density – very
far from a generic state. If it could be shown that eternal inflation began
from generic initial data, this objection would be largely overcome.
Presumably the resulting multiverse would be time-symmetric on large scales,
as in [34, 12, 25].
It is possible that considering the entire multiverse along a single time
slice is illegitimate, and we should follow the philosophy of horizon
complemenarity and only consider spacetime patches that are observable by a
single worldline. This approach would run into severe problems with Boltzmann
brains if our current de Sitter vacuum is long-lived [30, 31, 32, 33].
Alternatively, we might argue that the phase space is infinitely big, and
there is no sensible way to talk about probabilities. That may ultimately be
true, but represents an abandonment of any hope of explaining cosmological
fine-tuning via inflation, rather than a defense of the strategy.
Analogous concerns apply to cyclic cosmologies [35]. Here, conditions similar
to our observable universe happen multiple times, separated primarily in time
rather than in space. But the burden still remains to show that the
conjectured evolution would proceed from generic initial data. The fact that
the multiverse is not time-symmetric (the arrow of time points in a consistent
direction from cycle to cycle) makes this seem unlikely.
## References
* [1] A. H. Guth, Phys. Rev. D 23, 347 (1981).
* [2] A. D. Linde, Phys. Lett. B 108, 389 (1982).
* [3] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
* [4] R. Penrose, in S. W. Hawking and W. Israel. General Relativity: An Einstein Centenary Survey. Cambridge University Press. pp. 581 638. (1979).
* [5] G. W. Gibbons, S. W. Hawking and J. M. Stewart, Nucl. Phys. B 281, 736 (1987).
* [6] S. W. Hawking and D. N. Page, Nucl. Phys. B 298, 789 (1988).
* [7] D. H. Coule, Class. Quant. Grav. 12, 455 (1995) [arXiv:gr-qc/9408026].
* [8] W. G. Unruh, In *Princeton 1996, Critical dialogues in cosmology* 249-264.
* [9] S. Hollands and R. M. Wald, Gen. Rel. Grav. 34, 2043 (2002) [arXiv:gr-qc/0205058].
* [10] L. Kofman, A. Linde and V. F. Mukhanov, JHEP 0210, 057 (2002) [arXiv:hep-th/0206088].
* [11] S. Hollands and R. M. Wald, arXiv:hep-th/0210001.
* [12] S. M. Carroll and J. Chen, arXiv:hep-th/0410270.
* [13] G. W. Gibbons and N. Turok, arXiv:hep-th/0609095.
* [14] A. Linde, arXiv:0705.0164 [hep-th].
* [15] T. Vachaspati and M. Trodden, Phys. Rev. D 61, 023502 (1999) [arXiv:gr-qc/9811037].
* [16] S. D. Mathur, J. Phys. Conf. Ser. 140, 012009 (2008) [arXiv:0803.3727 [hep-th]].
* [17] C. A. Egan and C. H. Lineweaver, Astrophys. J. 710, 1825 (2010) [arXiv:0909.3983 [astro-ph.CO]].
* [18] B. Greene, K. Hinterbichler, S. Judes and M. K. Parikh, arXiv:0911.0693 [hep-th].
* [19] J. Garriga and V. F. Mukhanov, Phys. Lett. B 458, 219 (1999) [arXiv:hep-th/9904176].
* [20] K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. 65, 3233 (1990).
* [21] K. Freese, C. Savage and W. H. Kinney, Int. J. Mod. Phys. D 16, 2573 (2008) [arXiv:0802.0227 [hep-ph]].
* [22] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203 (1992).
* [23] V. F. Mukhanov, _Physical Foundations of Cosmology_ (2005), Cambridge University Press.
* [24] M. Trodden and S. M. Carroll, arXiv:astro-ph/0401547.
* [25] J. B. Hartle, S. W. Hawking and T. Hertog, Phys. Rev. D 77, 123537 (2008) [arXiv:0803.1663 [hep-th]].
* [26] A. Vilenkin, Phys. Rev. D 27, 2848 (1983).
* [27] A. D. Linde, Mod. Phys. Lett. A 1, 81 (1986).
* [28] A. D. Linde, Phys. Lett. B 175, 395 (1986).
* [29] A. H. Guth, J. Phys. A 40, 6811 (2007) [arXiv:hep-th/0702178].
* [30] L. Dyson, M. Kleban and L. Susskind, JHEP 0210, 011 (2002) [arXiv:hep-th/0208013].
* [31] A. Albrecht and L. Sorbo, Phys. Rev. D 70, 063528 (2004) [arXiv:hep-th/0405270].
* [32] D. N. Page, Phys. Rev. D 78, 063535 (2008) [arXiv:hep-th/0610079].
* [33] R. Bousso and B. Freivogel, JHEP 0706, 018 (2007) [arXiv:hep-th/0610132].
* [34] A. Aguirre and S. Gratton, Phys. Rev. D 67, 083515 (2003) [arXiv:gr-qc/0301042].
* [35] P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 126003 (2002) [arXiv:hep-th/0111098].
|
arxiv-papers
| 2010-07-08T16:55:24 |
2024-09-04T02:49:11.511144
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sean M. Carroll and Heywood Tam",
"submitter": "Sean Carroll",
"url": "https://arxiv.org/abs/1007.1417"
}
|
1007.1495
|
# An ab initio design of cluster-assembled silicon nanotubes
Lingju Guo Xiaohong Zheng Chunsheng Liu Zhi Zeng Corresponding author.
E-mail: zzeng@theory.issp.ac.cn Key Laboratory of Materials Physics,
Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031,
P.R. China
###### Abstract
Density functional calculations were performed to systematically study a
series of finite and infinite cluster-assembled silicon nanotubes (SiNTs).
One-dimensional SiNTs can be prepared by proper assembly of hydrogenated cage-
like silicon clusters to form semiconductors with a large band gap, and their
electronic properties can be accurately tuned by transition metal doping in
the center of the tubes. Specifically, doping with Fe made the SiNTs metallic
and magnetic materials. More interestingly, a metal to half-metal transition
was observed with increasing tube radius in Fe-doped SiNTs, which demonstrates
that SiNTs doped with magnetic elements may find important applications in
spintronics.
Keywords: Hydrogenated silicon clusters, HOMO-LUMO gap order, cluster-
assembled nanotube, magnetic properties, metal half-metal transition
###### pacs:
73.22.-f, 36.40.Cg, 61.46.Fg, 75.75.+a
## I INTRODUCTION
Since the discovery and application of carbon fullerenes and carbon
nanotubes(CNTs) fullerene , stable cage and tube-like structures have
attracted a great deal of attention. These structures have generated great
interest in creating analogous structures from other elements that are
suitable for applications in nanodevices. Silicon and carbon are members of
the same group in the periodic table, suggesting a potential ability to form
similar structures. Furthermore, due to the fundamental importance of silicon
in present-day integrated circuits, substantial efforts have focused on
investigating nano-scale forms of silicon, both for the purpose of further
miniaturizing the current microelectronic devices and in the hopes of
unveiling new properties that often arise at the nano-scale levelJPCM_16_1373
. However, it is difficult to form cages or tubes like carbon fullerenes or
nanotubes purely with Si atoms because silicon does not favor the $sp$2
hybridization that carbon does. Carbon normally forms strong $\pi$ bonds
through $sp^{2}$ hybridization, which can facilitate the formation of two-
dimensional spherical cages (or planar structures such as benzene and
graphene). Silicon, on the other hand, usually forms covalent $\sigma$ bonds
through $sp^{3}$ hybridization, which favor a three dimensional diamond-like
structure.
Fortunately, Si cage clusters can be synthesized by adding suitable foreign
atoms to terminate the dangling Si bonds that inherently arise in cage-like
networks. Many researchersjctn257 ; cms1 ; prb195417 reported that transition
metal(TM) atoms are the most suitable element for cage formation due to their
$d$ band features. In addition, rare earth atoms have been doped into silicon
cagesjcp084711 ; epjd343 ; prb125411 ; prb115429 . Another way to stabilize
the Si cage is to use hydrogens to terminate the cluster surfaceprb075402 ;
prb155425 , which is similar to the dodecahedral C20H20.
Several hollow and nonhollow silicon nanotube structures have been proposed
and theoretically characterized in recent years pnas2664 ; jmst127 ; prl265502
; nl301 ; nl1243 ; nano109 ; jmc555 ; njp78 ; prl146802 ; cpl81 ; prb075420 ;
prb195426 ; jpcb7577 ; jpcb8605 ; prb9994 ; jpcc5598 ; prb11593 ; prb205315 ;
prb075328 ; ss257 ; prb193409 ; jpcc16840 ; prl1958 ; pssr7 ; prb155432 ;
prl792 ; jpcc1234 . These structures, which were proposed based on intuition
or the behavior of similar materials, include the following:
(1) Regular polyhedron stacking nanotubes pnas2664 ,
(2) Surface capped polygonal prism stacking nanotubes jmst127 ,
(3) Polycrystalline forms of nanowires prl265502 ,
(4) Metal encapsulated polygonal prism nanotubes nl301 ; nl1243 ; nano109 ;
jmc555 ; njp78 ; prl146802 ,
(5) Carbon nanotube like structures cpl81 ; prb075420 ; prb195426 ; jpcb7577 ;
jpcb8605 ; prb9994 ; jpcc5598 ,
(6) Fullerene-based structures prb11593 ; prb205315 ,
(7) Metal centered fullerene-based structures prb075328 ,
(8) Hydrogenated single-wall silicon CNT like nanotubes ss257 ; prb193409 ;
jpcc16840 ,
(9) Exohydrogenated fullerene-like structures prl1958 ; pssr7 ,
(10) Multiwall nanotubes prb155432 ; prl792 ; jpcc1234 .
In the cases of (1), (3), (5) and (6), similar to pure silicon cages, it was
reported to be difficult to form hollow single wall Si nanotubes because of
the $sp^{3}$ nature of Si atom. Cases (4) and (7) proposed that metal doping
may be a good way to terminate the dangling bond of silicon bond in the tube,
but this approach was limited to tubes with very small radii($R$$\leq$1nm).
However, if the lateral surfaces of the dangling bonds are terminated by
hydrogens, the resulting cage-like silicon clusters may be perfect building
blocks for Si nanotubes. Theoretically, every Si atom has three neighboring Si
atoms in the nanotube, and these have an $sp$3 type bond nature with one H
terminating the dangling bond. Moreover, the success of this approach is
suggested by the fact that in experiments, the surfaces of Silicon
nanowires(SiNWs) and silicon nanotubes(SiNTs) are always passivated by
hydrogen atoms science1874 ; jpcb8605 or by silicon oxide layers am1219 ;
am1172 ; am564 ; prl116102 .
This work presents a theoretical study of hydrogenated cluster assembled
silicon nanotubes using density functional theory(DFT) calculation. A series
of finite and infinite silicon nanotubes assembled by hydrogenated cage-like
clusters was obtained. Additionally, the electronic and magnetic properties of
SiNTs can be accurately tuned by doping impurities at the center of the hollow
tube. One interesting result was that a metal to half-metal transition was
observed in Fe-doped SiNTs with the increasing of tube radius.
## II COMPUTATIONAL DETAILS AND MODEL DESIGN
All theoretical computations were performed with the DFT approach implemented
in the Dmol3 package jcp92 ; jcp113 , using all electron treatment and the
double numerical basis including the $d$-polarization function(DNP)jcp92 . The
exchange-correlation interaction was treated within the generalized gradient
approximation(GGA) using BLYP functional. Self-consistent field calculations
were performed with a convergence criterion of 2$\times$10-5 Hartree on total
energy.
The $Si_{n}H_{n}$ ($n$=16, 20, 24, 28, 32) clusters were optimized first, and
some of the initial structures in our work were based on the results reported
in Refs. prb075402, & prb155425, . After obtaining stable single clusters, we
took these clusters as basic units (keep them as original) and stacked the
clusters with a certain $n$ along the axis of symmetry to construct finite
nanotubes.
For infinite one dimensional nanotubes, the periodically repeated units were
placed in supercells. The supercell is cubic in geometry with the dimensions
25Å$\times$25Å$\times$$L$z, where the direction of $L$z was defined as the
axial direction of the nanotubes. Periodic boundary conditions were employed
along the nanowire axis to create, in effect, continuous wires. Meanwhile, a
sufficiently large vacuum region was introduced along the other directions is
configured between the wires.
## III RESULTS AND DISCUSSIONS
### III.1 Structures of $Si_{n}H_{n}$ clusters and finite nanotubes
The fully optimized structures of $Si_{n}H_{n}$ ($n$=16, 20, 24, 28, 32)
clusters that have been fully optimized are shown in Fig. 1. It was very
interesting to see that all these structures shared the following common
characteristics: 1. All of them were fullerene-like hollow structures. 2. Each
Si atom had three Si neighbors, with one H atom saturating the dangling bond
of each silicon atom outside the cage to fulfill an $sp^{3}$ type
hybridization bond. 3. All structures consisted of $\frac{n}{2}$ pentagons and
two other polyhedrons at the two ends, with the edge number of these two
polyhedrons as $\frac{n}{4}$. Meanwhile, these two polyhedrons were parallel
to each other and there was a relative angle of $\frac{4\pi}{n}$ between them.
Thus each vertex atom of one polyhedron fell exactly in the middle of one edge
in the other polyhedron. Specifically, for $Si_{20}H_{20}$, the cage was
composed of 12 pentagons, which was very similar to the structure of carbon
fullerene $C_{20}H_{20}$. Structures of $Si_{16}H_{16}$, $Si_{24}H_{24}$ and
$Si_{28}H_{28}$ have been widely discussed in very recent years prb075402 ;
prb155425 and the structural information we obtained was consistent with these
reports.
Taking these original clusters original as basic units, we stacked them along
the central axis of the cage to form finite nanotubes. The two adjacent cages
shared the same polyhedron. We noted that for the shared polyhedron, there was
no need for have hydrogen saturation because each Si atom already had 4 Si
neighbors and thus the $sp^{3}$ hybridization bond type was fulfilled. The
molecular formula of the finite tube was classified as
$Si_{m(3k+1)}H_{2m(k+1)}$, where the measure of radius $m$ was the number of
atoms of one shared polyhedron and the measure of length $k$ was the number of
repeated units. After full optimization, for one certain value $m$($m$=4, 5,
6, 7), and for $k$ range from 2 to 4 concerned in the present work, the tube
was always straight and stable. Furthermore, if the number of repeated units
$k$ was fixed, the length of the tubes decreased with the increasing $m$. The
angles of H-Si-Si and Si-Si-Si inside the repeated units were all about 109∘,
which were very close to the 109.5∘ of $sp^{3}$ , but the Si-Si-Si angle
(angle $\alpha$ in Fig. 2 (b)) between two units became smaller and smaller
with the increasing of tube radii (from 127.8∘ of $m$=4 to 99.0∘ of $m$=8).
Even though, the growth direction and the achievable length of the nanotubes
are the main experimental concerns, previous theoretical studies have paid
little attention to these issues. Here, we chose $Si_{5(3k+1)}H_{10(k+1)}$
nanotubes to examine if straight tubes were more stable than bent nanotubes.
As shown in Fig. 2, $Si_{50}H_{40}$ ($m$=5, $k$=3)(Fig. 2 (a)) had an isomer
(a1) (Fig. 2 ($a_{1}$)) that was bent to 120∘ from 180∘, but the total energy
(ET) of ($a_{1}$) was 0.02eV higher than that of the straight tube ($a$).
Likewise, $Si_{65}H_{50}$ ($m$=5, $k$=4) has two isomers $b_{1}$(Fig. 2
($b_{1}$)) and $b_{2}$ (Fig. 2($b_{2}$)), where the structure $b_{1}$ was a
tube in which the unit at the end was bent, and structure $b_{2}$ was
distorted further. These isomers were less stable because the total energy of
($E_{T}$) was higher than that of the linear structure by 0.021$eV$ or
0.034$eV$, respectively. For hydrogenate silicon tubes, if one tube was bent
by an angle, the distance between H atoms of adjacent units would decrease and
the Coulomb repulsion between them would makes the total energy greater than
the straight energy. Consequently, the linear structure was relatively more
stable than bent structures.
### III.2 ELECTRONIC STRUCTURES
In order to measure the relative stability of the tubes as well as the
influence of the length and width, we calculated the averaged binding energy
($E_{b}$) and dissociation energies ($DE$) of these tubes. The $E_{b}$ and DE
for finite tubes were defined by the following formulae:
$\begin{split}E_{b}({k})=\\{m(3k+1)E_{T}&(Si)+2m(k+1)E_{T}(H)-E_{T}[Si_{3k+1}H_{2m(k+1)}]\\}/{(5mk+3m)}\\\
&(m=4-8;~{}k=1-4)\end{split}$ (1)
$\begin{split}\ DE({k})&=E_{T}(SiH)-E_{T}[Si_{3k+1}H_{2m(k+1)}]\\\
&(m=4-8;~{}k=1-4~{})\end{split}$ (2)
where $E_{T}(Si)$, $E_{T}(H)$, $E_{T}(SiH)$ and
$E_{T}$[$Si_{m(3k+1)}H_{2m(k+1)}$] represent the total energies of the Si
atom, H atom, SiH dimer and the $Si_{m(3k+1)}H_{2m(k+1)}$ tube, respectively.
As illustrated in Fig. 3 (a), the binding energy $E_{b}(k)$ of the finite tube
increases gradually as the length increased for a certain $m$. This
correlation indicates that the tube became increasingly stable as it became
longer. On the other hand, the stability of the tubes did not depend linearly
on the tube’s radius, which was represented by $m$. For any $k$, the most
stable tubes were at $m$=5 or $m$=6.
DE is the energy needed to remove a Si-H dimer from the end of a tube, and
this parameter provided another method to probe the tube stability. The curves
of DE (Fig. 3 (b)) indicate that it was more difficult to remove a Si-H dimer
from the tubes with $m$=5 and 6 because these two specific tubes satisfied
$sp^{3}$ hybridization well. This is another indication of the fact that tubes
with $m$=5 or 6 are the most stable.
In addition, Fig. 3 (c) plots the variation of the highest occupied molecular
orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gaps of the
finite tubes. The quantum confinement concept demands a larger gap for a
smaller size. The authors in Ref. prb075402, ever argued that the quantum
confinement concept was not fully applicable to saturated systems such as
$Si_{n}H_{n}$ cages. Our work also provides evidence on the limitation of the
quantum confinement concept in interpreting the gaps of hydrogenated silicon
nanotubes.
As seen in Fig. 3 (c), when $m$ is fixed, the gaps between HOMO-LUMO became
smaller with increasing length. However, there was a big jump in the gap from
a single cluster to a two-unit tube. This difference could come from a hybrid
bond around the Sim cylinder joints which was different from the fully
hydrogenated silicon atoms in a single cluster. Moreover, the Coulomb
repulsion between hydrogen atoms of an adjacent unit could play a role in
defining gaps, as explained below.
The tubes of $Si_{4(3k+1)}H_{2\times 4(k+1)}$, $Si_{5(3k+1)}H_{2\times
5(k+1)}$ and $Si_{6(3k+1)}H_{2\times 6(k+1)}$ had similar HOMO-LUMO gaps due
to analogous structural parameters including bond angle, length and width. The
gaps of larger radii tubes $Si_{7(3k+1)}H_{2\times 7(k+1)}$ and
$Si_{8(3k+1)}H_{2\times 8(k+1)}$ tubes were much smaller than those of the
smaller radii tubes $Si_{4(3k+1)}H_{2\times 4(k+1)}$, $Si_{5(3k+1)}H_{2\times
5(k+1)}$ and $Si_{6(3k+1)}H_{2\times 6(k+1)}$. Similar to Ref. prb075402, , we
estimated the effective volume per electron $V_{eff}^{(e)}$ by drawing two
concentric cylinders with $R_{H}$ and $R_{Si}$ in Fig. 2 (b) up to the layer
of hydrogen and silicon.
The $V_{eff}^{(e)}$ was defined as:
$V_{eff}^{(e)}=\frac{1}{N_{e}}\times\pi\times
L~{}[(\frac{1}{2}R_{H})^{2}-(\frac{1}{2}R_{Si})^{2}]$ (3)
where $N_{e}$ was the number of electron and $L$ was the length of the
cylinder (Fig. 2 (b)).
$V_{eff}^{(e)}$ is the measures of electron ”confinement”, which would lead to
the same order of gap performance according to the quantum confinement
concept. Hence, in our work, the order of effective volume per electron of
$V_{eff}^{(e)}$[$m$=7] $<$ $V_{eff}^{(e)}$[$m$=6] $<$ $V_{eff}^{(e)}$[$m$=8]
$<$ $V_{eff}^{(e)}$[$m$=4] $<$ $V_{eff}^{(e)}$[$m$=5], as in Table 1, would
have resulted in a HOMO-LUMO gap order of $\Delta_{G}$ [$m$=7]$>$ $\Delta_{G}$
[$m$=6] $>$ $\Delta_{G}$ [$m$=8] $>$ $\Delta_{G}$ [$m$=4] $>$ $\Delta_{G}$
[$m$=5]. However, as shown in Fig. 3(c) and Table 1, the HOMO-LUMO gap order
was $\Delta_{G}$ [$m$=6] $>$ $\Delta_{G}$ [$m$=4] $>$ $\Delta_{G}$ [$m$=5] $>$
$\Delta_{G}$ [$m$=7] $>$ $\Delta_{G}$ [$m$=8]. The HOMO-LUMO gap order only
followed the rule of quantum confinement for small radii tubes($m$=4, 5, 6).
The limitation of quantum confinement for large radii tubes ($m$=7, 8) may
arise from the fact that these tubes have a smaller distance between the H-H
bonds of neighboring units($D_{H}$ in Fig. 2 (b)). For example, $D_{H}$ of
4.206Å for $m$=4 compared to that of 2.723Å or 2.567Å for $m$=7 or 8. That is
to say, for one-dimensional finite hydrogenated silicon nanotubes, quantum
confinement only worked for small radii tubes.
To summarize, in addition to the quantum confinement, the effects of suitable
bond angle($\angle\alpha$ in Fig. 2 (b)), and Coulomb repulsion between H
atoms also played important roles in affecting the HOMO-LUMO gaps of
$Si_{m(3k+1)}H_{2m(k+1)}$ systems.
### III.3 INFINITE TUBES
The increased stability of finite nanotubes with an increasing number of units
allowed us to examine further the stability of the infinite nanotubes
specifical with a stoichiometric $Si_{6m}H_{4m}$. This infinite tube was built
from finite $Si_{m(3k+1)}H_{2m(k+1)}$ by removing a $Si_{m}$$H_{m}$ on one
side and $m$ H atoms on the other side to form a repeated unit. Fig. 4 shows
two repeated cells of the infinite tubes with different radii. All of these
tubes had the stacking of SiH cages in common, with the wire axis lying in the
center. The axis passed through the centers of buckled $Si_{m}$ rings, and two
adjacent cages shared one ring.
Full structure relaxation indicated that the infinite nanotubes had similar
geometric structures to finite ones, but the tube lengths had slightly
changed. The lenths of $Si_{6\times 4}H_{4\times 4}$ and $Si_{6\times
5}H_{4\times 5}$ tubes became 0.08Å and 0.11Å longer than the finite ones, but
for $Si_{6\times 6}H_{4\times 6}$, $Si_{6\times 7}H_{4\times 7}$ and
$Si_{6\times 7}H_{4\times 7}$ the lengths became 0.075Å, 0.212Å and 0.365Å
shorter. The tube widths and Si-H bond lengths were almost the same as for the
finite ones.
The binding energies of the infinite tube shown in Table 2 were slightly
larger than those of finite tubes, which meant that it was possible to
synthesize tubes long enough. The silicon tube had a relatively large band
gap($\Delta_{g}$) (2.296eV-3.009eV), which implied that it was a wide gap
semiconductor. The band gap of the tube was inversely proportional to the
radius.
The band gaps of carbon nanotubes always decrease in an oscillatory behavior
with increasing radius because of the $\pi^{\ast}$ and $\sigma$ hybridization
with a small radius and a large curvature. However, for exhydrogenated single-
wall carbon nanotubes(SWCNT), the gap decreases monotonously prb075404 . This
difference can be easily understood by the fact that silicon atoms have
$sp^{3}$-type bond properties. The band-gap trend of silicon nanotubes is
similar to that of hydrogenated SWCNTs as described in other reports prb075328
; prb193409 ; prb155435 on Si nanotubes or nanowires.
### III.4 IRON DOPED INFINITE TUBES
Endohedral doping is an effective way to tune the properties of cage or tube-
like clusters. Therefore, we next investigated the effect of endohedral doping
on the geometric stability, conductivity and magnetism of the silicon
nanotubes. Specially, Fe doping was systematically analyzed. The stability was
robust when Fe atoms were inserted at the center of the hollow tube(Fig. 4).
However, the lattice underwent a very small expansion and exiting large band
gap in silicon nanotube disappeared. Subsequently, the doped tube turned into
a metal.
In previous reports, many types of metal atoms were inserted at the center of
the silicon hollow nano-clusters cms1 ; prb195417 or nanotubes nl1243 ; nl301
; prb075328 ; jmc555 , but the primary role of the doping atom was to saturate
the dangling bond of silicon cages or tubes. Therefore, the central atom had a
strong hybridization with the exterior silicon cage, and the magnetic moments
always became very small, or at times completely quenched. However, in the
case presented here, the outside cage was saturated by hydrogen atoms. Thus,
the Fe atom had only a weak interaction with the silicon atoms. As seen in
Table 2, the doped tube always maintained a relatively large magnetic moment
that originated from the $d$-electron of the Fe atom, and the variation of the
Fe moment increased as the radius increased.
The most intriguing phenomenon occurred when the radii of the Fe-doped
hydrogenated silicon tubes increased: a metal to half-metal transition was
observed from the spin resolved density of states (DOS) for the $\alpha$ and
$\beta$ channels of nanotubes, as shown in Fig. 5. From $m$=4 to $m$=6 (Fig.
5(a)–(c)), both the $\alpha$ spin and the $\beta$ spin contributed to the DOS
at the Fermi level. With the increase in $m$, the DOS peak of the $\alpha$
spin at Fermi level shifted down, while the peak of the $\beta$ spin below the
Fermi level shifted up towards the Fermi level. When it comes to $m$=7 (Fig. 5
(d))and $m$=8 (Fig. 5 (e)), the $\alpha$ spin states disappeared at the Fermi
level, and a band gap appeared for this spin channel. For the $\beta$ spin
channel, a DOS peak still emerged at the Fermi level. In other words, the
nanotube turned into an insulator for $\alpha$ spin and a conductor for the
$\beta$ spin. From the spin-resolved DOS contributed by Fe, Si and H atoms
shown in Fig. 5(f), we found that the centered Fe atoms mainly contributed to
the total number of $\beta$ spin states, and Si atoms made only very modest
contributions to the states at the Fermi level.
The above phenomenon resulted from the hybridization between Fe and silicon
atoms. For small radii of $m$=4, Fe atoms have a stronger hybridization with
the outside silicon atoms, but the hybridization becomes smaller as the radius
increases. When $m$=7 and 8, the coupling between the Fe atom and the tube
becomes so weak that the Fe atom is almost an isolated atom. Thus, the atomic
properties of Fe atom were recovered to a large extent. The electron
configuration of Fe is 3$d$64$s$2. According to Pauli exclusion and Hund’s
rule, five 3$d$ electrons would occupy five $d$ orbitals with $\alpha$ spin,
while the last 3$d$ electron will occupy $d$ orbitals with $\beta$ spin. When
such atoms are weakly coupled together to form a linear chain, highly spin-
polarized behavior can result.
Such behavior was also been predicted in Durgun’s workprl256806 which found
that hydrogen passivated silicon nanowires could exhibit the half-metallic
state when decorated with specific transition metal(TM) atoms. This situation
was very similar to the half-metallicity introduced by the weakly coupled
carbon atomic chain in single-walled carbon nanotubesapl163105 . In this case,
the carbon atoms in the chain have a very weak interactions with their
neighbors and the atomic behavior is largely recovered.
The large magnetic moments of the Fe doped silicon nanotubes and their half-
metallic behavior are promising for application of these materials in magnetic
devices and spintronic applications.
## IV CONCLUSIONS
A series of finite and infinite hydrogenated silicon nanotubes were
systematically studied by performing first-principles DFT-GGA calculations.
Our results demonstrated that one-dimensional SiNTs $Si_{m(3k+1)}H_{2m(k+1)}$
can be built by stacking $Si_{k}H_{k}$ clusters along the central axis of the
cage. The finite tubes had large HOMO-LUMO gaps. When $m$=5 and 6, the tubes
were more stable than other sizes because these tubes had the most suitable
bond angles.
Infinite tubes had similar geometric structures to the finite tubes and
presented wide gap semiconductivity due to their large band gaps. The band gap
decreased monotonously as the radius increased. Endohedral Fe doping greatly
modified the properties of the silicon nanotubes. For example, Fe doping can
tune the silicon nanotube from semiconductor to metal. Additionally, a metal
to half-metal transition was observed in Fe-doped SiNTs with an increasing
tube radius. These silicon nanotubes could be useful for nanoelectronic
devices. In particular, a half-metallicity of the Fe-doped silicon nanotubes
may find important application in spintronics.
## V ACKNOWLEDGMENTS
This work was supported by the National Science Foundation of China under
Grants No. 10774148 and No. 10904148, the special Funds for Major State Basic
Research Project of China(973) under grant No. 2007CB925004, 863 Project,
Knowledge Innovation Program of the Chinese Academy of Sciences, and Director
Grants of CASHIPS. Some of the calculations were performed at the Center for
Computational Science of CASHIPS and at the Shanghai Supercomputer Center.
## References
* (1) (a) S. Iijima, Nature(London) 354, 56 (1991). (b) R. E. Smalley and B. I. Yakobson, Sol. Stat. Comm. 107, 597 (1998). (c) H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, Nature 318, 162 (1985). (d) K. Hedberg, L. Hedberg, D. S. Bethune, C. A. Brown, H. C. Dorn, R. D. Johnson and M. De Veries, Science 254, 410 (1991).
* (2) F. Rosei, J. Phys.: Condens. Matter 16 S1373 (2004).
* (3) J. G. Han and F. Hagelberg, J. Comput. Theor. Nanosci. 6, 257, (2009) references therein.
* (4) V. Kumar, comp. mater. sci. 36, 1 (2006).
* (5) L. J. Guo, G. F. Zhao, Y. Z. Gu, X. Liu and Z. Zeng, Phys. Rev. B 77, 195417 (2008).
* (6) Q. Peng and J. Shen, J. Chem. Phys. 128, 084711 (2008).
* (7) T. T. Cao, X. J. Feng, L. X. Zhao, X. Liang, Y. M. Lei and Y. H. Luo, Eur. Phys. J. D 49, 343 (2008).
* (8) V. Kumar, A. K. Singh and Y. Kawazoe, Phys. Rev. B 74, , 125411 (2006).
* (9) A. K. Singh, V. Kumar and Y. Kawazoe, Phys. Rev. B 71, 115429 (2006).
* (10) A. D. Zdetsis, Phys. Rev. B 76, 075402 (2007).
* (11) V. Kumar, Y. Kawazoe, Phys. Rev. B 75, 155425 (2007).
* (12) J. Bai, X. C. Zeng, H. Tanaka and J. Y. Zeng, Proc. Natl. Acad. Sci. USA 101, 2664 (2004).
* (13) B. X. li and P. L. Cao, J. Mol. Struct.:THEOCHEM 679, 127 (2004).
* (14) I. Ponomareva, M. Menon, D. Srivastava and A. N. Andriotis, Phys. Rev. Lett. 95, 265502 (2005).
* (15) M. Menon, A. N. Andriotis and G. E. Froudakis, Nano Lett. 2, 301(2002).
* (16) A. K. Singh, V. Kumar, T. M. Briere and Y. Kawazoe, Nano Lett. 2, 1243 (2002).
* (17) J. Bai and X. C. Zeng, NANO: Brief Reports and Reviews 2, 109 (2007).
* (18) A. K. Singh, V. Kumar, T. M. Briere and Y.Kawazoe, J. Mater. Chem. 14, 555 (2004).
* (19) A. N. Andriotis, G. Mpourmpakis, G. E. Froudakis, and M. Menon, New J. Phys. 4, 78 (2002).
* (20) A. K. Singh, T. M. Briere, V. Kumar and Y. Kawazoe, Phys. Rev. Lett. 91, 146802 (2003).
* (21) M. Zhang, Y. H. Han, Q. J. Zang, Z. M. Su and R. S. Wang, Chem. Phys. Lett. 379, 81 (2003).
* (22) E. Durgun, S. Tongay and S. Ciraci, Phys. Rev. B 72, , 075420 (2005).
* (23) X. B. Yang and J. Ni, Phys. Rev. B 72, 195426 (2005).
* (24) A. S. Bamard and S. P. Russo, J. Phys. Chem. B 107, 7577 (2003).
* (25) R. Q. Zhang, H. L. Lee, W. K. Li and B. K. Teo, J. Phys. Chem. B 109, 8605 (2005).
* (26) Solange B. Fagan, R. J. Baierle, R. Mota Silva, Antonio J. R. da Silva and A. Fazzio Phys. Rev. B 61, 9994 (2000).
* (27) J. H. Lan, D. J. Cheng, D. P. Cao and W. C. Wang, J. Phys. Chem. C 112, 5598 (2008).
* (28) B. Marsen and K. Sattler Phys. Rev. B 60,11593 (1999).
* (29) A. Palaria, G. Klimeck and A. Strachan Phys. Rev. B 78, 205315 (2008).
* (30) S. Sirichantaropass, V. M. Garcia-Suarez and C. J. Lambert, Phys. Rev. B 75, 075328 (2008).
* (31) O. Ponomarenko, M. W. Randny and P. V. Smith, Sur. Sci. 562, 257 (2004).
* (32) G. Seifert, Th. Köhler, H. M. Urbassek, E. Hern$\acute{a}$ndez and Th. Frauenheim, Phys. Rev. B 63, 193409 (2001).
* (33) T. He, M. W. Zhao, W. F. Li, C. Song, X. H. Lin, X. D. Liu, Y. Y. Xia and L. M. Mei, J. Phys. Chem. C 111,16840 (2007).
* (34) U. Landman, R. N. Barnett, A. G. Scherbakov and P. Avouris, Phys. Rev. Lett. 85,1958 (2000).
* (35) M. Durandurdu, Phys. Stat. Sol.(b) 243, , No2, R7 (2006).
* (36) B. H. Yan, G. Zhou, J. Wu, W. H. Duan and B. L. Gu, Phys. Rev. B 73, 155432 (2006).
* (37) M. Menon and E. Richter, Phys. Rev. Lett. 83,792 (1999).
* (38) M. W. Zhao, R. Q. Zhang, Y. Y. Xia, C. Song and S. T. Lee, J. Phys. Chem. C 111, 1234 (2007).
* (39) D. D. D. Ma, C. S. Lee, F. C. K. Au, S. Y. Tong and S. T. Lee, Science 299, 1874 (2003).
* (40) J. Sha, J. Niu, X. Ma, J. Xu, X. Zhang, Q. Yang and D. Yang, Adv. Mater. 14, 1219 (2002).
* (41) S. Y. Jeong, J. Y. Kim, H.D. Yang, B.N. Yoon, S. H. Choi, H. K. Kang, C. W. Yang and Y. H. Lee, Adv. Mater. 15, 1172 (2003).
* (42) Y. W. Chen, Y. H. Tang, L. Z. Pei and C. Guo, Adv. Mater. 17, 564 (2005).
* (43) Y. H. Tang, L. Z. Pei, Y. W.Chen and C. Guo, Phys. Rev. Lett. 95, 116102 (2005).
* (44) DMOL is a density-functional theory package distributed by MSI B. Delley, J. Chem. Phys. 92, 508 (1990).
* (45) B. Delley, J. Chem. Phys. 113, 7756 (2000).
* (46) T. Yildirim, O. ;Gülseren and S. Ciraci, Phys. Rev. B 64, 075404(2001).
* (47) M. F. Ng, L. P. Zhou, S. W. Yang, L. Y. Sim, V. B. C. Tan and P. Wu, Phys. Rev. B 76, 155435 (2007).
* (48) E. Durgun, D. Cakir, N. Akman and S. Ciraci, Phys. Rev. Lett. 99, 256806 (2007).
* (49) X. P. Yang and J. M. Dong, Appl. Phys. Lett. 86, 163105 (2005).
## VI Figure and Table captions
Table 1. HOMO-LUMO gap($\Delta_{G}$) and effective volume per electron
$V_{eff}^{(e)}$ of $Si_{m(3\times 4+1)}H_{2m(4+1)}$ ($m$=4, 5, 6, 7, 8) tubes.
Table 2. Binding energy ($E_{b}$) of infinite SiH tubes and Fe-doped SiH
tubes, the band gap ($\Delta_{g}$)of SiH tubes, and the spin moments($M$) per
Fe atom and of Fe-doped SiH tubes.
$Fig$. 1. (Color online) Top view of the optimized $Si_{n}H_{n}$ clusters (the
1st row) and side view of cluster-assembled nanotubes
$Si_{m(3k+1)}H_{2m(k+1)}$ (all the rest).
$Fig$. 2. (Color online) Structures of $Si_{5(3k+1)}H_{3\times 5(k+1)}$
($k$=3, 4) tubes.
$Fig$. 3. (Color online) The averaged binding energies, dissociation energies
and HOMO-LUMO gaps of finite $Si_{m(3k+1)}H_{2m(k+1)}$ tubes.
$Fig$. 4. (Color online) Side views of two unit cells of $Si_{6m}H_{4m}$
nanotubes, (a) pristine SiH nanotubes, (b) Fe-doped SiH nanotubes, where the
dark atoms in the center of the tube represent Fe atoms.
$Fig$. 5. (Color online) $\alpha$ and $\beta$ spin resolved DOS of Fe-doped
SiH nanotubes($Si_{6m}H_{4m}$). (a)–(e): total $\alpha$ and $\beta$ spin DOS
of $Si_{6\times 4}H_{4\times 4}$, $Si_{6\times 5}H_{4\times 5}$, $Si_{6\times
6}H_{4\times 6}$, $Si_{6\times 7}H_{4\times 7}$, and $Si_{6\times 8}H_{4\times
8}$, (f): partial $\alpha$ and $\beta$ spin DOS of $Si_{6\times 7}H_{4\times
7}$.
Table 1: Guo $et$ $al$. $Si_{m(3k+1)}H_{2m(k+1)}$ | $\Delta_{G}$ (eV) | $V_{eff}^{(e)}$ (Å3/electron)
---|---|---
$Si_{52}H_{40}$ ($m$=4, $k$=4) | 2.934 | 6.748
$Si_{65}H_{50}$ ($m$=5, $k$=4) | 2.882 | 6.760
$Si_{78}H_{60}$ ($m$=6, $k$=4) | 3.006 | 6.653
$Si_{91}H_{70}$ ($m$=7, $k$=4) | 2.592 | 6.632
$Si_{104}H_{80}$ ($m$=8, $k$=4) | 2.257 | 6.681
Table 2: Guo $et$ $al$. | | | | $M$ ($\mu_{B}$)
---|---|---|---|---
m | $E_{b}$(SiH) (eV) | $E_{b}$(FeSiH) (eV) | $\Delta_{g}$ (eV) | per Fe | total
4 | 3.476 | 3.348 | 3.009 | 2.822 | 6.900
5 | 3.482 | 3.411 | 2.765 | 3.183 | 8.000
6 | 3.500 | 3.424 | 2.921 | 3.395 | 8.000
7 | 3.479 | 3.415 | 2.557 | 3.538 | 8.000
8 | 3.448 | 3.387 | 2.296 | 3.722 | 7.967
Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
|
arxiv-papers
| 2010-07-09T02:09:49 |
2024-09-04T02:49:11.527949
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lingju Guo, Xiaohong Zheng, Chunsheng Liu and Zhi Zeng",
"submitter": "Lingju Guo",
"url": "https://arxiv.org/abs/1007.1495"
}
|
1007.1717
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ A note on interval edge-
colorings of graphsR.R. Kamalian, P.A. Petrosyan
# A note on interval edge-colorings of graphs
R.R. Kamalian, P.A. Petrosyan[MCSD] email: rrkamalian@yahoo.com.email:
pet_petros@{ipia.sci.am, ysu.am, yahoo.com} Institute for Informatics and
Automation Problems,
National Academy of Sciences, 0014, Armenia Department of Applied Mathematics
and Informatics,
Russian-Armenian State University, 0051, Armenia Department of Informatics
and Applied Mathematics,
Yerevan State University, 0025, Armenia
###### Abstract
An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an
interval $t$-coloring if for each $i\in\\{1,2,\ldots,t\\}$ there is at least
one edge of $G$ colored by $i$, and the colors of edges incident to any vertex
of $G$ are distinct and form an interval of integers. In this paper we prove
that if a connected graph $G$ with $n$ vertices admits an interval
$t$-coloring, then $t\leq 2n-3$. We also show that if $G$ is a connected
$r$-regular graph with $n$ vertices has an interval $t$-coloring and $n\geq
2r+2$, then this upper bound can be improved to $2n-5$.
Keywords: edge-coloring, interval coloring, bipartite graph, regular graph
## 1 Introduction
All graphs considered in this paper are finite, undirected, and have no loops
or multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges
of $G$, respectively. An $(a,b)$-biregular bipartite graph $G$ is a bipartite
graph $G$ with the vertices in one part all having degree $a$ and the vertices
in the other part all having degree $b$. A partial edge-coloring of $G$ is a
coloring of some of the edges of $G$ such that no two adjacent edges receive
the same color. If $\alpha$ is a partial edge-coloring of $G$ and $v\in V(G)$,
then $S\left(v,\alpha\right)$ denotes the set of colors of colored edges
incident to $v$.
An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an
interval $t$-coloring if for each $i\in\\{1,2,\ldots,t\\}$ there is at least
one edge of $G$ colored by $i$, and the colors of edges incident to any vertex
of $G$ are distinct and form an interval of integers. A graph $G$ is interval
colorable, if there is $t\geq 1$ for which $G$ has an interval $t$-coloring.
The set of all interval colorable graphs is denoted by $\mathfrak{N}$. For a
graph $G\in\mathfrak{N}$, the greatest value of $t$ for which $G$ has an
interval $t$-coloring is denoted by $W\left(G\right)$.
The concept of interval edge-coloring was introduced by Asratian and Kamalian
[2]. In [2, 3] they proved the following theorem.
###### Theorem 1
If $G$ is a connected triangle-free graph and $G\in\mathfrak{N}$, then
$W(G)\leq|V(G)|-1$.
In particular, from this result it follows that if $G$ is a connected
bipartite graph and $G\in\mathfrak{N}$, then $W(G)\leq|V(G)|-1$. It is worth
noting that for some families of bipartite graphs this upper bound can be
improved. For example, in [1] Asratian and Casselgren proved the following
###### Theorem 2
If $G$ is a connected $(a,b)$-biregular bipartite graph with $|V(G)|\geq
2(a+b)$ and $G\in\mathfrak{N}$, then
$W(G)\leq|V(G)|-3$.
For general graphs, Kamalian proved the following
###### Theorem 3
[6]. If $G$ is a connected graph and $G\in\mathfrak{N}$, then
$W(G)\leq 2|V(G)|-3$.
The upper bound of Theorem 3 was improved in [5].
###### Theorem 4
[5]. If $G$ is a connected graph with $|V(G)|\geq 3$ and $G\in\mathfrak{N}$,
then
$W(G)\leq 2|V(G)|-4$.
On the other hand, in [7] Petrosyan proved the following theorem.
###### Theorem 5
For any $\varepsilon>0$, there is a graph $G$ such that $G\in\mathfrak{N}$ and
$W\left(G\right)\geq\left(2-\varepsilon\right)\left|V\left(G\right)\right|$.
For planar graphs, the upper bound of Theorem 3 was improved in [4].
###### Theorem 6
[4]. If $G$ is a connected planar graph and $G\in\mathfrak{N}$, then
$W(G)\leq\frac{11}{6}|V(G)|$.
In this note we give a short proof of Theorem 3 based on Theorem 1. We also
derive a new upper bound for the greatest possible number of colors in
interval edge-colorings of regular graphs.
## 2 Main results
* Proof of Theorem 3.
Let $V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and $\alpha$ be an interval
$W(G)$-coloring of the graph $G$. Define an auxiliary graph $H$ as follows:
$V(H)=U\cup W$, where
$U=\\{u_{1},u_{2},\ldots,u_{n}\\}$, $W=\\{w_{1},w_{2},\ldots,w_{n}\\}$ and
$E(H)=\left\\{u_{i}w_{j},u_{j}w_{i}|~{}v_{i}v_{j}\in E(G),1\leq i\leq n,1\leq
j\leq n\right\\}\cup\\{u_{i}w_{i}|~{}1\leq i\leq n\\}$.
Clearly, $H$ is a connected bipartite graph with $|V(H)|=2|V(G)|$.
Define an edge-coloring $\beta$ of the graph $H$ in the following way:
(1)
$\beta(u_{i}w_{j})=\beta(u_{j}w_{i})=\alpha(v_{i}v_{j})+1$ for every edge
$v_{i}v_{j}\in E(G)$,
(2)
$\beta(u_{i}w_{i})=\max S(v_{i},\alpha)+2$ for $i=1,2,\ldots,n$.
It is easy to see that $\beta$ is an edge-coloring of the graph $H$ with
colors $2,3,\ldots,W(G)+2$ and $\min S(u_{i},\beta)=\min S(w_{i},\beta)$ for
$i=1,2,\ldots,n$. Now we present an interval $(W(G)+2)$-coloring of the graph
$H$. For that we take one edge $u_{i_{0}}w_{i_{0}}$ with $\min
S(u_{i_{0}},\beta)=\min S(w_{i_{0}},\beta)=2$, and recolor it with color $1$.
Clearly, such a coloring is an interval $(W(G)+2)$-coloring of the graph $H$.
Since $H$ is a connected bipartite graph and $H\in\mathfrak{N}$, by Theorem 1,
we have
$W(G)+2\leq|V(H)|-1=2|V(G)|-1$, thus
$W(G)\leq 2|V(G)|-3$.
$\square$
###### Theorem 7
If $G$ is a connected $r$-regular graph with $|V(G)|\geq 2r+2$ and
$G\in\mathfrak{N}$, then
$W(G)\leq 2|V(G)|-5$.
* Proof.
In a similar way as in the prove of Theorem 3, we can construct an auxiliary
graph $H$ and to show that this graph has an interval $(W(G)+2)$-coloring.
Next, since $H$ is a connected $(r+1)$-regular bipartite graph with
$|V(H)|\geq 2(2r+2)$ and $H\in\mathfrak{N}$, by Theorem 2, we have
$W(G)+2\leq|V(H)|-3=2|V(G)|-3$, thus
$W(G)\leq 2|V(G)|-5$.
$\square$
## References
* [1] A.S. Asratian, C.J. Casselgren, On interval edge colorings of $(\alpha,\beta)$-biregular bipartite graphs, Discrete Mathematics 307 (2006) 1951-1956.
* [2] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph (in Russian), Appl. Math. 5 (1987) 25-34.
* [3] A.S. Asratian, R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994) 34-43.
* [4] M.A. Axenovich, On interval colorings of planar graphs, Congr. Numer. 159 (2002) 77-94.
* [5] K. Giaro, M. Kubale, M. Malafiejski, Consecutive colorings of the edges of general graphs, Discrete Math. 236 (2001) 131-143.
* [6] R.R. Kamalian, Interval edge-colorings of graphs, Doctoral Thesis, Novosibirsk, 1990.
* [7] P.A. Petrosyan, Interval edge-colorings of complete graphs and $n$-dimensional cubes, Discrete Mathematics 310 (2010) 1580-1587.
|
arxiv-papers
| 2010-07-10T12:25:17 |
2024-09-04T02:49:11.540437
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R.R. Kamalian, P.A. Petrosyan",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1007.1717"
}
|
1007.1807
|
# Fiber-diffraction Interferometer using Coherent Fiber Optic Taper
Hagyong Kihm∗ and Yun-Woo Lee Center for Space Optics, Korea Research
Institute of Standards and Science, 1 Doryong-dong, Yuseong-gu, Daejeon
305-340, South Korea hkihm@kriss.re.kr
###### Abstract
We present a fiber-diffraction interferometer using a coherent fiber optic
taper for optical testing in an uncontrolled environment. We use a coherent
fiber optic taper and a single-mode fiber having thermally-expanded core. Part
of the measurement wave coming from a test target is condensed through a fiber
optic taper and spatially filtered from a single-mode fiber to be reference
wave. Vibration of the cavity between the target and the interferometer probe
is common to both reference and measurement waves, thus the interference
fringe is stabilized in an optical way. Generation of the reference wave is
stable even with the target movement. Focus shift of the input measurement
wave is desensitized by a coherent fiber optic taper.
###### pacs:
07.60.Ly
††: Meas. Sci. Technol.
interferometers: Article preparation, IOP journals
## 1 Introduction
We describe a new fiber-diffraction interferometer using a coherent fiber
optic taper for stabilizing interference fringes in an uncontrolled
environment. Manufacturing technology of optical components such as lenses and
mirrors has evolved from the traditional polishing process to the automated
machining process. Diamond turning machine is now able to achieve sub-$\mu$m
accuracy over a few hundreds of millimeters in diameter. And injection molding
technology produces high quality optical components with unprecedented yield
rate. Testing those optics depends on optical interferometers due to their
fast 2 dimensional measurement and quality inspection at the operational
wavelength of the test optics. But the interferometer is so vulnerable to the
external vibration that isolating the test table from the machining bed is
essential in most cases. This fact deters the interferometer system from being
united with the machining center as a truly repeatable feedback sensor.
Nowadays, researches on vibration insensitive (anti-vibration) interferometers
have been spurred to meet those industrial demands, including large scale
optics used for ignition facility and telescope optics [1]. Several techniques
are used for desensitizing the interferometer to the external vibration and
can be categorized into 3 classes; common-path configuration, vibration
feedback control, and spatial phase shifting for real time inspection.
Common-path enables the vibration of test optics common to both reference and
measurement waves, whereby interference fringe between them looks static.
Point diffraction interferometer by Smartt and Steel [2] is an example, which
makes reference wave from the focused measurement wave through a pinhole.
Lateral shearing interferometer is a common-path interferometer and optical
pick-up lenses in production line were tested by Cho and Kim [3]. Scatter
plate interferometer is also a common-path interferometer with high immunity
to external vibration [4].
Another technique for anti-vibration is direct feedback control. Yoshino and
Yamaguchi [5] implemented a closed-loop phase shifting Fizeau interferometer
where optical phases are detected by a two-frequency optical heterodyne
method. Twymann-Green interferometer with a single point detector and an
electro-optic modulator was also proposed [6].
Last category of anti-vibration technique is a single-shot interferometer with
spatial phase shifting. Phase shifting is necessary to enhance measurement
accuracy and this is accomplished spatially as opposed to temporal phase
shifting techniques. Smythe and Moore [7] used polarization beam splitters and
waveplates to acquire 4 phase-shifted fringes. Millerd et al. [8] used
holographic elements and polarizers to obtain 4 phase-shifted fringe.
Pixelated phase-mask can also be used for spatial phase shifting [9]. Spatial
carrier phase shifting technique is widely used due to its simple and easy
embodiment [10]. By introducing linear tilt phase term in reference or test
beam, a spatially modulated fringe can be obtained and analyzed with several
algorithms such as sinusoidal fitting [11] and Fourier analysis [12].
Aforementioned anti-vibration interferometers of common-path, closed-loop
feedback control, and spatial phase shifting are vying each other with their
relative advantages and disadvantages. Closed-loop feedback cannot control
high frequency vibrations over the bandwidth of control loop and the actual
implementation is difficult. Spatial phase shifting suffers the same problem
with high frequency vibrations because the detector frame rate and shutter
speed is generally limited. And highly repeatable measurement is practically
impossible between successive tests, which lowers the system reliability.
Common-path interferometers are ideal for stabilizing fringes in principle,
but they usually lack real time capability due to inherent temporal phase
shifting principles.
Combining common-path configuration with a spatial phase shifting technique
can be a promising solution to anti-vibration interferometry. For examples,
Kwon [13] made three phase-shifted fringes with a phase grating and a pinhole
achieving a common-path real time interferometer. Millerd et al. [14] from 4D
Technology Corporation combined a point diffraction interferometer with their
spatial phase shifting interferometer. Both interferometers use pinholes and
their diffracting fields as reference waves. Recently, Kihm and et al. [15]
reported this type of interferometer using a single-mode fiber, but it suffers
from difficulties of focusing aberrant measurement beam into the small core of
a single mode fiber. Defocus or lateral vibration of the target severely
affects beam intensity out of the single-mode fiber. Therefore fringe
visibility changes when large vibration is involved. This paper overcomes that
weakness and introduces a new type of interferometer.
The main idea of the research is condensing measurement wave through a
coherent fiber optic taper (FOT) and a single-mode fiber (SMF) with thermally-
expanded core (TEC) to make reference wave. Vibration of the cavity between
the target and the interferometer probe is common to both reference and
measurement waves, thus the interference fringe is stabilized in an optical
way. Generation of the reference wave is stable even with the target movement.
Focus shift of the input measurement wave is desensitized by an FOT. The
uncertainly of measurement results can be lowered due to highly repeatable
performance even with external vibrations. Principles will be explained in
Sec. 2 and experimental results will be detailed in Sec. 3 followed by
conclusions in Sec. 4.
## 2 Principles
The fiber-diffraction interferometer using an FOT is shown in Fig. 1. Any type
of laser, which is linearly polarized and spatially coherent, can be used as a
light source. Continuous wave lasers like He-Ne lasers or super luminescent
diodes (SLD) of short coherence length can be used for general purposes. Pulse
lasers could be used for stroboscopic inspection. The laser is filtered
through a pinhole and collimated by a lens becoming well-defined plane wave.
Half-wave plate 1 (HWP1) rotates the polarization angle of the beam and
controls the amount of reflected beam at the polarization beam splitter 1
(PBS1). This eventually adjusts the brightness of interference fringes. A
quarter-wave plate (QWP) with its fast axis aligned at $45^{\circ}\/$ makes
the beam circularly-polarized. Objective lens (OL) or null lens in case of
aspheric targets forms the beam wavefront fit to the target surface for
reciprocation. The reflected measurement wave, which is circularly polarized,
is then linearly polarized in orthogonal direction after the QWP and passes
through the PBS1. HWP2 rotates the polarization angle of the measurement wave,
thus controls the split ratio between two arms at PBS2. The reference arm is
composed of a focusing lens, an FOT and an SMF with TEC to make spatially
filtered wavefront. Corner cube (CC) in the measurement arm translates to
compensate optical path length difference between two arms. PBS3 combines
those two beams and polarizer (P) with $45^{\circ}\/$ filters them in diagonal
direction making interference fringes at the detector.
Figure 1: Principles of fiber-diffraction interferometer using an FOT: HWP,
half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; OL,
objective lens; CC, corner cube; P, polarizer.
When phase shifting is required for wavefront analysis, detectors equipped
with a spatial phase shifter [8, 9, 15] could be used. Translation of CC for
temporal phase shifting might be adopted in vibration-isolated environment. In
this paper, we focus mainly on the verification of using an FOT in fiber-
diffraction interferometry. Also we assume that optical parts comprising the
interferometer are fixed as a single body. Relative motion or vibration of the
test surface doesn’t affect the alignment of the interferometer components.
This can be justified in general optical testing environment where unstable
cavity between the probe and a target is the major contributor lowering
repeatability.
Fringe stabilization is possible even with external vibrations by making
reference wave out of measurement wave. Shared vibrant phase which is mainly
piston motion cancel out and the fringes look static. A pinhole has been used
for this purpose in point diffraction interferometry [2, 16]. They use a
sidelobe of focused measurement wave and tilted fringes are inevitable.
Demodulating fringes [11, 12] lowers measurement accuracy and high frequency
features cannot be recovered. An SMF can be used due to the high quality
wavefront output and easy embodiment [15]. But the difficulty of focusing
aberrant measurement wave into the small core of a fiber poses doubt about
practical uses. Increasing coupling efficiency of a laser source into an SMF
has been a major research activity in optical communication, and thermally
expanding the core(TEC) of a fiber is an example [17].
In this research we propose to use an FOT as a beam coupler for
interferometric uses. An FOT consists of a large number of optical fibers
fused together to form a coherent bundle. The bundle is heat formed, resulting
in variation of its diameter from one end to the other. The magnification of a
taper is simply the ratio of the diameter of the end faces, which is generally
2–5$\times$ [18]. The light transmission of an FOT is given in terms of
internal transmission of glass core, Fresnel reflection at the end faces, and
the ratio of core area to the total area termed packaging fraction (PF). The
transmission $T$ can be expressed as
$T=\mathrm{PF}t_{f}\exp\left(-\beta_{\lambda}L\right)$ (1)
, where $t_{f}$ is the Fresnel transmission factor, $\beta_{\lambda}\/$ is the
absorption coefficient of the core glass, and $L$ is the length of the taper
[19]. The light transmission can be increased by anti-reflection (AR) coating
on input and output ports of the taper. A pinhole or a coated mask on the
small end face blocks unwanted spurious modes [20] and passes only a single-
mode by point diffraction.
Immunity to the focus shift at the input port is achieved by condensing light
through an FOT. The input end face of a taper is placed near the aperture stop
of a focusing objective lens. This slightly defocused input beam looks static
at the output even when the vibration of the target changes the direction
(tilt) and divergence angle (focus) of the measurement wave. High numerical
aperture (NA) of the FOT, which is 1, captures almost every incoming field
from the objective lens. The beam spot size at the output port is reduced
according to the reduction ratio of the taper. Multi-mode fields at the
output, however, should be filtered into a single-mode to be used as reference
wave [21, 22]. A TEC-processed SMF is used to combine individual fibers at the
output of the FOT. Then, the reference wave from the SMF is quite stable in
its amplitude while carrying vibrant phase motion of the target.
Considering the power of available lasers and sensitivity of detectors,
transmission loss through an FOT and a TEC-processed SMF is not a problem in
interferometric applications [23]. The following section explains the actual
implementation and verifies the use of an FOT in fiber-diffraction
interferometry.
## 3 Experimental Results and Discussion
Reference wave is made by fiber diffraction from the end face of an SMF.
Focusing aberrant measurement wave into the small core of SMF is difficult
when the test optic has vibrational motions. Axial motion makes the beam
defocused at the input port. Lateral or tilt motion shifts the beam focus and
lowers the coupling efficiency more severely. When large vibration is involved
at the cavity between the test optic and the interferometer probe, the
amplitude of reference wave will fluctuate accordingly. Thus we cannot get
stable fringes enough for practical phase measurement. The objective of this
research is to get stable fringes immune to the focus shift of the measurement
wave.
Figure 2: Coupling efficiency test with respect to input focus shift: (a)
single-mode fiber (SMF) only, (b) thermally-expanded core (TEC) fiber only,
(c) fiber-optic taper (FOT) and SMF, (d) FOT and TEC fiber; PD, photo-
detector.
We verified the improvement of using an FOT by comparative experiments shown
in Fig. 2. The laser is focused into an SMF and a photo-detector (PD) captures
the output intensity to examine the coupling efficiency. The laser is rotated
to simulate tilt motions of the measurement wave. We tested with 4 different
set-ups; (a) using an SMF, (b) a TEC fiber, (c) an FOT and an SMF, and (d) an
FOT and a TEC fiber. Fig. 2(a) is a conventional method using an SMF in fiber-
diffraction interferometry. TEC fiber in Fig. 2(b) has a longitudinal
variation of the core diameter. Mode-field diameter of the input end becomes
10 $\mu$m from 4 $\mu$m after the TEC process. This makes the coupling less
sensitive to the beam focus. We used an FOT from Schott [18]. The diameter of
a large end is 25 mm and the small end is 8 mm. The magnification is 3.1:1,
which is the ratio of end face diameters. The element size of each fiber at
large end side is 6 $\mu$m and the ratio of core and cladding area is 1:1. The
PF in Eq. 1 is lower than 50% due to the extramural absorption (EMA) which
eliminates stray light through the cladding [24]. The refractive index of the
core is 1.810 and NA of a fiber at the end face is 1 from manufacturer’s
specification [18].
Figure 3: Coupled intensity outputs from the experiments shown in Fig. 2
The output is a convolution process between the focused laser and the input
modal field of fibers and the FOT. The NA of the SMF and the TEC fiber is
0.12, while the NA of the FOT is 1. The mode-field diameters of the SMF and
the TEC fiber are 4 $\mu$m and 10 $\mu$m respectively. In case of (c) and (d)
in Fig. 2, mode-field diameter depends on the magnifications of the lens and
the FOT. We used a lens with 1$\times$ magnification and an FOT with
3.1$\times$ magnification. The input mode-field diameters of Fig. 2(c) and
Fig. 2(d) are 12 $\mu$m and 31 $\mu$m respectively.
Fig. 3 shows the experimental results with the set-up shown in Fig. 2. Coupled
intensity output from the PD is normalized in each case. Immunity to the focus
shift can be compared by their relative output profiles, not their maximum
values. If the plot is steep in a small input variation, the sensitivity to
the focus shift is high. The SMF is the most sensitive as shown in Fig. 3(a).
The TEC fiber in Fig. 3(b) is relatively insensitive to the focus shift due to
the enlarged mode-field diameter. The FOT has a large NA accepting almost
every incoming field, but the EMA makes the beam fluctuate as shown in Fig.
3(c). This problem can be minimized by using a TEC fiber as in Fig. 3(d). The
FOT’s large NA makes the output robust to the focus shift and the TEC fiber’s
large mode-field diameter mitigates EMA effect. About 5 times improvement is
observed in terms of acceptance angle when we compare $1/e^{2}$ intensity
outputs of Fig. 3(a) and Fig. 3(d).
Figure 4: Experimental set-up of the fiber-diffraction interferometer using an
FOT in Fig. 1
Fig. 4 shows an experimental set-up of the fiber-diffraction interferometer
using an FOT and a TEC SMF. Optical components are arranged similar to the
layout of Fig. 1 except for the test surface. We used a flat mirror instead of
a spherical mirror. The flat mirror is suitable for evaluating immunity to the
focus shift due to tilt motions. The TEC SMF is bent $90\,^{\circ}$ to filter
out unwanted multi-modes effectively within a short optical path. Fig. 5(a)
shows an interferogram when the test surface has a tilt angle of
$0.02\,^{\circ}$. Fig. 5(b) was obtained when the test surface was rotated by
$0.2\,^{\circ}$. Fringe visibility is comparable to that of Fig. 5(a). This is
expected in Fig. 3(d) where the coupled intensity output is over 80 % of its
maximum value at the angle $0.2\,^{\circ}$. The fiber-diffraction
interferometer using an SMF [15] cannot obtain fringes like Fig. 5(b). As
evident in Fig. 3(a), the coupled intensity output from an SMF is almost zero
at the angle $0.2\,^{\circ}$. This result proves the performance improvement
with the use of an FOT in fiber-diffraction interferometry.
Figure 5: Captured fringes from the set-up in Fig. 4 when the test surface is
rotated by the angle (a) $0.02\,^{\circ}$ and (b) $0.2\,^{\circ}$
## 4 Conclusions
We proposed a new fiber-diffraction interferometer using a coherent fiber
optic taper for stabilizing fringes in vibrational optical testing
environment. Reference wave is made by condensing measurement wave through a
coherent fiber optic taper and a single-mode fiber with thermally expanded
core. Experimental comparison proved the superior coupling efficiency of the
proposed method. Combining this technique with a spatial phase shifter will
increase measurement repeatability as well as freezing vibrations, whereby
ideal vibration insensitive interferometer could be realized. Cascading
multiple fiber optic tapers to study coupling efficiency and comparing with
the state of the art commercial interferometers will be the future works.
## 5 References
## References
* [1] J. C. Wyant. Dynamic interferometry. Optics and Photonics News, 14:36–41, 2003.
* [2] R. N. Smartt and W. H. Steel. Theory and application of point-diffraction interferometers. Jap. J. Appl. Phys., 14 (Supplement 14-1):351–356, 1975.
* [3] W.-J. Cho and S.-W. Kim. Stable lateral-shearing interferometer for production-line inspection of lenses. Opt. Eng., 36:896–900, 1997.
* [4] M. B. North-Morris, J. VanDelden, and J. C. Wyant. Phase-shifting birefringent scatterplate interferometer. Appl. Opt., 41(4):668–677, 2002.
* [5] T. Yoshino and H. Yamaguchi. Closed-loop phase-shifting interferometry with a laser diode. Opt. Lett., 23(20):1576–1578, 1998.
* [6] I. Yamaguchi, J.-Y. Liu, and J.-I. Kato. Active phase-shifting interferometers for shape and deformation measurements. Opt. Eng., 35(10):2930–2937, 1996.
* [7] R. Smythe and R. Moore. Instantaneous phase measuring interferometry. Opt. Eng., 23:361–364, 1984.
* [8] J. E. Millerd and N. J. Brock. Methods and apparatus for splitting, imaging and measuring wavefronts in interferometry. U.S. Patent 6,304,330, October 16, 2001.
* [9] J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant. Pixelated phase-mask dynamic interferometer. Proc. SPIE, 5531:304–314, 2004.
* [10] M. Melozzi, L. Pezzati, and A. Mazzoni. Vibration-insensitive interferometer for on-line measurements. Appl. Opt., 34(25):5595–5601, 1995.
* [11] P. L. Ransom and J. V. Kokal. Interferogram analysis by a modified sinusoid fitting technique. Appl. Opt., 25(22):4199–4204, 1986.
* [12] M. Takeda, H. Ina, and S. Kobayashi. Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. J. Opt. Soc. Am., 72(1):156–160, 1982.
* [13] O. Y. Kwon. Multichannel phase-shifted interferometer. Opt. Lett., 9(2):59–61, 1984.
* [14] J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant. Instantaneous phase-shift point-diffraction interferometer. Proc. SPIE, 5531:264–272, 2004.
* [15] H. Kihm and S.-W. Kim. Fiber-diffraction interferometer for vibration desensitization. Opt. Lett., 30(16):2059–2061, 2005.
* [16] H. Medecki, E. Tejnil, K. A. Goldberg, and J. Bokor. Phase-shifting point diffraction interferometer. Opt. Lett., 21(19):1526–1528, 1996.
* [17] H. Hanafusa, M. Horiguchi, and J. Noda. Thermally-diffused expanded core fibers for low-loss and inexpensive photonic components. Electron. Lett., 27:1968–1969, 1991.
* [18] SCHOTT North America, Inc. 555 Taxter Road Elmsford, NY 10523 USA. http://www.us.schott.com/lightingimaging/english/. Accessed on Jul. 2010.
* [19] E. Peli and W. P. Siegmund. Fiber-optic reading magnifiers for the visually impaired. J. Opt. Soc. Am. A, 12(10):2274–2285, 1995.
* [20] K. Shi, F. G. Omenetto, and Z. Liu. Supercontinuum generation in an imaging fiber taper. Opt. Express, 14(25):12359–12364, 2006.
* [21] Y.-F. Li and J. W. Y. Lit. Transmission properties of a multimode optical-fiber taper. J. Opt. Soc. Am. A, 2(3):462–468, 1985.
* [22] S. G. Leon-Saval, T. A. Birks, J. Bland-Hawthorn, and M. Englund. Multimode fiber devices with single-mode performance. Opt. Lett., 30(19):2545–2547, 2005.
* [23] A. Kosterin, V. Temyanko, M. Fallahi, and M. Mansuripur. Tapered fiber bundles for combining high-power diode lasers. Appl. Opt., 43(19):3893–3900, 2004.
* [24] W. P. Siegmund. Fiber optical image transfer device having a multiplicity of light absorbing elements. U.S. Patent 3,247,756, April 26, 1966.
|
arxiv-papers
| 2010-07-11T23:54:55 |
2024-09-04T02:49:11.550143
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hagyong Kihm and Yun-Woo Lee",
"submitter": "Hagyong Kihm",
"url": "https://arxiv.org/abs/1007.1807"
}
|
1007.1878
|
# Non-commutative $L_{p}$-spaces associated with a Maharam trace
Vladimir Chilin and Botir Zakirov Vladimir Chilin
Department of Mathematics, National University of Uzbekistan
Vuzgorodok, 100174 Tashkent, Uzbekistan chilin@ucd.uz Botir Zakirov
Tashkent Railway Engineering Institute
Odilhodjaev str. 1, 100167 Tashkent, Uzbekistan botirzakirov@list.ru
###### Abstract.
Non-commutative $L_{p}$-spaces $L^{p}(M,\Phi)$ associated with the Maharam
trace are defined and their dual spaces are described.
Mathematics Subject Classification (2000). 28B15, 46L50
Keywords: von Neumann algebra, measurable operator, Dedekind complete Riesz
space, integration with respect to a vector-valued trace.
## 1\. Introduction
Development of the theory of integration for measures $\mu$ with the values in
Dedekind complete Riesz spaces has inspired the study of $(bo)$-complete
lattice-normed spaces $L^{p}(\mu)$ (see, for example, [1], 6.1.8). Note that,
if the measure $\mu$ satisfies the Maharam property, then the spaces
$L^{p}(\mu)$ are Banach-Kantorovich.
The existence of center-valued traces on finite von Neumann algebras naturally
leads to a study of the integration for traces with the values in a complex
Dedekind complete Riesz space $F_{\mathbb{C}}=F\oplus iF.$ For commutative von
Neumann algebras, the development of $F_{\mathbb{C}}$-valued integration is a
part of the study of the properties of order continuous positive maps of Riesz
spaces, for which we refer to the treatise by A.G. Kusraev [1]. The operators
possessing the Maharam property provide important examples of such mappings,
while the $L^{p}$-spaces associated with such operators are non-trivial
examples of Banach-Kantorovich Riesz spaces.
Let $M$ be a non-commutative von Neumann algebra, $F_{\mathbb{C}}$ a von
Neumann subalgebra in the center of $M$, and let $\Phi:M\to F_{\mathbb{C}}$ be
a trace such that $\Phi(zx)=z\Phi(x)$ for all $z\in F_{\mathbb{C}},~{}x\in M.$
Then the non-commutative $L^{p}$-space $L^{p}(M,\Phi)$ is a Banach-Kantorovich
space [2], [3], and the trace $\Phi$ satisfies the Maharam property, that is,
if $0\leq z\leq\Phi(x),~{}z\in F_{\mathbb{C}},~{}0\leq x\in M,$ then there
exists $0\leq y\leq x$ such that $\Phi(y)=z$ (compare with [1], 3.4.1).
In [4], a faithful normal trace $\Phi$ on $M$ with the values in an arbitrary
complex Dedekind complete Riesz space was considered. In particular, a
complete description of such traces in the case when $\Phi$ is a Maharam trace
was given. In the same paper, utilizing the locally measure topology on the
algebra $S(M)$ of all measurable operators affiliated with $M,$ the Banach-
Kantorovich space $L^{1}(M,\Phi)\subset S(M)$ was constructed and a version of
Radon-Nikodym-type theorem for Maharam traces was established.
In the present article, we define a new class of Banach-Kantorovich spaces,
non-commutative $L_{p}$-spaces $L^{p}(M,\Phi)$ associated with a Maharam
trace; also, we give a description of their dual spaces. We use the
terminology and results of the theory of von Neumann algebras ([5], [6]), the
theory of measurable operators ([7], [8]), and of the theory of Dedekind
complete Riesz space and Banach-Kantorovich spaces ([1]).
## 2\. Preliminaries
Let $X$ be a vector space over the field $\mathbb{C}$ of complex numbers, and
let $F$ be a Riesz space. A mapping $\|\cdot\|:X\to F$ is said to be a vector
($F$-valued) norm if it satisfies the following axioms:
1. (1)
$\|x\|\geq 0,$ $\|x\|=0$ $\Leftrightarrow$ $x=0$ ($x\in X$);
2. (2)
$\|\lambda x\|=|\lambda|\ \|x\|$ ($\lambda\in\mathbb{C},$ $x\in X$);
3. (3)
$\|x+y\|\leq\|x\|+\|y\|$ ($x,y\in X$).
A norm $\|\cdot\|$ is called decomposable if the following property holds:
###### Property 1.
If $f_{1},f_{2}\geq 0$ and $\|x\|=f_{1}+f_{2},$ then there exist
$x_{1},x_{2}\in X$ such that $x=x_{1}+x_{2}$ and $\|x_{k}\|=f_{k}$ ($k=1,2$).
If property 1 is valid only for disjoint elements $f_{1},f_{2}\in F,$ the norm
is called disjointly decomposable or, briefly, d-decomposable.
The pair $(X,\|\cdot\|)$ is called a lattice-normed space (shortly, LNS). If
the norm $\|\cdot\|$ is decomposable (d-decomposable), then so is the space
$(X,\|\cdot\|).$
A net $\\{x_{\alpha}\\}_{\alpha\in A}\subset X$ $(bo)$\- converges to $x\in X$
if the net $\\{\|x_{\alpha}-x\|\\}_{\alpha\in A}$ $(o)$-converges to zero in
the Riesz space $F.$ A net $\\{x_{\alpha}\\}_{\alpha\in A}$ is said to be a
$(bo)$\- Cauchy net if
$\sup\limits_{\alpha,\beta\geq\gamma}\|x_{\alpha}-x_{\beta}\|\downarrow 0.$ An
LNS is called $(bo)$\- complete if any $(bo)$-Cauchy net $(bo)$-converges. A
Banach-Kantorovich space (shortly, BKS) is a d-decomposable $(bo)$-complete
LNS. It is well known that every BKS is a decomposable LNS.
Let $F$ be a Dedekind complete Riesz space, and let $F_{\mathbb{C}}=F\oplus
iF$ be the complexification of $F.$ If $z=\alpha+i\beta\in
F_{\mathbb{C}},~{}\alpha,\beta\in F,$ then $\overline{z}:=\alpha-i\beta,$ and
$|z|:=\sup\\{Re(e^{i\theta}z):0\leq\theta<2\pi\\}$ (see[1], 1.3.13).
Let $(X,\|\cdot\|_{X})$ be the BKS over $F.$ A linear operator $T:X\to
F_{\mathbb{C}}$ is said to be $F$-bounded if there exists $0\leq c\in F$ such
that $|T(x)|\leq c\|x\|_{X}$ for all $x\in X.$ For any $F$-bounded operator
$T,$ define the element $\|T\|=\sup\\{|T(x)|:x\in X,$
$\|x\|_{X}\leq\mathbf{1}_{F}\\},$ which is called the abstract $F$-norm of the
operator $T$ ([1], 4.1.3). It is known that $|T(x)|\leq\|T\|\,\|x\|_{X}$ for
all $x\in X$ ([1], 4.1.1).
The set $X^{*}$ of all $F$-bounded linear mappings from $X$ into
$F_{\mathbb{C}}$ is called the $F$-dual space to the BKS $X.$ For $T,S\in
X^{*},$ we set $(T+S)(x)=Tx+Sx,$ $(\lambda T)(x)=\lambda Tx,$ where $x\in
X,~{}\lambda\in\mathbb{C}.$ It is clear that $X^{*}$ is a linear space with
respect to the introduced algebraic operations. Moreover, $(X^{*},\|\cdot\|)$
is a BKS ([1], 4.2.6).
Let $H$ be a Hilbert space, let $B(H)$ be the $*$-algebra of all bounded
linear operators on $H,$ and let $\mathbf{1}$ be the identity operator on $H.$
Given a von Neumann algebra $M$ acting on $H,$ denote by $Z(M)$ the center of
$M$ and by $P(M)$ the lattice of all projections in $M$. Let $P_{fin}(M)$ be
the set of all finite projections in $M.$
A densely-defined closed linear operator $x$ (possibly unbounded) affiliated
with $M$ is said to be _measurable_ if there exists a sequence
$\\{p_{n}\\}_{n=1}^{\infty}\subset P(M)$ such that $p_{n}\uparrow\mathbf{1}$,
$p_{n}(H)\subset\mathfrak{D}(x)$ and $p_{n}^{\bot}=\mathbf{1}-p_{n}\in
P_{fin}(M)$ for every $n=1,2,\ldots$ (here $\mathfrak{D}(x)$ is the domain of
$x$). Let us denote by $S(M)$ the set of all measurable operators.
Let $x,y$ be measurable operators. Then $x+y,~{}xy$ and $x^{*}$ are densely-
defined and preclosed. Moreover, the closures $\overline{x+y}$ (strong sum),
$\overline{xy}$ (strong product) and $x^{*}$ are also measurable, and $S(M)$
is a $*$-algebra with respect to the strong sum, strong product, and the
adjoint operation (see [7]). For any subset $E\subset S(M)$ we denote by
$E_{h}$ (resp. $E_{+}$ ) the set of all self-adjoint (resp. positive )
operators from $E.$
For $x\in S(M)$ let $x=u|x|$ be the polar decomposition, where
$|x|=(x^{*}x)^{\frac{1}{2}},$ $u$ is a partial isometry in $B(H).$ Then $u\in
M$ and $|x|\in S(M).$ If $x\in S_{h}(M)$ and $\\{E_{\lambda}(x)\\}$ are the
spectral projections of $x,$ then $\\{E_{\lambda}(x)\\}\subset P(M).$
Let $M$ be a commutative von Neumann algebra. Then $M$ is $*$-isomorphic to
the $*$-algebra $L^{\infty}(\Omega,\Sigma,\mu)$ of all essentially bounded
complex measurable functions with the identification almost everywhere, where
$(\Omega,\Sigma,\mu)$ is a measurable space. In addition $S(M)\cong
L^{0}(\Omega,\Sigma,\mu),$ where $L^{0}(\Omega,\Sigma,\mu)$ is the $*$-algebra
of all complex measurable functions with the identification almost everywhere
[7].
The locally measure topology $t(M)$ on $L^{0}(\Omega,\Sigma,\mu)$ is by
definition the linear (Hausdorff) topology whose fundamental system of
neighborhoods of $0$ is given by
$W(B,\varepsilon,\delta)=\\{f\in\ L^{0}(\Omega,\,\Sigma,\,\mu)\colon\hbox{
there exists a set }\ E\in\Sigma,\mbox{ such that}$ $\ E\subseteq B,\
\mu(B\setminus E)\leqslant\delta,\ f\chi_{E}\in
L^{\infty}(\Omega,\Sigma,\mu),\
\|f\chi_{E}\|_{{L_{\infty}}(\Omega,\Sigma,\mu)}\leqslant\varepsilon\\}.$
Here $\varepsilon,\ \delta$ run over all strictly positive numbers and
$B\in\Sigma$, $\mu(B)<\infty.$ It is known that $(S(M),t(M))$ is a complete
topological $*$-algebra.
It is clear that zero neighborhoods $W(B,\varepsilon,\delta)$ are closed and
have the following property: if $f\in W(B,\varepsilon,\delta),\,g\in
L^{\infty}(\Omega,\Sigma,\mu),\|g\|_{L{\infty}(\Omega,\Sigma,\mu)}\leq 1,$
then $gf\in W(B,\varepsilon,\delta).$
A net $\\{f_{\alpha}\\}$ converges locally in measure to $f$ (notation:
$f_{\alpha}\stackrel{{\scriptstyle t(M)}}{{\longrightarrow}}f$) if and only if
$f_{\alpha}\chi_{B}$ converges in $\mu$-measure to $f\chi_{B}$ for each
$B\in\Sigma$ with $\mu(B)<\infty$. If $M$ is $\sigma$-finite then there exists
a faithful finite normal trace $\tau$ on $M.$ In this case, the topology
$t(M)$ is metrizable, and convergence $f_{n}\stackrel{{\scriptstyle
t(M)}}{{\longrightarrow}}f$ is equivalent to convergence in trace $\tau$ of
the sequence $f_{n}$ to $f.$
Let now $M$ be an arbitrary finite von Neumann algebra, $\Phi_{M}:M\to Z(M)$
be a center-valued trace on $M$ ([5], 7.11). Let $Z(M)\cong
L^{\infty}(\Omega,\Sigma,\mu).$ The locally measure topology $t(M)$ on $S(M)$
is the linear (Hausdorff) topology whose fundamental system of neighborhoods
of $0$ is given by
$V(B,\varepsilon,\delta)=\\{x\in S(M)\colon\ \mbox{there exists }\ p\in
P(M),z\in P(Z(M))$ $\mbox{ such that}\ xp\in
M,\|xp\|_{M}\leqslant\varepsilon,\ z^{\bot}\in W(B,\varepsilon,\delta),\
\Phi_{M}(zp^{\bot})\leqslant\varepsilon z\\},$
where $\|\cdot\|_{M}$ is the $C^{*}$-norm in $M.$ It is known that
$(S(M),t(M))$ is a complete topological $*$-algebra [9].
From ([8], §3.5) we have the following criterion for convergence in the
topology $t(M).$
###### Proposition 2.1.
A net $\\{x_{\alpha}\\}_{\alpha\in A}\subset S(M)$ converges to zero in the
topology $t(M)$ if and only if
$\Phi_{M}(E^{\bot}_{\lambda}(|x_{\alpha}|)\stackrel{{\scriptstyle
t(M)}}{{\longrightarrow}}0$ for any $\lambda>0.$
Let $M$ be an arbitrary von Neumann algebra, and let $F$ be a Dedekind
complete Riesz space. An $F_{\mathbb{C}}$-valued trace on the von Neumann
algebra $M$ is a linear mapping $\Phi:M\to F_{\mathbb{C}}$ with
$\Phi(x^{*}x)=\Phi(xx^{*})\geq 0$ for all $x\in M.$ It is clear that
$\Phi(M_{h})\subset F,~{}\Phi(M_{+})\subset F_{+}=\\{a\in F:a\geq 0\\}.$ A
trace $\Phi$ is said to be faithful if the equality $\Phi(x^{*}x)=0$ implies
$x=0,$ normal if $\Phi(x_{\alpha})\uparrow\Phi(x)$ for every $x_{\alpha},x\in
M_{h},~{}x_{\alpha}\uparrow x.$
If $M$ is a finite von Neumann algebra, then its canonical center-valued trace
$\Phi_{M}:M\to Z(M)$ is an example of a $Z(M)$-valued faithful normal trace.
Let us list some properties of the trace $\Phi:M\to F_{\mathbb{C}}.$
###### Proposition 2.2.
([4]) $(i)$ Let $x,y,a,b\in M.$ Then
$\Phi(x^{*})=\overline{\Phi(x)},$ $\Phi(xy)=\Phi(yx),$
$\Phi(|x^{*}|)=\Phi(|x|),$
$|\Phi(axb)|\leq\|a\|_{M}\|b\|_{M}\Phi(|x|);$
$(ii)$ If $\Phi$ is a faithful trace, then $M$ is finite;
$(iii)$ If $x_{n},x\in M$ and $\|x_{n}-x\|_{M}\to 0,$ then
$|\Phi(x_{n})-\Phi(x)|$ relative uniform converges to zero;
$(iv)$ $\Phi(|x+y|)\leq\Phi(|x|)+\Phi(|y|)$ for all $x,y\in M.$
The trace $\Phi:M\to F_{\mathbb{C}}$ possesses the Maharam property if for any
$x\in M_{+},~{}0\leq f\leq\Phi(x),~{}f\in F,$ there exists $y\in M_{+},$
$y\leq x$ such that $\Phi(y)=f.$ A faithful normal $F_{\mathbb{C}}$-valued
trace $\Phi$ with the Maharam property is called a Maharam trace (compare with
[1], III, 3.4.1). Obviously, any faithful finite numerical trace on $M$ is a
$\mathbb{C}$-valued Maharam trace.
Let us give another examples of Maharam traces. Let $M$ be a finite von
Neumann algebra, let $\mathscr{A}$ be a von Neumann subalgebra in $Z(M),$ and
let $T:Z(M)\to\mathscr{A}$ be an injective linear positive normal operator. If
$f\in S(\mathscr{A})$ is a reversible positive element, then
$\Phi(T,f)(x)=fT(\Phi_{M}(x))$ is an $S(\mathscr{A})$-valued faithful normal
trace on $M.$ In addition, if $T(ab)=aT(b)$ for all $a\in\mathscr{A},b\in
Z(M),$ then $\Phi(T,f)$ is a Maharam trace on $M.$
If $\tau$ is a faithful normal finite numerical trace on $M$ and
$\dim(Z(M))>1,$ then $\Phi(x)=\tau(x)\mathbf{1}$ is a $Z(M)$-valued faithful
normal trace, which does not possess the Maharam property (see [4]).
Let $F$ have a weak order unit $\mathbf{1}_{F}.$ Denote by $B(F)$ the complete
Boolean algebra of unitary elements with respect to $\mathbf{1}_{F},$ and let
$Q$ be the Stone compact space of the Boolean algebra $B(F).$ Let
$C_{\infty}(Q)$ be the Dedekind complete Riesz space of all continuous
functions $a:Q\to[-\infty,+\infty]$ such that $a^{-1}(\\{\pm\infty\\})$ is a
nowhere dense subset of $Q.$ We identify $F$ with the order-dense ideal in
$C_{\infty}(Q)$ containing algebra $C(Q)$ of all continuous real functions on
$Q.$ In addition, $\mathbf{1}_{F}$ is identified with the function equal to 1
identically on $Q$ ([1], 1.4.4).
We need the following theorem from [4].
###### Theorem 2.3.
Let $\Phi$ be an $F_{\mathbb{C}}$-valued Maharam trace on a von Neumann
algebra $M.$ Then there exists a von Neumann subalgebra $\mathscr{A}$ in
$Z(M),$ a $*$-isomorphism $\psi$ from $\mathscr{A}$ onto the $*$-algebra
$C(Q)_{\mathbb{C}},$ a positive linear normal operator $\mathscr{E}$ from
$Z(M)$ onto $\mathscr{A}$ with
$\mathscr{E}(\mathbf{1})=\mathbf{1},~{}\mathscr{E}^{2}=\mathscr{E},$ such that
$1)$ $\Phi(x)=\Phi(\mathbf{1})\psi(\mathscr{E}(\Phi_{M}(x)))$ for all $x\in
M;$
$2)$ $\Phi(zy)=\Phi(z\mathscr{E}(y))$ for all $z,y\in Z(M);$
$3)$ $\Phi(zy)=\psi(z)\Phi(y)$ for all $z\in\mathscr{A},$ $y\in M.$
Due to Theorem 2.3, the $*$-algebra $\mathscr{B}=C(Q)_{\mathbb{C}}$ is a
commutative von Neumann algebra, and $*$-algebra $C_{\infty}(Q)_{\mathbb{C}}$
is identified with the $*$-algebra $S(\mathscr{B}).$ It is clear that the
$*$-isomorphism $\psi$ from $\mathscr{A}$ onto $\mathscr{B}$ can be extended
to a $*$-isomorphism from $S(\mathscr{A})$ onto $S(\mathscr{B}).$ We denote
this mapping also by $\psi.$
Let $\Phi$ be a $S(\mathscr{B})$-valued Maharam trace on a von Neumann algebra
$M.$ A net $\\{x_{\alpha}\\}\subset S(M)$ converges to $x\in S(M)$ with
respect to the trace $\Phi$ (notation:
$x_{\alpha}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}x$) if
$\Phi(E_{\lambda}^{\bot}(|x_{\alpha}-x|))\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0$ for all $\lambda>0.$
###### Proposition 2.4.
([4]) $x_{\alpha}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}x$ iff
$x_{\alpha}\stackrel{{\scriptstyle t(M)}}{{\longrightarrow}}x.$
An operator $x\in S(M)$ is said to be $\Phi$-integrable if there exists a
sequence $\\{x_{n}\\}\subset M$ such that
$x_{n}\stackrel{{\scriptstyle\Phi}}{{\to}}x$ and
$\|x_{n}-x_{m}\|_{\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0$ as $n,m\to\infty.$
Let $x$ be a $\Phi$-integrable operator from $S(M).$ Then there exists a
$\widehat{\Phi}(x)\in S(\mathscr{B})$ such that
$\Phi(x_{n})\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}\widehat{\Phi}(x).$ In addition
$\widehat{\Phi}(x)$ does not depend on the choice of a sequence
$\\{x_{n}\\}\subset M,$ for which
$x_{n}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}x,$
$\Phi(|x_{n}-x_{m}|)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0$ [4]. It is clear that each operator
$x\in M$ is $\Phi$-integrable and $\widehat{\Phi}(x)=\Phi(x).$
Denote by $L^{1}(M,\Phi)$ the set of all $\Phi$-integrable operators from
$S(M).$ If $x\in S(M)$ then $x\in L^{1}(M,\Phi)$ iff $|x|\in L^{1}(M,\Phi),$
in addition $|\widehat{\Phi}(x)|\leq\widehat{\Phi}(|x|)$ [4]. For any $x\in
L^{1}(M,\Phi),$ set $\|x\|_{1,\Phi}=\widehat{\Phi}(|x|).$ It is known that
$L^{1}(M,\Phi)$ is a linear subspace of $S(M),$ $ML^{1}(M,\Phi)M\subset
L^{1}(M,\Phi),$ and $x^{*}\in L^{1}(M,\Phi)$ for all $x\in L^{1}(M,\Phi)$ [4].
Moreover, the following theorem is true.
###### Theorem 2.5.
([4]) $(i)$ $(L^{1}(M,\Phi),\|\cdot\|_{1,\Phi})$ is a Banach-Kantorovich
space;
$(ii)$ $S(\mathscr{A})L^{1}(M,\Phi)\subset L^{1}(M,\Phi),$ in addition
$\widehat{\Phi}(zx)=\psi(z)\widehat{\Phi}(x)$ for all $z\in
S(\mathscr{A}),~{}x\in L^{1}(M,\Phi).$
## 3\. $L_{p}$-spaces associated with a Maharam trace
Let $\mathscr{B}$ be a commutative von Neumann algebra, which is
$*$-isomorphic to a von Neumann subalgebra $\mathscr{A}$ in $Z(M),$ and let
$\Phi:M\to S(\mathscr{B})$ be a Maharam trace on $M$ (see Theorem 2.3). For
any $p>1,$ set $L^{p}(M,\Phi)=\\{x\in S(M):|x|^{p}\in L^{1}(M,\Phi)\\}$ and
$\|x\|_{p,\Phi}=\widehat{\Phi}(|x|^{p})^{\frac{1}{p}}.$ It is clear that
$M\subset L^{p}(M,\Phi).$
Let $e$ be a nonzero projection in $\mathscr{B},$ and put
$\Phi_{e}(a)=\Phi(a)e,~{}a\in M.$ A mapping $\Phi_{e}:M\to S(\mathscr{B}e)$ is
a normal (not necessarily faithful) $S(\mathscr{B}e)$-valued trace on $M.$
Denote by $s(\Phi_{e}):=\mathbf{1}-\sup\\{p\in P(M):\Phi_{e}(p)=0\\}$ the
support of the trace $\Phi_{e}.$ It is clear that $s(\Phi_{e})\in P(Z(M))$ and
$\Phi_{e}(a)=\Phi(as(\Phi_{e}))$ is a faithful normal $S(\mathscr{B}e)$-valued
trace on $Ms(\Phi_{e})$ (compare [5], 5.15). Moreover $\Phi_{e}$ possesses the
Maharam property.
If $e$ and $g$ are orthogonal nonzero projections in $P(\mathscr{B}),$ then
$\Phi_{g}(s(\Phi_{e}))=\Phi(s(\Phi_{e}))g=\Phi_{e}(\mathbf{1})g=\Phi(\mathbf{1})eg=0,$
i.e. $s(\Phi_{e})s(\Phi_{g})=0.$ Let $\\{e_{i}\\}_{i\in I}$ be a family of
nonzero mutually orthogonal projections in $P(\mathscr{B})$ with
$\sup\limits_{i\in I}e_{i}=\mathbf{1}_{\mathscr{B}},$ where
$\mathbf{1}_{\mathscr{B}}$ is the unit of the algebra $\mathscr{B}.$ If
$z=\mathbf{1}-\sup\limits_{i\in I}s(\Phi_{e_{i}})$ then
$\Phi(z)e_{i}=\Phi_{e_{i}}(z)=0$ for all $i\in I.$ Therefore $\Phi(z)=0,$ i.e.
$z=0,$ or $\sup\limits_{i\in I}s(\Phi_{e_{i}})=\mathbf{1}.$
Further, we need the following
###### Proposition 3.1.
Let $x\in S(M)$ and let $\\{e_{i}\\}_{i\in I}$ be the family of nonzero
mutually orthogonal projections in $P(\mathscr{B})$ with $\sup_{i\in
I}e_{i}=\mathbf{1}_{\mathscr{B}}.$ Then $x\in L^{p}(M,\Phi)$ if and only if
$xs(\Phi_{e})\in L^{p}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}})$ for all $i\in I.$ In
addition $\|x\|_{p,\Phi}e_{i}=\|xs(\Phi_{e_{i}})\|_{p,\Phi_{e_{i}}}.$
###### Proof.
Let $x\in L^{p}(M,\Phi),$ $a_{n}=E_{n}(|x|^{p})|x|^{p}$ where $E_{n}(|x|^{p})$
is the spectral projection of $|x|^{p}$ corresponding to the interval
$(-\infty,n].$ It is clear that
$a_{n}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}|x|^{p}$ and
$\Phi(|a_{n}-a_{m}|)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0$ as $n,m\to\infty.$ Hence,
$a_{n}s(\Phi_{e_{i}})\stackrel{{\scriptstyle\Phi_{e_{i}}}}{{\longrightarrow}}|x|^{p}s(\Phi_{e_{i}})$
(see Proposition 2.4). In addition, from the inequality
$\Phi_{e_{i}}(|a_{n}s(\Phi_{e_{i}})-a_{m}s(\Phi_{e_{i}})|)=\Phi(|a_{n}-a_{m}|s(\Phi_{e_{i}}))\leq\Phi(|a_{n}-a_{m}|),$
we have
$\Phi_{e_{i}}(|a_{n}s(\Phi_{e_{i}})-a_{m}s(\Phi_{e_{i}})|\stackrel{{\scriptstyle
t(\mathscr{B}e_{i})}}{{\longrightarrow}}0.$ This means that
$|xs(\Phi_{e_{i}})|^{p}=|x|^{p}s(\Phi_{e_{i}})\in
L^{1}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}})$ and
$\|xs(\Phi_{e_{i}})\|_{p,\Phi_{e_{i}}}=\widehat{\Phi}_{e_{i}}(|x|^{p}s(\Phi_{e_{i}}))^{\frac{1}{p}}=(\widehat{\Phi}(|x|^{p})e_{i})^{\frac{1}{p}}=\|x\|_{p,\Phi}e_{i}.$
Conversely, let $xs(\Phi_{e_{i}})\in L^{p}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}})$ for
all $i\in I.$ Set
$a_{n,i}=E_{n}(|xs(\Phi_{e_{i}})|^{p})|xs(\Phi_{e_{i}})|^{p}.$ It is clear
that $a_{n,i}\uparrow|xs(\Phi_{e_{i}})|^{p}=|x|^{p}s(\Phi_{e_{i}})$ as
$n\to\infty$ for any fixed $i\in I.$ Therefore $a_{n,i}\stackrel{{\scriptstyle
t(Ms(\Phi_{e_{i}}))}}{{\longrightarrow}}|x|^{p}s(\Phi_{e_{i}}),$
$\Phi_{e_{i}}(|a_{n,i}-a_{m,i}|)\stackrel{{\scriptstyle
t(\mathscr{B}e_{i})}}{{\longrightarrow}}0$ as $n,m\to\infty.$ Since
$0\leq\Phi(\sqrt{a_{n,i}}a_{m,j}\sqrt{a_{n,i}})=\Phi(a_{n,i}a_{m,j})\leq\|a_{m,j}\|_{M}\Phi(a_{n,i})=\|a_{m,j}\|_{M}\Phi(a_{n,i})e_{i}$
and $\Phi(a_{n,i}a_{m,j})\leq\|a_{n,i}\|_{M}\Phi(a_{m,j})e_{j},$ we have
$\Phi(a_{n,i}a_{m,j})=0.$ Hence, $a_{n,i}a_{m,j}=0$ for all $n,m,~{}i\neq j.$
Since $0\leq a_{n,i}\leq ns(\Phi_{e_{i}}),$
$s(\Phi_{e_{i}})s(\Phi_{e_{j}})=0,$ $i\neq j,$ there is an $x_{n}\in M_{+}$
such that $x_{n}s(\Phi_{e_{i}})=a_{n,i}.$ Using the equality
$\sup\limits_{i\in I}s(\Phi_{e_{i}})=\mathbf{1},$ we obtain
$x_{n}\stackrel{{\scriptstyle t(M)}}{{\longrightarrow}}|x|^{p}$ ([10]),
moreover $\Phi(|x_{n}-x_{m}|)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0.$ Therefore $x\in L^{p}(M,\Phi).$ ∎
Similar to in the case of the space $L^{1}(M,\Phi),$ the subset
$L^{p}(M,\Phi)$ is invariant with respect to the action of involution in
$S(M).$ The following proposition is devoted to this fact.
###### Proposition 3.2.
If $x\in L^{p}(M,\Phi),$ then $x^{*}\in L^{p}(M,\Phi)$ and
$\|x\|_{p,\Phi}=\|x^{*}\|_{p,\Phi}.$
###### Proof.
Let $x=u|x|$ be the polar decomposition of $x.$ Since an algebra $M$ has a
finite type, we can suppose that $u$ is a unitary operator in $M.$ For each
$y\in S(M),$ we set $U(y)=uyu^{*}.$ Then the mapping $U:S(M)\to S(M)$ is a
$*$-isomorphism, and therefore $U(\varphi(y))=\varphi(U(y))$ for any
continuous function $\varphi:[0,+\infty)\to[0,+\infty)$ and $y\in S_{+}(M)$
[10]. If $\varphi(t)=t^{p},$ $p>1,$ $t\geq 0,$ and $y\in S_{+}(M)$ then
$uy^{p}u^{*}=(uyu^{*})^{p}.$ In particular, we obtain the equality
$|x^{*}|^{p}=u|x|^{p}u^{*}.$ Hence, $x^{*}\in L^{p}(M,\Phi).$ Moreover
$\|x^{*}\|_{p,\Phi}=\widehat{\Phi}(|x^{*}|^{p})^{\frac{1}{p}}=\widehat{\Phi}(u|x|^{p}u^{*})^{\frac{1}{p}}=\widehat{\Phi}(|x|^{p})^{\frac{1}{p}}=\|x\|_{p,\Phi}.$
∎
Now we need a version of the Hölder inequality for Maharam traces. In the
proof of this inequality for numerical traces, properties of decreasing
rearrangements of integrable operators are used [11]. For Maharam traces such
theory of decreasing rearrangements does not exact. Therefore we use another
approach connected with the concept of a bitrace on a $C^{*}$-algebra.
Let $\mathscr{N}$ be a $C^{*}$-algebra. A function
$s:\mathscr{N}\times\mathscr{N}\to\mathbb{C}$ is called a bitrace on
$\mathscr{N}$ ([12], 6.2.1) if the following relations hold:
$(i)$ $s(x,y)$ is positively defined sesquilinear Hermitian form on
$\mathscr{N};$
$(ii)$ $s(x,y)=s(x^{*},y^{*})$ for all $x,y\in\mathscr{N};$
$(iii)$ $s(zx,y)=s(x,z^{*}y)$ for all $x,y,z\in\mathscr{N};$
$(iv)$ for any $z\in\mathscr{N},$ the mapping $x\to zx$ is continuous on
$(\mathscr{N},\|\cdot\|_{s})$ where $\|x\|_{s}=\sqrt{s(x,x)},$
$x\in\mathscr{N};$
$(v)$ the set $\\{xy:x,y\in\mathscr{N}\\}$ is dense in
$(\mathscr{N},\|\cdot\|_{s}).$
If $\mathscr{N}$ has a unit, then condition $(v)$ holds automatically.
Let us list examples of bitraces associated with the Maharam trace.
Let $M$ be a von Neumann algebra, let $\Phi:M\to S(\mathscr{B})$ be a Maharam
trace and let $Q=Q(P(\mathscr{B}))$ be the Stone compact space of the Boolean
algebra $P(\mathscr{B}).$ We claim that
$s(\Phi(\mathbf{1}))=\mathbf{1}_{\mathscr{B}}.$ If it is not the case, then
$e=\mathbf{1}_{\mathscr{B}}-s(\Phi(\mathbf{1}))\neq 0$ and $z=\psi^{-1}(e)\neq
0$ where $\psi$ is a $*$-isomorphism from Theorem 2.3. By Theorem 2.5$(ii),$
we have $\Phi(z)=e\Phi(\mathbf{1})=0,$ which contradicts to the faithfulness
of the trace $\Phi.$ Thus, $s(\Phi(\mathbf{1}))=\mathbf{1}_{\mathscr{B}},$ and
therefore the following elements are defined: $(\Phi(\mathbf{1}))^{-1}\in
S_{+}(\mathscr{B})$ and $(\Phi(\mathbf{1}))^{-1}\Phi(x)\in C(Q)$ where $x\in
M.$ For any $t\in Q,$ set $\varphi_{t}(x)=(\Phi(\mathbf{1})^{-1}\Phi(x))(t).$
It is clear that $\varphi_{t}$ is a finite numerical trace on $M.$ The
function $s_{t}(x,y)=\varphi_{t}(y^{*}x)=\varphi_{t}(xy^{*})$ is a bitrace on
$M.$ In fact, the conditions $(i)-(iii)$ are obvious. $(iv)$ follows from the
inequality
$\|zx\|_{s_{t}}=\sqrt{\varphi_{t}((zx)^{*}(zx))}=\sqrt{\varphi_{t}(x^{*}z^{*}zx)}\leq\|z\|_{M}\|x\|_{s_{t}}.$
Let $s(x,y)$ be an arbitrary bitrace on a von Neumann algebra $M.$ Set
$N_{s}=\\{x\in M:s(x,x)=0\\}.$ It follows from ([12], 6.2.2) that $N_{s}$ is a
self-adjoint two-sided ideal in $M.$ We consider the factor-space $M/N_{s}$
with the scalar product $([x],[y])_{s}=s(x,y)$ where $[x],[y]$ are the
equivalence classes from $M/N_{s}$ with representatives $x$ and $y,$
respectively. Denote by $(H_{s},(\cdot,\cdot)_{s})$ the Hilbert space which is
the completion of $(M/N_{s},(\cdot,\cdot)_{s}).$ By the formula
$\pi_{s}(x)([y])=[xy],~{}x,y\in M,$ one defines a $*$-homomorphism
$\pi_{s}:M\to B(H_{s}).$ In addition
$\pi_{s}(\mathbf{1}_{M})=\mathbf{1}_{B(H_{s})}.$
Denote by $U_{s}(M)$ the von Neumann subalgebra in $B(H_{s})$ generated by
operators $\pi_{s}(x),$ i.e. $U_{s}(M)$ is the closure of the $*$-subalgebra
$\pi_{s}(M)$ in $B(H_{s})$ with respect to the weak operator topology.
According to ([13], s. 85-88), there exists a faithful normal semifinite
numerical trace $\tau_{s}$ on $(U_{s}(M))_{+}$ such that
$\tau_{s}(\pi(x^{2}))=([x],[x])=s(x,x)$ for all $x\in M_{+}.$ If $\varphi$ is
a trace on $M$ and $s(x,y)=\varphi(y^{*}x)$ then
$\tau_{s}(\pi_{s}(x^{2}))=\varphi(x^{2})$ for all $x\in M_{+}.$ This means
that $\tau_{s}(\pi_{s}(x))=\varphi(x)$ for any $x\in M_{+}.$ In addition, if
$\varphi(\mathbf{1}_{M})<\infty,$ then
$\tau_{s}(\mathbf{1}_{B(H_{s})})<\infty.$ Consequently, $\tau_{s}$ is a
faithful normal finite trace on $U_{s}(M).$
###### Theorem 3.3.
Let $\Phi$ be a $S(\mathscr{B})$-valued Maharam trace on the von Neumann
algebra $M,$ $p,q>1,$ $\frac{1}{p}+\frac{1}{q}=1.$ If $x\in L^{p}(M,\Phi),$
$y\in L^{q}(M,\Phi),$ then $xy\in L^{1}(M,\Phi)$ and
$\|xy\|_{1,\Phi}\leq\|x\|_{p,\Phi}\|y\|_{q,\Phi}.$
###### Proof.
We consider the bitrace $s_{t}(x,y)=\varphi_{t}(y^{*}x)$ on $M$ where
$\varphi_{t}(x)=((\Phi(\mathbf{1}))^{-1}\Phi(x))(t),~{}t\in
Q(P(\mathscr{B})).$ Denote by $\tau_{t}$ a faithful normal finite trace on
$(U_{s_{t}}(M))_{+}$ such that $\tau_{t}(\pi_{s_{t}}(x))=\varphi_{t}(x)$ for
all $x\in M_{+}.$ Since the trace $\tau_{t}$ is finite,
$\tau_{t}(\pi_{s_{t}}(x)=\varphi_{t}(x)$ for any $x\in M.$ Let
$L^{p}(U_{s_{t}}(M),\tau_{t})$ be the non-commutative $L^{p}$-space associated
with the numerical trace $\tau_{t}.$ It follows from [11] that
$\|\pi_{s_{t}}(x)\pi_{s_{t}}(y)\|_{1,\tau_{t}}\leq\|\pi_{s_{t}}(x)\|_{p,\tau_{t}}\|\pi_{s_{t}}(y)\|_{q,\tau_{t}},$
i.e.
$\tau_{t}(|\pi_{s_{t}}(xy)|)\leq\tau_{t}(|\pi_{s_{t}}(x)|^{p})^{\frac{1}{p}}\tau_{t}(|\pi_{s_{t}}(y)|^{q})^{\frac{1}{q}}.$
Since $\pi_{s_{t}}(|x|)=|\pi_{s_{t}}(x)|,~{}x\in M,$ we get
$\pi_{s_{t}}(|x|^{p})=(\pi_{s_{t}}(|x|))^{p}$ ([12], 1.5.3).
Thus,
$\tau_{t}(\pi_{s_{t}}(|xy|))\leq\tau_{t}(\pi_{s_{t}}(|x|^{p}))^{\frac{1}{p}}\tau_{t}(\pi_{s_{t}}(|y|^{q}))^{\frac{1}{q}},$
i.e.
$\varphi_{t}(|xy|)\leq\varphi_{t}(|x|^{p})^{\frac{1}{p}}\varphi_{t}(|y|^{q})^{\frac{1}{q}},$
or
$(\Phi(\mathbf{1}))^{-1}\Phi(|xy|)(t)\leq[((\Phi(\mathbf{1}))^{-1}\Phi(|x|^{p}))(t)]^{\frac{1}{p}}[((\Phi(\mathbf{1}))^{-1}\Phi(|y|^{q}))(t)]^{\frac{1}{q}}$
for all $t\in Q(P(\mathscr{B})).$ This means that
$(\Phi(\mathbf{1}))^{-1}\Phi(|xy|)\leq[((\Phi(\mathbf{1}))^{-1}\Phi(|x|^{p}))]^{\frac{1}{p}}[((\Phi(\mathbf{1}))^{-1}\Phi(|y|^{q}))]^{\frac{1}{q}}.$
Multiplying this inequality by $\Phi(\mathbf{1}),$ we get
$\|xy\|_{1,\Phi}\leq\|x\|_{p,\Phi}\|y\|_{q,\Phi}.$
Let now $x\in L^{p}_{+}(M,\Phi),$ $y\in L^{q}_{+}(M,\Phi).$ We claim that
$xy\in L^{1}(M,\Phi).$ Set $a_{n}=E_{n}(x)x,~{}b_{n}=E_{n}(y)y.$ We have
$a_{n},b_{n}\in M_{+}$ and $a_{n}\uparrow x,~{}b_{n}\uparrow y,$ in
particular,
$a_{n}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}x,~{}b_{n}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}y.$
Hence, $a_{n}b_{n}\in M$ and
$a_{n}b_{n}\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}xy.$ In addition,
$\|a_{n}b_{n}-a_{m}b_{m}\|_{1,\Phi}\leq\|a_{n}b_{n}-a_{n}b_{m}\|_{1,\Phi}+\|a_{n}b_{m}-a_{m}b_{m}\|_{1,\Phi}\leq\|a_{n}\|_{p,\Phi}\|b_{n}-b_{m}\|_{q,\Phi}+\|a_{n}-a_{m}\|_{p,\Phi}\|b_{m}\|_{q,\Phi}.$
Since
$\|a_{n}\|_{p,\Phi}\leq\|x\|_{p,\Phi},~{}\|b_{m}\|_{q,\Phi}\leq\|y\|_{q,\Phi},$
and for $n>m,$
$\|a_{n}-a_{m}\|^{p}_{p,\Phi}=\widehat{\Phi}(x^{p}E_{n}(x)E_{m}^{\bot}(x))\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0,$
$\|b_{n}-b_{m}\|^{q}_{q,\Phi}=\widehat{\Phi}(y^{q}E_{n}(y)E_{m}^{\bot}(y))\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0,$ we get
$\|a_{n}b_{n}-a_{m}b_{m}\|_{1,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0$ as $n,m\to\infty.$ This means that
$xy\in L^{1}(M,\Phi)$ and $\|a_{n}b_{n}-xy\|_{1,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0.$ The inequality
$|\|xy\|_{1,\Phi}-\|a_{n}b_{n}\|_{1,\Phi}|\leq\|xy-a_{n}b_{n}\|_{1,\Phi}$
implies $\|a_{n}b_{n}\|_{1,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}\|xy\|_{1,\Phi}.$ Since
$\|a_{n}b_{n}\|_{1,\Phi}\leq\|a_{n}\|_{p,\Phi}\|b_{n}\|_{q,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}\|x\|_{p,\Phi}\|y\|_{q,\Phi},$
we obtain $\|xy\|_{1,\Phi}\leq\|x\|_{p,\Phi}\|y\|_{q,\Phi}.$
If $x\in L^{p}(M,\Phi)$ is arbitrary, $y\in L^{q}_{+}(M,\Phi)$ and $x=u|x|$ is
the polar decomposition of $x$ with the unitary $u\in M,$ then $xy=u(|x|y)\in
L^{1}(M,\Phi)$ and
$\|xy\|_{1,\Phi}=\||x|y\|_{1,\Phi}\leq\|x\|_{p,\Phi}\|y\|_{q,\Phi}.$
Let now $x\in L^{p}(M,\Phi),$ $y\in L^{q}(M,\Phi)$ be arbitrary and let
$y^{*}=v|y^{*}|$ be the polar decomposition of $y^{*}$ with the unitary $v\in
M.$ According to Proposition 3.2, $|y^{*}|\in L^{q}(M,\Phi)$ and
$\|y^{*}\|_{q,\Phi}=\|y\|_{q,\Phi}.$ Therefore $xy=(x|y^{*}|)v^{*}\in
L^{1}(M,\Phi)$ and
$\|xy\|_{1,\Phi}=\|x|y^{*}|\|_{1,\Phi}\leq\|x\|_{p,\Phi}\||y^{*}|\|_{q,\Phi}=\|x\|_{p,\Phi}\|y\|_{q,\Phi}.$
∎
###### Theorem 3.4.
Let $\Phi,M,p,$ and $q$ be the same as in Theorem 3.3. If $x\in S(M),$ $xy\in
L^{1}(M,\Phi)$ for all $y\in L^{q}(M,\Phi)$ and the set
$D(x)=\\{|\widehat{\Phi}(xy)|:y\in L^{q}(M,\Phi),$
$\|y\|_{q,\Phi}\leq\mathbf{1}_{\mathscr{B}}\\}$ is bounded in
$S_{h}(\mathscr{B}),$ then $x\in L^{p}(M,\Phi)$ and $\|x\|_{p,\Phi}=\sup
D(x).$
###### Proof.
Let $x\neq 0,$ and let $x=u|x|$ be the polar decomposition of $x$ with the
unitary $u\in M.$ Set
$y_{n}=|x|^{p-1}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)u^{*},~{}n=1,2,\dots$ It
is clear that $y_{n}\in M$ and
$xy_{n}=u|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)u^{*}=uE_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)u^{*}\geq
0.$
On the other hand,
$|y_{n}|^{2}=uE_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)|x|^{2p-2}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)u^{*}=$
$uE_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)|x|^{\frac{2p}{q}}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)u^{*},$
and therefore $0\leq|y_{n}|^{q}=(|y_{n}|^{2})^{\frac{q}{2}}=xy_{n},$ in
particular, $\|y_{n}\|_{q,\Phi}=\Phi(xy_{n})^{\frac{1}{q}}.$
Since $xy_{n}\stackrel{{\scriptstyle
t(M)}}{{\longrightarrow}}u|x|^{p}u^{*}\neq 0,$ we have $xy_{n}\neq 0$ for
all$n\geq n_{0}.$ Set $e_{n}=s(\Phi(xy_{n})$ as $n\geq n_{0}.$ Since
$S_{h}(\mathscr{B})=C_{\infty}(Q(P(\mathscr{B})),$ there exists a unique
$b_{n}\in S_{+}(\mathscr{B})e_{n}$ such that $b_{n}\Phi(xy_{n})=e_{n}.$ It is
clear that $b_{n}^{\frac{1}{q}}\Phi^{\frac{1}{q}}(xy_{n})=e_{n}.$ If
$z_{n}=\psi^{-1}(e_{n}),~{}a_{n}=\psi^{-1}(b_{n}^{\frac{1}{q}})\in
S(\mathscr{A}z_{n}),$ then by theorem 2.5$(ii),$ $a_{n}y_{n}\in L^{q}(M,\Phi)$
and
$\|a_{n}y_{n}\|^{q}_{q,\Phi}=\widehat{\Phi}(a_{n}^{q}|y_{n}|^{q})=b_{n}\widehat{\Phi}(xy_{n})=e_{n}\leq\mathbf{1}_{\mathscr{B}}.$
Hence, $|\widehat{\Phi}(a_{n}xy_{n})|=|\widehat{\Phi}(x(a_{n}y_{n}))|\leq\sup
D(x)$ for all $n\geq n_{0}.$ On the other hand,
$\widehat{\Phi}(a_{n}xy_{n})=b_{n}^{\frac{1}{q}}\widehat{\Phi}(xy_{n})=(b_{n}\widehat{\Phi}(xy_{n}))^{\frac{1}{q}}\widehat{\Phi}(xy_{n})^{1-\frac{1}{q}}=\widehat{\Phi}(xy_{n})^{\frac{1}{p}}=$
$\widehat{\Phi}(u|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)u^{*})^{\frac{1}{p}}=\widehat{\Phi}(|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|))^{\frac{1}{p}}.$
Since
$(|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|))\uparrow|x|^{p},~{}|x|^{p}(E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|)\in
M_{+}$ and
$\widehat{\Phi}(|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|))\leq(\sup
D(x))^{p},$ we have $|x|^{p}\in L^{1}(M,\Phi)$ and
$\widehat{\Phi}(|x|^{p})=\sup_{n\geq
1}\widehat{\Phi}(|x|^{p}E_{n}(|x|)E^{\bot}_{\frac{1}{n}}(|x|))$ [14]. This
means that $x\in L^{p}(M,\Phi)$ and $\|x\|_{p,\Phi}\leq\sup D(x).$ Theorem 3.3
implies $\sup D(x)\leq\|x\|_{p,\Phi},$ and therefore $\|x\|_{p,\Phi}=\sup
D(x).$ ∎
With the help of Theorem 3.4, it is not difficult to show that $L^{p}(M,\Phi)$
is disjointly decomposable LNS over $S_{h}(\mathscr{B})$ for all $p>1.$
###### Theorem 3.5.
$(i)$ $L^{p}(M,\Phi)$ is a linear subspace in $S(M),$ and $\|\cdot\|_{p,\Phi}$
is the disjointly decomposable $S_{h}(\mathscr{B})$-valued norm on
$L^{p}(M,\Phi);$
$(ii)$ $ML^{p}(M,\Phi)M\subset L^{p}(M,\Phi),$ and
$\|axb\|_{p,\Phi}\leq\|a\|_{M}\|b\|_{M}\|x\|_{p,\Phi}$ for all $a,b\in
M,~{}x\in L^{p}(M,\Phi);$
$(iii)$ If $0\leq x\leq y\in L^{p}(M,\Phi),~{}x\in S(M),$ then $x\in
L^{p}(M,\Phi)$ and $\|x\|_{p,\Phi}\leq\|y\|_{p,\Phi}.$
###### Proof.
$(i)$ It is clear that $\lambda x\in L^{p}(M,\Phi)$ and $\|\lambda
x\|_{p,\Phi}=|\lambda|\|x\|_{p,\Phi}$ for all $x\in
L^{p}(M,\Phi),~{}\lambda\in\mathbb{C}.$ Moreover, $\|x\|_{p,\Phi}\geq 0$ and
$\widehat{\Phi}(|x|^{p})=\|x\|^{p}_{p,\Phi}=0$ if and only if $x=0.$
We claim that $x+y\in L^{p}(M,\Phi)$ and
$\|x+y\|_{p,\Phi}\leq\|x\|_{p,\Phi}+\|y\|_{p,\Phi}$ for each $x,y\in
L^{p}(M,\Phi).$ By theorem 3.3, $(x+y)z=xz+yz\in L^{1}(M,\Phi)$ for all $z\in
L^{q}(M,\Phi),$ in addition
$|\widehat{\Phi}((x+y)z)|\leq|\widehat{\Phi}(xz)|+|\widehat{\Phi}(yz)|.$
If $\|z\|_{q,\Phi}\leq\mathbf{1}_{\mathscr{B}},$ then by theorem 3.4,
$|\widehat{\Phi}((x+y)z)|\leq\|x\|_{p,\Phi}+\|y\|_{p,\Phi}.$
Using Theorem 3.4 again, we obtain $x+y\in L^{p}(M,\Phi)$ and
$\|x+y\|_{p,\Phi}\leq\|x\|_{p,\Phi}+\|y\|_{p,\Phi}.$ Thus, $L^{p}(M,\Phi)$ is
a linear subspace in $S(M),$ and $\|\cdot\|_{p,\Phi}$ is a
$S_{h}(\mathscr{B})$-valued norm on $L^{p}(M,\Phi).$
Let us now show that the norm $\|\cdot\|_{p,\Phi}$ is $d$-decomposable. It is
known [4] that, if $x\in L^{1}(M,\Phi),~{}\|x\|_{1,\Phi}=f_{1}+f_{2},$ where
$f_{1},f_{2}\in S_{+}(\mathscr{B}),~{}f_{1}f_{2}=0,$ then, setting
$x_{i}=xp_{i}$ for $p_{i}=\psi^{-1}(s(f_{i})),~{}i=1,2,$ we get
$x=x_{1}+x_{2}$ and $\|x_{i}\|_{\Phi}=f_{i},~{}i=1,2.$
Let $y\in L_{+}^{p}(M,\Phi),~{}\|y\|_{p,\Phi}=g_{1}+g_{2}$ where
$g_{1},g_{2}\in S_{+}(\mathscr{B}),~{}g_{1}g_{2}=0,$ i.e.
$\|y^{p}\|_{1,\Phi}=\|y\|^{p}_{p,\Phi}=g_{1}^{p}+g_{2}^{p}.$ Set
$q_{i}=\psi^{-1}(s(g_{i}^{p}))\in P(\mathscr{A})\subset P(Z(M))$ and
$y_{i}=yq_{i}.$ Then $y_{i}^{p}=y^{p}q_{i}$ and using [4] for
$x=y^{p},~{}f_{i}=g_{i}^{p},~{}i=1,2$ we obtain that
$y^{p}q_{1}+y^{p}q_{2}=y^{p}$ and $\|yq_{i}\|_{p,\Phi}=g_{i},~{}i=1,2.$ Since
$q_{1}q_{2}=0,~{}q_{1},q_{2}\in P(Z(M)),$ we have $yq_{1}+yq_{2}=y.$
Let now $y$ be an arbitrary element from $L^{p}(M,\Phi)$ and let $y=u|y|$ be
the polar decomposition of $y$ with the unitary $u\in M.$ Let
$\||y|\|_{p,\Phi}=\|y\|_{p,\Phi}=f_{1}+f_{2}$ where $f_{1},f_{2}\in
S_{+}(\mathscr{B}),~{}f_{1}f_{2}=0.$ It follows from above that for
$q_{i}=\psi^{-1}(s(f_{i}^{p}))\in P(\mathscr{A}),$ we have
$|y|=|y|q_{1}+|y|q_{2}$ $\||y|q_{i}\|_{p,\Phi}=f_{i}.$ Consequently,
$y=u|y|=u|y|q_{1}+u|y|q_{2}=yq_{1}+yq_{2}$ and
$\|yq_{i}\|_{p,\Phi}=\||yq_{i}|\|_{p,\Phi}=\||y|q_{i}\|_{p,\Phi}=f_{i},~{}i=1,2.$
Hence, the norm $\|\cdot\|_{p,\Phi}$ is $d$-decomposable.
$(ii)$ Let $v$ be a unitary operator in $M,$ $x\in L^{p}(M,\Phi).$ Then
$|vx|=(x^{*}v^{*}vx)^{\frac{1}{2}}=|x|,$ and therefore $vx\in L^{p}(M,\Phi).$
Since any operator $a\in M$ is a linear combination of four unitary operators,
we have $ax\in L^{p}(M,\Phi),$ due to $(i).$
We claim that $\|ax\|_{p,\Phi}\leq\|a\|_{M}\|x\|_{p,\Phi}$ for $a\in M,~{}x\in
L^{p}(M,\Phi).$ Let $\nu$ be a faithful normal semifinite numerical trace on
$\mathscr{B}.$ If for some $a\in M,$ $x\in L^{p}(M,\Phi)$ the previous
inequality is not true, then there are $\varepsilon>0,~{}0\neq e\in
P(\mathscr{B}),~{}\nu(e)<\infty$ such that
$e\|ax\|_{p,\Phi}\geq e\|a\|_{M}\|x\|_{p,\Phi}+\varepsilon e.$
By the formula
$\tau(b)=\nu(e\Phi(b)(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(|x|^{p}))^{-1}),b\in
Ms(\Phi_{e})$
one defines a faithful normal finite numerical trace on $Ms(\Phi_{e}).$ If
$z=\psi^{-1}(e)\in P(\mathscr{A}),$ then
$\Phi_{e}(\mathbf{1}-z)=(\mathbf{1}_{\mathscr{B}}-e)e\Phi(\mathbf{1})=0,$ i.e.
$s(\Phi_{e})\leq z.$ Since
$\Phi(z-s(\Phi_{e}))=\Phi(z(\mathbf{1}-s(\Phi_{e}))=e\Phi(\mathbf{1}-s(\Phi_{e}))=0,$
we get $z=s(\Phi_{e}).$ We consider the $L^{p}$-space
$L^{p}(Ms(\Phi_{e}),\tau)$ associated with the numerical trace $\tau,$ and let
us show that $xz\in L^{p}(Ms(\Phi_{e}),\tau).$ Let $x_{n}=E_{n}(|x|)|x|.$ It
is clear that $0\leq x_{n}^{p}z\uparrow|x|^{p}z,$ moreover
$\tau(x_{n}^{p}z)=\nu(e\Phi(x_{n}^{p}z)(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(|x|^{p}))^{-1})\leq\nu(e)<\infty.$
Hence, $|xz|^{p}=|x|^{p}z\in L^{p}(Ms(\Phi_{e}),\tau)$ and
$\|xz\|^{p}_{p,\tau}=\lim\limits_{n\to\infty}\|x_{n}^{p}z\|^{p}_{p,\tau}=\nu(e\widehat{\Phi}(|x|^{p}z)(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(|x|^{p}))^{-1}).$
Thus, if $a\in M$ then $axz\in L^{p}(Ms(\Phi_{e}),\tau),$ in addition
$\|a\|_{M}\|xz\|^{p}_{p,\tau}\geq\|axz\|^{p}_{p,\tau}=\nu(e\widehat{\Phi}(|axz|^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(|x|^{p}))^{-1})=$
$\nu(e\|ax\|_{p,\Phi}^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(|x|^{p}))^{-1})\geq$
$\nu(e(\|a\|_{M}\|x\|_{p,\Phi}+\varepsilon)^{p}(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(|x|^{p}))^{-1})>\|a\|_{M}^{p}\|xz\|_{p,\tau}^{p},$
which is not the case. Consequently,
$\|ax\|_{p,\Phi}\leq\|a\|_{M}\|x\|_{p,\Phi}.$
If $b\in M,~{}x\in L^{p}(M,\Phi),$ then by Proposition 3.2 and from above, we
have $b^{*}x^{*}\in L^{p}(M,\Phi).$ Using Proposition 3.2 again, we obtain
$xb=(b^{*}x^{*})^{*}\in L^{p}(M,\Phi)$ and
$\|xb\|_{p,\Phi}=\|b^{*}x^{*}\|_{p,\Phi}\leq\|b^{*}\|_{M}\|x^{*}\|_{p,\Phi}=\|b\|_{M}\|x\|_{p,\Phi}.$
$(iii)$ Let $0\leq x\leq y\in L^{p}(M,\Phi),~{}x\in S(M).$ It follows from
([8], §2.4) that $\sqrt{x}=a\sqrt{y}$ where $a\in M$ with $\|a\|_{M}\leq 1.$
Hence, $x=\sqrt{x}(\sqrt{x})^{*}=aya^{*}\in L^{p}(M,\Phi)$
$\|x\|_{p,\Phi}\leq\|a\|_{M}\|a^{*}\|_{M}\|y\|_{p,\Phi}\leq\|y\|_{p,\Phi}.$
∎
Using the Hölder inequality and the $(bo)$-completeness of the space
$(L^{1}(M,\Phi),\|\cdot\|_{\Phi})$ we can establish the $(bo)$-completeness of
the space $(L^{p}(M,\Phi),\|\cdot\|_{p,\Phi}).$
###### Theorem 3.6.
Let $\Phi,M,p$ be the same as in Theorem 3.3. Then
$(L^{p}(M,\Phi),\|\cdot\|_{p,\Phi})$ is the Banach-Kantorovich space.
###### Proof.
First, we assume that $\mathscr{B}$ is a $\sigma$-finite von Neumann algebra.
Then there exists a faithful normal finite numerical trace $\nu$ on
$\mathscr{B}.$ The numerical function
$\tau(a)=\nu(\Phi(a)(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1}))^{-1})$ is a
faithful normal finite trace on $M.$ Moreover, the topology $t(M)$ coincides
with topology of convergence in measure $t_{\tau}$ in $(S(M),\tau)$ ([8],
§3.5).
Let $\\{x_{\alpha}\\}_{\alpha\in A}\subset(L^{p}(M,\Phi),\|\cdot\|_{p,\Phi})$
be an $(bo)$-Cauchy net i.e.
$b_{\gamma}=\sup\limits_{\alpha,\beta\geq\gamma}\|x_{\alpha}-x_{\beta}\|_{p,\Phi}\downarrow
0.$ According to the Hölder inequality, for each $x\in L^{p}(M,\Phi)$ we have
$x\in L^{1}(M,\Phi)$ and
$\|x\|_{1,\Phi}=\widehat{\Phi}(|x|\mathbf{1})\leq(\Phi(\mathbf{1}))^{\frac{1}{q}}\|x\|_{p,\Phi}.$
In particular, the set
$\\{\|x_{\alpha}-x_{\beta}\|_{1,\Phi}\\}_{\alpha,\beta\geq\gamma}$ is bounded
in $S_{h}(\mathscr{B}),$ and
$\sup\limits_{\alpha,\beta\geq\gamma}\|x_{\alpha}-x_{\beta}\|_{1,\Phi}\leq(\Phi(\mathbf{1}))^{\frac{1}{q}}b_{\gamma}$
for all $\gamma\in A.$ Consequently [4], there exists $x\in L^{1}(M,\Phi)$
such that
$\|x_{\alpha}-x\|_{1,\Phi}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}0$ in
particular, $x_{\alpha}\stackrel{{\scriptstyle t_{\tau}}}{{\longrightarrow}}x$
and $y_{\alpha}=|x_{\alpha}-x_{\beta}|\stackrel{{\scriptstyle
t_{\tau}}}{{\longrightarrow}}|x-x_{\beta}|.$ Since the function
$\varphi(t)=t^{p}$ is continuous on $[0,\infty),$ the operator function
$y\longmapsto y^{p}$ is continuous on $(S_{+}(M),t_{\tau})$ [15]. Hence,
$0\leq y_{\alpha}^{p}\stackrel{{\scriptstyle
t_{\tau}}}{{\longrightarrow}}|x-x_{\beta}|^{p},$ in addition
$\widehat{\Phi}(y_{\alpha}^{p})=\|x_{\alpha}-x_{\beta}\|^{p}_{p,\Phi}\leq
b_{\gamma}^{p}.$ Using the of Fatou’s theorem [14], we obtain
$|x-x_{\beta}|^{p}\in L^{1}(M,\Phi)$ and
$\widehat{\Phi}(|x-x_{\beta}|^{p})\leq b_{\gamma}^{p}.$ Thus,
$(x-x_{\beta})\in L^{p}(M,\Phi)$ for all $\beta\geq\gamma$ and
$\sup\limits_{\beta\geq\gamma}\|x-x_{\beta}\|_{p,\Phi}\leq
b_{\gamma}\downarrow 0.$ This means that $x\in L^{p}(M,\Phi),$ and
$\|x_{\alpha}-x\|_{p,\Phi}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}0.$
Now let $\mathscr{B}$ be an arbitrary von Neumann algebra ( not necessarily
$\sigma$-finite), and let $\\{x_{\alpha}\\}\subset L^{p}(M,\Phi)$ be a
$(bo)$-Cauchy net. It follows from the above that there exists $x\in
L^{1}(M,\Phi)$ such that
$\|x_{\alpha}-x\|_{1,\Phi}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}0.$
In particular $x_{\alpha}\stackrel{{\scriptstyle t(M)}}{{\longrightarrow}}x.$
Let $\nu$ be a faithful normal semifinite numerical trace on $\mathscr{B},$
and let $\\{e_{i}\\}_{i\in I}$ be the family of nonzero mutually orthogonal
projections in $\mathscr{B}$ such that $\sup\limits_{i\in
I}e_{i}=\mathbf{1}_{\mathscr{B}},$ and $\nu(e_{i})<\infty$ for all $i\in I.$
It is clear that $\\{x_{\alpha}s(\Phi_{e_{i}})\\}_{\alpha\in A}$ is a
$(bo)$-Cauchy net in $L^{p}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}}).$ Since the algebra
$\mathscr{B}e_{i}$ is $\sigma$-finite, from the above there exists $x_{i}\in
L^{p}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}})$ such that
$\|x_{i}-x_{\alpha}s(\Phi_{e_{i}})\|_{p,\Phi_{e_{i}}}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}0.$
In particular, $x_{\alpha}s(\Phi_{e_{i}})\stackrel{{\scriptstyle
t(M)}}{{\longrightarrow}}x_{i}=x_{i}s(\Phi_{e_{i}}).$ On the other hand,
convergence $x_{\alpha}\stackrel{{\scriptstyle t(M)}}{{\longrightarrow}}x$
implies $x_{\alpha}s(\Phi_{e_{i}})\stackrel{{\scriptstyle
t(M)}}{{\longrightarrow}}xs(\Phi_{e_{i}}).$ Thus,
$xs(\Phi_{e_{i}})=x_{i}s(\Phi_{e_{i}})$ for all $i\in I.$ By Proposition 3.1,
we have $x\in L^{p}(M,\Phi)$ and
$\|x-x_{\alpha}\|_{p,\Phi}e_{i}=\|xs(\Phi_{e_{i}})-x_{\alpha}s(\Phi_{e_{i}})\|_{p,\Phi_{e_{i}}}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}0$
for all $i\in I$ and therefore
$\|x-x_{\alpha}\|_{p,\Phi}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}0.$ ∎
###### Proposition 3.7.
If $\\{x_{\alpha}\\}_{\alpha\in A}\subset L^{p}_{h}(M,\Phi)$ and
$x_{\alpha}\downarrow 0,$ then $\|x_{\alpha}\|_{p,\Phi}\downarrow 0.$
###### Proof.
Let $\nu$ be a faithful normal semifinite numerical trace on $\mathscr{B}.$ If
$b=\inf_{\alpha\in I}\|x_{\alpha}\|_{p,\Phi}\neq 0,$ then there are
$\varepsilon>0,$ $0\neq e\in P(\mathscr{B})$ with $\nu(e)<\infty$ such that
$e\|x_{\alpha}\|_{p,\Phi}\geq eb\geq\varepsilon e$ for all $\alpha\in A.$ Put
$\Phi_{e}(x)=e\Phi(x),~{}x\in M,$ and
$\tau(y)=\nu(\Phi(y)(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x_{\alpha_{0}}^{p}))^{-1}),$
$y\in Ms(\Phi_{e}),$ where $\alpha_{0}$ is a fixed element from $A.$ Let us
prove that $L^{p}(Ms(\Phi_{e}),\tau)\subset L^{p}(Ms(\Phi_{e}),\Phi_{e})$ and
$\|x\|^{p}_{p,\tau}=\nu(\widehat{\Phi}(|x|^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1})$
for all $x\in L^{p}(Ms(\Phi_{e}),\tau).$ It is sufficient to consider the case
where $x\in L^{p}_{+}(Ms(\Phi_{e}),\tau).$ Set $x_{n}=E_{n}(x)xs(\Phi_{e}).$
It is clear that $x_{n}\in(Ms(\Phi_{e}))_{+},~{}x_{n}^{p}\uparrow
x^{p},~{}x_{n}^{p}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}x^{p},$ and
therefore $x^{p}_{n}\stackrel{{\scriptstyle t(M)}}{{\longrightarrow}}x^{p}.$
Moreover, $\Phi(|x_{n}^{p}-x_{m}^{p}|)=\Phi(x^{p}E_{n}(x)E_{m}^{\bot}(x))$ as
$m<n.$ Since
$\nu(e\Phi(|x_{n}^{p}-x_{m}^{p}|)(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1})=\|x_{n}^{p}-x_{m}^{p}\|_{1,\tau}=\|x^{p}E_{n}(x)E_{m}^{\bot}(x)\|_{1,\tau}\to
0$ as $n,m\to\infty,$ we get
$\Phi(|x_{n}^{p}-x_{m}^{p}|)=e\Phi(|x_{n}^{p}-x_{m}^{p}|)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0.$ This means that $x^{p}\in
L^{1}(M,\Phi)$ and $\Phi(x^{p}_{n})\uparrow\widehat{\Phi}(x^{p}),$ i.e. $x\in
L^{p}(Ms(\Phi_{e}),\Phi_{e})$ $\|x\|_{p,\Phi_{e}}=\sup_{n\geq
1}(\Phi(x_{n}^{p}))^{\frac{1}{p}}.$ Using the inequality
$\nu(\Phi(x_{n}^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1})=\tau(x_{n}^{p})\leq\tau(x^{p})$
we obtain that
$\widehat{\Phi}(x^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1}\in
L_{1}(\mathscr{B},\nu)$ and
$\nu(\widehat{\Phi}(x^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1})=\sup_{n\geq
1}\tau(x^{p}_{n})=\tau(x^{p}),$
i.e.
$\|x\|_{p,\tau}=\nu(\widehat{\Phi}(x^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1}).$
Since $\\{x_{\alpha}\\}\subset L^{p}(M,\Phi),$ we have that
$x_{\alpha}s(\Phi_{e})\in L^{p}(Ms(\Phi_{e}),\Phi_{e}),$ moreover
$x_{\alpha}s(\Phi_{e})\downarrow 0.$ Let us show that
$x=x_{\alpha_{0}}s(\Phi_{e})\in L^{p}(Ms(\Phi_{e}),\tau).$ As above, we
consider $x_{n}=E_{n}(x)x.$ Since
$0\leq\Phi(x_{n}^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1}\uparrow\widehat{\Phi}(x^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1}\leq
e,$
we get $\tau(x^{p}_{n})\leq\nu(e)<\infty.$ Consequently, $x\in
L^{p}(Ms(\Phi_{e}),\tau).$ The inequality $0\leq x_{\alpha}\leq
x_{\alpha_{0}},$ for $\alpha\geq\alpha_{0}$ implies $x_{\alpha}s(\Phi_{e})\in
L^{p}(Ms(\Phi_{e}),\tau)$ (see Theorem 3.5$(iii)$). Since
$x_{\alpha}s(\Phi_{e})\downarrow 0$ and the norm $\|\cdot\|_{p,\tau}$ is order
continuous, we have $\|x_{\alpha}s(\Phi_{e})\|_{p,\tau}\downarrow 0,$ i.e.
$\nu(e\widehat{\Phi}(x_{\alpha}^{p})(\mathbf{1}_{\mathscr{B}}+\Phi(\mathbf{1})+\widehat{\Phi}(x^{p}_{\alpha_{0}}))^{-1})\downarrow
0.$ Hence, $e\widehat{\Phi}(x_{\alpha})^{p}\downarrow 0,$ which contradicts to
the inequality $e\Phi(x_{\alpha}^{p})\geq\varepsilon^{p}e.$ ∎
## 4\. Duality for spaces $\mathbf{L^{p}(M,\Phi)}$
Let us start with the following property of $L^{p}$-spaces $L^{p}(M,\Phi).$
###### Proposition 4.1.
If $x\in L^{p}(M,\Phi),$ $y\in L^{q}(M,\Phi),$ $\frac{1}{p}+\frac{1}{q}=1,$
$p,q>1,$ then $xy,yx\in L^{1}(M,\Phi)$ and
$\widehat{\Phi}(xy)=\widehat{\Phi}(yx).$
###### Proof.
Without loss of generality, we can take $x\geq 0,~{}y\geq 0.$ It follows from
Theorem 3.3 that $xy\in L^{1}(M,\Phi).$ Hence, $yx=y^{*}x^{*}=(xy)^{*}\in
L^{1}(M,\Phi)$ and
$\widehat{\Phi}(yx)=\widehat{\Phi}((xy)^{*})=\overline{\widehat{\Phi}(xy)}.$
Let $x_{n}=xE_{n}(x),$ $y_{n}=yE_{n}(y).$ Then $x_{n},y_{n}\in M_{+}$ and
$\|x-x_{n}\|_{p,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0,~{}\|y-y_{n}\|_{q,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0.$ Using the inequalities
$|\widehat{\Phi}(xy)-\Phi(x_{n}y_{n})|\leq|\widehat{\Phi}(xy)-\widehat{\Phi}(x_{n}y)|+|\widehat{\Phi}(x_{n}y)-\Phi(x_{n}y_{n})|\leq\|x-x_{n}\|_{p,\Phi}\|y\|_{q,\Phi}+\|x_{n}\|_{p,\Phi}\|y-y_{n}\|_{q,\Phi},$
we obtain $\Phi(x_{n}y_{n})\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}\widehat{\Phi}(xy).$ Since
$\Phi(x_{n}y_{n})=\Phi(\sqrt{x_{n}}y_{n}\sqrt{x_{n}})\geq 0$ for all $n,$ we
get $\widehat{\Phi}(xy)\geq 0.$ Therefore
$\widehat{\Phi}(xy)=\overline{\widehat{\Phi}(xy)}=\widehat{\Phi}(yx).$ ∎
Let $L^{p}(M,\Phi)^{*}$ be a BKS of all $S_{h}(\mathscr{B})$-bounded linear
mappings from $L^{p}(M,\Phi)$ into $S(\mathscr{B}),$ i.e. $S_{h}(\mathscr{B})$
is the dual space to the BKS $L^{p}(M,\Phi).$ It is clear that any
$S_{h}(\mathscr{B})$-bounded linear operator $T$ is a continuous mapping from
$(L^{p}(M,\Phi),\|\cdot\|_{p,\Phi})$ into $(S(\mathscr{B}),t(\mathscr{B})),$
i.e., if $x_{\alpha},$ $x\in L^{p}(M,\Phi),$ and
$\|x_{\alpha}-x\|_{p,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0,$ then
$Tx_{\alpha}\stackrel{{\scriptstyle t(\mathscr{B})}}{{\longrightarrow}}Tx.$
###### Proposition 4.2.
(compare with [1], 5.1.9). Let $T\in L^{p}(M,\Phi)^{*},$
$\psi:S(\mathscr{A})\to S(\mathscr{B})$ be a $*$-isomorphism from Theorem
2.5$(ii).$ Then $T(ax)=\psi(a)T(x)$ for all $a\in S(\mathscr{A}),~{}x\in
L^{p}(M,\Phi).$
###### Proof.
By theorem 2.5$(ii),$ for each $z\in P(\mathscr{A}),~{}x\in L^{p}(M,\Phi)$ we
have
$\|zx\|_{p,\Phi}=\widehat{\Phi}(z|x|^{p})^{\frac{1}{p}}=\psi(z)\widehat{\Phi}(|x|^{p})^{\frac{1}{p}}=\psi(z)\|x\|_{p,\Phi}.$
Since $T\in L^{p}(M,\Phi)^{*},$ $|Tx|\leq c\|x\|_{p,\Phi}$ for some $c\in
S_{+}(\mathscr{B})$ and all $x\in L^{p}(M,\Phi).$ Hence
$|T(zx)|\leq\psi(z)c\|x\|_{p,\Phi},$ i.e. the support $s(T(zx))$ is majorized
by the projection $\psi(z).$ Multiplying the equality
$T(x)=T(zx)+T((\mathbf{1}-z)x)$ by $\psi(z),$ we obtain
$\psi(z)T(x)=\psi(z)T(zx)=T(zx).$
If $a=\sum_{i=1}^{n}\lambda_{i}z_{i}$ is a simple element from
$S(\mathscr{A}),$ where $\lambda_{i}\in\mathbb{C},~{}z_{i}\in
P(\mathscr{A}),~{}i=1,\dots,n,$ then
$T(ax)=\sum_{i=1}^{n}\lambda_{i}T(z_{i}x)=(\sum_{i=1}^{n}\lambda_{i}\psi(z_{i}))T(x)=\psi(a)T(x).$
Let $a$ be an arbitrary element from $S(\mathscr{A})$ and let $\\{a_{n}\\}$ be
a sequence of simple elements from $S(\mathscr{A})$ such that
$a_{n}\stackrel{{\scriptstyle t(\mathscr{A})}}{{\longrightarrow}}a.$ Then
$0\leq\psi(|a_{n}-a|)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0,~{}\psi(a_{n})\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}\psi(a),$ and
$\|a_{n}x-ax\|_{p,\Phi}=\widehat{\Phi}(|(a_{n}-a)x|^{p})^{\frac{1}{p}}=\widehat{\Phi}(|a_{n}-a|^{p}|x|^{p})^{\frac{1}{p}}=\psi(|a_{n}-a|)\|x\|_{p,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0.$
Since $T$ is continuous, $\psi(a_{n})T(x)=T(a_{n}x)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}T(ax).$ Due to the convergence
$\psi(a_{n})T(x)\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}\psi(a)T(x),$ the proof is complete. ∎
Now we pass to description of the $S_{h}(\mathscr{B})$-dual space
$L^{p}(M,\Phi)^{*}.$
###### Theorem 4.3.
Let $\Phi$ be an $S(\mathscr{B})$-valued Maharam trace on the von Neumann
algebra $M,$ $p,q>1,$ $\frac{1}{p}+\frac{1}{q}=1.$
$(i)$ If $y\in L^{q}(M,\Phi),$ then the linear mapping
$T_{y}(x)=\widehat{\Phi}(xy),~{}x\in L^{p}(M,\Phi),$ is
$S(\mathscr{B})$-bounded and $\|T_{y}\|=\|y\|_{q,\Phi}.$
$(ii)$ If $T\in L^{p}(M,\Phi)^{*},$ then there exists a unique $y\in
L^{q}(M,\Phi)$ such that $T=T_{y}.$
###### Proof.
$(i)$ By the Hölder inequality (theorem 3.3), $xy\in L^{1}(M,\Phi)$ for all
$x\in L^{p}(M,\Phi)$ and
$|T_{y}(x)|=|\widehat{\Phi}(xy)|\leq\|y\|_{q,\Phi}\|x\|_{p,\Phi}.$ Hence,
$T_{y}$ is $S_{h}(\mathscr{B})$-bounded linear mapping from $L^{p}(M,\Phi)$
into $S(\mathscr{B}).$ Due to Proposition 4.1 and Theorem 3.4 we have
$\|T_{y}\|=\sup\\{|\widehat{\Phi}(yx)|:x\in
L^{p}(M,\Phi),~{}\|x\|_{p,\Phi}\leq\mathbf{1}_{\mathscr{B}}\\}=\|y\|_{q,\Phi}.$
$(ii)$ Since $s(\Phi(\mathbf{1}))=\mathbf{1}_{\mathscr{B}},$ we can define the
element $b=(\Phi(\mathbf{1}))^{-1}\in S_{+}(\mathscr{A}).$ If
$\Phi_{1}(x)=b\Phi(x),~{}x\in M,$ then $L^{p}(M,\Phi_{1})=L^{p}(M,\Phi)$ and
$\|x\|_{p,\Phi_{1}}=b^{\frac{1}{p}}\|x\|_{p,\Phi}$ for all $x\in
L^{p}(M,\Phi).$ Therefore, one can take
$\Phi(\mathbf{1})=\mathbf{1}_{\mathscr{B}}.$
Let $T\in L^{p}(M,\Phi)^{*}.$ We choose $a\in S_{+}(\mathscr{B})$ with
$a\|T\|=s(\|T\|).$ Set $T_{1}(x)=aT(x),~{}x\in L^{p}(M,\Phi).$ It is clear
that $T_{1}\in L^{p}(M,\Phi)^{*}$ and
$\|T_{1}\|=a\|T\|=s(\|T\|)\leq\mathbf{1}_{\mathscr{B}}.$ If we show that there
exists $y_{1}\in L^{q}(M,\Phi)$ such that $T_{1}x=\Phi(xy_{1}),$ then by
virtue of Proposition 4.2,
$Tx=\|T\|T_{1}(xy_{1})=T(x(\psi^{-1}(\|T\|)y_{1}))=T(xy)$ where
$y=\psi^{-1}(\|T\|)y_{1}\in L^{q}(M,\Phi).$ Thus, one can also take that
$\|T\|\leq\mathbf{1}_{\mathscr{B}}.$
At first, we assume that the algebra $\mathscr{B}$ is $\sigma$-finite. Let
$\nu$ be a faithful normal finite numerical trace on $\mathscr{B}.$ Since
$|\Phi(x)|\leq\|x\|_{M}\Phi(\mathbf{1})\leq\|x\|_{M}\mathbf{1}_{\mathscr{B}},~{}x\in
M,$ we get $\Phi(x)\in L^{1}(\mathscr{B},\nu).$ Consider on $M$ the faithful
normal finite trace $\tau(x)=\nu(\Phi(x)),~{}x\in M.$ Using the same trick as
in the proof of Proposition 3.7, we can show that $L^{p}(M,\tau)\subset
L^{p}(M,\Phi)$ and
$\tau(|x|^{p})=\|x\|^{p}_{p,\tau}=\nu(\widehat{\Phi}(|x|^{p}))$ for all $x\in
L^{p}(M,\tau).$ Since
$|T(x)|\leq\|x\|_{p,\Phi}=(\widehat{\Phi}(|x|^{p}))^{\frac{1}{p}},$ we have
$T(x)\in L^{1}(\mathscr{B},\nu)$ for all $x\in L^{p}(M,\tau).$
We define on $L^{p}(M,\tau)$ the linear $\mathbb{C}$-valued functional
$f(x)=\nu(Tx),$ $x\in L^{p}(M,\tau).$ Since
$|f(x)|\leq\nu(|T(x)|)\leq\nu(\widehat{\Phi}(|x|^{p})^{\frac{1}{p}}\mathbf{1}_{\mathscr{B}})\leq(\nu(\widehat{\Phi}(|x|^{p})))^{\frac{1}{p}}(\nu(\mathbf{1}_{\mathscr{B}}))^{\frac{1}{q}}=(\nu(\mathbf{1}_{\mathscr{B}}))^{\frac{1}{q}}\|x\|_{p,\tau}$
for all $x\in L^{p}(M,\tau),$ we have that $f$ is a bounded linear functional
on $(L^{p}(M,\tau),\|\cdot\|_{p,\tau}).$ Hence there exists an operator $y\in
L^{q}(M,\tau)\subset L^{q}(M,\Phi)$ such that $f(x)=\tau(xy)$ for all $x\in
L^{p}(M,\tau)$ [11]. We claim that $\tau(xy)=\nu(\widehat{\Phi}(xy))$ for all
$x\in L^{p}(M,\tau).$ Let us remind that
$\tau(|z|^{p})=\nu(\widehat{\Phi}(|z|^{p}))$ for all $z\in L^{p}(M,\tau).$ If
$z\in L^{1}_{+}(M,\tau),$ then $z^{\frac{1}{p}}\in L_{+}^{p}(M,\tau),$ and
therefore $\tau(z)=\nu(\widehat{\Phi}(z).$ Hence,
$\tau(z)=\nu(\widehat{\Phi}(z))$ for all $z\in L^{1}(M,\tau),$ in particular,
$\tau(xy)=\nu(\widehat{\Phi}(xy))$ where $x\in L^{p}(M,\tau).$ Thus,
$\nu(T(x))=f(x)=\tau(xy)=\nu(\widehat{\Phi}(xy))$ for all $x\in
L^{p}(M,\tau).$
Let $T(x)-\widehat{\Phi}(xy)=v|T(x)-\widehat{\Phi}(xy)|$ be the polar
decomposition of the element $(T(x)-\widehat{\Phi}(xy))\in S(\mathscr{B})$ and
take $a=\psi^{-1}(v^{*}).$ Since
$0=\nu(T(ax)-\widehat{\Phi}(axy))=\nu(v^{*}(T(x)-\widehat{\Phi}(xy)))=\nu(|T(x)-\widehat{\Phi}(xy)|),$
we have $T(x)=\widehat{\Phi}(xy)$ for all $x\in L^{p}(M,\tau).$
Let $x\in L_{+}^{p}(M,\Phi),x_{n}=xE_{n}(x).$ Then
$\|x_{n}-x\|_{p,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0$ and therefore
$T(x_{n})\stackrel{{\scriptstyle t(\mathscr{B})}}{{\longrightarrow}}T(x)$ and
$|\widehat{\Phi}(x_{n}y)-\widehat{\Phi}(xy)|\leq\|x_{n}-x\|_{p,\Phi}\|y\|_{q,\Phi}\stackrel{{\scriptstyle
t(\mathscr{B})}}{{\longrightarrow}}0.$ Since
$T(x_{n})=\widehat{\Phi}(x_{n}y),$ $T(x)=\widehat{\Phi}(xy),$ i.e. $T=T_{y}.$
If $z$ is another element from $L^{q}(M,\Phi)$ with $T(x)=\widehat{\Phi}(xz),$
$x\in L^{p}(M,\Phi),$ then $\widehat{\Phi}(x(y-z))=0$ for all $x\in
L^{p}(M,\Phi).$ Taking $x=u^{*}$ where $u$ is the unitary operator from the
polar decomposition $y-z=u|y-z|,$ we obtain $\widehat{\Phi}(|y-z|)=0,$ i.e.
$y=z.$
Now let $\mathscr{B}$ be a general (not necessarily a $\sigma$-finite) von
Neumann algebra. Let $\nu$ be a faithful normal semifinite numerical trace on
$\mathscr{B},$ and let $\\{e_{i}\\}_{i\in I}$ be a family of nonzero mutually
orthogonal projections in $\mathscr{B}$ with $\sup\limits_{i\in
I}e_{i}=\mathbf{1}_{\mathscr{B}}$ and $\nu(e_{i})<\infty$ for all $i\in I.$ It
is clear that $\mathscr{B}e_{i}$ is a $\sigma$-finite algebra and
$\Phi_{e_{i}}(x)=e_{i}\Phi(x)$ is $S(\mathscr{B}e_{i})$-valued Maharam trace
on $Ms(\Phi_{e_{i}}).$ Since $T\in L^{p}(M,\Phi)^{*},$ $T_{i}(x)=e_{i}T(x)$ is
$S_{h}(\mathscr{B}e_{i})$-bounded linear mapping onto
$L^{p}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}}).$ By virtue of what we proved above,
there exists the unique $y_{i}\in L^{q}(Ms(\Phi_{e_{i}}),\Phi_{e_{i}}),$ such
that
$e_{i}T(xs(\Phi_{e_{i}}))=\widehat{\Phi_{e_{i}}}(xs(\Phi_{e_{i}})y_{i})=e_{i}\widehat{\Phi}(xs(\Phi_{e_{i}})y_{i})$
for all $x\in L^{p}(M,\Phi),~{}i\in I.$ Moreover,
$\|y_{i}\|_{q,\Phi}=\|T_{i}\|=\|T\|e_{i}.$ Since $\sup\limits_{i\in
I}s(\Phi_{e_{i}})=\mathbf{1},$ $\\{s(\Phi_{e_{i}})\\}_{i\in I}\subset P(Z(M)$
and $s(\Phi_{e_{i}})s(\Phi_{e_{j}})=0$ as $i\neq j,$ there exists a unique
$y\in S(M)$ such that $ys(\Phi_{e_{i}})=y_{i}.$ We have
$e_{i}\widehat{\Phi}(|y|^{q})=\widehat{\Phi}(|y_{i}|^{q})=\|T\|^{q}e_{i}$ for
all $i\in I.$ Hence, $y\in L^{q}(M,\Phi)$ $\|y\|_{q,\Phi}=\|T\|$ (see
Proposition 3.1). In addition
$e_{i}\widehat{\Phi}(xy)=\widehat{\Phi_{e_{i}}}(xs(\Phi_{e_{i}})y_{i})=e_{i}T(xs(\Phi_{e_{i}}))=e_{i}T(x),$
for all $i\in I,$ i.e. $T_{y}(x)=\widehat{\Phi}(xy)=T(x),~{}x\in
L^{p}(M,\Phi).$ ∎
###### Corollary 4.4.
The BKS $L^{p}(M,\Phi)^{*}$ is isometric to the space
$(L^{q}(M,\Phi),\|\cdot\|_{q,\Phi}).$
## References
* [1] Kusraev A.G., Dominanted Operators, Mathematics and its Applications, 519, Kluwer Academic Publishers, Dordrecht, 2000\. 446 p.
* [2] Ganiev I.G., Chilin V.I. Measurable bundles of non-commutative $L^{p}$-spaces associated with center-valued trace // Mat. Trudy, 4(2001) No 2. p. 27–41. (Russian).
* [3] Chilin V.I., Katz A.A. On abstract characterization of non-commutative $L^{p}$-spaces associated with center-valued trace // MFAT 2005. v. 11., No 4. p. 346–355.
* [4] Chilin V., Zakirov B. Maharam traces on von Neumann algebras // arXiv.math.OA: 0905.2857v1.
* [5] Stratila S., Zsido L. Lectures on von Neumann algebras, England Abacus Press, 1975. 477 p.
* [6] Takesaki M. Theory of operator algebras I. New York: Springer, 1979. 415 p.
* [7] Segal I.E. A non-commutative extension of abstract integration // Ann. Math. 1953. No 57. p. 401–457.
* [8] Muratov M.A., Chilin V.I. Algebras of measurable and locally measurable operators. Kyiv, Pratsi In-ty matematiki NAN Ukraini. 2007. V. 69. 390 p. (Russian).
* [9] Yeadon F.J. Convergence of measurable operators // Proc. Camb. Phil. Soc. 1973. v. 74. p. 257–268.
* [10] Zakirov B.S. Homomorphisms of algebras of locally measurable operators //VMJ ( accepted, 2009) (Russian).
* [11] Yeadon F.J. Non-commutative $L^{p}$-spaces // Math. Proc. Camb. Phil. Soc. – 1975. – V. 77. – P. 91–102.
* [12] Dixmier J. Les $C^{*}$-algebres et leurs representations – Paris: Gauthier - Villars Editeur, 1969.
* [13] Dixmier J. Les algebras d’operateurs dans l’espace Hilbertien (Algebres de von Neunann). Paris: Gauthier-Villars, 1969. –367 p.
* [14] Zakirov B.S., Chilin V.I. Non-commutative integration for traces with values in complex Kantorovich-Pinsker spaces // Izv. VUZov. Mathematika ( accepted, 2009) (Russian).
* [15] Tikhonov O.Y. Continuity of operator functions on a von Neumann algebra with respect to topology of convergence in measure. // Izv. VUZov. Mathematika. 1987. No 1. p. 77-79. (Russian).
* [16] Vladimirov D.A., Boolean Algebras, Nauka, Moscow, 1969. 319 p. (Russian).
* [17] Akemann C.A., Andersen T., Pedersen G.K. Triangle inequalities in operator algebras // Linear and Multilinear Algebra, 1982, v. 11,2 p. 167-178.
* [18] Kusraev A.G., Vector Duality and its Applications, Nauka, Novosibirsk, 1985. 256 p. (Russian).
|
arxiv-papers
| 2010-07-12T11:58:51 |
2024-09-04T02:49:11.559046
|
{
"license": "Public Domain",
"authors": "Vladimir Chilin, Botir Zakirov",
"submitter": "Aleksej Ber",
"url": "https://arxiv.org/abs/1007.1878"
}
|
1007.2008
|
# Vicious Lévy flights
Igor Goncharenko and Ajay Gopinathan School of Natural Sciences, University
of California, Merced, California, 95343, USA
###### Abstract
We study the statistics of encounters of Lévy flights by introducing the
concept of vicious Lévy flights - distinct groups of walkers performing
independent Lévy flights with the process terminating upon the first encounter
between walkers of different groups. We show that the probability that the
process survives up to time $t$ decays as $t^{-\alpha}$ at late times. We
compute $\alpha$ up to the second order in $\varepsilon$-expansion, where
$\varepsilon=\sigma-d$, $\sigma$ is the Lévy exponent and $d$ is the spatial
dimension. For $d=\sigma$, we find the exponent of the logarithmic decay
exactly. Theoretical values of the exponents are confirmed by numerical
simulations.
###### pacs:
64.60.ae, 64.60.F-, 05.40.Jc, 64.60.Ht
Diffusive processes with long range jumps play an important role in many
physical, chemical and biological phenomena. A Lévy flight is an example of
such a process where the probability distribution of the length of an
individual step, $r$, is governed by the power-law $r^{-d-\sigma}$, where $d$
is the dimension of the space and $\sigma$ is the Lévy exponent. Smaller
values of $\sigma$ therefore produce longer range jumps while for $\sigma\geq
2$, the mean jump length is finite and simple diffusive behavior is recovered.
Lévy flights have been used to describe a wide range of processes including
epidemic spreading, transcription factor proteins binding to DNA, kinetic
Ising models with long-range interactions, foraging animals and light
propagation in disordered optical materials Klaft2 ; DNA ; MenyOdor ; Berg ;
VishNat ; Barth . While individual Lévy flights have been studied in great
detail, the same is not true if we consider several distinct groups of Lévy
flights. One could, for example, be interested in the statistics of encounters
between members of different groups. This question is relevant for processes
where the outcome depends on the occurrence of such encounters. Examples
include sharks and other marine animals searching for prey Sims , chemical
reactions in turbulent environments Klaft1 , electron-hole recombination in
disordered media Shles and even male spider-monkeys encountering their mates
or other aggressive males in the forest Ramos .
In this Letter we compute the survival probability, i.e. the probability that
no two members of different groups of Lévy flights have met up to time $t$.
For the case of simple diffusion with exactly one particle in each group, this
corresponds to the classic problem of Gaussian vicious walks Fisher , i.e.
walks that are prohibited from being on the same site at the same time, but
remain independent otherwise. Here we generalize this concept to groups of
Lévy flights under the same constraints. We term them vicious Lévy flights
(VLF). We consider $p$ sets of particles with $n_{i}$ particles in each set,
$i=1\dots p$, that are driven by Lévy noise on the $d$ dimensional regular
lattice. A pairwise interset short-range (delta-function) interaction is
introduced to guarantee that trajectories which continue beyond an
intersection are discarded, i.e. have zero statistical weight. This terminates
the process at the first encounter between members of different groups.
Particles belonging to the same set do not interact. We note that Lévy flights
are allowed to jump over each other, unlike ordinary random walks which can
only jump to neighboring sites and can not intersect with the vicious
constraint. In $d=1$ this means that the ordering is preserved for vicious
walks but not for VLF. For simplicity we assume that Lévy exponents for all
flights are the same. Generalization to the case of different Lévy exponents
will be done elsewhere. At time $t=0$ all particles start in the vicinity of
the origin. We are interested in the survival probability of this system at
late times.
We start with a field theoretic formulation of the problem. Methods to
formulate field theories for such stochastic systems are well established Doi
; Peliti . Specifically for Gaussian vicious walks, such a formulation exists
Cardy and the form of the action is known. We can adapt the action to our
case by replacing the Laplacian $\nabla^{2}$ with the operator
$\nabla^{\sigma}$ that generates long-range jumps. This gives
$\mathcal{S}(\phi_{i},\phi^{\dagger}_{i})=\int
dtd^{d}x\sum_{i=1}^{p}[\phi^{\dagger}_{i}\partial_{t}\phi_{i}+\phi^{\dagger}_{i}\nabla^{\sigma}\phi_{i}]+\sum\limits_{1\leq
i<j\leq
p}\lambda_{ij}\phi^{\dagger}_{i}(t,x)\phi_{i}(t,x)\phi^{\dagger}_{j}(t,x)\phi_{j}(t,x).$
(1)
where $\phi_{i}(x,t)$ are $p$ complex order parameters corresponding to $p$
different sets of equivalent Lévy flights and $\lambda_{ij}$ are coupling
constants corresponding to interset interactions. The non-intersection
property of VLF arises from the choice $\lambda_{ij}\to\infty$ but we will
show that to leading order the survival probability does not depend on the
particular value of these coupling constants. This action is also similar to
the action for the reaction-diffusion problem with long-range interactions
Jans ; Hinr . Power counting shows that the upper critical dimension for the
above field theory is $d_{c}=\sigma$ for $\sigma<2$ and $d_{c}=2$ for
$\sigma\geq 2$. VLF exhibit different phases (see inset Figure 1) depending on
the values of $d$ and $\sigma$. In the mean field phase for $d>d_{c}$ (Region
I), the survival probability of VLF is non-zero at infinite time because the
walks become non-recurrent and particles can avoid each other for all time.
For $\sigma\geq 2$ (Region II) VLF reproduce Gaussian vicious walks. For
$d<\sigma<2$ (Region III) we expect fluctuations to play an important role. In
this phase, we will obtain the critical behavior of the survival probability
using $\varepsilon$-expansion ($\varepsilon=\sigma-d$) around mean field
theory in two-loop approximation.
Figure 1: a. $\ln(S)$ vs $\ln(t)$ for two identical VLF in $d=1$. $\sigma$
values from top tp bottom are $1.1,1.3,1.5,1.7,2.5$ respectively. Symbols
represent simulation data and solid lines correspond to best fit lines to the
late time data. Inset: Domains of VLF exponents in the $\sigma-d$ plane.
We now turn to the renormalization group analysis. The propagator given by (1)
is $\Gamma^{(1,1)}_{j}(s,k)=(s+k^{\sigma})^{-1}$. The particular form of the
vertex in (1) leads to the fact that there are no diagrams which dress the
propagator. This implies that the bare propagator is the exact propagator for
the theory. The proper vertex is defined by factoring out external legs from
the ordinary 4-point Green’s function of (1)
$G^{(2,2)}_{ij}(s_{l},k_{l};\lambda)$. Here $\lambda=\\{\lambda_{ij}\\}$ is
the collection of coupling constants and $(s_{l},k_{l})$ for $l=1\dots 4$ are
the energy (Laplace image of time) and momenta respectively Bronzan . This
yields
$\Gamma^{(2,2)}_{ij}(s_{l},k_{l};\lambda)=\frac{G^{(2,2)}_{ij}(s_{l},k_{l};\lambda)}{\prod_{m=1}^{4}\Gamma^{(1,1)}(s_{m},k_{m})}.$
(2)
The renormalized coupling, $\lambda_{Rij}$, is the value of the proper vertex
at $(s_{l}=\mu,k_{l}=0),$ for all $l$, where $\mu$ is a renormalization group
flow scaling parameter. It is possible to sum all diagrams in the series with
the result
$\lambda_{Rij}=\lambda_{ij}(1+\lambda_{ij}I_{1})^{-1},$ (3)
where $I_{1}=(2\pi)^{-d}\int d^{d}q(2\mu+2q^{\sigma})^{-1}$. The value of the
integral $I_{1}$ is given by
$I_{1}=A\mu^{-\varepsilon/\sigma}\varepsilon^{-1}$, where
$A=\frac{\varepsilon\Gamma(d/\sigma)\Gamma(\varepsilon/\sigma)}{2^{d}\pi^{d/2}\sigma\Gamma(d/2)}=\frac{2^{-\sigma}\pi^{-\sigma/2}}{\Gamma(\sigma/2)}+O(\varepsilon),$
(4)
is the geometric factor at the leading order in $\varepsilon=\sigma-d$. By
introducing the redefined coupling constant
$g_{Rij}=\lambda_{Rij}\mu^{-\varepsilon/\sigma}$, we obtain the following
renormalization group flow equations (see Appendix A for details):
$\mu\frac{\partial
g_{Rij}}{\partial\mu}=(-\varepsilon+Ag_{Rij})g_{Rij}/\sigma.$ (5)
The fixed point of this flow is $g_{*}=\varepsilon A^{-1}$. We note that this
value of the fixed point is exact to all orders since all diagrams were taking
into account in (3). The stability of the fixed point follows from the fact
that $-\partial\beta_{ij}/\partial
g_{Rij}|_{g_{Rij}=\varepsilon/A}=-\varepsilon/\sigma<0$ in the VLF region,
where $\beta_{ij}=\mu\partial g_{Rij}/\partial\mu$ is renormalization group
beta function.
Figure 2: a. 1-loop diagram corresponding to the integral $I_{1}$ b., c.
2-loop intgrals corresponding to $I_{1}^{2}$ and $I_{2}$ respectively.
We now consider the survival probability which is defined as the correlation
function Cardy
$S(t;\lambda)=\int\prod_{i=1}^{p}\prod_{\alpha_{i}=1}^{n_{i}}d^{d}x_{i,\alpha_{i}}\langle\phi_{i}(t,x_{i,\alpha_{i}})(\phi^{\dagger}_{i}(0,0))^{n_{i}}\rangle,$
(6)
with the measure $\int{\cal D}\phi^{\dagger}{\cal D}\phi\exp[-\mathcal{S}]$.
The Feynman diagram of (6) at zero order is a vertex with $2N$ external legs.
Similar to the case of the $(2,2)$-vertex it is convenient to work with the
truncated correlation function
$\Gamma(s_{l},k_{l};\lambda)=S(s_{l},k_{l};\lambda)/\prod^{2N}_{m=1}\Gamma^{(1,1)}(s_{m},k_{m})$.
The finite renormalized truncated correlation function
$\Gamma_{R}(s_{l},k_{l};\lambda_{R},\mu)$, where
$\lambda_{R}=\\{\lambda_{Rij}\\}$ is a collection of renormalized coupling
constants, is related to the bare truncated correlation function by
$\Gamma_{R}(s_{l},k_{l};\lambda_{R},\mu)=Z(\lambda,\mu)\Gamma(s_{l},k_{l};\lambda)$,
where $Z(\lambda,\mu)$ is the scaling function. From this one obtains the
renormalization group equation for $\Gamma_{R}(s_{l},k_{l};\lambda_{R},\mu)$
using the chain rule
$(\mu\frac{\partial}{\partial\mu}+\beta_{ij}\frac{\partial}{\partial
g_{Rij}}+\gamma)\Gamma_{R}(s_{l},k_{l};\lambda_{R},\mu)=0,$ (7)
where $\gamma=\mu\partial\ln Z/\partial\mu.$ At the fixed point (7) reduces to
$(\partial/\partial\ln(\mu)+\gamma_{*})\Gamma_{R}(s,\mu)=0$ whose solution is
$\Gamma_{R}\sim\exp(\int_{0}^{\mu}\gamma_{*}d(\ln(\mu^{\prime}))),$ (8)
where $\gamma_{*}=\mu\partial\ln Z/\partial\mu|_{g_{Rij}=g_{*}}$. Since
$\gamma_{*}$ is constant at the fixed point we have
$\Gamma_{R}\sim\mu^{-\gamma_{*}}$. The fact that the dimensions of field and
the action are $[\phi^{\dagger}]=[\phi]=k^{d/2}$ and $[\mathcal{S}]=1$ implies
that $[S]=1$. Thus it follows that the survival probability can only be a
function of the dimensionless product $\mu t$. From this one infers that the
asymptotic behavior of the survival probability is $S(t)\sim t^{-\gamma_{*}}$
which gives $\alpha=\gamma_{*}$. In order to find $Z$ one uses a normalization
condition on $\Gamma_{R}$ that fixes the value of $Z$. This can be chosen as
$\Gamma_{R}(s_{l},k_{l};\lambda_{R},\mu)=1$ when $s_{l}=\mu$ and $k_{l}=0$ for
all $l$. This implies that $Z=\Gamma(\mu,0;\lambda)^{-1}$.
$\Gamma(\mu,0;\lambda)$ can be expressed as a series, up to two-loop order, of
the integrals corresponding to the diagrams shown in Fig.2 with appropriate
combinatorial factors originating from the number of distinct ways in which
propagators can be assigned to the same diagram. $I_{1}$ was evaluated before.
The integral corresponding to the diagram 2c is
$I_{2}=\int\frac{d^{d}kd^{d}q}{(2\mu+2k^{\sigma})(3\mu+q^{\sigma}+k^{\sigma}+|k+q|^{\sigma})}$
(9)
This can be evaluated using Mellin-Barnes representation Smir which replaces
the sum in the denominator of $|k+q|^{\sigma}$ and $q^{\sigma}$ by the product
of these terms raised to some power. The result for $I_{2}$ up to the leading
order reads
$I_{2}=\frac{2^{-2\sigma}\pi^{-\sigma}}{\Gamma(\sigma/2)^{2}}\mu^{-2\varepsilon/\sigma}\left(\frac{1}{2\varepsilon^{2}}+\frac{2(-\log(3/4)/4+C)}{\sigma\varepsilon}\right),$
(10)
where $C=[\psi^{(0)}(\sigma/2)+\log(4\pi)]/2$ and $\psi^{(0)}(x)$ is standard
digamma function. The details of the calculations are summarized in Appendix
B. Knowing $I_{1},I_{2}$ and the appropriate combinatorial factors allows us
to evaluate $\Gamma(\mu,0;\lambda)$ and therefore $Z(\mu,\lambda)$ as a
series. Differentiating $\ln Z$ with respect to $\mu$ and substituting
$\lambda$ with $\lambda_{R}$ (inverting (3)) and then taking the value at the
fixed point gives us the survival probability exponent $\alpha=\gamma_{*}$,
with the final result (see Appendix C):
$\alpha=\sum_{1\leq i<j\leq
p}n_{i}n_{j}\varepsilon/\sigma+\ln(3/4)Q\varepsilon^{2}/\sigma^{2},$ (11)
where $Q=6\sum_{1\leq i<j<k\leq p}n_{i}n_{j}n_{k}+\sum_{1\leq i<j\leq
p}n_{i}n_{j}(n_{i}+n_{j}-2)$. At the critical dimension $d=d_{c}=\sigma$ we
see that the fixed point coincides with the Gaussian point. The interaction
becomes marginal in the renormalization group sense. Equation (5) then yields
the flow equations for the running coupling constant
$xd\bar{g}_{ij}(x)/dx=A\bar{g}^{2}_{ij}(x)/\sigma,$ (12)
with initial condition $\bar{g}_{ij}(1)=g_{Rij}.$ Solving this and
substituting the result into (8) we get $S(t)=(\ln t)^{-\alpha_{l}},$ where
$\alpha_{l}=\sum_{1\leq i<j\leq p}n_{i}n_{j}.$ (13)
Figure 3: $\alpha$ as a function of $\sigma$ for $d=1$. Symbols with error
bars represent simulation data corresponding from top to bottom to $N=2,3,4$
VLF respectively. Lines correspond to 1-loop approximation from formula (11)
for $1<\sigma<2$. For $\sigma<1$, $\sigma\geq 2$ lines represent the mean
field and Gaussian exponents respectively. Inset: Same simulation data
compared to 2-loop approximation from (11) Figure 4: $\alpha$ vs $\sigma$ for
predator and prey problem in $d=1$. Symbols represent simulation data for 4
predators and 1 prey (circles) and 3 predators and 2 prey (squares). Lines are
2-loop approximation from formula (11).
Now we describe the details of the numerical simulation that we used to
confirm our results in $d=1$. At $t=0$ we start with $N=\sum_{i=1}^{p}n_{i}$
particles belonging to $p$ distinct sets placed equidistantly on the lattice.
At each time step we generate $N$ random variables, $x_{j}$, drawn from the
uniform distribution on the interval $(0,1)$. Each particle jumps a distance
$l_{j}=x_{j}^{-1/\sigma}$ with equal probability to the left or to the right.
This procedure generates an independent Lévy flight trajectory for each
particle. The process stops whenever particles from different sets land on the
same site. We perform $\sim 10^{5}$ iterations for each set of parameter
values. The survival probability $S(t)$ is defined by the number of processes
that survived beyond time $t$ divided by the total number of iterations.
Figure 1 shows the plot of the survival probability as a function of time for
$N=2$ for different values of $\sigma$. It is clear that at late times $\ln
S(t)$ is linear in $\ln t$ verifying our predicted power-law decay $S(t)\sim
t^{-\alpha}$. The critical exponent $\alpha$ is evaluated from the slope of
the best fit line of the late time data.
We first consider systems with exactly one particle in each set. Figure 3
shows the value of exponent $\alpha$ for various values of $\sigma$ with the
total number of particles $N=2,3$ and $4$ in $d=1$. Values of $\sigma\geq 2$
will reproduce Gaussian vicious walks and we therefore expect our exponents to
approach the exact Fisher exponents Fisher as seen in Figure 3. For two VLF
higher loop corrections are absent (see (11)) and the one-loop result is an
exact result in agreement with the simulation. It is interesting to note that
the survival probability for the $N=2$ case is equivalent to the first return
probability of a single Lévy flight to the origin after time $t$ which scales
as $t^{-1+d/\sigma}$ Klaft3 and matches our results. For $\sigma<1$, or
$d>d_{c}$, we expect mean field behavior where survival probability at late
times approaches a non-zero value implying that there is a finite probability
that Lévy flights with $\sigma<1$ will never find each other. The fact that
the survival probability tends to a constant at late times is reflected in the
small values of $\alpha$ for $\sigma=0.9$. For $1<\sigma<2$, the mean field
behavior is incorrect and we expect the fluctuations to shift the decay
exponent to some non-trivial value. For $\sigma$ close to one, the 1-loop
result is in good agreement with the simulation. For larger values of
$\sigma$, the 2-loop corrections perform better (see Fig.3 inset). It is to be
noted that the discrepancy between theory and simulation becomes large for
higher values of $N$ because the combinatorial factors in (11) become large
and we therefore need to keep higher order terms in $\varepsilon$ for the same
degree of accuracy. It is interesting that the 1-loop approximation works
reasonably well over the entire range of $1<\sigma\leq 2$ simply because the
1-loop term in (11) happens to give the exact Fisher result if we set
$\sigma=2$. We notice that in all cases the value of the survival probability
exponent $\alpha$ increases with $\sigma$ starting from $\alpha\sim 0$ at
$\sigma<1$ and rising to the value of the Fisher exponents for the equivalent
Gaussian vicious walks. This is in contrast to diffusion-annihilation
reactions with long-range jumps where the density of reactants decays faster
for smaller values of $\sigma$ Hinr .
We now consider a system that consists of 2 sets of identical VLF with
different numbers of independent particles in each set. We shall call one set
predators and the other set prey. Figure 4 compares the values of the 2-loop
exponents to the simulation results for various values of $\sigma$ for two
different cases: 4 predators - 1 prey and 3 predators - 2 prey. Similar to the
previous case we have mean field and Gaussian behaviors for $\sigma<1$ and
$\sigma\geq 2$ respectively. The Gaussian case is also known as the lion-lamb
problem and has been studied before Redner . Unlike the lion-lamb problem,
however, our results do not depend on the initial ordering of predators and
prey because ordering is not preserved for VLF. For a given $\sigma$ and total
number of predators plus prey, the number of potentially lethal encounters is
maximized when the difference between the number of predators and prey is the
smallest implying that the survival probability will decay faster as seen in
Fig.4.
Our results suggest that it is interesting to solve the problem in the general
case of particles with different diffusion constants and Lévy exponents. The
predictive power of the $\varepsilon$-expansion for VLF, that we have
demonstrated, should be useful in many applications of practical importance.
Examples include the optimization of the predator-prey search Vish1 or
trapping probabilities Hub . Generalization to the case of intelligent
predators, i.e. interacting with a prey by means of the long-range potential,
may lead to different critical behavior Gonch ; SatMaj1 ; SatMaj2 . Simple
diffusion processes in power-law small world networks are effectively Lévy
flights Kozma with the exponent $\sigma$ controlling the distribution of
long-range links. Our work could be used to understand what network structure,
or what $\sigma$, would optimize the search and how much more efficient
several independent searchers will be.
The authors would like to acknowledge UC Merced start-up funds and a James S.
McDonnell Foundation Award for Studying Complex Systems.
## I Appendix A
Here we derive the 1-loop integral
$I_{1}=I_{1}(\mu)=(2\pi)^{-d}\int d^{d}k(2\mu+2k^{\sigma})^{-1}.$ (14)
We will use dimensional regularization. First we notice that there is no angle
dependence under the integral thus one can integrate out $d-1$ angle variable
and use alpha representation:
$X^{-\lambda}=\Gamma(\lambda)^{-1}\int_{0}^{+\infty}d\alpha\alpha^{\lambda-1}\exp(-\alpha
X)$ (15)
to handle 1d momenta integral:
$I_{1}=\frac{S_{d}}{2}\int\limits_{0}^{+\infty}\frac{k^{d-1}dk}{\mu+k^{\sigma}}=\frac{S_{d}}{2}\int\limits_{0}^{+\infty}\int\limits_{0}^{+\infty}d\alpha
dkk^{d-1}\exp(-\alpha\mu-\alpha
k^{\sigma})=\frac{S_{d}}{2\sigma}\Gamma(d/\sigma)\int\limits_{0}^{+\infty}d\alpha\alpha^{-d/\sigma}\exp(-\alpha\mu),$
(16)
where $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the area of the d-dimensional unit
sphere. After taking the integral over $\alpha$ one has
$I_{1}=\frac{S_{d}}{2\sigma}\Gamma(d/\sigma)\Gamma(\varepsilon/\sigma)\mu^{-\varepsilon/\sigma}=A\mu^{-\varepsilon/\sigma}\varepsilon^{-1}+O(\varepsilon^{0}),$
(17)
where $A$ has been defined by the formula (4).
Now we show details of deriving renormalization group flow equations (5). We
start with equation (3) and express $\lambda_{ij}$ in terms of
$\lambda_{Rij}$. The result reads
$\lambda_{ij}=\frac{\lambda_{Rij}}{1-\lambda_{Rij}I_{1}}.$ (18)
Multiplying left and right hand side of the last equation on
$\mu^{-\varepsilon/\sigma}$ and redefining the coupling constant
$g_{Rij}=\mu^{-\varepsilon/\sigma}\lambda_{Rij}$ we infer that
$\mu^{-\varepsilon/\sigma}\lambda_{ij}=g_{Rij}/(1-g_{Rij}A\varepsilon^{-1}).$
(19)
Differentiating left and right hand side of (19) with
$\mu\frac{\partial}{\partial\mu}$ we obtain
$(-\varepsilon/\sigma)\mu^{-\varepsilon/\sigma}\lambda_{ij}=-\beta_{ij}g^{-2}_{Rij}/(g^{-1}_{Rij}-A\varepsilon^{-1})^{2}$
(20)
Now we substitute (19) into (20) and find beta function up to second order in
small $\varepsilon$ and $g_{Rij}$ expansion:
$\beta_{ij}=(-\varepsilon g_{Rij}+Ag^{2}_{Rij})/\sigma+O(\varepsilon g^{2})$
(21)
## II Appendix B
Here we derive the 2-loop integral
$I_{2}=I_{2}(\mu)=(2\pi)^{-2d}\int
d^{d}kd^{d}q[(2\mu+2k^{\sigma})(3\mu+k^{\sigma}+q^{\sigma}+|k+q|^{\sigma})]^{-1}.$
(22)
The term $|k+q|^{\sigma}$ leads to the appearance of angle integration.
Nevertheless it is possible to avoid angle integration. The key idea is to use
Mellin-Barnes representation Smir :
$\frac{1}{(X+Y)^{\lambda}}=\int_{-i\infty}^{+i\infty}\frac{dz}{2\pi
i}\frac{Y^{z}}{X^{\lambda+z}}\frac{\Gamma(\lambda+z)\Gamma(-z)}{\Gamma(\lambda)}$
(23)
Applying MB formula twice we split the sum of two terms containing $q$
integration into the factor of these terms raised to some power:
$\displaystyle I_{2}$
$\displaystyle=\int\frac{d^{d}kd^{d}q}{2(2\pi)^{2d}}\int\limits_{-i\infty}^{+i\infty}\frac{dz}{2\pi
i}\frac{\Gamma(1+z_{1})\Gamma(-z_{1})}{\mu+k^{\sigma}}\frac{(3\mu+k^{\sigma}+q^{\sigma})^{z_{1}}}{|k+q|^{\sigma(1+z_{1})}}$
$\displaystyle=\int\frac{d^{d}kd^{d}q}{2(2\pi)^{2d}}\int\limits_{-i\infty}^{+i\infty}\frac{dz_{1}dz_{2}}{(2\pi
i)^{2}}\frac{\Gamma(1+z_{1})\Gamma(-z_{1}+z_{2})\Gamma(-z_{2})}{\mu+k^{\sigma}}\frac{(3\mu+k^{\sigma})^{z_{2}}}{|k+q|^{\sigma(1+z_{1})}q^{\sigma(-z_{1}+z_{2})}}\,,$
(24)
Now integral over $q$ becomes standard:
$I_{q}=\int\frac{d^{d}q}{(q^{2})^{a_{1}}((k+q)^{2})^{a_{2}}}=\pi^{d/2}k^{d-2(a_{1}+a_{2})}\frac{\Gamma(a_{1}+a_{2}-d/2)\Gamma(d/2-a_{1})\Gamma(d/2-a_{2})}{\Gamma(a_{1})\Gamma(a_{2})\Gamma(d-a_{1}-a_{2})},$
(25)
where $a_{1}=\sigma(-z_{1}+z_{2})/2$ and $a_{2}=\sigma(1+z_{1})/2$. Thus we
will be left with integral over $k$ of the form:
$I_{k}=\int\frac{d^{d}kk^{-\varepsilon-\sigma
z_{2}}(3\mu+k^{\sigma})}{2\mu+2k^{\sigma}}$ (26)
The function under the integral does not depend on the angle and therefore
$I_{k}$ can be cast into one dimensional integral over momenta:
$I_{k}=\frac{S_{d}}{2\sigma}\int\limits_{0}^{+\infty}dkk^{-2\varepsilon/\sigma-
z_{2}}\frac{(3\mu+k)^{z_{2}}}{\mu+k}$ (27)
We will compute this integral using alpha representation.
$I_{k}=\frac{S_{d}}{2\sigma}\int\limits_{0}^{+\infty}dkd\alpha_{1}d\alpha_{2}\frac{\alpha_{1}^{-z_{2}-1}k^{-2\varepsilon/\sigma-
z_{2}}}{\Gamma(-z_{2})}\exp(-3\mu\alpha_{1}-\alpha_{1}k-\alpha_{2}\mu-\alpha_{2}k)$
(28)
After momenta integration we obtain
$I_{k}=\frac{S_{d}\Gamma(1-2\varepsilon/\sigma-
z_{2})}{2\sigma\Gamma(-z_{2})}\int\limits_{0}^{+\infty}d\alpha_{1}d\alpha_{2}(\alpha_{1}+\alpha_{2})^{2\varepsilon/\sigma+z_{2}-1}\alpha_{1}^{-z_{2}-1}\exp(-3\mu\alpha_{1}-\alpha_{2}\mu)$
(29)
First we will take care the integral over $\alpha_{2}$. We do substitution
$\tilde{\alpha}_{2}=\alpha_{1}+\alpha_{2}$
$\displaystyle I_{k}$ $\displaystyle=\frac{S_{d}\Gamma(1-2\varepsilon/\sigma-
z_{2})}{2\sigma\Gamma(-z_{2})}\int\limits_{0}^{+\infty}d\alpha_{1}\alpha_{1}^{-z_{2}-1}e^{-2\mu\alpha_{1}}\int\limits_{\alpha_{1}}^{+\infty}d\tilde{\alpha}_{2}\tilde{\alpha}_{2}^{2\varepsilon/\sigma+z_{2}-1}e^{-\tilde{\alpha}_{2}\mu}$
$\displaystyle=\frac{S_{d}\Gamma(1-2\varepsilon/\sigma-
z_{2})\mu^{-z_{2}-2\varepsilon/\sigma}}{2\sigma\Gamma(-z_{2})}\int\limits_{0}^{+\infty}d\alpha_{1}\alpha_{1}^{-z_{2}-1}e^{-2\mu\alpha_{1}}\Gamma(2\varepsilon/\sigma+z_{2},\alpha_{1}\mu)\,,$
(30)
where $\Gamma(\lambda,x)$ is incomplete gamma function. The value of the last
integral can be found in GR . The final result or $I_{k}$ reads
$I_{k}=\frac{S_{d}}{2\sigma}\frac{\Gamma(1-2\varepsilon/\sigma-
z_{2})}{\Gamma(1-z_{2})}\Gamma(2\varepsilon/z_{2})\mu^{-2\varepsilon/\sigma}3^{-2\varepsilon/\sigma}\,_{2}F_{1}(1,2\varepsilon/\sigma,1-z_{2},2/3)$
(31)
Inserting (25) and (31) into (24) we infer
$\displaystyle I_{2}$
$\displaystyle=\frac{S_{d}\pi^{d/2}}{2\sigma(2\pi)^{d}}\frac{\Gamma(\sigma/2-\varepsilon/2)}{\Gamma(-\varepsilon/\sigma-1)^{2}}\mu^{-2\varepsilon/\sigma}3^{-2\varepsilon/\sigma}\Gamma(2\varepsilon/\sigma)\int\limits_{-i\infty}^{+i\infty}\frac{dz_{1}dz_{2}}{(2\pi
i)^{2}}\,_{2}F_{1}(1,2\varepsilon/\sigma,1-z_{2},2/3)\frac{\Gamma(1-2\varepsilon/\sigma-
z_{2})}{-z_{2}}$
$\displaystyle\frac{\Gamma(1+z_{1})}{\Gamma(\sigma(1+z_{1})/2)}\frac{\Gamma(-z_{1}+z_{2})}{\Gamma(\sigma(-z_{1}+z_{2})/2)}\frac{\Gamma(\varepsilon/2+\sigma
z_{2}/2)}{\Gamma(\sigma/2-\varepsilon-\sigma
z_{2}/2)}\Gamma(-\varepsilon/2-\sigma
z_{1}/2)\Gamma(-\sigma(-z_{1}+z_{2})/2-\varepsilon/2+\sigma/2)\,.$ (32)
First we sum over all poles of $\Gamma(-\varepsilon/2-\sigma z_{1}/2)$ and
then over pole at $z_{2}=0$. The result reads
$\displaystyle I_{2}$
$\displaystyle=\frac{S_{d}\pi^{d/2}}{2\sigma(2\pi)^{d}}\mu^{-2\varepsilon/\sigma}3^{-2\varepsilon/\sigma}\Gamma(2\varepsilon/\sigma)\,_{2}F_{1}(1,2\varepsilon/\sigma,1,2/3)\frac{\Gamma(1-2\varepsilon/\sigma)}{\Gamma(\sigma/2-\varepsilon)}\Gamma(\varepsilon/2)$
$\displaystyle\sum\limits^{+\infty}_{n=0}\frac{(-1)^{n}}{n!}\frac{\Gamma(1-\varepsilon/\sigma+2n/\sigma)}{\Gamma(\sigma(1-\varepsilon/\sigma+2n/\sigma)/2)}\frac{\Gamma(\varepsilon/\sigma-2n/\sigma)}{\Gamma(\sigma(\varepsilon/\sigma-2n/\sigma)/2)}\Gamma(-\varepsilon+n\sigma/2)\,.$
(33)
We will look the final result in the form
$I_{2}=\mu^{-2\varepsilon/\sigma}e^{-B\varepsilon}(c_{-2}\varepsilon^{-2}+c_{-1}\varepsilon^{-1})$
(34)
To obtain the divergent part of $I_{2}$ it is convenient to use MATHEMATICA.
The result for coefficients $c_{-2}$ and $c_{-1}$ are given by the formula
(10).
## III Appendix C
Here we present the derivation of formula (11). Expanding scaling function
$\ln(Z)$ at two-loop order and Cardy one can infer that
$\displaystyle\ln(Z)$ $\displaystyle=\sum\limits_{1\leq i<j\leq
p}\lambda_{ij}n_{i}n_{j}I_{1}-\frac{1}{2}\left(\sum\limits_{1\leq i<j\leq
p}\lambda_{ij}n_{i}n_{j}I_{1}\right)^{2}-\sum\limits_{1\leq i<j\leq
p}\lambda^{2}_{ij}n_{i}n_{j}I^{2}_{1}-\frac{1}{2}\sum\limits_{1\leq i<j\leq
p}\lambda^{2}_{ij}n^{2}_{i}n^{2}_{j}I^{2}_{1}$
$\displaystyle-\sum\limits_{1\leq i<j<k<l\leq
p}(\lambda_{ij}\lambda_{kl}+\lambda_{ik}\lambda_{jl}+\lambda_{il}\lambda_{jk})n_{i}n_{j}n_{k}n_{l}I^{2}_{1}+\frac{1}{2}\sum\limits_{1\leq
i<j\leq p}\lambda^{2}_{ij}n_{i}n_{j}(n_{i}+n_{j}-2)I^{2}_{1}$
$\displaystyle+\sum\limits_{1\leq i<j<k\leq
p}(\lambda_{ij}\lambda_{ik}+\lambda_{ij}\lambda_{jk}+\lambda_{ik}\lambda_{jk})n_{i}n_{j}n_{k}I^{2}_{1}-2\sum\limits_{1\leq
i<j<k\leq
p}(\lambda_{ij}\lambda_{ik}+\lambda_{ij}\lambda_{jk}+\lambda_{ik}\lambda_{jk})n_{i}n_{j}n_{k}I_{2}$
$\displaystyle-\sum\limits_{1\leq i<j<k\leq
p}(\lambda_{ij}\lambda_{ik}n^{2}_{i}n_{j}n_{k}+\lambda_{ij}\lambda_{jk}n_{i}n^{2}_{j}n_{k}+\lambda_{ik}\lambda_{jk}n_{i}n_{j}n^{2}_{k})I^{2}_{1}$
$\displaystyle-\sum\limits_{1\leq i<j\leq
p}\lambda^{2}_{ij}n_{i}n_{j}(n_{i}+n_{j}-2)I_{2}+\frac{1}{2}\sum\limits_{1\leq
i<j\leq p}\lambda^{2}_{ij}n_{i}n_{j}I^{2}_{1}\,.$ (35)
By the definition $\gamma=\mu\frac{\partial\ln(Z)}{\partial\mu}$. After
differentiation we use the formula
$\lambda_{ij}\mu^{-\varepsilon/\sigma}=g_{Rij}+Ag^{2}_{Rij}/\varepsilon$,
which one can infer from (19), and the integral expansions (10) and
$I_{1}^{2}=\frac{2^{-2\sigma}\pi^{-\sigma}}{\varepsilon^{2}\Gamma(\sigma/2)^{2}}+\frac{2^{-2\sigma}\pi^{-\sigma}}{\varepsilon\Gamma(\sigma/2)^{2}}[\ln(4\pi)+\psi^{(0)}(\sigma/2)]+O(\varepsilon^{0})$
(36)
to derive the following result
$\displaystyle\gamma$ $\displaystyle=-\frac{1}{\sigma}\sum\limits_{1\leq
i<j\leq p}n_{i}n_{j}g_{Rij}-\frac{1}{\varepsilon\sigma}\sum\limits_{1\leq
i<j\leq p}n_{i}n_{j}g^{2}_{Rij}+\frac{2}{\varepsilon\sigma}\sum\limits_{1\leq
i<j\leq p}n_{i}n_{j}g^{2}_{Rij}-\frac{1}{\varepsilon\sigma}\sum\limits_{1\leq
i<j\leq p}n_{i}n_{j}g^{2}_{Rij}$ $\displaystyle+\sum\limits_{1\leq i<j<k\leq
p}n_{i}n_{j}n_{k}(g_{Rij}g_{Rik}+g_{Rij}g_{Rjk}+g_{Rik}g_{Rjk})\left(\frac{2}{\sigma^{2}}\frac{2^{-2\sigma}\pi^{-\sigma}}{\Gamma(\sigma/2)^{2}}\ln(3/4)\right)$
$\displaystyle+\sum\limits_{1\leq i<j\leq
p}g^{2}_{Rij}n_{i}n_{j}(n_{i}+n_{j}-2)\left(\frac{1}{\sigma^{2}}\frac{2^{-2\sigma}\pi^{-\sigma}}{\Gamma(\sigma/2)^{2}}\ln(3/4)\right)\,.$
(37)
The critical exponent is the value of this expression evaluated at the fixed
point $g_{Rij}=\varepsilon$. It easy to see that the resut is equivalent to
(11).
## References
* (1) R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
* (2) M. A. Lomholt, B. van den Broek, S.-M. J. Kalisch, G. J. L. Wuite and R. Metzler, PNAS 106, 8204 (2009).
* (3) N. Menyhard and G. Odor J. Phys. A: Math. Gen. 28, 4505 (1995).
* (4) B. Bergersen and Z. Rácz, Phys. Rev. Lett. 67, 3047 (1991).
* (5) P. Barthelemy, J. Bertolotti and D. S. Wiersma, Nature 453, 495 (2008).
* (6) G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince and H. E. Stanley Nature 381, 413 (1996).
* (7) D. W. Sims et al, Nature 451, 1098 (2008).
* (8) M. F. Shlesinger, B. J. West and J. Klafter, Phys. Rev. Lett. 58, 1100 (1987).
* (9) M. F. Shlesinger, Nature 411, 641 (2001).
* (10) G. Ramos-Fernandez, J. L. Mateos, O. Miramontes, G. Cocho, H. Larralde and B. Ayala-Orozco, Behav. Ecol. Sociobiol. 55, 223 (2004).
* (11) M. E. Fisher, J. Stat. Phys. 34, 667 (1984).
* (12) M. Doi, J. Phys. A: Math. Gen. 9, 1465 (1976).
* (13) L. Peliti, J. Physique 46, 1469 (1985).
* (14) J. Cardy and M. Katori, J. Phys. A 36, 609 (2003).
* (15) H. K. Janssen and O. Stenull, Phys. Rev. E 78, 061117 (2008).
* (16) H. Hinrichsen, J. Stat. Mech.: Theor. Exp. P07066 (2007).
* (17) J. B. Bronzan and J. W. Dash, Phys. Rev. D 10, 4208 (1974).
* (18) V. A. Smirnov, _Evaluating Feynman Integrals_ , (Springer, 2004).
* (19) R. Metzler and J. Klafter, J. Phys. A 37, R161 (2004).
* (20) S. Redner, P. L. Krapivsky, Am. J. Phys. 67, 1277 (1999).
* (21) F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra, and G. M. Viswanathan, Phys. Rev. Lett. 88, 097901 (2002).
* (22) R. F. Kayser and J. B. Hubbard Phys. Rev. Lett. 51, 79 (1983)
* (23) I. Goncharenko and A. Gopinathan; arXiv:1003.5970.
* (24) G. Schehr, S. N. Majumdar, A. Comtet and J. Randon-Furling, Phys. Rev. Lett. 101, 150601 (2008).
* (25) C. Nadal and S. N. Majumdar, Phys. Rev. E 79, 061117 (2009).
* (26) B. Kozma, M. B. Hastings and G. Korniss, Phys. Rev. Lett. 95, 018701 (2005).
* (27) I. S. Gradshtein I. M. Ryjik _Tables of Integrals_ , (Academic Press, 1965).
|
arxiv-papers
| 2010-07-12T23:47:52 |
2024-09-04T02:49:11.573738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Igor Goncharenko, Ajay Gopinathan",
"submitter": "Igor Goncharenko",
"url": "https://arxiv.org/abs/1007.2008"
}
|
1007.2123
|
# Comparative Studies of Programming Languages, COMP6411 Lecture Notes,
Revision 1.9
Joey Paquet Serguei A. Mokhov (Eds.)
## Preface
Lecture notes for the Comparative Studies of Programming Languages course,
COMP6411, taught at the Department of Computer Science and Software
Engineering, Faculty of Engineering and Computer Science, Concordia
University, Montreal, QC, Canada. These notes include a compiled book of
primarily related articles from the Wikipedia, the Free Encyclopedia [24], as
well as Comparative Programming Languages book [7] and other resources,
including our own. The original notes were compiled by Dr. Paquet [14]
###### Contents
1. 1 Brief History and Genealogy of Programming Languages
1. 1.1 Introduction
1. 1.1.1 Subreferences
2. 1.2 History
1. 1.2.1 Pre-computer era
2. 1.2.2 Subreferences
3. 1.2.3 Early computer era
4. 1.2.4 Subreferences
5. 1.2.5 Modern/Structured programming languages
3. 1.3 References
2. 2 Programming Paradigms
1. 2.1 Introduction
2. 2.2 History
1. 2.2.1 Low-level: binary, assembly
2. 2.2.2 Procedural programming
3. 2.2.3 Object-oriented programming
4. 2.2.4 Declarative programming
3. 3 Program Evaluation
1. 3.1 Program analysis and translation phases
1. 3.1.1 Front end
2. 3.1.2 Back end
2. 3.2 Compilation vs. interpretation
1. 3.2.1 Compilation
2. 3.2.2 Interpretation
3. 3.2.3 Subreferences
3. 3.3 Type System
1. 3.3.1 Type checking
4. 3.4 Memory management
1. 3.4.1 Garbage collection
2. 3.4.2 Manual memory management
4. 4 Programming Languages Popularity
1. 4.1 langpop.com
1. 4.1.1 Yahoo Search
2. 4.1.2 Job Postings
3. 4.1.3 Books
4. 4.1.4 Open source code repositories
5. 4.1.5 Discussion sites
2. 4.2 www.tiobe.com
1. 4.2.1 Top 20 programming languages (March 2010)
2. 4.2.2 Long-term trend
3. 4.2.3 Categories of programming languages
5. 5 Programming Languages Performance Ranking
1. 5.1 Execution time
2. 5.2 Memory consumption
3. 5.3 Source code size
4. 5.4 Overall results
## Chapter 1 Brief History and Genealogy of Programming Languages
### 1.1 Introduction
A programming language is an artificial language designed to express
computations that can be performed by a machine, particularly a computer.
Programming languages can be used to create programs that control the behavior
of a machine, to express algorithms precisely, or as a mode of human
communication. The earliest programming languages predate the invention of the
computer, and were used to direct the behavior of mechanical machines such as
player pianos. Thousands of different programming languages have been created,
mainly in the computer field, with many more being created every year. Most
programming languages describe computation in an imperative style, i.e., as a
sequence of commands, although some languages, such as those that support
functional programming or logic programming, use alternative forms of
description.
#### 1.1.1 Subreferences
1. 1.
http://en.wikipedia.org/wiki/Programming_language
### 1.2 History
#### 1.2.1 Pre-computer era
##### Analytical engine
The analytical engine, an important step in the history of computers, was the
design of a mechanical general-purpose computer by the British mathematician
Charles Babbage. It was first described in 1837. Because of financial,
political, and legal issues, the engine was never built. In its logical design
the machine was essentially modern, anticipating the first completed general-
purpose computers by about 100 years. The input (programs and data) was to be
provided to the machine via punched cards, a method being used at the time to
direct mechanical looms such as the Jacquard loom. For output, the machine
would have a printer, a curve plotter and a bell. The machine would also be
able to punch numbers onto cards to be read in later. It employed ordinary
base-10 fixed-point arithmetic. There was to be a store (i.e., a memory)
capable of holding 1,000 numbers of 50 decimal digits each (ca. 20.7kB). An
arithmetical unit (the “mill”) would be able to perform all four arithmetic
operations, plus comparisons and optionally square roots. Like the central
processing unit (CPU) in a modern computer, the mill would rely upon its own
internal procedures, to be stored in the form of pegs inserted into rotating
drums called “barrels,” in order to carry out some of the more complex
instructions the user’s program might specify. The programming language to be
employed by users was akin to modern day assembly languages. Loops and
conditional branching were possible and so the language as conceived would
have been Turing-complete long before Alan Turing’s concept. Three different
types of punch cards were used: one for arithmetical operations, one for
numerical constants, and one for load and store operations, transferring
numbers from the store to the arithmetical unit or back. There were three
separate readers for the three types of cards.
#### 1.2.2 Subreferences
1. 1.
http://en.wikipedia.org/wiki/Analytical_engine
#### 1.2.3 Early computer era
##### Plankalk l
Plankalk l is a computer language developed for engineering purposes by Konrad
Zuse. It was the first high-level non-von Neumann programming language to be
designed for a computer and was designed between 1943 and 1945. Also, notes
survive with scribblings about such a plan calculation dating back to 1941.
Plankalk l was not published at that time owing to a combination of factors
such as conditions in wartime and postwar Nazi Germany. By 1946, Zuse had
written a book on the subject but this remained unpublished. In 1948 Zuse
published a paper about the Plankalk l in the “Archiv der Mathematik” but
still did not attract much feedback - for a long time to come programming a
computer would only be thought of as programming with machine code. The
Plankalk l was eventually more comprehensively published in 1972 and the first
compiler for it was implemented in 1998. Another independent implementation
followed in the year 2000 by the Free University of Berlin. Plankalk l drew
comparisons to APL and relational algebra. It includes assignment statements,
subroutines, conditional statements, iteration, floating point arithmetic,
arrays, hierarchical record structures, assertions, exception handling, and
other advanced features such as goal-directed execution. Thus, this language
included many of the syntactical elements of structured programming languages
that would be invented later, but it failed to be recognized widely.
#### 1.2.4 Subreferences
1. 1.
http://en.wikipedia.org/wiki/Plankalk%C3%BCl
##### Short Code
Short Code was one of the first higher-level languages ever developed for an
electronic computer. Unlike machine code, Short Code statements represented
mathematic expressions rather than a machine instruction. Short Code was
proposed by John Mauchly in 1949 and originally known as Brief Code. William
Schmitt implemented a version of Brief Code in 1949 for the BINAC computer,
though it was never debugged and tested. The following year Schmitt
implemented a new version of Brief Code for the UNIVAC I where it was now
known as Short Code. While Short Code represented expressions, the
representation itself was not direct and required a process of manual
conversion. Elements of an expression were represented by two-character codes
and then divided into 6-code groups in order to conform to the 12 byte words
used by BINAC and Univac computers. Along with basic arithmetic, Short Code
allowed for branching and calls to a library of functions. The language was
interpreted and ran about 50 times slower than machine code.
1. 1.
http://en.wikipedia.org/wiki/Short_Code_%28computer_language%29
##### A-0
The A-0 system (Arithmetic Language version 0), written by Grace Hopper in
1951 and 1952 for the UNIVAC I, was the first compiler ever developed for an
electronic computer. The A-0 functioned more as a loader or linker than the
modern notion of a compiler. A program was specified as a sequence of
subroutines and arguments. The subroutines were identified by a numeric code
and the arguments to the subroutines were written directly after each
subroutine code. The A-0 system converted the specification into machine code
that could be fed into the computer a second time to execute the program. The
A-0 system was followed by the A-1, A-2, A-3 (released as ARITH-MATIC), AT-3
(released as MATH-MATIC) and B-0 (released as FLOW-MATIC).
1. 1.
http://en.wikipedia.org/wiki/A-0_System
#### 1.2.5 Modern/Structured programming languages
##### Fortran
Fortran (previously FORTRAN) is a general-purpose, procedural, imperative
programming language that is especially suited to numeric computation and
scientific computing. Originally developed by IBM in the 1950s for scientific
and engineering applications, Fortran came to dominate this area of
programming early on and has been in continual use for over half a century in
computationally intensive areas such as numerical weather prediction, finite
element analysis, computational fluid dynamics (CFD), computational physics,
and computational chemistry. It is one of the most popular languages in the
area of high-performance computing and is the language used for programs that
benchmark and rank the world’s fastest supercomputers.
Fortran (a blend derived from The IBM Mathematical Formula Translating System)
encompasses a lineage of versions, each of which evolved to add extensions to
the language while usually retaining compatibility with previous versions.
Successive versions have added support for processing of character-based data
(FORTRAN 77), array programming, modular programming and object-based
programming (Fortran 90 / 95), and object-oriented and generic programming
(Fortran 2003).
1. 1.
http://en.wikipedia.org/wiki/Fortran
##### Algol
ALGOL (short for ALGOrithmic Language) is a family of imperative computer
programming languages originally developed in the mid 1950s which greatly
influenced many other languages and became the de facto way algorithms were
described in textbooks and academic works for almost the next 30 years. It was
designed to avoid some of the perceived problems with FORTRAN and eventually
gave rise to many other programming languages (including BCPL, B, Pascal,
Simula and C). ALGOL introduced code blocks and was the first language to use
begin and end pairs for delimiting them.
John Backus developed the Backus normal form method of describing programming
languages specifically for ALGOL 58. It was revised and expanded by Peter Naur
for ALGOL 60, and at Donald Knuth’s suggestion renamed Backus Naur Form.
Niklaus Wirth based his own ALGOL W on ALGOL 60 before moving to develop
Pascal.
1. 1.
http://en.wikipedia.org/wiki/ALGOL
##### LISP
LISP is a family of computer programming languages with a long history and a
distinctive, fully parenthesized syntax. Originally specified in 1958, Lisp is
the second-oldest high-level programming language in widespread use today;
only Fortran is older. Like Fortran, Lisp has changed a great deal since its
early days, and a number of dialects have existed over its history. Today, the
most widely known general-purpose Lisp dialects are Clojure, Common Lisp and
Scheme. Lisp was originally created as a practical mathematical notation for
computer programs, influenced by the notation of Alonzo Church’s lambda
calculus. It quickly became the favored programming language for artificial
intelligence (AI) research. As one of the earliest programming languages, Lisp
pioneered many ideas in computer science, including tree data structures,
automatic storage management, dynamic typing, and the self-hosting compiler.
The name LISP derives from “LISt Processing”. Linked lists are one of Lisp
languages’ major data structures, and Lisp source code is itself made up of
lists. As a result, Lisp programs can manipulate source code as a data
structure, giving rise to the macro systems that allow programmers to create
new syntax or even new domain-specific programming languages embedded in Lisp.
1. 1.
http://en.wikipedia.org/wiki/Lisp_%28programming_language%29
##### Cobol
COBOL is one of the oldest programming languages. Its name is an acronym for
COmmon Business-Oriented Language, defining its primary domain in business,
finance, and administrative systems for companies and governments. A
specification of COBOL was initially created during the second half of 1959 by
Grace Hopper. The specifications were to a great extent inspired by the FLOW-
MATIC language invented by Grace Hopper, commonly referred to as “the mother
of the COBOL language”. Since 1959 COBOL has undergone several modifications
and improvements. In an attempt to overcome the problem of incompatibility
between different versions of COBOL, the American National Standards Institute
(ANSI) developed a standard form of the language in 1968. This version was
known as American National Standard (ANS) COBOL. In 1974, ANSI published a
revised version of (ANS) COBOL, containing a number of features that were not
in the 1968 version. In 1985, ANSI published still another revised version
that had new features not in the 1974 standard. The language continues to
evolve today. The COBOL 2002 standard includes support for object-oriented
programming and other modern language features.
1. 1.
http://en.wikipedia.org/wiki/COBOL
##### Simula
Simula is a name for two programming languages, Simula I and Simula 67,
developed in the 1960s at the Norwegian Computing Center in Oslo, by Ole-Johan
Dahl and Kristen Nygaard. Syntactically, it is a fairly faithful superset of
Algol 60. Simula 67 introduced objects, classes, subclasses, virtual methods,
coroutines, discrete event simulation, and features garbage collection.
Simula is considered the first object-oriented programming language. As its
name implies, Simula was designed for doing simulations, and the needs of that
domain provided the framework for many of the features of object-oriented
languages today. Simula has been used in a wide range of applications such as
simulating VLSI designs, process modeling, protocols, algorithms, and other
applications such as typesetting, computer graphics, and education. Since
Simula-type objects are reimplemented in C++, Java and C# the influence of
Simula is often understated. The creator of C++, Bjarne Stroustrup, has
acknowledged that Simula 67 was the greatest influence on him to develop C++,
to bring the kind of productivity enhancements offered by Simula to the raw
computational speed offered by lower level languages like BCPL.
1. 1.
http://en.wikipedia.org/wiki/Simula
##### Basic
In computer programming, BASIC (an acronym for Beginner’s All-purpose Symbolic
Instruction Code) is a family of high-level programming languages. The
original BASIC was designed in 1964 by John George Kemeny and Thomas Eugene
Kurtz at Dartmouth College in New Hampshire, USA to provide computer access to
non-science students. At the time, nearly all use of computers required
writing custom software, which was something only scientists and
mathematicians tended to be able to do. The language and its variants became
widespread on microcomputers in the late 1970s and 1980s. The language was
based partly on FORTRAN II and partly on ALGOL 60.
It was intended to address the complexity issues of older languages with a new
language design specifically for the new class of users that the then new
time-sharing systems allowed that is, a less technical user who did not have
the mathematical background of the more traditional users and was not
interested in acquiring it. Being able to use a computer to support teaching
and research was quite novel at the time.
The eight original design principles of BASIC were:
* •
Be easy for beginners to use.
* •
Be a general-purpose programming language.
* •
Allow advanced features to be added for experts (while keeping the language
simple for beginners).
* •
Be interactive.
* •
Provide clear and friendly error messages.
* •
Respond quickly for small programs.
* •
Not to require an understanding of computer hardware.
* •
Shield the user from the operating system.
BASIC remains popular to this day in a handful of highly modified dialects and
new languages influenced by BASIC such as Microsoft Visual Basic. As of 2006,
59% of developers for the .NET platform used Visual Basic .NET as their only
language.
1. 1.
http://en.wikipedia.org/wiki/BASIC
##### ISWIM
ISWIM [9] is an abstract computer programming language (or a family of
programming languages) devised by Peter J. Landin and first described in his
article, The Next 700 Programming Languages, published in the Communications
of the ACM in 1966 [9]. The acronym stands for “If you See What I Mean”.
Although not implemented, it has proved very influential in the development of
programming languages, especially functional programming languages such as
SASL, Miranda, ML, Haskell and their successors. A notationally distinctive
feature of ISWIM is its use of where clauses. An ISWIM program is a single
expression qualified by where clauses (auxiliary definitions including
equations among variables), conditional expressions and function definitions.
With CPL, ISWIM was one of the first programming languages to use where
clauses, which are used widely in functional and declarative programming
languages.
1. 1.
http://en.wikipedia.org/wiki/ISWIM
##### Pascal
Pascal is an influential imperative and procedural programming language,
designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and
efficient language intended to encourage good programming practices using
structured programming and data structuring.
Pascal is based on the ALGOL programming language and named in honor of the
French mathematician and philosopher Blaise Pascal. Wirth subsequently
developed the Modula-2 and Oberon, languages similar to Pascal. Before, and
leading up to Pascal, Wirth developed the language Euler, followed by Algol-W.
Initially, Pascal was largely, but not exclusively, intended to teach students
structured programming. Generations of students have used Pascal as an
introductory language in undergraduate courses. A derivative known as Object
Pascal was designed for object oriented programming.
1. 1.
http://en.wikipedia.org/wiki/Pascal_%28programming_language%29
##### Smalltalk
Smalltalk is an object-oriented, dynamically typed, reflective programming
language. Smalltalk was created as the language to underpin the “new world” of
computing exemplified by “human computer symbiosis.” It was designed and
created in part for educational use at the Learning Research Group of Xerox
PARC by Alan Kay, Dan Ingalls, Adele Goldberg, Ted Kaehler, Scott Wallace, and
others during the 1970s.
The language was first generally released as Smalltalk-80 and has been widely
used since. Smalltalk-like languages are in continuing active development, and
have gathered loyal communities of users around them. ANSI Smalltalk was
ratified in 1998 and represents the standard version of Smalltalk.
One of the early versions, Smalltalk-76, was one of the first programming
languages to be implemented along with a development environment featuring
most of the now familiar tools, including a class library code browser/editor.
Smalltalk-80 added metaclasses, to help maintain the “everything is an object”
(except private instance variables) paradigm by associating properties and
behavior with individual classes. Smalltalk programs are usually compiled to
bytecode, which is then interpreted by a virtual machine or dynamically
translated into machine-native code as in just-in-time compilation.
Smalltalk has influenced the wider world of computer programming in four main
areas. It inspired the syntax and semantics of other computer programming
languages. Secondly, it was a prototype for a model of computation known as
message passing. Thirdly, its WIMP GUI inspired the windowing environments of
personal computers in the late twentieth and early twenty-first centuries, so
much so that the windows of the first Macintosh desktop look almost identical
to the MVC windows of Smalltalk-80. Finally, the integrated development
environment was the model for a generation of visual programming tools that
look like Smalltalk’s code browsers and debuggers.
As in other object-oriented languages, the central concept in Smalltalk-80
(but not in Smalltalk-72) is that of an object. An object is always an
instance of a class. Classes are “blueprints” that describe the properties and
behavior of their instances. For example, a Window class would declare that
windows have properties such as the label, the position and whether the window
is visible or not. The class would also declare that instances support
operations such as opening, closing, moving and hiding. Each particular Window
object would have its own values of those properties, and each of them would
be able to perform operations defined by its class.
A Smalltalk object can do exactly three things:
1. 1.
Hold state (references to other objects).
2. 2.
Receive a message from itself or another object.
3. 3.
In the course of processing a message, send messages to itself or another
object.
The state an object holds is always private to that object. Other objects can
query or change that state only by sending requests (messages) to the object
to do so. Any message can be sent to any object: when a message is received,
the receiver determines whether that message is appropriate. (Alan Kay has
commented that despite the attention given to objects, messaging is the most
important concept in Smalltalk.)
Smalltalk is a “pure” object-oriented programming language, meaning that,
unlike Java and C++, there is no difference between values which are objects
and values which are primitive types. In Smalltalk, primitive values such as
integers, booleans and characters are also objects, in the sense that they are
instances of corresponding classes, and operations on them are invoked by
sending messages. A programmer can change the classes that implement primitive
values, so that new behavior can be defined for their instances–for example,
to implement new control structures–or even so that their existing behavior
will be changed. This fact is summarized in the commonly heard phrase “In
Smalltalk everything is an object”. Since all values are objects, classes
themselves are also objects. Each class is an instance of the metaclass of
that class. Metaclasses in turn are also objects, and are all instances of a
class called Metaclass. Code blocks are also objects.
Smalltalk-80 is a totally reflective system, implemented in Smalltalk-80
itself. Smalltalk-80 provides both structural and computational reflection.
Smalltalk is a structurally reflective system whose structure is defined by
Smalltalk-80 objects. The classes and methods that define the system are
themselves objects and fully part of the system that they help define. The
Smalltalk compiler compiles textual source code into method objects, typically
instances of CompiledMethod. These get added to classes by storing them in a
class’s method dictionary. The part of the class hierarchy that defines
classes can add new classes to the system. The system is extended by running
Smalltalk-80 code that creates or defines classes and methods. In this way a
Smalltalk-80 system is a “living” system, carrying around the ability to
extend itself at run time.
Since the classes are themselves objects, they can be asked questions such as
“what methods do you implement?” or “what fields/slots/instance variables do
you define?”. So objects can easily be inspected, copied, (de)serialized and
so on with generic code that applies to any object in the system.
Smalltalk-80 also provides computational reflection, the ability to observe
the computational state of the system. In languages derived from the original
Smalltalk-80 the current activation of a method is accessible as an object
named via a keyword, thisContext. By sending messages to thisContext a method
activation can ask questions like “who sent this message to me”. These
facilities make it possible to implement co-routines or Prolog-like back-
tracking without modifying the virtual machine.
When an object is sent a message that it does not implement, the virtual
machine sends the object the doesNotUnderstand: message with a reification of
the message as an argument. The message (another object, an instance of
Message) contains the selector of the message and an Array of its arguments.
In an interactive Smalltalk system the default implementation of
doesNotUnderstand: is one that opens an error window reporting the error to
the user. Through this and the reflective facilities the user can examine the
context in which the error occurred, redefine the offending code, and
continue, all within the system, using Smalltalk-80’s reflective facilities.
1. 1.
http://en.wikipedia.org/wiki/Smalltalk
##### C
C is a general-purpose computer programming language developed in 1972 by
Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix
operating system.
Although C was designed for implementing system software, it is also widely
used for developing portable application software. C is one of the most
popular programming languages. It is widely used on many different software
platforms, and there are few computer architectures for which a C compiler
does not exist. C has greatly influenced many other popular programming
languages, most notably C++, which originally began as an extension to C.
C is an imperative (procedural) systems implementation language. It was
designed to be compiled using a relatively straightforward compiler, to
provide low-level access to memory, to provide language constructs that map
efficiently to machine instructions, and to require minimal run-time support.
C was therefore useful for many applications that had formerly been coded in
assembly language. Despite its low-level capabilities, the language was
designed to encourage machine-independent programming. A standards-compliant
and portably written C program can be compiled for a very wide variety of
computer platforms and operating systems with little or no change to its
source code. The language has become available on a very wide range of
platforms, from embedded microcontrollers to supercomputers.
C also exhibits the following more specific characteristics:
* •
lack of nested function definitions
* •
variables may be hidden in nested blocks
* •
partially weak typing; for instance, characters can be used as integers
* •
low-level access to computer memory by converting machine addresses to typed
pointers
* •
function and data pointers supporting ad hoc run-time polymorphism
* •
array indexing as a secondary notion, defined in terms of pointer arithmetic
* •
a preprocessor for macro definition, source code file inclusion, and
conditional compilation
* •
complex functionality such as I/O, string manipulation, and mathematical
functions consistently delegated to library routines
* •
A relatively small set of reserved keywords
C does not have some features that are available in some other programming
languages:
* •
No direct assignment of arrays or strings (copying can be done via standard
functions; assignment of objects having struct or union type is supported)
* •
No automatic garbage collection
* •
No requirement for bounds checking of arrays
* •
No operations on whole arrays
* •
No syntax for ranges, such as the A..B notation used in several languages
* •
Prior to C99, no separate Boolean type (zero/nonzero is used instead)[6]
* •
No functions as parameters (only function and variable pointers)
* •
No exception handling; standard library functions signify error conditions
with the global errno variable and/or special return values
* •
Only rudimentary support for modular programming
* •
No compile-time polymorphism in the form of function or operator overloading
* •
Only rudimentary support for generic programming
* •
Very limited support for object-oriented programming with regard to
polymorphism and inheritance
* •
Limited support for encapsulation
* •
No native support for multithreading and networking
* •
No standard libraries for computer graphics and several other application
programming needs
1. 1.
http://en.wikipedia.org/wiki/C_%28programming_language%29
##### Prolog
Prolog is a general purpose logic programming language associated with
artificial intelligence and computational linguistics. Prolog has its roots in
formal logic, and unlike many other programming languages, Prolog is
declarative: The program logic is expressed in terms of relations, represented
as facts and rules. Execution is triggered by running queries over these
relations. The language was first conceived by a group around Alain Colmerauer
in Marseille, France, in the early 1970s and the first Prolog system was
developed in 1972 by Alain Colmerauer and Phillipe Roussel. Prolog was one of
the first logic programming languages, and remains among the most popular such
languages today, with many free and commercial implementations available.
While initially aimed at natural language processing, the language has since
then stretched far into other areas like theorem proving, expert systems,
games, automated answering systems, ontologies and sophisticated control
systems. Modern Prolog environments support the creation of graphical user
interfaces, as well as administrative and networked applications.
1. 1.
http://en.wikipedia.org/wiki/Prolog
##### ML
ML is a general-purpose functional programming language developed by Robin
Milner and others in the late 1970s at the University of Edinburgh, whose
syntax is inspired by ISWIM. Historically, ML stands for metalanguage: it was
conceived to develop proof tactics in the LCF theorem prover (whose language,
pplambda, a combination of the first-order predicate calculus and the simply
typed polymorphic lambda-calculus, had ML as its metalanguage). It is known
for its use of the Hindley-Milner type inference algorithm, which can
automatically infer the types of most expressions without requiring explicit
type annotations. ML is often referred to as an impure functional language,
because it does not encapsulate side-effects, unlike purely functional
programming languages such as Haskell.
Features of ML include a call-by-value evaluation strategy, first class
functions, automatic memory management through garbage collection, parametric
polymorphism, static typing, type inference, algebraic data types, pattern
matching, and exception handling.
Unlike Haskell, ML uses eager evaluation, which means that all subexpressions
are always evaluated. However, lazy evaluation can be achieved through the use
of closures. Thus one can create and use infinite streams as in Haskell,
however, their expression is comparatively indirect.
Today there are several languages in the ML family; the two major dialects are
Standard ML (SML) and Caml, but others exist, including F# – a language which
Microsoft supports for their .NET platform. Ideas from ML have influenced
numerous other languages, like ML≤ [6], Haskell, Cyclone, and Nemerle.
ML’s strengths are mostly applied in language design and manipulation
(compilers, analyzers, theorem provers [15, 13]), but it is a general-purpose
language also used in bioinformatics, financial systems, and applications
including a genealogical database, a peer-to-peer client/server program, etc.
1. 1.
http://en.wikipedia.org/wiki/ML_%28programming_language%29
##### Scheme
Scheme is one of the two main dialects of the programming language Lisp.
Unlike Common Lisp, the other main dialect, Scheme follows a minimalist design
philosophy specifying a small standard core with powerful tools for language
extension. Its compactness and elegance have made it popular with educators,
language designers, programmers, implementors, and hobbyists, and this diverse
appeal is seen as both a strength and, because of the diversity of its
constituencies and the wide divergence between implementations, one of its
weaknesses.
Scheme was developed at the MIT AI Lab by Guy L. Steele and Gerald Jay Sussman
who introduced it to the academic world via a series of memos, now referred to
as the Lambda Papers, over the period 1975-1980. Scheme had a significant
influence on the effort that led to the development of its sister, Common
LISP.
1. 1.
http://en.wikipedia.org/wiki/Scheme_%28programming_language%29
##### C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-
purpose programming language. It is regarded as a middle-level language, as it
comprises a combination of both high-level and low-level language features. It
was developed by Bjarne Stroustrup starting in 1979 at Bell Labs as an
enhancement to the C programming language and originally named “C with
Classes”. It was renamed C++ in 1983.
C++ is widely used in the software industry, and remains one of the most
popular languages ever created. Some of its application domains include
systems software, application software, device drivers, embedded software,
high-performance server and client applications, and entertainment software
such as video games. Several groups provide both free and proprietary C++
compiler software, including the GNU Project, Microsoft, Intel, Borland and
others.
The language began as enhancements to C, first adding classes, then virtual
functions, operator overloading, multiple inheritance, templates, and
exception handling among other features.
Stroustrup began work on “C with Classes” in 1979. The idea of creating a new
language originated from Stroustrup’s experience in programming for his Ph.D.
thesis. Stroustrup found that Simula had features that were very helpful for
large software development, but the language was too slow for practical use,
while BCPL (the ancestor of C) was fast but too low-level to be suitable for
large software development. Remembering his Ph.D. experience, Stroustrup set
out to enhance the C language with Simula-like features. C was chosen because
it was general-purpose, fast, portable and widely used. Besides C and Simula,
some other languages that inspired him were ALGOL 68, Ada, CLU and ML. At
first, the class, derived class, strong type checking, inlining, and default
argument features were added to C via Cfront. The first commercial release
occurred in October 1985.[3] In 1983, the name of the language was changed
from C with Classes to C++ (++ being the increment operator in C and C++). New
features were added including virtual functions, function name and operator
overloading, references, constants, user-controlled free-store memory control,
improved type checking, and BCPL style single-line comments with two forward
slashes (//). In 1985, the first edition of The C++ Programming Language was
released, providing an important reference to the language, since there was
not yet an official standard. Release 2.0 of C++ came in 1989. New features
included multiple inheritance, abstract classes, static member functions,
const member functions, and protected members. In 1990, The Annotated C++
Reference Manual was published. This work became the basis for the future
standard. Late addition of features included templates, exceptions,
namespaces, new casts, and a Boolean type.
As the C++ language evolved, a standard library also evolved with it. The
first addition to the C++ standard library was the stream I/O library which
provided facilities to replace the traditional C functions such as printf and
scanf. Later, among the most significant additions to the standard library,
was the Standard Template Library. C++ continues to be used and is one of the
preferred programming languages to develop professional applications. The
language has gone from being mostly Western to attracting programmers from all
over the world.
Bjarne Stroustrup describes some rules that he uses for the design of C++:
* •
C++ is designed to be a statically typed, general-purpose language that is as
efficient and portable as C
* •
C++ is designed to directly and comprehensively support multiple programming
styles (procedural programming, data abstraction, object-oriented programming,
and generic programming)
* •
C++ is designed to give the programmer choice, even if this makes it possible
for the programmer to choose incorrectly
* •
C++ is designed to be as compatible with C as possible, therefore providing a
smooth transition from C
* •
C++ avoids features that are platform specific or not general purpose
* •
C++ does not incur overhead for features that are not used (the “zero-overhead
principle”)
* •
C++ is designed to function without a sophisticated programming environment
Stroustrup also mentions that C++ was always intended to make programming more
fun and that many of the double meanings in the language are intentional, such
as the name of the language.
1. 1.
http://en.wikipedia.org/wiki/C%2B%2B
##### Ada
Ada is a structured, statically typed, imperative, wide-spectrum, and object-
oriented high-level computer programming language, extended from Pascal and
other languages. It was originally designed by a team led by Jean Ichbiah of
CII Honeywell Bull under contract to the United States Department of Defense
(DoD) from 1977 to 1983 to supersede the hundreds of programming languages
then used by the DoD. Ada is strongly typed and compilers are validated for
reliability in mission-critical applications, such as avionics software. Ada
was named after Ada Lovelace (1815 1852), who is often credited as being the
first computer programmer.
1. 1.
http://en.wikipedia.org/wiki/Ada_%28programming_language%29
##### Perl
Perl is a high-level, general-purpose, interpreted, dynamic programming
language. Perl was originally developed by Larry Wall, a linguist working as a
systems administrator for NASA, in 1987, as a general-purpose Unix scripting
language to make report processing easier. Since then, it has undergone many
changes and revisions and become widely popular amongst programmers. Larry
Wall continues to oversee development of the core language, and its upcoming
version, Perl 6. Perl borrows features from other programming languages
including C, shell scripting (sh), AWK, and sed [11]. The language provides
powerful text processing facilities without the arbitrary data length limits
of many contemporary Unix tools, facilitating easy manipulation of text files.
It is also used for graphics programming, system administration, network
programming, applications that require database access and CGI programming on
the Web.
1. 1.
http://en.wikipedia.org/wiki/Perl
##### Python
Python is a general-purpose high-level programming language. Its design
philosophy emphasizes code readability. Python claims to “[combine] remarkable
power with very clear syntax”, and its standard library is large and
comprehensive. Its use of indentation for block delimiters is unusual among
popular programming languages.
Python supports multiple programming paradigms (primarily object oriented,
imperative, and functional) and features a fully dynamic type system and
automatic memory management, similar to Perl, Ruby, Scheme, and Tcl. Like
other dynamic languages, Python is often used as a scripting language.
The language has an open, community-based development model managed by the
non-profit Python Software Foundation, which maintains the de facto definition
of the language in CPython, the reference implementation.
Python is a multi-paradigm programming language. Rather than forcing
programmers to adopt a particular style of programming, it permits several
styles: object-oriented programming and structured programming are fully
supported, and there are a number of language features which support
functional programming and aspect-oriented programming (including by
metaprogramming and by magic methods). Many other paradigms are supported
using extensions, such as pyDBC and Contracts for Python which allow Design by
Contract.
Python is often used as a scripting language for web applications, and has
seen extensive use in the information security industry. Python has been
successfully embedded in a number of software products as a scripting
language, including in finite element method software such as Abaqus, 3D
animation packages such as Maya [4], MotionBuilder, Softimage, Cinema 4D,
BodyPaint 3D, modo, and Blender [5], and 2D imaging programs like GIMP,
Inkscape, Scribus, and Paint Shop Pro. It has even been used in several
videogames.
For many operating systems, Python is a standard component; it ships with most
Linux distributions, with NetBSD, and OpenBSD, and with Mac OS X.
Among the users of Python are YouTube and the original BitTorrent client.
Large organizations that make use of Python include Google, Yahoo!, CERN and
NASA.
1. 1.
http://en.wikipedia.org/wiki/Python_%28programming_language%29
##### Haskell
Haskell is a standardized, general-purpose purely functional programming
language, with non-strict semantics and strong static typing. It is named
after logician Haskell Curry.
Haskell is a purely functional language, which means that in general,
functions in Haskell do not have side effects. There is a distinct type for
representing side effects, orthogonal to the type of functions. A pure
function may return a side effect which is subsequently executed, modeling the
impure functions of other languages. Haskell has a non-strict semantics. Most
implementations of Haskell use lazy evaluation. Haskell has a strong, static,
type system based on Hindley-Milner type inference. Haskell’s principal
innovation in this area is to add type classes, which were originally
conceived as a principled way to add overloading to the language, but have
since found many more uses
Following the release of Miranda by Research Software Ltd, in 1985, interest
in lazy functional languages grew. By 1987, more than a dozen non-strict,
purely functional programming languages existed. Of these, Miranda was the
most widely used, but was not in the public domain. At the conference on
Functional Programming Languages and Computer Architecture (FPCA ’87) in
Portland, Oregon, a meeting was held during which participants formed a strong
consensus that a committee should be formed to define an open standard for
such languages. The committee’s purpose was to consolidate the existing
functional languages into a common one that would serve as a basis for future
research in functional-language design. The first version of Haskell (”Haskell
1.0”) was defined in 1990. The committee’s efforts resulted in a series of
language definitions. In late 1997, the series culminated in Haskell 98,
intended to specify a stable, minimal, portable version of the language and an
accompanying standard library for teaching, and as a base for future
extensions.
1. 1.
http://en.wikipedia.org/wiki/Haskell_%28programming_language%29
##### Java
Java is a programming language originally developed by James Gosling at Sun
Microsystems and released in 1995 as a core component of Sun Microsystems’
Java platform. The language derives much of its syntax from C and C++ but has
a simpler object model and fewer low-level facilities. Java applications are
typically compiled to bytecode (class file) that can run on any Java Virtual
Machine (JVM) regardless of computer architecture. Java is general-purpose,
concurrent, class-based, and object-oriented, and is specifically designed to
have as few implementation dependencies as possible. It is intended to let
application developers “write once, run anywhere”.
There were five primary goals in the creation of the Java language:
* •
It should be “simple, object oriented, and familiar”.
* •
It should be “robust and secure”.
* •
It should be “architecture neutral and portable”.
* •
It should execute with “high performance”.
* •
It should be “interpreted, threaded, and dynamic”.
Programs written in Java have a reputation for being slower and requiring more
memory than those written in some other languages. However, Java programs’
execution speed improved significantly with the introduction of Just-in-time
compilation in 1997/1998 for Java 1.1, the addition of language features
supporting better code analysis, and optimizations in the Java Virtual Machine
itself, such as HotSpot becoming the default for Sun’s JVM in 2000.
Java uses an automatic garbage collector to manage memory in the object
lifecycle. The programmer determines when objects are created, and the Java
runtime is responsible for recovering the memory once objects are no longer in
use. Once no references to an object remain, the unreachable memory becomes
eligible to be freed automatically by the garbage collector. Something similar
to a memory leak may still occur if a programmer’s code holds a reference to
an object that is no longer needed, typically when objects that are no longer
needed are stored in containers that are still in use. If methods for a
nonexistent object are called, a “null pointer exception” is thrown.
One of the ideas behind Java’s automatic memory management model is that
programmers be spared the burden of having to perform manual memory
management. In some languages memory for the creation of objects is implicitly
allocated on the stack, or explicitly allocated and deallocated from the heap.
Either way, the responsibility of managing memory resides with the programmer.
If the program does not deallocate an object, a memory leak occurs. If the
program attempts to access or deallocate memory that has already been
deallocated, the result is undefined and difficult to predict, and the program
is likely to become unstable and/or crash. This can be partially remedied by
the use of smart pointers, but these add overhead and complexity. Note that
garbage collection does not prevent ’logical’ memory leaks, i.e. those where
the memory is still referenced but never used. Garbage collection may happen
at any time. Ideally, it will occur when a program is idle. It is guaranteed
to be triggered if there is insufficient free memory on the heap to allocate a
new object; this can cause a program to stall momentarily. Explicit memory
management is not possible in Java.
Java does not support C/C++ style pointer arithmetic, where object addresses
and unsigned integers (usually long integers) can be used interchangeably.
This allows the garbage collector to relocate referenced objects, and ensures
type safety and security. As in C++ and some other object-oriented languages,
variables of Java’s primitive data types are not objects. This was a conscious
decision by Java’s designers for performance reasons. Because of this, Java
was not considered to be a pure object-oriented programming language. However,
as of Java 5.0, autoboxing enables programmers to proceed as if primitive
types are instances of their wrapper classes.
1. 1.
http://en.wikipedia.org/wiki/Java_%28programming_language%29
### 1.3 References
1. 1.
http://en.wikipedia.org/wiki/Timeline_of_programming_languages
2. 2.
http://merd.sourceforge.net/pixel/language-study/diagram.html
3. 3.
http://www.levenez.com/lang/
4. 4.
http://oreilly.com/pub/a/oreilly/news/languageposter_0504.html
5. 5.
http://www.digibarn.com/collections/posters/tongues/ComputerLanguagesChart-
med.png
## Chapter 2 Programming Paradigms
### 2.1 Introduction
A programming paradigm is a fundamental style of computer programming.
(Compare with a methodology, which is a style of solving specific software
engineering problems.) Paradigms differ in the concepts and abstractions used
to represent the elements of a program (such as objects, functions, variables,
constraints, etc.) and the steps that compose a computation (assignation,
evaluation, continuations, data flows, etc.).
A programming paradigm can be understood as an abstraction of a computer
system, for example the von Neumann model used in traditional sequential
computers. For parallel computing, there are many possible models typically
reflecting different ways processors can be interconnected. The most common
are based on shared memory, distributed memory with message passing, or a
hybrid of the two. In object-oriented programming, programmers can think of a
program as a collection of interacting objects, while in functional
programming a program can be thought of as a sequence of stateless function
evaluations. When programming computers or systems with many processors,
process-oriented programming allows programmers to think about applications as
sets of concurrent processes acting upon logically shared data structures.
Just as different groups in software engineering advocate different
methodologies, different programming languages advocate different programming
paradigms. Some languages are designed to support one particular paradigm
(Smalltalk supports object-oriented programming, Haskell supports functional
programming), while other programming languages support multiple paradigms
(such as Object Pascal, C++, C#, Visual Basic, Common Lisp, Scheme, Perl,
Python, Ruby, Oz and F Sharp).
A multi-paradigm programming language is a programming language that supports
more than one programming paradigm. The design goal of such languages is to
allow programmers to use the best tool for a job, admitting that no one
paradigm solves all problems in the easiest or most efficient way. An example
is Oz, which has subsets that are a logic language (Oz descends from logic
programming), a functional language, an object-oriented language, a dataflow
concurrent language, and more. Oz was designed over a ten-year period to
combine in a harmonious way concepts that are traditionally associated with
different programming paradigms.
### 2.2 History
#### 2.2.1 Low-level: binary, assembly
Initially, computers were hard-wired or soft-wired and then later programmed
using binary code that represented control sequences fed to the computer CPU.
This was difficult and error-prone. Programs written in binary are said to be
written in machine code, which is a very low-level programming paradigm. Hard-
wired, soft-wired, and binary programming are considered first generation
languages.
To make programming easier, assembly languages were developed. These replaced
machine code functions with mnemonics and memory addresses with symbolic
labels. Assembly language programming is considered a low-level paradigm
although it is a “second generation” paradigm. Even assembly languages of the
1960s actually supported library COPY and quite sophisticated conditional
macro generation and pre-processing capabilities. They also supported modular
programming features such as CALL (subroutines), external variables and common
sections (globals), enabling significant code re-use and isolation from
hardware specifics via use of logical operators as READ/WRITE/GET/PUT.
Assembly was, and still is, used for time critical systems and frequently in
embedded systems. Assembly languages are considered second generation
languages.
1. 1.
http://en.wikipedia.org/wiki/Machine_code
2. 2.
http://en.wikipedia.org/wiki/Low-level_programming_language
3. 3.
http://en.wikipedia.org/wiki/Assembly_programming
#### 2.2.2 Procedural programming
Procedural programming can sometimes be used as a synonym for imperative
programming (specifying the steps the program must take to reach the desired
state), but can also refer (as in this article) to a programming paradigm,
derived from structured programming, based upon the concept of the procedure
call. Procedures, also known as routines, subroutines, methods, or functions
(not to be confused with mathematical functions, but similar to those used in
functional programming) simply contain a series of computational steps to be
carried out. Any given procedure might be called at any point during a
program’s execution, including by other procedures or itself. A procedural
programming language provides a programmer a means to define precisely each
step in the performance of a task. The programmer knows what is to be
accomplished and provides through the language step-by-step instructions on
how the task is to be done. Using a procedural language, the programmer
specifies language statements to perform a sequence of algorithmic steps.
Procedural programming is often a better choice than simple sequential or
unstructured programming in many situations which involve moderate complexity
or which require significant ease of maintainability.
Possible benefits:
* •
The ability to re-use the same code at different places in the program without
copying it.
* •
An easier way to keep track of program flow than a collection of “GOTO” or
“JUMP” statements (which can turn a large, complicated program into spaghetti
code).
* •
The ability to be strongly modular or structured.
Thus, procedural programming allows for ”’modularity”’, which is generally
desirable, especially in large, complicated programs. Inputs are usually
specified syntactically in the form of arguments and the outputs delivered as
return values. ”’Scoping”’ is another technique that helps keep procedures
strongly modular. It prevents a procedure from accessing the variables of
other procedures (and vice-versa), including previous instances of itself such
as in recursion, without explicit authorization. Because of the ability to
specify a simple interface, to be self-contained, and to be reused, procedures
are a convenient vehicle for making pieces of code written by different people
or different groups, including through programming ”’libraries”’.
The focus of procedural programming is to break down a programming task into a
collection of variables, data structures, and subroutines, whereas in object-
oriented programming it is to break down a programming task into objects with
each “object” encapsulating its own data and methods (subroutines). The most
important distinction is whereas procedural programming uses procedures to
operate on data structures, object-oriented programming bundles the two
together so an “object” operates on its “own” data structure.
The earliest imperative languages were the machine languages of the original
computers. In these languages, instructions were very simple, which made
hardware implementation easier, but hindered the creation of complex programs.
FORTRAN, developed by John Backus at IBM starting in 1954, was the first major
programming language to remove the obstacles presented by machine code in the
creation of complex programs. FORTRAN was a compiled language that allowed
named variables, complex expressions, subprograms, and many other features now
common in imperative languages. The next two decades saw the development of a
number of other major high-level imperative programming languages. In the late
1950s and 1960s, ALGOL was developed in order to allow mathematical algorithms
to be more easily expressed, and even served as the operating system’s target
language for some computers. COBOL (1960) and BASIC (1964) were both attempts
to make programming syntax look more like English. In the 1970s, Pascal was
developed by Niklaus Wirth, and C was created by Dennis Ritchie while he was
working at Bell Laboratories. For the needs of the United States Department of
Defense, Jean Ichbiah and a team at Honeywell began designing Ada in 1978.
1. 1.
http://en.wikipedia.org/wiki/Procedural_programming
2. 2.
http://en.wikipedia.org/wiki/Imperative_programming
#### 2.2.3 Object-oriented programming
Object-oriented programming (OOP) is a programming paradigm that uses
“objects” data structures consisting of data fields and methods together with
their interactions to design applications and computer programs. Programming
techniques may include features such as data abstraction, encapsulation,
modularity, polymorphism, and inheritance. Though it was invented with the
creation of the [[Wikipedia:Simula—Simula]] language in 1965, it was not
commonly used in mainstream software application development until the early
1990s. Many modern programming languages now support OOP.
There is still some controversy by notable programmers such as Alexander
Stepanov, Richard Stallman and others, concerning the efficacy of the OOP
paradigm versus the procedural paradigm. The necessity of every object to have
associative methods leads some skeptics to associate OOP with software bloat.
Polymorphism was developed as one attempt to resolve this dilemma.
Object-oriented programming is characterized by a group of inter-related
fundamental concepts:
* •
“Class”: Defines the abstract characteristics of a thing (object), including
the thing’s characteristics (its attributes, fields or properties) and the
thing’s behaviors (the things it can do, or methods, operations or features).
One might say that a class is a blueprint or factory that describes the nature
of something. For example, the class Dog would consist of traits shared by all
dogs, such as breed and fur color (characteristics), and the ability to bark
and sit (behaviors). Classes provide modularity and structure in an object-
oriented computer program. A class should typically be recognizable to a non-
programmer familiar with the problem domain, meaning that the characteristics
of the class should make sense in context. Also, the code for a class should
be relatively self-contained (generally using encapsulation). Collectively,
the properties and methods defined by a class are called members.
* •
“Object”: An individual of a class. The class Dog defines all possible dogs by
listing the characteristics and behaviors they can have; the object Lassie is
one particular dog, with particular versions of the characteristics. A Dog has
fur; Lassie has brown-and-white fur. One can have an instance of a class or a
particular object. The instance is the actual object created at runtime. In
programmer jargon, the Lassie object is an instance of the Dog class. The set
of values of the attributes of a particular object is called its state. The
object consists of state and the behaviour that’s defined in the object’s
class.
* •
“Method”: An object’s abilities. In language, methods (sometimes referred to
as “functions”) are verbs. Lassie, being a Dog, has the ability to bark. So
bark() is one of Lassie’s methods. She may have other methods as well, for
example sit() or eat() or walk() or saveTimmy(). Within the program, using a
method usually affects only one particular object; all Dogs can bark, but you
need only one particular dog to do the barking.
* •
“Message passing”: The process by which an object sends data to another object
or asks the other object to invoke a method. Also known to some programming
languages as interfacing. For example, the object called Breeder may tell the
Lassie object to sit by passing a “sit” message which invokes Lassie’s “sit”
method. The syntax varies between languages, for example: [Lassie sit] in
Objective-C. In Java, code-level message passing corresponds to “method
calling”. Some dynamic languages use double-dispatch or multi-dispatch to find
and pass messages.
* •
“Inheritance”: “Subclasses” are more specialized versions of a class, which
inherit attributes and behaviors from their parent classes, and can introduce
their own. For example, the class Dog might have sub-classes called Collie,
Chihuahua, and GoldenRetriever. In this case, Lassie would be an instance of
the Collie subclass. Suppose the Dog class defines a method called bark() and
a property called furColor. Each of its sub-classes (Collie, Chihuahua, and
GoldenRetriever) will inherit these members, meaning that the programmer only
needs to write the code for them once. Each subclass can alter its inherited
traits. For example, the Collie subclass might specify that the default
furColor for a collie is brown-and-white. The Chihuahua subclass might specify
that the bark() method produces a high pitch by default. Subclasses can also
add new members. The Chihuahua subclass could add a method called tremble().
So an individual chihuahua instance would use a high-pitched bark() from the
Chihuahua subclass, which in turn inherited the usual bark() from Dog. The
chihuahua object would also have the tremble() method, but Lassie would not,
because she is a Collie, not a Chihuahua. In fact, inheritance is an “a… is a”
relationship between classes, while instantiation is an “is a” relationship
between an object and a class: a Collie is a Dog (“a… is a”), but Lassie is a
Collie (“is a”). Thus, the object named Lassie has the methods from both
classes Collie and Dog. Multiple inheritance is inheritance from more than one
ancestor class, neither of these ancestors being an ancestor of the other. For
example, independent classes could define Dogs and Cats, and a Chimera object
could be created from these two which inherits all the (multiple) behavior of
cats and dogs. This is not always supported, as it can be hard both to
implement and to use well.
* •
“Abstraction”: Abstraction is simplifying complex reality by modeling classes
appropriate to the problem, and working at the most appropriate level of
inheritance for a given aspect of the problem. For example, Lassie the Dog may
be treated as a Dog much of the time, a Collie when necessary to access
Collie-specific attributes or behaviors, and as an Animal (perhaps the parent
class of Dog) when counting Timmy’s pets. Abstraction is also achieved through
Composition. For example, a class Car would be made up of an Engine, Gearbox,
Steering objects, and many more components. To build the Car class, one does
not need to know how the different components work internally, but only how to
interface with them, i.e., send messages to them, receive messages from them,
and perhaps make the different objects composing the class interact with each
other.
* •
“Encapsulation”: Encapsulation conceals the functional details of a class from
objects that send messages to it. For example, the Dog class has a bark()
method. The code for the bark() method defines exactly how a bark happens
(e.g., by inhale() and then exhale(), at a particular pitch and volume).
Timmy, Lassie’s friend, however, does not need to know exactly how she barks.
Encapsulation is achieved by specifying which classes may use the members of
an object. The result is that each object exposes to any class a certain
interface – those members accessible to that class. The reason for
encapsulation is to prevent clients of an interface from depending on those
parts of the implementation that are likely to change in the future, thereby
allowing those changes to be made more easily, that is, without changes to
clients. Members are often specified as public, protected or private,
determining whether they are available to all classes, sub-classes or only the
defining class. Some languages go further: Java uses the default access
modifier to restrict access also to classes in the same package, C# and VB.NET
reserve some members to classes in the same assembly using keywords internal
(C#) or Friend (VB.NET), and Eiffel and C++ allow one to specify which classes
may access any member.
* •
“Polymorphism”: Polymorphism allows the programmer to treat derived class
members just like their parent class’ members. More precisely, Polymorphism in
object-oriented programming is the ability of objects belonging to different
data types to respond to method calls of methods of the same name, each one
according to an appropriate type-specific behavior. One method, or an operator
such as +, -, or *, can be abstractly applied in many different situations. If
a Dog is commanded to speak(), this may elicit a bark(). However, if a Pig is
commanded to speak(), this may elicit an oink(). They both inherit speak()
from Animal, but their derived class methods override the methods of the
parent class; this is ”’Overriding Polymorphism”’. ”’Overloading
Polymorphism”’ is the use of one method signature, or one operator such as
“+”, to perform several different functions depending on the implementation.
The “+” operator, for example, may be used to perform integer addition, float
addition, list concatenation, or string concatenation. Any two subclasses of
Number, such as Integer and Double, are expected to add together properly in
an OOP language. The language must therefore overload the addition operator,
“+”, to work this way. This helps improve code readability. How this is
implemented varies from language to language, but most OOP languages support
at least some level of overloading polymorphism. Many OOP languages also
support ”’parametric polymorphism”’, where code is written without mention of
any specific type and thus can be used transparently with any number of new
types. The use of pointers to a superclass type later instantiated to an
object of a subclass is a simple yet powerful form of polymorphism, such as
used in C++.
The 1980s saw a rapid growth in interest in object-oriented programming. These
languages were imperative in style, but added features to support objects. The
last two decades of the 20th century saw the development of a considerable
number of such programming languages. Smalltalk-80, originally conceived by
Alan Kay in 1969, was released in 1980 by the Xerox Palo Alto Research Center.
Drawing from concepts in another object-oriented language – Simula (which is
considered to be the world’s first object-oriented programming language,
developed in the late 1960s) Bjarne Stroustrup designed C++, an object-
oriented language based on C. C++ was first implemented in 1985. In the late
1980s and 1990s, the notable imperative languages drawing on object-oriented
concepts were Perl, released by Larry Wall in 1987; Python, released by Guido
van Rossum in 1990; PHP, released by Rasmus Lerdorf in 1994; Java, first
released by Sun Microsystems in 1994 and Ruby, released in 1995 by Yukihiro
Matsumoto.
1. 1.
http://en.wikipedia.org/wiki/Object-oriented_programming
2. 2.
http://en.wikipedia.org/wiki/Class_(computer_science)
3. 3.
http://en.wikipedia.org/wiki/Instance_(programming)
4. 4.
http://en.wikipedia.org/wiki/Method_(computer_science)
5. 5.
http://en.wikipedia.org/wiki/Message_passing
6. 6.
http://en.wikipedia.org/wiki/Inheritance_(object-oriented_programming)
7. 7.
http://en.wikipedia.org/wiki/Abstraction_(computer_science)
8. 8.
http://en.wikipedia.org/wiki/Encapsulation_(computer_science)
9. 9.
http://en.wikipedia.org/wiki/Polymorphism_in_object-oriented_programming
##### Aspect-oriented programming
Aspect-oriented programming (AOP) is a programming paradigm in which secondary
or supporting functions are isolated from the main program’s business logic.
It aims to increase modularity by allowing the separation of cross-cutting
concerns, forming a basis for aspect-oriented software development.
Aspect-oriented programming entails breaking down a program into distinct
parts (so-called concerns, cohesive areas of functionality). All programming
paradigms support some level of grouping and encapsulation of concerns into
separate, independent entities by providing abstractions (e.g. procedures,
modules, classes, methods) that can be used for implementing, abstracting and
composing these concerns. But some concerns defy these forms of implementation
and are called crosscutting concerns because they “cut across” multiple
abstractions in a program. Logging is a common example of a crosscutting
concern because a logging strategy necessarily affects every single logged
part of the system. Logging thereby crosscuts all logged classes and methods.
All AOP implementations have some crosscutting expressions that encapsulate
each concern in one place. The difference between implementations lies in the
power, safety, and usability of the constructs provided. For example,
interceptors that specify the methods to intercept express a limited form of
crosscutting, without much support for type-safety or debugging. AspectJ [3]
has a number of such expressions and encapsulates them in a special class
type, called an aspect. For example, an aspect can alter the behavior of the
base code (the non-aspect part of a program) by applying advice (additional
behavior) at various join points (points in a program) specified in a
quantification or query called a pointcut (that detects whether a given join
point matches). An aspect can also make binary-compatible structural changes
to other classes, like adding members or parents.
The following are some standard terminology used in Aspect-oriented
programming:
* •
Cross-cutting concerns: Even though most classes in an OO model will perform a
single, specific function, they often share common, secondary requirements
with other classes. For example, we may want to add logging to classes within
the data-access layer and also to classes in the UI layer whenever a thread
enters or exits a method. Even though the primary functionality of each class
is very different, the code needed to perform the secondary functionality is
often identical.
* •
Advice: This is the additional code that you want to apply to your existing
model. In our example, this is the logging code that we want to apply whenever
the thread enters or exits a method.
* •
Pointcut: This is the term given to the point of execution in the application
at which cross-cutting concern needs to be applied. In our example, a pointcut
is reached when the thread enters a method, and another pointcut is reached
when the thread exits the method.
* •
Aspect: The combination of the pointcut and the advice is termed an aspect.
Programmers need to be able to read code and understand what is happening in
order to prevent errors. Even with proper education, understanding
crosscutting concerns can be difficult without proper support for visualizing
both static structure and the dynamic flow of a program. Visualizing
crosscutting concerns is just beginning to be supported in IDEs, as is support
for aspect code assist and refactoring. Given the power of AOP, if a
programmer makes a logical mistake in expressing crosscutting, it can lead to
widespread program failure. Conversely, another programmer may change the join
points in a program e.g., by renaming or moving methods in ways that were not
anticipated by the aspect writer, with unintended consequences. One advantage
of modularizing crosscutting concerns is enabling one programmer to affect the
entire system easily; as a result, such problems present as a conflict over
responsibility between two or more developers for a given failure. However,
the solution for these problems can be much easier in the presence of AOP,
since only the aspect need be changed, whereas the corresponding problems
without AOP can be much more spread out.
1. 1.
http://en.wikipedia.org/wiki/Aspect-oriented_programming
2. 2.
http://en.wikipedia.org/wiki/Aspect-Oriented_Software_Development
3. 3.
http://www.eclipse.org/aspectj/
4. 4.
http://www.sable.mcgill.ca/publications/techreports/sable-tr-2004-2.pdf
##### Reflection-oriented programming
Reflection-oriented programming, or reflective programming, is a functional
extension to the object-oriented programming paradigm. Reflection-oriented
programming includes self-examination, self-modification, and self-
replication. However, the emphasis of the reflection-oriented paradigm is
dynamic program modification, which can be determined and executed at runtime.
Some imperative approaches, such as procedural and object-oriented programming
paradigms, specify that there is an exact predetermined sequence of operations
with which to process data. The reflection-oriented programming paradigm,
however, adds that program instructions can be modified dynamically at runtime
and invoked in their modified state. That is, the program architecture itself
can be decided at runtime based upon the data, services, and specific
operations that are applicable at runtime.
Programming sequences can be classified in one of two ways, atomic or
compound. Atomic operations are those that can be viewed as completing in a
single, logical step, such as the addition of two numbers. Compound operations
are those that require a series of multiple atomic operations.
A compound statement, in classic procedural or object-oriented programming,
can lose its structure once it is compiled. The reflective programming
paradigm introduces the concept of meta-information, which keeps knowledge of
program structure. Meta-information stores information such as the name of the
contained methods, the name of the class, the name of parent classes, and/or
what the compound statement is supposed to do. Using this stored information,
as an object is consumed (processed), it can be reflected upon to find out the
operations that it supports. The operation that issues in the required state
via the desired state transition can be chosen at run-time without hard-coding
it.
Reflection can be used for observing and/or modifying program execution at
runtime. A reflection-oriented program component can monitor the execution of
an enclosure of code and can modify itself according to a desired goal related
to that enclosure. This is typically accomplished by dynamically assigning
program code at runtime [8].
Reflection can also be used to adapt a given program to different situations
dynamically. For example, consider an application that uses two different
classes X and Y interchangeably to perform similar operations. Without
reflection-oriented programming, the application might be hard-coded to call
method names of class X and class Y. However, using the reflection-oriented
programming paradigm, the application could be designed and written to utilize
reflection in order to invoke methods in classes X and Y without hard-coding
method names. Reflection-oriented programming almost always requires
additional knowledge, framework, relational mapping, and object relevance in
order to take advantage of more generic code execution. Hard-coding can be
avoided to the extent that reflection-oriented programming is used. Reflection
is also a key strategy for metaprogramming.
A language supporting reflection provides a number of features available at
runtime that would otherwise be very obscure or impossible to accomplish in a
lower-level language. Some of these features are the abilities to:
* •
Discover and modify source code constructions (such as code blocks, classes,
methods, protocols, etc.) as a first-class object at runtime.
* •
Convert a string matching the symbolic name of a class or function into a
reference to or invocation of that class or function.
* •
Evaluate a string as if it were a source code statement at runtime.
* •
Create a new interpreter for the language’s bytecode to give a new meaning or
purpose for a programming construct.
These features can be implemented in different ways. Compiled languages rely
on their runtime system to provide information about the source code. A
compiled Objective-C executable, for example, records the names of all methods
in a block of the executable, providing a table to correspond these with the
underlying methods (or selectors for these methods) compiled into the program.
In a compiled language that supports runtime creation of functions, such as
Common Lisp, the runtime environment must include a compiler or an
interpreter. Reflection can be implemented for languages not having built-in
reflection facilities by using a program transformation system to define
automated source code changes.
1. 1.
http://en.wikipedia.org/wiki/Reflection_%28computer_science%29
#### 2.2.4 Declarative programming
Declarative programming is a programming paradigm that expresses the logic of
a computation without describing its control flow. Many languages applying
this style attempt to minimize or eliminate side effects by describing what
the program should accomplish, rather than describing how to go about
accomplishing it. This is in contrast with imperative programming, which
requires an explicitly provided algorithm. In declarative programming, the
program is structured as a collection of properties to find in the expected
result, not as a procedure to follow. For example, given a database or a set
of rules, the computer tries to find a solution matching all the desired
properties. Common declarative languages include those of regular expressions,
logic programming, and functional programming. The archetypical example of a
declarative language is the fourth generation language SQL.
Declarative programming is often defined as any style of programming that is
not imperative. A number of other common definitions exist that attempt to
give the term a definition other than simply contrasting it with imperative
programming. For example:
* •
A program that describes what computation should be performed and not how to
compute it
* •
Any programming language that lacks side effects (or more specifically, is
referentially transparent)
* •
A language with a clear correspondence to mathematical logic.
##### Functional programming
Functional programming is a programming paradigm that treats computation as
the evaluation of mathematical functions and avoids state changes and mutable
data. It emphasizes the application of functions, in contrast to the
imperative programming style, which emphasizes changes in state. Functional
programming has its roots in the lambda calculus, a formal system developed in
the 1930s to investigate function definition, function application, and
recursion. Many functional programming languages can be viewed as elaborations
on the lambda calculus.
In practice, the difference between a mathematical function and the notion of
a “function” used in imperative programming is that imperative functions can
have side effects, changing the value of already calculated variables. Because
of this they lack referential transparency, i.e. the same language expression
can result in different values at different times depending on the state of
the executing program. Conversely, in functional code, the output value of a
function depends only on the arguments that are input to the function, so
calling a function f twice with the same value for an argument x will produce
the same result f(x) both times. Eliminating side-effects can make it much
easier to understand and predict the behavior of a program, which is one of
the key motivations for the development of functional programming.
Functional programming languages, especially purely functional ones, have
largely been emphasized in academia rather than in commercial software
development. However, prominent functional programming languages such as
Scheme, Erlang, OCaml, and Haskell, have been used in industrial and
commercial applications by a wide variety of organizations. Functional
programming also finds use in industry through domain-specific programming
languages like R (statistics), Mathematica (symbolic math), J and K (financial
analysis), and XSLT (XML). Widespread declarative domain specific languages
like SQL and Lex/Yacc, use some elements of functional programming, especially
in eschewing mutable values. Spreadsheets can also be viewed as functional
programming languages.
Programming in a functional style can also be accomplished in languages that
aren’t specifically designed for functional programming. For example, the
imperative Perl programming language has been the subject of a book describing
how to apply functional programming concepts. Javascript, one of the most
widely employed languages today, incorporates functional programming
capabilities.
John Backus presented the FP programming language in his 1977 Turing Award
lecture “Can Programming Be Liberated From the von Neumann Style? A Functional
Style and its Algebra of Programs”. He defines functional programs as being
built up in a hierarchical way by means of “combining forms” that allow an
“algebra of programs”; in modern language, this means that functional programs
follow the principle of compositionality. Backus’s paper popularized research
into functional programming, though it emphasized function-level programming
rather than the lambda-calculus style which has come to be associated with
functional programming.
In the 1970s the ML programming language was created by Robin Milner at the
University of Edinburgh, and David Turner developed initially the language
SASL at the University of St. Andrews and later the language Miranda at the
University of Kent. ML eventually developed into several dialects, the most
common of which are now Objective Caml and Standard ML. Also in the 1970s, the
development of the Scheme programming language (a partly-functional dialect of
Lisp), as described in the influential “Lambda Papers’ and the 1985 textbook
“Structure and Interpretation of Computer Programs’, brought awareness of the
power of functional programming to the wider programming-languages community.
The Haskell programming language was released in the late 1980s in an attempt
to gather together many ideas in functional programming research.
###### Higher-order functions
Most functional programming languages use higher-order functions, which are
functions that can either take other functions as arguments or return them as
results (the differential operator $d/dx$ that produces the derivative of a
function f is an example of this in calculus).
Higher-order functions are closely related to first-class functions, in that
higher-order functions and first-class functions both allow functions as
arguments and results of other functions. The distinction between the two is
subtle: “higher-order” describes a mathematical concept of functions that
operate on other functions, while “first-class” is a computer science term
that describes programming language entities that have no restriction on their
use (thus first-class functions can appear anywhere in the program that other
first-class entities like numbers can, including as arguments to other
functions and as their return values).
###### Pure functions
Purely functional functions (or expressions) have no memory or I/O side
effects. They represent a function whose valuation depends only on the value
of the parameters it is given. This means that pure functions have several
useful properties, many of which can be used to optimize the code:
* •
If the result of a pure expression is not used, it can be removed without
affecting other expressions.
* •
If a pure function is called with parameters that cause no side-effects, the
result is constant with respect to that parameter list (sometimes called
referential transparency), i.e. if the pure function is again called with the
same parameters, the same result will be returned (this can enable caching
optimizations).
* •
If there is no data dependency between two pure expressions, then their order
can be reversed, or they can be performed in parallel and they cannot
interfere with one another (in other terms, the evaluation of any pure
expression is thread-safe and enables parallel execution).
* •
If the entire language does not allow side-effects, then any evaluation
strategy can be used; this gives the compiler freedom to reorder or combine
the evaluation of expressions in a program.
The notion of pure function is central to code optimization in compilers.
While most compilers for imperative programming languages detect pure
functions, and perform common-subexpression elimination for pure function
calls, they cannot always do this for pre-compiled libraries, which generally
do not expose this information, thus preventing optimizations that involve
those external functions. Some compilers, such as gcc [21], add extra keywords
for a programmer to explicitly mark external functions as pure, to enable such
optimizations. Fortran 95 allows functions to be designated “pure” in order to
allow such optimizations.
###### Recursion
Iteration in functional languages is usually accomplished via recursion.
Recursion may require maintaining a stack, and thus may lead to inefficient
memory consumption, but tail recursion can be recognized and optimized by a
compiler into the same code used to implement iteration in imperative
languages. The Scheme programming language standard requires implementations
to recognize and optimize tail recursion. Tail recursion optimization can be
implemented by transforming the program into continuation passing style during
compilation, among other approaches. Common patterns of recursion can be
factored out using higher order functions, catamorphisms and anamorphisms,
which “folds” and “unfolds” a recursive function call nest by using higher-
order functions being the most obvious examples.
###### Eager vs Lazy evaluation
Functional languages can be categorized by whether they use strict (eager) or
non-strict (lazy) evaluation, concepts that refer to how function arguments
are processed when an expression is being evaluated. The technical difference
is in the denotational semantics of expressions containing failing or
divergent computations. Under strict evaluation, the evaluation of any term
containing a failing subterm will itself fail. For example, the expression
print length([2+1, 3*2, 1/0, 5-4])
will fail under eager evaluation because of the division by zero in the third
element of the list. Under lazy evaluation, the length function will return
the value 4 (the length of the list), since evaluating it will not attempt to
evaluate the terms making up the list. In brief, eager evaluation always fully
evaluates function arguments before invoking the function. Lazy evaluation
does not evaluate function arguments unless their values are required to
evaluate the function call itself. The usual implementation strategy for lazy
evaluation in functional languages is graph reduction, a technique first
developed by Chris Wadsworth in 1971. Lazy evaluation is used by default in
several pure functional languages, including Miranda, Clean and Haskell.
###### Type system
Especially since the development of Hindley-Milner (later Damas-Milner) type
inference in the 1970s, functional programming languages have tended to use
typed lambda calculus, as opposed to the untyped lambda calculus used in Lisp
and its variants (such as Scheme). Type inference, or implicit typing, refers
to the ability to deduce automatically the type of a value in a programming
language. It is a feature present in some strongly statically typed languages.
It is often characteristic of but not limited to functional programming
languages in general. Some languages that include type inference are: Ada,
BitC, Boo, C# 3.0, Cayenne, Clean, Cobra, D, Delphi, Epigram, F#, Haskell,
haXe, JavaFX Script, ML, Mythryl, Nemerle, OCaml, Oxygene, Scala, and Visual
Basic .NET 9.0. This feature is planned for Fortress, C++0x and Perl 6. The
ability to infer types automatically makes many programming tasks easier,
leaving the programmer free to omit type annotations while maintaining some
level of type safety.
###### Functional programming in non-functional languages
It is possible to employ a functional style of programming in languages that
are not traditionally considered functional languages. Some non-functional
languages have borrowed features such as higher-order functions, and list
comprehensions from functional programming languages. This makes it easier to
adopt a functional style when using these languages. Functional constructs
such as higher-order functions and lazy lists can be obtained in C++ via
libraries, such as in FC++ [12, 16]. In C, function pointers can be used to
get some of the effects of higher-order functions. For example the common
function map can be implemented using function pointers. In C# version 3.0 and
higher, lambda functions can be employed to write programs in a functional
style. Many object-oriented design patterns are expressible in functional
programming terms: for example, the Strategy pattern simply dictates use of a
higher-order function, and the Visitor pattern roughly corresponds to a
Catamorphism, or fold.
##### Logic programming
The logic programming paradigm views computation as automated reasoning over a
corpus of knowledge. Facts about the problem domain are expressed as logic
formulas, and programs are executed by applying inference rules over them
until an answer to the problem is found, or the collection of formulas is
proved inconsistent.
Logic programming refers to a paradigm that uses a form of symbolic logic as a
programming language, such as first-order logic. Imperative programming and
functional programming are essentially about implementing mappings (i.e.
functions). Having implemented a mapping M, we can make requests like the
following:
* •
Given $A$, determine the value of $M(A)$
A request like this will always have a single answer. Logic programming is
based on the notion that a program implements a relation rather than a
mapping. Consider two sets of values S and T. R is a relation between S and T
if, for every x in S and y in T, R(x,y) is either true or false. If R(x,y) is
true, we say that R holds between x and y. Logic programming is about
implementing relations. Having implemented a relation, we can make requests
like:
1. 1.
Given $A$ and $B$, determine whether $R(A,B)$ is true
2. 2.
Given $A$, find all $B$s such that $R(A,B)$ is true
3. 3.
Given $B$, find all $A$s such that $R(A,B)$ is true
4. 4.
Find all $A$s and $B$s for which $R(A,B)$ is true
A request like (1) will have a single answer, yes or no, but the other
requests (2)(3) and (4) might have many answers or none. These requests are
characteristic of logic programming, and explain why it is potentially higher
level than imperative or functional programming.
Robert Kowalski, one of the fathers of logic programming and automatic theorem
proving, introduced the property of logic programming models as
Logic Programming = Logic Statements + Control Strategy
where Logic Statements consist of a set of rules and facts expressing the
relationships between objects. Control Strategy is known as “resolution” or
the sequence of steps the deduction system chooses to answer a question from
the logic system expressed in the program. In other words, it refers to how a
logic language computes a response to a question. This indicates that the
original set of logic statements represent a knowledge base of the
computation, and the deductive system provides the control by which a new
logic statement may be derived. This property indicates that in the logic
programming system, a programmer needs to be concerned about the logic system
in question, but the control system is the evaluation strategy implemented by
the program evaluator and can be ignored from his consideration.
###### Evaluation
In a logic programming system, there is no indication in a program about how a
particular goal might be proved from a given set of predicates. Instead, the
evaluation strategy involves the use of “inference rules” that allow it to
construct a new set of logical statements that are proved to be true from a
given set of original logical statements that are already true. This result
indicates that the new set of logical statements (the original ones and the
inferred ones) can be viewed as representing the potential computation of all
logical consequences of a set of original statements. Hence, the essence of a
logic program is that from a collection of logical statements, known facts and
rules, a desired fact, known as a query or goal, might be proved to be true by
applying the inference rules. The two primary operations used by inference
rules to derive new facts are “resolution” and “unification”.
1. 1.
http://en.wikipedia.org/wiki/Logic_programming
## Chapter 3 Program Evaluation
In order to be executed, computer programs need to be translated into machine-
understandable code. In order to achieve this, the program to be executed must
first be analyzed to determine first if it is a valid program from the point
of view of its lexical conventions (lexical analysis), syntactical form
(syntactical analysis), and semantic validity (semantic analysis). During
analysis, the various components will correctly report errors found, allowing
the programmer to understand the location and nature of the errors found.
After the program has been confirmed to be valid, it is translated (code
generation) and possibly optimized (code optimization) before execution.
### 3.1 Program analysis and translation phases
#### 3.1.1 Front end
The front end analyzes the source code to build an internal representation of
the program, called the intermediate representation. It also manages the
symbol table, a data structure mapping each symbol in the source code to
associated information such as location, type and scope. This is done over
several phases, which includes some of the following:
##### Lexical analysis
In computer science, lexical analysis is the process of converting a sequence
of characters into a sequence of tokens. A program or function which performs
lexical analysis is called a lexical analyzer, lexer or scanner. A lexer often
exists as a single function, which is called by the parser. The lexical
specification of a programming language is defined by a set of rules which
defines the lexer, which are understood by a lexical analyzer generator such
as lex [19]. The lexical analyzer (either generated automatically by a tool
like lex, or hand-crafted) reads in a stream of characters, identifies the
lexemes in the stream, categorizes them into tokens, and outputs a token
stream. This is called “tokenizing.” If the lexer finds an invalid token, it
will report an error.
##### Syntactic analysis
Syntax analysis involves parsing the token sequence to identify the syntactic
structure of the program. The parser’s output is some form of intermediate
representation of the program’s structure, typically a parse tree, which
replaces the linear sequence of tokens with a tree structure built according
to the rules of a formal grammar, which define the language’s syntax. This is
usually done with reference to a context-free grammar, which recursively
defines components that can make up an expression and the order in which they
must appear. The parse tree is often analyzed, augmented, and transformed by
later phases in the compiler. Parsers are written by hand or generated by
parser generators, such as Yacc, Bison [20, 10], or JavaCC [23].
##### Semantic analysis
Semantic analysis is the phase in which the compiler adds semantic information
to the parse tree and builds the symbol table. This phase performs semantic
checks such as type checking (checking for type errors), or object binding
(associating variable and function references with their definitions), or
definite assignment (requiring all local variables to be initialized before
use), rejecting incorrect programs or issuing warnings. Semantic analysis
usually requires a complete parse tree, meaning that this phase logically
follows the parsing phase, and logically precedes the code generation phase,
though it is often possible to fold multiple phases into one pass over the
code in a compiler implementation. Not all rules defining programming
languages can be expressed by context-free grammars alone, for example
semantic validity such as type validity and proper declaration of identifiers.
These rules can be formally expressed with attribute grammars that implement
attribute migration across syntax tree nodes when necessary.
#### 3.1.2 Back end
The term back end is sometimes confused with code generator because of the
overlapped functionality of generating assembly code. Some literature uses
middle end to distinguish the generic analysis and optimization phases in the
back end from the machine-dependent code generators. The main phases of the
back end include the following:
##### Analysis
This is the gathering of program information from the intermediate
representation derived by the front end. Typical analyzes are data flow
analysis to build use-define chains, dependence analysis, alias analysis,
pointer analysis, etc. Accurate analysis is the basis for any compiler
optimization. The call graph and control flow graph are usually also built
during the analysis phase.
##### Optimization
The intermediate language representation is transformed into functionally
equivalent but faster (or smaller) forms. Popular optimizations are inline
expansion, dead code elimination, constant propagation, loop transformation,
register allocation or even automatic parallelization.
##### Code generation
The transformed intermediate language is translated into the output language,
usually the native machine language of the system. This involves resource and
storage decisions, such as deciding which variables to fit into registers and
memory and the selection and scheduling of appropriate machine instructions
along with their associated addressing modes.
### 3.2 Compilation vs. interpretation
#### 3.2.1 Compilation
A compiler is a computer program (or set of programs) that transforms source
code written in a computer language (the source language) into another
computer language (the target language, often having a binary form known as
object code). The most common reason for wanting to transform source code is
to create an executable program.
The name “compiler” is primarily used for programs that translate source code
from a high-level programming language to a lower level language (e.g.,
assembly language or machine code). A program that translates from a low level
language to a higher level one is a decompiler. A program that translates
between high-level languages is usually called a language translator, source
to source translator, or language converter. A language rewriter is usually a
program that translates the form of expressions without a change of language.
While a typical compiler outputs machine code from its final pass, there are
several other types:
##### Source-to-source compiler
A source-to-source compiler is a type of compiler that takes a high level
language as its input and outputs a high-level language. For example, an
automatic parallelizing compiler will frequently take in a high-level language
program as an input and then transform the code and annotate it with parallel
code annotations (e.g. OpenMP) or language constructs (e.g. Fortran’s DOALL
statements).
##### Stage compiler
A “stage compiler” is a compiler that compiles to assembly language of a
theoretical machine, like most Prolog implementations that use what is known
as the “Warren Abstract Machine” (or WAM), designed by David Warren in 1983.
Bytecode compilers for Java (executed on a Java Virtual Machine (JVM)) and
Python (executed on the CPython virtual machine), and many more are also a
subtype of this. In a sense, languages compiled in this manner are evaluated
in a hybrid compilation/interpretation mode, where the source code is compiled
into byte code, which is then interpreted at runtime.
##### Dynamic compilation
Dynamic compilation is a process used by some programming language
implementations to gain performance during program execution. Although the
technique originated in the Sun’s Self programming language, the best-known
language that uses this technique is Java. It allows optimizations to be made
that can only be known at runtime. Runtime environments using dynamic
compilation typically have programs run slowly for the first few minutes, and
then after that, most of the compilation and recompilation is done and it runs
faster as execution goes. Due to this initial performance lag, dynamic
compilation is undesirable in certain cases. In most implementations of
dynamic compilation, some optimizations that could be done at the initial
compile time are delayed until further compilation at runtime, causing further
unnecessary slowdowns. Just-in-time compilation is a form of dynamic
compilation.
In web development in this category, for example, JSP [18] pages and the
associated tag libraries e.g. within the Tomcat [2] container get first
automatically compiled into Java servlets (Java classes implemented the
Servlet API [17]) that then dynamically compiled using the standard Java
compiler into .class files.
##### Just-in-time compiler
Just-in-time compilation (JIT), also known as dynamic translation, is a
technique for improving the runtime performance of a computer program. JIT
builds upon two earlier ideas in run-time environments: bytecode compilation
and dynamic compilation. It converts code at runtime prior to executing it
natively, for example bytecode into native machine code. The performance
improvement over interpreters originates from caching the results of
translating blocks of code, and not simply reevaluating each line or operand
each time it is met, such as in classical interpretation. It also has
advantages over statically compiling the code at development time, as it can
recompile the code if this is found to be advantageous, and may be able to
enforce security guarantees. Thus JIT can combine some of the advantages of
interpretation and static (ahead-of-time) compilation. Several modern runtime
environments, such as Microsoft’s .NET Framework and most implementations of
Java, rely on JIT compilation for high-speed code execution.
In a bytecode-compiled system, source code is translated to an intermediate
representation known as bytecode . Bytecode is not the machine code for any
particular computer, and may be portable among computer architectures. The
bytecode may then be interpreted by, or run on, a virtual machine. A just-in-
time compiler can be used as a way to speed up execution of bytecode. At the
time the bytecode is run, the just-in-time compiler will compile some or all
of it to native machine code for better performance. This can be done per-
file, per-function or even on any arbitrary code fragment; the code can be
compiled when it is about to be executed (hence the name “just-in-time”).
In contrast, a traditional interpreted virtual machine will simply interpret
the bytecode, generally with much lower performance. Some interpreters even
interpret source code, without the step of first compiling to bytecode, with
even worse performance. Statically compiled code or native code is compiled
prior to deployment. A dynamic compilation environment is one in which the
compiler can be used during execution. For instance, most Common LISP systems
have a compile function which can compile new functions created at runtime.
This provides many of the advantages of JIT, but the programmer, rather than
the runtime system, is in control of what parts of the code are compiled. This
can also compile dynamically generated code, which can, in many scenarios,
provide substantial performance advantages over statically compiled code, as
well as over most JIT systems.
A common goal of using JIT techniques is to reach or surpass the performance
of static compilation, while maintaining the advantages of bytecode
interpretation: much of the “heavy lifting” of parsing the original source
code and performing basic optimization is often handled at compile time, prior
to deployment: compilation from bytecode to machine code is much faster than
compiling from source. The deployed bytecode is portable, unlike native code.
Compilers from bytecode to machine code are easier to write, because the
portable bytecode compiler has already done much of the analysis work.
JIT techniques generally offer far better performance than interpreters. In
addition, it can in some or many cases offer better performance than static
compilation, as many optimizations are only feasible at run-time, such as:
* •
The compilation can be optimized to the targeted CPU and the operating system
model where the application runs.
* •
The system is able to collect statistics about how the program is actually
running in the environment it is in, and it can rearrange and recompile for
optimum performance.
* •
The system can do global code optimizations (e.g. inlining of library
functions) without losing the advantages of dynamic linking and without the
overheads inherent to static compilers and linkers.
* •
Although this is possible with statically compiled garbage collected
languages, a bytecode system can more easily rearrange memory for better cache
utilization.
##### Ahead-of-time compile
Ahead-of-time compilation (AOT) refers to the act of compiling an intermediate
language, such as Java bytecode or .NET Common Intermediate Language
(CIL[CIL]), into a system-dependent binary. Most languages that can be
compiled to an intermediate language (such as bytecode) take advantage of
just-in-time compilation. JIT compiles intermediate code into binary code for
a native run while the intermediate code is executing, which may decrease an
application’s performance. Ahead-of-time compilation eliminates the need for
this step by performing the compilation before execution rather than during
execution.
#### 3.2.2 Interpretation
An interpreted language is a programming language whose programs are not
directly executed by the host cpu but rather executed (or said to be
interpreted) by a software program known as an interpreter. The source code of
the program is often translated to a form that is more convenient to
interpret, which may be some form of machine language for a virtual machine
(such as Java’s bytecode). Theoretically, any language may be compiled or
interpreted, so this designation is applied purely because of common
implementation practice and not some underlying property of a language.
Many languages have been implemented using both compilers and interpreters,
including LISP, Pascal, C, BASIC, and Python. While Java is translated to a
form that is intended to be interpreted, just-in-time compilation is often
used to generate machine code at run time. The Microsoft’s .NET languages
compile to CIL (Microsoft’s Common Intermediate Language) which is often then
compiled into native machine code; however, there is a virtual machine capable
of interpreting CIL. Many LISP implementations can freely mix interpreted and
compiled code. These implementations also use a compiler that can translate
arbitrary source code at runtime to machine code.
In the early days of computing, language design was heavily influenced by the
decision to use compilation or interpretation as a mode of execution. For
example, some compiled languages require that programs must explicitly state
the data-type of a variable at the time it is declared or first used while
some interpreted languages take advantage of the dynamic aspects of
interpretation to make such declarations unnecessary. For example, Smalltalk
80, which was designed to be interpreted at run-time, allows generic Objects
to dynamically interact with each other.
Initially, interpreted languages were compiled line-by-line; that is, each
line was compiled as it was about to be executed, and if a loop or subroutine
caused certain lines to be executed multiple times, they would be recompiled
every time. This has become much less common. Most so-called interpreted
languages use an intermediate representation, which combines both compilation
and interpretation. In this case, a compiler may output some form of bytecode
or threaded code, which is then executed by a bytecode interpreter. Examples
include Python, Java, and Ruby. The intermediate representation can be
compiled once and for all (as in Java), each time before execution (as in Perl
or Ruby), or each time a change in the source is detected before execution (as
in Python).
Interpreting a language gives implementations some additional flexibility over
compiled implementations. Features that are often easier to implement in
interpreters than in compilers include (but are not limited to):
* •
platform independence (Java’s bytecode, for example)
* •
reflection and reflective usage of the evaluator
* •
dynamic typing
* •
smaller executable program size (since implementations have flexibility to
choose the instruction code)
* •
dynamic scoping
The main disadvantage of interpreting is a much slower speed of program
execution compared to direct machine code execution on the host CPU. A
technique used to improve performance is just-in-time compilation which
converts frequently executed sequences of interpreted instruction to host
machine code.
#### 3.2.3 Subreferences
1. 1.
http://en.wikipedia.org/wiki/Semantic_analysis_(computer_science)
2. 2.
http://en.wikipedia.org/wiki/Lexical_analysis
3. 3.
http://en.wikipedia.org/wiki/Parsing
4. 4.
http://en.wikipedia.org/wiki/Interpreted_language
5. 5.
http://en.wikipedia.org/wiki/Dynamic_compilation
6. 6.
http://en.wikipedia.org/wiki/Just-in-time_compilation
7. 7.
http://en.wikipedia.org/wiki/Compiler
### 3.3 Type System
A type system is a framework for classifying programming languages’ phrases
according to the kinds of values they compute. A type system associates a type
with each computed value. By examining the flow of these values, a type system
attempts to prove that no type errors can occur in a given program. The type
system in question determines what constitutes a type error, but a type system
generally seeks to guarantee that operations expecting a certain kind of value
are not used with values for which that operation makes no sense.
Technically, assigning data types (i.e. typing) gives meaning to collections
of bits in the computer’s memory. Types usually have associations either with
values in memory or with objects such as variables. Because any value simply
consists of a sequence of bits in a computer, hardware makes no intrinsic
distinction even between memory addresses, instruction code, characters,
integers and floating-point numbers, being unable to discriminate between them
based on bit pattern alone. Associating a sequence of bits and a type informs
programs and programmers how that sequence of bits should be understood, i.e.
it gives semantic information to the values manipulated by a computer program.
Major functions provided by type systems include:
* •
Safety – Use of types may allow a compiler to detect meaningless or probably
invalid code. For example, we can identify an expression:
3 / ‘‘Hello, World’’
as invalid (depending on the language) because the rules of arithmetic do not
specify how to divide an integer by a string. As discussed below, strong
typing offers more safety, but generally does not guarantee complete safety.
* •
Optimization – Static type-checking may provide useful compile-time
information. For example, if a type requires that a value must align in memory
at a multiple of 4 bytes, the compiler may be able to use more efficient
machine instructions.
* •
Documentation – In more expressive type systems, types can serve as a form of
documentation, since they can illustrate the intent of the programmer. For
instance, timestamps may be represented as integers, but if a programmer
declares a function as returning a timestamp type rather than merely an
integer type, this documents part of the meaning of the function.
* •
Abstraction – Types allow programmers to think about programs at a higher
level than the bit or byte, not bothering with low-level implementation. For
example, programmers can think of a string as a collection of character values
instead of as a mere array of bytes. Or, types can allow programmers to
express the interface between two subsystems. This helps localize the
definitions required for interoperability of the subsystems and prevents
inconsistencies when those subsystems communicate.
#### 3.3.1 Type checking
The process of verifying and enforcing the constraints of types – type
checking – may occur either at compile-time (a static check) or run-time (a
dynamic check). If a language specification requires its typing rules strongly
(i.e. more or less allowing only those automatic type conversions which do not
lose information), one can refer to the process as strongly typed, if not, as
weakly typed.
##### Static typing
A programming language is said to use static typing when type checking is
performed during compile-time as opposed to run-time. Statically typed
languages include Ada, C, C++, C#, F#, Java, Fortran, Haskell, ML, Pascal,
Perl, Objective-C and Scala. Static typing is a limited form of program
verification. It allows many type errors to be caught early in the development
cycle. Static type checkers evaluate only the type information that can be
determined at compile time, but are able to verify that the checked conditions
hold for all possible executions of the program, which eliminates the need to
repeat type checks every time the program is executed. Program execution may
also be made more efficient (i.e. faster or taking reduced memory) by omitting
runtime type checks and enabling other optimizations.
Because they evaluate type information during compilation, and therefore lack
type information that is only available at run-time, static type checkers are
conservative. They will reject some programs that may be well-behaved at run-
time, but that cannot be statically determined to be well-typed. For example,
even if an expression <complex test> always evaluates to true at run-time, a
program containing the code
if <complex test> then 42 else <type error>
will be rejected as ill-typed, because a static analysis cannot determine that
the else branch won’t be taken. The conservative behaviour of static type
checkers is advantageous when <complex test> evaluates to false infrequently:
a static type checker can detect type errors in rarely used code paths.
Note that the most widely used statically typed languages are not formally
type safe. They have “loopholes” in the programming language specification
enabling programmers to write code that circumvents the verification performed
by a static type checker. For example, most C-style languages have type
coercion, and Haskell has such features as unsafePerformIO: such operations
may be unsafe at runtime, in that they can cause unwanted behaviour due to
incorrect typing of values when the program runs.
##### Dynamic typing
A programming language is said to be dynamically typed when the majority of
its type checking is performed at run-time as opposed to at compile-time. In
dynamic typing, values have types but variables do not; that is, a variable
can refer to a value of any type. Dynamically typed languages include Erlang,
Groovy, JavaScript, LISP, Objective-C, Perl (with respect to user-defined
types but not built-in types), PHP, Prolog, Python, Ruby and Smalltalk.
Compared to static typing, dynamic typing can be more flexible (e.g. by
allowing programs to generate types and functionality based on run-time data),
though at the expense of fewer a priori typing guarantees. This is because a
dynamically typed language accepts and attempts to execute some programs which
may be ruled as invalid by a static type checker.
Dynamic typing may result in runtime type errors that is, at runtime, a value
may have an unexpected type, and an operation nonsensical for that type is
applied. This operation may occur long after the place where the programming
mistake was made that is, the place where the wrong type of data passed into a
place it should not have. This may make the bug difficult to locate.
Dynamically typed language systems, compared to their statically typed
cousins, make fewer “compile-time” checks on the source code (but will check,
for example, that the program is syntactically correct). Run-time checks can
potentially be more sophisticated, since they can use dynamic information as
well as any information that was present during compilation. On the other
hand, runtime checks only assert that conditions hold in a particular
execution of the program, and these checks are repeated for every execution of
the program.
Development in dynamically typed languages is often supported by programming
practices such as unit testing. Testing is a key practice in professional
software development, and is particularly important in dynamically typed
languages. In practice, the testing done to ensure correct program operation
can detect a much wider range of errors than static type-checking, but
conversely cannot search as comprehensively for the errors that both testing
and static type checking are able to detect.
##### Duck typing
Duck typing is a style of dynamic typing in which an object’s current set of
methods and properties determines the valid semantics, rather than its
inheritance from a particular class or implementation of a specific interface.
The name of the concept refers to the duck test, attributed to American writer
James Whitcomb Riley, which may be phrased as follows:
> “when I see a bird that walks like a duck and swims like a duck and quacks
> like a duck, I call that bird a duck.”
In duck typing, one is concerned with just those aspects of an object that are
used, rather than with the type of the object itself. For example, in a non-
duck-typed language, one can create a function that takes an object of type
Duck and calls that object’s walk() and quack() methods. In a duck-typed
language, the equivalent function would take an object of any type and call
that object’s walk() and quack() methods. If the object does not have the
methods that are called then the function signals a run-time error. It is this
action of any object having the correct walk() and quack() methods being
accepted by the function that evokes the quotation and hence the name of this
form of typing.
Duck typing is aided by habitually not testing for the type of arguments in
method and function bodies, relying on documentation, clear code, and unit
testing to ensure correct use. Users of statically typed languages new to
dynamically typed languages are usually tempted to add such static (before
run-time) type checks, defeating the benefits and flexibility of duck typing,
and constraining the language’s dynamism.
##### Structural type system
A structural type system (or property-based type system) is a major class of
type systems, in which type compatibility and equivalence are determined by
the type’s structure, and not through explicit declarations. Structural
systems are used to determine if types are equivalent, as well as if a type is
a subtype of another. It contrasts with nominative systems (see Section
3.3.1), where comparisons are based on explicit declarations or the names of
the types, and duck typing (see Section 3.3.1), in which only the part f the
structure accessed at runtime is checked for compatibility.
In structural typing, two objects or terms are considered to have compatible
types if the types have identical structure. Depending on the semantics of the
language, this generally means that for each feature within a type, there must
be a corresponding and identical feature in the other type. Some languages may
differ on the details (such as whether the features must match in name).
ML and Objective Caml are examples of structurally-typed languages. C++
template functions exhibit structural typing on type arguments.
In languages, which support subtype polymorphism, a similar dichotomy can be
formed based on how the subtype relationship is defined. One type is a subtype
of another if and only if it contains all the features of the base type (or
subtypes thereof); the subtype may contain additional features (such as
members not present in the base type, or stronger invariants).
A pitfall of structural typing versus nominative typing is that two separately
defined types intended for different purposes, each consisting of a pair of
numbers, could be considered the same type by the type system, simply because
they happen to have identical structure. One way this can be avoided is by
creating one algebraic data type for one use of the pair and another algebraic
data type for the other use.
##### Nominative type system
A nominal or nominative type system (or name-based type system) is a major
class of type system, in which compatibility and equivalence of data types is
determined by explicit declarations and/or the name of the types. Nominative
systems are used to determine if types are equivalent, as well as if a type is
a subtype of another. It contrasts with structural systems, where comparisons
are based on the structure of the types in question and do not require
explicit declarations.
Nominal typing means that two variables are type-compatible if and only if
their declarations name the same type. For example, in C, two struct types
with different names are never considered compatible, even if they have
identical field declarations. However, C also allows a typedef declaration,
which introduces an alias for an existing type. These are merely syntactical
and do not differentiate the type from its alias for the purpose of type
checking.
In a similar fashion, nominal subtyping means that one type is a subtype of
another if and only if it is explicitly declared to be so in its definition.
Nominally-typed languages typically enforce the requirement that declared
subtypes be structurally compatible (though Eiffel allows non-compatible
subtypes to be declared). However, subtypes which are structurally compatible
“by accident”, but not declared as subtypes, are not considered to be
subtypes. C, C++, C# and Java all primarily use both nominal typing and
nominal subtyping.
Some nominally-subtyped languages, such as Java and C#, allow classes to be
declared final (or sealed in C# terminology), indicating that no further
subtyping is permitted.
Nominal typing is useful at preventing accidental type equivalence, and is
considered to have better type-safety than structural typing. The cost is a
reduced flexibility, as, for example, nominal typing does not allow new super-
types to be created without modification of the existing subtypes.
1. 1.
http://en.wikipedia.org/wiki/Type_system
2. 2.
http://en.wikipedia.org/wiki/Duck_typing
3. 3.
http://en.wikipedia.org/wiki/Dynamic_typing
4. 4.
http://en.wikipedia.org/wiki/Type_safety
5. 5.
http://en.wikipedia.org/wiki/Structural_type_system
6. 6.
http://en.wikipedia.org/wiki/Nominative_type_system
### 3.4 Memory management
Memory management is the act of managing computer memory. In its simpler
forms, this involves providing ways to allocate portions of memory to programs
at their request, and freeing it for reuse when no longer needed. The
management of main memory is critical to the computer system.
Garbage collection is the automated allocation, and deallocation of computer
memory resources for a program. This is generally implemented at the
programming language level and is in opposition to manual memory management,
the explicit allocation and deallocation of computer memory resources.
#### 3.4.1 Garbage collection
Garbage collection is a form of automatic memory management. It is a special
case of resource management, in which the limited resource being managed is
memory. The garbage collector attempts to reclaim garbage, or memory occupied
by objects that are no longer in use by the program. Garbage collection was
invented by John McCarthy around 1959 to solve problems in Lisp.
Garbage collection is often portrayed as the opposite of manual memory
management, which requires the programmer to specify which objects to
deallocate and return to the memory system. However, many systems use a
combination of the two approaches, and other techniques such as stack
allocation and region inference using syntactical program blocks can carve off
parts of the problem.
Garbage collection does not traditionally manage limited resources other than
memory that typical programs use, such as network sockets, database handles,
user interaction windows, and file and device descriptors. Methods used to
manage such resources, particularly destructors, may suffice as well to manage
memory, leaving no need for garbage collection. Some garbage collection
systems allow such other resources to be associated with a region of memory
that, when collected, causes the other resource to be reclaimed; this is
called finalization. Finalization may introduce complications limiting its
usability, such as intolerable latency between disuse and reclaim of
especially limited resources.
A finalizer is a special method that is executed when an object is garbage
collected. It is similar in function to a destructor. In less technical terms,
a finalizer is a piece of code that ensures that certain necessary actions are
taken when an acquired resource (such as a file or access to a hardware
device) is no longer being used. This could be closing the file or signaling
to the operating system that the hardware device is no longer needed. However,
as noted below, finalizers are not the preferred way to accomplish this and
for the most part serve as a fail-safe. Unlike destructors, finalizers are not
deterministic. A destructor is run when the program explicitly frees an
object. A finalizer, by contrast, is executed when the internal garbage
collection system frees the object. Programming languages which use finalizers
include Java and C#. In C#, and a few others which support finalizers, the
syntax for declaring a finalizer mimics that of destructors in C++.
Due to the lack of programmer control over their execution, it is usually
recommended to avoid finalizers for any but the most trivial operations. In
particular, operations often performed in destructors are not usually
appropriate for finalizers. For example, destructors are often used to free
expensive resources such as file or network handles. If placed in a finalizer,
the resources may remain in use for long periods of time after the program is
finished with them. Instead, most languages encourage the dispose pattern
whereby the object has a method to clean up the object’s resources, leaving
the finalizer as a fail-safe in the case where the dispose method doesn’t get
called. The C# language supports the dispose pattern explicitly, via the
IDisposable interface.
The basic principles of garbage collection are:
1. 1.
Find data objects in a program that cannot be accessed in the future
2. 2.
Reclaim the resources used by those objects
By making manual memory deallocation unnecessary (and often forbidding it),
garbage collection frees the programmer from having to worry about releasing
objects that are no longer needed, which can otherwise consume a significant
amount of design effort. It also aids programmers in their efforts to make
programs more stable, because it prevents several classes of runtime errors.
For example, it prevents dangling pointer errors, where a reference to a
deallocated object is used. The pointer still points to the location in memory
where the object or data was, even though the object or data has since been
deleted and the memory may now be used for other purposes. This can, and often
does, lead to storage violation errors that are extremely difficult to
resolve.
Many computer languages require garbage collection, either as part of the
language specification (e.g., Java, C#, and most scripting languages) or
effectively for practical implementation (e.g., formal languages like lambda
calculus); these are said to be garbage collected languages. Other languages
were designed for use with manual memory management, but have garbage
collected implementations available (e.g., C, C++). Some languages, like Ada,
Modula-3, and C++ allow both garbage collection and manual memory management
to co-exist in the same application by using separate heaps for collected and
manually managed objects; others, like D, are garbage collected but allow the
user to manually delete objects and also entirely disable garbage collection
when speed is required.
###### Benefits
Garbage collection frees the programmer from manually dealing with memory
allocation and deallocation. As a result, certain categories of bugs are
eliminated or substantially reduced:
* •
Dangling pointer bugs, which occur when a piece of memory is freed while there
are still pointers to it, and one of those pointers is then used.
* •
Double free bugs, which occur when the program attempts to free a region of
memory that is already free.
* •
Certain kinds of memory leaks, in which a program fails to free memory
occupied by objects that will not be used again, leading, over time, to memory
exhaustion.
###### Disadvantages
Typically, garbage collection has certain disadvantages.
* •
Garbage collection is a process that consumes limited computing resources in
deciding what memory is to be freed and when, reconstructing facts that may
have been known to the programmer. The penalty for the convenience of not
annotating memory usage manually in the code is overhead leading, potentially,
to decreased performance.
* •
The point when the garbage is actually collected can be unpredictable,
resulting in delays scattered throughout a session. Unpredictable delays can
be unacceptable in real-time environments such as device drivers, or in
transaction processing. Recursive algorithms that take advantage of automatic
storage management often delay automatic release of stack objects until after
the final call has completed, causing increased memory requirements.
* •
Memory may leak despite the presence of a garbage collector if references to
unused objects are not themselves manually disposed of. Researchers draw a
distinction between “physical” and “logical” memory leaks. In a physical
memory leak, the last pointer to a region of allocated memory is removed, but
the memory is not freed. In a logical memory leak, a region of memory is still
referenced by a pointer, but is never actually used.[4] Garbage collectors
generally can do nothing about logical memory leaks.
* •
Perhaps the most significant problem is that programs that rely on garbage
collectors often exhibit poor locality (interacting badly with cache and
virtual memory systems), occupying more address space than the program
actually uses at any one time, and touching otherwise idle pages. These may
combine in a phenomenon called thrashing, in which a program spends more time
copying data between various grades of storage than performing useful work.
They may make it impossible for a programmer to reason about the performance
effects of design choices, making performance tuning difficult. They can lead
garbage-collecting programs to interfere with other programs competing for
resources.
* •
The execution of a program using a garbage collector is not deterministic. An
object which becomes eligible for garbage collection will usually be cleaned
up eventually, but there is no guarantee when (or even if) that will happen.
Most run-time systems using garbage collectors require manual deallocation of
limited non-memory resources (using finalizers), as an automatic deallocation
during the garbage collection phase may run too late or in the wrong
circumstances. Also, the performance impact caused by the garbage collector is
seemingly random and hard to predict, leading to program execution that non-
deterministic.
Generally speaking, higher-level programming languages are more likely to have
garbage collection as a standard feature. In languages that do not have built
in garbage collection, it can often be added through a library, as with the
Boehm garbage collector for C and C++. This approach is not without drawbacks,
such as changing object creation and destruction mechanisms.
Most functional programming languages, such as ML, Haskell, and APL, have
garbage collection built in. Lisp, which introduced functional programming, is
especially notable for introducing this mechanism.
Other dynamic languages, such as Ruby (but not Perl, or PHP, which use
reference counting), also tend to use garbage collection. Object-oriented
programming languages such as Smalltalk and Java usually provide integrated
garbage collection. A notable exception is C++ which, uniquely, relies on
destructors [22].
1. 1.
http://en.wikipedia.org/wiki/Garbage_collection_%28computer_science%29
2. 2.
http://en.wikipedia.org/wiki/Finalizer
#### 3.4.2 Manual memory management
Manual memory management refers to the usage of manual instructions by the
programmer to identify and deallocate unused objects, or garbage. Up until the
mid 1990s, the majority of programming languages used in industry supported
manual memory management. Today, however, languages with garbage collection
are becoming increasingly popular; the main manually-managed languages still
in widespread use today are C and C++.
Most programming languages use manual techniques to determine when to allocate
a new object from the free store. C uses the malloc() function; C++ and Java
use the new operator; determination of when an object ought to be created is
trivial and unproblematic. The fundamental issue is determination of when an
object is no longer needed (ie. is garbage), and arranging for its underlying
storage to be returned to the free store so that it may be re-used to satisfy
future memory requests. In manual memory allocation, this is also specified
manually by the programmer; via functions such as free() in C, or the delete
operator in C++. Manual memory management is known to enable several major
classes of bugs into a program, when used incorrectly:
* •
When an unused object is never released back to the free store, this is known
as a memory leak. In some cases, memory leaks may be tolerable, such as a
program which “leaks” a bounded amount of memory over its lifetime, or a
short-running program which relies on an operating system to deallocate its
resources when it terminates. However, in many cases memory leaks occur in
long-running programs, and in such cases an unbounded amount of memory is
leaked. When this occurs, the size of the available free store continues to
decrease over time; when it finally is exhausted the program then crashes.
* •
When an object is deleted more than once, or when the programmer attempts to
release a pointer to an object not allocated from the free store, catastrophic
failure of the dynamic memory management system can result. The result of such
actions can include heap corruption, premature destruction of a different (and
newly-created) object which happens to occupy the same location in memory as
the multiply-deleted object, and other forms of undefined behavior.
* •
Pointers to deleted objects become wild pointers if used post-deletion;
attempting to use such pointers can result in difficult-to-diagnose bugs.
Languages which exclusively use garbage collection are known to avoid the last
two classes of defects. Memory leaks can still occur (and bounded leaks
frequently occur with generational or conservative garbage collection), but
are generally less severe than memory leaks in manual memory allocation
schemes.
Manual memory management has one correctness advantage, which comes into play
when objects own scarce system resources (like graphics resources, file
handles, or database connections) which must be relinquished when an object is
destroyed. Languages with manual management, via the use of destructors, can
arrange for such actions to occur at the precise time of object destruction.
In C++, this ability is put to further use to automate memory deallocation
within an otherwise-manual framework, use of the auto ptr template in the
language’s standard library to perform memory management is a common paradigm.
Many advocates of manual memory management argue that it affords superior
performance when compared to automatic techniques such as garbage collection.
Manual allocation does not suffer from the long “pause” times often associated
with garbage collection (although modern garbage collectors have collection
cycles which are often not noticeable), and manual allocation frequently has
superior locality of reference. This is especially an issue in real time
systems, where unbounded collection cycles are generally unacceptable. Manual
allocation is also known to be more appropriate for systems where memory is a
scarce resource.
On the other hand, manual management has documented performance disadvantages:
* •
Calls to delete and such incur an overhead each time they are made, this
overhead can be amortized in garbage collection cycles. This is especially
true of multithreaded applications, where delete calls must be synchronized.
* •
The allocation routine may be more complicated, and slower. Some garbage
collection schemes, such as those with heap compaction, can maintain the free
store as a simple array of memory (as opposed to the complicated
implementations required by manual management schemes).
1. 1.
http://en.wikipedia.org/wiki/Manual_memory_management
## Chapter 4 Programming Languages Popularity
Since the beginning of the computer, thousands of programming languages have
been designed. However, a relatively few proportion of these were actually put
to practical use in the industry. Some languages have been developed and/or
adopted by the academic community for teaching purposes but did not really
come into practical use either. Due to their adoption in University computer
science programs, such languages can boast great popularity, but still failed
to have become popular for extensive industrial software development. Many
factors or indicators can be used to measure the popularity of programming
languages. Here we discuss various surveys that have been made to measure the
popularity of programming languages.
### 4.1 langpop.com
The methodology and results presented here are based on:
http://www.langpop.com/. It uses various web sites as a data source to extract
indicators of programming languages’ popularity rankings.
#### 4.1.1 Yahoo Search
Arguably, the popularity of a programming language can be measured by the
number of web pages discussing it. Yahoo provides an API for doing customized
searches on its web pages indexing database. Searching for various programming
languages’ names, http://www.langpop.com/ arrives to the conclusion that the
programming languages most referred to on the web are: C++, C, Java, and PHP,
as shown on the extracted figure:
#### 4.1.2 Job Postings
Since industry demand is probably a more relevant indicator of a language’s
popularity, one might search a job offerings database and search for specific
programming languages being asked for in job offerings. Based on a search of
craigslist.org, http://www.langpop.com/ arrives to the conclusion that PHP,
SQL, C, C++, Java, JavaScript, and C# are the most in-demand languages, as
shown on the extracted figure:
#### 4.1.3 Books
Arguably, the more a programming language is popular, the more books are being
sold explaining how to use it. In http://www.langpop.com/, a search is made to
a bookstore database to figure out what languages are more popular as books’
topics. It arrives to the conclusion that Java, C++, Visual Basic, C#, and C
are the most popular from this regard, as shown on the extracted figure:
#### 4.1.4 Open source code repositories
The open source code development community provides a public a free source of
information that can be used to measure the popularity of programming
languages effectively used by developers. In http://www.langpop.com/, a search
is made on various open source code repositories and the programming language
used in each open source project is extracted.
##### freshmeat.org
While searching the freshmeat.net open source community web site, it showed
that C, Java, C++, PHP, Perl, and Python are the most popular, as shown in the
extracted figure:
##### Google code search
Doing a similar search on the Google code web site
(http://www.google.com/codesearch), it was extracted that (as for the results
for freshmeat.org) C, Java, C++, PHP, Perl, and Python are the most popular
languages.
##### Ohloh
Doing yet another similar search on the Ohloh web site (www.ohloh.net) [1], it
was extracted that, quite differently from the two previous results, Java, C,
C++, JavaScript, Shell, Python, and PHP are the most popular languages, as
shown in the figure:
#### 4.1.5 Discussion sites
Another relevant source of information to measure the relative popularity of
programming languages is public discussion sites, where programmers discuss
technical issues among themselves. In http://www.langpop.com/, the following
discussion sites were searched for: lambda-the-ultimate.org, slashdot.org,
programming.reddit.com, and freenode.net. Overall, the results of these
searches revealed that Java, C, C++, C#, PHP and Python are the languages that
the programmers communities are the most discussing among themselves, as shown
in the extracted figure:
.
Notably in the search on public discussion sites, lambda-the-ultimate.org is a
site that is very popular particularly with the academia circles. Searches on
this site has revealed a different taste in academia, showing that languages
like Haskell, Lisp, Scheme, Python, Erlang, Ruby, Scala, Smalltalk, and Ada
are popular discussion topics in academic circles, as shown in the extracted
figure:
.
### 4.2 www.tiobe.com
The results presented here are based on a monthly survey done by Tiobe
Software (www.tiobe.com). As stated on the “Tiobe index” web page updated
monthly, the TIOBE Programming Community index gives an indication of the
popularity of programming languages. The index is updated once a month. The
ratings are based on the number of skilled engineers world-wide, courses and
third party vendors. The popular search engines Google, MSN, Yahoo!, Wikipedia
and YouTube are used to calculate the ratings. Observe that the TIOBE index is
not about the best programming language or the language in which most lines of
code have been written. The data presented here is as of March 2010.
#### 4.2.1 Top 20 programming languages (March 2010)
According to the TIOBE index (March 2010), the top 20 most popular programming
languages are depicted as in the extracted figure:
.
#### 4.2.2 Long-term trend
As the TIOBE index is calculated monthly, and has been there for a long time,
it is very interesting to see long-term trends in the history of programming
languages popularity, as depicted in the extracted figure:
.
#### 4.2.3 Categories of programming languages
The TIOBE index also enables the categorization of popular programming
languages. The results are as depicted in the extracted figure:
.
## Chapter 5 Programming Languages Performance Ranking
In order to perform a relative evaluation of programming languages other than
based on their popularity, the execution performance of programs can be
exercised and compared. This requires the design of algorithmically comparable
solutions in various languages and executing these programs in the same
execution environment.
Many such empirical studies have been made to compare programming languages.
Notably, the Computer Languages Benchmark Game [1]
(http://shootout.alioth.debian.org/) provides a dynamic web site demonstrating
extensive empirical results enabling the comparison of various programming
languages using a wide variety of benchmarking programs.
Among the most revealing comparisons that can be found on this site is the
following: [2] (http://shootout.alioth.debian.org/u32q/which-language-is-
best.php). It enables the comparison of programming languages according to
performance criteria such as: execution time, memory consumption, and source
code size.
### 5.1 Execution time
### 5.2 Memory consumption
### 5.3 Source code size
### 5.4 Overall results
## Bibliography
* [1] Jason Allen, Scott Collison, Robin Luckey, and Andy Verprauskus. Ohloh. ohloh.net, 2008. http://www.ohloh.net/, viewed in January 2008.
* [2] Apache Foundation. Apache Jakarta Tomcat. [online], 1999–2010. http://jakarta.apache.org/tomcat/index.html.
* [3] AspectJ Contributors. AspectJ: Crosscutting Objects for Better Modularity. eclipse.org, 2007. http://www.eclipse.org/aspectj/.
* [4] Autodesk. Maya. [digital], 2008–2010. autodesk.com.
* [5] Blender Foundation. Blender. [online], 2008–2010. http://www.blender.org.
* [6] Fancois Bourdoncle and Stephan Merz. On the integration of the functional programming, class-based object-oriented programming, and multi-methods. Technical report, Centre de Mathemathiques Appliquees, Ecole des Mines de Paris and Institut fur Informatik, Technische Universitat Munchen, October 1996.
* [7] Robert G. Clark. Comparative Programming Languages. Addison-Wesley, 3 edition, November 2000. ISBN: 978-0201710120.
* [8] Dale Green. Java reflection API. Sun Microsystems, Inc., 2001–2005. http://java.sun.com/docs/books/tutorial/reflect/index.html.
* [9] P. J. Landin. The next 700 programming languages. Communications of the ACM, 9(3):157–166, 1966.
* [10] Kenneth C. Louden. Compiler Construction: Principles and Practice. PWS Publishing Company, 1997. ISBN 0-564-93972-4.
* [11] Lee E. McMahon, Paolo Bonzini, Aur’elio M. Jargas, Eric Pement, Tilmann Bitterberg, Yao-Jen Chang, Yiorgos Adamopoulos, et al. sed – stream editor for filtering and transforming text. ftp://ftp.gnu.org/pub/gnu/sed and http://sed.sf.net, 1973–2006. http://sed.sourceforge.net/, last viewed May 2008.
* [12] Brian McNamara and Yannis Smaragdakis. Functional programming in C++ using the FC++ library. SIGPLAN Notices, 36(4):25–30, 2001.
* [13] Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL: A Proof Assistant for Higher-Order Logic, volume 2283\. Springer-Verlag, November 2007. http://www.in.tum.de/~nipkow/LNCS2283/, last viewed: December 2007\.
* [14] Joey Paquet. Course notes for COMP6411, winter 2010. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada, 2010. [online].
* [15] Lawrence C. Paulson and Tobias Nipkow. Isabelle: A generic proof assistant. University of Cambridge and Technical University of Munich, 2010. http://isabelle.in.tum.de/, last viewed: February 2010.
* [16] Yannis Smaragdakis and Brian McNamara. FC++: Functional tools for object-oriented tasks. Softw., Pract. Exper., 32(10):1015–1033, 2002.
* [17] Sun Microsystems, Inc. Java servlet technology. [online], 1994–2005. http://java.sun.com/products/servlets.
* [18] Sun Microsystems, Inc. JavaServer pages technology. [online], 2001–2005. http://java.sun.com/products/jsp/.
* [19] Various Contributors and the Flex Project. flex: The fast lexical analyzer. [online], 1987–2008. http://flex.sourceforge.net/, viewed in January 2008.
* [20] Various Contributors and the GNU Project. Bison – GNU parser generator.
* [21] Various Contributors and the GNU Project. GNU Compiler Collection (GCC). [online], 1988–2009. http://gcc.gnu.org/onlinedocs/gcc/.
* [22] Emil Vassev and Joey Paquet. Aspects of memory management in Java and C++. In Hamid R. Arabnia and Hassan Reza, editors, Software Engineering Research and Practice, pages 952–958. CSREA Press, 2006.
* [23] Sreenivasa Viswanadha and Contributors. Java compiler compiler (JavaCC) - the Java parser generator. [online], 2001–2008. https://javacc.dev.java.net/.
* [24] Jimmy Wales, Larry Sanger, and other authors from all over the world. Wikipedia: The free encyclopedia. [online], Wikimedia Foundation, Inc., 2001–2010. http://wikipedia.org.
## Index
* .NET
* JIT §3.2.1
* Ada §1.2.5
* Ahead-of-time compile §3.2.1
* Alan Kay §2.2.3
* AOP §2.2.3
* API
* aspect §2.2.3
* bark() 3rd item, 5th item, 5th item, 5th item, 5th item, 7th item, 7th item, 8th item
* CALL §2.2.1
* COPY §2.2.1
* DOALL §3.2.1
* Duck §3.3.1
* else §3.3.1
* false §3.3.1
* free() §3.4.2
* GET §2.2.1
* IDisposable §3.4.1
* inhale() 7th item
* malloc() §3.4.2
* new §3.4.2
* printf §1.2.5
* PUT §2.2.1
* quack() §3.3.1, §3.3.1, §3.3.1
* READ §2.2.1
* scanf §1.2.5
* speak() 8th item, 8th item, 8th item
* unsafePerformIO §3.3.1
* walk() §3.3.1, §3.3.1, §3.3.1
* where §1.2.5, §1.2.5, §1.2.5
* WRITE §2.2.1
* Aspect-oriented programming §2.2.3
* AspectJ §2.2.3
* Assembly §2.2.1
* Back end §3.1.2
* BASIC §1.2.5
* bytecode §3.2.1
* C §1.2.5, §1.2.5, §1.2.5, §1.2.5, §2.2.3, §2.2.4, §3.2.2, §3.3.1, §3.3.1, §3.4.1, §3.4.1, §3.4.2, §3.4.2, §4.1.1, §4.1.2, §4.1.3, §4.1.4
* C# §1.2.5, §2.1, 7th item, §2.2.4, §2.2.4, §3.3.1, §3.3.1, §3.3.1, §3.4.1, §3.4.1, §3.4.1, §4.1.2, §4.1.3
* C++ 1st item, 2nd item, 3rd item, 4th item, 5th item, 6th item, 7th item, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §2.1, 7th item, 8th item, §2.2.3, §2.2.4, §2.2.4, §3.3.1, §3.3.1, §3.3.1, 3rd item, §3.4.1, §3.4.1, §3.4.1, §3.4.1, §3.4.2, §3.4.2, §4.1.1, §4.1.2, §4.1.3, §4.1.4
* Code generation §3.1.2
* Common LISP §3.2.1
* Compilation §3.2.1
* Compilation vs. interpretation §3.2
* David Warren §3.2.1
* Duck typing §3.3.1
* Dynamic compilation §3.2.1
* Dynamic typing §3.3.1
* embedded systems §2.2.1
* FC++ §2.2.4
* Files
* .class §3.2.1
* Fortran §3.2.1, §3.3.1
* Front end §3.1.1
* Garbage collection §3.4.1
* Benefits §3.4.1
* Disadvantages §3.4.1
* hard-wired §2.2.1
* Haskell §1.2.5, §1.2.5, §2.2.4, §2.2.4, §2.2.4, §2.2.4, §3.3.1, §3.3.1, §3.4.1
* History §1.2
* Analytical engine §1.2.1
* Early computer era §1.2.3
* Modern/Structured programming languages §1.2.5
* Pre-computer era §1.2.1
* Interpretation §3.2.2
* ISWIM §1.2.5
* Java §1.2.5, §1.2.5, 4th item, 7th item, §2.2.3, §2.2.4, 1st item, §3.2.1, §3.2.1, §3.2.1, §3.2.1, §3.2.1, §3.2.2, §3.2.2, §3.2.2, §3.3.1, §3.3.1, §3.3.1, §3.4.1, §3.4.1, §3.4.1, §3.4.2, §4.1.1, §4.1.2, §4.1.3, §4.1.4
* JIT §3.2.1
* JIT §3.2.1
* .NET §3.2.1
* Java §3.2.1
* JSP §3.2.1
* Just-in-time compiler §3.2.1
* JVM §3.2.1
* Lexical analysis §3.1.1
* LISP §1.2.5, §1.2.5, §3.2.2, §3.3.1
* low-level §2.2.1
* Mac OS X §1.2.5
* Memory management §3.4
* ML≤ §1.2.5
* Nominative type system §3.3.1
* OpenMP §3.2.1
* Optimization §3.1.2
* Pascal §1.2.5, §3.2.2
* Performance Chapter 5
* Perl §1.2.5, §1.2.5, §2.1, §2.2.3, §2.2.4, §2.2.4, §3.2.2, §3.3.1, §3.3.1, §3.4.1
* PHP §2.2.3, §3.3.1, §3.4.1, §4.1.1, §4.1.2, §4.1.4
* Plankalk l §1.2.3
* Popularity Chapter 4
* Program analysis and translation phases §3.1
* Program Evaluation Chapter 3
* Prolog §1.2.5, §3.2.1
* Python §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §2.1, §2.2.3, §3.2.1, §3.2.2, §3.2.2, §3.3.1
* Ruby §3.2.2
* Scheme §1.2.5, §1.2.5, §1.2.5, §1.2.5, §1.2.5, §2.1, §2.2.4, §2.2.4, §2.2.4, §2.2.4
* security §3.2.1
* Self §3.2.1
* Semantic analysis §3.1.1
* Short Code §1.2.4
* Smalltalk §1.2.5
* 80 §3.2.2
* Source-to-source compiler §3.2.1
* Stage compiler §3.2.1
* Static typing §3.3.1
* Structural type system §3.3.1
* symbol table §3.1.1
* Syntactic analysis §3.1.1
* Tools
* gcc §2.2.4
* JavaCC §3.1.1
* lex §3.1.1, §3.1.1
* sed §1.2.5
* Type checking §3.3.1
* Type System §3.3
* WAM §3.2.1
* Warren Abstract Machine §3.2.1
|
arxiv-papers
| 2010-07-12T17:22:54 |
2024-09-04T02:49:11.586170
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joey Paquet and Serguei A. Mokhov",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/1007.2123"
}
|
1007.2371
|
∎
11institutetext: K. Lange 22institutetext: Departments of Biomathematics,
Human Genetics, and Statistics, University of California, Los Angeles, CA
90095-1766, USA.
22email: klange@ucla.edu 33institutetext: H. Zhou 44institutetext: Department
of Statistics, North Carolina State University, 2311 Stinson Drive, Campus Box
8203, Raleigh, NC 27695-8203, USA.
44email: huazhou@ucla.edu
# MM Algorithms for Geometric and Signomial Programming††thanks: Research was
supported by United States Public Health Service grants GM53275 and MH59490.
Kenneth Lange Hua Zhou
(Received: date / Accepted: date)
###### Abstract
This paper derives new algorithms for signomial programming, a generalization
of geometric programming. The algorithms are based on a generic principle for
optimization called the MM algorithm. In this setting, one can apply the
geometric-arithmetic mean inequality and a supporting hyperplane inequality to
create a surrogate function with parameters separated. Thus, unconstrained
signomial programming reduces to a sequence of one-dimensional minimization
problems. Simple examples demonstrate that the MM algorithm derived can
converge to a boundary point or to one point of a continuum of minimum points.
Conditions under which the minimum point is unique or occurs in the interior
of parameter space are proved for geometric programming. Convergence to an
interior point occurs at a linear rate. Finally, the MM framework easily
accommodates equality and inequality constraints of signomial type. For the
most important special case, constrained quadratic programming, the MM
algorithm involves very simple updates.
###### Keywords:
arithmetic-geometric mean inequality global convergence MM algorithm parameter
separation penalty method
###### MSC:
90C25 26D07
††journal: Under review
## 1 Introduction
As a branch of convex optimization theory, geometric programming is next in
line to linear and quadratic programming in importance boyd04 ; ecker80 ;
peressini88 ; peterson76 . It has applications in chemical equilibrium
problems passy68 , structural mechanics ecker80 , integrated circuit design
hershenson01 , maximum likelihood estimation mazumdar83 , stochastic processes
feigin81 , and a host of other subjects ecker80 . Geometric programming deals
with posynomials, which are functions of the form
$\displaystyle f(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{i}^{\alpha_{i}}.$ (1)
Here the index set $S\subset\mathbb{R}^{n}$ is finite, and all coefficients
$c_{\boldsymbol{\alpha}}$ and all components $x_{1},\ldots,x_{n}$ of the
argument $\boldsymbol{x}$ of $f(\boldsymbol{x})$ are positive. The possibly
fractional powers $\alpha_{i}$ corresponding to a particular
$\boldsymbol{\alpha}$ may be positive, negative, or zero. For instance,
$x_{1}^{-1}+2x_{1}^{3}x_{2}^{-2}$ is a posynomial on $\mathbb{R}^{2}$. In
geometric programming we minimize a posynomial $f(\boldsymbol{x})$ subject to
posynomial inequality constraints of the form $u_{j}(\boldsymbol{x})\leq 1$
for $1\leq j\leq q$, where the $u_{j}(\boldsymbol{x})$ are again posynomials.
In some versions of geometric programming, equality constraints of posynomial
type are permitted boyd07 .
A signomial function has the same form as the posynomial (1), but the
coefficients $c_{\boldsymbol{\alpha}}$ are allowed to be negative. A signomial
program is a generalization of a geometric program, where the objective and
constraint functions can be signomials. From a computational point of view,
signomial programming problems are significantly harder to solve than
geometric programming problems. After suitable change of variables, a
geometric program can be transformed into a convex optimization problem and
globally solved by standard methods. In contrast, signomials may have many
local minima. Wang et al. WangZhangShen02SP recently derived a path algorithm
for solving unconstrained signomial programs.
The theory and practice of geometric programming has been stable for a
generation, so it is hard to imagine saying anything novel about either. The
attractions of geometric programming include its beautiful duality theory and
its connections with the arithmetic-geometric mean inequality. The present
paper derives new algorithms for both geometric and signomial programming
based on a generic device for iterative optimization called the MM algorithm
hunter04 ; lange00 . The MM perspective possesses several advantages. First it
provides a unified framework for solving both geometric and signomial
programs. The algorithms derived here operate by separating parameters and
reducing minimization of the objective function to a sequence of one-
dimensional minimization problems. Separation of parameters is apt to be an
advantage in high-dimensional problems. Another advantage is ease of
implementation compared to competing methods of unconstrained geometric and
signomial programming WangZhangShen02SP . Finally, straightforward
generalizations of our MM algorithms extend beyond signomial programming.
We conclude this introduction by sketching a roadmap to the rest of the paper.
Section 2 reviews the MM algorithm. Section 3 derives MM algorithm for
unconstrained signomial program from two simple inequalities. The behavior of
the MM algorithm is illustrated on a few numerical examples in Section 4.
Section 5 extends the MM algorithm for unconstrained problems to the
constrained cases using the penalty method. Section 6 specializes to linearly
constrained quadratic programming on the positive orthant. Convergence results
are discussed in Section 7.
## 2 Background on the MM Algorithm
The MM principle involves majorizing the objective function
$f(\boldsymbol{x})$ by a surrogate function
$g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ around the current iterate
$\boldsymbol{x}_{m}$ (with $i$th component $x_{mi}$) of a search. Majorization
is defined by the two conditions
$\displaystyle f(\boldsymbol{x}_{m})$ $\displaystyle=$ $\displaystyle
g(\boldsymbol{x}_{m}\mid\boldsymbol{x}_{m})$ (2) $\displaystyle
f(\boldsymbol{x})$ $\displaystyle\leq$ $\displaystyle
g(\boldsymbol{x}\mid\boldsymbol{x}_{m})\>,\quad\quad\boldsymbol{x}\neq\boldsymbol{x}_{m}.$
In other words, the surface $\boldsymbol{x}\mapsto
g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ lies above the surface
$\boldsymbol{x}\mapsto f(\boldsymbol{x})$ and is tangent to it at the point
$\boldsymbol{x}=\boldsymbol{x}_{m}$. Construction of the majorizing function
$g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ constitutes the first M of the MM
algorithm.
The second M of the algorithm minimizes the surrogate
$g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ rather than $f(\boldsymbol{x})$. If
$\boldsymbol{x}_{m+1}$ denotes the minimizer of
$g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$, then this action forces the descent
property $f(\boldsymbol{x}_{m+1})\leq f(\boldsymbol{x}_{m})$. This fact
follows from the inequalities
$\displaystyle f(\boldsymbol{x}_{m+1})\leq
g(\boldsymbol{x}_{m+1}\mid\boldsymbol{x}_{m})\leq
g(\boldsymbol{x}_{m}\mid\boldsymbol{x}_{m})=f(\boldsymbol{x}_{m}),$
reflecting the definition of $\boldsymbol{x}_{m+1}$ and the tangency
conditions (2). The descent property lends the MM algorithm remarkable
numerical stability. Strictly speaking, it depends only on decreasing
$g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$, not on minimizing
$g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$.
## 3 Unconstrained Signomial Programming
The art in devising an MM algorithm revolves around intelligent choice of the
majorizing function. For signomial programming problems, fortunately one can
invoke two simple inequalities. For terms with positive coefficients
$c_{\boldsymbol{\alpha}}$, we use the arithmetic-geometric mean inequality
$\displaystyle\prod_{i=1}^{n}z_{i}^{\alpha_{i}}$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n}\frac{\alpha_{i}}{\|\boldsymbol{\alpha}\|_{1}}z_{i}^{\|\boldsymbol{\alpha}\|_{1}}$
(3)
for nonnegative numbers $z_{i}$ and $\alpha_{i}$ and $\ell_{1}$ norm
$\|\boldsymbol{\alpha}\|_{1}=\sum_{i=1}^{n}|\alpha_{i}|$ steele04 . If we make
the choice $z_{i}=x_{i}/x_{mi}$ in inequality (3), then the majorization
$\displaystyle\prod_{i=1}^{n}x_{i}^{\alpha_{i}}$ $\displaystyle\leq$
$\displaystyle\left(\prod_{i=1}^{n}x_{mi}^{\alpha_{i}}\right)\sum_{i=1}^{n}\frac{\alpha_{i}}{\|\boldsymbol{\alpha}\|_{1}}\left(\frac{x_{i}}{x_{mi}}\right)^{\|\boldsymbol{\alpha}\|_{1}},$
(4)
emerges, with equality when $\boldsymbol{x}=\boldsymbol{x}_{m}$. We can
broaden the scope of the majorization (4) to cases with $\alpha_{i}<0$ by
replacing $z_{i}$ by the reciprocal ratio $x_{mi}/x_{i}$ whenever
$\alpha_{i}<0$. Thus, for terms
$c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{i}^{\alpha_{i}}$ with
$c_{\boldsymbol{\alpha}}>0$, we have the majorization
$\displaystyle c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{i}^{\alpha_{i}}\leq
c_{\boldsymbol{\alpha}}\left(\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\right)\sum_{i=1}^{n}\frac{|\alpha_{i}|}{\|\boldsymbol{\alpha}\|_{1}}\left(\frac{x_{i}}{x_{mi}}\right)^{\|\boldsymbol{\alpha}\|_{1}\text{sgn}(\alpha_{i})},$
where $\mathop{\rm sgn}\nolimits(\alpha_{i})$ is the sign function.
The terms $c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{i}^{\alpha_{i}}$ with
$c_{\boldsymbol{\alpha}}<0$ are handled by a different majorization. Our point
of departure is the supporting hyperplane minorization
$\displaystyle z$ $\displaystyle\geq$ $\displaystyle 1+\ln z$
at the point $z=1$. If we let $z=\prod_{i=1}^{n}(x_{i}/x_{mi})^{\alpha_{i}}$,
then it follows that
$\displaystyle\prod_{i=1}^{n}x_{i}^{\alpha_{i}}$ $\displaystyle\geq$
$\displaystyle\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\left(1+\sum_{i=1}^{n}\alpha_{i}\ln
x_{i}-\sum_{i=1}^{n}\alpha_{i}\ln x_{mi}\right)$ (5)
is a valid minorization in $\boldsymbol{x}$ around the point
$\boldsymbol{x}_{m}$. Multiplication by the negative coefficient
$c_{\boldsymbol{\alpha}}$ now gives the desired majorization. The surrogate
function separates parameters and is convex when all of the $\alpha_{i}$ are
positive.
In summary, the objective function (1) is majorized up to an irrelevant
additive constant by the sum
$\displaystyle g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}g_{i}(x_{i}\mid\boldsymbol{x}_{m})$ $\displaystyle
g_{i}(x_{i}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S_{+}}c_{\boldsymbol{\alpha}}\Bigg{(}\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\Bigg{)}\frac{|\alpha_{i}|}{\|\boldsymbol{\alpha}\|_{1}}\left(\frac{x_{i}}{x_{mi}}\right)^{\|\boldsymbol{\alpha}\|_{1}\text{sgn}(\alpha_{i})}$
$\displaystyle+\sum_{\boldsymbol{\alpha}\in
S_{-}}c_{\boldsymbol{\alpha}}\Bigg{(}\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\Bigg{)}\alpha_{i}\ln
x_{i},$
where $S_{+}=\\{\boldsymbol{\alpha}:c_{\boldsymbol{\alpha}}>0\\}$, and
$S_{-}=\\{\boldsymbol{\alpha}:c_{\boldsymbol{\alpha}}<0\\}$. To guarantee that
the next iterate is well defined and occurs on the interior of the parameter
domain, it is helpful to assume for each $i$ that at least one
$\boldsymbol{\alpha}\in S_{+}$ has $\alpha_{i}$ positive and at least one
$\boldsymbol{\alpha}\in S_{+}$ has $\alpha_{i}$ negative. Under these
conditions each $g_{i}(x_{i}\mid\boldsymbol{x}_{m})$ is coercive and attains
its minimum on the open interval $(0,\infty)$.
Minimization of the majorizing function is straightforward because the
surrogate functions $g_{i}(x_{i}\mid\boldsymbol{x}_{m})$ are univariate
functions. The derivative of $g_{i}(x_{i}\mid\boldsymbol{x}_{m})$ with respect
to its left argument equals
$\displaystyle g_{i}^{\prime}(x_{i}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S_{+}}c_{\boldsymbol{\alpha}}\Bigg{(}\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\Bigg{)}\alpha_{i}x_{i}^{-1}\left(\frac{x_{i}}{x_{mi}}\right)^{\|\boldsymbol{\alpha}\|_{1}\mathop{\rm
sgn}\nolimits(\alpha_{i})}$ $\displaystyle\hskip
36.135pt+\sum_{\boldsymbol{\alpha}\in
S_{-}}c_{\boldsymbol{\alpha}}\Bigg{(}\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\Bigg{)}\alpha_{i}x_{i}^{-1}$
Assuming that the exponents $\alpha_{i}$ are integers, this is a rational
function of $x_{i}$, and once we equate it to 0, we are faced with solving a
polynomial equation. This task can be accomplished by bisection or by Newton’s
method.
In a geometric program, the function
$g_{i}^{\prime}(x_{i}\mid\boldsymbol{x}_{m})$ has a single root on the
interval $(0,\infty)$. For a proof of this fact, note that making the standard
change of variables $x_{i}=e^{y_{i}}$ eliminates the positivity constraint
$x_{i}>0$ and renders the transformed function
$h_{i}(y_{i}\mid\boldsymbol{x}_{m})=g_{i}(x_{i}\mid\boldsymbol{x}_{m})$
strictly convex. Because $|\alpha_{i}|\mathop{\rm
sgn}\nolimits(\alpha_{i})^{2}=|\alpha_{i}|$, the second derivative
$\displaystyle h_{i}^{\prime\prime}(y_{i}\mid\boldsymbol{x}_{m})$
$\displaystyle=$ $\displaystyle\sum_{\boldsymbol{\alpha}\in
S_{+}}c_{\boldsymbol{\alpha}}\Bigg{(}\prod_{j=1}^{n}x_{mj}^{\alpha_{j}}\Bigg{)}\frac{|\alpha_{i}|\cdot\|\boldsymbol{\alpha}\|_{1}}{x_{mi}^{\|\boldsymbol{\alpha}\|_{1}\mathop{\rm
sgn}\nolimits(\alpha_{i})}}e^{\|\boldsymbol{\alpha}\|_{1}\mathop{\rm
sgn}\nolimits(\alpha_{i})y_{i}}$
is positive. Hence, $h_{i}(y_{i}\mid\boldsymbol{x}_{m})$ is strictly convex
and possesses a unique minimum point. These arguments yield the even sweeter
dividend that the MM iteration map is continuously differentiable. From the
vantage point of the implicit function theorem hoffman75 , the stationary
condition $h_{i}^{\prime}(y_{m+1,i}\mid\boldsymbol{x}_{m})=0$ determines
$y_{m+1,i}$, and consequently $x_{m+1,i}$, in terms of $\boldsymbol{x}_{m}$.
Observe here that $h_{i}^{\prime\prime}(y_{mi}\mid\boldsymbol{x}_{m})\neq 0$
as required by the implicit function.
It is also worth pointing out that even more functions can be brought under
the umbrella of signomial programming. For instance, majorization of the
functions $-\ln f(\boldsymbol{x})$ and $\ln f(\boldsymbol{x})$ is possible for
any posynomial
$f(\boldsymbol{x})=\sum_{\boldsymbol{\alpha}}c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{i}^{\alpha_{i}}$.
In the first case,
$\displaystyle-\ln f(\boldsymbol{x})$ $\displaystyle\leq$
$\displaystyle-\sum_{\boldsymbol{\alpha}}\frac{a_{m\boldsymbol{\alpha}}}{b_{m}}\Big{[}\sum_{i=1}^{n}\alpha_{i}\ln
x_{i}+\ln\Big{(}\frac{c_{\boldsymbol{\alpha}}b_{m}}{a_{m\boldsymbol{\alpha}}}\Big{)}\Big{]}$
(7)
holds for
$a_{m\boldsymbol{\alpha}}=c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{mi}^{\alpha_{i}}$
and $b_{m}=\sum_{\boldsymbol{\alpha}}a_{m\boldsymbol{\alpha}}$ because
Jensen’s inequality applies to the convex function $-\ln t$. In the second
case, the supporting hyperplane inequality applied to the convex function
$-\ln t$ implies
$\displaystyle\ln f(\boldsymbol{x})$ $\displaystyle\leq$ $\displaystyle\ln
f(\boldsymbol{x}_{m})+\frac{1}{f(\boldsymbol{x}_{m})}\Big{[}f(\boldsymbol{x})-f(\boldsymbol{x}_{m})\Big{]}.$
This puts us back in the position of needing to majorize a posynomial, a
problem we have already discussed in detail. By our previous remarks, the
coefficients $c_{\boldsymbol{\alpha}}$ can be negative as well as positive in
this case. Similar majorizations apply to any composition $\phi\circ
f(\boldsymbol{x})$ of a posynomial $f(\boldsymbol{x})$ with an arbitrary
concave function $\phi(y)$.
## 4 Examples of Unconstrained Minimization
Our first examples demonstrate the robustness of the MM algorithms in
minimization and illustrate some of the complications that occur. In each case
we can explicitly calculate the MM updates. To start, consider the posynomial
$\displaystyle f_{1}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\frac{1}{x_{1}^{3}}+\frac{3}{x_{1}x_{2}^{2}}+x_{1}x_{2}$
with the implied constraints $x_{1}>0$ and $x_{2}>0$. The majorization (4)
applied to the third term of $f_{1}(\boldsymbol{x})$ yields
$\displaystyle x_{1}x_{2}$ $\displaystyle\leq$ $\displaystyle
x_{m1}x_{m2}\left[\frac{1}{2}\left(\frac{x_{1}}{x_{m1}}\right)^{2}+\frac{1}{2}\left(\frac{x_{2}}{x_{m2}}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{x_{m2}}{2x_{m1}}x_{1}^{2}+\frac{x_{m1}}{2x_{m2}}x_{2}^{2}.$
Applied to the second term of $f_{1}(\boldsymbol{x})$ using the reciprocal
ratios, it gives
$\displaystyle{3\over x_{1}x_{2}^{2}}$ $\displaystyle\leq$
$\displaystyle{3\over x_{m1}x_{m2}^{2}}\left[{1\over 3}\left({x_{m1}\over
x_{1}}\right)^{3}+{2\over 3}\left({x_{m2}\over x_{2}}\right)^{3}\right]$
$\displaystyle=$ $\displaystyle{x_{m1}^{2}\over x_{m2}^{2}}{1\over
x_{1}^{3}}+{2x_{m2}\over x_{m1}}{1\over x_{2}^{3}}.$
The sum $g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ of the two surrogate
functions
$\displaystyle g_{1}(x_{1}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle{1\over x_{1}^{3}}+{x_{m1}^{2}\over x_{m2}^{2}}{1\over
x_{1}^{3}}+{x_{m2}\over 2x_{m1}}x_{1}^{2}$ $\displaystyle
g_{2}(x_{2}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle{2x_{m2}\over x_{m1}}{1\over x_{2}^{3}}+{x_{m1}\over
2x_{m2}}x_{2}^{2}$
majorizes $f_{1}(\boldsymbol{x})$. If we set the derivatives
$\displaystyle g_{1}^{\prime}(x_{1}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle-{3\over x_{1}^{4}}-{x_{m1}^{2}\over x_{m2}^{2}}{3\over
x_{1}^{4}}+{x_{m2}\over x_{m1}}x_{1}$ $\displaystyle
g_{2}^{\prime}(x_{1}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle-{6x_{m2}\over x_{m1}}{1\over x_{2}^{4}}+{x_{m1}\over
x_{m2}}x_{2}$
of each of these equal to 0, then the updates
$\displaystyle x_{m+1,1}$ $\displaystyle=$
$\displaystyle\sqrt[5]{3\left({x_{m1}^{2}\over
x_{m2}^{2}}+1\right){x_{m1}\over x_{m2}}},\quad\quad
x_{m+1,2}\;\;\,=\;\;\,\sqrt[5]{6{x_{m2}^{2}\over x_{m1}^{2}}}$
solve the minimization step of the MM algorithm. It is also obvious that the
point $\boldsymbol{x}=(\sqrt[5]{6},\sqrt[5]{6})^{t}$ is a fixed point of the
updates, and the reader can check that it minimizes $f_{1}(\boldsymbol{x})$.
It is instructive to consider the slight variations
$\displaystyle f_{2}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle{1\over
x_{1}x_{2}^{2}}+x_{1}x_{2}^{2}$ $\displaystyle f_{3}(\boldsymbol{x})$
$\displaystyle=$ $\displaystyle{1\over x_{1}x_{2}^{2}}+x_{1}x_{2}$
of this objective function. In the first case, the reader can check that the
MM algorithm iterates according to
$\displaystyle x_{m+1,1}$ $\displaystyle=$
$\displaystyle\sqrt[3]{{x_{m1}^{2}\over x_{m2}^{2}}},\quad\quad
x_{m+1,2}\;\;\,=\;\;\,\sqrt[3]{{x_{m2}\over x_{m1}}}.$
In the second case, it iterates according to
$\displaystyle x_{m+1,1}$ $\displaystyle=$
$\displaystyle\sqrt[5]{{x_{m1}^{3}\over x_{m2}^{3}}},\quad\quad
x_{m+1,2}\;\;\,=\;\;\,\sqrt[5]{2{x_{m2}^{2}\over x_{m1}^{2}}}.$
The objective function $f_{2}(\boldsymbol{x})$ attains its minimum value
whenever $x_{1}x_{2}^{2}=1$. The MM algorithm for $f_{2}(\boldsymbol{x})$
converges after a single iteration to the value 2, but the converged point
depends on the initial point $\boldsymbol{x}_{0}$. The infimum of
$f_{3}(\boldsymbol{x})$ is 0. This value is attained asymptotically by the MM
algorithm, which satisfies the identities $x_{m1}x_{m2}^{3/2}=2^{3/10}$ and
$x_{m+1,2}=2^{2/25}x_{m2}$ for all $m\geq 1$. These results imply that
$x_{m1}$ tends to 0 and $x_{m2}$ to $\infty$ in such a manner that
$f_{3}(\boldsymbol{x}_{m})$ tends to 0. One could not hope for much better
behavior of the MM algorithm in these two examples.
The function
$\displaystyle f_{4}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
x_{1}^{2}x_{2}^{2}-2x_{1}x_{2}x_{3}x_{4}+x_{3}^{2}x_{4}^{2}\;\;=\;\;(x_{1}x_{2}-x_{3}x_{4})^{2}$
is a signomial but not a posynomial. The surrogate function (3) reduces to
$\displaystyle g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle\frac{x_{m2}^{2}}{2x_{m1}^{2}}x_{1}^{4}+\frac{x_{m1}^{2}}{2x_{m2}^{2}}x_{2}^{4}+\frac{x_{m4}^{2}}{2x_{m3}^{2}}x_{3}^{4}+\frac{x_{m3}^{2}}{2x_{m4}^{2}}x_{4}^{4}$
$\displaystyle-2x_{m1}x_{m2}x_{m3}x_{m4}(\ln x_{1}+\ln_{2}+\ln x_{3}+\ln
x_{4})$
with all variables separated. The MM updates
$\displaystyle x_{m+1,1}$ $\displaystyle=$
$\displaystyle\sqrt[4]{\frac{x_{m1}^{3}x_{m3}x_{m4}}{x_{m2}}},\quad\quad
x_{m+1,2}\;\;=\;\;\sqrt[4]{\frac{x_{m2}^{3}x_{m3}x_{m4}}{x_{m1}}}$
$\displaystyle x_{m+1,3}$ $\displaystyle=$
$\displaystyle\sqrt[4]{\frac{x_{m3}^{3}x_{m1}x_{m2}}{x_{m4}}},\quad\quad
x_{m+1,4}\;\;=\;\;\sqrt[4]{\frac{x_{m4}^{3}x_{m1}x_{m2}}{x_{m3}}}$
converge in a single iteration to a solution of $f_{4}(\boldsymbol{x})=0$.
Again the limit depends on the initial point.
The function
$\displaystyle f_{5}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}-\ln(x_{1}+x_{2}+x_{3})$
is more complicated than a signomial. It also is unbounded because the point
$\boldsymbol{x}$ with components $x_{1}=m$ and $x_{2}=x_{3}=1/m$ satisfies
$f_{5}(\boldsymbol{x})=2+m^{-2}-\ln(m+2/m)$. According to the majorization
(7), an appropriate surrogate is
$\displaystyle g(\boldsymbol{x}\mid\boldsymbol{x}_{m})$ $\displaystyle=$
$\displaystyle\Big{(}\frac{x_{m2}}{2x_{m1}}+\frac{x_{m3}}{2x_{m1}}\Big{)}x_{1}^{2}+\Big{(}\frac{x_{m1}}{2x_{m2}}+\frac{x_{m3}}{2x_{m2}}\Big{)}x_{2}^{2}+\Big{(}\frac{x_{m1}}{2x_{m3}}+\frac{x_{m2}}{2x_{m3}}\Big{)}x_{3}^{2}$
$\displaystyle-\frac{x_{m1}}{x_{m1}+x_{m2}+x_{m3}}\ln
x_{1}-\frac{x_{m2}}{x_{m1}+x_{m2}+x_{m3}}\ln x_{2}$
$\displaystyle-\frac{x_{m3}}{x_{m1}+x_{m2}+x_{m3}}\ln x_{3}$
up to an irrelevant constant. The MM updates are
$\displaystyle x_{m+1,i}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{x_{mi}^{2}}{(\sum_{j\neq
i}x_{mj})(x_{m1}+x_{m2}+x_{m3})}}.$
If the components of the initial point coincide, then the iterates converge in
a single iteration to the saddle point with all components equal to
$1/\sqrt{6}$. Otherwise, it appears that $f_{5}(\boldsymbol{x}_{m})$ tends to
$-\infty$.
The following objective functions
$\displaystyle f_{6}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
x_{1}^{2}x_{2}^{6}+x_{1}^{2}x_{2}^{4}-2x_{1}^{2}x_{2}^{3}-x_{1}^{2}x_{2}^{2}+5.25x_{1}x_{2}^{3}$
$\displaystyle-2x_{1}^{2}x_{2}+4.5x_{1}x_{2}^{2}+3x_{1}^{2}+3x_{1}x_{2}-12.75x_{1}$
$\displaystyle f_{7}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{10}x_{i}^{4}+2\sum_{i=1}^{9}x_{i}^{2}\sum_{j=i+1}^{10}x_{j}^{2}+(10^{-5}-0.5)\sum_{i=1}^{10}x_{i}^{2}$
$\displaystyle-(2\times 10^{-5})\sum_{i=7}^{10}x_{i}+\frac{1}{16}$
$\displaystyle f_{8}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
x_{1}x_{3}^{2}x_{6}^{-1}x_{7}^{-1}+x_{1}^{2}x_{3}^{-1}x_{5}^{-2}x_{6}^{-1}x_{7}$
$\displaystyle+x_{1}^{3}x_{2}^{2}x_{5}^{-2}x_{6}^{2}+x_{2}^{-1}x_{4}^{-1}x_{6}^{2}+x_{3}x_{5}^{3}x_{6}^{-3}$
$\displaystyle f_{9}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
x_{1}x_{4}^{2}+x_{2}x_{3}+x_{1}x_{2}x_{3}x_{4}^{2}+x_{1}^{-1}x_{4}^{-2}$
from the reference WangZhangShen02SP are intended for numerical illustration.
Table 1 lists initial conditions, minimum points, minimum values, and number
of iterations until convergence under the MM algorithm. Convergence is
declared when the relative change in the objective function is less than a
pre-specified value $\epsilon$, in other words, when
$\displaystyle\frac{f(\boldsymbol{x}_{m})-f(\boldsymbol{x}_{m+1})}{|f(\boldsymbol{x}_{m})|+1}$
$\displaystyle\leq$ $\displaystyle\epsilon.$
Optimization of the univariate surrogate functions easily succumbs to Newton’s
method. The MM algorithm takes fewer iterations to converge than the path
algorithm for all of the test functions mentioned in WangZhangShen02SP except
$f_{6}(\boldsymbol{x})$. Furthermore, the MM algorithm avoids calculation of
the gradient and Hessian and requires no matrix decompositions or selection of
tuning constants.
As Section 7 observes, MM algorithms typically converge at a linear rate.
Although slow convergence can occur for functions such as the test function
$f_{6}(\boldsymbol{x})$, there are several ways to accelerate an MM algorithm.
For example, our published quasi-Newton acceleration ZhouAlexanderLange09QN
often reduces the necessary number of iterations by one or two orders of
magnitude. Figure 1 shows the progress of the MM iterates for the test
function $f_{6}(\boldsymbol{x})$ with and without quasi-Newton acceleration.
Under a convergence criterion of $\epsilon=10^{-9}$ and $q=1$ secant
condition, the required number of iterations falls to 30; under the same
convergence criterion and $q=2$ secant conditions, the required number of
iterations falls to 12. It is also worth emphasizing that separation of
parameters enables parallel processing in high-dimensional problems. We have
recently argued ZhouLangeSuchard09MM-GPU that the best approach to parallel
processing is through graphics processing units (GPUs). These cheap hardware
devices offer one to two orders of magnitude acceleration in many MM
algorithms with parameters separated.
Fun | Type | Initial Point $\boldsymbol{x}_{0}$ | Min Point | Min Value | Iters ($10^{-9}$)
---|---|---|---|---|---
$f_{1}$ | P | (1,2) | (1.4310,1.4310) | 3.4128 | 38
$f_{2}$ | P | (1,2) | (0.6300,1.2599) | 2.0000 | 2
$f_{3}$ | P | (1,1) | diverges | 0.0000 |
$f_{4}$ | S | (0.1,0.2,0.3,0.4) | (0.1596,0.3191,0.1954,0.2606) | 0.0000 | 3
$f_{5}$ | G | (1,1,1) | (0.4082,0.4082,0.4082) | 0.2973 | 2
| | (1,2,3) | diverges | $-\infty$ |
$f_{6}$ | S | (1,1) | (2.9978,0.4994) | -14.2031 | 558
$f_{7}$ | S | $(1,\ldots,10)$ | $0.0255\boldsymbol{x}_{0}$ | 0.0000 | 18
$f_{8}$ | P | $(1,\ldots,7)$ | diverges | 0.0000 |
$f_{9}$ | P | (1,2,3,4) | (0.3969,0.0000,0.0000,1.5874) | 2.0000 | 7
Table 1: Numerical examples of unconstrained signomial programming. Test
functions $f_{4}(\boldsymbol{x})$, $f_{6}(\boldsymbol{x})$,
$f_{7}(\boldsymbol{x})$, $f_{8}(\boldsymbol{x})$ and $f_{9}(\boldsymbol{x})$
are taken from WangZhangShen02SP . P: posynomial; S: signomial; G: general
function.
$\begin{array}[]{cc}\includegraphics[width=166.2212pt]{Test01-Surface.eps}&\includegraphics[width=166.2212pt]{Test01-Contour-q0.eps}\\\
\includegraphics[width=166.2212pt]{Test01-Contour-q1.eps}&\includegraphics[width=166.2212pt]{Test01-Contour-q2.eps}\end{array}$
Figure 1: Upper left: The test function $f_{6}(\boldsymbol{x})$. Upper right:
558 MM iterates. Lower left: 30 accelerated MM iterates ($q=1$ secant
conditions). Lower right: 12 accelerated MM iterates ($q=2$ secant
conditions).
## 5 Constrained Signomial Programming
Extending the MM algorithm to constrained geometric and signomial programming
is challenging. Box constraints $a_{i}\leq x_{i}\leq b_{i}$ are consistent
with parameter separation as just developed, but more complicated posynomial
constraints that couple parameters are not. Posynomial inequality constraints
take the form
$\displaystyle h(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\beta}}d_{\boldsymbol{\beta}}\prod_{i=1}^{n}x_{i}^{\beta_{i}}\;\;\leq\;\;1.$
The corresponding equality constraint sets $h(\boldsymbol{x})=1$. We propose
handling both constraints by penalty methods. Before we treat these matters in
more depth, let us relax the positivity restrictions on the
$d_{\boldsymbol{\beta}}$ but enforce the restriction $\beta_{i}\geq 0$. The
latter objective can be achieved by multiplying $h(\boldsymbol{x})$ by
$x_{i}^{\max_{\boldsymbol{\beta}}\\{-\beta_{i},0\\}}$ for all $i$. If we
subtract the two sides of the resulting equality, then the equality constraint
$h(\boldsymbol{x})=1$ can be rephrased as
$r(\boldsymbol{x})=\sum_{\boldsymbol{\gamma}}e_{\boldsymbol{\gamma}}\prod_{i=1}^{n}x_{i}^{\gamma_{i}}=0$,
with no restriction on the signs of the $e_{\boldsymbol{\gamma}}$ but with the
requirement $\gamma_{i}\geq 0$ in effect. For example, the equality constraint
$\displaystyle\frac{1}{x_{1}}+\frac{x_{1}}{x_{2}^{2}}$ $\displaystyle=$
$\displaystyle 1$
becomes
$\displaystyle x_{1}^{2}+x_{2}^{2}-x_{1}x_{2}^{2}$ $\displaystyle=$
$\displaystyle 0.$
In the quadratic penalty method NocedalWrightBook ; RuszczynskiBook with
objective function $f(\boldsymbol{x})$ and a single equality constraint
$r(\boldsymbol{x})=0$ and a single inequality constraint
$s(\boldsymbol{x})\leq 0$, one minimizes the sum
$f_{\lambda}(\boldsymbol{x})=f(\boldsymbol{x})+\lambda
r(\boldsymbol{x})^{2}+\lambda s(\boldsymbol{x})_{+}^{2}$, where
$s(\boldsymbol{x})_{+}=\max\\{s(\boldsymbol{x}),0\\}$. As the penalty constant
$\lambda$ tends to $\infty$, the solution vector $\boldsymbol{x}_{\lambda}$
typically converges to the constrained minimum. In the revised objective
function, the term $r(\boldsymbol{x})^{2}$ is a signomial whenever
$r(\boldsymbol{x})$ is a signomial. For example, in our toy problem the choice
$r(\boldsymbol{x})=x_{1}^{2}+x_{2}^{2}-x_{1}x_{2}^{2}$ has square
$\displaystyle r(\boldsymbol{x})^{2}$ $\displaystyle=$ $\displaystyle
x_{1}^{4}+x_{2}^{4}+x_{1}^{2}x_{2}^{4}+2x_{1}^{2}x_{2}^{2}-2x_{1}x_{2}^{4}-2x_{1}^{3}x_{2}^{2}.$
Of course, the powers in $r(\boldsymbol{x})$ can be fractional here as well as
integer. The term $s(\boldsymbol{x})_{+}^{2}$ is not a signomial and must be
subjected to the majorization
$\displaystyle
s(\boldsymbol{x})_{+}^{2}\leq\begin{cases}[s(\boldsymbol{x})-s(\boldsymbol{x}_{m})]^{2}&s(\boldsymbol{x}_{m})<0\\\
s(\boldsymbol{x})^{2}&s(\boldsymbol{x}_{m})\geq 0\end{cases}$
to achieve this status. In practice, one does not need to fully minimize
$f_{\lambda}(\boldsymbol{x})$ for any fixed $\lambda$. If one increases
$\lambda$ slowly enough, then it usually suffices to merely decrease
$f_{\lambda}(\boldsymbol{x})$ at each iteration. The MM algorithm is designed
to achieve precisely this goal. Our exposition so far suggests that we
majorize $r(\boldsymbol{x})^{2}$, $s(\boldsymbol{x})^{2}$, and
$[s(\boldsymbol{x})-s(\boldsymbol{x}_{m})]^{2}$ in exactly the same manner
that we majorize $f(\boldsymbol{x})$. Separation of parameters generalizes,
and the resulting MM algorithm keeps all parameters positive while permitting
pertinent parameters to converge to 0. Section 7 summarizes some of the
convergence properties of this hybrid procedure.
The quadratic penalty method traditionally relies on Newton’s method to
minimize the unconstrained functions $f_{\lambda}(\boldsymbol{x})$.
Unfortunately, this tactic suffers from roundoff errors and numerical
instability. Some of these problems disappear with the MM algorithm. No matrix
inversions are involved, and iterates enjoy the descent property. Ill-
conditioning does cause harm in the form of slow convergence, but the
previously mentioned quasi-Newton acceleration largely remedies the situation
ZhouAlexanderLange09QN . As an alternative to quadratic penalties, exact
penalties take the form $\lambda|r(\boldsymbol{x})|+\lambda
s(\boldsymbol{x})_{+}$. Remarkably, the exact penalty method produces the
constrained minimum, not just in the limit, but for all finite $\lambda$
beyond a certain point. Although this desirable property avoids the numerical
instability encountered in the quadratic penalty method, the kinks in the
objective functions $f(\boldsymbol{x})+\lambda|r(\boldsymbol{x})|+\lambda
s(\boldsymbol{x})_{+}$ are a nuisance. We will demonstrate in a future paper
how to harness the MM algorithm to exact penalization.
## 6 Nonnegative Quadratic Programming
As an illustration of constrained signomial programming, consider quadratic
programming over the positive orthant. Let
$\displaystyle f(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{x}^{t}\boldsymbol{Q}\boldsymbol{x}+{\bf
c}^{t}\boldsymbol{x}$
be the objective function, $\boldsymbol{E}\boldsymbol{x}=\boldsymbol{d}$ the
linear equality constraints, and
$\boldsymbol{A}\boldsymbol{x}\leq\boldsymbol{b}$ the linear inequality
constraints. The symmetric matrix $\boldsymbol{Q}$ can be negative definite,
indefinite, or positive definite. The quadratic penalty method involves
minimizing the sequence of penalized objective functions
$\displaystyle f_{\lambda}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{x}^{t}\boldsymbol{Q}\boldsymbol{x}+\boldsymbol{c}^{t}\boldsymbol{x}+\frac{\lambda}{2}\|(\boldsymbol{A}\boldsymbol{x}-\boldsymbol{b})_{+}\|_{2}^{2}+\frac{\lambda}{2}\|\boldsymbol{E}\boldsymbol{x}-\boldsymbol{d}\|_{2}^{2}$
as $\lambda$ tends to $\infty$. Based on the obvious majorization
$\displaystyle x_{+}^{2}$ $\displaystyle\leq$
$\displaystyle\begin{cases}(x-x_{m})^{2}&x_{m}<0\\\ x^{2}&x_{m}\geq
0\end{cases},$
the term $\|(\boldsymbol{A}\boldsymbol{x}-\boldsymbol{b})_{+}\|_{2}^{2}$ is
majorized by
$\|\boldsymbol{A}\boldsymbol{x}-\boldsymbol{b}-\boldsymbol{r}_{m}\|_{2}^{2}$,
where
$\displaystyle\boldsymbol{r}_{m}$ $\displaystyle=$
$\displaystyle\min\\{\boldsymbol{A}\boldsymbol{x}_{m}-\boldsymbol{b},\bf
0\\}.$
A brief calculation shows that $f_{\lambda}(\boldsymbol{x})$ is majorized by
the surrogate function
$\displaystyle g_{\lambda}(\boldsymbol{x}\mid\boldsymbol{x}_{m})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{x}^{t}\boldsymbol{H}_{\lambda}\boldsymbol{x}+\boldsymbol{v}_{\lambda
m}^{t}\boldsymbol{x}$
up to an irrelevant constant, where $\boldsymbol{H}_{\lambda}$ and
$\boldsymbol{v}_{\lambda m}$ are defined by
$\displaystyle\boldsymbol{H}_{\lambda}$ $\displaystyle=$
$\displaystyle\boldsymbol{Q}+\lambda(\boldsymbol{A}^{t}\boldsymbol{A}+\boldsymbol{E}^{t}\boldsymbol{E})$
$\displaystyle\boldsymbol{v}_{\lambda m}$ $\displaystyle=$
$\displaystyle\boldsymbol{c}-\lambda\boldsymbol{A}^{t}(\boldsymbol{b}+\boldsymbol{r}_{m})-\lambda\boldsymbol{E}^{t}\boldsymbol{d}.$
It is convenient to assume that the diagonal coefficients
$\frac{1}{2}h_{\lambda ii}$ appearing in the quadratic form
$\frac{1}{2}\boldsymbol{x}^{T}\boldsymbol{H}_{\lambda}\boldsymbol{x}$ are
positive. This is generally the case for large $\lambda$. One can handle the
off-diagonal term $h_{\lambda ij}x_{i}x_{j}$ by either the majorization (4) or
the majorization (5) according to the sign of $h_{\lambda ij}$. The reader can
check that the MM updates reduce to
$\displaystyle x_{m+1,i}$ $\displaystyle=$
$\displaystyle\frac{x_{mi}}{2}\left[-\frac{v_{\lambda mi}}{h_{\lambda
mi}^{+}}+\sqrt{\left(\frac{v_{\lambda mi}}{h_{\lambda
mi}^{+}}\right)^{2}-4\frac{h_{\lambda mi}^{-}}{h_{\lambda mi}^{+}}}\,\right],$
(8)
where
$\displaystyle h_{\lambda mi}^{+}$ $\displaystyle=$
$\displaystyle\sum_{j:h_{\lambda ij}>0}h_{\lambda ij}x_{mj},\quad\quad
h_{\lambda mi}^{-}=\sum_{j:h_{\lambda ij}<0}h_{\lambda ij}x_{mj}.$
When $h_{\lambda mi}^{-}=0$, the update (8) collapses to
$\displaystyle x_{m+1,i}$ $\displaystyle=$ $\displaystyle
x_{mi}\max\Big{\\{}-\frac{v_{\lambda mi}}{h_{\lambda mi}^{+}},0\Big{\\}}.$ (9)
To avoid sticky boundaries, we replace 0 in equation (9) by a small positive
constant $\epsilon$ such as $10^{-9}$. Sha et al. Sha derived the update (8)
for $\lambda=0$ ignoring the constraints
$\boldsymbol{E}\boldsymbol{x}=\boldsymbol{d}$ and
$\boldsymbol{A}\boldsymbol{x}\leq\boldsymbol{b}$.
For a numerical example without equality constraints take
$\displaystyle f_{10}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\frac{1}{2}x_{1}^{2}+x_{2}^{2}-x_{1}x_{2}-2x_{1}-6x_{2}$
$\displaystyle\boldsymbol{A}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&1\\\ -1&2\\\
2&1\end{pmatrix},\quad\quad\boldsymbol{b}\;\;\,=\;\;\,\begin{pmatrix}2\\\ 2\\\
3\end{pmatrix}.$
The minimum occurs at the point $(2/3,4/3)^{t}$. Table 2 lists the number of
iterations until convergence and the converged point
$\boldsymbol{x}_{\lambda}$ for the sequence of penalty constants
$\lambda=2^{k}$. The quadratic program
$\displaystyle f_{11}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle-8x_{1}-16x_{2}+x_{1}^{2}+4x_{2}^{2}$
$\displaystyle\boldsymbol{A}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&1\\\
1&0\end{pmatrix},\quad\quad\boldsymbol{b}\;\;\,=\;\;\,\begin{pmatrix}4\\\
3\end{pmatrix}$
converges much more slowly. Its minimum occurs at the point $(2.4,1.6)^{t}$.
Table 3 lists the numbers of iterations until convergence with $(q=1$) and
without ($q=0$) acceleration and the converged point
$\boldsymbol{x}_{\lambda}$ for the same sequence of penalty constants
$\lambda=2^{k}$. Fortunately, quasi-Newton acceleration compensates for ill
conditioning in this test problem.
$\log_{2}{\lambda}$ | Iters | $\boldsymbol{x}_{\lambda}$
---|---|---
0 | 8 | (0.9503,1.6464)
1 | 6 | (0.8580,1.5164)
2 | 5 | (0.8138,1.4461)
3 | 23 | (0.7853,1.4067)
4 | 32 | (0.7264,1.3702)
5 | 31 | (0.6967,1.3518)
6 | 30 | (0.6817,1.3426)
7 | 29 | (0.6742,1.3380)
8 | 28 | (0.6704,1.3356)
9 | 26 | (0.6686,1.3345)
10 | 25 | (0.6676,1.3339)
11 | 23 | (0.6671,1.3336)
12 | 22 | (0.6669,1.3335)
13 | 21 | (0.6668,1.3334)
14 | 19 | (0.6667,1.3334)
15 | 18 | (0.6667,1.3334)
16 | 16 | (0.6667,1.3333)
17 | 15 | (0.6667,1.3333)
Table 2: Iterates from the quadratic penalty method for the test function $f_{10}(\boldsymbol{x})$. The convergence criterion for the inner loops is $10^{-9}$. $\log_{2}{\lambda}$ | Iters ($q=0$) | Iters ($q=1$) | $\boldsymbol{x}_{\lambda}$
---|---|---|---
0 | 18 | 5 | (3.0000,1.8000)
1 | 2 | 2 | (2.8571,1.7143)
2 | 56 | 6 | (2.6667,1.6667)
3 | 97 | 5 | (2.5455,1.6364)
4 | 167 | 5 | (2.4762,1.6190)
5 | 312 | 5 | (2.4390,1.6098)
6 | 541 | 6 | (2.4198,1.6049)
7 | 955 | 5 | (2.4099,1.6025)
8 | 1674 | 4 | (2.4050,1.6012)
9 | 2924 | 3 | (2.4025,1.6006)
10 | 4839 | 3 | (2.4013,1.6003)
11 | 7959 | 4 | (2.4006,1.6002)
12 | 12220 | 4 | (2.4003,1.6001)
13 | 17674 | 4 | (2.4002,1.6000)
14 | 21739 | 3 | (2.4001,1.6000)
15 | 20736 | 3 | (2.4000,1.6000)
16 | 8073 | 3 | (2.4000,1.6000)
17 | 111 | 3 | (2.4000,1.6000)
18 | 6 | 4 | (2.4000,1.6000)
19 | 5 | 2 | (2.4000,1.6000)
20 | 3 | 2 | (2.4000,1.6000)
21 | 2 | 2 | (2.4000,1.6000)
Table 3: Iterates from the quadratic penalty method for the test function
$f_{11}(\boldsymbol{x})$. The convergence criterion for the inner loops is
$10^{-16}$.
## 7 Convergence
As we have seen, the behavior of the MM algorithm is intimately tied to the
behavior of the objective function $f(\boldsymbol{x})$. For the sake of
simplicity, we now restrict attention to unconstrained minimization of
posynomials and investigate conditions guaranteeing that $f(\boldsymbol{x})$
possesses a unique minimum on its domain. Uniqueness is related to the strict
convexity of the reparameterization
$\displaystyle h(\boldsymbol{y})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}\boldsymbol{y}}$
of $f(\boldsymbol{x})$, where
$\boldsymbol{\alpha}^{t}\boldsymbol{y}=\sum_{i=1}^{n}\alpha_{i}y_{i}$ is the
inner product of $\boldsymbol{\alpha}$ and $\boldsymbol{y}$ and
$x_{i}=e^{y_{i}}$ for each $i$. The Hessian matrix
$\displaystyle d^{2}h(\boldsymbol{y})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}\boldsymbol{y}}\boldsymbol{\alpha}\boldsymbol{\alpha}^{t}$
of $h(\boldsymbol{y})$ is positive semidefinite, so $h(\boldsymbol{y})$ is
convex. If we let $T$ be the subspace of $\mathbb{R}^{n}$ spanned by
$\\{\boldsymbol{\alpha}\\}_{\boldsymbol{\alpha}\in S}$, then
$h(\boldsymbol{y})$ is strictly convex if and only if $T=\mathbb{R}^{n}$.
Indeed, suppose the condition holds. For any $\boldsymbol{v}\neq{\bf 0}$, we
then must have $\boldsymbol{\alpha}^{t}\boldsymbol{v}\neq 0$ for some
$\boldsymbol{\alpha}\in S$. It follows that
$\displaystyle\boldsymbol{v}^{t}d^{2}h(\boldsymbol{y})\boldsymbol{v}\;\;=\;\;\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}\boldsymbol{y}}(\boldsymbol{\alpha}^{t}\boldsymbol{v})^{2}\;\;>\;\;0,$
and $d^{2}h(\boldsymbol{y})$ is positive definite. Conversely, suppose
$T\neq\mathbb{R}^{n}$, and take $\boldsymbol{v}\neq{\bf 0}$ with
$\boldsymbol{\alpha}^{t}\boldsymbol{v}=0$ for every $\boldsymbol{\alpha}\in
S$. Then $h(\boldsymbol{y}+t\boldsymbol{v})=h(\boldsymbol{y})$ for every
scalar $t$, which is incompatible with $h(\boldsymbol{y})$ being strictly
convex.
Strict convexity guarantees uniqueness, not existence, of a minimum point.
Coerciveness ensures existence. The objective function $f(\boldsymbol{x})$ is
coercive if $f(\boldsymbol{x})$ tends to $\infty$ whenever any component of
$\boldsymbol{x}$ tends to 0 or $\infty$. Under the reparameterization
$x_{i}=e^{y_{i}}$, this is equivalent to $h(\boldsymbol{y})=f(\boldsymbol{x})$
tending to $\infty$ as $\|\boldsymbol{y}\|_{2}$ tends to $\infty$. A necessary
and sufficient condition for this to occur is that
$\max_{\boldsymbol{\alpha}\in S}\boldsymbol{\alpha}^{t}\boldsymbol{v}>0$ for
every $\boldsymbol{v}\neq{\bf 0}$. For a proof, suppose the contrary condition
holds for some $\boldsymbol{v}\neq{\bf 0}$. Then it is clear that
$h(t\boldsymbol{v})$ remains bounded above by $h({\bf 0})$ as the scalar $t$
tends to $\infty$. Conversely, if the stated condition is true, then the
function $q(\boldsymbol{y})=\max_{\boldsymbol{\alpha}\in
S}\boldsymbol{\alpha}^{t}\boldsymbol{y}$ is continuous and achieves its
minimum of $d>0$ on the sphere
$\\{\boldsymbol{y}\in\mathbb{R}^{n}:\|\boldsymbol{y}\|_{2}=1\\}$. It follows
that $q(\boldsymbol{y})\geq d\|\boldsymbol{y}\|_{2}$ and that
$\displaystyle h(\boldsymbol{y})\;\;\geq\;\;\max_{\boldsymbol{\alpha}\in
S}\\{c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}y}\\}\;\;\geq\;\;\left(\min_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}\right)e^{d\|\boldsymbol{y}\|_{2}}.$
This lower bound shows that $h(\boldsymbol{y})$ is coercive.
The coerciveness condition is hard to apply in practice. An equivalent
condition is that the origin ${\bf 0}$ belongs to the interior of the convex
hull of the set $\\{\boldsymbol{\alpha}\\}_{\boldsymbol{\alpha}\in S}$. It is
straightforward to show that the negations of these two conditions are
logically equivalent. Thus, suppose $q(\boldsymbol{v})=\max_{\alpha\in
S}\boldsymbol{\alpha}^{t}\boldsymbol{v}\leq 0$ for some
$\boldsymbol{v}\neq{\bf 0}$. Every convex combination
$\sum_{\boldsymbol{\alpha}}p_{\boldsymbol{\alpha}}\boldsymbol{\alpha}$ then
satisfies
$\left(\sum_{\boldsymbol{\alpha}}p_{\boldsymbol{\alpha}}\boldsymbol{\alpha}\right)^{t}\boldsymbol{v}\leq
0$. If the origin is in the interior of the convex hull, then
$\epsilon\boldsymbol{v}$ is also for every sufficiently small $\epsilon>0$.
But this leads to the contradiction
$\epsilon\boldsymbol{v}^{t}\boldsymbol{v}=\epsilon\|\boldsymbol{v}\|_{2}^{2}\leq
0$. Conversely, suppose ${\bf 0}$ is not in the interior of the convex hull.
According to the separating hyperplane theorem for convex sets, there exists a
unit vector $\boldsymbol{v}$ with $\boldsymbol{v}^{t}\boldsymbol{\alpha}\leq
0=\boldsymbol{v}^{t}{\bf 0}$ for every $\boldsymbol{\alpha}\in S$. In other
words, $q(\boldsymbol{v})\leq 0$. The convex hull criterion is easier to
check, but it is not constructive. In simple cases such as the objective
function $f_{1}(\boldsymbol{x})$ where the power vectors are
$\boldsymbol{\alpha}=(-3,0)^{t}$, $\boldsymbol{\alpha}=(-1,-2)^{t}$, and
$\boldsymbol{\alpha}=(1,1)^{t}$, it is visually obvious that the origin is in
the interior of their convex hull.
One can also check the criterion $q(\boldsymbol{v})>0$ for all
$\boldsymbol{v}\neq{\bf 0}$ by solving a related geometric programming
problem. This problem consists in minimizing the scalar $t$ subject to the
inequality constraints $\boldsymbol{\alpha}^{t}\boldsymbol{y}\leq t$ for all
$\boldsymbol{\alpha}\in S$ and the nonlinear equality constraint
$\|\boldsymbol{y}\|_{2}^{2}=1$. If $t_{\mbox{\scriptsize min}}\leq 0$, then
the original criterion fails.
In some cases, the objective function $f(\boldsymbol{x})$ does not attain its
minimum on the open domain
$\mathbb{R}_{>0}^{n}=\\{\boldsymbol{x}:x_{i}>0,1\leq i\leq n\\}$. This
condition is equivalent to the corresponding function $\ln h(\boldsymbol{y})$
being unbounded below on $\mathbb{R}^{n}$. According to Gordon’s theorem
borwein00 ; lange04 , this can happen if and only if ${\bf 0}$ is not in the
convex hull of the set $\\{\boldsymbol{\alpha}\\}_{\boldsymbol{\alpha}\in S}$.
Alternatively, both conditions are equivalent to the existence of a vector
$\boldsymbol{v}$ with $\boldsymbol{\alpha}^{t}\boldsymbol{v}<0$ for all
$\boldsymbol{\alpha}\in S$. For the objective function
$f_{3}(\boldsymbol{x})$, the power vectors are
$\boldsymbol{\alpha}=(-1,-2)^{t}$ and $\boldsymbol{\alpha}=(1,1)^{t}$. The
origin $(0,0)^{t}$ does not lie on the line segment between them, and the
vector $(-3/2,1)^{t}$ forms a strictly oblique angle with each. As predicted,
$f_{3}(\boldsymbol{x})$ does not attain its infimum on $\mathbb{R}_{>0}^{n}$.
The theoretical development in reference lange04 demonstrates that the MM
algorithm converges at a linear rate to the unique minimum point of the
objective function $f(\boldsymbol{x})$ when $f(\boldsymbol{x})$ is coercive
and its convex reparameterization $h(\boldsymbol{y})$ is strictly convex. The
theory does not cover other cases, and it would be interesting to investigate
them. The general convergence theory of MM algorithms lange04 states that
five properties of the objective function $f(\boldsymbol{x})$ and MM
algorithmic map $\boldsymbol{x}\mapsto M(\boldsymbol{x})$ guarantee
convergence to a stationary point of $f(\boldsymbol{x})$: (a)
$f(\boldsymbol{x})$ is coercive on its open domain; (b) $f(\boldsymbol{x})$
has only isolated stationary points; (c) $M(\boldsymbol{x})$ is continuous;
(d) $\boldsymbol{x}^{*}$ is a fixed point of $M(\boldsymbol{x})$ if and only
if $\boldsymbol{x}^{*}$ is a stationary point of $f(\boldsymbol{x})$; and (e)
$f[M(\boldsymbol{x}^{*})]\geq f(\boldsymbol{x}^{*})$, with equality if and
only if $\boldsymbol{x}^{*}$ is a fixed point of $M(\boldsymbol{x})$. For a
general signomial program, items (a) and (b) are the hardest to check. Our
examples provide some clues.
The standard convergence results for the quadratic penalty method are covered
in the references lange04 ; NocedalWrightBook ; RuszczynskiBook . To summarize
the principal finding, suppose that the objective function $f(\boldsymbol{x})$
and the constraint functions $r_{i}(\boldsymbol{x})$ and
$s_{i}(\boldsymbol{x})$ are continuous and that $f(\boldsymbol{x})$ is
coercive on $\mathbb{R}_{>0}^{n}$. If $\boldsymbol{x}_{\lambda}$ minimizes the
penalized objective function
$\displaystyle f_{\lambda}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
f(\boldsymbol{x})+\lambda\sum_{i}r_{i}(\boldsymbol{x})^{2}+\lambda\sum_{j}s_{j}(\boldsymbol{x})_{+}^{2},$
and $\boldsymbol{x}_{\infty}$ is a cluster point of $\boldsymbol{x}_{\lambda}$
as $\lambda$ tends to $\infty$, then $\boldsymbol{x}_{\infty}$ minimizes
$f(\boldsymbol{x})$ subject to the constraints. In this regard observe that
the coerciveness assumption on $f(\boldsymbol{x})$ implies that the solution
set $\\{\boldsymbol{x}_{\lambda}\\}_{\lambda}$ is bounded and possesses at
least one cluster point. Of course, if the solution set consists of a single
point, then $\boldsymbol{x}_{\lambda}$ tends to that point.
## 8 Discussion
The current paper presents novel algorithms for both geometric and signomial
programming. Although our examples are low dimensional, the previous
experience of Sha et al. Sha offers convincing evidence that the MM algorithm
works well for high-dimensional quadratic programming with nonnegativity
constraints. The ideas pursued here – the MM principle, separation of
variables, quasi-Newton acceleration, and penalized optimization – are
surprisingly potent in large-scale optimization. The MM algorithm deals with
the objective function directly and reduces multivariate minimization to a
sequence of one-dimensional minimizations. The MM updates are simple to code
and enjoy the crucial descent property. Treating constrained signomial
programming by the penalty method extends the MM algorithm even further.
Quadratic programming with linear equality and inequality constraints is the
most important special case of constrained signomial programming. Our new MM
algorithm for constrained quadratic programming deserves consideration in
high-dimensional problems. Even though MM algorithms can be notoriously slow
to converge, quasi-Newton acceleration can dramatically improve matters.
Acceleration involves no matrix inversion, only matrix times vector
multiplication. Finally, it is worth keeping in mind that parameter separated
algorithms are ideal candidates for parallel processing.
Because geometric programs are ultimately convex, it is relatively easy to
pose and check sufficient conditions for global convergence of the MM
algorithm. In contrast it is far more difficult to analyze the behavior of the
MM algorithm for signomial programs. Theoretical progress will probably be
piecemeal and require problem-specific information. A major difficulty is
understanding the asymptotic nature of the objective function as parameters
approach 0 or $\infty$. Even in the absence of theoretical guarantees, the
descent property of the MM algorithm makes it an attractive solution technique
and a diagnostic tool for finding counterexamples. Some of our test problems
expose the behavior of the MM algorithm in non-standard situations. We welcome
the help of the optimization community in unraveling the mysteries of the MM
algorithm in signomial programming.
## References
* (1) J.M. Borwein and A.S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer-Verlag, New York, 2000.
* (2) S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi , A tutorial on geometric programming. Optimization and Engineering, 8:67–127, 2007.
* (3) S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
* (4) J.G. Ecker, Geometric programming: methods, computations and applications, SIAM Review 22 (1980) 338–362.
* (5) P.D. Feigin and U. Passy, The geometric programming dual to the extinction probability problem in simple branching processes, Annals Prob 9 (1981) 498–503.
* (6) M. del Mar Hershenson, S.P. Boyd, and T.H. Lee, Optimal design of a CMOS op-amp via geometric programming, IEEE Trans Computer-Aided Design 20 (2001) 1–21.
* (7) K. Hoffman, Analysis in Euclidean Space, Prentice-Hall, Englewood Cliffs, NJ, 1975.
* (8) D.R. Hunter and K. Lange, A tutorial on MM algorithms, Amer Statistician 58 (2004) 30–37.
* (9) K. Lange, Optimization, Springer-Verlag, New York, 2004.
* (10) K. Lange, D.R. Hunter, and I. Yang, Optimization transfer using surrogate objective functions (with discussion), J Comput Graphical Stat, 9 (2000) 1–59.
* (11) M. Mazumdar and T.R. Jefferson, Maximum likelihood estimates for multinomial probabilities via geometric programming, Biometrika 70 (1983) 257–261.
* (12) J. Nocedal and S.J. Wright. (1999) Numerical Optimization, Springer.
* (13) U. Passy and D.J. Wilde, A geometric programming algorithm for solving chemical equilibrium problems, SIAM J Appl Math, 16 (1968) 363–373.
* (14) A.L. Peressini, F.E. Sullivan, and J.J. Uhl Jr. The Mathematics of Nonlinear Programming, Springer-Verlag, New York, 1988.
* (15) E.L. Peterson, Geometric programming, SIAM Review 18 (1976) 338-362.
* (16) A. Ruszczynski. Optimization, 2006, Princeton University Press.
* (17) F. Sha, L.K. Saul, and D.D. Lee. Multiplicative updates for nonnegative quadratic programming in support vector machines. In S. Becker, S. Thrun, and K. Obermayer (eds.), Advances in Neural Information Processing Systems 15, pages 1065-1073. MIT Press: Cambridge, MA.
* (18) J.M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Inequalities, Cambridge University Press and the Mathematical Association of America, Cambridge, 2004.
* (19) Y. Wang, K. Zhang, and P. Shen. (2002) A new type of condensation curvilinear path algorithm for unconstrained generalized geometric programming, Math. Comput. Modelling 35: 1209–1219.
* (20) H. Zhou, D. Alexander, and K.L. Lange. (2009) A quasi-Newton acceleration method for high-dimensional optimization algorithms, Statistics and Computing, DOI 10.1007/s11222-009-9166-3
* (21) H. Zhou, K.L. Lange, and M.A. Suchard. (2009) Graphical processing units and high-dimensional optimization. arXiv:1003.3272v1
|
arxiv-papers
| 2010-07-14T16:30:00 |
2024-09-04T02:49:11.611479
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kenneth Lange, Hua Zhou",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1007.2371"
}
|
1007.2450
|
# Directional Statistics on Permutations
Sergey M. Plis The Mind Research Network splis@mrn.org , Terran Lane The
University of New Mexico terran@cs.unm.edu and Vince D. Calhoun The Mind
Research Network vcalhoun@mrn.org
###### Abstract.
Distributions over permutations arise in applications ranging from multi-
object tracking to ranking of instances. The difficulty of dealing with these
distributions is caused by the size of their domain, which is factorial in the
number of considered entities ($n!$). It makes the direct definition of a
multinomial distribution over permutation space impractical for all but a very
small $n$. In this work we propose an embedding of all $n!$ permutations for a
given $n$ in a surface of a hypersphere defined in $\mathbbm{R}^{(n-1)^{2}}$.
As a result of the embedding, we acquire ability to define continuous
distributions over a hypersphere with all the benefits of directional
statistics. We provide polynomial time projections between the continuous
hypersphere representation and the $n!$-element permutation space. The
framework provides a way to use continuous directional probability densities
and the methods developed thereof for establishing densities over
permutations. As a demonstration of the benefits of the framework we derive an
inference procedure for a state-space model over permutations. We demonstrate
the approach with applications.
supported by NIH under grant number NCRR 1P20 RR021938
## 1\. Introduction
Since the inception of the field of computer science, there has been a strong
dichotomy between optimization in continuous spaces (such as
$\mathbbm{R}^{d}$) and combinatorial spaces (such as the space of permutations
on $d$ objects). While there are computationally hard problems in both kinds
of spaces, combinatorial spaces are far more often the villain. It seems as if
nearly all interesting learning, optimization, and representation problems in
combinatorial spaces are NP-complete in the best case. Bayesian inference in
the space of permutations, for example, is an important, yet frustratingly
difficult problem [5].
We feel that a key factor at the heart of this dichotomy is that combinatorial
spaces are far more _unstructured_ than the familiar continuous spaces. A
priori, combinatorial spaces are simply sets of objects, with no relationship
among them. Compare this to, say, Euclidean $d$-space, which comes equipped
with a topology, continuity, completeness, compact subsets, a metric, an inner
product, and so on [10]. On these properties are built the entire
infrastructure of analysis, including tools like the derivative [6]. In turn,
the derivative is at the heart of most optimization techniques and
representations such as the Fourier basis. Essentially, the last four
centuries of mathematics has been developing tools for representation and
optimization in continuous spaces. Combinatorial spaces, on the other hand,
have been burdened with fewer assumptions, but endowed with fewer advantages.
One strategy for working with combinatorial spaces is to embed them into
continuous spaces and work there with powerful analytic tools. This trick has
proven to be powerful in, for example, continuous relaxations of integer
programming problems [3]. It has enjoyed relatively less penetration in
machine learning, however. And where versions of it have appeared [2, 9], the
connection to the topology and analytic properties of the embedding space is
typically not made explicit, nor fully exploited.
In this paper, we demonstrate the power of the embedding approach by
developing a fast, accurate approach to Bayesian inference over permutations.
Arising in tasks such as object tracking [5] or ranking [9], this problem is
challenging because of the factorially-large number of parameters in an exact
representation of a general probability distribution in this space. Prior
approaches have worked by approximating a general probability distribution
with a restricted set of basis functions [4], or by embedding the permutation
space only implicitly, and working with a heuristically chosen probability
distribution [9].
The paper follows the hierarchical structure of our main contributions, where
each level of the hierarchy is split into theoretical observations and
developments that make these observations practical:
* •
Theoretical observations: we demonstrate an embedding of the $n!$ permutation
set onto the surface of a hypersphere $\mathbbm{S}^{d}$ centered at the origin
in $\mathbbm{R}^{d+1}$ with $d=(n-1)^{2}-1$.
* –
Observations: we propose a hypersphere embedding of permutations.
* –
Practical results: we develop polynomial time transformations between the
discrete $n!$ permutation space and its continuous hypersphere representation.
* •
Results that allow practical use of the theory: we demonstrate a bridge
between directional statistics [8] and permutation sets that leads to
efficient inference.
* –
Observations: we propose the von Mises-Fisher density over permutations.
* –
Practical results: we develop efficient inference over permutations in a
state-space model.
* *
We employ analytical product and marginalization operations.
* *
We show efficient transformation of partially observed permutations onto the
surface of the hypersphere $\mathbbm{S}^{d}$.
## 2\. Embedding permutations onto a hypersphere surface
Among many representations of permutations in this work we are interested in
the $n\times n$ permutation matrix representation $\mathbf{P}$. Note that
nowhere in the paper we are going to use this as the permutation operator,
which is the usual intension of the matrix representation of permutations. The
permutation matrix representation is a square bistochastic matrix with entries
$\mathbf{P}_{ij}\in\\{0,1\\}$, serves more as an easy to interpret guide and a
way to establish some required properties than an expression for a linear
operator, whereas we interpret it in the rest of the paper merely as a vector
in $\mathbbm{R}^{n^{2}}$. To avoid notation clutter we treat all the matrices
further in the paper as vectors in $\mathbbm{R}^{n^{2}}$ omitting the special
vector stacking operation symbols (such as $vec\left(\cdot\right)$), unless
specified otherwise.
### 2.1. Representation
In this section we will show how a permutation set with $n!$ elements can be
embedded onto the surface of a $(n-1)^{2}$ dimensional hypersphere.
Our representation takes advantage of the geometry of the Birkhoff polytope
and in part relies on the Birkhoff-von Neumann theorem [11], which we state
here without proof.
###### Theorem 1.
All $n\times n$ permutation matrices in $\mathbbm{R}^{n^{2}}$ are extreme
points of a convex $(n-1)^{2}$ dimensional polytope, which is the convex hull
of all bistochastic matrices.
Next, we formulate a lemma that the rest of the section is based on:
###### Lemma 1.
Extreme points of the Birkhoff polytope are located on the surface of a radius
$\sqrt{n-1}$ hypersphere clustered around the center of mass of all $n!$
permutations.
###### Proof.
To show that the statement is valid we first compute the center of mass and
then show that each permutation is located at an equal distance from this
center. The center of mass for all the permutations on $n$ objects is defined
in $\mathbbm{R}^{n^{2}}$ as $c_{M}=\frac{1}{n!}\sum_{k=1}^{n!}\mathbf{P}_{k}$.
We observe that the number of permutation matrices for which
$\mathbf{P}_{11}=1$ is $(n-1)!$, which follows from the effective removal of
the first row and column of an $n\times n$ matrix caused by the assignment.
Thus, $\sum\mathbf{P}_{11}=(n-1)!$ which, following the same reasoning, is
true for any $\mathbf{P}_{ij}$ and leads to
(1) $\displaystyle c_{M}$
$\displaystyle=\frac{1}{n!}(n-1)!\mathbbm{1}=\frac{1}{n}\mathbbm{1}$
To see that all permutations are equidistant from the center of mass, we
observe that $\|\mathbbm{1}-\mathbf{P}\|_{2}=\sqrt{n^{2}-n}$ for any
$\mathbf{P}$. With this observation we can compute the radius of the sphere:
(2) $\displaystyle r_{s}$
$\displaystyle=\left\|\frac{1}{n}\mathbbm{1}-\mathbf{P}\right\|_{2}=\sqrt{(n^{2}-n)\frac{1}{n^{2}}+n(\frac{1}{n}-1)^{2}}=\sqrt{n-1}$
∎
To show that the hypersphere of Lemma 1 is embedded into a space of lower
dimension than $\mathbbm{R}^{n^{2}}$ we observe the following. With respect to
the original formulation of permutations in $\mathbbm{R}^{n^{2}}$, all of the
permutations are located on the intersection of a hypersphere centered at the
origin with $\sqrt{n}$ radius and a hypersphere of Lemma 1. This intersection
is still a hypersphere only with dimension lowered by one. The following lemma
allows us to get the dimension of this hypersphere down to the one of Theorem
1.
###### Lemma 2.
All permutations $\mathbf{P}$ are located on the intersection of $2n-1$
hyperplanes, i.e., in $(n-1)^{2}$-dimensional affine subspace of
$\mathbbm{R}^{n^{2}}$.
###### Proof.
Let us denote by $\mathbf{W}_{i,\mbox{\boldmath$1$}}$ an $n\times n$ matrix
with all elements except a single $i^{th}$ row of ones set to zero and
likewise $\mathbf{W}_{\mbox{\boldmath$1$},i}$ for columns. Observe that:
(3)
$\displaystyle\begin{split}vec\left(\mathbf{W}_{i,\mbox{\boldmath$1$}}\right)^{T}vec\left(\mathbf{P}\right)&=1\end{split}\begin{split}vec\left(\mathbf{W}_{\mbox{\boldmath$1$},i}\right)^{T}vec\left(\mathbf{P}\right)&=1\end{split}$
for any permutation matrix111In fact, for any bistochastic matrix, as implied
by Theorem 1. It follows, that all permutations are located at an intersection
of $2n$ hyperplanes defined by their normals:
$\mathbf{W}_{\mbox{\boldmath$1$},i}$ and $\mathbf{W}_{i,\mbox{\boldmath$1$}}$,
with $n\in\\{1\ldots n\\}$, and having bias of 1. This set is, however, not
independent, because any $\mathbf{W}_{i,\mbox{\boldmath$1$}}$ can be expressed
by a linear combination of the other $2n-1$ vectors by setting weights of
$\mathbf{W}_{j\neq i,\mbox{\boldmath$1$}}$ to $-1$ and weights of
$\mathbf{W}_{\mbox{\boldmath$1$},i}$ to 1 for $i,j\in\\{1\ldots n\\}$. This
leads to $2n-1$ hyperplanes whose intersection forms the space in which the
hypersphere containing the Birkhoff-polytope is located. Thus, the dimension
of the space containing the polytope is $n^{2}-2n+1=(n-1)^{2}$. ∎
All permutation matrices on $n$ objects belong to the surface of a radius
$\sqrt{n-1}$ hypersphere, $\mathbbm{S}^{d}$, in $\mathbbm{R}^{(n-1)^{2}}$ as
established by Lemmas 1 and 2. We do not rigorously show here, but assume that
by inherent symmetry in the structure of permutation matrices they are
distributed evenly across the surface of $\mathbbm{S}^{d}$.
### 2.2. Transformations
The representation of the previous section allows us to define and manipulate
probability density functions on $\mathbbm{S}^{d}$ using approaches of
continuous mathematics and only then transforming quantities of interest back
to the discrete $n!$ permutation space. This is useful when there is a way to
efficiently transform elements of one space to the other. Next we show how
this can be achieved in polynomial time.
The key components posing difficulties are discrete vs. continuous space, and
the requirement of $\mathbbm{S}^{d}$ to be origin-centered (required for
Section 3). The former poses a considerably more challenging problem than the
latter and absence of both would reduce the required transformations to a
simple change of basis between $\mathbbm{R}^{n^{2}}$ and
$\mathbbm{R}^{(n-1)^{2}}$. We develop the transformations in the proof to the
following lemma.
###### Lemma 3.
There exist polynomial time transformations between the discrete $n!$
permutation space and the surface of the origin-centered $(n-1)^{2}$
dimensional hypersphere of radius $\sqrt{n-1}$.
###### Proof.
The transformation from a permutation space to $\mathbbm{S}^{d}$ requires only
a short sequence of linear operations as it is made clear by lemmas of Section
2.1:
1. (1)
Shift the permutation matrix $\mathbf{P}$ by $\frac{1}{n}\mathbbm{1}$ to put
the center of mass at the origin.
2. (2)
Change the basis by projecting into the $\mathbbm{R}^{(n-1)^{2}}$ subspace
orthogonal to $\mathbf{W}_{\mbox{\boldmath$1$},i}$ and
$\mathbf{W}_{i,\mbox{\boldmath$1$}}$.
Since there are $(n-1)^{2}$ basis vectors of length $n^{2}$, the projection
operation takes $O(n^{4})$. Note that the basis can be obtained by the QR
factorization, which is $O(n^{6})$ in this case, but needs to be computed only
once for a given $n$.
Transforming an arbitrary point from $\mathbbm{S}^{d}$ to the permutation
space is more challenging. Now we have to linearly transform the point from
$\mathbbm{S}^{d}$ to $\mathbbm{R}^{n^{2}}$ and then among $n!$ possibilities
find a permutation, that is the closest, in $L_{2}$ sense, to a given point.
The transformation is easily done by inverting the order of operations for
going from $\mathbbm{R}^{n^{2}}$ to $\mathbbm{S}^{d}$, which amounts to
$O(n^{4})$ operations. Let us show how to efficiently find a permutation
matrix closest to a transformed point.
Given an arbitrary point $\mathbf{T}^{\mathbbm{S}}$ in $\mathbbm{R}^{n^{2}}$,
which corresponds to a point on $\mathbbm{S}^{d}$, as indicated by the
superscript, we introduce a matrix $\mathbf{D}$ where
(4) $\displaystyle\mathbf{D}_{ij}=(\mathbf{T}_{ij}^{\mathbbm{S}}-1)^{2}$
Finding the permutation $\mathbf{P}^{\mathbbm{S}}$ closest to
$\mathbf{T}^{\mathbbm{S}}$ amounts to finding $\mathbf{P}^{\mathbbm{S}}$ that
minimizes $\sum_{ij}\mathbf{D}_{ij}\mathbf{P}_{ij}$. This is the same as
matching every column and each row to a single counterpart so that the sum of
matching weights (elements of $\mathbf{D}$) is minimal. In this case,
$\mathbf{D}$ is an $n\times n$ edge-weight matrix for a $2n$ node bipartite
graph with $n$ elements per partition. This is the familiar minimum weighted
bipartite matching problem [13]. This observation allows us to apply a minimum
weighted bipartite matching algorithm [13] and obtain a permutation
$\mathbf{P}^{\mathbbm{S}}$ closest to $\mathbf{T}^{\mathbbm{S}}$. The running
time of the fastest general algorithms for solving this problem is
$O(n^{2}\log{n}+n^{2}e)$, where $e$ is the number of edges in the bipartite
graph. Since the number of edges in our case is always $n$, the running time
effectively becomes $O(n^{3})$. However it is dominated by the time of
projecting a point from $\mathbbm{S}^{d}$ to $\mathbbm{R}^{n^{2}}$, which is
$O(n^{4})$ as shown before. ∎
Coupling the probability representations to the transformation operations
bridges the gap between the discrete, combinatorial space of permutations and
the continuous, low-dimensional hypersphere. This allows us to lift the large
body of results developed for directional statistics [8] directly to
permutation inference.
## 3\. Directional statistics
A number of probability density functions on $\mathbbm{S}^{d}$ have been
developed in the field of directional statistics [8]. A detailed account is
given for an interested reader in [8, Chapter 9]. The directional statistics
framework allows us to define quite general classes of density functions over
permutations. In the rest of the paper, we use one of the basic models to
demonstrate the usefulness of our representation and the model as well.
### 3.1. von Mises-Fisher distribution
This is a $m$-variate von Mises-Fisher222Sometimes also called the Langevin
distribution. (vMF) distribution of a $m$-dimensional vector $x$, where
$\|\mbox{\boldmath$\mu$}\|=1$, $\mbox{\boldmath$\kappa$}\geq 0$ and $m\geq 2$:
(5)
$\displaystyle\begin{split}f(\mbox{\boldmath$x$}|\mbox{\boldmath$\mu$},\kappa)&=Z_{m}\left(\kappa\right)e^{\kappa\mbox{\boldmath$\mu$}^{T}\mbox{\boldmath$x$}}\end{split}\begin{split}\hskip
3.61371pt&\mbox{with normalization term}\hskip
10.84006pt\end{split}\begin{split}Z_{m}\left(\kappa\right)&=\frac{\kappa^{m/2-1}}{(2\pi)^{m/2}\bm{I}_{m/2-1}(\kappa)},\end{split}$
where $\bm{I}_{r}(\cdot)$ is the $r^{th}$ order modified Bessel function of
the first kind and $\kappa$ is called the concentration parameter. Examples of
samples from the distribution on $\mathbbm{S}^{2}$ are shown in Figure 1.
Figure 1. Samples of the von Mises-Fisher density function on
$\mathbbm{S}^{2}$ for random $\mu$ and $\kappa$.
In terms of a pdf on permutations the vMF establishes a distance-based model,
where distances are geodesic on $\mathbbm{S}^{d}$. The advantage of the
formulation in a continuous space is the ability to apply a range of
operations on the pdf and still end up with the result on $\mathbbm{S}^{d}$.
This advantage is realized in the inference procedures which we establish
next.
### 3.2. Efficient inference in a state space model
The results presented above establish a framework in which it is possible to
define and manage in reasonable time probability densities over permutations.
An important application of this framework is in the probabilistic data
association (PDA) [12]. In PDA we are interested in maintaining links between
objects and tracks under the noisy tracking conditions. Ignoring the
underlying position estimation problem we focus on the part related to the
identity management, as in [5], which boils down to tracking a hidden
permutation (identity assignment) under a noisy observed assignment.
In order to perform identity tracking of permutations in the context of
recursive Bayesian filtering (which we are going to do) we need to define the
following components:
1. (1)
A transition model, $P\left(\mathbf{X}_{t}|\mathbf{X}_{t-1}\right)$;
2. (2)
An observation model, $P\left(\mathbf{Y}_{t}|\mathbf{X}_{t}\right)$ where
$\mathbf{Y}_{t}$ is the noisy observation of the hidden permutation matrix
$\mathbf{X}_{t}$;
3. (3)
A way to perform the following operations:
(6) multiplication: $\displaystyle P(\mathbf{X}_{t}|\mathbf{Y}_{t})\propto
P(\mathbf{Y}_{t}|\mathbf{X}_{t})P(\mathbf{X}_{t}|\mathbf{Y}_{t-1})$ (7)
marginalization: $\displaystyle P(\mathbf{X}_{t}|\mathbf{Y}_{t-1})=\int
P(\mathbf{X}_{t}|\mathbf{X}_{t-1})P(\mathbf{X}_{t-1}|\mathbf{Y}_{t-1})d\mathbf{X}_{t-1}$
Avoiding transformation overhead we restrict all of the above to
$\mathbbm{S}^{d}$. Hence, $\mathbf{X}$ and $\mathbf{Y}$ are $\mathbbm{S}^{d}$
representations of their respective hidden and observed permutations. We
define both transition and observation models as vMF functions centered at the
true permutation. Due to similarity of the vMF model to the multivariate
Gaussian density, it seems natural to view this recursive filter as an analogy
of the Kalman filter. In this view, the result of this sections is porting a
widely successful tracking model to the discrete $n!$ permutation space.
To further stress the analogy with the Kalman filter, we show that projection
operation can be computed analytically in a closed form and marginalization
operation can be efficiently approximated with good accuracy [1, 8]. For
observation model $P(\mathbf{Y}_{t}|\mathbf{X}_{t})\propto
vMF(\mathbf{Y}_{t},\kappa_{obs})$ and posterior model
$P(\mathbf{X}_{t}|\mathbf{Y}_{t-1})\propto
vMF(\mbox{\boldmath$\mu$}_{pos},\kappa_{pos})$ the multiplication operation
results in a vMF for $P(\mathbf{X}_{t}|\mathbf{Y}_{t})$ parametrized as
(8)
$\displaystyle\begin{split}\mbox{\boldmath$\mu$}_{t}&=\frac{1}{\kappa}\left(\kappa_{obs}\mathbf{Y}_{t}+\kappa_{pos}\mbox{\boldmath$\mu$}_{pos}\right)\end{split}$
$\displaystyle\begin{split}\kappa_{t}&=\|\kappa_{obs}\mathbf{Y}_{t}+\kappa_{pos}\mbox{\boldmath$\mu$}_{pos}\|.\end{split}$
In the case of a vMF transition model, the marginalization can be performed
with a reasonable accuracy and speed using the fact that a vMF can be
approximated by an angular Gaussian and performing analytical convolution of
angular Gaussian with subsequent projection back to vMF space [8]. Resulting
vMF $P(\mathbf{X}_{t}|\mathbf{Y}_{t-1})$ is parametrized as:
(9)
$\displaystyle\begin{split}\mbox{\boldmath$\mu$}&=\mathbf{X}_{t-1}+\mbox{\boldmath$\mu$}_{pos}\end{split}\begin{split}\kappa&=A_{d}^{-1}(A_{d}(\kappa_{pos})A_{d}(\kappa_{tr}))\end{split}$
$\displaystyle\begin{split}A_{d}(\kappa)&=\frac{\bm{I}_{d/2}(\kappa)}{\bm{I}_{d/2-1}(\kappa)}\end{split}$
The ratio of modified Bessel functions required for this approach can be
efficiently computed with high accuracy by using Lentz method based on
evaluating continued fractions [7].
#### 3.2.1. Partial observations
Figure 2. An example of a fallback to a lower dimensional permutation space
when a partial observation becomes available.
Analytical computation of the Bayesian recursive filtering presented above
relies on the fact that permutations are observed completely. In tracking
problems that would mean the algorithm has to receive observations (up to
noise) of identities of every tracked object. This is a rare setting and most
commonly observations are available only partially.
When a partial observation of $o$ objects becomes available, the dimension of
the unknown part of $\mathbf{Y}$ is reduced from $n^{2}$ to $(n-o)^{2}$. The
mechanism of this is shown in Figure 2, where circles indicate two observed
objects and squares indicate the unknown parts of $\mathbf{P}$. The unknown
part of the representation of $\mathbf{P}$ on $\mathbbm{S}^{d}$ needs to be
marginalized out to obtain the likelihood used in (6). Figure 2 shows that
this marginalization is straightforward in $\mathbbm{R}^{n^{2}}$ space.
Unfortunately, to implement (7), we need to marginalize on the surface of the
sphere, $\mathbbm{S}^{d}\subset\mathbbm{R}^{(n-1)^{2}}$ – a much more
difficult task.
Denoting the orthogonal part of the basis in $\mathbbm{R}^{n^{2}}$ that
represents the $\mathbbm{R}^{(n-1)^{2}}$ subspace by an $n^{2}\times(n-1)^{2}$
matrix $\mathbf{Q}$, we project into this subspace by:
(10) $\displaystyle vec\left(\mathbf{Y}\right)$
$\displaystyle=\mathbf{Q}^{T}vec\left(\mathbf{P-\frac{1}{n}\mathbbm{1}}\right).$
In the case of a partial observation, we know which elements of the vector
being projected are consistent with the observation and are not going to
change and which elements can have any possible value. This allows us to split
the resulting vector $\mathbf{Y}$ into
(11) $\displaystyle\mathbf{Y}=\mathbf{Y}_{*}+\mathbf{Y}_{?},$
where $\mathbf{Y}*$ and $\mathbf{Y}_{?}$ respectively denote the observed and
unobserved parts.
The likelihood with the unknown observations marginalized out becomes:
(12)
$\displaystyle\frac{1}{Z}\int_{\mathbf{Y}_{?}}e^{\kappa_{1}\mathbf{Y}_{*}^{T}\mbox{\boldmath$x$}+\kappa_{1}\mathbf{Y}_{?}^{T}\mbox{\boldmath$x$}}d\mathbf{Y}_{?}$
$\displaystyle=\frac{1}{Z}e^{\kappa_{1}\mathbf{Y}_{*}^{T}\mbox{\boldmath$x$}}\int_{\mathbf{Y}_{?}}e^{\kappa_{1}\mathbf{Y}_{?}^{T}\mbox{\boldmath$x$}}d\mathbf{Y}_{?}$
Some details make computing the integral in (12) not totally trivial:
$\mbox{\boldmath$x$},\mathbf{Y}_{*}$, and $\mathbf{Y}_{?}$ are of different
length; although $x$ is fixed, $\mathbf{Y}_{*}$ and $\mathbf{Y}_{?}$ are not
allowed to take any possible angle in $\mathbbm{R}^{(n-1)^{2}}$. We omit the
details of the derivation dealing with these difficulties and just state the
parameters of the resulting vMF likelihood function:
(13)
$\displaystyle\begin{split}\mbox{\boldmath$\mu$}&=\frac{\mathbf{Y}_{*}}{\|\mathbf{Y}_{*}\|_{2}}\end{split}\begin{split}\kappa&=\|\kappa_{1}\mathbf{Y}_{*}\|_{2}\end{split}$
Thus, in the case of vMF we can execute a recursive Bayesian filter using only
analytical computation even in the cases when only partially observed data is
available. This makes the state space model applicable in a much wider range
of scenarios than our initial model presented in Section LABEL:sec:cfo.
## 4\. Experiments
(a) (b)
Figure 3. Average error of a random hidden permutation inference from 100
(partial) noisy observations on 25 and 50 objects simulated datasets. Runs
were repeated 10 times with a different permutation.
To demonstrate correctness of our approach, we show inference of a fixed
hidden permutation from its noisy partial observations. Figure 3 shows results
of this inference on dataset of 25 and 50 objects. In these first, synthetic
data, experiments, we first randomly chose a true (hidden) permutation,
$\mathbf{P}_{true}$. We controlled both observation noise
($\nu\in{0.1,0.2,...,0.9}$) and fraction of objects missing from observations
($m\in{0\%,20\%,40\%,60\%}$). Noisy observations were drawn from
vMF($\mathbf{P}_{true}$,$\kappa_{\nu}$), where $\kappa_{\nu}$ was chosen to
achieve $\nu$ fraction of incorrectly observed object identities. The final
observation, $\mathbf{P}_{m}$, was generated by hiding $m$ percent of entries
from the noisy observation matrix, chosen uniformly at random without
replacement. Figure 3 shows that our representation of the $n!$ discrete
permutation space is functional and the approach can gracefully handle large
number of objects, partial observations and observation noise.
(a) (b)
Figure 4. Tracking error on the air traffic control dataset for 6 and 10
planes as a function of observation noise shown as the fraction of incorrectly
reported planes. Separate plots show error for partial observations when a
fraction of object identities is unobserved.
The above simulation was generated with the noise model used by the inference
and did not have a temporal component, although it was applied to a really
large state space. Next we show experiments on a tracking dataset with a non-
vMF transition model. We use a dataset of planar locations of aircraft within
a 30 mile diameter of John F. Kennedy airport of New York. The data, in
streaming format, is available at http://www4.passur.com/jfk.html. The
complexity of the plane routes and frequent crossings of tracks in the planar
projection make this an interesting dataset for identity tracking. Identity
tracking results on this dataset, in the context of the symmetric semigroup
approach to permutation inference, were previously reported in [5].
Replicating the task reported in [5], we show results on tracking datasets of
6 and 10 flights, dropping the 15 flights dataset (but see below).
The dataset comes prelabeled, but the uncertainty is introduced by randomly
swapping identities of flights $i$ and $j$ at their respective locations
$\mbox{\boldmath$x$}_{i}$ and $\mbox{\boldmath$x$}_{j}$ with probability
$p_{swap}exp(-\|\mbox{\boldmath$x$}_{j}(t)-\mbox{\boldmath$x$}_{i}(t)\|^{2}/(2s^{2}))$,
where $p_{swap}=0.1$ and $s=0.1$ are strength and scale parameters
respectively.
We then generated observation and hidden identity noise in the same way as for
the prior experiment. Figure 4 shows results of applying our identity tracking
method to the air traffic control dataset for various levels of observation
noise and amount of missing identity observations. It is difficult to compare
the performance to the method of [5] applied to the same dataset, since it is
not clear how observation noise levels correspond to each other. However,
error values reported in [5] were 0.12 to 0.17 on the 6 flights dataset and
0.2 to 0.32 on the 10 flights dataset. This is comparable to what we get with
our approach for observation error below 50%, even when 60% of the flight
identities are unobserved. Results of the application of our state space model
to this dataset indicate robustness of the model to the choice of the
transition model, which was different from the generative model of our
tracking inference engine.
(a) (b)
Figure 5. Tracking error on a football visual surveillance dataset for 41
players as a function of observation noise. Different missing data fractions
are shown.
Due to the unmanageable size of the factorial space in identity tracking
problems, even the powerful and efficient methods based on Fourier
representation of permutations do not report results on more than 11 [4] or 15
[5] simultaneously tracked objects. The results of Figure 3 show that our
approach can handle large numbers of objects, and Figure 4 demonstrated
comparable accuracy on the air traffic control dataset. Next we show results
on 41 objects from a visual surveillance dataset available from
http://vspets.visualsurveillance.org/. Figure 5 shows an example of the
underlying data and results of the identity tracking. The problem is similar
to the above air traffic control: we have added uncertainty to the players,
identities using the same exponential proximity model as before. Further,
unlike the air traffic domain, here there are very few time steps that do
_not_ involve an identity swap. This kind of situation is difficult for
recursive Bayesian filtering in general. However, our approach handles the
situation and produces reasonable results with acceptable error rate – indeed,
quite a good error rate, considering the size of the state space.
## 5\. Conclusions
The main result of this work is embedding permutations into a continuous
manifold, thus lifting a body of results from directional statistics field [8]
to the fields of ranking, identity tracking and others, where permutations
play essential role. Among many potential applications of this embedding we
have chosen probabilistic identity tracking and were able to set up a state-
space model with efficient recursive Bayesian filter that produced results
comparable with the state of the art techniques very efficiently even on a
very large datasets that pose difficulties to existing methods. There remains
much to be done in this direction. However, a simple model, that can be
thought of as a continuous generalization of the Mallows model [9, 2],
equipped with results from the field of directional statistics has efficiently
produced results of a reasonable accuracy. This is promising and encourages
further development of more complicated probability distributions for
permutations: further exploration of the exponential family already developed
in the field [8] as well as developing more complex representations using
spherical harmonics representations.
## Acknowledgments
We thank Risi Kondor for being impressively responsive and generously
providing the air traffic control dataset. This work was supported by NIH
under grant number NCRR 1P20 RR021938. Dr. Lane’s work was supported by NSF
under Grant No. 0705681.
## References
* [1] A. Chiuso and G. Picci. Visual tracking of points as estimation on the unit sphere. The confluence of vision and control, pages 90–105, 1998.
* [2] M. A. Fligner and J. S. Verducci. Distance based ranking models. Journal of the Royal Statistical Society. Series B (Methodological), 48(3):359–369, 1986.
* [3] R. E. Gomory. Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society, 64(5):275–278, 1958\.
* [4] J. Huang, C. Guestrin, and L. Guibas. Fourier theoretic probabilistic inference over permutations. Journal of Machine Learning (JMLR), 10:997–1070, May 2009.
* [5] R. Kondor, A. Howard, and T. Jebara. Multi-object tracking with representations of the symmetric group. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, March 2007.
* [6] E. Kreyszig. Introductory Functional Analysis with Applications. John Wiley & Sons, 1978.
* [7] W. J. Lentz. Generating bessel functions in mie scattering calculations using continued fractions. Appl. Opt, 15:668–671, 1976.
* [8] K. V. Mardia and P. E. Jupp. Directional Statistics. John Wiley and Sons Ltd., 2 edition, 2000.
* [9] M. Meila, K. Phadnis, A. Patterson, and J. Bilmes. Consensus ranking under the exponential model. In Proceedings of the 23rd Annual Conference on Uncertainty in Artificial Intelligence (UAI), pages 285–294, Corvallis, Oregon, 2007. AUAI Press.
* [10] J. R. Munkres. Topology: A First Course. Prentice Hall, 1975.
* [11] T. E. S. Raghavan R. B. Bapat. Nonnegative Matrices and Applications. Encyclopedia of mathematics and its applications. cambridge University Press, 1997.
* [12] C. Rasmussen and G. D. Hager. Probabilistic data association methods for tracking complex visual objects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(6):560–576, 2001.
* [13] D. B. West. Introduction to graph theory. Prentice Hall Upper Saddle River, NJ, 2001.
|
arxiv-papers
| 2010-07-14T22:59:19 |
2024-09-04T02:49:11.621382
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sergey M. Plis and Terran Lane and Vince D. Calhoun",
"submitter": "Sergey Plis",
"url": "https://arxiv.org/abs/1007.2450"
}
|
1007.2462
|
# QCD inspired relativistic bound state model and meson structures
Shun-Jin Wang sjwang@home.swjtu.edu.cn School of Physics and Technology,
Sichuan University, Chengdu, 610064, PR China Jun Tao taojscu@gmail.com
School of Physics and Technology, Sichuan University, Chengdu, 610064, PR
China Xiao-Bo Guo School of Science, Southwest University of Science and
Technology, Mianyang 621010, PR China Lei Li School of Science, Southwest
University of Science and Technology, Mianyang 621010, PR China
###### Abstract
A QCD inspired relativistic effective Hamiltonian model for the bound states
of mesons has been constructed, which integrates the advantages of several QCD
effective Hamiltonian models. Based on light-front QCD effective Hamiltonian
model, the squared invariant mass operator of meson is used as the effective
Hamiltonian. The model has been improved significantly in four major aspects:
i) it is proved that in center of mass frame and in internal coordinate
Hilbert subspace, the total angular momentum $J$ of meson is conserved and the
mass eigen equation can be expressed in total angular momentum representation
and in terms of a set of coupled radial eigen equations for each $J$; ii)
Based on lattice QCD results, a relativistic confining potential is introduced
into the effective interaction and the excited states of mesons can be well
described; iii) an SU(3) flavor mixing interaction is introduced
phenomenologically to describe the flavor mixing mesons and the mass eigen
equations contain the coupling among different flavor components; iv) the mass
eigen equations are of relativistic covariance and the coupled radial mass
eigen equations take full account of $L-S$ coupling and tensor interactions.
The model has been applied to describe the whole meson spectra of about 265
mesons with available data, and the mass eigen equations have been solved
nonperturbatively and numerically. The agreement of the calculated masses,
squared radii, and decay constants with data is quite well. For the mesons
whose mass data have large experimental uncertainty, the model produces
certain mass values for test. For some mesons whose total angular momenta and
parity are not assigned experimentally, the model gives a prediction of the
spectroscopic configuration${}^{2S+1}L_{J}$. The connection between our model
and the recent low energy QCD issues-the infrared conformal scaling invariance
and holographic QCD hadron models is discussed.
###### pacs:
12.38.Lg, 11.10.Ef, 14.40.-n
## I Introduction
To study hadronic properties at low energy scales, nonperturbative effects
must be taken into accountWilson . To describe mesons and baryons, there are
several main approaches: coupled Bethe-Salpeter(BS) and Dyson-Schwinger(DS)
equation approach, relativistic constituent quark model based on Bethe-
Salpeter equation(BSE), relativistic string Hamiltonian approach, and
holographic light-front QCD approach. In the coupled Bethe-Salpeter and Dyson-
Schwinger equation approach by P. Maris, P. Tandy, L. Kaptari et al.MarisTandy
, the dressed quark propagators are assumed to have time like complex mass
poles where the absence of real mass poles simulates quark confinement; the BS
kernel is approximated by ladder rainbow truncation with two-parameter
infrared structure. The approach contains four parameters in u-d-s quark
sector and is consistent with quark and gluon confinement. Besides, it has the
feature of preserving the relevant Ward identity and generating Dynamical
chiral symmetry breaking. The vector mesons $\rho,\phi$, and $K^{*}$ are
studied in detail, the calculated masses of $\rho,\phi$,and $K^{*}$ mesons and
decay constants $f_{\rho},f_{\phi}$, and $f_{K^{*}}$ are within $5\%$ and
$10\%$ of the data respectively. Moreover, the ground-state spectra of light-
quark mesons are also studied and a good description of flavor-octet
pseudoscalar, vector, and axial-vector meson spectrum is obtained. The
applicable domain of ladder truncation and the relative importance of various
components of the two-body BS amplitude are also explored. However heavy quark
mesons are not investigated and the number of mesons treated are not too many.
R. Alkofer, P. Watson, and H. Weigel AlkoferMeigel follow the same approach,
scalar and pseudoscalar, vector and axial vector mesons are studied. A similar
approach is pursued by P. Jain and MunczekMunczek , about 50 mesons are
investigated and the results are in good agreement with experiments. But heavy
quarks are analyzed by non-relativistic dynamics. It should be noted that in
contrary to Hamiltonian dynamics which works with wave functions that are not
manifestly covariant quantities, the above BSE/DSE approaches emphasize the
relativistic covariant aspect of the formalism and invariant quantities are
studied.
The constituent quark model(CQM) works surprisingly well for most of the
observed hadronic states Isgur01 ; Hersbach01 . However, as a phenomenological
theory, there are still some problems and puzzles that need to be clarified
and understoodGodfrey . One of the most important problems is relativistic
effect. To solve the relativistic covariant problem of CQM, the relativistic
constituent quark model based on Bethe-Salpeter equation was proposed by B.
Metsch et al. in Bonn Group Metsch . In this approach, the meson and baryon
Hamiltonians are extracted from Bethe-Salpeter equation and the relativistic
covariant constituent quark models for mesons and baryons are constructed.
Based on Dirac structure of the two-body effective interactions, two types of
models( A and B) are constructed. This approach addresses hadron mass spectra
from ground state to 3GeV, light-flavor mesons, scalar excitations, linear
Regge trajectory, pseudoscalar mixing, and parity doublet(for baryons). In
this approach, the Dyson-Schwinger equation(DSE) is approximated by
parametrization of infrared effective gluon propagator, the interaction kernel
of BSE is given by single gluon exchange(OGE) and the confinement is
parameterized by a string-like potential ( having two versions defined by
Dirac structures A and B ). The instanton-induced spin-flavor dependent
interaction is also included in the BSE kernel. The mass spectra up to 3 GeV,
electroweak and strong-decay properties are calculated with 7 to 9 parameters.
About 60 scalar and pseudo-scalar, vector and axial vector, and some tensor
mesons with J=0,1,2 are calculated by models A and B, and compared to Godfrey-
Isgur’s calculation and experimental data ( the deviation seems large but the
errors are not indicated ). Due to the Dirac structure of the effective
interactions, spin-spin and spin-orbital interactions are included. Besides,
heavy mesons are not treated.
The relativistic string Hamiltonian approach was proposed by A.M. Badalian et
al.Badalian . The merit of this approach is that the quark-anti-quark
interaction and the confinement are generated by the relativistic string (
through Nambu-Goto action for QCD vacuum fluctuation) which leads to a large
reduction of the number of model parameters. After quantizing the action by
path integral, they construct a Hamiltonian with a linear confining potential
and hyperfine quark-anti-quark interactions. Using only one parameter of
string tension, they study the systematic property of orbital excitations and
rotation of mesons. The linear Regge trajectory relation between squared mass
and orbital angular momentum is produced nicely and in agreement with the data
for about 40 mesons. The relativistic string Hamiltonian approach is spin-
independent. In the lowest order, this approach doesn’t contain spin-spin,
spin-orbital, and tensor interactions, thus it can produce the spin averaged
mass spectra for mesons. However, to include the higher order effects by
perturbation method, the hyperfine spin-dependent interactions could be
obtained.
The holographic light-front QCD approach by S. J. Brodsky and G. F. de
Teramond et al. Brod is based on light-front QCD and AdS/CFT correspondence.
The AdS/CFT correspondence between string theory in AdS space and conformal
field theories in physical space-time leads to an analytic, semi-classical
model for strongly-coupled QCD, which has scale invariance and dimensional
counting at short distances and color confinement at large distances. This
correspondence also provides AdS/CFT or holographic QCD predictions for the
analytic form of the frame-independent light-front wave functions (LFWFs) and
masses of mesons and baryons. Recently, Brodsky $et\ al.$Brod have found that
the transverse separation of quarks within hadron is related to holographic
coordinate (the fifth dimensional z-coordinate) in AdS/CFT correspondence, the
mass eigen equation of meson in light-front effective Hamiltonian approach
corresponds to the equation of motion for the holographic field of effective
gravity field of super string in AdS space at low energy limit. Recently, they
have modified the gravitation background by using a positive-sign dilaton
metric to generate confinement and break conformal symmetry. In the meanwhile,
the chiral symmetry is broken and a mass scale is introduced to simulate the
effect. Based on AdS/CFT correspondence, the holographic light-front QCD model
yields a first order description of some hadronic spectra. This model is quite
appealing and promising, since it has established a profound relationship
between super string theory and QCD in low energy limit. In this model, very
few parameters (cutoff parameter $\Lambda_{QCD}$) are used to obtain the
spectra for both mesons and baryons, such as $\pi$, $\rho$, and $\Delta$,
etc., which fit the experimental data wellBrod . However, for the large body
of mesons, only few of them are described properly and a large part of mesons
are still left over. Besides, in its preset form the full spin interactions
are not treated properly although it has potential to describe spin
splittings.
The light-front formalismDirac provides a convenient nonperturbative
framework for the relativistic description of hadrons in terms of quark and
gluon degrees of freedomMa . Some fundamental nonperturbative light-front QCD
approaches are available, such as light-front Bethe-Salpeter
approachKisslinger , holographic light-front QCD modelBrod , and light-front
Hamiltonian methodBPP . The light-front Bethe-Salpeter approach has been
proposed by Kisslinger et al. to study pion form factor and the transition
from non-perturbative to perturbative QCD calculation of pion form factor.
Like the B-S approach of instant form, the equation of motion for light-front
B-S wave function should be solved together with Schwinger-Dyson equation for
dressed quark propagator, vertex, and self-energy, and the model parameters
include confining potential strengths, and others for parametrization of the
BS Kernel and the running quark masses. An interesting conclusion drawn from
the study of this approach is that the perturbative QCD calculation works at
the energy of 4-5 GeV, much lower than that explored previously by the instant
form of QCD.
The effective light-front QCD Hamiltonian theory proposed by Brodsky and Pauli
BPP is an attempt to describe the hadron structure as a bound constituent
quark system in terms of Fock-space for the light-front wave-function. The
effective Hamiltonian of the approach has been constructed recursively from
the larger valence quark and anti-quark Fock sectors and reduced to the lowest
valence quark-anti-quark sectorZhou . Because of some unique features,
particularly the apparent simplicity of the light-front vacuum, this model is
a promising approach to the bound-state problem of relativistic composite
systems. Within the framework of the discretized light-front QCD, Pauli
$et\,al.$ have derived non-perturbatively an effective light-front Hamiltonian
for mesons, which acts only on the $q\bar{q}$ sectorPauli01 ; Pauli02 . The
mass eigen equations of mesons are formulated in momentum-helicity
representation which hinders its solution in total angular momentum
representation. Besides, in this effective Hamiltonian, confining potentials
and flavor mixing interactions are lacking, so that the excited states of
mesons and flavor diagonal light mesons can not be treated properlyWang .
In order to apply the approach to describe mesons in whole $q\bar{q}$ sector,
essential changes are needed. First we have proved that in center of mass
frame (rest frame) and in internal coordinate Hilbert subspace, the total
angular momentum of the meson system is conserved(see Appendix A and B ). Then
we are working in center of mass frame and in internal coordinate Hilbert
subspace and make the following three significant improvements on the model:
(1) transforming mass eigen equations from momentum-spin representation to
total angular momentum representation and establishing a set of coupled radial
mass eigen equations for each total angular momentum ; (2) introducing a
relativistic confining potential into the effective meson interaction
phenomenologically based on lattice QCD results ; (3) including an SU(3)
flavor-mixing interaction in the model phenomenologically and obtaining a set
of coupled radial eigen equations for different flavor components. In having
done above, finally we have a complete QCD inspired relativistic bound state
model for mesons on the whole $q\bar{q}$ sector. This model has been applied
to about 265 mesons with available data and with total angular momentum from
$J=0$ to $6$. The mass spectra, squared radii, and decay constants are
calculated, and the calculated results are in good agreement with the data.
While the most important physical results have been reported briefly in a
short letterPLB , the present article will provide detailed information and
solid foundation of the model for completion
This paper is organized as follows. In Sec. ii@ the QCD inspired relativistic
bound state model for mesons is described and the relativistic mass eigen
equations for bound states with any total angular momentum are derived. In
Sec. iii@ based on lattice QCD results, a relativistic confining potential in
momentum space is introduced in the effective interaction of mesons. The
effective interaction is extended to include an SU(3) flavor mixing
interaction in Sec. iv@. In Sec. v@ we present the numerical solutions for 265
mesons including both flavor-off and flavor diagonal mesons with $J=0-6$.
Sec.vi@ is an analysis of the results obtained. Finally, conclusion and
discussion are given in Sec. vii@. The four Appendices are for clarifying some
important issues and for the derivation of key equations.
## II Description of the model
For convenience, Brodsky and Pauli defined a light-front Lorentz invariant
Hamiltonian BPP
$\displaystyle H_{\mathrm{LC}}\equiv
P^{\mu}P_{\mu}=P^{-}P^{+}-{\bm{P}}^{2}_{\perp}=\hat{M}_{0}^{2},$ (1)
The relativistic bound state problem in front form can be solved by solving
the light-front mass eigen equation:
$\displaystyle H_{\mathrm{LC}}|\Psi\rangle=M_{0}^{2}|\Psi\rangle\ .$ (2)
If one disregards possible zero modes and works in the light-front gauge, this
equation can be solved in terms of a complete set of Fock states
$|\mu_{n}\rangle$:
$\displaystyle\sum_{n^{\prime}}\\!\int\\!d[\mu^{\prime}_{n^{\prime}}]\langle\mu_{n}|H_{\mathrm{LC}}|\mu^{\prime}_{n^{\prime}}\rangle\langle\mu^{\prime}_{n^{\prime}}|\Psi\rangle=M_{0}^{2}\langle\mu_{n}|\Psi\rangle\
.$ (3)
For a meson, the ket $|\Psi\rangle$ holds:
$\displaystyle|\Psi_{\mathrm{meson}}\rangle=\sum_{i}\Psi_{q\bar{q}}(x_{i},\vec{k}_{\\!\perp
i},\lambda_{i})|q\bar{q}\rangle$ (4)
$\displaystyle+\sum_{i}\Psi_{gg}(x_{i},\vec{k}_{\\!\perp
i},\lambda_{i})|gg\rangle$
$\displaystyle+\sum_{i}\Psi_{q\bar{q}g}(x_{i},\vec{k}_{\\!\perp
i},\lambda_{i})|q\bar{q}g\rangle$
$\displaystyle+\sum_{i}\Psi_{q\bar{q}q\bar{q}}(x_{i},\vec{k}_{\\!\perp
i},\lambda_{i})|q\bar{q}q\bar{q}\rangle$ $\displaystyle+...$
Within the framework of discrete quantization of light-front QCD, infinite
dimensional Fock space has been truncated at a proper cutoff energy and the
energy truncation plays a role of renormalization in discrete light-front QCD.
By Tamm-Dancoff projection method and resolvent technique, the equation of
motion in a larger Fock space of multi-particles can be reduced to that in a
smaller one with an effective interaction to account for the effect of the
projected out part of the Fock space. The reduction and projection procedure
can be carried out recursively, finally the effective Hamiltonian and its
eigen equation on $q\bar{q}$ sector can be obtained. For flavor off-diagonal
mesons, disregarding the zero modes and the two-gluon annihilation effect,
Pauli et al. has obtained the effective mass eigen equation for mesons in
light-front relative momentum coordinate space Pauli01 ; Pauli02
$\displaystyle M_{0}^{2}\langle
x,\vec{k}_{\\!\perp};\lambda_{q},\lambda_{\bar{q}}|\psi\rangle=$ (5)
$\displaystyle\left[{\overline{m}^{\,2}_{q}+\vec{k}_{\\!\perp}^{\,2}\over
x}+{\overline{m}^{\,2}_{\bar{q}}+\vec{k}_{\\!\perp}^{\,2}\over
1-x}\right]\langle
x,\vec{k}_{\\!\perp};\lambda_{q},\lambda_{\bar{q}}|\psi\rangle$
$\displaystyle-{4\over
3}{m_{1}m_{2}\over\pi^{2}}\sum_{\lambda_{q}^{\prime},\lambda_{\bar{q}}^{\prime}}\\!\int\\!\frac{dx^{\prime}d^{2}\vec{k}_{\\!\perp}^{\prime}\,R(x^{\prime},k_{\perp}^{\prime})}{\sqrt{x(1-x)x^{\prime}(1-x^{\prime})}}$
$\displaystyle{\overline{\alpha}(Q)\over
Q^{2}}\,S_{\lambda_{q}\lambda_{\bar{q}};\lambda_{q}^{\prime}\lambda_{\bar{q}}^{\prime}}\,\langle
x^{\prime},\vec{k}_{\\!\perp}^{\prime};\lambda_{q}^{\prime},\lambda_{\bar{q}}^{\prime}|\psi\rangle\
,$
This is a relativistic covariant mass eigen equation for mesons in center of
mass fame and in internal Hilbert subspace. However the equation of motion is
written in relative momentum and helicity representation, and the momentum-
helicity plane wave function contains all possible components of partial waves
of the spin spherical harmonic functions $\Phi_{JlsM}$, the total angular
momentum $J$ and its $z$-component $M$ are not conserved.
Despite this, Trittmann and PauliPauli03 found an appropriate method which
can calculate the eigenvalue spectrum separately for each $J_{z}=M$. To do so,
they transformed the light-front coordinate $x$ back to the coordinate $k_{3}$
by Terent’ev transformationTere , and used a unitary transformation to
transform the Lepage-Brodsky spinors to the Bjorken-Drell spinorsPauli04 .
Then the mass eigen equation (5) becomesPauli05 :
$\displaystyle\left[M_{0}^{2}-\left(E_{1}(k)+E_{2}(k)\right)^{2}\right]\varphi_{s_{1}s_{2}}(\bm{k})$
(6) $\displaystyle=$ $\displaystyle\sum_{s_{1}^{\prime}s_{2}^{\prime}}\int
d^{3}\bm{k^{\prime}}U_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\bm{k};\bm{k^{\prime}})\varphi_{s_{1}^{\prime}s_{2}^{\prime}}(\bm{k}^{\prime}),$
This integration equation is written in momentum-spin representation in terms
of internal relative momenta of two quarks, spin singlet and triplet are
mixed. For the same reason as discussed above, the momentum-spin plane wave
does not conserve $J$ and $M$. As noted in RefBPP , in general it is difficult
to explicitly compute the total angular momentum of a bound state by using
light-front quantization. However, as addressed in Introduction, in the center
of mass frame and in internal Hilbert subspace, the total angular momentum is
conserved. This makes it possible to solve the mass eigen equation in total
angular momentum representation (see Appendices B,C,D ).
Since in center of mass frame and in internal Hilbert subspace, the total
angular momentum $J^{2}$ and $J_{z}$ are conserved, we can transform the mass
eigen equation (6) from momentum-spin representation to total angular momentum
representation and establish the mass eigen equation for each $J$. Expanding
the momentum-spin plane wave function in terms of the spin spherical harmonic
functions $\Phi_{JslM}(\Omega_{k},s_{1},s_{2})$ and projecting out the spin
and angular part of the wave function in $|JslM\rangle$ subspace by the
projecting operation,
$\Big{\langle}\sum_{m\mu}\sum_{s_{1}s_{2}}\langle
lms\mu|JM\rangle\langle\frac{1}{2}s_{1}\frac{1}{2}s_{2}|s\mu\rangle
Y_{lm}(\Omega_{k})\chi(s_{1}),\chi(s_{2})\Big{|},$ (7)
we obtain the mass eigen equation for the radial wave function of $R_{Jsl}(k)$
(see Appendix C).
$\displaystyle\left[M_{0}^{2}-\left(E_{1}(k)+E_{2}(k)\right)^{2}\right]R_{Jsl}(k)$
$\displaystyle=$
$\displaystyle\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\int
k^{\prime
2}dk^{\prime}U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})R_{Js^{\prime}l^{\prime}}(k^{\prime}).$
This is a set of coupled equations for radial functions $R_{Jsl}(k)$ of
different partial waves and of spin singlet and triplets, coupled by the
tensor potential and by the relativistic spin-orbital potential. In this case,
the eigen wave functions $R_{Jsl}(k)$ has the conventional definition and
physical meaning. The bound states of mesons can be described concisely by the
spectroscopic symbol of ${}^{2S+1}L_{J}$.
The kernel $U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})$ can be written as
(see Appendix D ),
$\displaystyle
U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})=\sum_{mm^{\prime}}\sum_{s_{1}s_{2}}\sum_{s_{1}^{\prime}s_{2}^{\prime}}\int\int
d\Omega_{k}d\Omega_{k^{\prime}}$ $\displaystyle\mbox{}\times\langle
Y_{lm}(\Omega_{k})|U_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\bm{k},\bm{k}^{\prime})|Y_{l^{\prime}m^{\prime}}(\Omega_{k^{\prime}})\rangle$
$\displaystyle\mbox{}\times\langle
lms\mu|JM\rangle\textstyle\langle\frac{1}{2}s_{1}\frac{1}{2}s_{2}|s\mu\rangle\langle
l^{\prime}m^{\prime}s^{\prime}\mu^{\prime}|JM\rangle\textstyle\langle\frac{1}{2}s_{1}^{\prime}\frac{1}{2}s_{2}^{\prime}|s^{\prime}\mu^{\prime}\rangle.$
The above kernel $U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})$ contains
different kinds of central potentials, relativistic spin-orbit coupling
potentials, and tensor potentials changing $l$ by $\Delta l=\pm 2$ and mixing
spin singlet and triplets (see Appendix D ).
## III Introducing a Confining potential
Quark confinement is one of the fundamental problems in QCD for hadronic
physics. The confinement and the spontaneous breaking of chiral symmetry are
key ingredients for solving the low-energy hadronic bound states from QCD, but
none of them has been completely understood and solved. Numerical results show
that the effective light-front Hamiltonian model proposed by Pauli et al.
without confining potentials can well describe the ground states but can not
apply to the radial excited states of mesons. To describe the excited states
properly, the confining potential must be included in the modelWang .
Fortunately, we can refer to the constituent quark model which is successful
due to the inclusion of a phenomenological confining potential in some wayLQCD
. The key idea of this model consists in the introduction of a linear
confining potential in coordinate space based on the numerical calculations of
lattice QCD, and this non-relativistic confining potential can be generalized
to relativistic form.
In nonrelativistic quark models the confining potential in configuration space
is,
$\displaystyle V_{\mathrm{con}}(r)=\lambda r+c\ ,$ (10)
where $\lambda$ is the strength of the linear interaction, and $c$ is a
constant irrelevant in the present case and omitted hereafter. By Fourier
transformation, the counterpart of the linear term $\lambda r$ in momentum
space is obtained,
$\displaystyle V_{\mathrm{lin}}(\bm{q})\sim-\frac{1}{|{\bm{q}}|^{4}}\ ,$
$\displaystyle\bm{q}=\bm{k}-\bm{k}^{\prime}\ .$ (11)
At the point of $\bm{q}=0$, the singularity indicates that the directly
transformed result of linear potential could not be described correctly in
momentum space, which results in an ill-defined bound state equationGross .
However, some different methods were employed to solve this problem for the
relativistic case. In the present paper, the correct form for
$V_{\mathrm{lin}}(\bm{q})$ is constructed by introducing a small parameter
$\eta$:
$\displaystyle V(\bm{q})$ $\displaystyle=$ $\displaystyle\lim_{\eta\rightarrow
0}{\lambda\over
2\pi^{2}}\frac{\partial^{2}}{\partial\\!\eta^{2}}\left[{1\over|\bm{q}|^{2}+\eta^{2}}\right]$
(12)
The relativistic linear potential in momentum space $V_{\mathrm{lin}}(Q)$ is a
direct generalization of the nonrelativistic one, just replacing the
nonrelativistic $|\bm{q}|^{2}$ in (12) by the relativistic $Q^{2}$, which has
the following specification in Sommerer01 ; Hersbach01 ,
$\displaystyle Q^{2}=(\mathbf{k}-\mathbf{k^{\prime}})^{2}+\varpi^{2}$ (13)
and
$\displaystyle\varpi^{2}=(E_{1}-E_{1}^{\prime})(E_{2}-E_{2}^{\prime})$ (14)
Then the form of relativistic confining potential is,
$\displaystyle V_{\mathrm{con}}(Q)$ $\displaystyle=$
$\displaystyle\lim_{\eta\rightarrow 0}{\lambda\over
2\pi^{2}}\frac{\partial^{2}}{\partial\\!\eta^{2}}\left[{1\over
Q^{2}+\eta^{2}}\right]$ (15)
Obviously, this confining potential is Lorentz covariant and can be used in
either spin system or non-spin system.
Now as the relativistic confining potential $V_{\mathrm{con}}(Q)$ is included
in the interaction, one has the new kernel
$U_{s_{1}s_{2};s^{\prime}_{1}s^{\prime}_{2}}(\textbf{k},\textbf{k}^{\prime})$,
$\displaystyle
U_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\textbf{k},\textbf{k}^{\prime})=$
(16)
$\displaystyle\frac{4}{3}\frac{m_{1}m_{2}}{\pi^{2}}\sqrt{(\frac{1}{E_{1}}\\!+\\!\frac{1}{E_{2}})(\frac{1}{E_{1}^{\prime}}\\!+\\!\frac{1}{E_{2}^{\prime}})}\bar{u}(\mathbf{k},s_{1})\bar{u}(-\mathbf{k},s_{2})$
$\displaystyle\times[\gamma^{(1)}_{\mu}\cdot\gamma^{(2)\mu}V_{V}+\mathbb{I}^{(1)}\cdot\mathbb{I}^{(2)}V_{S}]u(\mathbf{k^{\prime}},s_{1}^{\prime})u(-\mathbf{k^{\prime}},s_{2}^{\prime}).$
(17)
The scalar and vector interaction potentials read
$\displaystyle V_{V}$ $\displaystyle=$
$\displaystyle-\frac{\bar{\alpha}(Q)}{Q^{2}}-\frac{3}{4}\epsilon\>V_{\mathrm{con}}(Q)$
$\displaystyle V_{S}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}(1-\epsilon)V_{\mathrm{con}}(Q)\,$ (18)
where $\epsilon$ represents the scalar-vector mixing of the confining
potential.
## IV Including a flavor mixing interaction
It is extremely difficult to derive a simple form of flavor mixing interaction
in the above effective Hamiltonian from light-front QCD at present. However,
without flavor mixing potential, one can not deal with the flavor diagonal
mesons such as $\pi^{0}$, $\rho^{0}$, and $f_{0}$, etc. In the fundamental
hadronic theory, the quarks of u, d, and s have an approximate $SU(3)$
symmetry. Due to this symmetry, the quarks fields transform each other under
the $SU(3)$ transformation Weinberg ,
$\left(\begin{array}[]{c}u\\\ d\\\ s\end{array}\right)\longrightarrow exp\
[i\sum_{a}(\theta_{a}^{V}T_{a}+\theta_{a}^{A}T_{a}\gamma_{5})]\left(\begin{array}[]{c}u\\\
d\\\ s\end{array}\right)$
where $T_{a}$ are Gell-Mann Matrices. For convenience of numerical
calculation, we introduce phenomenologically a simple flavor mixing
interaction as follows,
$\displaystyle V_{f}=$
$\displaystyle\gamma_{0}\left[T^{+}_{ud}(1)T^{+}_{ud}(2)+T^{-}_{ud}(1)T^{-}_{ud}(2)\right]$
$\displaystyle+$
$\displaystyle\delta_{0}\left[T^{+}_{us}(1)T^{+}_{us}(2)+T^{-}_{us}(1)T^{-}_{us}(2)\right.$
$\displaystyle+$
$\displaystyle\left.T^{+}_{ds}(1)T^{+}_{ds}(2)+T^{-}_{ds}(1)T^{-}_{ds}(2)\right]$
(19)
where $\gamma_{0}$ and $\delta_{0}$ are the strengths of flavor-mixing
interaction, the index $1$ and $2$ denote the quark and anti-quark in meson,
respectively. The flavor SU(3) wave functions and generators are defined as,
$\displaystyle|u\rangle=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
\end{array}\right),|\bar{u}\rangle=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
\end{array}\right);|d\rangle=\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right)$ (29)
$\displaystyle|\bar{d}\rangle=\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right),|s\rangle=\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
\end{array}\right)|\bar{s}\rangle=\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
\end{array}\right)$ (39) $\displaystyle
T_{ud}^{+}=(T_{ud}^{-})^{\dagger}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right),$ (43) $\displaystyle
T_{us}^{+}=(T_{us}^{-})^{\dagger}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right)$ (47) $\displaystyle
T_{ds}^{+}=(T_{ds}^{-})^{\dagger}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&0&0\\\ \end{array}\right),$ (51)
The action of the flavor mixing interaction on flavor wave function is as
follows,
$\displaystyle V_{f}|u\bar{u}\rangle$
$\displaystyle=\gamma_{0}|d\bar{d}\rangle+\delta_{0}|s\bar{s}\rangle$ (52)
$\displaystyle V_{f}|d\bar{d}\rangle$
$\displaystyle=\gamma_{0}|u\bar{u}\rangle+\delta_{0}|s\bar{s}\rangle$ (53)
$\displaystyle V_{f}|s\bar{s}\rangle$
$\displaystyle=\delta_{0}|d\bar{d}\rangle+\delta_{0}|u\bar{u}\rangle$ (54)
Combining this interaction with the precious one in equations (18) , we have a
set of flavor-coupled radial eigen equations for the flavor components of up,
down, and strange quarks,
$\displaystyle\left[M_{0}^{2}-\left(E_{1}(k)+E_{2}(k)\right)^{2}\right]R_{Jsl}^{p_{1}p_{2}}(k)$
$\displaystyle=$
$\displaystyle\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\sum_{p^{\prime}_{1}p^{\prime}_{2}}\int
k^{\prime
2}dk^{\prime}U_{Jsl;s^{\prime}l^{\prime}}^{p_{1}p_{2},p^{\prime}_{1}p^{\prime}_{2}}(k;k^{\prime})R_{Js^{\prime}l^{\prime}}^{p^{\prime}_{1}p^{\prime}_{2}}(k^{\prime}).$
The interaction kernel including the flavor-mixing interaction is
$\displaystyle
U_{Jsl;s^{\prime}l^{\prime}}^{p_{1}p_{2};p^{\prime}_{1}p^{\prime}_{2}}(k,k^{\prime})=\sum_{mm^{\prime}}\sum_{\mu\mu^{\prime}}\sum_{s_{1}s_{2}}\sum_{s^{\prime}_{1}s^{\prime}_{2}}\int\int{d}\Omega_{k}\Omega_{k^{\prime}}$
$\displaystyle\times{Y^{*}_{lm}(\Omega_{k})}W^{p_{1}p_{2};p^{\prime}_{1}p^{\prime}_{2}}_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\textbf{k},\textbf{k}^{\prime})Y_{l^{\prime}m^{\prime}}(\Omega_{k^{\prime}})$
(56)
$\displaystyle\langle{lms\mu}|JM\rangle\langle{\frac{1}{2}s_{1}\frac{1}{2}s_{2}}|s\mu\rangle\langle{l^{\prime}m^{\prime}s^{\prime}\mu^{\prime}}|JM\rangle\langle{\frac{1}{2}s^{\prime}_{1}\frac{1}{2}s^{\prime}_{2}}|s^{\prime}\mu^{\prime}\rangle,$
where
$W^{p_{1}p_{2};p^{\prime}_{1}p^{\prime}_{2}}_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\textbf{k},\textbf{k}^{\prime})$
is defined as
$\displaystyle
W^{p_{1}p_{2};p^{\prime}_{1}p^{\prime}_{2}}_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\textbf{k},\textbf{k}^{\prime})=\frac{4}{3}\frac{m_{p_{1}}m_{p_{2}}}{\pi^{2}}\sqrt{\left(\frac{1}{E_{1}}+\frac{1}{E_{2}}\right)\left(\frac{1}{E^{\prime}_{1}}+\frac{1}{E^{\prime}_{2}}\right)}\times$
$\displaystyle{\bar{u}(p_{1},\textbf{k},s_{1})}\bar{u}(p_{2},-\textbf{k},s_{2})\left[\gamma^{\mu}(p_{1})\cdot\gamma_{\mu}(p_{2})V_{V}+I^{(1)}\cdot{I}^{(2)}V_{S}\right]\times$
$\displaystyle\left[I_{f}+V_{f}\right]{{u}^{\prime}(p^{\prime}_{1},\textbf{k}^{\prime},s^{\prime}_{1})}{u}^{\prime}(p^{\prime}_{2},-\textbf{k}^{\prime},s^{\prime}_{2}).$
(57)
where
$p_{1}p_{2},p^{\prime}_{1}p^{\prime}_{2}=\\{u\bar{u},d\bar{d},s\bar{s}\\}$,
$I_{f}$ is the identity operator in flavor space, $V_{V}$ and $V_{S}$ are the
vector potential and scalar potential, respectively. For the flavor mixing
mesons with total angular momentum $J$, the mass eigen equations are described
explicitly by the following set of flavor-coupled equations
$\displaystyle\left[M_{0}^{2}-\left(E_{u}(k)+E_{\bar{u}}(k)\right)^{2}\right]R^{u\bar{u}}_{Jsl}(k)$
(58a)
$\displaystyle=\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\int{k^{\prime
2}}dk^{\prime}\left(U_{Jsl;s^{\prime}l^{\prime}}^{u\bar{u};u\bar{u}}(k,k^{\prime})R^{u\bar{u}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})\right.+$
$\displaystyle\left.\gamma_{0}U_{Jsl;s^{\prime}l^{\prime}}^{u\bar{u};d\bar{d}}(k,k^{\prime})R^{d\bar{d}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})+\delta_{0}U_{Jsl;s^{\prime}l^{\prime}}^{u\bar{u};s\bar{s}}(k,k^{\prime})R^{s\bar{s}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})\right)$
$\displaystyle\left[M_{0}^{2}-\left(E_{d}(k)+E_{\bar{d}}(k)\right)^{2}\right]R^{d\bar{d}}_{Jsl}(k)$
(58b)
$\displaystyle=\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\int{k^{\prime
2}}dk^{\prime}\left(\gamma_{0}U_{Jsl;s^{\prime}l^{\prime}}^{d\bar{d};u\bar{u}}(k,k^{\prime})R^{u\bar{u}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})\right.+$
$\displaystyle\left.U_{Jsl;s^{\prime}l^{\prime}}^{d\bar{d};d\bar{d}}(k,k^{\prime})R^{d\bar{d}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})+\delta_{0}U_{Jsl;s^{\prime}l^{\prime}}^{d\bar{d};s\bar{s}}(k,k^{\prime})R^{s\bar{s}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})\right)$
$\displaystyle\left[M_{0}^{2}-\left(E_{s}(k)+E_{\bar{s}}(k)\right)^{2}\right]R^{s\bar{s}}_{Jsl}(k)$
(58c)
$\displaystyle=\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\int{k^{\prime
2}}dk^{\prime}\left(\delta_{0}U_{Jsl;s^{\prime}l^{\prime}}^{s\bar{s};u\bar{u}}(k,k^{\prime})R^{u\bar{u}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})\right.+$
$\displaystyle\left.\delta_{0}U_{Jsl;s^{\prime}l^{\prime}}^{s\bar{s};d\bar{d}}(k,k^{\prime})R^{d\bar{d}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})+U_{Jsl;s^{\prime}l^{\prime}}^{s\bar{s};s\bar{s}}(k,k^{\prime})R^{s\bar{s}}_{Js^{\prime}l^{\prime}}(k,k^{\prime})\right)$
## V Numerical Solutions
In the above equations, $J,s,l$ denote total angular momentum, total spin, and
total orbital angular momentum, respectively. Different mesons can be
classified by the spectroscopic symbol ${}^{2S+1}L_{J}$(or their combination),
which is equivalent to the symbol $J^{PC}$. The space parity and charge
conjugation parity are denoted as,
$\displaystyle P$ $\displaystyle=$ $\displaystyle(-1)^{L+1}$ $\displaystyle C$
$\displaystyle=$ $\displaystyle(-1)^{L+S}.$ (59)
In the present model, the mass eigen value problem of mesons is described by a
set of coupled integration equations, and the interaction includes a quark-
anti-quark one gluon exchange potential $V_{OGE}$ , a confining potential
$V_{con}$, and a flavor mixing interaction $V_{f}$. If the confining potential
has a pure iso-scalar structure( $\epsilon\\!=\\!0$) and the flavor mixing
interaction is omitted, the interaction contains two parameters: the effective
coupling constant $\bar{\alpha}$ and the confining potential strength
$\lambda$. Besides, the flavor mixing interaction has two parameters, and the
constituent quark masses are also indispensable parameters to describe
spontaneous chiral symmetry breaking.
The numerical solution of the eigen equations can be obtained by
discretization of integration equation $(8)$ or $(31a-c)$, and the integration
equations are transformed into matrix equations. The 4-fold integral of the
kennel is completed by the integration technique of spherical harmonic
functions and the angular momentum algebra, while the integration over $k$ is
performed by using Gauss-Legendre quadratures. The integration region
$k\in[0,\infty)$ is projected onto the finite interval $x\in[-1,1]$ by
$x\\!=\\!\frac{k-1}{k+1}$. The radial mass eigen equation (8) is discretized
as follows:
$\displaystyle\left[M_{0}^{2}-\big{(}E_{1}(k_{i})+E_{2}(k_{i})\big{)}^{2}\right]R_{Jsl}(k_{i})$
$\displaystyle=$
$\displaystyle\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\sum_{j=1}^{N}U_{sl;s^{\prime}l^{\prime}}^{J}(k_{i};k_{j})R_{Js^{\prime}l^{\prime}}(k_{j})k_{j}^{2}w_{j}\
,$
where $w_{j}$ is the weight of integration. Before diagonalizing this matrix
equations, special care should be taken for two kinds of singularities: the
singularity at infinite $k$ and the singularity as $k=k^{\prime}$ inside the
region of integration. The first one has been solved by the projection of the
region $k\in[0,\infty)$ onto the finite interval $x\in[-1,1]$, and the second
one is treated by infrared singularity treatment. The detailed procedures of
calculation can be found in Ref.Numer ; Wang .
The parameters of the model are determined from best fit to experimental data.
In this paper, a purely scalar confining potential($\epsilon=0$) is used.
Reproducing the masses of $\pi^{0}$, $\pi^{\pm}$, and $\pi(1300)$, we can
determine $\bar{\alpha}$, $\lambda$, and the masses of up and down quarks.
Then by reproducing the masses of $K^{\pm}$, $D^{0}$, and $B^{\pm}$, the mass
parameters of strange, charm, and bottom quarks are obtained. The parameters
of flavor mixing interaction are determined by the best fit to the data of
flavor diagonal mesons. From all the available data of mesons Data with
$J=0-6$ ( 12 mesons are left for future study: including 6 exotic mesons and 6
mesons without any information about their $J,s,L$ ), we have obtained an
appropriate set of 6 parameters for flavor off-diagonal mesons:
$\bar{\alpha}=0.2574$, $\lambda=0.92\times 10^{4}MeV^{2}$,
$m_{u/d}=0.297GeV,m_{s}=0.418GeV,m_{c}=1.353GeV,m_{b}=4.447GeV;$ and for the
flavor diagonal mesons: $\gamma_{0}=0.1$ and $\delta_{0}=0.1$. The number of
the model parameters is minimum for this kind of semi-phenomenological models
and comparable to BSE and CQM. The masses and wave functions of scalar and
pseudoscalar, vector and axial-vector, tensor and pseudotensor mesons, and
others with $J=3-6$ have been calculated and compared with the experimental
data in the Table( including 265 mesons and anti-mesons: 123 (u,d)-light
mesons, 50 (s,u/d)-K mesons, 24 (c,u/d)-D mesons, 14 (s,c)-D mesons, 12
(b,u/d)-B mesons, 10 (s,b)-$B_{s}$ mesons, 2 (c,b)-$B_{c}$ mesons, 16
(c,$\bar{c}$) mesons, and 14 (b,$\bar{b}$ ) mesons ). It is remarkable that
among 265 mesons, 259 mesons are well described by this model within mass
error less than $23\%$.
In addition, the radial wave-functions of mesons in configuration space can be
obtained from the radial wave-functions in momentum space by Fourier
transformation(see Appendix C), then one can calculate the mean square radii
and the decay constants for some pseudoscalar mesons listed in Tab.1 and
compared with experimental data.
Table 1: The mean square radii and decay constants of some pseudoscalar mesons, compared with the experimental dataData . (Radii are given in $fm^{2}$ and decay constants are given in $MeV$) | $\quad\pi^{+}\quad$ | $\quad K^{+}\quad$ | $\quad D^{+}\quad$ | $\quad D_{s}\quad$ | $\quad B\quad$
---|---|---|---|---|---
$\langle r^{2}\rangle_{the}$ | 0.385 | 0.253 | 0.235 | - | -
$\langle r^{2}\rangle_{exp}$ | 0.452 | 0.314 | - | - | -
$f_{the}$ | 135.2 | 210.7 | 189.2 | 253.1 | 227.5
$f_{exp}$ | 130.4 | 155.5 | 205.8 | 273 | 216
Table 2: The pseudoscalar mesons mass spectra (in MeV). Meson | $I^{G}(J^{PC})$ | Exp(Mev) | Our’s(Mev) | err(%)
---|---|---|---|---
$\pi^{0}$ | $0^{-+}$ | 135 | 135 | 0
$\pi^{\pm}$ | $0^{-+}$ | 140 | 140 | 0
$\eta$ | $0^{-+}$ | 548 | 143 | 73
$\eta(958)$ | $0^{-+}$ | 958 | 690 | 27
$\eta(1295)$ | $0^{-+}$ | 1294 | 1258 | 2.8
$\pi(1300)^{\pm}$ | $0^{-+}$ | 1300$\pm$100 | 1408 | 0
$\pi(1300)^{0}$ | $0^{-+}$ | 1300$\pm$100 | 1350 | 0
$\eta(1405)$ | $0^{-+}$ | 1410 | 1652 | 17
$\eta(1475)$ | $0^{-+}$ | 1476 | 1700 | 15
$\eta(1760)$ | $0^{-+}$ | 1756 | 1769 | 0.8
$\pi(1800)^{\pm}$ | $0^{-+}$ | 1816 | 1454 | 19
$\pi(1800)^{0}$ | $0^{-+}$ | 1816 | 2096 | 15.4
$X(1835)$ | $0^{-+}?$ | 1833 | 2110 | 15.1
$\eta(2225)$ | $0^{-+}$ | 2220 | 2160 | 2.7
$K^{\pm}$ | $0^{-}$ | 494 | 494 | 0
$K^{0}$ | $0^{-}$ | 498 | 494 | 0.8
$K(1460)$ | $0^{-}$ | 1460 | 1522 | 4.2
$K(1830)$ | $0^{-}$ | 1830 | 1597 | 12.7
$D^{0}$ | $0^{-}$ | 1865 | 1931 | 3.5
$D^{\pm}$ | $0^{-}$ | 1869 | 1931 | 3.3
$D_{s}^{\pm}$ | $0^{-}$ | 1969 | 2001 | 1.6
$B^{0}$ | $0^{-}$ | 5279 | 5584 | 5.8
$B^{\pm}$ | $0^{-}$ | 5279 | 5584 | 5.8
$B_{s}^{0}$ | $0^{-}$ | 5367 | 5667 | 5.6
$B_{c}^{\pm}$ | $0^{-}$ | 6286 | 6342 | 0.9
$\eta_{c}(1S)$ | $0^{-+}$ | 2980 | 2980 | 0
$\eta_{c}(2S)$ | $0^{-+}$ | 3637 | 3533 | 2.9
$\eta_{b}(1S)$ | $0^{-+}$ | 9391 | 8800 | 6.3
Table 3: The scalar mesons mass spectra (in MeV). Meson | $J^{PC}$ | Exp(Mev) | Our’s(Mev) | err(%)
---|---|---|---|---
$f_{0}(600)$ | $0^{++}$ | 400-1200 | 736 | 0
$f_{0}(980)$ | $0^{++}$ | 980 | 994 | 1.4
$a_{0}(980)^{0}$ | $0^{++}$ | 985 | 1080 | 9.6
$a_{0}(980)^{\pm}$ | $0^{++}$ | 985 | 930 | 5.6
$f_{0}(1370)$ | $0^{++}$ | 1200-1500 | 1231 | 0
$a_{0}(1450)^{0}$ | $0^{++}$ | 1474 | 1333 | 9.5
$a_{0}(1450)^{\pm}$ | $0^{++}$ | 1474 | 1457 | 1.1
$f_{0}(1500)$ | $0^{++}$ | 1505 | 1522 | 1.1
$f_{0}(1710)$ | $0^{++}$ | 1724 | 1568 | 9.0
$f_{0}(2020)$ | $0^{++}$ | 1992 | 1606 | 19.4
$f_{0}(2100)$ | $0^{++}$ | 2103 | 1989 | 5.4
$f_{0}(2200)$ | $0^{++}$ | 2189 | 2026 | 7.4
$f_{0}(2330)$ | $0^{++}$ | 2321 | 2052 | 11.6
$K_{0}^{*}(800)$ | $0^{+}$ | 672 | 731 | 8.8
$K_{0}^{*}(1430)$ | $0^{+}$ | 1412 | 1535 | 8.7
$D_{0}^{*}(2400)^{0}$ | $0^{+}$ | 2352 | 2254 | 4.2
$D_{0}^{*}(2400)^{\pm}$ | $0^{+}$ | 2403 | 2254 | 6.2
$D_{s0}^{*}(2317)^{\pm}$ | $0^{+}$ | 2317 | 2169 | 6.4
$\chi_{c0}(1P)$ | $0^{++}$ | 3415 | 3352 | 1.8
$\chi_{b0}(1P)$ | $0^{++}$ | 9860 | 9860 | 0
$\chi_{b0}(2P)$ | $0^{++}$ | 10232 | 9990 | 2.3
Table 4: The axial vector meson mass spectra (in MeV). Meson | $J^{PC}$ | Exp(Mev) | Our’s(Mev) | err(%)
---|---|---|---|---
$h_{1}(1170)$ | $1^{+-}$ | 1170 | 1027 | 12.2
$b_{1}(1235)^{0}$ | $1^{+-}$ | 1230 | 1127 | 8.4
$b_{1}(1235)^{\pm}$ | $1^{+-}$ | 1229 | 1343 | 9.3
$a_{1}(1260)^{0}$ | $1^{++}$ | 1230 | 1276 | 3.7
$a_{1}(1260)^{\pm}$ | $1^{++}$ | 1230 | 1371 | 11.4
$f_{1}(1285)$ | $1^{++}$ | 1281 | 1295 | 1.1
$h_{1}(1380)$ | $1^{+-}$ | 1386 | 1301 | 6.1
$f_{1}(1420)$ | $1^{++}$ | 1426 | 1311 | 8.0
$f_{1}(1510)$ | $1^{++}$ | 1518 | 1419 | 6.5
$h_{1}(1595)$ | $1^{+-}$ | 1594 | 1495 | 6.2
$a_{1}(1640)^{0}$ | $1^{++}$ | 1647 | 1745 | 6.0
$a_{1}(1640)^{\pm}$ | $1^{++}$ | 1647 | 1724 | 4.7
$K_{1}(1270)$ | $1^{+}$ | 1273 | 1459 | 14.6
$K_{1}(1400)$ | $1^{+}$ | 1402 | 1484 | 5.8
$K_{1}(1650)$ | $1^{+}$ | 1650 | 1757 | 6.4
$D_{1}(2420)^{0}$ | $1^{+}$ | 2422 | 2400 | 0.9
$D_{1}(2420)^{\pm}$ | $1^{+}?$ | 2423 | 2400 | 0.9
$D_{1}(2430)^{0}$ | $1^{+}$ | 2427 | 2425 | 0.1
$D_{S1}(2460)^{\pm}$ | $1^{+}$ | 2460 | 2530 | 2.9
$D_{S1}(2536)^{\pm}$ | $1^{+}$ | 2535 | 2549 | 0.6
$B_{1}(5721)^{0}$ | $1^{+}$ | 5721 | 5666 | 1.0
$B_{S1}(5830)^{0}$ | $1^{+}$ | 5829 | 5800 | 0.5
$\chi_{c1}(1p)$ | $1^{++}$ | 3510 | 3504 | 0.2
$h_{c1}(1p)$ | $1^{+-}$ | 3526 | 3509 | 0.5
$\chi_{b1}(1p)$ | $1^{++}$ | 9892 | 10040 | 1.5
$\chi_{b1}(2p)$ | $1^{++}$ | 10255 | 10040 | 2.1
Table 5: The vector meson mass spectra (in MeV). Meson | $J^{PC}$ | Exp(Mev) | Our’s(Mev) | err(%)
---|---|---|---|---
$\rho(770)^{0}$ | $1^{--}$ | 775 | 1015 | 31
$\rho(770)^{\pm}$ | $1^{--}$ | 775 | 1239 | 60
$\omega(782)$ | $1^{--}$ | 783 | 1270 | 62
$\phi(1020)$ | $1^{--}$ | 1019 | 1334 | 31
$\omega(1420)$ | $1^{--}$ | 1425 | 1410 | 1.0
$\rho(1450)^{0}$ | $1^{--}$ | 1465 | 1636 | 11.6
$\rho(1450)^{\pm}$ | $1^{--}$ | 1465 | 1323 | 9.7
$\rho(1570)^{0}$ | $1^{--}$ | 1570 | 1641 | 4.5
$\rho(1570)^{\pm}$ | $1^{--}$ | 1570 | 1740 | 10.8
$\omega(1650)$ | $1^{--}$ | 1670 | 1675 | 0.3
$\phi(1680)$ | $1^{--}$ | 1680 | 1786 | 6.3
$\rho(1700)^{0}$ | $1^{--}$ | 1720 | 1836 | 6.7
$\rho(1700)^{\pm}$ | $1^{--}$ | 1700 | 1362 | 19.8
$\rho(1900)^{0}$ | $1^{--}$ | 1909 | 1996 | 4.6
$\rho(1900)^{\pm}$ | $1^{--}$ | 1909 | 1761 | 7.8
$\rho(2150)^{0}$ | $1^{--}$ | 2149 | 2087 | 2.9
$\rho(2150)^{\pm}$ | $1^{--}$ | 2149 | 2430 | 13.1
$K^{*}(892)$ | $1^{-}$ | 892 | 1345 | 50.1
$K^{*}(1410)$ | $1^{-}$ | 1414 | 1415 | 0.1
$K^{*}(1630)$ | $1^{-}?$ | 1629 | 1502 | 7.8
$K^{*}(1680)$ | $1^{-}$ | 1717 | 1531 | 10.8
$D^{*}(2007)^{0}$ | $1^{-}$ | 2007 | 2100 | 4.6
$D^{*}(2010)^{\pm}$ | $1^{-}$ | 2010 | 2126 | 5.8
$D^{*}(2640)$ | $1^{-}?$ | 2637 | 2403 | 8.9
$D_{S}^{*\pm}$ | $1^{-}?$ | 2112 | 2214 | 4.8
$D_{S1}(2700)^{\pm}$ | $1^{-}$ | 2690 | 2233 | 16.9
$B^{*}$ | $1^{-}$ | 5325 | 5518 | 3.6
$B_{S}^{*}$ | $1^{-}$ | 5413 | 5625 | 3.9
$J/\psi$ | $1^{--}$ | 3097 | 3284 | 6.0
$\psi(2S)$ | $1^{--}$ | 3686 | 3362 | 8.8
$\psi(3770)$ | $1^{--}$ | 3773 | 3684 | 2.3
$\psi(4040)$ | $1^{--}$ | 4039 | 3700 | 8.4
$\psi(4160)$ | $1^{--}$ | 4153 | 4197 | 1.1
$X(4260)$ | $1^{--}$ | 4263 | 4769 | 11.9
$X(4360)$ | $1^{--}$ | 4361 | 4783 | 9.7
$\psi(4415)$ | $1^{--}$ | 4421 | 5341 | 20.8
$X(4660)$ | $1^{--}$ | 4664 | 5418 | 16.2
$\gamma(1S)$ | $1^{--}$ | 9460 | 9693 | 2.4
$\gamma(2S)$ | $1^{--}$ | 10023 | 10013 | 0.1
$\gamma(3S)$ | $1^{--}$ | 10355 | 10060 | 2.8
$\gamma(4S)$ | $1^{--}$ | 10580 | 10765 | 1.7
$\gamma(10860)$ | $1^{--}$ | 10865 | 10783 | 0.8
$\gamma(11020)$ | $1^{--}$ | 11019 | 10861 | 1.4
Table 6: The tensor and pseudotensor meson mass (in MeV). Meson | $J^{PC}$ | Exp(MeV) | Theor(MeV) | err(%)
---|---|---|---|---
$\pi_{2}(1670)$ | $2^{-+}$ | 1672 | 1587 | 5.1
$\pi_{2}(1880)$ | $2^{-+}$ | 1895 | 1589 | 16.1
$\pi_{2}(2100)$ | $2^{-+}$ | 2090 | 1922 | 8.0
$K_{2}(1580)$ | $2^{-}$ | 1580 | 1530 | 3.2
$K_{2}(1770)$ | $2^{-}$ | 1773 | 1539 | 13.2
$K_{2}(1820)$ | $2^{-}$ | 1816 | 1763 | 2.9
$K_{2}(2250)$ | $2^{-}$ | 2247 | 1765 | 21.4
$\gamma(1D)$ | $2^{--}$ | 10161 | 10218 | 0.6
$a_{2}(1320)$ | $2^{++}$ | 1318 | 1421 | 7.8
$a_{2}(1700)$ | $2^{++}$ | 1723 | 1474 | 14.5
$K^{*}_{2}(1430)$ | $2^{+}$ | 1425 | 1531 | 7.4
$K^{*}_{2}(1980)$ | $2^{+}$ | 1973 | 1575 | 20.1
$D^{*}_{2}(2460)^{\pm}$ | $2^{+}$ | 2460 | 2456 | 0.1
$D^{*}_{2}(2460)^{0}$ | $2^{+}$ | 2462 | 2456 | 0.2
$D^{*}_{S2}(2573)^{\pm}$ | $2^{+}?$ | 2573 | 2580 | 0.3
$B^{*}_{J}(5732)$ | $2^{+}?$ | 5698 | 5706 | 0.1
$B^{*}_{2}(5747)^{0}$ | $2^{+}$ | 5743 | 5765 | 0.4
$B^{*}_{S2}(5840)^{0}$ | $2^{+}$ | 5840 | 5831 | 0.2
$B^{*}_{SJ}(5850)^{0}$ | $2^{+}?$ | 5853 | 5883 | 0.5
$\chi_{c2}(1P)$ | $2^{++}$ | 3556 | 3732 | 4.9
$\chi_{c2}(2P)$ | $2^{++}$ | 3929 | 3745 | 4.7
$\chi_{b2}(1P)$ | $2^{++}$ | 9912 | 10014 | 1.0
$\chi_{b2}(2P)$ | $2^{++}$ | 10269 | 10354 | 0.8
Table 7: The mesons of $J\geq 3$ (in MeV). Meson | $J^{PC}$ | Exp(MeV) | Theor(MeV) | err(%)
---|---|---|---|---
$\omega_{3}(1670)$ | $3^{--}$ | 1672 | 1677 | 0.3
$\rho_{3}(1690)^{\pm}$ | $3^{--}$ | 1688 | 1702 | 0.8
$\rho_{3}(1690)^{0}$ | $3^{--}$ | 1688 | 1697 | 0.5
$\phi_{3}(1850)$ | $3^{--}$ | 1854 | 1807 | 2.5
$\rho_{3}(1990)^{\pm}$ | $3^{--}$ | 1982 | 1795 | 9.4
$\rho_{3}(1990)^{0}$ | $3^{--}$ | 1982 | 1807 | 8.8
$\rho_{3}(2250)^{\pm}$ | $3^{--}$ | 2230 | 2660 | 19.2
$\rho_{3}(2250)^{0}$ | $3^{--}$ | 2230 | 1852 | 17.0
$K_{3}^{*}(1780)$ | $3^{-}$ | 2324 | 1777 | 23.5
$K_{3}(2320)$ | $3^{+}$ | 2324 | 1812 | 22.0
$a_{4}(2040)^{\pm}$ | $4^{++}$ | 2001 | 1745 | 12.7
$a_{4}(2040)^{0}$ | $4^{++}$ | 2001 | 1743 | 12.9
$f_{4}(2050)$ | $4^{++}$ | 2018 | 1865 | 7.6
$f_{4}(2300)$ | $4^{++}$ | 2300 | 2016 | 12.3
$K_{4}^{*}(2045)$ | $4^{+}$ | 2045 | 1827 | 10.6
$K_{4}(2500)$ | $4^{-}$ | 2490 | 1933 | 22.3
$\rho_{5}(2350)^{\pm}$ | $5^{--}$ | 2330 | 2292 | 1.6
$\rho_{5}(2350)^{0}$ | $5^{--}$ | 2330 | 2218 | 4.8
$K_{5}^{*}(2380)$ | $5^{-}$ | 2382 | 2352 | 1.3
$a_{6}(2450)^{\pm}$ | $6^{++}$ | $2450\pm 130$ | 2412 | 0
$a_{6}(2450)^{0}$ | $6^{++}$ | $2450\pm 130$ | 2423 | 0
$f_{6}(2510)$ | $6^{++}$ | $2465\pm 50$ | 2649 | 5.3
## VI Analysis of the results
In the above calculations, only one set of parameters are used, which deserves
discussion. The effective coupling strength or running coupling constant
$\bar{\alpha}$ and the related constituent quark masses have a great influence
on ground state of light mesons, such as $\pi$. The confining potential
strength $\lambda$ governs the quark confinement at large distances and has
strong influence on the excited states of light mesons and also on the spectra
of heavy mesons. From the recent experiments of hadron physics, we know that
the QCD coupling $\alpha(Q^{2})$ becomes large constant(not singular) in the
low momentum limit, which is called infrared conformal invariance Conf . This
experimental fact explains why our model with a set of constant parameters
works well to describe the structures of mesons in the energy region of
$0.14\text{GeV}\\!\rightarrow\\!10\text{GeV}$, and our results may be thought
of confirming the infrared conformal invariance feature of QCD on meson
sector.
For light scalar mesons such as $a_{0}$ ,$K_{0}^{*}$, etc., although the
structure of the scalar mesons remains a challenging puzzle, our model still
describes $a_{0}(980)$, $a_{0}(1450)$, $K_{0}^{*}(800)$, etc. quite well. For
heavy mesons, because of the large masses of heavy quarks, the effective
double-gluon-exchange interactions for off-diagonal heavy mesons are weak,
which makes the model applicable to them. Therefore, the calculated mass
spectra for the mesons of $u/d\bar{s}$, $u/d\bar{c}$, $s\bar{c}$, $c\bar{c}$,
$c\bar{b}$, $u/d\bar{b}$, $s\bar{b}$, and $b\bar{b}$ are in good agreement
with the data. However the meson $K^{*}(892)$ on $u/d\bar{s}$ sector with
larger error of $50.1\%$ needs special investigation(see below).
It should be noted that the $J$ and $P$ of $D_{s}^{*\pm}$ are not identified
by experiments, but their width and decay modes are observed and consistent
with the $1^{-}$ state. Nevertheless, our model provides a definite assignment
of $J=1$ and $P=-1$ for $D_{s}^{*\pm}$. A similar prediction of the
unidentified $J$ and $P$ is also made for other 8 mesons: $X(1835)$,
$D_{1}(2420)^{\pm}$, $K^{*}(1630)$, $D^{*}(2640)$, $D^{*}_{S2}(2573)^{\pm}$,
$B^{*}_{J}(5732)$, $B^{*}_{SJ}(5850)^{0}$, and $f_{J}(2220)$.
The 6 mesons with errors larger than $23\%$ provide some information. For the
vector mesons of $\eta,\eta^{\prime}(985),\rho(770)^{0}$, $\phi(1020)$, and
$\omega(782)$ on $u/d$ sector, and $K^{*}(892)$ on $(u/d)s$ sector , the large
discrepancy indicates that the structures of these mesons are special than
others and need a different set of parameters: indeed, as the set of
parameters are re-adjusted to the set of
($\alpha=0.4594,\gamma_{0}=0.58,\delta_{0}=0.74$) and with the others the
same, a better fit is found with errors less than $23\%$. Increase of the
effective interaction strengths implies that these vector mesons may have
strong coupling between $q\bar{q}$ and $qq\bar{q}\bar{q}$ subspaces and among
different flavor components.
## VII Conclusion and discussion
In conclusion, we have formulated the QCD inspired relativistic bound state
model for mesons and derived its mass eigen equations in total angular
momentum representation. It is proved that in center of mass frame and in
internal Hilbert subspace, total angular momentum of the meson system is
conserved. Moreover, by taking the advantages of other effective QCD
approaches Sommerer01 ; Hersbach01 , the model has been improved significantly
by introducing both a relativistic confining potential and an $SU(3)$ flavor
mixing interaction. The resulting radial mass eigen equations are solved
numerically and nonperturbatively, and 265 mesons including flavor off-
diagonal mesons and flavor diagonal ones with $J=0-6$ are calculated and
compared with experimental data. The calculated masses are in good agreement
with the data within the mean square root mass error of $14\%$, only 6 mesons
with mass error larger than $23\%$. Besides, the wave functions obtained from
the model also yield reasonable mean square radii and decay constants for some
pseudo scalar mesons. In view that the structure of the light scalar mesons is
still a subject of controversyDon , and the internal dynamics of heavy-light
mesons in the static limit is far more complicated than that of the heavy-
heavy onesDam , our model can be thought to be successful to describe a large
body of mesons.
The comparison of our model with other approaches is as follows:
1\. As Pauli’s model is concerned, we have improved the model significantly on
5 important points and make it a predictive and systematic model for mesons:
1) Proving that in internal Hilbert subspace, total angular momentum is
conserved; 2) establishing the mass eigen equations in total angular
representation for the first time; 3) introducing the relativistic confining
potential into the model, which is new and quite different from Pauli, and its
form taken from the Sommerer01 ; Hersbach01 ; 4) including the flavor mixing
interaction; 5) solving the mass equations for 265 mesons nonperturbatively
and the results are in good agreement with the data.
2\. Comparing to other BSE and CQM meson models, our model is novel in
following points: 1) The effective Hamiltonian is derived within the framework
of light-front QCD and the form (the spinor structure ) of the effective
interactions is fixed by the lowest order of light-front QCD. 2) The mass
eigen equation is for the squared rest mass, the separation between
kinematical energy operator and interaction operators is rigorous. 3)The
spinor structure of the effective interaction make it momentum-energy
dependent. 4) Also due to the spinor structure of the effective interactions,
the dynamics of spin-spin, spin-orbital, and tensor interactions ( especially
the spin singlet-triplet mxing and orbital angular momentum mixing ) are
included( see Appendix C,D ). 5) The predictive power and the descriptive
precision of the model are much better.
3\. Comparing to holographic light-font QCD model of Brodsky $et\ al.$Brod ,
our model has the following new aspects : 1) In the effective Hamiltonian of
mesons, the kinematical energy operator is identical for both holographic
light-front QCD model and our model, but the interaction terms are quite
different. 2) Holographic light-front QCD model does not specify the effective
interaction in detail, but just simulates confining potential by boundary
condition ( or harmonic oscillator potential), or recently by a positive-sign
dilaton metric to generate confinement and break conformal symmetry; instead,
our model provides a detailed semi-phenomenological effective interaction
including its spinor structure, the confining potential, and the flavor mixing
interactions. 3) Holographic light-front QCD model does not include spin-spin,
spin-orbital, and tensor interactions, the total angular momentum of the
system is not treated properly ( although it has potential to describe the
spin splittings ); in the contrary, our model specifies the spin interactions
and the spin dynamics is described fully in total angular momentum
representation. 4) Finally, our model has been applied to a larger number of
mesons ( 265 mesons identified experimentally ) with higher precision than
those of holographic light-front QCD model. In the above respects, our model
has provided a tentative and effective solution to the problems listed above
and the results are amazingly in good agreement with experimental data. In
this sense, our model can be considered to be of complementarity to and
refinement of the holographic light-front QCD model.
This work was supported in part by the National Natural Science Foundation of
China under grant Nos.10974137 and 10775100, and by the Fund of Theoretical
Nuclear Physics Center of HIRFL of China.
## Appendix A Dynamics in light front form and instant form in center of mass
frame and in internal Hilbert subspace
To avoid misunderstanding of light-front dynamics, we start from a discussion
of full contents of dynamics for both instant form(IF) and light front
form(LF). The content of dynamics should contain the following four aspects,
we list them for both dynamics of instant form and dynamics of light form as
follows.
### A.1 Full contents of dynamics in instant form
1) Definition of time $x^{0}=ct=t(c=1)$
2) Hamiltonian (energy) operator is defined as the time translation operator:
$\displaystyle i\hbar\frac{\partial}{\partial
x^{0}}\sim\hat{P}^{0}=\hat{H}=\hat{M}$ (61)
$\hat{M}$ is dynamical mass operator.
3) Dynamics
(i) Time evolution dynamics: equation of motion (Schr$\ddot{o}$dinger
equation),
$\displaystyle i\hbar\frac{\partial\Psi}{\partial
x^{0}}=\hat{H}\Psi=\hat{M}\Psi$ (62)
(ii) Stationary dynamics: for stationary solution ,
$\displaystyle\Psi(t)=e^{-iMt/\hbar}\Psi$ (63)
one has the Hamiltonian eigen equation
$\displaystyle\hat{H}\Psi=\hat{M}\Psi=M\Psi,$ (64)
where $M$ is the eigen value of $\hat{M}$.
4) Specification of dynamical operators and kinematical operators among
Poincare generators: 6 kinematical operators:
$\hat{P}^{i},\hat{J}^{i},(i=1,2,3)$; 4 dynamical operators:
$\hat{P}^{0},\hat{K}^{i},(i=1,2,3)$.
It should be noted that the dynamical operators contain interactions via the
Hamiltonian and Lorentz boost operators while the kinematical operators do
not. Consequently, the kinematical operators can be used to characterize the
state of the system as good quantum numbers according their algebraic
structure and the dynamical operators except the Hamiltonian operator can not
play such a role. It should be emphasized that the above specification is made
in whole Hilbert space of the states of composite systems. For a composite
many-body system, the whole Hilbert space of states can be factorized into two
parts: a) the center of mass motion characterized by its momentum $\vec{P}$ ,
and b) the internal motion characterized by internal quantum numbers and
($J,J^{3}$ ). Correspondingly, the Poincare operators contain two kinds of
operations, one on the subspace of center of mass motion and the other on the
subspace of internal motion. Since the center of mass motion can always be
separated from the internal motion, the state wave function of the composite
system $\Psi$ can be written as $\Psi=\Psi_{cm}\Psi_{inter}$, where the wave
function of center of mass motion is characterized by the center of mass
momentum, namely $\Psi_{cm}=\Psi_{\vec{P}}$ with
$\hat{P}^{i}\Psi_{\vec{P}}=P^{i}\Psi_{\vec{P}}$ , while the internal wave
function is characterized by internal quantum numbers and ($J,J^{3}$ ).
### A.2 Full contents of dynamics in light front form:
1) Definition of time $x^{+}$: $x^{+}=x^{0}+x^{3}$
2) Hamiltonian (”energy”) operator is defined as the time translation
operator:
$\displaystyle i\hbar\frac{\partial}{\partial x^{+}}\sim\hat{P}^{-}$ (65)
From
$\displaystyle\hat{P}^{+}\hat{P}^{-}-\hat{P}^{2}_{\bot}=\hat{P}_{0}\hat{P}^{0}-\hat{P}_{3}^{2}-\hat{P}_{1}^{2}-\hat{P}_{2}^{2}=\hat{M}_{0}^{2}$
(66)
$\hat{M}_{0}$ is rest mass operator; one has
$\displaystyle\hat{P}^{-}=\frac{1}{\hat{P}^{+}}(\hat{M}_{0}^{2}+\hat{P}_{\bot}^{2})$
(67)
3) Dynamics
(i) Time evolution dynamics: equation of motion (Schr$\ddot{o}$dinger
equation),
$\displaystyle i\hbar\frac{\partial}{\partial
x^{+}}\Psi=\frac{1}{\hat{P}^{+}}(\hat{M}_{0}^{2}+\hat{P}_{\bot}^{2})\Psi$ (68)
(ii) Stationary dynamics: for stationary solution
$\Psi(\frac{M^{2}}{P^{+}},P^{+},\vec{P}_{\bot},x^{+})$ with quantum numbers:
”energy” $E^{-}=\frac{M^{2}}{P^{+}}$ and momentum
$\vec{P}=(P^{+},\vec{P}_{\bot})$($E^{-}$ is the eigen value of $\hat{P}^{-}$,
$P^{+}$ and $\vec{P}_{\bot}$ are eigen values of $\hat{\vec{P}}^{+}$,
$\hat{\vec{P}}_{\bot}$),
$\displaystyle\Psi(x^{+})=e^{-iM^{2}x^{+}/P^{+}\hbar}\Psi(\frac{M^{2}}{P^{+}},P^{+},\vec{P}_{\bot})$
(69)
One has mass eigen equation:
$\displaystyle\frac{M^{2}}{P^{+}}\Psi(\frac{M^{2}}{P^{+}},P^{+},\vec{P}_{\bot})=\frac{1}{P^{+}}(\hat{M}_{0}^{2}+\hat{P}_{\bot}^{2})\Psi(\frac{M^{2}}{P^{+}},P^{+},\vec{P}_{\bot})$
(70)
or
$\displaystyle\hat{M}_{0}^{2}\Psi(\frac{M^{2}}{P^{+}},P^{+},\vec{P}_{\bot})=(M^{2}-\vec{P}_{\bot}^{2})\Psi(\frac{M^{2}}{P^{+}},P^{+},\vec{P}_{\bot})$
(71)
4) Specification of dynamical operators and kinematical operators among
Poincare generators: 7 kinematical operators: $\hat{P}^{+}$, $\hat{J}^{3}$,
$\hat{P}^{i}(i=1,2)$,$\hat{K}^{3}$, $\hat{E}^{1}=\hat{K}^{1}+\hat{J}^{2}$,
$\hat{E}^{2}=\hat{K}^{2}-\hat{J}^{1}$; 3 dynamical operators: $\hat{P}^{-}$,
$\hat{F}^{1}=\hat{K}^{1}-\hat{J}^{2}$, $\hat{F}^{2}=\hat{K}^{2}+\hat{J}^{1}$.
### A.3 Dynamics in center of mass frame and in internal Hilbert subspace for
both forms of dynamics
The internal structure of a composite system should be described in the rest
frame as well as in the corresponding internal Hilbert subspace. Since the
center of mass frame always follows the center of mass motion of the system
and the position of the center of mass of the system is at the origin of the
frame, the wave function of center of mass motion of the system should be
$\Psi_{\vec{P}=0}$, and the center of mass momentum and the center of mass
coordinates of the system should be zero, namely
$<\Psi_{\vec{P}=0}|\hat{P}^{i}|\Psi_{\vec{P}=0}>=0$ and
$<\Psi_{\vec{P}=0}|\hat{x}^{i}|\Psi_{\vec{P}=0}>=0$. In the center of mass
frame, the Hilbert subspace of center of mass motion is frozen to
$\Psi_{\vec{P}=0}$, the whole Hilbert space of states of the system is thus
projected onto the corresponding internal Hilbert subspace $\Psi_{inter}$.
Consequently, the dynamics of the composite system is reduced to the internal
dynamics. Projecting onto the frozen center of mass wave function and
integrating out the center of mass degrees of freedom, one obtain the Poincare
operators in the internal subspace $\Psi_{inter}$ as follows.
1) Four momentum and property of time in center of mass frame and in internal
Hilbert subspace.
In center of mass frame, the wave function of center of mass motion is :
$\Psi_{\vec{P}=0}$. The four momentum operator in internal Hilbert subspace
can be obtained by projecting out the center of mass degrees of freedom (
namely averaging over the center of mass wave function). Since
$\hat{P}^{0}=\hat{H}=(\hat{M}_{0}^{2}+\hat{\vec{P}}^{2})^{1/2}$ and
$\hat{P}^{i}\Psi_{\vec{P}=0}=0$, in internal Hilbert subspace, one has
$\hat{P}_{inter}^{i}=\left<\Psi_{\vec{P}=0}|\hat{P}^{i}|\Psi_{\vec{P}=0}\right>=0$
(72)
$\hat{P}_{inter}^{0}=\left<\Psi_{\vec{P}=0}|\hat{P}^{0}|\Psi_{\vec{P}=0}\right>=\hat{M}_{0}.$
(73)
Here $\hat{M}_{0}$ is the operator of rest mass of the system.
Thus in internal Hilbert subspace, the four momentum operators for instant
form read:
$\displaystyle\hat{P}_{inter}^{\mu}=(\hat{M}_{0},0,0,0)$ (74)
while four momentum operators for light front form are:
$\displaystyle\hat{P}_{inter}^{\mu}=(\hat{M}_{0},0,0,\hat{M}_{0}),$ (75)
From the above results , one has
$\displaystyle\hat{P}_{inter}^{-}=\hat{P}_{inter}^{0}=\hat{P}_{inter}^{+}=\hat{M}_{0},$
(76)
$\displaystyle i\hbar\frac{\partial}{\partial
x^{+}}=i\hbar\frac{\partial}{\partial
x^{0}}=i\hbar\frac{\partial}{\partial\tau},$ (77)
where $\tau$ is the proper time corresponding to the rest mass operator
$\hat{M}_{0}$. The last equation leads to
$\displaystyle x^{+}=\tau+\tau_{0},~{}x^{0}=\tau+\tau_{0}^{\prime}$ (78)
where $\tau_{0}$ and $\tau_{0}^{\prime}$ are constant shifts of proper time.
One can choose the start point of time such that
$\displaystyle\tau=0\rightarrow x^{+}=x^{0}=0$ (79)
This leads to
$\displaystyle\tau_{0}=\tau_{0}^{\prime}=0$ (80)
and
$\displaystyle x^{+}=x^{0}=\tau$ (81)
2) Dynamics in center of mass frame and in internal Hilbert subspace
(i) Time evolution dynamics: equations of motion in center of mass frame and
in internal Hilbert subspace
The Schr$\ddot{o}$dinger equations
$\displaystyle i\hbar\frac{\partial\Psi}{\partial x^{+}}=\hat{M}_{0}\Psi$ (82)
in light front form, and
$\displaystyle i\hbar\frac{\partial\Psi}{\partial x^{0}}\Psi=\hat{M}_{0}\Psi$
(83)
in instant form become the same
$\displaystyle i\hbar\frac{\partial\Psi}{\partial\tau}=\hat{M}_{0}\Psi$ (84)
(ii) Stationary dynamics: mass (energy) eigen equations in center of mass
frame and in internal Hilbert subspace.
The mass eigen equations
$\displaystyle\hat{M}_{0}\Psi=M_{0}\Psi$ (85)
in instant form where $M_{0}$ is the eigen value of $\hat{M}_{0}$, and
$\displaystyle\hat{M}_{0}^{2}\Psi=M_{0}^{2}\Psi$ (86)
in light front form are also the same because multiplying $\hat{M}_{0}$ on the
first equation leads to the second one.
3) Kinematical and dynamical operators in center of mass frame and in internal
Hilbert subspace
Projecting onto internal Hilbert subspace, the kinematical and dynamical
operators can be obtained from the following calculation. From the results of
(A12-A15) and
$\displaystyle\hat{J}^{i}=\hat{J}_{cm}^{i}+\hat{J}_{inter}^{i},$ (87)
$\displaystyle\hat{J}_{cm}^{1}=\hat{x}^{2}\hat{P}^{3}-\hat{P}^{2}\hat{x}^{3},cyclic,$
(88)
one obtain
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{J}_{cm}^{i}|\Psi_{\vec{P}=0}\right>=0,$
(89)
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{J}^{i}|\Psi_{\vec{P}=0}\right>=\hat{J}_{inter}^{i}$
(90)
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{K}^{i}|\Psi_{\vec{P}=0}\right>=\left<\Psi_{\vec{P}=0}|\hat{x}^{0}\hat{P}^{i}-\hat{x}^{i}\hat{P}^{0}|\Psi_{\vec{P}=0}\right>,$
(91)
$\displaystyle=\left<\Psi_{\vec{P}=0}|\hat{x}^{i}|\Psi_{\vec{P}=0}\right>\hat{M}_{0}=0,$
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{E}^{1}|\Psi_{\vec{P}=0}\right>=\left<\Psi_{\vec{P}=0}|\hat{K}^{1}+\hat{J}^{2}|\Psi_{\vec{P}=0}\right>=\hat{J}_{inter}^{2}$
(92)
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{E}^{2}|\Psi_{\vec{P}=0}\right>=\left<\Psi_{\vec{P}=0}|\hat{K}^{2}-\hat{J}^{1}|\Psi_{\vec{P}=0}\right>=-\hat{J}_{inter}^{1}$
(93)
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{F}^{1}|\Psi_{\vec{P}=0}\right>=\left<\Psi_{\vec{P}=0}|\hat{K}^{1}-\hat{J}^{2}|\Psi_{\vec{P}=0}\right>=-\hat{J}_{inter}^{2}$
(94)
$\displaystyle\left<\Psi_{\vec{P}=0}|\hat{F}^{2}|\Psi_{\vec{P}=0}\right>=\left<\Psi_{\vec{P}=0}|\hat{K}^{2}+\hat{J}^{1}\hat{P}^{0}|\Psi_{\vec{P}=0}\right>=\hat{J}_{inter}^{1}$
(95)
From the above results, one obtains the same reduced and degenerated
kinematical and dynamical operators for both forms of dynamics in internal
Hilbert subspace as follows: kinematical operators:
$\hat{J}_{inter}^{i}(i=1,2,3)$; dynamical operator:
$\hat{P}_{inter}^{0}=\hat{P}_{inter}^{-}=\hat{M}_{0}$.
The above results tell that in center of mass frame and in internal Hilbert
subspace, light front time and instant time, light front dynamics and instant
dynamics, light front angular momentum and instant angular momentum are
identical.
### A.4 Conclusion
In general frames and in whole Hilbert space, both forms of dynamics are quite
different. However, in center of mass frame and in internal Hilbert subspace,
the two forms of dynamics are reduced to the identical internal dynamics.
There is a dilemma in this paper at first glance: our model begins with a
light front QCD model, but the final form of our model possesses the feature
of instant dynamics of QCD. Is it of LF dynamics or IF dynamics? The solution
to the dilemma is given in this Appendix, the answer is that in center of mass
frame and in internal Hilbert subspace, the reduced internal dynamics of both
forms are identical.
Therefore, our model contains ingredients of both the instant form and light
front form of QCD, it can be called as QCD inspired effective Hamiltonian
meson model.
## Appendix B Conservation of total angular momentum in internal Hilbert
subspace
The reduction of angular momentum operators in internal Hilbert subspace can
be discussed in an alternative manner and the results are the same as that in
Appendix A.
A relativistic dynamical system has inhomogeneous Lorentz symmetry defined by
the $Poincar\acute{e}\ algebra$: $P^{\mu}$ is energy-momentum vector, and
$M^{\mu\nu}$ is used to describes the rotational and boost transformations. In
instant form, the angular momentum and boost vectors are given as:
$M^{ij}=\epsilon_{ijk}J^{k}$ and $M^{0i}=K^{i}$
Now define the ”quasi angular momentum” operators in the light-front form:
$\displaystyle\mathcal{J}^{3}=J^{3}+\frac{\varepsilon_{ij}{\bm{E}}_{\perp}^{i}{\bm{P}}_{\perp}^{j}}{P^{+}}\
,$ $\displaystyle\mathcal{J}^{\perp
i}=M_{0}^{-1}\varepsilon_{ij}(\frac{1}{2}({\bm{F}}_{\perp}^{j}P^{+}-{\bm{E}}_{\perp}^{j}P^{-})-K^{3}{\bm{P}}_{\perp}^{j}$
$\displaystyle+\mathcal{J}^{3}\varepsilon_{jl}{\bm{P}}_{\perp}^{l}),(i,j=1,2).$
(96)
It is easy to prove that they satisfy the SU(2) algebra:
$\displaystyle[\mathcal{J}^{i},\mathcal{J}^{j}]=i\epsilon_{ijk}\mathcal{J}^{k}$
(97)
It is very useful to define a ‘light-front Hamiltonian’ as the operator:
$\displaystyle
H_{LC}=P^{\mu}P_{\mu}=P^{-}P^{+}-\vec{P_{\bot}}^{2}=\hat{M}_{0}^{2}$ (98)
$H_{LC}$ commutes with the quasi angular momentum operators :
$\displaystyle[H_{LC},\vec{\mathcal{J}}]=0.$ (99)
In principle, one could label the eigen states as
$|M,P^{+},\vec{P_{\bot}},\vec{\mathcal{J}}^{2},\vec{\mathcal{J}}_{3}\rangle$,
since $\mathcal{J}_{3}$ is kinematical. However, $\vec{\mathcal{J}_{\bot}}$ is
dynamical and depends on the interactions. Thus it is generally difficult to
explicitly compute the total spin $\vec{\mathcal{J}}$ of a state using light-
front quantization. Fortunately, in center-of-mass frame and in internal
Hilbert subspace, by using the results of Appendix A, one has the following
equations,
$\displaystyle\left<\Psi_{\vec{P}=0}|\mathcal{J}^{3}|\Psi_{\vec{P}=0}\right>=J_{inter}^{3}\
,$
$\displaystyle\left<\Psi_{\vec{P}=0}|\mathcal{J}^{i}|\Psi_{\vec{P}=0}\right>=J_{inter}^{i}$
(100) $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(i,j=1,2)\ \,.$
Therefore, in internal Hilbert subspace, the quasi angular momentum operators
$\mathcal{J}_{inter}^{i}$ are identical to the total angular momentum
operators $J_{inter}^{i}(i=1,2,3)$, the total angular momentum is conserved,
and the eigen equation of the Hamiltonian $H_{LC}$ of the internal dynamics
can be solved in the total angular momentum representation.
## Appendix C Derivation of the radial mass eigen equations in total angular
momentum representation
According to Pauli et al., the effective mass eigen equation of mesons of
light-front QCD in center of mass frame and in internal Hilbert subspace
reads:
$\displaystyle\left[M_{0}^{2}-\left(E_{1}(k)+E_{2}(k)\right)^{2}\right]\varphi_{s_{1}s_{2}}(\bm{k})$
(101) $\displaystyle=$ $\displaystyle\sum_{s_{1}^{\prime}s_{2}^{\prime}}\int
d^{3}\bm{k}U_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\bm{k};\bm{k^{\prime}})\varphi_{s_{1}^{\prime}s_{2}^{\prime}}(\bm{k}^{\prime}),$
where
$U_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}=\displaystyle\frac{4m_{s}}{3\pi^{2}}\displaystyle\frac{\overline{\alpha}(Q)}{Q^{2}}R(Q)\displaystyle\frac{S_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}}{\sqrt{A(k)A(k^{\prime})}}$
(102)
with
$\displaystyle S_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}$ $\displaystyle=$
$\displaystyle[\overline{u}(\bm{k},s_{1})\gamma_{\mu}(1)u(\bm{k^{\prime}},s_{1}^{\prime})]$
(103) $\displaystyle\times$
$\displaystyle[\overline{v}(-\bm{k},s_{2})\gamma^{\mu}(2)v(-\bm{k^{\prime}},s_{2}^{\prime})]$
and
$\displaystyle\displaystyle\frac{1}{A(k)}$ $\displaystyle=$ $\displaystyle
m_{r}\left(\displaystyle\frac{1}{E_{1}(k)}+\displaystyle\frac{1}{E_{2}(k)}\right),\
$ $\displaystyle m_{s}$ $\displaystyle=$ $\displaystyle m_{1}+m_{2},\
m_{r}=\displaystyle\frac{m_{1}m_{2}}{m_{1}+m_{2}},\ $ $\displaystyle Q$
$\displaystyle=$ $\displaystyle Q(\bm{k};\bm{k}^{\prime}).$ (104)
Equation (A1) can be written as Schrödinger equation in the light front QCD,
$\widehat{H}\Psi_{\text{meson}}=M_{0}^{2}\Psi_{\text{meson}}$ (105)
The general eigen wave function $\Psi_{\text{meson}}$ of meson can be
expressed in momentum-spin representation,
$\Psi_{\text{meson}}=\sum_{s_{1},s_{2}}\int
d^{3}\bm{k}\varphi_{s_{1}s_{2}}(\bm{k})\left|\chi(s_{1})\chi(s_{2})\cdot\bm{k}\right\rangle.$
(106)
Here basis of the momentum-spin representation are
$\langle\bm{r}|\chi(s_{1})\chi(s_{2})\cdot\bm{k}\rangle=\displaystyle\frac{1}{(2\pi\hbar)^{3/2}}\chi(s_{1})\chi(s_{2})e^{i\bm{k}\cdot\bm{r}},$
(107)
where the spin wave functions and their orthogonal conditions read
$\chi(+\frac{1}{2})=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),\
\chi(-\frac{1}{2})=\left(\begin{array}[]{c}0\\\ 1\end{array}\right),$ (108)
$\langle\chi(s_{1})|\chi(s_{2})\rangle=\delta_{s_{1}s_{2}}.$ (109)
The orthogonal conditions of the spinors are
$\displaystyle\langle\bm{k}\cdot\overline{u}(\bm{k},s_{1})\overline{v}(-\bm{k},s_{2})|u(\bm{k}^{\prime},s_{1}^{\prime})v(-\bm{k}^{\prime},s_{2}^{\prime})\cdot\bm{k}^{\prime}\rangle$
(110) $\displaystyle=$
$\displaystyle\delta^{(3)}(\bm{k}-\bm{k}^{\prime})\delta_{s_{1}s_{1}^{\prime}}\delta_{s_{2}s_{2}^{\prime}},$
$\displaystyle\overline{u}(\bm{k},s_{1})u(\bm{k},s_{1}^{\prime})$
$\displaystyle=$ $\displaystyle\delta_{s_{1}s_{1}^{\prime}},$ (111)
$\displaystyle\overline{v}(-\bm{k},s_{2})v(-\bm{k},s_{2}^{\prime})$
$\displaystyle=$ $\displaystyle\delta_{s_{2}s_{2}^{\prime}},$ (112)
and the completeness conditions read,
$\displaystyle\sum_{s}u(\bm{k},s)\overline{u}(\bm{k},s)$ $\displaystyle=$
$\displaystyle\frac{1}{2m}\left(\gamma_{\mu}k_{1}^{\mu}+m\right),$ (113)
$\displaystyle\sum_{s}v(-\bm{k},s)\overline{v}(-\bm{k},s)$ $\displaystyle=$
$\displaystyle\frac{1}{2m}\left(\gamma_{\mu}k_{2}^{\mu}-m\right),$ (114)
where $k_{1}^{\mu}=(E_{1}(k),\bm{k})$, and $k_{2}^{\mu}=(E_{2}(k),-\bm{k})$.
According to the Dirac form of quantum mechanics, in the eigen equation (105),
the Dirac form of the Hamiltonian operator is
$\widehat{H}=\widehat{E}+\widehat{U},$ (115)
where
$\displaystyle\widehat{E}$ $\displaystyle=$ $\displaystyle\int
d^{3}\bm{k}\displaystyle\left[E_{1}(k)+E_{2}(k)\right]^{2}$ (116)
$\displaystyle\mbox{}\times\sum_{s_{1}s_{2}}|\chi(s_{1})\chi(s_{2})\cdot\bm{k}\rangle\langle\bm{k}\cdot\chi(s_{1})\chi(s_{2})|,$
and
$\displaystyle\widehat{U}=\int
d^{3}\bm{k}d^{3}\bm{k}^{\prime}\sum_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}U(k,k^{\prime})$
(117)
$\displaystyle\mbox{}\times\left[\overline{u}(\bm{k},s_{1})\overline{v}(-\bm{k},s_{2})(\gamma_{\mu}(1)\gamma^{\mu}(2))u(\bm{k}^{\prime},s_{1}^{\prime})v(-\bm{k}^{\prime},s_{2}^{\prime})\right]$
$\displaystyle\mbox{}\times|\chi(s_{1})\chi(s_{2})\cdot\bm{k}\rangle\langle\bm{k}^{\prime}\cdot\chi(s_{1}^{\prime})\chi(s_{2}^{\prime})|,$
with the definition,
$U(k,k^{\prime})\equiv\displaystyle\frac{1}{3m_{r}\pi^{2}}\displaystyle\frac{\overline{\alpha}(Q)}{Q^{2}}R(Q)\displaystyle\frac{1}{\sqrt{A(k)A(k^{\prime})}}.$
(118)
In the above equation, as done by Pauli et al.Pauli03 , the light front $k-$
space has been transformed back to the Lab $k-$ space by the Terent’ev
transformation, and Lepage-Brodsky (helicity) spinors have been transformed to
the Bjorken-Drell (spin) spinors.
Using eqs.(106, 115-118) and projecting equation (105) onto the subspace
$|\chi(s_{1})\chi(s_{2})\cdot\bm{k}\rangle$, we recover the equation (101),
indicating that the Dirac Form of the eigen equation (105) is equivalent that
of (101).
Since $(E_{1}(k)+E_{2}(k))^{2}$ and the interaction kernal operator
$\widehat{U}[\bm{k},\bm{k}^{\prime};\bm{\sigma}(1),\bm{\sigma}(2)]$ are scalar
(see Appendix D, discussion below eq.(142)), $\widehat{H}$ is rotational
invariant with respect to the total angular momentum
$\bm{J}_{i}=\bm{l}_{i}+\bm{s}_{i}^{1}+\bm{s}_{i}^{2}=\bm{l}_{i}+\bm{s}_{i}$,
$[\widehat{H},\bm{J}_{i}]=0$. That means the total angular momentum
$\widehat{\bm{J}}^{2}$ and $\widehat{J}_{z}$ are conserved. Based on this
point, the wave function of the meson system can be written in total angular
representation as follows,
$\Psi_{meson}(k,\Omega_{k},s)=\sum_{J,M}\sum_{l=|J-s|}^{J+s}\sum_{s=0,1}R_{Jsl}(k)\Phi_{JslM}(\Omega_{k},s),$
(119)
were the total angular momentum eigen functions $\Phi_{JslM}$ of
$\\{\widehat{\bm{J}}^{2},\widehat{J}_{z},\widehat{\bm{s}}^{2},\widehat{\bm{l}}^{2}\\}$
are,
$\Phi_{JslM}(\Omega_{k},s)=\sum_{m\mu}\langle lms\mu|JM\rangle
Y_{lm}(\Omega_{k})\chi_{s\mu}(12),$ (120)
the eigen wave functions of spin singlet and triplet read as,
$\chi_{s\mu}(12)=\sum_{s_{1}s_{2}}\textstyle\langle\frac{1}{2}s_{1}\frac{1}{2}s_{2}|s\mu\rangle\chi(s_{1})\chi(s_{2}).$
(121)
By virtue of the Fourier transformation in spherical coordinates, from the
eigen wave function in the momentum radial $k-$space, one can obtain the
corresponding wave function in the configuration radial $r-$ space,
$\displaystyle\Psi_{JM}(r,\Omega_{r},s)$ $\displaystyle=$ $\displaystyle\int
d\bm{k}^{3}\Psi_{JM}(k,\Omega_{k},s)e^{i\bm{k}\cdot\bm{r}}$ (122)
$\displaystyle=$
$\displaystyle\sum_{l=|J-s|}^{J+s}\sum_{s=0,1}\sum_{l^{\prime},m^{\prime}}\int
k^{2}dkR_{Jsl}(k)J_{l}(kr)$ $\displaystyle\int
d\Omega_{k}\Phi_{JslM}(\Omega_{k},s)Y^{*}_{l^{\prime}m^{\prime}}(\Omega_{k})Y_{l^{\prime}m^{\prime}}(\Omega_{r})$
$\displaystyle=$
$\displaystyle\sum_{l=|J-s|}^{J+s}\sum_{s=0,1}R_{Jsl}(r)\Phi_{JslM}(\Omega_{r},s),$
where
$\displaystyle\Phi_{JslM}(\Omega_{r},s)$ $\displaystyle=$
$\displaystyle\sum_{m\mu}\langle lms\mu|JM\rangle
Y_{lm}(\Omega_{r})\chi_{s\mu}(12),$ $\displaystyle R_{Jsl}(r)$
$\displaystyle=$ $\displaystyle\int k^{2}dkR_{Jsl}(k)J_{l}(kr),$
$\displaystyle J(kr)$ $\displaystyle=$ $\displaystyle\sqrt{4\pi(2l+1)}\ i^{l}\
j_{l}(kr).$ (123)
$j_{l}(kr)$ is the spherical Bessel function of order $l$.
Using the expression (119) of the wave function $\Psi_{meson}$, projecting the
mass eigen equation (105) onto the $\Phi_{JslM}$ subspace from the left, and
integrating out the spin and angular part of the wave function, we obtain the
eigen equations for the radial wave functions $R_{Jsl}(k)$,
$\displaystyle\left[M_{0}^{2}-\left(E_{1}(k)+E_{2}(k)\right)^{2}\right]R_{Jsl}(k)$
$\displaystyle=$
$\displaystyle\sum_{l^{\prime}=|J-s^{\prime}|}^{J+s^{\prime}}\sum_{s^{\prime}=0,1}\int
k^{\prime
2}dk^{\prime}U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})R_{Js^{\prime}l^{\prime}}(k^{\prime}),$
where the kernel $U_{Sl;S^{\prime}l^{\prime}}^{J}(k;k^{\prime})$ is defined
as,
$\displaystyle
U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})=\sum_{mm^{\prime}}\sum_{s_{1}s_{2}}\sum_{s_{1}^{\prime}s_{2}^{\prime}}\int\int
d\Omega_{k}d\Omega_{k^{\prime}}$ $\displaystyle\mbox{}\times\langle
Y_{lm}(\Omega_{k})|U_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}(\textbf{k},\textbf{k}^{\prime})|Y_{l^{\prime}m^{\prime}}(\Omega_{k^{\prime}})\rangle$
$\displaystyle\mbox{}\times\langle
lms\mu|JM\rangle\textstyle\langle\frac{1}{2}s_{1}\frac{1}{2}s_{2}|s\mu\rangle\langle
l^{\prime}m^{\prime}s^{\prime}\mu^{\prime}|JM\rangle\textstyle\langle\frac{1}{2}s_{1}^{\prime}\frac{1}{2}s_{2}^{\prime}|s^{\prime}\mu^{\prime}\rangle.$
This is a set of coupled equations for the radial functions $R_{Jsl}(k)$ that
have different partial waves, spin singlet and triplet coupled by the tensor
potentials and by the relativistic spin-orbital potential (see below).
## Appendix D Calculation of the interaction kernel in total angular momentum
representation
The quark and anti-quark spinors are given in the Bjørken-Drell
representation,
$\displaystyle u(\bm{k},s=+\textstyle\frac{1}{2})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\sqrt{2m_{1}(E_{1}+m_{1})}}\left(\begin{array}[]{c}E_{1}+m_{1}\\\
0\\\ k_{z}\\\ k_{l}\end{array}\right),$ $\displaystyle
u(\bm{k},s=-\textstyle\frac{1}{2})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\sqrt{2m_{1}(E_{1}+m_{1})}}\left(\begin{array}[]{c}0\\\
E_{1}+m_{1}\\\ k_{r}\\\ -k_{z}\end{array}\right),$ $\displaystyle
v(-\bm{k},s=+\textstyle\frac{1}{2})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\sqrt{2m_{2}(E_{2}+m_{2})}}\left(\begin{array}[]{c}-k_{z}\\\
-k_{l}\\\ E_{2}+m_{2}\\\ 0\end{array}\right),$ $\displaystyle
v(-\bm{k},s=-\textstyle\frac{1}{2})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\sqrt{2m_{2}(E_{2}+m_{2})}}\left(\begin{array}[]{c}-k_{r}\\\
k_{z}\\\ 0\\\ E_{2}+m_{2}\\\ \end{array}\right),$
where
$\displaystyle k_{l,r}$ $\displaystyle=$ $\displaystyle k_{x}\pm
ik_{y}=k\sin\theta_{k}e^{\pm i\varphi_{k}}=k\sqrt{\frac{8\pi}{3}}Y_{1\pm
1}(\theta_{k},\varphi_{k}),$ $\displaystyle k_{z}$ $\displaystyle=$
$\displaystyle
k\cos\theta_{k}=k\sqrt{\frac{4\pi}{3}}Y_{10}(\theta_{k},\varphi_{k}).$ (130)
Defining the spherical spinors
$\Phi_{\frac{1}{2}s}^{A}(\Omega_{k})=\sum_{m\nu}\textstyle\langle
1m\frac{1}{2}\nu|\frac{1}{2}s\rangle
Y_{00}(\Omega_{k})\chi(\nu)=\displaystyle\frac{1}{\sqrt{4\pi}}\chi(s),$ (131)
$\Phi_{\frac{1}{2}s}^{B}(\Omega_{k})=\sum_{m\nu}\textstyle\langle
1m\frac{1}{2}\nu|\frac{1}{2}s\rangle
Y_{1m}(\Omega_{k})\chi(\nu)=\displaystyle\frac{1}{\sqrt{4\pi}}\sigma_{k}\chi(s),$
(132)
where $\sigma_{k}=(\bm{\sigma}\cdot\bm{k})/k$ and
$\Omega_{k}=(\theta_{k},\varphi_{k})$($\sigma_{k}$ is pseudo scalar ), the
spinors can be re-expressed as
$\displaystyle u(\bm{k},s)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\phantom{-}A_{1}(k)\Phi_{\frac{1}{2}s}^{A}(\Omega_{k})\\\
\phantom{-}B_{1}(k)\Phi_{\frac{1}{2}s}^{B}(\Omega_{k})\end{array}\right)\ ,$
(135) $\displaystyle v(-\bm{k},s)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-B_{2}(k)\Phi_{\frac{1}{2}s}^{B}(\Omega_{k})\\\
\phantom{-}A_{2}(k)\Phi_{\frac{1}{2}s}^{A}(\Omega_{k})\end{array}\right)\ ,$
(138)
where
$A_{i}(k)=\sqrt{\frac{2\pi(E_{i}+m_{i})}{m_{i}}},\ B_{i}(k)=\sqrt{\frac{2\pi
k^{2}}{m_{i}(E_{i}+m_{i})}}\ .$ (139)
The spin factor $S_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}$ of the
interaction can be written as
$\displaystyle S_{s_{1}s_{2};s_{1}^{\prime}s_{2}^{\prime}}$ $\displaystyle=$
$\displaystyle\left[\overline{u}(\bm{k},s_{1})\gamma_{0}(1)u(\bm{k^{\prime}},s_{1}^{\prime})\right]\left[\overline{v}(-\bm{k},s_{2})\gamma_{0}(2)v(-\bm{k^{\prime}},s_{2}^{\prime})\right]-\left[\overline{u}(\bm{k},s_{1})\gamma_{i}(1)u(\bm{k^{\prime}},s_{1}^{\prime})\right]\left[\overline{v}(-\bm{k},s_{2})\gamma_{i}(2)v(-\bm{k^{\prime}},s_{2}^{\prime})\right]$
(140) $\displaystyle=$
$\displaystyle\left[A_{1}^{\ast}(k)A_{1}(k^{\prime})+B_{1}^{\ast}(k)B_{1}(k^{\prime})\big{\langle}\Phi_{\frac{1}{2}s_{1}}^{B}(\Omega_{k})\big{|}\Phi_{\frac{1}{2}s_{1}^{\prime}}^{B}(\Omega_{k^{\prime}})\big{\rangle}\right]\left[A_{2}^{\ast}(k)A_{2}(k^{\prime})+B_{2}^{\ast}(k)B_{2}(k^{\prime})\big{\langle}\Phi_{\frac{1}{2}s_{2}}^{B}(\Omega_{k})\big{|}\Phi_{\frac{1}{2}s_{2}^{\prime}}^{B}(\Omega_{k^{\prime}})\big{\rangle}\right]$
$\displaystyle\mbox{}+\left[A_{1}^{\ast}(k)B_{1}(k^{\prime})\big{\langle}\Phi_{\frac{1}{2}s_{1}}^{A}(\Omega_{k})\sigma_{i}\big{|}\Phi_{\frac{1}{2}s_{1}^{\prime}}^{B}(\Omega_{k^{\prime}})\big{\rangle}+B_{1}^{\ast}(k)B_{1}(k^{\prime})\big{\langle}\Phi_{\frac{1}{2}s_{1}}^{B}(\Omega_{k})\big{|}\sigma_{i}\Phi_{\frac{1}{2}s_{1}^{\prime}}^{A}(\Omega_{k^{\prime}})\big{\rangle}\right]$
$\displaystyle\mbox{}\times\left[B_{2}^{\ast}(k)A_{2}(k^{\prime})\big{\langle}\Phi_{\frac{1}{2}s_{1}}^{B}(\Omega_{k})\big{|}\sigma_{i}\Phi_{\frac{1}{2}s_{1}^{\prime}}^{A}(\Omega_{k^{\prime}})\big{\rangle}+A_{2}^{\ast}(k)B_{2}(k^{\prime})\big{\langle}\Phi_{\frac{1}{2}s_{2}}^{A}(\Omega_{k})\sigma_{i}\big{|}\Phi_{\frac{1}{2}s_{2}^{\prime}}^{B}(\Omega_{k^{\prime}})\big{\rangle}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi}}\Big{\langle}\chi(s_{1})\chi(s_{2})\Big{|}\Big{\\{}\left[A_{1}^{\ast}(k)A_{1}(k^{\prime})+B_{1}^{\ast}(k)B_{1}(k^{\prime})\sigma_{k}(1)\sigma_{k^{\prime}}(1)\right]\left[A_{2}^{\ast}(k)A_{2}(k^{\prime})+B_{2}^{\ast}(k)B_{2}(k^{\prime})\sigma_{k}(2)\sigma_{k^{\prime}}(2)\right]$
$\displaystyle\hskip
79.6678pt\mbox{}+\left[A_{1}^{\ast}(k)B_{1}(k^{\prime})\bm{\sigma}(1)\sigma_{k^{\prime}}(1)+B_{1}^{\ast}(k)A_{1}(k^{\prime})\sigma_{k}(1)\bm{\sigma}(1)\right]$
$\displaystyle\hskip
79.6678pt\mbox{}\cdot\left[B_{2}^{\ast}(k)A_{2}(k^{\prime})\sigma_{k}(2)\bm{\sigma}(2)+A_{2}^{\ast}(k)B_{2}(k^{\prime})\bm{\sigma}(2)\sigma_{k^{\prime}}(2)\right]\Big{\\}}\Big{|}\chi(s_{1}^{\prime})\chi(s_{2}^{\prime})\Big{\rangle}.$
The kernel $U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})$ can be rewritten as
$U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})=\big{\langle}\Phi_{JslM}(\Omega_{k},s)\big{|}\widehat{U}[\bm{k},\bm{k}^{\prime};\bm{\sigma}(1),\bm{\sigma}(2)]\big{|}\Phi_{Js^{\prime}l^{\prime}M}(\Omega_{k^{\prime}},s^{\prime})\big{\rangle},$
(141)
where the interaction operator in momentum and spin space is
$\displaystyle\widehat{U}[\bm{k},\bm{k}^{\prime};\bm{\sigma}(1),\bm{\sigma}(2)]=\frac{U(k,k^{\prime})}{\sqrt{4\pi}}\\!\\!\\!$
$\displaystyle\Big{\\{}$
$\displaystyle\\!\\!\left[A_{1}^{\ast}(k)A_{1}(k^{\prime})+B_{1}^{\ast}(k)B_{1}(k^{\prime})\sigma_{k}(1)\sigma_{k^{\prime}}(1)\right]\left[A_{2}^{\ast}(k)A_{2}(k^{\prime})+B_{2}^{\ast}(k)B_{2}(k^{\prime})\sigma_{k}(2)\sigma_{k^{\prime}}(2)\right]$
(142) $\displaystyle\hskip
79.6678pt\mbox{}+\left[A_{1}^{\ast}(k)B_{1}(k^{\prime})\bm{\sigma}(1)\sigma_{k^{\prime}}(1)+B_{1}^{\ast}(k)A_{1}(k^{\prime})\sigma_{k}(1)\bm{\sigma}(1)\right]$
$\displaystyle\hskip
79.6678pt\mbox{}\cdot\left[B_{2}^{\ast}(k)A_{2}(k^{\prime})\sigma_{k}(2)\bm{\sigma}(2)+A_{2}^{\ast}(k)B_{2}(k^{\prime})\bm{\sigma}(2)\sigma_{k^{\prime}}(2)\right]\Big{\\}}.$
Since $\sigma_{k}$ and $\sigma_{k^{\prime}}$ are pseudo scalar, $k$,
$k^{\prime}$, $\sigma_{k}$ $\sigma_{k^{\prime}}$, and
$\bm{\sigma}(1)\cdot\bm{\sigma}(2)$ are scalar, the above interaction kernel
operator $\widehat{U}[\bm{k},\bm{k}^{\prime};\bm{\sigma}(1),\bm{\sigma}(2)]$
is scalar.
From the last expression of the kernel
$U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})$, we could see that the first
term contributes to different kinds of central potentials and relativistic
spin-orbit coupling potentials, the second term contributes to the tensor
potentials changing $l$ by $\Delta l=\pm 2$ and mixing spin singlet and
triplet.
If $m_{1}=m_{2}$ and the tensor potentials are neglected, $l$ and $s$ are
conserved and the interaction kernel becomes diagonal in $l$ and $s$
representation,
$U_{sl;s^{\prime}l^{\prime}}^{J}(k;k^{\prime})=U_{sl;sl}^{J}(k;k^{\prime})\delta_{ll^{\prime}}\delta_{ss^{\prime}}=U_{Jsl}(k;k^{\prime})\delta_{ll^{\prime}}\delta_{ss^{\prime}}.$
(143)
## References
* (1) K. G. Wilson, T. S. Walhout,A. Harindranath, W. M. Zhang, R. J. Perry, and S. D. Glazek, Phys. Rev. D 49 6720 (1994);
M. M. Brisudov$\acute{a}$, R. J. Perry and K. G. Wilson, Phys. Rev. L 78 1227
(1997).
* (2) M.S.Bhagwatt, M.A. Pichowsky, P. C. Tandy, Phys.Rev.D67,054019(2008); P. Maris and C. D. Roberts, J. Mod. Phys 12 297 (2003); A. Holl, C. D. Roberts, and S. V. Wright, arXiv:nucl-th/0601071; M.S.Bhagwatt, P. C. Tandy, arXiv:nucl-th/0601020 ; P. Maris, C. D. Roberts and P. C. Tandy, Physics Letters B 420 3 (1998)
* (3) R. Alkofer, P. Watson, and H. Weigel Phys. Rev. D 65 094026 (2002)
* (4) P. Jain and H. J. Munczek, Phys. Rev. D 48 5403 (1993)
* (5) S. Godfrey and N. Isgur, Phys. Rev. D 32 189 (1985)
* (6) H. Hersbach, Phys. Rev. C 50 2562 (1994).
* (7) S. Godfrey and J. Napolitano, Rev. Mod. Phys. 71 1411 (1999)
* (8) B. Metsch, arXiv:hep-ph/0403118,
B. Metsch, arXiv:hep-ph/0308294;
R. Ricken, M. Koll, D. Merten, B.Ch. Metsch, and H.R. Petry, Eur. Phys. J. A 9
221 (2000)
* (9) A. M. Badalian, B. L. G. Bakker, Yu. A. Simonov, Phys. Rev. D 66 034026 (2002); A. M. Badalian, Yu. A. Simonov, M. A. Trusov, Phys. Rev. D 77 074017 (2008); V.L.Mogunov, A.V.Nefediev, Y.A.Simonov, Phys.Lett.B459 , 653(1999); A. M. Badalian, B. L. G. Bakker, I. V. Danilkin, arXiv:0911.4634; A. M. Badalian, B. L. G. Bakker, arXiv:hep-ph/0202246.
* (10) G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 94 201601 (2005); Phys. Rev. Lett. 102 081601 (2009); S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 96 201601 (2006); Phys. Rev. D 77 056007 (2008); arXiv:0802.0514; arXiv:0804.3562; G. F. de Teramond and S. J. Brodsky, arXiv:0909.3900; S. F. Brodsky, G. F. Teramond, and A. Geur, arXiv:1002.v660v1, arXiv:1007.5385.
* (11) P. A. M. Dirac, Rev. Mod. Phys. 21 392(1949).
* (12) W. Qian and B. Q. Ma, Phys. Rev. D 78 074002 (2008).
* (13) L.S. Kisslinger, Int.J.Mod.Phys. E 13 375-424 (2004)
J. H. O. Sales, T.Frederico, B. V. Carlson and P. U. Sauer, Phys. Rev. C 61
044003 (2000)
* (14) S. J. Brodsky, H. C. Pauli and S. S. Pinsky , Phys. Rep. 301 299 (1998).
* (15) H. C. Pauli, Nucl. Phys. B (Proc. Suppl.) 90 259 (2000); T. Frederico, H. C. Pauli and S. G. Zhou, Phys. Rev. D 66 054007 (2002)
* (16) H. C. Pauli, Eur.Phys.J.C7 289(1998).
* (17) H. C. Pauli, Nucl. Phys. B (Proc. Suppl.) 90 154 (2000)
* (18) S. J. Wang, L. Li, and G. B. Zhang, Chin. Phys. Lett 23 1426 (2006);
L. Li, S. J. Wang, S. G. Zhou, and G. B. Zhang, Chin. Phys. Lett 24 374
(2007).
* (19) S. J. Wang, J. Tao, X. B. Guo, and L. Li, submitted to Phys.Lett.B.
* (20) U. Trittmann and H. C. Pauli, Nucl. Phys. B (Proc. Suppl.) 90 161 (2000)
* (21) V. B. Berestetskii, M. V. Terentev, Sov. J. Nucl. Phys. 24 5 1976; M. Sawicki, Phys. Rev. D 32 2666 (1985)
* (22) H. C. Pauli, Nucl. Phys. B (Proc. Suppl.) 108 251 (2002)
* (23) H. C. Pauli, Nucl. Phys. B (Proc. Suppl.) 108 273 (2002).
* (24) J. D. Stack, Phys. Rev. D 27 412 (1983)
Ding H D, Phys. Rev. D 44 2200 (1991) 189
Hasenfratz A and Hasenfratz P , Annu. Rev. Nucl. Part. Sci. 35 559 (1985)
* (25) F. Gross and J. Milana, Phys. Rev. D 43 2401 (1991)
* (26) A. J. Sommerer, J. R. Spence, and J. P. Vary, Phys. Rev. C 49 513 (1994)
* (27) S. Weinberg, The Quantum Theory of Fields ii@ (Cambridge, 2001), p.226.
* (28) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge University Press, Cambridge, 1992)
* (29) C Amsler, A Masoni, Phys. Lett. B 667 1(2008)
* (30) S.J.Brodsky, S.Menke, C.Merino, and J.Rathsman, Phys.Rev. D67, 055008(2003);
M.Baldicchi and G.M.Prosperi, Phys.Rev.D 66 074008(2002).
* (31) A. Donnachie, Yu. S. Kalashnikova, Phys. Rev. C 78 064603 (2008).
* (32) D. Becirevic, E. Chang, A. LeYaouanc, Phys. Rev. D 80 034504 (2009).
|
arxiv-papers
| 2010-07-15T01:51:52 |
2024-09-04T02:49:11.629621
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shun-Jin Wang, Jun Tao, Xiao-Bo Guo and Lei Li",
"submitter": "Wang Shun-Jin",
"url": "https://arxiv.org/abs/1007.2462"
}
|
1007.2701
|
# Additional symmetries of constrained CKP and BKP hierarchies
Kelei Tian†, Jingsong He∗‡, Jipeng Cheng† and Yi Cheng † Department of
Mathematics, University of Science and Technology of China, China
‡ Department of Mathematics, Ningbo University, China
###### Abstract.
The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP)
hierarchies are given by their actions on the Lax operators, and their actions
on the eigenfunction and adjoint eigenfunction $\\{\Phi_{i},\Psi_{i}\\}$ are
presented explicitly. Furthermore, we show that acting on the space of the
wave operator, $\partial_{k}^{*}$ forms new centerless $W^{cC}_{1+\infty}$ and
$W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$ respectively. In
order to define above symmetry flows $\partial_{k}^{*}$ of the cCKP and cBKP
hierarchies, two vital operators $Y_{k}$ are introduced to revise the
additional symmetry flows of the CKP and BKP hierarchies.
###### Key words and phrases:
Keywords. constrained CKP hierarchy, constrained BKP hierarchy, additional
symmetry.
∗Corresponding author. email: hejingsong@nbu.edu.cn
PACS (2003). 02.30.Ik.
Mathematics Subject Classification (2000). 17B80, 37K05, 37K10.
## 1\. Introduction
The research on Kadomtsev-Petviashvili (KP) hierarchy [1, 2] is one of the
most important topics in the development of the theory of integrable systems.
A specific interesting aspect on this topic is additional symmetry[3, 4, 5, 6,
7, 8]. 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}$[5, 8]. More 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 [9, 10, 11, 12, 13]. In fact, there is a parallel research line on
additional symmetries of evolution equations and their Lie algebraic
structures, particularly for $1+1$ dimensional integrable equations [14], and
the corresponding symmetries are called $\tau$ -symmetries and related Lax
operators are constructed pretty systematically [15, 16].
It is well known that we can get different sub-hierarchies of the KP by
different reduction conditions on Lax operator $L$. The first two important
sub-hierarchies are CKP hierarchy [17] through a restriction $L^{*}=-L$ and
BKP hierarchy[1] through a restriction $L^{*}=-\partial L\partial^{-1}$. So
the additional symmetries of the CKP (or BKP) hierarchy have been constructed
[18, 19, 20, 21] and shown to be a new infinite dimensional algebra
$W^{C}_{1+\infty}$ (or $W^{B}_{1+\infty}$), which is subalgebra of
$W_{1+\infty}$. The third sub-hierarchy is constrained KP (cKP) hierarchy [22,
23, 24], which is obtained by setting a special form of Lax operator
$L=\partial+\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i},$ where $\Phi_{i}$ is
the eigenfunction and $\Psi_{i}$ is the adjoint eigenfunction of the cKP
hierarchy. This is a relative new reduction condition and is different from
the reduction conditions of previous two sub-hierarchies. Hence it is very
natural to explore the additional symmetries of the cKP hierarchy. However,
additional symmetry flows of the KP hierarchy break the special form of the
Lax operator for the cKP hierarchy, and so it is highly non-trivial to find a
suitable form of additional symmetry flow for this sub-hierarchy. To this end,
it is well defined[25] by means of a crucial modification of the corresponding
additional symmetry flows of the KP hierarchy. Note that a new and complicated
operator $X_{k}$ plays a very important role in this procedure.
Furthermore, the fourth sub-hierarchy and fifth sub-hierarchy are constrained
CKP (cCKP)[26] and constrained BKP (cBKP) hierarchy[27], which are obtained by
putting the combination of the previous mentioned reduction conditions, i.e.,
CKP and cKP conditions, BKP and cKP conditions. Some results associated cCKP
and cBKP hierarchies have already been reported, for example, a Gramm-type
$\tau$-function [26] of the cCKP hierarchy and a Pfaffian-form $\tau$-function
[27] of the cBKP are also constructed from the vacuum solution, the
dimensional reductions of the CKP and BKP hierarchies are presented in [28].
Moreover, determinant representations of the $\tau$ function, which are
generated by the the gauge transformations, of the cCKP and cBKP hierarchies
are obtained [29]. However, for our best knowledge, there is no any results on
the additional symmetries of the cBKP and cCKP hierarchies. So, we shall fill
the gap in this paper by constructing additional symmetries for the two new
sub-hierarchies and their actions on eigenfunctions and adjoint
eigenfunctions. The basic idea is to revise the additional symmetry flows of
CKP and BKP hierarchies such that the new symmetry flows $\partial_{k}^{*}$
will preserve the form of Lax operator of the cKP hierarchy. This will be
realized by introducing two vital operators $Y_{k}$ in sequel.
The paper is organized as follows. We first recall some basic results of the
KP hierarchy, CKP hierarchy, BKP hierarchy and the constrained KP hierarchy in
Section 2. The main results are stated and proved in Sections 3 and 4, which
are the additional symmetries and its action on the eigenfunction and adjoint
eigenfunction of the cCKP and cBKP hierarchies, and we further show that
acting on the space of the wave operator, $\partial_{k}^{*}$ forms new
centerless $W^{cC}_{1+\infty}$ and $W^{cB}_{1+\infty}$-subalgebra of
centerless $W_{1+\infty}$ respectively. Section 5 is devoted to conclusions
and discussions.
## 2\. background on KP hierarchy
In this section we first give a brief introduction of KP hierarchy based on
[1, 2]. Let the pseudo-differential operator
$L=\partial+u_{1}\partial^{-1}+u_{2}\partial^{-2}+u_{3}\partial^{-3}+\cdots$
(2.1)
be a Lax operator of the KP hierarchy, which is described by the associated
Lax equations
$\dfrac{\partial L}{\partial{t_{n}}}=[B_{n},L],\ n=1,2,3,\cdots,$ (2.2)
where $B_{n}=(L^{n})_{+}=\sum\limits_{k=0}^{n}a_{k}\partial^{k}$ denotes the
non-negative powers of $\partial$ in $L^{n}$, $\partial=\partial/\partial x$,
$u_{i}=u_{i}(x=t_{1},t_{2},t_{3},\cdots)$. The other notation
$L^{n}_{-}=L^{n}-L^{n}_{+}$ will be used in the paper. The Lax operator $L$ in
eq.(2.1) can be generated by dressing operator
$S=1+\sum_{k=1}^{\infty}s_{k}\partial^{-k}$ in the following way
$L=S\partial S^{-1}.$ (2.3)
The dressing operator $S$ satisfies Sato equation
$\dfrac{\partial S}{\partial t_{n}}=-(L^{n})_{-}S,\quad n=1,2,3,\cdots.$ (2.4)
The wave function $w(t,z)$ of the KP hierarchy is defined by
$w(t,z)=Se^{\xi(t,z)},$ (2.5)
where $\xi(t,z)=t_{1}z+t_{2}z^{2}+t_{3}z^{3}+\cdots+t_{n}z^{n}\cdots.$ The
wave function satisfies the equations
$L^{k}w(t,z)=z^{k}w(t,z),\dfrac{\partial}{\partial
t_{n}}w=B_{n}w,k\in\mathbb{Z},n\in\mathbb{N}.$ (2.6)
Moreover $w(t,z)$ has a very simple expression by $\tau$ function of the KP
hierarchy
$w(t,z)=\frac{\tau(t_{1}-\frac{1}{z},t_{2}-\frac{1}{2z^{2}},t_{3}-\frac{1}{3z^{3}},\cdots)}{\tau(t_{1},t_{2},t_{3},\cdots)}e^{\xi(t,z)}.$
(2.7)
Beside the existence of the Lax operator, wave function, $\tau$ function for
the KP hierarchy, another important property is the Zakharov-Shabat equation
and associated linear equation. In other words, KP hierarchy also has an
alternative expression, i.e.,
$\displaystyle\frac{\partial B_{m}}{\partial t_{n}}-\frac{\partial
B_{n}}{\partial t_{m}}+[B_{m},B_{n}]=0,\qquad m,n=1,2,3,\cdots.$ (2.8)
The eigenfunction $\Phi=\Phi(t_{1},t_{2},t_{3},\cdots)$ and adjoint
eigenfunction $\Psi=\Psi(t_{1},t_{2},t_{3},\cdots)$ of KP hierarchy associated
to (2.8) are defined by
$\dfrac{\partial\Phi}{\partial t_{n}}=(B_{n}\Phi),\
\dfrac{\partial\Psi}{\partial t_{n}}=-(B_{n}^{*}\Psi),$ (2.9)
where the symbol $*$ is the formal adjoint operation. For an arbitrary pseudo-
differential operator $P=\sum\limits_{i}p_{i}\partial^{i}$,
$P^{*}=\sum\limits_{i}(-1)^{i}\partial^{i}p_{i}$, and $(AB)^{*}=B^{*}A^{*}$
for pseudo-differential operators $A$ and $B$. For example,
$\partial^{*}=-\partial$, $(\partial^{-1})^{*}=-\partial^{-1}$.
To construct additional symmetries associated with the KP hierarchy, let us
introduce $\Gamma$ and Orlov-Shulman’s $M$ operator as
$\Gamma=\sum\limits_{i=1}^{\infty}it_{i}\partial^{i-1}$ and $M=S\Gamma
S^{-1}.$ Dressing $[\partial_{k}-\partial^{k},\Gamma]=0$ gives
$[\partial_{k}-B_{k},M]=0,$ i.e.
$\partial_{k}M=[B_{k},M],$ (2.10)
and then
$\partial_{k}(M^{m}L^{n})=[B_{k},M^{m}L^{n}].$ (2.11)
Thus, the additional flows are defined by
$\dfrac{\partial S}{\partial t_{m,n}^{*}}=-(M^{m}L^{n})_{-}S,$ (2.12)
or equivalently
$\dfrac{\partial L}{\partial t_{m,n}^{*}}=-[(M^{m}L^{n})_{-},L].$ (2.13)
Here $(M^{m}L^{n})_{-}$ serves as the generator of the additional flows along
the new time variables $t_{m,n}^{*}$. The additional flows
${\partial_{m,n}^{*}}=\dfrac{\partial}{\partial t_{m,n}^{*}}$ commute with the
hierarchy, i.e. $[\partial_{m,n}^{*},\partial_{k}]=0$ but do not commute with
each other, they indeed determine symmetries. Therefore these flows
$\partial_{m,n}^{*}$ are also called additional symmetry flows, and they forms
a centerless $W_{1+\infty}$ algebra[5, 8] acting on the spaces of the Lax
operator $L$ and wave operator $S$.
The CKP hierarchy is a reduction of the KP hierarchy through the constraint on
$L$ given by eq.(2.1) as
$L^{*}=-L,$ (2.14)
then $L$ is called the Lax operator of the CKP hierarchy, and the associated
Lax equation of the CKP hierarchy is
$\dfrac{\partial L}{\partial t_{n}}=[B_{n},L],n=1,3,5,\cdots.$ (2.15)
which compresses all even flows, i.e. the Lax equation of the CKP hierarchy
has only odd flows. Additional symmetry flows [18] for the CKP hierarchy are
defined as
$\dfrac{\partial L}{\partial{t^{*}_{m,l}}}=-[(A_{m,l})_{-},L],$ (2.16)
where $A_{m,l}$ for the CKP hierarchy should satisfy
$A_{m,l}^{*}=-A_{m,l},$ (2.17)
we can assume $A_{m,l}$ be
$A_{m,l}=M^{m}L^{l}-(-1)^{l}L^{l}M^{m},$ (2.18)
where
$M=S(\sum\limits_{i=0}^{\infty}(2i+1)t_{2i+1}\partial^{2i})S^{-1}.$ (2.19)
Acting on the space of the wave operator $S$, $\partial_{m,l}^{*}$ of the CKP
hierarchy forms a new centerless $W^{C}_{1+\infty}$\- subalgebra of centerless
$W_{1+\infty}$.
If Lax operator $L$ given by eq.(2.1) satisfies
$L^{*}=-\partial L\partial^{-1},$ (2.20)
then $L$ is called the Lax operator of the BKP hierarchy, the Lax equation of
the BKP hierarchy also has only odd flows
$\dfrac{\partial L}{\partial t_{n}}=[B_{n},L],n=1,3,5,\cdots.$ (2.21)
Additional symmetry flows [19, 20] for the BKP hierarchy are given by
$\dfrac{\partial L}{\partial{t^{*}_{m,l}}}=-[(A_{m,l})_{-},L],$ (2.22)
where $A_{m,l}$ for the BKP hierarchy should satisfy the constraint equation
$A_{m,l}^{*}=-\partial A_{m,l}\partial^{-1},$ (2.23)
then $A_{m,l}$ for the BKP hierarchy could be chosen as
$A_{m,l}=M^{m}L^{l}-(-1)^{l}L^{l-1}M^{m}L,$ (2.24)
where
$M=S(\sum\limits_{i=0}^{\infty}(2i+1)t_{2i+1}\partial^{2i})S^{-1}.$ (2.25)
Acting on the space of the wave operator $S$, $\partial_{m,l}^{*}$ of the BKP
hierarchy forms another new centerless $W^{B}_{1+\infty}$ \- subalgebra of
centerless $W_{1+\infty}$.
We now turn to the cKP hierarchy [22, 23, 24]. The Lax operator $L$ of the
constrained KP hierarchy is given by
$L=\partial+\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i},$ (2.26)
where $\Phi_{i}$ ($\Psi_{i}$) is the (adjoint) eigenfunctions of this
hierarchy, and the corresponding Lax equation is formulated as
$\dfrac{\partial L}{\partial t_{n}}=[B_{n},L],n=1,2,3,\cdots.$ (2.27)
In particular, we stress that eigenfunctions and adjoint eigenfunctions
{$\Phi_{i},\Psi_{i}$} are important dynamical variables in cKP hierarchy, so
it is necessary to find the action of the correct additional symmetry flows on
them. Here the correct flow $\partial_{\tau}$ means its action on $L$ of the
cKP hierarchy should be the form of[25]
$(\partial_{\tau}L)_{-}=\sum_{i=1}^{m}(\tilde{A}\partial^{-1}\Psi_{i}+\Phi_{i}\partial^{-1}\tilde{B}),$
(2.28)
which will result in its action on ($\partial_{\tau}\Phi_{i}=\tilde{A}$) and
($\partial_{\tau}\Psi_{i}=\tilde{B}$). However, in general, the additional
symmetry flows of the KP hierarchy acting on $L$ of the cKP hierarchy are not
the form of eq.(2.28). In other words, these flows do not preserve the form of
the Lax operator of the cKP hierarchy, the fact shows additional symmetry of
the KP hierarchy is not consistent with cKP reduction condition automatically.
Therefore, the additional symmetry flows have to be revised according to the
analysis above, and then the correct additional symmetry flows of the cKP
hierarchy are given by[25]
$\partial^{*}_{k}L=[-(ML^{k})_{-}+X_{k-1},L],k=0,1,2,3,\cdots,$ (2.29)
where
$X_{k-1}=0,k=0,1,2$
and
$X_{k-1}=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(j-\dfrac{1}{2}(k-2))L^{k-2-j}(\Phi_{i})\partial^{-1}(L^{*})^{j}(\Psi_{i}),k\geq
3.$ (2.30)
Furthermore[25],
$\displaystyle\partial_{k}^{*}\Phi_{i}$
$\displaystyle=\frac{k}{2}L^{k-1}(\Phi_{i})+X_{k-1}(\Phi_{i})+(A_{1,k})_{+}(\Phi_{i}),$
(2.31) $\displaystyle\partial_{k}^{*}\Psi_{i}$
$\displaystyle=\frac{k}{2}L^{*(k-1)}(\Psi_{i})-X_{k-1}^{*}(\Psi_{i})-(A_{1,k})_{+}^{*}(\Psi_{i}).$
(2.32)
## 3\. Additional symmetries of constrained CKP hierarchy
Let us consider the constrained CKP hierarchy, which is the C-type sub-
hierarchy of cKP hierarchy. The Lax operator $L$ of the cCKP hierarchy [26] is
given by
$L=\partial+\sum_{i=1}^{m}(q_{i}\partial^{-1}r_{i}+r_{i}\partial^{-1}q_{i}),$
(3.1)
where $q_{i}$ and $r_{i}$ are eigenfunctions. The corresponding Lax equation
of the cCKP hierarchy is defined by
$\dfrac{\partial L}{\partial t_{n}}=[B_{n},L],n=1,3,5,\cdots.$ (3.2)
For $k=1,3,5,\cdots$, we first try to calculate the original additional
symmetry flows of the CKP hierarchy as
$\dfrac{\partial L}{\partial{t^{*}_{1,k}}}=-[(A_{1,k})_{-},L],$ (3.3)
where $A_{1,k}=ML^{k}-(-1)^{k}L^{k}M.$ Thus
$(\dfrac{\partial
L}{\partial{t^{*}_{1,k}}})_{-}=[(A_{1,k})_{+},L]_{-}+2(L^{k})_{-}.$ (3.4)
In order to get its action on $q_{i}$ and $r_{i}$, we need the form of
$(L^{k})_{-}$.
Lemma 3.1. The Lax operator $L$ of the constrained CKP hierarchy given by
eq.(3.1) satisfies the relation
$(L^{k})_{-}=\sum_{i=1}^{m}\sum_{j=0}^{k-1}(L^{k-j-1}(q_{i})\partial^{-1}(L^{*})^{j}(r_{i})+L^{k-j-1}(r_{i})\partial^{-1}(L^{*})^{j}(q_{i})),k=1,3,5,\cdots$
(3.5)
where
$L(q_{i})=L_{+}(q_{i})+\sum_{j=1}^{m}(q_{j}\partial_{x}^{-1}(r_{j}q_{i})+r_{j}\partial_{x}^{-1}(q_{j}q_{i})).$
However, $(L^{k})_{-}$ is not in the form of
$(\partial_{\tau}L)_{-}=\sum_{i=1}^{m}((\partial_{\tau}q_{i})\partial^{-1}r_{i}+(\partial_{\tau}r_{i})\partial^{-1}q_{i}+q_{i}\partial^{-1}(\partial_{\tau}r_{i})+r_{i}\partial^{-1}(\partial_{\tau}q_{i})),$
(3.6)
as we expected for a correct flows. Here correctness means we can get the
actions on $\partial_{\tau}q_{i}$ and $\partial_{\tau}r_{i}$ from eq.(3.6).
This shows that eq.(3.4) can not imply its action on eigenfunctions
$\partial^{*}_{1,k}q_{i}$ and $\partial^{*}_{1,k}r_{i}$.
Therefore, in order to get correct additional flows of the cCKP hierarchy, we
revise eq.(3.3) and then define new flows by
$\partial_{k}^{*}L=[-(A_{1,k})_{-}+Y_{k},L],k=1,3,5,\cdots,$ (3.7)
where $A_{1,k}=ML^{k}-(-1)^{k}L^{k}M$ and $Y_{k}$ is introduced such that the
left hand side of eq.(3.7) will be the form of eq.(3.6). Next, we shall prove
new flows $\partial_{k}^{*}$ are correct additional symmetry flows of the cCKP
hierarchy. First of all, $Y_{k}$ is discussed, which is crucial to our
purpose.
Lemma 3.2. For the cCKP hierarchy, the CKP reduction condition infers a
constraint on $Y_{k}$,
$Y_{k}^{*}=-Y_{k},\ k=1,3,5,\cdots.$ (3.8)
Proof. The action of the additional flows $\partial_{k}^{*}$ on the adjoint
Lax operator $L^{*}$ of the constrained CKP hierarchy can be obtained by two
different ways. We first computet it by a formal adjoint operation on both
sides of eq. (3.7), i.e.
$\partial_{k}^{*}L^{*}=([-(A_{1,k})_{-}+Y_{k},L])^{*}=[L^{*},-(A_{1,k})_{-}^{*}+Y_{k}^{*}],k=1,3,5,\cdots.$
(3.9)
The another way is to do a derivative with respect to $t_{k}^{*}$ on $L^{*}$
and use CKP reduction condition $L^{*}=-L$, then
$\partial_{k}^{*}L^{*}=-\partial_{k}^{*}L=-[-(A_{1,k})_{-}+Y_{k},L]=[L^{*},(A_{1,k})_{-}-Y_{k}],k=1,3,5,\cdots.$
(3.10)
Comparing eq.(3.9) and eq.(3.10), we have
$Y_{k}^{*}=-Y_{k},\ k=1,3,5,\cdots,$
with the help of $A_{1,k}^{*}=-A_{1,k}$. $\square$
Thus we set $Y_{k}$ as
$\displaystyle Y_{1}$ $\displaystyle=0,$ (3.11) $\displaystyle Y_{k}$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(L^{k-2-j}(q_{i})\partial^{-1}(L^{*})^{j}(r_{i})+L^{k-2-j}(r_{i})\partial^{-1}(L^{*})^{j}(q_{i})),k\geq
3,$ (3.12)
and shall show $Y_{k}$ satisfy the constraint given by eq.(3.8) .
Lemma 3.3.
$Y_{k}=-Y_{k}^{*}.$ (3.13)
Proof. $Y_{1}^{*}=Y_{1}=0$ is obvious, for $k=3,5,7,\cdots$, we have
$\displaystyle Y_{k}^{*}$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(-(L^{*})^{j}(r_{i})\partial^{-1}L^{k-2-j}(q_{i})-(L^{*})^{j}(q_{i})\partial^{-1}L^{k-2-j}(r_{i}))$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(-(-1)^{j}L^{j}(r_{i})\partial^{-1}(-1)^{k-2-j}(L^{*})^{k-2-j}(q_{i})$
$\displaystyle-(-1)^{j}L^{j}(q_{i})\partial^{-1}(-1)^{k-2-j}(L^{*})^{k-2-j}(r_{i}))$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(L^{j}(r_{i})\partial^{-1}(L^{*})^{k-2-j}(q_{i})+L^{j}(q_{i})\partial^{-1}(L^{*})^{k-2-j}(r_{i}))$
$\displaystyle=-\sum_{i=1}^{m}\sum_{l=0}^{k-2}(2l-(k-2))(L^{k-2-l}(q_{i})\partial^{-1}(L^{*})^{l}(r_{i})+L^{k-2-l}(r_{i})\partial^{-1}(L^{*})^{l}(q_{i}))$
$\displaystyle=-Y_{k}$
In the fourth step we let $l=k-2-j$. $\square$
Furthermore, in order to calculate $[Y_{k},L]$ in the eq.(3.7), the following
lemma is necessary.
Lemma 3.4. For the Lax operator $L$ of the cCKP hierarchy in eq.(3.1) and a
pseudo-different operator $X=\sum_{k=1}^{l}M_{k}\partial^{-1}N_{k}$, the
equation
$\displaystyle[X,L]_{-}$
$\displaystyle=\sum_{k=1}^{l}(-L(M_{k})\partial^{-1}N_{k}+M_{k}\partial^{-1}L^{*}(N_{k}))$
$\displaystyle+\sum_{i=1}^{m}(X(q_{i})\partial^{-1}r_{i}+X(r_{i})\partial^{-1}q_{i}-q_{i}\partial^{-1}X^{*}(r_{i})-r_{i}\partial^{-1}X^{*}(q_{i}))$
holds.
Based on the above lemma, we have
$[Y_{k},L]_{-}=-2(L^{k})_{-}+k\sum_{i=1}^{m}(L^{k-1}(q_{i})\partial^{-1}r_{i}+L^{k-1}(r_{i})\partial^{-1}q_{i})$
$\hskip
17.07182pt+k\sum_{i=1}^{m}(q_{i}\partial^{-1}L^{*(k-1)}(r_{i})+r_{i}\partial^{-1}L^{*(k-1)}(q_{i}))$
$+\sum_{i=1}^{m}(Y_{k}(q_{i})\partial^{-1}r_{i}+Y_{k}(r_{i})\partial^{-1}q_{i})$
$-\sum_{i=1}^{m}(q_{i}\partial^{-1}Y_{k}^{*}(r_{i})+r_{i}\partial^{-1}Y_{k}^{*}(q_{i})).$
(3.14)
We are now in a position to calculate the explicit form of the right hand side
of the new additional flow give by eq.(3.7).
Theorem 3.1. The additional flows acting on the eigenfunction $q_{i}$ and
$r_{i}$ of the constrained CKP hierarchy are
$\partial_{k}^{*}q_{i}=kL^{k-1}(q_{i})+Y_{k}(q_{i})+(A_{1,k})_{+}(q_{i}),k=1,3,5,\cdots$
(3.15)
and
$\partial_{k}^{*}r_{i}=kL^{k-1}(r_{i})+Y_{k}(r_{i})+(A_{1,k})_{+}(r_{i}),k=1,3,5,\cdots.$
(3.16)
Proof. From the revised definition of the additional symmetry flows in
eq.(3.7) of the cCKP hierarchy, by a short calculation, we have
$\partial_{k}^{*}L_{-}=[-(A_{1,k})_{-}+Y_{k},L]_{-}=[(A_{1,k})_{+},L]_{-}+2(L^{k})_{-}+[Y_{k},L]_{-}.$
Using eq.(3.14) and the technical identity
$[K,f\partial^{-1}g]_{-}=K(f)\partial^{-1}g-f\partial^{-1}K^{*}(g)$ (where
$f,$ $g$ are arbitrary functions and $K$ is a purely differential operator),
note that $k=1,3,5,\cdots,$ it is followed by
$\displaystyle\partial_{k}^{*}L_{-}$
$\displaystyle=\sum_{i=1}^{m}((kL^{k-1}(q_{i})+Y_{k}(q_{i})+(A_{1,k})_{+}(q_{i}))\partial^{-1}r_{i}$
$\displaystyle+(kL^{k-1}(r_{i})+Y_{k}(r_{i})+(A_{1,k})_{+}(r_{i}))\partial^{-1}q_{i}$
$\displaystyle+q_{i}\partial^{-1}(k(L^{*})^{k-1}(r_{i})-Y_{k}^{*}(r_{i})-(A_{1,k})_{+}^{*}(r_{i}))$
$\displaystyle+r_{i}\partial^{-1}(k(L^{*})^{k-1}(q_{i})-Y_{k}^{*}(q_{i})-(A_{1,k})_{+}^{*}(q_{i}))$
thus
$\partial_{k}^{*}q_{i}=kL^{k-1}(q_{i})+Y_{k}(q_{i})+(A_{1,k})_{+}(q_{i})=k(L^{*})^{k-1}(q_{i})-Y_{k}^{*}(q_{i})-(A_{1,k})_{+}^{*}(q_{i}),$
$\partial_{k}^{*}r_{i}$ is also obtained at the same time. $\square$
Corollary 3.1. The additional flows act on wave operator $S$ of the
constrained CKP hierarchy as
$\partial_{k}^{*}S=(-(A_{1,k})_{-}+Y_{k})S,k=1,3,5,\cdots.$ (3.17)
Theorem 3.2. The additional flows ${\partial_{k}^{*}}$ commute with the
constrained CKP hierarchy flows
$\partial_{t_{2n+1}}=\dfrac{\partial}{\partial{t_{2n+1}}}$, i.e.
$[{\partial_{k}^{*}},\partial_{t_{2n+1}}]=0.$ (3.18)
Thus $\partial_{k}^{*}(k=1,3,5,\cdots)$ are indeed the additional symmetry
flows of the cCKP hierarchy.
Proof. The proof starts with the definition
$[\partial_{k}^{*},\partial_{t_{2n+1}}]S=\partial_{k}^{*}(\partial_{t_{2n+1}}S)-\partial_{t_{2n+1}}(\partial_{k}^{*}S),$
and using the action of the additional flows, we get
$\displaystyle[\partial_{k}^{*},\partial_{t_{2n+1}}]S$ $\displaystyle=$
$\displaystyle-\partial_{k}^{*}\left(L^{2n+1}_{-}S\right)+\partial_{t_{2n+1}}\left(((A_{1,k})_{-}-Y_{k})S\right)$
$\displaystyle=$
$\displaystyle-(\partial_{k}^{*}L^{2n+1})_{-}S-(L^{2n+1})_{-}(\partial_{k}^{*}S)+\partial_{t_{2n+1}}((A_{1,k})_{-}-Y_{k})S$
$\displaystyle+$ $\displaystyle((A_{1,k})_{-}-Y_{k})(\partial_{t_{2n+1}}S).$
Taking eq.(3.17) of Corollary 1 into the above formula, it is not difficult to
compute
$\displaystyle[\partial_{k}^{*},\partial_{t_{2n+1}}]S$ $\displaystyle=$
$\displaystyle[(A_{1,k})_{-}-Y_{k},L^{2n+1}]_{-}S+(L^{2n+1})_{-}((A_{1,k})_{-}-Y_{k})S$
$\displaystyle+$
$\displaystyle[(L^{2n+1})_{+},(A_{1,k})_{-}-Y_{k}]_{-}S-((A_{1,k})_{-}-Y_{k})(L^{2n+1})_{-}S$
$\displaystyle=$
$\displaystyle[((A_{1,k})_{-}-Y_{k}),L^{2n+1}]_{-}S-[(A_{1,k})_{-}-Y_{k},L^{2n+1}_{+}]_{-}S+[L^{2n+1}_{-},(A_{1,k})_{-}-Y_{k}]S$
$\displaystyle=$
$\displaystyle[(A_{1,k})_{-}-Y_{k},L^{2n+1}_{-}]_{-}S+[L^{2n+1}_{-},(A_{1,k})_{-}-Y_{k}]S=0$
We have used the fact that
$[L^{2n+1}_{+},(A_{m,l})-Y_{k}]_{-}=[L^{2n+1}_{+},((A_{1,k})_{-}-Y_{k})]_{-}$
in the second step of the above derivation, since $(Y_{k})_{-}=Y_{k}$ and
$[L^{2n+1}_{+},(A_{m,l})_{+}]_{-}=0$. The last equality holds by virtue of
$(P_{-})_{-}=P_{-}$ for arbitrary pseduo-differential operator $P$. $\square$
Taking into account of $Y_{k}^{*}=-Y_{k},\ k=1,3,5,\cdots$, we can give the
next theorem by a straightforward and tedious calculation.
Theorem 3.3. Acting on the space of the wave operator $S$ of the constrained
CKP hierarchy, $\partial_{k}^{*}$ forms a new centerless
$W^{cC}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$.
## 4\. Additional symmetries of constrained BKP hierarchy
We now turn to the case of the constrained BKP hierarchy, this follows in a
similar way of the case of the constrained CKP hierarchy. The Lax operator $L$
of the constrained BKP hierarchy [27] is given by
$L=\partial+\sum_{i=1}^{m}(q_{i}\partial^{-1}r_{i,x}-r_{i}\partial^{-1}q_{i,x}),$
(4.1)
where the $q_{i}$ and $r_{i}$ are eigenfunctions. The corresponding Lax
equation of the constrained BKP hierarchy is defined by
$\dfrac{\partial L}{\partial t_{n}}=[B_{n},L],n=1,3,5,\cdots.$ (4.2)
For $k=1,3,5,\cdots$, at first we calculate the original additional symmetry
flows of the BKP hierarchy as
$\dfrac{\partial L}{\partial{t^{*}_{1,k}}}=-[(A_{1,k})_{-},L],$ (4.3)
where $A_{1,k}=M^{m}L^{l}-(-1)^{l}L^{l-1}M^{m}L.$ Thus
$(\dfrac{\partial
L}{\partial{t^{*}_{1,k}}})_{-}=[(A_{1,k})_{+},L]_{-}+2(L^{k})_{-}.$ (4.4)
For the cBKP hierarchy, the action of one flow $\partial_{\tau}$ on the
eigenfunctions $(\partial_{\tau}q_{i})$ and $(\partial_{\tau}r_{i})$ may be
derived from its action on the $L$ if
$(\partial_{\tau}L)_{-}=\sum_{i=1}^{m}((\partial_{\tau}q_{i})\partial^{-1}r_{i,x}-(\partial_{\tau}r_{i})\partial^{-1}q_{i,x}+q_{i}\partial^{-1}(\partial_{\tau}r_{i,x})-r_{i}\partial^{-1}(\partial_{\tau}q_{i,x})).$
(4.5)
At the same time, $(\partial_{\tau}q_{i,x})$ should be consistent with
$(\partial_{\tau}q_{i})$, $(\partial_{\tau}r_{i,x})$ should be consistent with
$(\partial_{\tau}r_{i})$. However, the following lemma shows that
$\partial_{t^{*}_{1,k}}$ is not the case due to the form of $(L^{k})_{-}$.
Lemma 4.1. The Lax operator $L$ of the constrained BKP hierarchy given by
eq.(4.1) satisfies the relation
$(L^{k})_{-}=\sum_{i=1}^{m}\sum_{j=0}^{k-1}(L^{k-j-1}(q_{i})\partial^{-1}(L^{*})^{j}(r_{i,x})-L^{k-j-1}(r_{i})\partial^{-1}(L^{*})^{j}(q_{i,x})),k=1,3,5,\cdots$
(4.6)
where
$L(q_{i})=L_{+}(q_{i})+\sum_{j=1}^{m}(q_{j}\partial_{x}^{-1}(r_{j,x}q_{i})-r_{j}\partial_{x}^{-1}(q_{j,x}q_{i})).$
According to the analysis above, we may revise the flows in eq.(4.3) to define
a correct additional symmetry flows of the cBKP hierarchy. Similar to the case
of cCKP hierarchy, we define a new additional flow
$\partial_{k}^{*}L=[-(A_{1,k})_{-}+Y_{k},L],k=1,3,5,\cdots,$ (4.7)
where $A_{1,k}=M^{m}L^{l}-(-1)^{l}L^{l-1}M^{m}L$ is the generator of the
additional symmetry of the BKP hierarchy. We shall show in this section that
$\partial_{k}^{*}$ is a correct additional symmetry flow for the cBKP
hierarchy. First of all, by a similar discussion as the case of the cCKP
hierarchy, we have the following lemma.
Lemma 4.2. For the cBKP hierarchy, the BKP reduction condition infers a
constraint on $Y_{k}$ as
$Y_{k}^{*}=-\partial Y_{k}\partial^{-1},\ k=1,3,5,\cdots$ (4.8)
Thus we can set
$\displaystyle Y_{1}$ $\displaystyle=0,$ (4.9) $\displaystyle Y_{k}$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(L^{k-2-j}(q_{i})\partial^{-1}(L^{*})^{j}(r_{i,x})-L^{k-2-j}(r_{i})\partial^{-1}(L^{*})^{j}(q_{i,x})),k\geq
3.$ (4.10)
The following lemma shows that $\\{Y_{k},\ k=1,3,5,\cdots\\}$ satisfy the
property we need.
Lemma 4.3.
$Y_{k}^{*}=-\partial Y_{k}\partial^{-1}.$ (4.11)
Proof. $Y_{1}^{*}=Y_{1}=0$ is trivial. For $k=3,5,7,\cdots$, we have
$\displaystyle\partial Y_{k}\partial^{-1}$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(\partial
L^{k-2-j}(q_{i})\partial^{-1}(L^{*})^{j}(r_{i,x})\partial^{-1}-\partial
L^{k-2-j}(r_{i})\partial^{-1}(L^{*})^{j}(q_{i,x})\partial^{-1})$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))((-1)^{k-2-j}(L^{*})^{k-2-j}(q_{i,x})(-1)^{j}(L^{j}(r_{i})\partial^{-1}-\partial^{-1}L^{j}(r_{i}))$
$\displaystyle-(-1)^{k-2-j}(-L^{*})^{k-2-j}(r_{i,x})(-1)^{j}(L^{j}(q_{i})\partial^{-1}-\partial^{-1}L^{j}(q_{i})))$
$\displaystyle=-\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))((L^{*})^{k-2-j}(q_{i,x})L^{j}(r_{i})\partial^{-1}-(L^{*})^{k-2-j}(q_{i,x})\partial^{-1}L^{j}(r_{i})$
$\displaystyle-(L^{*})^{k-2-j}(r_{i,x})L^{j}(q_{i})\partial^{-1}+(L^{*})^{k-2-j}(r_{i,x})\partial^{-1}L^{j}(q_{i}))$
$\displaystyle=-\sum_{i=1}^{m}\sum_{l=0}^{k-2}(2l-(k-2))(-(L^{*})^{l}(r_{i,x})\partial^{-1}L^{k-2-l}(q_{i})+(L^{*})^{l}(q_{i,x})\partial^{-1}L^{k-2-l}(r_{i}))$
$\displaystyle-\sum_{i=1}^{m}\sum_{j=0}^{k-2}(2j-(k-2))(-1)^{k-2-j}(\partial
L^{k-2-j}(q_{i})L^{j}(r_{i})\partial^{-1}-\partial
L^{k-2-j}(r_{i})L^{j}(q_{i})\partial^{-1})$ $\displaystyle=-Y_{k}^{*}$
we have used the identity
$\partial^{-1}f_{x}\partial^{-1}=f\partial^{-1}-\partial^{-1}f$ as
$f=(L^{*})^{j}(r_{i,x})$ and $f=(L^{*})^{j}(q_{i,x})$ in the derivation.
$\square$
In order to get the explicit form of the right hand side of eq.(4.7), the
following lemma is necessary.
Lemma 4.4. For the Lax operator $L$ of the cBKP hierarchy and
$X=\sum_{k=1}^{l}M_{k}\partial^{-1}N_{k}$,
$\displaystyle[X,L]_{-}$
$\displaystyle=\sum_{k=1}^{l}(-L(M_{k})\partial^{-1}N_{k}+M_{k}\partial^{-1}L^{*}(N_{k}))$
$\displaystyle+\sum_{i=1}^{m}(X(q_{i})\partial^{-1}r_{i,x}-X(r_{i})\partial^{-1}q_{i,x}-q_{i}\partial^{-1}X^{*}(r_{i,x})+r_{i}\partial^{-1}X^{*}(q_{i,x}))$
holds.
Applying the above lemma we conclude that
$[Y_{k},L]_{-}=-2(L^{k})_{-}+k\sum_{i=1}^{m}(L^{k-1}(q_{i})\partial^{-1}r_{i,x}-L^{k-1}(r_{i})\partial^{-1}q_{i,x})$
$\hskip
22.76228pt+k\sum_{i=1}^{m}(q_{i}\partial^{-1}L^{*(k-1)}(r_{i,x})-r_{i}\partial^{-1}L^{*(k-1)}(q_{i,x}))$
$+\sum_{i=1}^{m}(Y_{k}(q_{i})\partial^{-1}r_{i,x}-Y_{k}(r_{i})\partial^{-1}q_{i,x})$
$-\sum_{i=1}^{m}(-q_{i}\partial^{-1}Y_{k}^{*}(r_{i,x})+r_{i}\partial^{-1}Y_{k}^{*}(q_{i,x})).$
Theorem 4.1. For the cBKP hierarchy, the additional flows defined by eq.(4.7)
acting on the eigenfunction $q_{i}$ and $r_{i}$ are
$\displaystyle\partial_{k}^{*}q_{i}\ \
=kL^{k-1}(q_{i})+Y_{k}(q_{i})-(A_{1,k})_{+}(q_{i}),$ (4.12)
$\displaystyle\partial_{k}^{*}r_{i}\ \
=kL^{k-1}(r_{i})+Y_{k}(r_{i})+(A_{1,k})_{+}(r_{i}),$ (4.13)
$\displaystyle\partial_{k}^{*}q_{i,x}=-kL^{*(k-1)}(q_{i,x})-Y_{k}^{*}(q_{i,x})+(A_{1,k})_{+}^{*}(q_{i,x}),$
(4.14)
$\displaystyle\partial_{k}^{*}r_{i,x}=-kL^{*(k-1)}(r_{i,x})-Y_{k}^{*}(r_{i,x})-(A_{1,k})_{+}^{*}(r_{i,x}),k=1,3,5,\cdots.$
(4.15)
Proof. From the revised definition of the additional flows in eq.(4.7) of the
cBKP hierarchy, by a short calculation, we have
$\displaystyle\partial_{k}^{*}L_{-}$
$\displaystyle=[-(A_{1,k})_{-}+Y_{k},L]_{-}$
$\displaystyle=[(A_{1,k})_{+},L]_{-}+2(L^{k})_{-}+[Y_{k},L]_{-}$
$\displaystyle=\sum_{i=1}^{m}(kL^{k-1}(q_{i})+Y_{k}(q_{i})-(A_{1,k})_{+}(q_{i}))\partial^{-1}r_{i,x}$
$\displaystyle-(kL^{k-1}(r_{i})+Y_{k}(r_{i})+(A_{1,k})_{+}(r_{i}))\partial^{-1}q_{i,x}$
$\displaystyle+q_{i}\partial^{-1}(-kL^{*(k-1)}(r_{i,x})-Y_{k}^{*}(r_{i,x})-(A_{1,k})_{+}^{*}(r_{i,x}))$
$\displaystyle-
r_{i}\partial^{-1}(-kL^{*(k-1)}(q_{i,x})-Y_{k}^{*}(q_{i,x})+(A_{1,k})_{+}^{*}(q_{i,x}))$
thus
$\partial_{k}^{*}q_{i}\ \ =kL^{k-1}(q_{i})+Y_{k}(q_{i})-(A_{1,k})_{+}(q_{i}),$
the other three identities could be also obtained in the same way. $\square$
Remark 4.1. By a simple calculation, $\partial_{k}^{*}q_{i,x}$ and
$\partial_{k}^{*}r_{i,x}$ are not essential and necessary, because it can be
obtained from $\partial_{k}^{*}q_{i}$ and $\partial_{k}^{*}r_{i}$,
respectively.
Corollary 4.1. The additional flows act on wave operator $S$ of the
constrained BKP hierarchy as
$\partial_{k}^{*}S=(-(A_{1,k})_{-}+Y_{k})S,k=1,3,5,\cdots.$ (4.16)
Theorem 4.2. The additional flows ${\partial_{k}^{*}}$ commute with
constrained BKP hierarchy $\dfrac{\partial}{\partial{t_{2n+1}}}$,
i.e.
$[{\partial_{k}^{*}},\partial_{t_{2n+1}}]=0.$ (4.17)
This theorem shows $\partial_{k}^{*}(k=1,3,5,\cdots)$ are indeed the
additional symmetry flows of the cBKP hierarchy.
Using the identity $Y_{k}^{*}=-\partial Y_{k}\partial^{-1},\ k=1,3,5,\cdots$,
we can present the next theorem omitting the proof.
Theorem 4.3. Acting on the space of the wave operator $S$ of the constrained
BKP hierarchy, $\partial_{k}^{*}$ forms new centerless
$W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$.
## 5\. Conclusions and Discussions
To summarize, we have constructed the additional symmetries of the constrained
CKP hierarchy in eq.(3.7) and theorem 3.2. The additional flows action on the
eigenfunction of the cCKP hierarchy are given in theorem 3.1. Acting on the
space of the wave operator $S$ of the cCKP hierarchy, $\partial_{k}^{*}$ forms
a new centerless $W^{cC}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$
algebra in theorem 3.3. Similarly, the conclusions for the constrained BKP
hierarchy are obtained using the analogous technique in the case of
constrained CKP hierarchy. The main results of the cBKP hierarchy are
presented in eq.(4.7), theorem 4.1, 4.2 and 4.3. Our results show that the
cCKP and cBKP hierarchies have, indeed, some different properties for
additional symmetry comparing with the KP, BKP, CKP and constrained KP
hierarchies. For example, the definitions of additional symmetry flows for the
cCKP and cBKP hierarchies are different from the above hierarchies, the
revised operators ${Y_{k}}$ of the cCKP and cBKP hierarchies are different
from the corresponding operator $X_{k}$ of the cKP hierarchy given in [25].
Moreover, we also would like to point out that ${Y_{k}}$ of the cCKP and cBKP
hierarchies are distinct. We can use the similar technique without essential
difficulty to construct additional symmetries of the case as $L^{k}=B_{k}+\sum
q_{i}\partial^{-1}r_{i}+r_{i}\partial^{-1}q_{i},k>1$, the results will be
given in the future.
Acknowledgments This work was supported by the National Natural Science
Foundation of China (Grant No.10971109, 10971209), and the Program for New
Century Excellent Talents in University (Grant No.NCET-08-0515).
## References
* [1] Date E, Kashiwara M, Jimbo M, Miwa T. Nonlinear Integrable Systems- Classical and Quantum Theory. Singapore: World Scientific, 1983, 39-119
* [2] Dickey L A. Soliton Equations and Hamiltonian Systems (2nd Edition). Singapore: World Scientific, 2003
* [3] Orlov A Yu, Schulman E I. Additional symmetries of integrable equations and conformal algebra reprensentaion, Lett Math Phys, 1986, 12: 171–179
* [4] Moerbke P Van. Integrable foundation of string theory, in Lectures on integrable systems, P.163-267. Singapore: World Scientific, 1994
* [5] Adler M, Shiota T, Moerbeke P van. A Lax representation for the Vertex operator and the central extension. Comm Math Phys, 1995, 171: 547–588
* [6] Adler M, Shiota T, Moerbke P Van. From the $w_{\infty}$-algebra to its central extension: a $\tau$-function approach, Phys Lett A, 1994, 194: 33–43
* [7] Dickey L A. On additional symmetries of the KP hierarchy and Sato’s Backlund transformation. Comm Math Phys, 1995, 167: 227–233
* [8] Dickey L A. Additional symmetries of KP, Grassmannian, and the string equation. Mod Phys Lett A, 1993, 8: 1259–1272
* [9] Morozov A. Integrability and matrix models. Phys Usp, 1994, 37: 1–55
* [10] Mironov A. WDVV equations in Seiberg-Witten theory and associative algebras. Nuclear Phys B Proc Suppl, 1998, 61A: 177–185
* [11] Alexandrov A, Mironov A, Morozov A. Solving Virasoro constraints in matrix models. Fortschr Phys, 2005, 53: 512–521
* [12] Mironov A, Morozov A. Virasoro constraints for Kontsevich-Hurwitz partition function. J High Ener Phys, 2009, 02: 024, 51pages
* [13] Aratyn H, Gomes J F, Zimerman A H. Integrable hierarchy for multidimensional Toda equations and topological-anti-topological fusion. J Geom Phys, 2003, 46: 21–47
* [14] Ma W X. K symmetries and $\tau$ symmetries of evolution equations and their Lie algebras. J Phys A: Math Gen, 1990, 23: 2707–2716
* [15] Ma W X. K symmetries and $\tau$ symmetries of evolution equations and their Lie algebras. J Phys A: Math Gen, 1992, 25: 5329–5343
* [16] Ma W X. Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations. J Math Phys, 1992, 33: 2464–2476
* [17] Date E, Kashiwara M, Jimbo M, Miwa T. KP hierarchy of Orthogonal symplectic type–transformation groups for soliton equations VI. J Phys Soc Japan, 1981, 50: 3813–3818
* [18] He J S, Tian K L, Foerster A, Ma W X. Additional symmetries and string equation of the CKP hierarchy. Lett Math Phys, 2007, 81: 119–134
* [19] Takasaki K. Quasi-classical limit of BKP hierarchy and W-infity symmetries. Lett Math Phys, 1993, 28: 177–185
* [20] Tu M H. On the BKP hierarchy: Additional symmetries, Fay identity and Adler-Shiota- van Moerbeke formula. Lett Math Phys, 2007 81: 91–105
* [21] Shen H F, Tu M H. On the string equation of the BKP hierarchy. Int J Mode Phys A, 2009, V24, 22: 4193–4208
* [22] Konopelchenko B G, Sidorenko J, Strampp W. $(1+1)$-dimensional integrable systems as symmetry constraints of $(2+1)$-dimensional systems. Phys Lett A, 1991, 157: 17–21
* [23] Cheng Y, Li Y S. The constraint of the Kadomtsev-Petviashvili equation and its special solutions. Phys Lett A, 1991, 157: 22–26
* [24] Cheng Y. Constraints of the Kadomtsev-Petviashvili hierarchy. J Math Phys, 1992, 33: 3774–3782
* [25] Aratyn H, Nissimov E, Pacheva,S. Virasoro symmetry of constrained of constrained KP Hierarchies, Phys Lett A, 1997, 228: 164–175
* [26] Loris I. On reduced CKP equations. Inverse Problems, 1999, 15: 1099–1109
* [27] Loris I. Symmetry reductions of the BKP hierarchy. J Math Phys, 1999, 40: 1420–1431
* [28] Loris I. Dimensional reductions of BKP and CKP hierarchies. J Phys A, 2001, 34: 3447–3459
* [29] He J S, Wu Z W, Cheng Y. Gauge transformations for the constrained CKP and BKP hierarchies. J Math Phys, 2007, 48: 113519, 16pages
|
arxiv-papers
| 2010-07-16T06:12:45 |
2024-09-04T02:49:11.646073
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kelei Tian, Jingsong He, Jipeng Cheng, Yi Cheng",
"submitter": "Jingsong He",
"url": "https://arxiv.org/abs/1007.2701"
}
|
1007.2709
|
11institutetext: Tianshu Luo, Yimu Guo 22institutetext: Institute of Solid
Mechanics, Department of Applied Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China 22email: ltsmechanic@zju.edu.cn
33institutetext: Yimu Guo 44institutetext: Institute of Solid Mechanics,
Department of Applied Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China 44email: guoyimu@zju.edu.cn
# An Examination of the Time-Centered Difference Scheme for Dissipative
Mechanical Systems from a Hamiltonian Perspective
Tianshu Luo Yimu Guo
(Received: date / Accepted: date)
###### Abstract
In this paper, we have proposed an approach to observe the time-centered
difference scheme for dissipative mechanical systems from a Hamiltonian
perspective and to introduce the idea of symplectic algorithm to dissipative
systems. The dissipative mechanical systems discussed in this paper are finite
dimensional. This approach is based upon a proposition: for any
nonconservative classical mechanical system and any initial condition, there
exists a conservative one; the two systems share one and only one common phase
curve; the Hamiltonian of the conservative system is the sum of the total
energy of the nonconservative system on the aforementioned phase curve and a
constant depending on the initial condition. Hence, this approach entails
substituting an infinite number of conservative systems for a dissipative
mechanical system corresponding to varied initial conditions. One key way we
use to demonstrate these viewpoints is that by the Newton-Laplace principle
the nonconservative force can be reasonably assumed to be equal to a function
of a component of generalized coordinates $q_{i}$ along a phase curve, such
that a nonconservative system can be reformulated as countless conservative
systems. The advantage of this approach is such that there is no need to
change the definition of canonical momentum and the motion is identical to
that of the original system. Therefore, first we utilize the time-centered
difference scheme directly to solve the original system, after which we
substitute the numerical solution for the analytical solution to construct a
conservative force equal to the dissipative force on the phase curve, such
that we would obtain a substituting conservative system numerically. Finally,
we use the time-centered scheme to integrate the substituting system
numerically. We will find an interesting fact that the latter solution
resulting from the substituting system is equivalent to that of the former.
Indeed, there are two transition matrices within time grid points: the first
one is unsymplectic and the second symplectic. In fact, the time-centered
scheme for dissipative systems can be thought of as an algorithm that
preserves the symplectic structure of the substituting conservative systems.
In addition, via numerical examples we find that the time-centered scheme
preserves the total energy of dissipative systems. According to such
behaviors, we might explain why some algorithms, e.g., the time-centered Euler
scheme, are better than other unsymplectic algorithms for dissipative systems,
such that we might choose better algorithms and introduce the idea of this
paper to more symplectic algorithms.
###### Keywords:
Hamiltonian, dissipation, non-conservative system, damping, symplectic
algorithm
††journal: Acta Mechanica Sinica
## 1 Introduction
FengFeng1985 ; Feng1989 ; Feng1990 ; Feng1991 , ZhongZhong2005 ; Zhong2009 ;
Gao2009 and MarsdenMarsden1998 have developed a series of symplectic
algorithms for conservative systems and proposed common theories for the
construction of symplectic algorithms.
FengFeng1985 has investigated some existing old numerical algorithms from a
Hamiltonian perspective, such as the simplest symplectic algorithm, the Euler
time-centered difference scheme. He have explained why for a conservative
system the Euler time-centered difference scheme is more accurate than other
unsymplectic schemes from a Hamiltonian perspective. Because the time-centered
difference scheme is derived from Hamilton’s equation, the algorithm can
preserve symplectic structure and mechanical energy. The mechanical energy-
preserving behavior was proven by Xingxingyufeng2007 , who found that for
dissipative problems this algorithm has good mechanical energy-preserving
characteristics. We will state the reason using the Hamiltonian description of
dissipative mechanical systems, which are finite dimensional in this paper,
and we will then apply the idea of symplectic algorithms to the analyses of
the time-centered difference scheme for dissipative systems. Nevertheless,
since Hamilton originated Hamilton equations of motion and Hamiltonian
formalism, it has been stated in most classical textbooks that the Hamiltonian
formalism focuses on solving conservative problems.
If one needs to apply symplectic algorithms to dissipative systems, one must
convert a dissipative system into a Hamiltonian system or find some
relationship between the dissipative system and a conservative one. An
attemptKane2000 has been made to apply symplectic algorithms directly to
dissipative systems. The transition matrix between phase variable vectors must
be unsymplectic. Symplectic algorithms called variational integratorsKane2000
were derived from discretization of the variational principle for conservative
systems
$\delta\int_{a}^{b}L(q(t),\dot{q}(t))\mathrm{d}t=0,$
which is equivalent to Hamilton’s equation. Therefore, the algorithms called
variational integrators are natively symplectic. But the direct variational
integrators for dissipative systems were derived from discretization of the
Lagrange-d’Alembert principle
$\delta\int_{a}^{b}L(q(t),\dot{q}(t))\mathrm{d}t+\int_{a}^{b}F(q(t),\dot{q}(t))\delta
q\mathrm{d}t=0,$
where $F(q(t),\dot{q}(t))$ is a nonconservative force. Obviously Hamilton’s
equation cannot be derived from the Lagrange-d’Alembert principle.
ZhongZhong2009 considered that discretization of Hamilton’s equations or the
Hamiltonian least action variational principle can lead to symplectic
transition matrices. Therefore, the variational integrators applied directly
to dissipative systems cannot be considered as symplectic schemes.
ZhongZhong2005 attempted to convert a damping Duffing equation into a
conservative system by redefining the position variable $q$ and the canonical
momentum $p$ and then applying the time-fem method to this conservative
system. His method is to multiply $q$ by the reciprocal of the amplitude decay
coefficient such that the momentum is redefined. The characteristic of the
original dissipative system has changed entirely.
Although several other approaches have been proposed to represent dissipative
systems as Hamiltonian formalism, these approaches might not be accepted by
researchers in geometrical mechanics. For instance, MorrisonMorrison2006593 ;
RevModPhys.70.467 and SalmonRickSalmon1988 focused on the conservative
system or some special dissipative systems, e.g. an oscillator with gyroscopic
damping. MorrisonMorrison2006593 wrote, ’The ideal fluid description is one
in which viscosity or other phenomenological terms are neglected. Thus, as is
the case for systems governed by Newton’s second law without dissipation, such
fluid descriptions posses Lagrangian and Hamiltonian descriptions.’ If there
had existed an approach appropriate for representing an oscillator with
damping as Hamiltonian formalism, these researchers would have attempted to
extend the Hamiltonian description to other dissipative problems.
Marsden Marsden2007 and other researchers applied the equations as below to
the problem of stability of dissipative systems
$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q_{i}}+\bm{F}\left(\frac{\partial{r}}{\partial q_{i}}\right)$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}},$ (1)
where the position vector $r$ depends on the canonical variable $\\{q,p\\}$,
i.e. $r(q,p)$, $H$ denotes Hamiltonian, and
$\bm{F}(\partial{r}/\partial{q_{i}})$ denotes a generalized force in the
direction $i$, $i=1,\dots,n$. Marsden considered that Eqs.(1) was composed of
a conservative part and a non-conservative part. Eq.(1) apparently is not a
Hamilton’s equation but only a representation of dissipative mechanical
systems in the phase space. Although one can utilize the approaches discussed
in some papers to convert Eq.(1) into a Hamiltonian system, one must first
change the definition of the canonical momentum of the system. If one uses
symplectic algorithms to solve the Hamiltonian system, the numerical solution
will lose the physical characteristics of the original system, because the
phase flow of the original system is different from that of the new system. We
need a Hamiltonian system that shares common phase flow or solution with the
original system. But this demand cannot be satisfied, because it conflicts
with Louisville’s theorem. Therefore, we would have to attempt to find some
relationship between dissipative systems and conservative ones, such that we
can introduce the concept of symplectic algorithms to dissipative problems and
explain the time-centered difference scheme from Hamiltonian viewpoints.
Based on Eq.(1), in this paper we will attempt to demonstrate that a
dissipative mechanical system shares a single common phase curve with a
conservative system. In the light of this property, we will propose an
approach to substitute a group of conservative systems for a dissipative
mechanical system. In the following section, we will illustrate the
relationship between a dissipative mechanical system and a conservative one.
## 2 Relationship between a Dissipative Mechanical System and a Conservative
One
### 2.1 A Proposition
Under general circumstances, the force $\bm{F}$ is a damping force that
depends on the variable set
$q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n}$. $F_{i}$ denotes the
components of the generalized force $\bm{F}$.
$F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})=\bm{F}\left(\frac{\partial{r}}{\partial
q_{i}}\right).$ (2)
Thus we can reformulate Eq.(1) as follows:
$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q_{i}}+F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}}.$ (3)
Suppose the Hamiltonian quantity of a conservative system without damping is
$\hat{H}$. Thus we may write a Hamilton’s equation of the conservative system
:
$\displaystyle\dot{p}_{i}$ $\displaystyle=$
$\displaystyle-\frac{\partial{\hat{H}}}{\partial q_{i}}$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$
$\displaystyle\frac{\partial\hat{H}}{\partial p_{i}}.$ (4)
We do not intend to change the definition of momentum in classical mechanics,
but we do require that a special solution of Eq.(4) is the same as that of
Eq.(3). We may therefore assume a phase curve $\gamma$ of Eq.(3) coincides
with that of Eq.(4). The phase curve $\gamma$ corresponds to an initial
condition $q_{i0},p_{i0}$. Consequently by comparing Eq.(3) and Eq.(4), we
have
$\displaystyle\left.\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right|_{\gamma}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial H}{\partial
q_{i}}\right|_{\gamma}-\left.F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right|_{\gamma}$
$\displaystyle\left.\frac{\partial{\hat{H}}}{\partial{p_{i}}}\right|_{\gamma}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial H}{\partial
p_{i}}\right|_{\gamma},$ (5)
where
$\left.\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right|_{\gamma},\left.\frac{\partial
H}{\partial
q_{i}}\right|_{\gamma},\left.\frac{\partial{\hat{H}}}{\partial{p_{i}}}\right|_{\gamma}and\left.\frac{\partial
H}{\partial p_{i}}\right|_{\gamma}$ denote the values of these partial
derivatives on the phase curve $\gamma$ and
$\left.F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right|_{\gamma}$
denotes the value of the force $F_{i}$ on the phase curve $\gamma$. In
classical mechanics the Hamiltonian $H$ of a conservative mechanical system is
mechanical energy and can be written as:
$H=\int_{\gamma}\left(\frac{\partial{H}}{\partial{q_{i}}}\right)\mathrm{d}q_{i}+\int_{\gamma}\left(\frac{\partial
H}{\partial p_{i}}\right)\mathrm{d}p_{i}+const_{1},$ (6)
where $const_{1}$ is a constant that depends on the initial condition
described above. If $q_{i}=0,p_{i}=0$, then $const_{1}=0$. The Einstein
summation convention has been used this section. Thus an attempt has been made
to find $\left.\hat{H}\right|_{\gamma}$ through line integral along the phase
curve $\gamma$ of the dissipative system
$\displaystyle\int_{\gamma}\frac{\partial{\hat{H}}}{\partial{q_{i}}}\mathrm{d}q_{i}$
$\displaystyle=$ $\displaystyle\int_{\gamma}\left[\frac{\partial H}{\partial
q_{i}}-F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right]\
\mathrm{d}q_{i}$ $\displaystyle\int_{\gamma}\frac{\partial\hat{H}}{\partial
p_{i}}\mathrm{d}p_{i}$ $\displaystyle=$
$\displaystyle\int_{\gamma}\frac{\partial H}{\partial p_{i}}\mathrm{d}p_{i}.$
(7)
Analogous to Eq.(6), we have
$\left.\hat{H}\right|_{\gamma}=\int_{\gamma}\frac{\partial{\hat{H}}}{\partial{q_{i}}}\mathrm{d}q_{i}+\int_{\gamma}\frac{\partial{\hat{H}}}{\partial
p_{i}}\mathrm{d}p_{i}+const_{2},$ (8)
where $const_{2}$ is a constant which depends on the initial condition.
Substituting Eq.(6)(7) into Eq.(8), we have
$\left.\hat{H}\right|_{\gamma}=H-\int_{\gamma}F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\mathrm{d}q_{i}+const.$
(9)
where $const=const_{2}-const_{1}$, and $H=\left.H\right|_{\gamma}$ because $H$
is mechanical energy of the nonconservative system(3). According to the
physical meaning of Hamiltonian, $const_{1}$, $const_{2}$ and $const$ are
added into Eq.(6)(8)(9) respectively such that the integral constant vanishes
in the Hamiltonian quantity. ArnoldArnold1997 had presented the Newton-
Laplace principle of determinacy as, ’This principle asserts that the state of
a mechanical system at any fixed moment of time uniquely determines all of its
(future and past) motion.’ In other words, in the phase space the position
variable and the velocity variable are determined only by the time $t$.
Therefore, we can assume that we have already a solution of Eq.(3)
$\displaystyle q_{i}$ $\displaystyle=$ $\displaystyle q_{i}(t)$
$\displaystyle\dot{q_{i}}$ $\displaystyle=$ $\displaystyle\dot{q_{i}}(t),$
(10)
where the solution satisfies the initial condition. We can divide the whole
time domain into a group of sufficiently small domains and in these domains
$q_{i}$ is monotone, and hence we can assume an inverse function $t=t(q_{i})$.
If $t=t(q_{i})$ is substituted into the nonconservative force
$\left.F_{i}\right|_{\gamma}$, we can assume that:
$\left.F_{i}(q_{1}(t(q_{i})),\cdots,q_{n}(t(q_{i})),\dot{q}_{1}(t(q_{i})),\cdots,\dot{q}_{n}(t(q_{i})))\right|_{\gamma}=\mathcal{F}_{i}(q_{i}),$
(11)
where $\mathcal{F}_{i}$ is a function of $q_{i}$ alone. In Eq.(11) the
function $F_{i}$ is restricted on the curve $\gamma$, such that a new function
$\mathcal{F}_{i}(q_{i})$ yields. Thus we have
$\displaystyle\int_{\gamma}F_{i}\mathrm{d}q_{i}$ $\displaystyle=$
$\displaystyle\int_{q_{i0}}^{q_{i}}\mathcal{F}_{i}(q_{i})\mathrm{d}q_{i}=W_{i}(q_{i})-W_{i}(q_{i0}).$
(12)
According to Eq.(12) the function $\mathcal{F}_{i}$ is path independent, and
therefore $\mathcal{F}_{i}$ can be regarded as a conservative force. For that
Eq.(11) represents an identity map from the nonconservative force $F$ on the
curve $\gamma$ to the conservative force $\mathcal{F}_{i}$ which is distinct
from $F_{i}$. Eq.(11) is tenable only on the phase curve $\gamma$.
Consequently the function form of $\mathcal{F}_{i}$ depends on the
aforementioned initial condition; from other initial conditions
$\mathcal{F}_{i}$ with different function forms will yield.
According to the physical meaning of Hamiltonian, $const$ is added to Eq.(9)
such that the integral constant vanishes in Hamiltonian quantity. Hence
$const=-W_{i}(q_{i0})$. Substituting Eq.(12) and $const=-W_{i}(q_{i0})$ into
Eq.(9), we have
$\left.\hat{H}\right|_{\gamma}=H-W_{i}(q_{i})$ (13)
where $-W_{i}(q_{i})$ denotes the potential of the conservative force
$\mathcal{F}_{i}$ and $W_{i}(q_{i})$ is equal to the sum of the work done by
the nonconservative force $F$ and $const$. In Eq.(13) $\hat{H}$ and $H$ are
both functions of $q_{i}$ and $W_{i}(q_{i})$ a function of $q_{i}$. Eq.(13)
and Eq.(9) can be thought of as a map from the total energy of the dissipative
system(3) to the Hamiltonian of the conservative system(4). Indeed,
$\left.\hat{H}\right|_{\gamma}$ and the total energy differ in the constant
$const=-W_{i}(q_{i0})$. When the conservative system takes a different initial
condition, if one does not change the function form of
$\left.\hat{H}\right|_{\gamma}$, one can consider
$\left.\hat{H}\right|_{\gamma}$ as a Hamiltonian quantity $\hat{H}$,
$\hat{H}=\left.\hat{H}\right|_{\gamma}=H-W_{i}(q_{i})$ (14)
and the conservative system(4) can be thought of as an entirely new
conservative system.
Based on the above, the following proposition is made:
###### Proposition 1
For any nonconservative classical mechanical system and any initial condition,
there exists a conservative one; the two systems share one and only one common
phase curve; the value of the Hamiltonian of the conservative system is equal
to the sum of the total energy of the nonconservative system on the
aforementioned phase curve and a constant depending on the initial condition.
###### Proof
First we must prove the first part of the Proposition 1, i.e. that a
conservative system with Hamiltonian presented by Eq.(14) shares a common
phase curve with the nonconservative system represented by Eq.(3). In other
words the Hamiltonian quantity presented by Eq.(14) satisfies Eq.(5) under the
same initial condition. Substituting Eq.(14) into the left side of Eq.(5), we
have
$\displaystyle\frac{\partial{\hat{H}(q_{i},p_{i})}}{\partial{q_{i}}}$
$\displaystyle=$ $\displaystyle\frac{\partial
H(q_{i},p_{i})}{\partial{q_{i}}}-\frac{\partial{W_{j}(q_{j})}}{\partial{q_{i}}}$
$\displaystyle\frac{\partial{\hat{H}(q_{i},p_{i})}}{\partial{p_{i}}}$
$\displaystyle=$ $\displaystyle\frac{\partial
H(q_{i},p_{i})}{\partial{p_{i}}}-\frac{\partial{W_{j}(q_{j})}}{\partial{p_{i}}}.$
(15)
It must be noted that although $q_{i}$ and $p_{i}$ are considered as distinct
variables in Hamilton’s mechanics, we can consider $q_{i}$ and $\dot{q_{i}}$
as dependent variables in the process of constructing of $\hat{H}$. At the
trajectory $\gamma$ we have
$\displaystyle\frac{\partial{{W_{j}(q_{j})}}}{\partial{q_{i}}}$
$\displaystyle=$
$\displaystyle\frac{\partial{(\int_{q_{j0}}^{q_{j}}\mathcal{F}_{j}(q_{j})\mathrm{d}q_{j}+W_{i}(q_{i0}))}}{\partial{q_{i}}}=\mathcal{F}_{i}(q_{i})$
$\displaystyle\frac{\partial{{W_{j}(q_{j})}}}{\partial{p_{i}}}$
$\displaystyle=0,$ (16)
where $\mathcal{F}_{i}(q_{i})$ is equal to the damping force $F_{i}$ on the
phase curve $\gamma$. Hence under the initial condition $q_{0},p_{0}$, Eq.(5)
is satisfied. As a result, we can state that the phase curve of Eq.(4)
coincides with that of Eq.(3) under the initial condition; and $\hat{H}$
represented by Eq.(14) is the Hamiltonian of the conservative system
represented by Eq.(4).
Then we must prove the second part of Proposition 1: the uniqueness of the
common phase curve.
We assume that eq.(4) shares two common phase curves, $\gamma_{1}$ and
$\gamma_{2}$, with eq.(3). Let a point of $\gamma_{1}$ at the time $t$ be
$z_{1}$, a point of $\gamma_{2}$ at the time $t$ $z_{2}$, and $g^{t}$ the
Hamiltonian phase flow of eq.(4). Suppose a domain $\Omega$ at $t$ which
contains only points $z_{1}$ and $z_{2}$, and $\Omega$ is not only a subset of
the phase space of the nonconservative system(3) but also that of the phase
space of the conservative system(4). Hence there exists a phase flow
$\hat{g}^{t}$ composed of $\gamma_{1}$ and $\gamma_{2}$, and $\hat{g}^{t}$ is
the phase flow of eq.(3) restricted by $\Omega$. According to the following
Louisville’s theoremArnold1978 :
###### Theorem 2.1
The phase flow of Hamilton’s equations preserves volume: for any region $D$ we
have
$volume\ of\ g^{t}D=volume\ of\ D$
where $g^{t}$ is the one-parameter group of transformations of phase space
$g^{t}:(p(0),q(0))\longmapsto:(p(t),q(t))$
$g^{t}$ preserves the volume of $\Omega$. This implies that the phase flow of
eq.(3) $\hat{g}^{t}$ preserves the volume of $\Omega$ too. But the system (3)
is not conservative, which conflicts with Louisville’s theorem; hence only a
phase curve of eq.(4) coincides with that of eq.(3).
∎∎
### 2.2 An Example in Vibration Mechanics
Take an $n$-dimensional oscillator with damping as an example, the governing
equation of which is as below:
$\ddot{\bm{q}}+\mathsfsl{C}\dot{\bm{q}}+\mathsfsl{K}\bm{q}=0,$ (17)
where $\bm{q}=\left[q_{1},\dots,q_{n}\right]^{T}$, superscript $T$ denotes a
matrix transpose,
$\mathsfsl{C}=\left[\begin{array}[]{ccc}C_{11}&\dots&C_{1n}\\\
\vdots&\ddots&\vdots\\\
C_{n1}&\dots&C_{nn}\end{array}\right],\mathsfsl{K}=\left[\begin{array}[]{ccc}K_{11}&\dots&K_{12}\\\
\vdots&\ddots&\vdots\\\ K_{21}&\dots&K_{22}\end{array}\right]$
, and $C_{ij}$ and $K_{ij}$ are constants.
It is complicated to solve Eq.(17). If Eq.(17) is higher dimensional, it is
almost impossible to solve Eq.(17) analytically. Therefore we assume that a
solution exists already.
$\bm{q}=\bm{q}(t)=\left[q_{1}(t),\dots,q_{n}(t)\right].$ (18)
Suppose a group of inverse functions
$t=t(q_{1}),\dots,t=t(q_{n}).$ (19)
As in Eq.(11) we can consider that the damping forces are equal to some
conservative force under an initial condition
$\begin{array}[]{ccc}c_{11}\dot{q}_{1}=\varrho_{11}(q_{1})&\dots&c_{1n}\dot{q}_{n}=\varrho_{1n}(q_{1})\\\
\vdots&\ddots&\vdots\\\
c_{n1}\dot{q}_{1}=\varrho_{21}(q_{n})&\dots&c_{nn}\dot{q}_{n}=\varrho_{nn}(q_{n}).\end{array}$
(20)
For convenience, these conservative forces can be thought of as elastic
restoring forces:
$\begin{array}[]{ccc}\varrho_{11}(q_{1})=\kappa_{11}(q_{1})q_{1}&\dots&\varrho_{1n}(q_{1})=\kappa_{1n}(q_{1})q_{1}\\\
\vdots&\ddots&\vdots\\\
\varrho_{n1}(q_{1})=\kappa_{n1}(q_{n})q_{n}&\dots&\varrho_{nn}(q_{n})=\kappa_{nn}(q_{n})q_{n}.\end{array}$
(21)
An equivalent stiffness matrix $\mathsfsl{\tilde{K}}$ is obtained, which is a
diagonal matrix
$\mathsfsl{\tilde{K}}_{ii}=\sum_{l=1}^{n}\kappa_{il}(q_{l}).$ (22)
Consequently an $n$-dimensional conservative system is obtained
$\bm{\ddot{q}}+(\mathsfsl{K}+\mathsfsl{\tilde{K}})\bm{q}=0$ (23)
which shares a common phase curve with the $n$-dimensional damping system(17).
The Hamiltonian of Eqs.(23) is
$\hat{H}=\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(24)
where $\bm{0}$ is a zero vector. $\hat{H}$ in Eq.(24) is the mechanical energy
of the conservative system(23), because
$\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}$ is a
potential function such that $\hat{H}$ doest not depend on any path.
### 2.3 Discussion
Based on the above, we can outline the relationship between a dissipative
mechanical system and a group of conservative systems by means of Fig. 1. The
relationship can be stated from two perspectives:
Figure 1: A Dissipative Mechanical System and Conservative Systems
If one explains the relationship from a geometrical perspective, one can
obtain Proposition 1. In this paper the conservative systems (4) and (23) are
called the substituting systems. Although a substituting system shares a
common phase curve with the original system, under other initial conditions
the substituting system exhibits different phase curves. Therefore the phase
flow of the substituting system differs from that of the original system, it
follows that the substituting systems is not equal to the original system.
According to Louisville’s theorem (2.1), the phase flow of the original
dissipative system Eq.(3) certainly does not preserve its phase volume, but
the phase flow of the substituting conservative Eq.(4) does.
One also could explain the relationship from a mechanical perspective. It is
known that there are non-conservative forces in a nonconservative system. The
total energy of the nonconservative system consists of the work done by
nonconservative forces. Hence the function form of the total energy depends on
a phase curve i.e. under an initial condition. If one constrains the total
energy function to a phase curve $\gamma$, the total energy function can be
converted into a function of $q,p$. One take $\hat{H}$ consisting of this new
function and a constant as a Hamiltonian quantity, such that a Hamilton’s
system (i.e., a conservative system) is obtained. Under the initial condition
mentioned above, the solution curve of the conservative system is the same as
that of the original nonconservative system; under other initial conditions
the solution curve of the conservative is different from that of the original
nonconservative system. Since one defines the forces(11,20,21,22) in the new
system, the Hamiltonian quantity of the conservative can be thought of as the
mechanical energy of the new conservative system as Eq.(24).
The Hamiltonians of the new conservative systems in general are not
analytically integrable, unless the original mechanical system is integrable.
The reason is that the work done by damping force depends on the phase curve.
If the system is integrable, then the phase curve can be explicitly written
out, the system has an analytical solution, and therefore the work done by
damping force can be explicitly integrated. Subsequently, the Hamiltonian
$\hat{H}$ can be explicitly expressed. Most systems do not have an analytical
solution. Despite this, the Hamilton quantity, coordinates and momentum must
satisfy Eq.(4) under a certain initial condition. Why had KleinKlein1928
written, ”Physicists can make use of these theories only very little, an
engineers nothing at all”? The answer: when one is seeking an analytical
solution to a classical mechanics problem by utilizing Hamiltonian formalism,
in fact one must inevitably convert the problem back to Newtonian formalism.
This means that an explicit form of Hamiltonian quantity is not necessary for
classical mechanics. What is important is the relationship between $q,p$ and
the Hamiltonian quantity embodied in the Hamilton’s Equation.
According to the conclusion above, we can consider the Euler time-centered
difference scheme for dissipative systems from a Hamiltonian perspective.
## 3 Discussion on the Euler Time-Centered Difference Scheme
### 3.1 Introducing the Euler Time-Centered Scheme for Conservative Systems
FengFeng1985 ; Feng1989 ; Feng1990 ; Feng1991 constructed a series of
symplectic difference schemes via two approaches: the first approach
discretizes Hamilton’s equation and utilizes the property of the Cayley
transform to demonstrate that the map $g:\bm{z}^{n}\rightarrow\bm{z}^{n+1}$ is
a symplectic map; the second approach is a so-called generating function
method. Feng had represented the first approach as below:
Suppose $H$ is a differentiable function of $2n$ variables
$p_{1},\cdots,p_{n},q_{1},\cdots,q_{n}$, the Hamilton’s equations are
represented as:
$\dot{\bm{p}}=-H_{q},\ \ \ \ \ \ \dot{\bm{q}}=H_{p},$ (25)
where $\bm{p}=[p_{1},\cdots,p_{n}],\bm{q}=[q_{1},\cdots,q_{n}],\ \
H_{q}=\partial H/\partial\bm{q}$ and $H_{p}=\partial H/\partial\bm{p}$. Let
$\bm{z}=\left[\bm{p},\bm{q}\right]^{T}$, and Eq. (25) can be further
represented as:
$\dot{\bm{z}}=\mathsfsl{J}^{-1}H_{z},$ (26)
where $H_{z}=\left[H_{q},H_{p}\right]^{T}$,
$\mathsfsl{J}=\left[\begin{array}[]{cc}O&\mathsfsl{I}_{n}\\\
-\mathsfsl{I}_{n}&O\\\ \end{array}\right],$ (27)
where $\mathsfsl{I}_{n}$ and $\mathsfsl{I}$ denote unity matrices. If $H$ is a
quadratic form:
$H(\bm{z})=\frac{1}{2}\bm{z}^{T}C\bm{z},\ \ \ \ C^{T}=C,$ (28)
then canonical equations(25,26) can be rewritten as:
$\frac{\mathrm{d}{\bm{z}}}{\mathrm{d}{t}}=\mathsfsl{B}\bm{z},\ \ \
\mathsfsl{B}=\mathsfsl{J}^{-1}\mathsfsl{C}.$ (29)
PaperFeng1990 gives the following definition:
###### Definition 1
A matrix $\mathsfsl{B}$ of order $2n$ is called infinitesimal symplectic if
$\mathsfsl{JB}+\mathsfsl{B^{T}J}=\mathsfsl{O},$
where $\mathsfsl{O}$ is a null matrix. All infinitesimal symplectic matrices
form a Lie algebra $sp(2n)$ with commutation operation
$[\mathsfsl{A},\mathsfsl{B}]=\mathsfsl{AB}-\mathsfsl{BA}$, and $sp(2n)$ is the
Lie algebra of Lie group $Sp(2n)$ known as the symplectic group.
According to this definition, $\mathsfsl{B}=\mathsfsl{J}^{-1}\mathsfsl{C}$ in
Eq.(29) can be considered as an infinitesimal symplectic matrix. In the
paperFeng1985 a number of symplectic schemes for Eq.(29) were proposed, one
of which is the Euler time-centered scheme:
$\frac{\bm{z}^{n+1}-\bm{z}^{n}}{\tau}=\mathsfsl{B}\frac{\bm{z}^{n+1}+\bm{z}^{n}}{2}.$
(30)
The transition $\bm{z}^{n}\rightarrow\bm{z}^{n+1}$ is given by the following
linear transformation $\mathsfsl{F}_{\tau}$ which coincides with its own
Jacobian
$\mathsfsl{F}_{\tau}=\phi(-\frac{\tau}{2}\mathsfsl{B})=(\mathsfsl{I}-\frac{\tau}{2}\mathsfsl{B})^{-1}(\mathsfsl{I}+\frac{\tau}{2}\mathsfsl{B}).$
(31)
PaperFeng1990 gives the following theorem:
###### Theorem 3.1
If $\mathsfsl{B}\in sp(2n)$ and $|\mathsfsl{I}+\mathsfsl{B}|\neq 0$, then
$\mathsfsl{F}=(\mathsfsl{I}+\mathsfsl{B})^{-1}(\mathsfsl{I}-\mathsfsl{B})\in
Sp(2n)$, the Cayley transform of $\mathsfsl{B}$.
According to Theorem 3.1, $\mathsfsl{F}_{\tau}$ can be considered as the
Cayley transform of the infinitesimal symplectic matrix $-\tau\mathsfsl{B}/2$,
and consequently $\mathsfsl{F}_{\tau}$ is a symplectic matrix, and
$\bm{z}^{n}\rightarrow\bm{z}^{n+1}$ is symplectic.
### 3.2 Apply the Time-centered Difference Scheme Indirectly to Damping
Oscillators
Figure 2: Flowchart of applying the time-centered difference scheme indirectly
to damping Oscillators in a time-step
Based on the discussion in Section 2.2, we have created a flow chart (Fig.2.)
that shows the process of applying the time-centered difference scheme
indirectly to the damping oscillator(17). We have defined a conservative
system through an equivalent stiffness matrix represented by Eq.(20)(21)(22).
If one needs to formulate the equivalent stiffness matrix in numerical
schemes, one must have a numerical solution at time $t^{k+1}$, such that an
equivalent elastic restoring force can be thought of as a function of
$(q_{i}^{k}+q_{i}^{k+1})/2$. Hence we first apply the time-centered
difference scheme directly to the original dissipative system. Consequently
the algorithm can be depicted by Fig. 2.
The first step is to apply the time-centered difference scheme directly to
Eq.(17) :
$\displaystyle\frac{\bm{q}^{k+1}-\bm{q}^{k}}{\tau}$ $\displaystyle=$
$\displaystyle\frac{\bm{p}^{k+1}+\bm{p}^{k}}{2}$
$\displaystyle\frac{\bm{p}^{k+1}-\bm{p}^{k}}{\tau}$ $\displaystyle=$
$\displaystyle-\mathsfsl{K}\left(\frac{\bm{q}^{k+1}+\bm{q}^{k}}{2}\right)-\mathsfsl{C}\frac{\mathrm{d}[(\bm{q}^{k}+\bm{q}^{k+1})/2]}{\mathrm{d}t}$
(32)
If we let
$\mathrm{d}[(\bm{q}^{k}+\bm{q}^{k+1})/2]/\mathrm{d}t=(\bm{q}^{k+1}-\bm{q}^{k})/\tau$
according to the definition of the time-centered difference scheme, Eq.(32)
can be represented as:
$\displaystyle\left[\begin{array}[]{c}\bm{q}^{k+1}\\\ \bm{p}^{k+1}\\\
\end{array}\right]=\mathsfsl{F}_{\tau}^{1}\left[\begin{array}[]{c}\bm{q}^{k}\\\
\bm{p}^{k}\\\ \end{array}\right],$ (37)
where the transition matrix is
$\displaystyle\mathsfsl{F}_{\tau}^{1}=\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\tau(\frac{1}{2}\cdot\mathsfsl{K}+\frac{1}{\tau}\cdot\mathsfsl{C})&\mathsfsl{I}\end{array}\right]^{-1}\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
-\tau(\frac{1}{2}\cdot\mathsfsl{K}-\frac{1}{\tau}\cdot\mathsfsl{C})&\mathsfsl{I}\\\
\end{array}\right]$ (42)
Obviously the difference scheme is not a symplectic scheme, because the
transition matrix $F_{\tau}^{1}$ is not a symplectic matrix. The prerequisite
propositionFeng1990 for the proof to this point is as the following:
###### Proposition 2
Matrix $\mathsfsl{S}=\mathsfsl{M}^{-1}\mathsfsl{N}\in Sp(2n)$, iff
$\mathsfsl{MJM^{T}}=\mathsfsl{NJN^{T}}$
The proof of this point is given as follows:
###### Proof
Since
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\tau(\frac{1}{2}\cdot\mathsfsl{K}+\frac{1}{\tau}\cdot\mathsfsl{C})&\mathsfsl{I}\end{array}\right]\mathsfsl{J}\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\tau(\frac{1}{2}\cdot\mathsfsl{K}+\frac{1}{\tau}\cdot\mathsfsl{C})\\\
\frac{\tau}{2}\cdot\mathsfsl{I}&\mathsfsl{I}\end{array}\right]$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&\frac{\tau^{2}}{2}(\frac{1}{2}\cdot\mathsfsl{K}+\frac{1}{\tau}\cdot\mathsfsl{C})+\mathsfsl{I}\\\
-\frac{\tau^{2}}{2}(\frac{1}{2}\cdot\mathsfsl{K}+\frac{1}{\tau}\cdot\mathsfsl{C})-\mathsfsl{I}&\mathsfsl{O}\end{array}\right]$
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
-\tau(\frac{1}{2}\cdot\mathsfsl{K}-\frac{1}{\tau}\cdot\mathsfsl{C})&\mathsfsl{I}\end{array}\right]\mathsfsl{J}\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&-\tau(\frac{1}{2}\cdot\mathsfsl{K}-\frac{1}{\tau}\cdot\mathsfsl{C})\\\
\frac{\tau}{2}\cdot\mathsfsl{I}&\mathsfsl{I}\end{array}\right]$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&\frac{\tau^{2}}{2}(\frac{1}{2}\cdot\mathsfsl{K}-\frac{1}{\tau}\cdot\mathsfsl{C})+\mathsfsl{I}\\\
-\frac{\tau^{2}}{2}(\frac{1}{2}\cdot\mathsfsl{K}-\frac{1}{\tau}\cdot\mathsfsl{C})-\mathsfsl{I}&\mathsfsl{O}\end{array}\right]$
and according to the Proposition 2
$\displaystyle F_{\tau}^{1}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\tau(\frac{1}{2}\cdot\mathsfsl{K}+\frac{1}{\tau}\cdot\mathsfsl{C})&\mathsfsl{I}\end{array}\right]^{-1}\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
-\tau(\frac{1}{2}\cdot\mathsfsl{K}-\frac{1}{\tau}\cdot\mathsfsl{C})&\mathsfsl{I}\\\
\end{array}\right]$ $\displaystyle\notin Sp(2n).$
$F_{\tau}^{1}$ is not a symplectic matrix. ∎∎
Utilizing the scheme described by Eq.(37) we can obtain the numerical solution
$\left[\bm{q}^{k+1},p^{k+1}\right]$ which is taken as a probe solution in lieu
of the analytical solution, and then execute the second step in Fig. 2. Let
$\displaystyle\mathsfsl{C}\cdot\mathrm{d}[(\bm{q}^{k}+\bm{q}^{k+1})/2]/\mathrm{d}t=\mathsfsl{\tilde{K}}([(\bm{q}^{k}+\bm{q}^{k+1})/2),$
$\displaystyle\mathsfsl{\tilde{K}}=\left[\begin{array}[]{ccc}\tilde{K}_{11}&\dots&0\\\
\vdots&\ddots&\vdots\\\ 0&\dots&\tilde{K}_{nn}\end{array}\right]$ (51)
$\displaystyle\tilde{K}_{ii}=\sum_{j=1}^{n}2C_{ij}(q^{k+1}_{j}-q^{k}_{j})/(\tau(q^{k+1}_{i}+q^{k}_{i})),$
where $\mathsfsl{\tilde{K}}$ is the numerical approximation of the equivalent
stiffness matrix(22). Thus we would obtain the numerical approximation of the
conservative system(23).
Then we consider to apply the time-centered scheme to the conservative
system.(23). Suppose the solution vector at the time $t^{k+1}$ is
$\tilde{\bm{z}}^{k+1}=\left[\tilde{\bm{q}}^{k+1},\tilde{\bm{p}}^{k+1}\right]$
According to the time-centered difference scheme defined by Eq.(28)(29)(30),
we have a time-centered scheme for the conservative system(23):
$\displaystyle\frac{\tilde{\bm{q}}^{k+1}-\bm{q}^{k}}{\tau}$ $\displaystyle=$
$\displaystyle\frac{\tilde{\bm{p}}^{k+1}+\bm{p}^{k}}{2}$
$\displaystyle\frac{\tilde{\bm{p}}^{k+1}-\bm{p}^{k}}{\tau}$ $\displaystyle=$
$\displaystyle-(\mathsfsl{K}+\mathsfsl{\tilde{K}})\left(\frac{\tilde{\bm{q}}^{k+1}+\bm{q}^{k}}{2}\right).$
(52)
The time-centered scheme above can be represented as
$\left[\begin{array}[]{c}\tilde{\bm{q}}^{k+1}\\\ \tilde{\bm{p}}^{k+1}\\\
\end{array}\right]=\mathsfsl{F}_{\tau}^{2}\left[\begin{array}[]{c}{\bm{q}}^{k}\\\
{\bm{p}}^{k}\\\ \end{array}\right].$ (53)
where
$\displaystyle\mathsfsl{F}_{\tau}^{2}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&-\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{I}\\\
\end{array}\right]^{-1}\cdot$ (59)
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{I}&\frac{\tau}{2}\cdot\mathsfsl{I}\\\
-\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{I}\\\
\end{array}\right]$
According to Eq.(53) and Eq.(59), the map
$\bm{z}^{k}\rightarrow\tilde{\bm{z}}^{k+1}$ must be symplectic, because
$\mathsfsl{F}_{\tau}^{2}$ must be a symplectic matrix. The proof of this point
is as follows:
###### Proof
$\displaystyle\mathsfsl{F}_{\tau}^{2}$ $\displaystyle=$
$\displaystyle\left(\mathsfsl{I}+\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&-\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{O}\end{array}\right]\right)^{-1}$
$\displaystyle\left(\mathsfsl{I}-\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&-\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{O}\end{array}\right]\right)$
Since
$\displaystyle\mathsfsl{J}\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&-\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{O}\end{array}\right]+$
$\displaystyle\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&-\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{O}\end{array}\right]^{T}\mathsfsl{J}=\mathsfsl{O}$
and according to Definition 1
$\left[\begin{array}[]{>{\displaystyle}c>{\displaystyle}c}\mathsfsl{O}&-\frac{\tau}{2}\cdot\mathsfsl{I}\\\
\frac{\tau}{2}(\mathsfsl{K}+\mathsfsl{\tilde{K}})&\mathsfsl{O}\end{array}\right]$
is an infinitesimal symplectic matrix. Therefore, according to Theorem 3.1,
$\mathsfsl{F}^{2}_{\tau}$ is a symplectic matrix.
Finally, by substituting $\bm{z}^{k}=\left[\bm{q}^{k},\bm{p}^{k}\right]$ and
Eqs.(51) into Eq.(53) and Eq.(59), we can obtain the numerical solution
$\tilde{\bm{z}}^{k+1}$. Then if we repeat the process that consists of
Eq.(37)(51)(53), we would get a series of numerical solutions for Eq.(17) that
are canonical for its substituting conservative system.
From the derivation above, we can obtain the numerical solution $\bm{z}^{k+1}$
by direct application to the dissipative system(17) and the numerical solution
$\tilde{\bm{z}}^{k+1}$ by indirect application. Although
$\mathsfsl{F}^{1}_{\tau}\neq\mathsfsl{F}^{2}_{\tau}$, from Eq.(37), Eq.(42),
Eq.(51) and Eq.(52) we can derive
$\displaystyle\left[\begin{array}[]{c}\tilde{\bm{q}}^{k+1}\\\
\tilde{\bm{p}}^{k+1}\\\ \end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}{\bm{q}}^{k+1}\\\ {\bm{p}}^{k+1}\\\
\end{array}\right]$ (68) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\dfrac{4\bm{q}^{k}-\tau^{2}\mathsfsl{K}\bm{q}^{k}+2\tau\mathsfsl{C}\bm{q}^{k}+4\tau\bm{p}^{k}}{4+\tau^{2}\mathsfsl{K}+2\tau\mathsfsl{C}}\\\
-\dfrac{4\tau\mathsfsl{K}\bm{q}^{k}+\tau^{2}\mathsfsl{K}\bm{p}^{k}+2\tau\mathsfsl{C}\bm{p}^{k}-4\bm{p}^{k}}{4+\tau^{2}\mathsfsl{K}+2\tau\mathsfsl{C}}\\\
\end{array}\right].$ (71)
According to Eq.(71), the result of the transition
$\bm{z}^{k}\rightarrow\bm{z}^{k+1}$ is exactly identical to that of
$\bm{z}^{k}\rightarrow\tilde{\bm{z}}^{k+1}$, and hence we have
$\bm{z}^{k+1}=\tilde{\bm{z}}^{k+1}$. It is a known fact, that the linear
transition matrix between vectors $\bm{z}^{k}$ and $\bm{z}^{k+1}$ should not
be unique; hence $F^{1}_{\tau}$ may not be equal to $F^{2}_{\tau}$. Therefore,
we can say that the time centered difference scheme(37) for a dissipative
mechanical system is in fact also a symplectic scheme for the substituting
conservative system. Consequently, one does not need to execute the second and
third steps in Fig. 2.
### 3.3 Numerical Examples
Figure 3: Total energy behavior of integrators for an one-dimensional damping
oscillator Figure 4: Total energy behavior of integrators for a two-
dimensional damping oscillator Figure 5: 1st displacement of a two-dimensional
damping oscillator
Although the time-centered difference scheme is used widely, the total energy
behavior of this scheme for dissipative systems has not been illustrated via
numerical examples. We now have two numerical examples to show the total
energy behavior for damping systems.
The first numerical example involves an one-dimensional damping oscillator
$\ddot{x}+2x+0.05\dot{x}=0$
with initial values $x_{0}=0.1,\ \ \dot{x}_{0}=0.2$. The time-centered scheme
with step-size=0.2 and the 4th-Runge-Kutta method with step-size=0.2 are used
to compute the one-dimensional problem. The total energy result is shown in
Fig. 3.
The second numerical example involves a two-dimensional damping oscillator
$\displaystyle\ddot{x_{1}}+3x_{1}+0.03\dot{x_{1}}-0.01\dot{x_{2}}=0$
$\displaystyle\ddot{x_{2}}+3x_{2}-0.01\dot{x_{1}}+0.01\dot{x_{2}}=0,$
with initial values $x_{1}|_{t=0}=0.1,\ \ \dot{x}_{1}|_{t=0}=0.1,\ \
x_{2}|_{t=0}=0.2,\ \ \dot{x}_{2}|_{t=0}=0.2$. The same method as earlier has
been employed to compute this problem. The numerical result of the 1st
displacement is shown in Fig.5, and the total energy result in Fig. 4.
As shown in Fig. 3 and Fig. 4, the Euler time-centered difference scheme can
preserve the total energy, the primary reason being is that the foundation of
the Euler time-centered difference scheme is a Hamilton’s system, the
Hamiltonian quantity of which is the sum of the total energy and a constant.
By comparing Fig. 4 and Fig. 5, one will find that the period of the result of
the time-centered scheme is shorter than that of the result of the Runge-Kutta
method. The reason is clear: the Runge-Kutta method would cause artificial
energy growth. This result can be considered as a generalization of the
conclusion in the paperxingyufeng2007 , which asserts that the Euler time-
centered scheme preserves the mechanical energy of conservative oscillators.
By comparing the results in this paper and the work in the paperKane2000 , we
can identify the difference. As presented in Fig. 6.1 and Fig. 6.2 in the
paper Kane2000 , ’The variational integrators simulate energy decay, unlike
standard methods such as Runge-Kutta.’, there is no accurate criterion of
numerical integration methods for dissipative system just as the accurate
criterion for conservative systems which is a horizontal line(see Fig. 4.1 and
Fig. 4.2 in the paperKane2000 ), its rationale has have been presented in
Sec.1.
## 4 conclusions
We can conclude that a dissipative mechanical system has such properties: for
any nonconservative classical mechanical system and any initial condition,
there exists a conservative one, the two systems share one and only one common
phase curve; the Hamiltonian of the conservative system is the sum of the
total energy of the nonconservative system on the aforementioned phase curve
and a constant depending on the initial condition. We can further conclude,
that a dissipative problem can be reformulated as an infinite number of non-
dissipative problems, one corresponding to each phase curve of the dissipative
problem. One can avoid having to change the definition of the canonical
momentum in the Hamilton formalism, because under a certain initial condition
the motion of one of the group of conservative systems is the same as the
original dissipative system.
From a Hamiltonian perspective, this paper has revealed that the transition
matrix of $\bm{z}^{k}\rightarrow\bm{z}^{k+1}$ is not unique, the transition
matrix may be non-symplectic or symplectic, the numerical solution of the
original dissipative obtained through the time-centered difference scheme
system is equal to that of the substituting conservative system obtained
through a time-centered difference scheme. In addition, we have discovered
that the time-centered scheme preserves the total energy of dissipative
systems. Based on the above, we might explain why the time-centered difference
scheme is more accurate than an unsymplectic one for dissipative problem. For
this reason we might be able to choose and construct better algorithms.
## References
* (1) Arnold., V.I.: Mathematical Methods of classical Mechanics, second edition. Springer-Verlag, Berlin (1978)
* (2) Arnold., V.I.: Mathematical aspects of classical and celestial mechanics. Springer-Verlag, Berlin (1997)
* (3) Feng, K.: On difference schemes and symplectic geometry. In: Ed. Feng Kang Proceeding of the 1984 Beijing Symposium on differential geometry and differential equations-computation of partial differential equations, pp. 42–58. Science Press, Beijing (1985)
* (4) Feng, K.: The hamiltonian way for computing hamiltonian dynamics. Math. Appl. 56, 17–35 (1991)
* (5) F.Klein: Entwickelung der Mathematik im 19 Jahrhundert. Teubner (1928)
* (6) Gao, Q., Zhong, W.: The symplectic and energy preserving method for the integration of hamilton system. JOURNAL OF DYNAMICS AND CONTRO 7, 193–199 (2009)
* (7) Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering 49(10), 1295–1325 (2000). URL http://dx.doi.org/10.1002/1097-0207(20001210)49:10¡1295::AID-NME993¿3.0.CO;2-W
* (8) Krechetnikov, R., Marsden, J.E.: Dissipation-induced instabilities in finite dimensions. Reviews of Modern Physics 79, 519–553 (2007). DOI 10.1103/RevModPhys.79.519
* (9) Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear pdes. Communications in Mathematical Physics 199(2), 351–395 (1998). URL www.scopus.com. Cited By (since 1996): 129
* (10) Morrison, P.: Hamiltonian fluid dynamics. In: J.P. Fran oise, G.L. Naber, , T.S. Tsun (eds.) Encyclopedia of Mathematical Physics, pp. 593–600. Academic Press, Oxford (2006). DOI DOI: 10.1016/B0-12-512666-2/00246-7. URL http://www.sciencedirect.com/science/article/B7T7D-4KF807K-7Y/2/a5086ccc96edae361e0a4c005562290a
* (11) Morrison, P.J.: Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70(2), 467–521 (1998). DOI 10.1103/RevModPhys.70.467
* (12) Salmon, R.: Hamiltonian fluid mechanics. Ann. Rev. Fluid Mechanics 20, 225–256 (1988)
* (13) Wanxie, G.Q.T.S.Z.H.Z.: Symplectic method based on dual variable principle and independent momentum at two ends. JOURNAL OF DYNAMICS AND CONTROL 7, 97–103 (2009)
* (14) Wu, H., Qin, M., Feng, K.: Construction of canonical difference schemes for hamiltonian formalism via generating functions. JCM 7(1), 71–96 (1989)
* (15) Wu, H., Qin, M., Feng, K.: Symplectic difference schemes for the linear hamiltonian canonical systems. JCM 8(4), 371–380 (1990)
* (16) Xing, Y., Yang, R.: Application of euler midpoint symplectic integration method for the solution of dynamic equilibrium equations. CHINESE JOURNAL OF THEORETICAL AND APPLIED MECHANICS 39(1) (2007)
* (17) Zhong, W., Yao, Z.: Time domain fem and symplectic conservation. Journal of Mechanical Strength 27, 178–183 (2005)
|
arxiv-papers
| 2010-07-16T07:32:20 |
2024-09-04T02:49:11.653187
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tianshu Luo and Yimu Guo",
"submitter": "Tianshu Luo",
"url": "https://arxiv.org/abs/1007.2709"
}
|
1007.2772
|
New Examples of Simple Jordan Superalgebras
over an Arbitrary Field of Characteristic Zero
V. N. Zhelyabin
00footnotetext: RFBR 09-01-00157, SB RAS grant, Development of Scientific
Potential of Higher School (Grant 2.1.1.419), Federal Program “Scientific and
Pedagogical Staff of Innovative Russia” 2009-2013 (State Contract No.
02.740.11.0429).
Abstract: An new example of a unital simple special Jordan superalgebra over
the field of real numbers was constructed in [10]. It turned out to be a
subsuperalgebra of the Jordan superalgebra of vector type $J(\Gamma,D)$, but
not isomorphic to a superalgebra of this type. Moreover, its superalgebra of
fractions is isomorphic to a Jordan superalgebra of vector type. A similar
example of a Jordan superalgebra over a field of characteristic 0 in which the
equation $t^{2}+1=0$ has no solutions was constructed in [12]. In this article
we present an example of a Jordan superalgebra with the same properties over
an arbitrary field of characteristic 0. A similar example of a superalgebra is
found in the Cheng–Kac superalgebra.
Keywords: Jordan superalgebra, $(-1,1)$-superalgebra, superalgebra of vector
type, differentially simple algebra, polynomial algebra, projective module
Jordan algebras and superalgebras constitute an important class of algebras in
ring theory. Simple Jordan superalgebras are studied in [1, 2, 3, 4, 5, 6, 7,
8].
The unital simple special Jordan superalgebras with the associative even part
$A$ and the odd part $M$ which is an associative $A$-module were described in
[9, 10]. The study in [9] was considerably influenced by [11], which described
the simple $(-1,1)$-superalgebras of characteristic $\neq 2,3$. In the Jordan
case, if a superalgebra is not the superalgebra of a nondegenerate bilinear
superform, then its even part $A$ is a differentially simple algebra with
respect to some set of derivations, and its odd part $M$ is a finitely
generated projective $A$-module of rank 1. Here, as for
$(-1,1)$-superalgebras, we define multiplication in $M$ using fixed finite
sets of derivations and elements of $A$. It turns out that every Jordan
superalgebra of this type is a subsuperalgebra of the superalgebra of vector
type $J(\Gamma,D)$. Under certain restrictions on $A$ the odd part $M$ is a
cyclic $A$-module, and consequently, the original Jordan superalgebra is
isomorphic to the superalgebra $J(\Gamma,D)$. For instance, if $A$ is a local
algebra then by the well-known Kaplansky theorem $M$ is free, and
consequently, it is a cyclic $A$-module. If the ground field is of
characteristic $p>2$ then [13] implies that $A$ is a local algebra; thus, $M$
is a cyclic $A$-module. If $A$ is the ring of polynomials in finitely many
variables then $M$ is free by [14], and consequently, it is a cyclic
$A$-module.
A natural question arose: is the original superalgebra isomorphic to
$J(\Gamma,D)$? Equivalently, is the odd part $M$ a cyclic $A$-module? Examples
are constructed in [10, 12] of unital simple special Jordan superalgebras with
certain associative even part and the odd part $M$ which is not free, i.e.,
not cyclic. In those examples the ground field is either the field of real
numbers or an arbitrary field of characteristic 0 in which the equation
$t^{2}+1=0$ has no solutions.
In this article we construct a similar example of a Jordan superalgebra over
an arbitrary field of characteristic 0, as well as an example of a simple
Jordan superalgebra which is a subsuperalgebra of the Cheng–Kac Jordan
superalgebra. Examples of these superalgebras answer a question of Cantarini
and Kac [8].
Take a field $F$ of characteristic not equal to 2. A superalgebra
$J=J_{0}+J_{1}$ is a ${\rm Z_{2}}$-graded $F$-algebra:
$J_{0}^{2}\subseteq J_{0},J_{1}^{2}\subseteq J_{0},J_{1}J_{0}\subseteq
J_{1},J_{0}J_{1}\subseteq J_{1}.$
Put $A=J_{0}$ and $M=J_{1}$. The spaces $A$ and $M$ are called the even and
odd parts of $J$. The elements of $A\cup M$ are called homogeneous. The
expression $p(x)$ with $x\in A\cup M$ means the parity of $x$: $p(x)=0$ for
$x\in A$ ($x$ is even) and $p(x)=1$ for $x\in M$ ($x$ is odd).
Given $x$ in $J$ denote by $R_{x}$ the operator of right multiplication by
$x$. A superalgebra $J$ is called a Jordan superalgebra if the homogeneous
elements satisfy the operator identities
$\displaystyle aR_{b}=(-1)^{p(a)p(b)}bR_{a},$ (1) $\displaystyle
R_{a^{2}}R_{a}=R_{a}R_{a^{2}},$ (2) $\displaystyle
R_{a}R_{b}R_{c}+(-1)^{p(a)p(b)+p(a)p(c)+p(b)p(c)}R_{c}R_{b}R_{a}+(-1)^{p(b)p(c)}R_{(ac)b}=$
$\displaystyle\phantom{R_{b}}R_{a}R_{bc}+(-1)^{p(a)p(b)}R_{b}R_{ac}+(-1)^{p(a)p(c)+p(b)p(c)}R_{c}R_{ab}.$
(3)
In every Jordan superalgebra, the homogeneous elements satisfy
$\displaystyle(x,tz,y)=(-1)^{p(x)p(t)}t(x,z,y)+(-1)^{p(y)p(z)}(x,t,y)z,$ (4)
where $(x,z,y)=(xz)y-x(zy)$ is the associator of $x$, $z$, and $y$.
Let us give some examples of Jordan superalgebras.
Take an associative $Z_{2}$-graded algebra $B=B_{0}+B_{1}$ with multiplication
$\ast$. Defining on the space $B$ the supersymmetric product
$a\circ_{s}b=\frac{1}{2}(a\ast b+(-1)^{p(a)p(b)}b\ast a),\mbox{ \ \ \ }a,b\in
B_{0}\cup B_{1},$
we obtain the Jordan superalgebra $B^{(+)s}$. A Jordan superalgebra $J=A+M$ is
called special whenever it embeds (as a $Z_{2}$-graded algebra) in the
superalgebra $B^{(+)s}$ for a suitable $Z_{2}$-graded associative algebra $B$.
The superalgebra of vector type $J(\Gamma,D)$. Take a commutative associative
$F$-algebra $\Gamma$ equipped with a nonzero derivation $D$. Denote by
$\overline{\Gamma}$ an isomorphic copy of the linear space $\Gamma$, and a
fixed isomorphism, by $a\mapsto\overline{a}$. On the direct sum
$J(\Gamma,D)=\Gamma+\overline{\Gamma}$ of linear spaces define a
multiplication ($\cdot$) as
$a\cdot b=ab,\mbox{ \ \ \ }a\cdot\overline{b}=\overline{ab},\mbox{ \ \
}\overline{a}\cdot b=\overline{ab},\mbox{ \ \ \
}\overline{a}\cdot\overline{b}=D(a)b-aD(b),$
where $a,b\in\Gamma$ and $ab$ is the product in $\Gamma$. Then $J(\Gamma,D)$
is a Jordan superalgebra with the even part $A=\Gamma$ and the odd part
$M=\overline{\Gamma}$. The superalgebra $J(\Gamma,D)$ is simple if and only if
$\Gamma$ is a $D$-simple algebra [15] (i.e., $\Gamma$ contains no proper
nonzero $D$-invariant ideals, and $\Gamma^{2}=\Gamma$).
Consider the associative superalgebra $B=M_{2}^{1,1}({\rm End}\,\Gamma)$ with
the even part
$B_{0}=\Big{\\{}\left(\begin{array}[]{cc}\phi&0\\\
0&\psi\end{array}\right),\text{ where }\phi,\psi\in{\rm End}\,\Gamma\Big{\\}}$
and the odd part
$B_{1}=\Big{\\{}\left(\begin{array}[]{cc}0&\phi\\\
\psi&0\end{array}\right)\text{ where }\phi,\psi\in{\rm End}\,\Gamma\Big{\\}}.$
It is shown in [16] that the mapping
$a+\overline{b}\mapsto\left(\begin{array}[]{cc}\text{\ \
}R_{a}&4R_{b}D+2R_{D(b)}\\\ -R_{b}&R_{a}\end{array}\right)$
is an embedding of $J(\Gamma,D)$ into $B^{(+)s}$. Consequently, the Jordan
superalgebra $J(\Gamma,D)$ is special.
The Kantor double $J(\Gamma,\\{\,,\\})$. Take an associative supercommutative
superalgebra $\Gamma=\Gamma_{0}+\Gamma_{1}$ with unit 1 equipped with a super-
skew-symmetric bilinear mapping $\\{\,,\\}:\Gamma\mapsto\Gamma$, which we call
the bracket. From $\Gamma$ and $\\{\,,\\}$ we can construct a superalgebra
$J(\Gamma,\\{\,,\\})$ as follows. Consider the direct sum
$J(\Gamma,\\{\,,\\})=\Gamma\oplus\Gamma x$ of linear spaces, where $\Gamma x$
is an isomorphic copy of $\Gamma$. Take two homogeneous elements $a$ and $b$
of $\Gamma$. The multiplication ($\cdot$) on $J(\Gamma,\\{\,,\\})$ is defined
as
$a\cdot b=ab,\mbox{ \ \ \ }a\cdot bx=(ab)x,\mbox{ \ \ \ }ax\cdot
b=(-1)^{p(b)}(ab)x,\mbox{ \ \ \ }ax\cdot bx=(-1)^{p(b)}\\{a,b\\}.$
Put $A=\Gamma_{0}+\Gamma_{1}x$ and $M=\Gamma_{1}+\Gamma_{0}x$. Then
$J(\Gamma,\\{\,,\\})=A+M$ is a $Z_{2}$-graded algebra.
Refer to $\\{\,,\\}$ as a Jordan bracket if $J(\Gamma,\\{\,,\\})$ is a Jordan
superalgebra. It is known (see [17]) that $\\{\,,\\}$ is a Jordan bracket if
and only if it satisfies
$\displaystyle\\{a,bc\\}=\\{a,b\\}c+(-1)^{p(a)p(b)}b\\{a,c\\}-\\{a,1\\}bc,$
(5)
$\displaystyle\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+(-1)^{p(a)p(b)}\\{b,\\{a,c\\}\\}+\\{a,1\\}\\{b,c\\}+$
$\displaystyle(-1)^{p(a)(p(b)+p(c))}\\{b,1\\}\\{c,a\\}+(-1)^{p(c)(p(a)+p(b))}\\{c,1\\}\\{a,b\\},$
(6) $\displaystyle\\{d,\\{d,d\\}\\}=\\{d,d\\}\\{d,1\\},$ (7)
where $a,b,c\in\Gamma_{0}\cup\Gamma_{1}$, and $d\in\Gamma_{1}$.
In particular, $J(\Gamma,D)$ is the algebra $J(\Gamma,\\{\,,\\})$ if
$\\{a,b\\}=D(a)b-aD(b).$
The next theorem is proved in [10].
Theorem. Take a simple special unital Jordan superalgebra $J=A+M$ whose even
part $A$ is an associative algebra, and whose odd part $M$ is an associative
$A$-module. If $J$ is not the superalgebra of a nondegenerate bilinear
superform then there exist $x_{1},\ldots,x_{n}\in M$ such that
$M=x_{1}A+\ldots+x_{n}A,$
and the product in $M$ satisfies
$\displaystyle ax_{i}\cdot bx_{j}=\gamma_{ij}ab+D_{ij}(a)b-aD_{ji}(b),\mbox{ \
\ }i,j=1,\ldots,n,$ (8)
where $\gamma_{ij}\in A$, and $D_{ij}$ is a derivation of $A$. The algebra $A$
is differentially simple with respect to the set of derivations
$\Delta\\{D_{ij}|i,j=1,\ldots,n\\}$. The module $M$ is a projective $A$-module
of rank 1. Moreover, $J$ is a subalgebra of the superalgebra $J(\Gamma,D)$.
In addition, [10] includes an example of a Jordan superalgebra over the field
of real numbers satisfying the hypotheses of the theorem which is not
isomorphic to $J(\Gamma,D)$. A similar example of a Jordan superalgebra over a
field of characteristic zero in which the equation $t^{2}+1=0$ has no
solutions is constructed in [12]. Let us give another example of this kind of
superalgebra over an arbitrary field of characteristic zero.
Fix an arbitrary field $F$ of characteristic 0. Consider the polynomial
algebra $F[x,y]$ in two variables $x$ and $y$. Denote by
$\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ the operators
of differentiation with respect to $x$ and $y$ on $F[x,y]$. Put
$D=2y^{3}\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}$ and
$f(x,y)=x^{2}+y^{4}-1$. Then $D$ is a derivation of $F[x,y]$, and
$D(f(x,y))=0$. Take the quotient algebra $\Gamma=F[x,y]/f(x,y)F[x,y]$ of
$F[x,y]$ by the ideal $f(x,y)F[x,y]$. It is clear that $D$ induces a
derivation of $\Gamma$, which we denote by $D$ as well. Identify the images of
$x$ and $y$ under the canonical homomorphism $F[x,y]\mapsto\Gamma$ with the
elements $x$ and $y$. Then $\Gamma=F[y]+xF[y]$, where $F[y]$ is the polynomial
ring in $y$.
Proposition 1. The algebra $\Gamma$ is differentially simple with respect to
$D$.
Proof. Suppose that $I$ is a nonzero $D$-invariant ideal of $\Gamma$. If
$f(y)\in F[y]$ and $f(y)\in I$ then $D(f(y))=-xf^{\prime}(y)\in I$, where
$f^{\prime}(y)$ is the derivative of $f(y)$ with respect to $y$. Then
$(1-y^{4})f^{\prime}(y)\in I$ and $D((1-y^{4})f^{\prime}(y))\in I$. Thus,
$-x(-4y^{3}f^{\prime}(y)+(1-y^{4})f^{\prime\prime}(y))\in I.$
This implies that $(1-y^{4})^{2}f^{\prime\prime}(y)\in I$. Continuing this
process, we deduce that $(1-y^{4})^{k}f^{(k)}(y)\in I$ for all $k$, where
$f^{(k)}(y)$ is the order $k$ derivative of $f(y)$. Consequently,
$(1-y^{4})^{k}\in I$ for some $k$. Take the smallest $k$ with
$z_{k}=(1-y^{4})^{k}\in I$. Then
$D(z_{k})=4kxy^{3}(1-y^{4})^{k-1}\in I.$
Thus,
$x(1-y^{4})^{k-1}=xz_{k}+\frac{1}{4k}yD(z_{k})\in I.$
Consequently,
$D(x(1-y^{4})^{k-1})=2y^{3}(1-y^{4})^{k-1}+(k-1)4y^{3}(1-y^{4})^{k-1}2(2k-1)y^{3}(1-y^{4})^{k-1}\in
I.$
This implies that $y^{3}(1-y^{4})^{k-1}\in I$ and $y^{4}(1-y^{4})^{k-1}\in I$.
Then,
$z_{k-1}=(1-y^{4})^{k}+y^{4}(1-y^{4})^{k-1}\in I.$
Therefore, we may assume that $F[y]\cap I=0$.
Suppose that $f(y)+xg(y)\in I$. Then
$(f(y)+xg(y))(f(y)-xg(y))=f(y)^{2}-(1-y^{4})g(y)^{2}\in I.$
By the argument above, $f(y)^{2}=(1-y^{4})g(y)^{2}$. Then, $1-y^{4}=h(y)^{2}$
for some $h(y)\in F[y]$, and we arrive at a contradiction.
Consequently, $\Gamma$ is a differentially simple algebra with respect to $D$.
$\Box$
Consider in $\Gamma$ the subalgebra $A$ generated by $1$, $y^{2}$, and $xy$.
Then,
$D(y^{2})=-2xy\in A\text{ and }D(xy)=3y^{4}-1\in A.$
Consequently, $D(A)\subseteq A$. Observe that $1,y^{2i},xy^{2i-1}$, where
$i=1,2,\dots$, constitute a linear basis for $A$. We can express every element
of $A$ as $f(y)+xyg(y)$ with $f(y),g(y)\in F[y^{2}]$.
Proposition 2. The algebra $A$ is differentially simple with respect to $D$.
Proof. Suppose that $I$ is a nonzero $D$-invariant ideal of $A$. If $f(y)\in
F[y^{2}]$ and $f(y)\in I$ then $xf^{\prime}(y)=-D(f(y))\in I$. Thus,
$(1-y^{4})yf^{\prime}(y)=(xy)(xf^{\prime}(y))\in I$. Since
$D(xf^{\prime}(y))=2y^{3}f^{\prime}(y)-(1-y^{4})f^{\prime\prime}(y)\in I,$
it follows that $(1-y^{4})^{2}f^{\prime\prime}(y)\in I$. An easy induction
implies that
$(1-y^{4})^{2k-1}yf^{(2k-1)}(y)\in I\text{\ \ and \ \
}(1-y^{4})^{2k}f^{(2k)}(y)\in I.$
This yields $(1-y^{4})^{2k}\in I$.
Take the smallest $k$ with $(1-y^{4})^{k}\in I$. Then,
$D((1-y^{4})^{k})=-4kxy^{3}(1-y^{4})^{k-1}\in I.$
Consequently,
$xy(1-y^{4})^{k-1}=xy(1-y^{4})^{k}+y^{2}(xy^{3}(1-y^{4})^{k-1})\in I.$
Thus,
$D(xy(1-y^{4})^{k-1})=(3y^{4}-1)(1-y^{4})^{k-1}+(k-1)4y^{4}(1-y^{4})^{k-1}((4k-1)y^{4}-1)(1-y^{4})^{k-1}\in
I.$
Then,
$(4k-2)(1-y^{4})^{k-1}=(4k-1)(1-y^{4})^{k}+((4k-1)y^{4}-1)(1-y^{4})^{k-1}\in
I.$
Therefore, we may assume that $F[y^{2}]\cap I=0$.
Suppose that $f(y)+xyg(y)\in I$. Then,
$f(y)^{2}-(1-y^{4})y^{2}g(y)^{2}=(f(y)+xyg(y))(f(y)-xyg(y))\in I.$
By the argument above, $f(y)^{2}-(1-y^{4})y^{2}g(y)^{2}=0$, and we arrive at a
contradiction since $\deg f(y)^{2}=4n$ but $\deg(1-y^{4})y^{2}g(y)^{2}=4m+6$.
Therefore, $A$ is a differentially simple algebra with respect to $D$. $\Box$
The subspace $M=xA+yA$ of $\Gamma$ is an associative $A$-module.
Proposition 3. The module $M$ is not a cyclic $A$-module.
Proof. Assuming the contrary, denote the generator of $M$ by $z$. Then
$z=xa+yb$ with $a,b\in A$, $x=zc$, and $y=zd$ for some $c,d\in A$. This
implies that
$xd=yc,$ (9) $x=x(ac+bd),y=y(ac+bd).$ (10)
We can write
$a=f_{0}+xyf_{1},b=g_{0}+xyg_{1},c=e_{0}+xye_{1},d=h_{0}+xyh_{1},$
where $f_{0},f_{1},g_{0},g_{1},e_{0},e_{1},h_{0},h_{1}$ are polynomials in
$F[y^{2}]$.
From (9) we deduce that
$h_{0}=y^{2}e_{1}\text{ \ \ and \ \ }e_{0}=(1-y^{4})h_{1}.$
From (10) we deduce that
$f_{0}e_{0}+(1-y^{4})y^{2}f_{1}e_{1}+g_{0}h_{0}+(1-y^{4})y^{2}g_{1}h_{1}=1,$
(11) $f_{0}e_{1}+f_{1}e_{0}+g_{0}h_{1}+g_{1}h_{0}=0.$ (12)
Denote by $(e_{1},h_{1})$ the greatest common divisor of $e_{1}$ and $h_{1}$.
Since $h_{0}=y^{2}e_{1}$ and $e_{0}=(1-y^{4})h_{1}$, by (11) we have
$1=(1-y^{4})f_{0}h_{1}+(1-y^{4})y^{2}f_{1}e_{1}+y^{2}g_{0}e_{1}+(1-y^{4})y^{2}g_{1}h_{1}=$
$(1-y^{4})(f_{0}+y^{2}g_{1})h_{1}+y^{2}((1-y^{4})f_{1}+g_{0})e_{1}.$
Consequently, $(e_{1},h_{1})=1$. By (12),
$(f_{0}+y^{2}g_{1})e_{1}+((1-y^{4})f_{1}+g_{0})h_{1}=0.$
This and $(e_{1},h_{1})=1$ imply that $f_{0}+y^{2}g_{1}=h_{1}u$, where $u\in
F[y]$. Then,
$h_{1}ue_{1}+((1-y^{4})f_{1}+g_{0})h_{1}=0.$
Thus,
$ue_{1}+((1-y^{4})f_{1}+g_{0})=0.$
By the argument above,
$1=(1-y^{4})(f_{0}+y^{2}g_{1})h_{1}+y^{2}((1-y^{4})f_{1}+g_{0})e_{1}=(1-y^{4})h_{1}^{2}u-y^{2}e_{1}^{2}u.$
Then, $u\in F$. Consequently,
$(1-y^{4})h_{1}^{2}u=1+y^{2}e_{1}^{2}u,$
which is impossible since on the left we have a polynomial of degree $4k+4$,
while on the right, of degree $4m+2$.
Therefore, $M$ is not a cyclic $A$-module. $\Box$
Put
$D_{11}=(1-y^{4})D,D_{12}=xyD,D_{22}=y^{2}D.$
Then $D_{11},D_{12},D_{22}$ are derivations of $A$.
Proposition 4. The algebra $A$ is differentially simple with respect to the
set of derivations $\Delta=\\{D_{11},D_{12},D_{22}\\}$.
Proof. Suppose that $I$ is an ideal of $A$ closed under $\Delta$. Then
$y^{2}D_{22}(I)\subseteq y^{2}I\subseteq I$. Since
$D=D_{11}+y^{2}D_{22},$
it follows that $D(I)\subseteq I$. By Proposition 2, either $I=0$ or $I=A$.
Consequently, $A$ is a differentially simple algebra with respect to
$\Delta=\\{D_{11},D_{12},D_{22}\\}$. $\Box$
Consider now the superalgebra $J(\Gamma,D)$. Proposition 1 implies that
$J(\Gamma,D)$ is a simple superalgebra. Consider its subspace
$J(A,\Delta)=A+\overline{M}.$
Recall that $A$ is the subalgebra of $\Gamma$ generated by $1$, $y^{2}$, and
$xy$, while $M=xA+yA$.
Given $a,b\in A$, in $J(\Gamma,D)$ we have
$\overline{xa}\cdot\overline{xb}=D(xa)xb-D(xb)xa=$
$D(x)axb+D(a)x^{2}b-D(x)xab-D(b)x^{2}a=D_{11}(a)b-aD_{11}(b)\in A.$
Similarly,
$\overline{ya}\cdot\overline{yb}=D(y)ayb+D(a)y^{2}b-D(y)yab-D(b)y^{2}a=D_{22}(a)b-aD_{22}(b)\in
A,$
$\overline{xa}\cdot\overline{yb}=D(x)ayb+D(a)xyb-D(y)xab-D(b)yxa=(1+y^{4})ab+D_{12}(a)b-aD_{12}(b)\in
A.$
Consequently, $J(A,\Delta)$ is a subsuperalgebra of $J(\Gamma,D)$. Thus,
$J(A,\Delta)$ is a Jordan superalgebra. Moreover, the odd elements in
$J(\Gamma,D)$ multiply according to (8), where
$\Delta=\\{D_{11},D_{12},D_{22}\\}$, and $\gamma_{12}=1+y^{4}$. By Proposition
3, $J(A,\Delta)$ is not isomorphic to a superalgebra of type
$J(\Gamma_{0},D_{0})$.
Verify that $J(A,\Delta)$ is a simple superalgebra. Suppose that $I$ is a
nonzero ${\rm Z_{2}}$-graded ideal of $J(A,\Delta)$. Then $I=I_{0}+I_{1}$,
where $I_{0}$ is an ideal of $A$. Given $r\in I_{0}$, we have
$D_{11}(r)=\overline{(xr)}\cdot\overline{x}=(r\cdot\overline{x})\cdot\overline{x}\in
I_{0}.$
Similarly, $D_{12}(r),D_{22}(r)\in I_{0}$. Consequently, $I_{0}$ is invariant
under the set of derivations $\Delta$. By Proposition 4, either $I_{0}=A$ or
$I_{0}=0$. If $I_{0}=A$ then $1\in I_{0}\subseteq I$ and $I=J(A,\Delta)$. If
$I_{0}=0$ then $I\subseteq\overline{M}$ and $I\cdot\overline{M}\subseteq
I_{0}=0$. It is clear that
$A=AD_{11}(A)+AD_{12}(A)+AD_{22}(A).$
Thus,
$1=\sum_{i}(a_{1i},\overline{x},\overline{x})b_{1i}+\sum_{i}(a_{2i},\overline{x},\overline{y})b_{2i}+\sum_{i}(a_{3i},\overline{y},\overline{y})b_{3i}$
for some elements $a_{1i}$, $a_{2i}$, $a_{3i}$, $b_{1i}$, $b_{2i}$, and
$b_{3i}$ of $A$. By (4) we deduce that $1\in(A,\overline{M},\overline{M})$ and
$I\cdot(A,\overline{M},\overline{M})\subseteq(A,I\cdot\overline{M},\overline{M})+(A,I,\overline{M})\cdot\overline{M}=0.$
Then, $I=0$. Consequently, $J(A,\Delta)$ is a simple superalgebra.
Let us summarize the argument as
Theorem 1. Take an arbitrary field $F$ of characteristic 0. Consider the
polynomial algebra $F[x,y]$ in two variables $x$ and $y$. Put
$f(x,y)=x^{2}+y^{4}-1$ and $D=2y^{3}\frac{\partial}{\partial
x}-x\frac{\partial}{\partial y}$. Put $\Gamma=F[x,y]/f(x,y)F[x,y]$. Then the
derivation $D$ induces a derivation of the algebra $\Gamma$, which we denote
by $D$ as well. Identify the images of $x$ and $y$ under the canonical
homomorphism $F[x,y]\mapsto\Gamma$ with the elements $x$ and $y$. Suppose that
$A$ is a subalgebra of $\Gamma$ generated by $1$, $y^{2}$, and $xy$, while
$M=xA+yA$. Put
$\Delta=\\{D_{11},D_{12},D_{22}\\},\text{ where
}D_{11}=(1-y^{4})D,D_{12}=xyD,D_{22}=y^{2}D.$
Then the subspace $J(A,\Delta)=A+\overline{M}$ is a subsuperalgebra of
$J(\Gamma,D)$, and the multiplication of odd elements in $J(A,\Delta)$ is
defined as
$\overline{xa}\cdot\overline{xb}=D_{11}(a)b-aD_{11}(b),\text{ \ \
}\overline{ya}\cdot\overline{yb}=D_{22}(a)b-aD_{22}(b),$
$\overline{xa}\cdot\overline{yb}=(1+y^{4})ab+D_{12}(a)b-aD_{12}(b).$
Moreover, $J(A,\Delta)$ is a simple superalgebra, and $\overline{M}$ is not a
cyclic $A$-module; i.e., $J(A,\Delta)$ is not isomorphic to a superalgebra of
vector type $J(\Gamma_{0},D_{0})$.
The Superalgebra of Type $JS(\Gamma,D)$. Take an associative supercommutative
superalgebra $\Gamma=\Gamma_{0}+\Gamma_{1}$ equipped with a nonzero odd
derivation $D$; i.e., $D(\Gamma_{i})\subseteq\Gamma_{(i+1)mod\,2}$ and
$D(ab)=D(a)b+(-1)^{p(a)}aD(b)$
for $a,b\in\Gamma_{0}\cup\Gamma_{1}$.
Put $A=\Gamma_{1}$, $M=\Gamma_{0}$, and $JS(\Gamma,D)=A+M$. Define on the
space $JS(\Gamma,D)$ the multiplication
$a\circ b=aD(b)+(-1)^{p(a)}D(a)b.$
Then $JS(\Gamma,D)$ is a Jordan superalgebra. If $JS(\Gamma,D)$ is a simple
superalgebra then $\Gamma$ is a differentially simple superalgebra (see [8]).
Proposition 5. The superalgebra $JS(\Gamma,D)$ is not unital.
Proof. Suppose that $e$ is the unit of $JS(\Gamma,D)$. Then $e\in
A\subseteq\Gamma_{1}$. Given $a\in JS(\Gamma,D)$, we have
$a=e\circ a=eD(a)+D(e)a.$
Since $\Gamma$ is supercommutative and $e\in\Gamma_{1}$, it follows that
$e=2eD(e)$ and $e^{2}=0$ in $\Gamma$. Consequently, $ea=eD(e)a=\frac{1}{2}ea$.
This implies that $e\Gamma=0$. Then, $e=2eD(e)=0$. $\Box$
Corollary 1. The superalgebra $J(A,\Delta)$ is not isomorphic to the
superalgebra $JS(\Gamma,D)$.
The Cheng–Kac superalgebra. Take an associative commutative $F$-algebra
$\Gamma$ equipped with a nonzero derivation $D$. Consider two direct sums
$J_{0}=\Gamma+w_{1}\Gamma+w_{2}\Gamma+w_{3}\Gamma$
and
$J_{1}=\overline{\Gamma}+x_{1}\overline{\Gamma}+x_{2}\overline{\Gamma}+x_{3}\overline{\Gamma}$
of linear spaces, where $\overline{\Gamma}$ is an isomorphic copy of $\Gamma$.
For $a,b\in\Gamma$ define a multiplication on the space $J_{0}$ by putting
$a\cdot b=ab,\,a\cdot w_{i}b=w_{i}ab,\,w_{1}a\cdot w_{1}b=w_{2}a\cdot
w_{2}b=ab,\,w_{3}a\cdot w_{3}b=-ab,$ $w_{i}a\cdot w_{j}b=0\text{ for }i\neq
j.$
Put $x_{i\times i}=0$, $x_{1\times 2}=-x_{2\times 1}=x_{3}$, $x_{1\times
3}=-x_{3\times 1}=x_{2}$, and $x_{2\times 3}=-x_{3\times 2}=-x_{1}$. Define a
bimodule action $J_{0}\times J_{1}\mapsto J_{1}$ by putting
$a\cdot\overline{b}=\overline{ab},\,a\cdot
x_{i}\overline{b}=x_{i}\overline{ab},\,w_{i}a\cdot\overline{b}=x_{i}\overline{D(a)b},\,w_{i}a\cdot
x_{j}\overline{b}=x_{i\times j}\overline{ab}.$
The bracket on $J_{1}$ is defined as
$\overline{a}\cdot\overline{b}=D(a)b-aD(b),\,\overline{a}\cdot
x_{i}\overline{b}=-w_{i}(ab),\,x_{i}\overline{a}\cdot\overline{b}=w_{i}(ab),\,x_{i}\overline{a}\cdot
x_{j}\overline{b}=0.$
Then the space $J=J_{0}+J_{1}$ with the multiplication
$(a_{0}+a_{1})\cdot(b_{0}+b_{1})=(a_{0}\cdot b_{0}+a_{1}\cdot
b_{1})+(a_{0}\cdot b_{1}+a_{1}\cdot b_{0})$
for $a_{0},b_{0}\in J_{0}$ and $a_{1},b_{1}\in J_{1}$ is an algebra, which is
denoted by $CK(\Gamma,D)$. It is known (see [5, 8]) that $CK(\Gamma,D)$ is a
Jordan superalgebra, which is simple if and only if $\Gamma$ is $D$-simple.
Suppose now that $\Gamma=F[x,y]/f(x,y)F[x,y]$, where $f(x,y)=x^{2}+y^{4}-1$
and $D=2y^{3}\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}$.
Consider the Jordan superalgebra $J(A,\Delta)=A+\overline{M}$ constructed
above. In $CK(\Gamma,D)$ consider the subspace
$GCK(A,\Delta)=A+w_{1}A+w_{2}A+w_{3}A+\overline{M}+x_{1}\overline{M}+x_{2}\overline{M}+x_{3}\overline{M}.$
In $\Gamma$ we have $M^{2}\subseteq A$. Thus, $GCK(A,\Delta)$ is a
subsuperalgebra of $CK(\Gamma,D)$. Consequently, $GCK(A,\Delta)$ is a Jordan
superalgebra with the even part $GCK(A,\Delta)_{0}=A+w_{1}A+w_{2}A+w_{3}A$ and
the odd part
$GCK(A,\Delta)_{1}=\overline{M}+x_{1}\overline{M}+x_{2}\overline{M}+x_{3}\overline{M}$.
Theorem 2. For an arbitrary field $F$ of characteristic zero $GCK(A,\Delta)$
is a simple unital Jordan superalgebra.
Proof. Suppose that $I=I_{0}+I_{1}$ is a nonzero ideal of $GCK(A,\Delta)$.
Then $K=A\cap I_{0}$ is an ideal of $A$, and
$(K,\overline{M},\overline{M})\subseteq K$. Thus, $K+K\cdot\overline{M}$ is an
ideal of $J(A,\Delta)$. If $K\neq 0$ then since $J(A,\Delta)$ is a simple
superalgebra, we have $1\in K$. Consequently, $I=GCK(A,\Delta)$.
Suppose that $A\cap I_{0}=0$ and take $r=a+w_{1}a_{1}+w_{2}a_{2}+w_{3}a_{3}\in
I_{0}$. Then $w_{2}(w_{2}(w_{1}r))=a_{1}\in A\cap I_{0}$. Consequently,
$a_{1}=0$. Similarly, $a_{2}=a_{3}=0$. Thus, $I_{0}=0$. This implies that
$I\subseteq GCK(A,\Delta)_{1}$ and $I\cdot GCK(A,\Delta)_{1}\subseteq
I_{0}=0$. Since $1\in(A,\overline{M},\overline{M})$, by (4) we deduce that
$I\cdot(A,\overline{M},\overline{M})\subseteq(A,I\cdot\overline{M},\overline{M})+(A,I,\overline{M})\cdot\overline{M}=0.$
Then, $I=0$. Consequently, $GCK(A,\Delta)$ is a simple superalgebra. $\Box$
I would like to take this chance to express by special gratitude to A. P.
Pozhidaev, whose comments helped to improve this article.
## References
* [1] V. G. Kac, Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras // Comm. in Algebra 5, 1375-1400, (1977).
* [2] I. L. Kantor, Jordan and Lie superalgebras determined by a Poisson algebra // The 2nd Siberian school ”Algebra and Analysis”, Tomsk (1989), 55-80.
* [3] I. P. Shestakov, Prime alternative superalgebras of arbitrary characteristic // Algebra and Logic, 36, N 6 (1997), 701-731.
* [4] E. Zelmanov, Semisimple finite dimensional Jordan superalgebras // in: Y. Fong, A.A. Mihalev, E. Zelmanov (Eds.), Lie Algebras and Related Topics, Springer, New York, (2000), 227-243.
* [5] C. Martinez and E. Zelmanov, Simple finite dimesional Jordan superalgebras of Prime Characteristic // Journal of Algebra v. 236, N 2, (2001), 575-629.
* [6] V. G. Kac, C. Martinez, E. Zelmanov, Graded simple Jordan superalgebras of growth one // Mem. Amer. Math. Soc. 711 (2001).
* [7] M. Racine and E. Zelmanov, Simple Jordan superalgebras with semisimple even part // Journal of Algebra v. 270, N 2, ( 2003), 374-444.
* [8] N. Cantarini, V. G.Kac, Classification of linearly compact simple Jordan and generalized Poisson superalgebras // Journal of Algebra v. 313, N 2, (2007), 100-124.
* [9] V. N. Zhelyabin, Simple special Jordan superalgebras with associative nil-semisimple even part // Algebra and Logic 41, 3 (2002), 276-310.
* [10] V. N. Zhelyabin, I. P. Shestakov, Simple special superalgebras with associative even part // Sib. Math. J., 45(5)(2004), 1046-1072.
* [11] I. P. Shestakov, Simple superalgebras of type $(-1,1)$ // Algebra and Logic 37, 6 (1998), 721-739.
* [12] V. N. Zhelyabin, Differential algebras and simple Jordan superalgebras // Mat. Tr. 12 (2009), no. 2, 41-51.
* [13] Shuen Yuan, Differentiable simple rings of prime characteristic // Duke Math. J.,V.31, N 4 (1964), 623-630.
* [14] A. A. Suslin, On the structure of the special linear group over polynomial rings // USSR Acad. Sci. Izvestiya, ser. Math. . 41, N 2, (1977), 235-252.
* [15] D. King and K. McCrimmon, The Kantor construction of Jordan superalgebras // Comm. in Algebra 20(1)(1992), 109-126.
* [16] K. McCrimmon, Speciality and nonspeciality of two Jordan superalgebras, // J. of Algebra 149(1992), 326-351.
* [17] D. King and K. McCrimmon, The Kantor doubling process revisited // Comm. in Algebra 23(1)(1995), 357-372.
ZHELYABIN Viktor Nikolaevich,
Sobolev Institute of Mathematics, RAS
4 Acad. Koptyug prospekt
Novosibirsk 630090
RUSSIA
phone +7(383)(363-45-57)
email: vicnic@math.nsc.ru
and
Novosibirsk State University
2 Pirogova str.
|
arxiv-papers
| 2010-07-16T13:25:25 |
2024-09-04T02:49:11.662562
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.N. Zhelyabin",
"submitter": "Victor Zhelyabin",
"url": "https://arxiv.org/abs/1007.2772"
}
|
1007.2828
|
# Swift Observations of the Be/X-ray Transient System 1A 1118–615
Lupin Chun-Che Lin1, Jumpei Takata2, Albert K. H. Kong3,4 and Chorng-Yuan
Hwang1
1Graduate Institute of Astronomy, National Central University, Jhongli 32001,
Taiwan
2Department of Physics, University of Hong Kong, PRC
3Institute of Astronomy and Department of Physics, National Tsing Hua
University, Hsinchu 30013, Taiwan
4Kenda Foundation Golden Jade Fellow E-mail: lupin@crab0.astr.nthu.edu.tw
(Feb 2010; ??? 2010)
###### Abstract
We report results of Swift observations for the high mass Be/X-ray binary
system 1A 1118–615, during an outburst stage in January, 2009 and at a flaring
stage in March, 2009. Using the epoch-folding method, we successfully detected
a pulsed period of 407.69(2) sec in the outburst of January and of 407.26(1)
sec after the flare detection in March. We find that the spectral detection
for the source during outburst can be described by a blackbody model with a
high temperature ($kT\sim 1-3$ keV) and a small radius ($R\sim 1$ km),
indicating that the emission results from the polar cap of the neutron star.
On the other hand, the spectra obtained after the outburst can further be
described by adding an additional component with a lower temperature ($kT\sim
0.1-0.2$ keV) and a larger emission radius ($R\sim 10-500$ km), which
indicates the emission from around the inner region of an accretion disk. We
find that the thermal emission from the hot spot of the accreting neutron star
dominates the radiation in outburst; the existence of both this X-ray
contribution and the additional soft component suggest that the polar cap and
the accretion disk emission might co-exist after the outburst. Because the
two-blackbody signature at the flaring stage is a unique feature of 1A
1118–615, our spectral results may provide a new insight to interpret the
X-ray emission for the accreting neutron star. The time separation between the
three main outbursts of this system is $\sim 17$ years and it might be related
to the orbital period. We derive and discuss the associated physical
properties by assuming the elongated orbit for this specific Be/X-ray
transient.
###### keywords:
X-rays: binaries — accretion, accretion discs — stars: emission-line, Be —
stars: neutron — pulsars: individual (1A 1118–615)
††pagerange: Swift Observations of the Be/X-ray Transient System 1A
1118–615–Swift Observations of the Be/X-ray Transient System 1A
1118–615††pubyear: 2010
## 1 Introduction
The transient X-ray source 1A 1118–615 was first discovered by Ariel V in
outburst during an observation (1974 December to 1975 January) of Cen X-3
(Eyles et al., 1975). A pulsation of 405.3(6) sec was also detected by Ives et
al. (1975) during this observation but it was wrongly attributed to an orbital
period with an assumption that 1A 1118–615 consists of two massive compact
objects. The optical counterpart was identified as the Be star He 3-640/WRA
793 (Chevalier & Ilovaisky, 1975) and is classified as an O9.5 III–Ve massive
star with a distance of $5\pm 2$ kpc (Janot-Pacheco et al., 1981). Long-term
and short timescale variability in the Balmer emissions of 1A 1118–615 has
been reported by Motch et al. (1988) and Polcaro et al. (1993), respectively .
This source is also the most extreme Be/X-ray transient system known in terms
of the strength of its Hα emission (Coe et al., 1994). Polcaro et al. (1993)
also identified a number of other lines (FeII, He${}_{\rm{I~{}and~{}II}}$,
SiII, CIII, NIII and a tentative identification of CrI); this indicates that
1A 1118–615 has an extremely unusual Be-type primary with complex emission and
absorption features.
The orbital period of 1A 1118–615 is not well studied. Although the X-ray flux
of the source is variable, it remains in the quiescent state for most of the
time. The Be/X-ray binary system underwent a flaring state detected by the
Burst and Transient Source Experiment (BASTE) on board the Compton Gamma-Ray
Observatory (CGRO) (Coe et al., 1994) and the WATCH experiment on board the
GRANAT satellite (Lund et al., 1992) in the late January of 1992. More
recently, 1A 1118–615 showed an outburst that triggered the Burst Alert
Telescope of Swift (Swift/BAT) on 2009 January 4 (Mangano et al., 2009).
Initial X-ray data analysis was done by Kong (2009) and Mangano (2009). 1A
1118–615 was detected flaring again during February and March of 2009 by both
the Joint European X-ray Telescope (JEM-X) and the soft gamma-ray imager
(ISGRI) instruments on the International Gamma-Ray Astrophysics Laboratory
(INTEGRAL) (Leyder et al., 2009). The X-Ray Telescope of Swift (Swift/XRT)
also made follow-up observations after the flaring in March. We here report
the Swift observations of 1A 1118–615 in January and March of 2009.
Table 1: Log of Swift observations for the PC and WT modes
ObsID | Epoch (UT) | Exposures (s) | Pulsed counts | Total countsa
---|---|---|---|---
| | (PC/WT) | (WT) | (PC/WT)
1 | 2009 Jan04 : 03:36 | 105/549 | 351 | —/3865
2 | 2009 Jan04 : 10:05 | —/314 | 209 | —/2061
3 | 2009 Jan04 : 13:21 | 5729/12818 | 18697 | 5729/127740
4 | 2009 Jan06 : 12:07 | 2595/1041 | 249 | 1969/5950
5 | 2009 Jan07 : 10:18 | 462/2746 | 3186 | —/35340
6 | 2009 Mar04 : 22:16 | 75/4270 | 1605 | —/9817
7 | 2009 Mar08 : 04:47 | —/5221 | 4593 | —/24195
8 | 2009 Mar08 : 15:51 | —/3275 | 2074 | —/13775
9 | 2009 Mar14 : 00:39 | —/5627 | 1506 | —/7249
10 | 2009 Mar14 : 16:24 | —/4109 | 971 | —/5177
a Only the photons observed within 0.3–10 keV were counted.
## 2 Observations and data analysis
We analyzed the X-ray data observed by Swift/XRT. These data were mainly
observed within two time intervals. From 2009 January 4, the Be/X-ray
transient was monitored by Swift/XRT for 4 days with a total exposure of 26
ks, which includes 17 ks in the Windowed Timing (WT) mode and 9 ks in the
Photon Counting (PC) mode during the outburst. INTEGRAL detected flares of 1A
1118–615 again from 2009 February 3 to March 3 for a total exposure of 270 ks.
The same field was extensively observed by INTEGRAL and Swift over the
following days, and our analyses focus on the observations made over an 11 day
period with Swift/XRT from 2009 March 4 with a total exposure of 22.6 ks (22.5
ks in the WT mode and only 75 s in the PC mode). Details of the observations
are listed in Table 1.
All the Swift/XRT data were processed with standard procedures (xrtpipeline
v.0.12.0), filtering, and screening criteria by using FTOOLS (v.6.5) in the
HEAsoft package (v.6.5.1). We obtained a refined X-ray position of the source
by the task “xrtcentroid” from the image of ObsID 1 (Table 1) investigated
with the PC mode. The source located at (J2000) R.A.=$11^{h}20^{m}57^{s}.8$,
decl.=$-61^{\circ}54^{\prime}57^{\prime\prime}.9$ with an uncertainty of
$3^{\prime\prime}.9$ (90% confidence level) is the only bright X-ray object in
the field (Kong, 2009), and the source in all the other PC images follow this
position well. We extracted the source counts from a circular region within a
radius of $47^{\prime\prime}$ (20 pixels) which is consistent with the 92% of
the encircle energy function. The background is obtained from a source-free
neighborhood of our target with the same extraction size.
Figure 1: Folded 0.3–10 keV Swift light curves of 1A 1118–615. The upper curve
shows the pulsation of 407.69(2) s in the January outburst with a pulsed
luminosity of $5.3\times 10^{35}$ ergs/s. The lower one shows the pulsation of
407.26(1) s after the flare detection in March with a pulsed luminosity of
$1.6\times 10^{35}$ ergs/s. The distance of the source is assumed to be 4 kpc
(Coe & Payne, 1985). Table 2: Best–fit spectral parameters for the non-thermal
spectroscopy of 1A 1118–615. The radius is measured from the normalization
factor for a source distance of 4 kpc and the flux is measured in the range
0.5–10 keV. $P_{F-test}$ is the chance probability that the improvement of the
fit, compared to the single blackbody model (shown in Table 3).
Observed Time | Jan. 04 | Jan. 04 | Jan. 04 | Jan. 04 | Jan. 06 | Jan. 06 | Jan. 07
---|---|---|---|---|---|---|---
/model | 03:36(WT) | 10:05(WT) | 13:21(PC) | 13:21(WT) | 12:07(PC) | 12:07(WT) | 10:18(WT)
PO | $N_{H}$ (1022 cm-2) | 2.40${}^{+0.36}_{-0.31}$ | 1.20${}^{+0.38}_{-0.31}$ | 1.40${}^{+0.22}_{-0.19}$ | 1.24a | 2.51${}^{+0.63}_{-0.57}$ | 2.40${}^{+0.29}_{-0.27}$ | 1.08a
$\Gamma$ | 1.09$\pm$0.14 | 0.36$\pm$-0.17 | 0.53${}^{+0.10}_{-0.09}$ | 0.66a | 0.43${}^{+0.20}_{-0.18}$ | 0.93$\pm$0.11 | 0.58a
$\chi^{2}_{\nu}$/d.o.f. | 1.82/68 | 1.61/36 | 1.75/100 | 3.77/755 | 0.86/34 | 1.64/104 | 2.05/514
$F_{\rm{PO}}$ ($10^{-10}$ erg cm-2 s-1) | 11.2 | 10.1 | 9.5 | 13.4 | 5.9 | 8.6 | 17.1
| $N_{H}$ (1022 cm-2) | 1.39${}^{+0.44}_{-0.39}$ | 1.23${}^{+0.94}_{-0.57}$ | 0.84${}^{+0.65}_{-0.37}$ | 0.59${}^{+0.10}_{-0.08}$ | – | 1.33${}^{+0.34}_{-0.30}$ | 0.42${}^{+0.11}_{-0.07}$
| $\Gamma$ | 6.42${}^{+1.77}_{-1.84}$ | 6.30${}^{+2.09}_{-2.14}$ | 4.79${}^{+5.20}_{-2.35}$ | 4.15${}^{+0.36}_{-0.37}$ | – | 6.92${}^{+1.94}_{-2.53}$ | 5.18${}^{+1.35}_{-0.04}$
PO | $kT_{\rm{PC}}$ (keV) | 1.65${}^{+0.11}_{-0.09}$ | 2.02${}^{+0.24}_{-0.21}$ | 2.10${}^{+0.14}_{-0.13}$ | 1.94${}^{+0.03}_{-0.02}$ | – | 1.81${}^{+0.09}_{-0.10}$ | 2.00$\pm$0.04
+ | $R_{\rm{PC}}$ (km) | 1.36${}^{+0.15}_{-0.13}$ | 0.99${}^{+0.19}_{-0.16}$ | 0.91${}^{+0.10}_{-0.08}$ | 1.21$\pm$0.02 | – | 1.05${}^{+0.10}_{-0.08}$ | 1.31$\pm$-0.04
BB | $\chi^{2}_{\nu}$/d.o.f. | 1.06/66 | 0.77/34 | 1.25/98 | 1.27/753 | – | 1.18/102 | 1.12/512
| $F_{\rm{PO+BB}}$ ($10^{-10}$ erg cm-2 s-1) | 14.2 | 14.0 | 8.5 | 11.2 | – | 9.6 | 14.1
| $P_{F-test}$ | 0.026 | 0.018 | 0.28 | 1.0$\times$10-10 | – | 0.065 | 1.3$\times$10-5
Observed Time | Mar. 04 | Mar. 08 | Mar. 08 | Mar. 14 | Mar. 14 |
/model | 22:16(WT) | 04:47(WT) | 15:51(WT) | 00:39(WT) | 16:24(WT) |
PO | $N_{H}$ (1022 cm-2) | 3.57a | 2.25a | 2.78a | 2.63a | 2.85a |
$\Gamma$ | 1.24a | 0.80a | 0.99a | 1.35a | 1.45a |
$\chi^{2}_{\nu}$/d.o.f. | 2.61/163 | 3.14/348 | 2.67/221 | 2.61/123 | 2.89/90 |
$F_{\rm{PO}}$ ($10^{-10}$ erg cm-2 s-1) | 4.0 | 7.0 | 6.6 | 1.9 | 1.9 |
| $N_{H}$ (1022 cm-2) | 2.71${}^{+0.41}_{-0.37}$ | 1.85${}^{+0.23}_{-0.20}$ | 2.14${}^{+0.28}_{-0.27}$ | 1.35${}^{+0.19}_{-0.18}$ | 2.17${}^{+0.51}_{-0.45}$ |
| $\Gamma$ | 8.33${}^{+0.92}_{-0.90}$ | 7.77$\pm$0.53 | 8.01${}^{+0.73}_{-0.71}$ | 9.50$\pm$2.12 | 8.11$\pm$1.15 |
PO | $kT_{\rm{PC}}$ (keV) | 1.58$\pm$ 0.06 | 1.81${}^{+0.04}_{-0.05}$ | 1.72${}^{+0.05}_{-0.06}$ | 1.47${}^{+0.06}_{-0.04}$ | 1.32${}^{+0.08}_{-0.07}$ |
+ | $R_{\rm{PC}}$ (km) | 0.87${}^{+0.07}_{-0.06}$ | 0.98$\pm$0.04 | 1.01${}^{+0.06}_{-0.05}$ | 0.66${}^{+0.04}_{-0.05}$ | 0.80$\pm$0.10 |
BB | $\chi^{2}_{\nu}$/d.o.f. | 1.12/161 | 1.08/346 | 0.95/219 | 1.02/121 | 1.54/88 |
| $F_{\rm{PO+BB}}$ ($10^{-10}$ erg cm-2 s-1) | 77.5 | 40.3 | 61.5 | 11.7 | 25.1 |
| $P_{F-test}$ | 1.2$\times$10-12 | 6.7$\times$10-33 | 1.1$\times$10-20 | 1.3$\times$10-11 | 3.4$\times$10-5 |
a The $\chi^{2}_{\nu}$ of the fit is larger than 2.0 and XSPEC can not show
the uncertainty of the parameter.
### 2.1 Timing Analysis
The Swift/XRT observations listed in Table 1 were performed with the PC and WT
modes. The data of the PC mode retain full imaging and spectroscopic
resolution but the time resolution is limited to 2.5 s. The data of the WT
mode only cover the central 8′ of the field of view and one dimensional
imaging is preserved with a better time resolution of 1.7 ms. The observations
we used for analysis were divided into two groups (January and March) to get
more counts for accurate temporal analysis. We also applied solar system
barycentric time correction with the task “hdaxbary” using JPL DE200 solar
system ephemeris. We restricted the data in the effective energy range (0.3–10
keV) of Swift/XRT and performed epoch-folding with the period centered at
405.3 sec (Nagase, 1989; Ives et al., 1975) using the resolution of 0.1 sec
with 200 trials in the close neighborhood ($\sim$ 395–415 sec). After getting
the possible signal around 407.0 sec, we then checked the period with more
delicate resolution (0.01 sec) in the range of $\pm$1.0 sec (406–408 sec). The
light curve was folded with 32 bins and the epoch zero (54835.16972 MJD) was
defined at the start of the good time interval (GTI) from the ObsID 1. The
most possible period in our detections is 407.69(2) sec at epoch 54837.01125
MJD with $\chi^{2}_{\nu}=152.0~{}\rm{for}~{}31~{}\rm{dof}$ and 407.26(1) sec
at epoch 54899.95535 MJD with
$\chi^{2}_{\nu}=52.6~{}\rm{for}~{}31~{}\rm{dof}$. (The reference epoch was
determined by the mid-point of the whole time span.) The corresponding random
probabilities of these two periods are close to 0. The uncertainty level of
the period was determined from the $\chi^{2}$ value using eq. 1 by Leahy
(1987)
$\dfrac{\sigma_{P}}{\Delta P}=0.71(\chi_{\nu}^{2}-1)^{-0.63}$ (1)
where ${\Delta P=\dfrac{P^{2}}{T}}$ is the inferred Fourier spacing and $\nu$
is the degree of freedom. We show the associated pulse profiles in Fig. 1. We
note that the normalized strength of the interpulse around phase 0.2 comparing
to the main pulse around phase 0.5 increased significantly but the third peak
around phase 0.8 appears to vanish entirely from January to March.
### 2.2 Spectral Analysis
Since the center of the source in all the PC mode observations is “pile-up”
that leads to detector saturation, the central source photons has already been
eliminated in the clean image. In order to avoid the pronounced depression of
counts in the center of our target, we therefore directly downloaded the
spectral products of PC mode observations from the UK Swift Science Data
Centre (www.swift.ac.uk/user$\\_$objects/). The WT mode observations were
obtained from the High Energy Astrophysics Science Archive Research Center and
proceeded to the spectral analysis with the standard data reduction. We
downloaded the response matrices (rmf) from the official Swift website. The
ancillary files (arf) are generated by the task “xrtmkarf” with the Swift/XRT
CALDB. Because of a strong absorption feature around 0.5 keV, which is
associated with the Oxygen edge (Cusumano & The Xrt Calibration Team, 2006),
we only fitted our spectra in the 0.5–10 keV band using XSPEC 12.5.1 and
rebinned the data with a minimum of 50 counts per bin to ensure $\chi^{2}$
statistics. We produced a phase-averaged spectrum for each PC and WT
observation with enough exposures to generate more than 30 degrees of freedom
in statistics for spectral fit. Among all the investigations, we applied a
single power-law model to represent the main non-thermal emission of the
accretion column/ outer gap (Table 2 & Table 4) and a single blackbody model
to indicate the thermal emission contributed by the hot spot of the neutron
star (Table 3 & Table 4). If the $\chi_{\nu}^{2}$ of the result is larger than
1, an additional component is required to improve the fit. An additional
blackbody feature was also included in the composite model when we take the
thermal surface emission of the neutron star (Table 2) or from the accretion
disk (Table 3) into the consideration. The best-fit spectral parameters of
different models are all given with the errors at 90% confidence level.
Table 3: Best–fit spectral parameters for the thermal spectroscopy of 1A
1118–615. PC and disk individually represent the related parameters of polar
cap and accretion disk. The radius is measured from the normalization factor
for a source distance of 4 kpc and the flux is measured in the range 0.5–10
keV. $P_{F-test}$ is the chance probability that the improvement of the fit,
compared to the single blackbody model.
Observed Time | Jan. 04 | Jan. 04 | Jan. 04 | Jan. 04 | Jan. 06 | Jan. 06 | Jan. 07
---|---|---|---|---|---|---|---
/model | 03:36(WT) | 10:05(WT) | 13:21(PC) | 13:21(WT) | 12:07(PC) | 12:07(WT) | 10:18(WT)
BB | $N_{H}$ (1022 cm-2) | 0.84${}^{+0.17}_{-0.14}$ | 0.47${}^{+0.21}_{-0.17}$ | 0.49${}^{+0.11}_{-0.10}$ | 0.36${}^{+0.01}_{-0.02}$ | 1.10${}^{+0.42}_{-0.38}$ | 0.97${}^{+0.15}_{-0.14}$ | 0.29$\pm$0.03
$kT_{\rm{PC}}$ (keV) | 1.77$\pm$0.09 | 2.27${}^{+0.23}_{-0.19}$ | 2.19${}^{+0.11}_{-0.10}$ | 2.01${}^{+0.01}_{-0.02}$ | 2.65${}^{+0.34}_{-0.28}$ | 1.89$\pm$0.09 | 2.05${}^{+0.04}_{-0.03}$
$R_{\rm{PC}}$ (km) | 1.21$\pm$0.09 | 0.83${}^{+0.10}_{-0.09}$ | 0.85${}^{+0.06}_{-0.05}$ | 1.15${}^{+0.01}_{-0.02}$ | 0.52${}^{+0.08}_{-0.07}$ | 0.98${}^{+0.06}_{-0.07}$ | 1.25${}^{+0.03}_{-0.02}$
$\chi^{2}_{\nu}$/d.o.f. | 1.14/68 | 0.92/36 | 1.25/100 | 1.35/755 | 0.81/34 | 1.22/104 | 1.17/514
$F_{\rm{BB}}$ ($10^{-10}$ erg cm-2 s-1) | 7.3 | 8.2 | 7.8 | 10.8 | 4.9 | 6.4 | 13.8
2BB | $N_{H}$ (1022 cm-2) | 1.29${}^{+0.35}_{-0.31}$ | – | 0.78${}^{+0.42}_{-0.30}$ | 0.52${}^{+0.07}_{-0.06}$ | – | 1.22${}^{+0.28}_{-0.22}$ | 0.36${}^{+0.06}_{-0.04}$
$kT_{\rm{PC}}(keV)$ | 1.66${}^{+0.10}_{-0.09}$ | – | 2.10${}^{+0.14}_{-0.12}$ | 1.94${}^{+0.03}_{-0.02}$ | – | 1.82${}^{+0.09}_{-0.10}$ | 2.01$\pm$0.04
$R_{\rm{PC}}$ (km) | 1.35${}^{+0.14}_{-0.12}$ | – | 0.91${}^{+0.09}_{-0.08}$ | 1.20$\pm$0.02 | – | 1.04${}^{+0.09}_{-0.08}$ | 1.29${}^{+0.04}_{-0.03}$
$kT_{\rm{disk}}(keV)$ | 0.14${}^{+0.05}_{-0.03}$ | – | 0.18${}^{+0.09}_{-0.15}$ | 0.16${}^{+0.02}_{-0.02}$ | – | 0.12${}^{+0.07}_{-0.03}$ | 0.10${}^{+0.03}_{-0.02}$
$R_{\rm{disk}}$ (km) | 106${}^{+220}_{-75}$ | – | 22${}^{+14}_{-21}$ | 24${}^{+9}_{-6}$ | – | 121${}^{+482}_{-101}$ | 109${}^{+158}_{-66}$
$\chi^{2}_{\nu}$/d.o.f. | 1.03/66 | – | 1.24/98 | 1.27/753 | – | 1.17/102 | 1.12/512
$F_{\rm{2BB}}$ ($10^{-10}$ erg cm-2 s-1) | 9.5 | – | 8.1 | 10.9 | – | 7.3 | 13.9
$P_{F-test}$ | 0.013 | – | 0.24 | 4.6$\times$10-11 | – | 0.046 | 1.5$\times$10-5
Observed Time | Mar. 04 | Mar. 08 | Mar. 08 | Mar. 14 | Mar. 14 |
/model | 22:16(WT) | 04:47(WT) | 15:51(WT) | 00:39(WT) | 16:24(WT) |
BB | $N_{H}$ (1022 cm-2) | 1.43${}^{+0.16}_{-0.15}$ | 0.76${}^{+0.08}_{-0.07}$ | 0.96$\pm$0.11 | 0.82${}^{+0.14}_{-0.13}$ | 0.92${}^{+0.18}_{-0.16}$ |
$kT_{\rm{PC}}(keV)$ | 1.77$\pm$0.06 | 2.05$\pm$0.05 | 1.94${}^{+0.06}_{-0.05}$ | 1.59$\pm$0.06 | 1.52${}^{+0.06}_{-0.07}$ |
$R_{\rm{PC}}$ (km) | 0.70$\pm$0.04 | 0.79${}^{+0.02}_{-0.03}$ | 0.81${}^{+0.03}_{-0.04}$ | 0.57${}^{+0.03}_{-0.04}$ | 0.60${}^{+0.05}_{-0.04}$ |
$\chi^{2}_{\nu}$/d.o.f. | 1.55/163 | 1.65/348 | 1.43/221 | 1.52/123 | 1.90/90 |
| $F_{\rm{BB}}$ ($10^{-10}$ erg cm-2 s-1) | 2.6 | 5.4 | 4.8 | 1.2 | 1.1 |
2BB | $N_{H}$ (1022 cm-2) | 2.50${}^{+0.34}_{-0.32}$ | 1.71$\pm$0.19 | 1.94${}^{+0.25}_{-0.23}$ | 1.21$\pm$0.17 | 2.06${}^{+0.45}_{-0.42}$ |
$kT_{\rm{PC}}(keV)$ | 1.59${}^{+0.07}_{-0.05}$ | 1.82${}^{+0.05}_{-0.04}$ | 1.73${}^{+0.06}_{-0.05}$ | 1.50${}^{+0.05}_{-0.06}$ | 1.33$\pm$0.07 |
$R_{\rm{PC}}$ (km) | 0.85$\pm$0.06 | 0.96$\pm$0.04 | 0.99${}^{+0.05}_{-0.06}$ | 0.64$\pm$0.04 | 0.79${}^{+0.09}_{-0.08}$ |
$kT_{\rm{disk}}$ (keV) | 0.14$\pm$0.02 | 0.14$\pm$0.01 | 0.14${}^{+0.01}_{-0.02}$ | 0.09${}^{+0.01}_{-0.02}$ | 0.14$\pm$0.02 |
$R_{\rm{disk}}$ (km) | 179${}^{+150}_{-81}$ | 152${}^{+69}_{-44}$ | 189${}^{+119}_{-73}$ | 481${}^{+749}_{-272}$ | 125${}^{+136}_{-68}$ |
$\chi^{2}_{\nu}$/d.o.f. | 1.12/161 | 1.12/346 | 0.96/219 | 1.05/121 | 1.52/88 |
$F_{\rm{2BB}}$ ($10^{-10}$ erg cm-2 s-1) | 6.6 | 8.6 | 9.1 | 2.9 | 3.1 |
$P_{F-test}$ | 2.2$\times$10-12 | 3.8$\times$10-30 | 3.6$\times$10-20 | 8.1$\times$10-11 | 1.9$\times$10-5 |
Figure 2: Spectral fits to blackbody models. (a).The left panel shows the fit
to the single blackbody model for the WT mode data of ObsID 4 and (b).the
right panel shows the fit to the composite blackbody model (BB+BB) for the WT
mode data of ObsID 8. The corresponding null hypothesis probabilities of these
two fits are all larger than 5% and both of the results are attributed to be
acceptable. The residuals in terms of sigmas with the error bars of size one
are also shown.
Figure 3: Spectral fits to blackbody models for 3 spectra observed in January
and one spectrum observed in March. The January spectra are fitted to a single
blackbody component; the March one is fitted to two blackbody components. All
of these data have relatively long exposures than the other observations and
the statistical results of these spectral fits are poor. Some strong absorbed
features (e.g. 1.5 keV of Aluminum, 1.8 keV of Silicon et al.) caused by the
uncertainty of the calibrations are clearly visible (the Oxygen edge at 0.5
keV had already been removed before performing the spectral analysis). There
is also the strong instrumental Nickel contamination present in the high
energy edge which sometimes is not fully subtracted. If all of these effects
can be completely removed, the results of these fits will have significant
improvements.
For all observations, a single power-law model can not provide a reasonable
fit as demonstrated in Table 2 (except for the observation of January 06). On
the other hand, a single blackbody model can provide an acceptable fit (Table
3 & (a) of Fig. 2) for the data observed in January when we took account the
systematic uncertainties caused by calibrations (Fig. 3). The spectra with
long exposures seem to have worse statistics than those with short exposures.
This might be caused by the complicated absorption features associated with
the response matrices (Cusumano & The Xrt Calibration Team, 2006). A composite
model with both power-law and blackbody can be fitted with the extracted
spectra in March; however, the photon index ($\Gamma\sim 7.5-9$) is too soft
compared with other X-ray binary pulsars (XBPs) and the inferred luminosity in
the range 0.3–10 keV will exceed the Eddington luminosity. A composite model
can also fit most of the spectra in January, but the associated photon index
of the power-law component is still too soft and the main contribution is
dominated by the blackbody component. We therefore ignore the contribution
from the power-law for all the spectra and only considered a single/composite
model with the blackbody component(s) to explain the spectral behavior.
Two-blackbody components can improve the spectral fit during the outburst. For
ObsID 3 and 5 in WT mode, an F-test shows that the additional soft component
during the outburst is significant at $>99\%$. But for the other 3
observations in January, an additional soft blackbody component does not
improve the fits significantly. In contrary, for the observations detected
after the flaring in March, a two-blackbody model fits significantly better
than a single blackbody one in Table 3 ((b) of Fig. 2). However, we note the
WT spectra in ObsID 1, 2 & 4 can be described by an acceptable single
blackbody model with much shorter exposures than the other observations. It is
unclear whether the single blackbody fits might be due to the larger
statistical uncertainty in shorter exposures. However, the single blackbody
component fits with long exposures in January are still much better than the
single blackbody component fits of March observations. This indicates the
variation of the spectral behavior might be real.
We also note that the fitting parameters of Jan 04 observations (ObsID 1 & 2)
are very different. This might be caused by the fact that the fitting
parameter of the hydrogen absorption is very sensitive to the soft band
spectrum if the the photon counts are low; a very small variation of the
spectrum at soft band will result in a complete different hydrogen absorption,
which will then cause a large variation in other fitting parameters. For the
spectra of Jan 04 observations (ObsID 1 & 2), if we fixed the hydrogen
absorption to be $\sim 6.4\times 10^{21}$ cm-2, the ObsID 1& 2 of the WT mode
observations can both be fitted by a single blackbody model with $kT=1.86\pm
0.07$ keV, R =$1.11\pm 0.05$ km with $\chi^{2}_{\nu}=1.21$ with 69 d.o.f and
$kT=2.16^{+0.16}_{-0.13}$ keV, R =$0.89\pm 0.07$ km with $\chi^{2}_{\nu}=0.95$
with 37 d.o.f. The variations of the parameters between these two spectra with
the fixed hydrogen absorption are much smaller than those listed in the Table
3. This indicates that the huge variations of the polar cap temperatures and
radius within the same day may not be real.
For all the spectra extracted in March, only the second spectrum of March 14
(ObsID 10) does not provide an acceptable fit with a two-blackbody model; this
might be due to the serious Nickel contamination at 7–8 keV that is not fully
subtracted from the calibration and many absorption features. Some of the
features are caused by the instrumental effects as shown by Cusumano & The Xrt
Calibration Team (2006); however, the true spectra of the source might be much
more complicated than our models and the instrumental effects probably do not
explain all the poor fits to the data.
Table 4: Best–fit spectral parameters for the pulsed spectroscopy of 1A
1118–615. The radius is measured from the normalization factor for a source
distance of 4 kpc and the flux is measured in the range 0.5–10 keV.
Observed Time | January | March
---|---|---
/model | (WT) | (WT)
PO | $N_{H}$ (1022 cm-2) | 0.64 (fixed) | 1.88 (fixed)
$\Gamma$ | 0.23$\pm$0.07 | 1.04${}^{+0.12}_{-0.13}$
$\chi^{2}_{\nu}$/d.o.f. | 1.23/314 | 1.28/156
$F_{\rm{PO}}$ ($10^{-10}$ erg cm-2 s-1) | 3.6 | 1.2
BB | $N_{H}$ (1022 cm-2) | 0.64 (fixed) | 1.88 (fixed)
$kT_{\rm{PC}}$ (keV) | 1.78${}^{+0.10}_{-0.09}$ | 1.16$\pm$0.07
$R_{\rm{PC}}$ (km) | 0.70$\pm$0.05 | 0.84${}^{+0.09}_{-0.08}$
$\chi^{2}_{\nu}$/d.o.f. | 0.76/314 | 0.74/156
$F_{\rm{BB}}$ ($10^{-10}$ erg cm-2 s-1) | 2.8 | 0.83
#### 2.2.1 Pulsed Spectral Analysis
To further study the emission from the neutron star, we also generated the
pulsed spectrum of 1A 1118–615 with the data observed in the WT mode. We
divided each observation into pulsed and unpulsed emission depending on their
pulsed phase shown in Fig. 1. The light curve of 1A 1118–615 was folded with
32 bins and we assign the phase from 0.375 to 0.90625 (12th – 29th of 32 bins)
to be the pulsed phase of the 407.69 s period and the phase from 0.125 to 0.75
(4th – 24th of 32 bins) to be the pulsed phase of the 407.26 s one. We also
used the same criterion to estimate all the pulsed photons in 0.3–10 keV for
each data listed in Table 1. To get enough counts for pulsed spectral fits,
all the pulsed spectra in January and March were combined into a January
spectrum and March one respectively. The associated ancillary response files
were also combined with weightings depending on the relative exposures (shown
in Table 1). The pulsed spectrum was then obtained by subtracting the unpulsed
emission from the pulsed emission. According to Table 3, the average hydrogen
absorption in the best fit to a single blackbody spectrum obtained at the
outburst stage in January is $\sim 6.4\times 10^{21}$ cm-2 and is $\sim
1.9\times 10^{22}$ cm-2 to a two blackbody spectrum after the flare detection
in March. In the fits to the pulsed spectra, we fixed these values to
associate with the pulsed-average spectra (Table 4). The difference of the
absorption can be caused by the possibility that the absorber is related to
the different amounts and/or the viewing angle of the material accreted by the
pulsar along the line-of-sight.
The pulsed spectra of January and March can not be well fitted by a single
power-law, and we can not find any evidence of the non-thermal emission from
each phase-averaged spectrum. However, a single blackbody model can be applied
to the pulsed spectra in January and March at 0.5–10 keV with
$kT_{\rm{BB}}=1.78^{+0.10}_{-0.09}$ keV, $\rm{R}_{BB}=0.70\pm 0.05$ km with
$\chi^{2}_{\nu}=0.76~{}\rm{for}~{}314~{}\rm{d.o.f}$ and $kT_{\rm{BB}}=1.16\pm
0.07$ keV, $\rm{R}_{BB}=0.84^{+0.09}_{-0.08}$ km with
$\chi^{2}_{\nu}=0.74~{}\rm{for}~{}156~{}\rm{d.o.f}$. The inferred pulsed flux
is $(2.8\pm 0.6)\times 10^{-10}$ ergs cm-2 s-1 in January and $(8.3\pm
2.6)\times 10^{-11}$ ergs cm-2 s-1 in March. The decrease of the luminosity
might be caused by the cooling of the surface temperature.
## 3 Discussion
According to Ariel V (Eyles et al., 1975), CGRO/BATSE (Coe et al., 1994) and
Swift/BAT observations, the time separation of $\sim$ 17 years between the
detection of each outburst for 1A 1118–615 suggests that the neutron star does
not interact violently with the massive companion during most of the orbit
(Villada et al., 1999). If the outburst is only caused by the approaching of
the neutron star to the periastron, the long time separations of the outbursts
indicate a very flat elliptical orbit.
However, the orbital periods for known Be/XBPs range from several days (A
0538–66; Johnston et al. 1980) to hundred of days (X1145–619; Watson et al.
1981) and no other convincing evidence can connect this long time separation
to be the orbital period of 1A 1118–615. The time separation of $\sim$ 1 month
between the outburst of 2009 January detected by Swift/XRT and the flare of
2009 February detected by INTEGRAL might also indicate a possible orbital
period. One major concern is that we would expect to have observed much more
flares during the interval of 17 years or at least more than one flares after
the main outburst if the orbital period was only around 1 month. In addition,
each main outburst detected every 17 years is the normal outbursts (type I,
1036 – 1037 erg/s). However, we would expect to observe a giant burst for a
long recurrence interval (type II, $>10^{37}$ erg/s; Caballero et al. 2008).
On the other hand, if the orbital period for 1A 1118–615 is 17 years, this
binary system will have the longest orbital period than all the other known
Be/XBPs shown by Corbet et al. (2009).
We would expect to observe some evidences of the passage of the neutron star
during the periastron in other wave bands. Coe et al. (1994) found that the
equivalent width of Hα varied on very short time scales around the outburst of
1992; this can be explained as the disruption of the circumstellar disk caused
by the passage of the neutron star during the periastron. This indicates that
the assumption of the 17-year orbital period is consistent with the Hα
equivalent width variation. Further optical investigations and the study of
high-energy emission mechanism of this Be/X-ray transient may help us to
verify the orbital period and solve the related mystery.
### 3.1 Observational features detected by Swift
The pulsation in January was measured during the outburst while in March it
was detected after the flare detection of February. The pulsations of 1A
1118–615, shown in Fig. 1, obviously demonstrate that the pulsar has spun-up
during the period between the outburst and flare events. The pulse periods do
not show variation during the different observations of January and March
respectively. As a result it is unclear whether the outburst or flare is
responsible for the spinning-up of the pulsar. But the change of the pulse
profile for 1A 1118–615 may relate to the decrease of the pulsed luminosity;
similar luminosity dependencies were also claimed for Cen X-3 (Nagase et al.,
1992), LMC X-4 (Levine et al., 1991) and EXO 2030+375 (Parmar et al., 1989).
In the evolution of the pulse profile for EXO 2030+375 following the first
outburst decay, an interpulse appeared and became stronger as the luminosity
decreased from the maximum and this trend continued until the main pulse had
become entirely disappeared. Then the profile became dominated by the
contribution from the interpulse. The same luminosity dependence was also
evident for 1A 1118–615. The pulse profiles of 1A 1118–615 show features
similar to those of EXO 2030+375 (Parmar et al., 1989); the pulsed luminosity
decreased and the normalized strength of the interpulse became significant
relative to the main pulse after the flaring. A long term survey is needed to
further examine the dependence of the pulse profile and X-ray luminosity.
According to Table 3, a single blackbody model can provide an acceptable fit
with a temperature of $kT>1$ keV and a small hot spot radius ($\sim 1$ km;
assuming a source distance of 4 kpc; Coe & Payne 1985) for most of the data
observed in the outburst. The thermal emission can be attributed to the
ionized photons or the collision of the heated gas; however, the most
plausible origin is from the surface (polar cap) emission of the neutron star
(La Palombara & Mereghetti, 2007; La Palombara et al., 2009). Hickox et al.
(2004) presented a clear physical picture to show the emitting process during
the accretion. In this picture, the radius of the accreting polar cap is $\sim
0.1R_{NS}$, where $R_{NS}$ is the radius of the neutron star, typically
assumed to be 10 km. In contrary to the spectra of most XBPs (Hickox et al.,
2004), we did not find any obvious evidence of a main non-thermal
contribution. Instead of X-rays emitted from the accretion column or outer
gap, we detected strong emission from a surface layer. This X-ray contribution
during the outburst can be originated from the thermal emission around the
compact source, which is similar to the polar cap emission of an isolated
neutron star.
The second thermal feature ($R\geq 100~{}\rm{km}$ & $kT<0.2~{}\rm{keV}$)
resembles the X-rays from the inner edge of an accretion disk (Hickox et al.,
2004). This additional soft characteristic may become an important component
when the pulsed flux of the source weakens. Such X-ray thermal contributions
are always observed in the bright XBPs with luminosity $>$ 1037-38 ergs/s
(Paul et al., 2002; Ramsay et al., 2002; Manousakis et al., 2009), but the
source luminosity of 1A 1118–615 in 0.5 – 10 keV observed by Swift/XRT is only
$\sim(0.6-1.7)\times 10^{36}$ ergs s-1. Our results might indicate that the
soft excess contributed by the emission from the inner disk can generally be
observed for the XBPs after flaring, not only for the bright targets. If the
variation of the second thermal component is real, our spectral analyses in
different time intervals may indicate that the surface (polar cap) emission of
the neutron star dominates the observed X-rays from 1A 1118–615 at the
beginning of outburst. The fact that the thermal emission contributed by the
accretion disk becomes significant at another specific time interval may
represent that a different violent radiation process starts in one month and
may be also associated with the flare detection. Although the average source
luminosity in different time intervals only changes by a small fraction, the
cooling of temperature in the similar polar cap area clearly exhibit a decay
of the surface emission from the neutron star. The decrease of the pulsed
emission derived from our phase-resolved spectrum also supports this
viewpoint.
### 3.2 Parameters of the binary system
The parameters of the neutron star and the properties of the orbital motion
for the binary system, 1A 1118–615, are not well understood. Furthermore, the
possible orbital period of $\sim 17$ yr is extremely large compared with 100
days for the typical Be/XBPs with long spin period. Therefore, it will be
worth examining the parameters of the neutron star and orbital motion for
further discussion on this unique high-mass X-ray binary system.
The increasing X-ray emission at the outburst stage in January of 2009 allowed
us to find a rotation period of $P\sim 407.69(2)$ sec. On the other hand, a
rotation period of $P\sim 409.2(8)$ sec was reported by Chandra observation on
August 21 of 2003 in the quiescence stage of the source (Rutledge et al.,
2007). Assuming that this spin up of the neutron star from its quiescence
stage is a result of a continuous accretion of matter onto the stellar surface
of the neutron star, we can estimate the accretion rate and the magnetic field
of the neutron star. The Chandra observation in 2003 detected an X-ray
luminosity of $L_{\rm{X}}\sim 1.8\times 10^{34}~{}\mathrm{erg/s}$ in the
0.5-10 keV energy bands. Assuming the X-ray emission is from the accretion
onto the stellar surface, the accretion rate at the surface is estimated as
(Campana et al., 1998)
$\displaystyle\dot{M}_{acc}$ $\displaystyle=$
$\displaystyle\frac{R_{NS}}{GM_{NS}}\eta_{BC}L_{X}$ (2) $\displaystyle\sim$
$\displaystyle 5\times
10^{13}\eta_{BC}M_{1.4}^{-1}R_{6}L_{X,34}~{}\mathrm{g/s}$
where $\eta_{BC}$ is the bolometric correction, $R_{NS}$ and $M_{NS}$ are
stellar radius of and the mass of the neutron star, respectively. In addition,
$L_{X,34}$ is the X-ray luminosity in units of $10^{34}~{}\mathrm{erg/s}$, and
$R_{6}$ is the stellar radius of the neutron star in units of $10^{6}$ cm and
$M_{1.4}$ is the mass of the neutron star in units of the solar mass.
We assume that the accretion rate is constant in the quiescent stage before
the onset of the outburst in 2009 January. Combining the estimated accretion
rate $\dot{M}_{acc}$ with conservation of the angular momentum that
$I\delta{\Omega}/\delta t=\dot{M}_{acc}\sqrt{GM_{NS}R_{m}}$ (3)
with $R_{m}$ being the magnetospheric radius where the magnetic pressure is
balanced with the pressure of the accretion disk, the strength of the magnetic
field of the neutron star is given by
$B\sim 2\times
10^{14}\eta_{BC}^{-3}M_{1.4}^{3/2}R_{6}^{-6}L_{X,34}^{-3}~{}\mathrm{G}$ (4)
where we used the momentum of inertia of $I=10^{45}~{}\mathrm{g\cdot cm^{3}}$.
The strength of the magnetic field can be estimated as $B\sim 2\times
10^{14}~{}\mathrm{G}$ for $M_{1.4}=1$ and $R=10^{6}$ cm. This value is also
similar to those extremely magnetized neutron stars (known as magnetar, Woods
& Thompson 2006) if the bolometric correction is $\eta_{BC}\sim 1$. On the
other hand, a typical strength of the magnetic field of a young neutron star
$B\sim 10^{12}~{}\mathrm{G}$ is expected if $\eta_{BC}\sim 5$. Accretion onto
the neutron star surface is permitted when the magnetospheric radius $R_{m}$
becomes smaller than the corotation radius,
$R_{cor}=(GM_{NS}P^{2}/4\pi^{2})^{1/3}$, where $P$ is the period of the
neutron star. This condition is satisfied when $\eta_{BC}\geq 2$.
It will be expected that the outburst occurs as a result of the interaction
between the equatorial disk of the Be star and the neutron star at the
periastron passage. Near the periastron, the velocity of the neutron star,
$v_{NS}$, will become $v_{NS}\la(GM_{Be}/R_{Be})^{1/2}(\sim
500~{}\mathrm{km/s})$, where we used stellar mass and radius of Be star of
$M_{Be}=17M_{\odot}$ and $R_{Be}=10R_{\odot}$, respectively (Sigut et al.,
2009). This implies that the velocity of the neutron star is in general larger
than the typical velocity for the matter of the equatorial disk of the Be
star, whose typical initial velocity is $<<1$ km/s (Marlborough et al., 1997;
Porter, 1998). The velocity of the neutron star dominates that of the matter
of the equatorial disk of the Be star near the periastron. Then the capture of
the matter from the equatorial disk of the Be star by the neutron star will
occur at a radius of
$r_{acc}\sim 2GM_{NS}/v_{r}^{2}\sim 3.8\times
10^{11}\left(\frac{M_{NS}}{1.4M_{\odot}}\right)\left(\frac{v_{NS}}{100~{}\mathrm{km/s}}\right)^{-2}~{}cm$
(5)
from the neutron star, where $v_{r}\sim v_{NS}$ is the velocity of the
material of the equatorial disk relative to the neutron star.
At the outburst stage in January of 2009, the average X-ray emission at 0.5–10
keV was measured with a luminosity of $L_{X}\sim 1.5\times
10^{36}~{}\mathrm{erg/s}$, indicating that the accretion rate onto the neutron
star is
$\dot{M}_{acc}\sim 6.4\times
10^{-10}\dfrac{\eta_{BC}}{5}M_{\odot}/\mathrm{yr}.$ (6)
This accretion rate for a young neutron star is about 2 orders of magnitude
smaller than typical mass loss rate ($\dot{M}_{w}\sim
10^{-7}~{}M_{\odot}/\mathrm{yr}$; Waters et al. 1988) of an equatorial disk
around a Be star. We can also estimate the accreting density of the
surrounding medium from Bondi accretion ($\dot{M_{acc}}=4\pi r^{2}_{acc}\rho
v_{r}$; where $\rho$ is the ambient density). Thus we obtain:
$\rho=\dfrac{\dot{M_{acc}}}{4\pi r_{acc}^{2}v_{r}}\sim 5\times
10^{-16}\eta_{BC}\left(\frac{M_{NS}}{1.4M_{\odot}}\right)^{-2}\left(\frac{v_{NS}}{100~{}\mathrm{km/s}}\right).~{}\mathrm{g/cm^{3}}$
(7)
This is several orders smaller than the typical value ($\rho\sim
10^{-10}-10^{-13}~{}\mathrm{g/cm^{3}}$) of the density at the inner region of
the disk (Sigut et al., 2009), implying that (1) the neutron star is likely
interacting with the outer edge of the Be star disk or (2) the disk of Be star
in 1A 1118–615 is relatively thin.
Our spectral analyses show one thermal polar cap component during the outburst
stage in January and show an additional thermal disk component after the
flaring in March. This might be explained as a result of the neutron star
approaching the equatorial disk of the Be star. When the neutron star first
approached the equatorial disk, the accretion disk of the neutron star could
have accumulated relatively little material, so the accretion disk would not
have significant emission and we could only detect the polar cap emission of
the neutron star. The flare of February might be caused by disk instabilities
of the accretion disk of the neutron star. This indicates that the neutron
star would have accreted significant amount of material from the equatorial
disk of the Be star during the periastron; this makes the thermal disk
component visible in the March observations after the February flare.
## 4 Conclusion
The Be/X-ray transient 1A 1118–615 has been observed in outburst by Swift
starting in January of 2009. Comparing with the epochs of two previous
outbursts for 1A 1118–615 detected by Ariel V and CGRO/BATSE, the time
separation of $\sim 17$ years might give a hint as the orbital period. On the
other hand, the time interval of $\sim 1$ month between the detection of the
outburst and the flaring in 2009 might also give another indication of the
orbital period; especially given the fact that the new Be/XBP (e.g. SAX
J2103.5+4545; Baykal et al., 2000) was discovered to have a rapid orbital
motion with a long spin period. We performed timing analysis on the Swift/XRT
archive data and the pulsations at 407.69(2) sec and 407.26(1) sec were
individually detected in January and March. 1A 1118–615 had obviously spun-up
due to the accretion from the massive primary between the January and March
observations.
We also investigated the X-ray spectral behavior during the outburst in
January and after another flare detection in March. Almost all the phase-
averaged spectra can be well-modeled by thermal emission. During the outburst,
we found that a single blackbody component with a higher temperature of $\sim
1$ keV and a small radius of $R\sim 1$ km provides a reasonable fit, and it
indicates that most of the observed X-rays are from the polar cap emission of
the compact star. After the flare detected by INTEGRAL, the following
Swift/XRT observations display a significant soft thermal excess above the
main surface emission from the neutron star in their spectral behavior. The
soft thermal excess shows a lower temperature of $\sim 0.1-0.2$ keV and a
larger emitting radius of $\sim 100$ km; this component might be connected to
the emission from the inner accretion disk of the binary system. In contrary
to other XBPs, we do not find a significant non-thermal component in the
spectrum of 1A 1118–615. Our results show that the thermal radiation from the
hot spot of neutron star and from the reprocessing of the dense material in
the inner disk provides the emission mechanism for 1A 1118–615 after the
outburst. The highest luminosity at 0.5–10 keV of this source detected after
the outburst is only $\sim 2.7\times 10^{36}$ ergs/s. Our results also
demonstrate that the soft excess contributed by the inner accretion disk may
not only be observed in bright XBPs ($L_{x}\ga 10^{37-38}$ ergs/s).
Hickox et al. (2004) gave a clear physical picture to explain the possible
origin of the soft excess. Our discoveries based on the spectral analysis may
provide a hint to examine the accreting phenomenon. We also derived the
physical properties of this accreting system using a simple model. The
accretion rate and the density of the circumstellar disk derived with the
assumption of the elongated orbit of 1A 1118–615 are lower than those of the
other general Be/XBPs. This indicates the neutron star might pass the outer
edge of the equatorial disk of the Be star.
## Acknowledgments
The authors appreciate an anonymous referee for his/her fruitful comments. We
thank Dr. Kwong-Sang Cheng for the fruitful discussion to this manuscript.
This research has made use of the data obtained through the High Energy
Astrophysics Science Archive Research Center Online Service, provided by the
NASA/Goddard Space Flight Center. This work was partially supported by the
National Science Council through grants NSC 98-2811-M-008-044. CYH
acknowledges support from the National Science Council through grants NSC
96-2112-M-008-017-MY3 and NSC 95-2923-M-008-001-MY3. AKHK acknowledges support
from the National Science Council through grants NSC 96-2112-M-007-037-MY3.
## References
* Baykal et al. (2000) Baykal A., Stark M. J., Swank J., 2000, ApJL, 544, L129
* Caballero et al. (2008) Caballero I., Santangelo A., Kretschmar P., Staubert R., Postnov K., Klochkov D., Camero-Arranz A., Finger M. H., Kreykenbohm I., Pottschmidt K., Rothschild R. E., Suchy S., Wilms J., Wilson C. A., 2008, A&A, 480, L17
* Campana et al. (1998) Campana S., Colpi M., Mereghetti S., Stella L., Tavani M., 1998, A&A Rev., 8, 279
* Chevalier & Ilovaisky (1975) Chevalier C., Ilovaisky S. A., 1975, IAU Circ., 2778, 1
* Coe & Payne (1985) Coe M. J., Payne B. J., 1985, Ap&SS, 109, 175
* Coe et al. (1994) Coe M. J., Roche P., Everall C., Fishman G. J., Hagedon K. S., Finger M., Wilson R. B., Buckley D. A. H., Shrader C., Fabregat J., Polcaro V. F., Giovannelli F., Villada M., 1994, A&A, 289, 784
* Corbet et al. (2009) Corbet R. H. D., Coe M. J., McGowan K. E., Schurch M. P. E., Townsend L. J., Galache J. L., Marshall F. E., 2009, in J. T. van Loon & J. M. Oliveira ed., IAU Symposium Vol. 256 of IAU Symposium, Properties of X-ray binaries in the Magellanic Clouds from RXTE and Chandra observations. pp 361–366
* Cusumano & The Xrt Calibration Team (2006) Cusumano G., The Xrt Calibration Team 2006, Nuovo Cimento B Serie, 121, 1463
* Eyles et al. (1975) Eyles C. J., Skinner G. K., Willmore A. P., Rosenberg F. D., 1975, Nature, 254, 577
* Hickox et al. (2004) Hickox R. C., Narayan R., Kallman T. R., 2004, ApJ, 614, 881
* Ives et al. (1975) Ives J. C., Sanford P. W., Bell Burnell S. J., 1975, Nature, 254, 578
* Janot-Pacheco et al. (1981) Janot-Pacheco E., Ilovaisky S. A., Chevalier C., 1981, A&A, 99, 274
* Johnston et al. (1980) Johnston M. D., Griffiths R. E., Ward M. J., 1980, Nature, 285, 26
* Kong (2009) Kong A. K. H., 2009, The Astronomer’s Telegram, 1889, 1
* La Palombara & Mereghetti (2007) La Palombara N., Mereghetti S., 2007, A&A, 474, 137
* La Palombara et al. (2009) La Palombara N., Sidoli L., Esposito P., Tiengo A., Mereghetti S., 2009, A&A, 505, 947
* Leahy (1987) Leahy D. A., 1987, A&A, 180, 275
* Levine et al. (1991) Levine A., Rappaport S., Putney A., Corbet R., Nagase F., 1991, ApJ, 381, 101
* Leyder et al. (2009) Leyder J., Walter R., Lubinski P., 2009, The Astronomer’s Telegram, 1949, 1
* Lund et al. (1992) Lund N., Brandt S., Castro-Tirado A. J., 1992, IAU Circ., 5448, 1
* Mangano (2009) Mangano V., 2009, The Astronomer’s Telegram, 1896, 1
* Mangano et al. (2009) Mangano V., Baumgartner W. H., Gehrels N., Guidorzi C., Holland S. T., Hoversten E. A., Immler S., Kennea J. A., La Parola V., Landsman W. B., Markwardt C. B., Marshall F. E., Parsons A. M., Schady P., 2009, GRB Coordinates Network, 8777, 1
* Manousakis et al. (2009) Manousakis A., Walter R., Audard M., Lanz T., 2009, A&A, 498, 217
* Marlborough et al. (1997) Marlborough J. M., Zijlstra J., Waters L. B. F. M., 1997, A&A, 321, 867
* Motch et al. (1988) Motch C., Pakull M. W., Janot-Pacheco E., Mouchet M., 1988, A&A, 201, 63
* Nagase (1989) Nagase F., 1989, PASJ, 41, 1
* Nagase et al. (1992) Nagase F., Corbet R. H. D., Day C. S. R., Inoue H., Takeshima T., Yoshida K., Mihara T., 1992, ApJ, 396, 147
* Parmar et al. (1989) Parmar A. N., White N. E., Stella L., 1989, ApJ, 338, 373
* Paul et al. (2002) Paul B., Nagase F., Endo T., Dotani T., Yokogawa J., Nishiuchi M., 2002, ApJ, 579, 411
* Polcaro et al. (1993) Polcaro V. F., Villada M., Giovannelli F., 1993, A&A, 273, L49+
* Porter (1998) Porter J. M., 1998, A&A, 336, 966
* Ramsay et al. (2002) Ramsay G., Zane S., Jimenez-Garate M. A., den Herder J., Hailey C. J., 2002, MNRAS, 337, 1185
* Rutledge et al. (2007) Rutledge R. E., Bildsten L., Brown E. F., Chakrabarty D., Pavlov G. G., Zavlin V. E., 2007, ApJ, 658, 514
* Sigut et al. (2009) Sigut T. A. A., McGill M. A., Jones C. E., 2009, ApJ, 699, 1973
* Villada et al. (1999) Villada M., Rossi C., Polcaro V. F., Giovannelli F., 1999, A&A, 344, 277
* Waters et al. (1988) Waters L. B. F. M., van den Heuvel E. P. J., Taylor A. R., Habets G. M. H. J., Persi P., 1988, A&A, 198, 200
* Watson et al. (1981) Watson M. G., Warwick R. S., Ricketts M. J., 1981, MNRAS, 195, 197
* Woods & Thompson (2006) Woods P. M., Thompson C., 2006, Soft gamma repeaters and anomalous X-ray pulsars: magnetar candidates. pp 547–586
|
arxiv-papers
| 2010-07-16T18:48:52 |
2024-09-04T02:49:11.670239
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lupin Chun-Che Lin, Jumpei Takata, Albert K. H. Kong and Chorng-Yuan\n Hwang",
"submitter": "Lupin Chun-Che Lin Lupin",
"url": "https://arxiv.org/abs/1007.2828"
}
|
1007.2935
|
# On the way to meet the experimental observation of persistent current in a
mesoscopic cylinder: A mean field study
Santanu K. Maiti santanu.maiti@saha.ac.in Theoretical Condensed Matter
Physics Division, Saha Institute of Nuclear Physics, Sector-I, Block-AF,
Bidhannagar, Kolkata-700 064, India Department of Physics, Narasinha Dutt
College, 129 Belilious Road, Howrah-711 101, India
###### Abstract
The behavior of persistent current in a mesoscopic cylinder threaded by an
Aharonov-Bohm flux $\phi$ is carefully investigated within a Hartree-Fock mean
field approach. We examine the combined effect of second-neighbor hopping
integral and Hubbard correlation on the enhancement of persistent current in
presence of disorder. A significant change in current amplitude is observed
compared to the traditional nearest-neighbor hopping model and the current
amplitude becomes quite comparable to experimental realizations. Our analysis
is found to exhibit several interesting results which have so far remained
unaddressed.
###### pacs:
73.23.-b, 73.23.Ra.
## I Introduction
Over the last many years appearance of persistent current in metallic single-
channel rings and multi-channel cylinders has drawn much attention in
theoretical as well as in experimental research. In mesoscopic regime where
dimensions of a system are comparable to mean free path of an electron the
phase coherence of electronic states is of fundamental importance and the
existence of dissipationless current in a mesoscopic conducting ring threaded
by an Aharonov-Bohm (AB) flux $\phi$ is a direct consequence of quantum phase
coherence. In this new quantum regime, two important aspects appear at low
temperatures and they are as follows.
$\bullet$ The phase coherence length $L_{\phi}$ i.e., the length scale for
which an electron maintains its phase memory, increases significantly with the
lowering of temperature and becomes comparable to the system size $L$.
$\bullet$ The energy levels of these small finite size systems are discrete.
These two are the most essential criteria for the existence of persistent
current in a small metallic ring/cylinder due to the application of an
external magnetic flux $\phi$. In the pioneering work of Büttiker, Imry and
Landauer butt , the appearance of persistent current in metallic rings has
been explored. Later, many excellent experiments levy ; chand ; mailly ; jari
; deb ; reu have been carried out in several ring and cylindrical geometries
to reveal the actual mechanisms of persistent current. Though much efforts
have been paid to study persistent current both theoretically cheu1 ; cheu2 ;
peeters1 ; peeters2 ; peeters3 ; mont ; mont1 ; alts ; von ; schm ; ambe ;
abra ; bouz ; giam ; yu ; belu ; ore ; xiao1 ; xiao2 ; san1 ; san2 ; san3 as
well as experimentally levy ; chand ; mailly ; jari ; deb ; reu , yet several
drawbacks still exist between the theory and experiment, and the full
knowledge about it in this scale is not well established even today.
The results of the single loop experiments are significantly different from
those for the ensemble of isolated loops. Persistent currents with expected
$\phi_{0}$ periodicity have been observed in isolated single Au rings chand
and in a GaAs-AlGaAs ring mailly . Levy et al. levy found oscillations with
period $\phi_{0}/2$ rather than $\phi_{0}$ in an ensemble of $10^{7}$
independent Cu rings. Similar $\phi_{0}/2$ oscillations were also reported for
an ensemble of disconnected $10^{5}$ Ag rings deb as well as for an array of
$10^{5}$ isolated GaAs-AlGaAs rings reu . In a recent experiment, Jariwala et
al. jari obtained both $\phi_{0}$ and $\phi_{0}/2$ periodic persistent
currents in an array of thirty diffusive mesoscopic Au rings. Except for the
case of the nearly ballistic GaAs-AlGaAs ring mailly , all the measured
currents are in general one or two orders of magnitude larger than those
expected from the theory.
Free electron theory predicts that at $T=0$, an ordered one-dimensional
metallic ring threaded by magnetic flux $\phi$ supports persistent current
with maximum amplitude $I_{0}=ev_{F}/L$, where $v_{F}$ is the Fermi velocity
and $L$ is the circumference of the ring. Metals are intrinsically disordered
which tends to decrease the persistent current, and the calculations show that
the disorder-averaged current $\langle I\rangle$ crucially depends on the
choice of the ensemble cheu2 ; mont ; mont1 . The magnitude of the current
$\langle I^{2}\rangle^{1/2}$ is however insensitive to the averaging issues,
and is of the order of $I_{0}l/L$, $l$ being the elastic mean free path of the
electrons. This expression remains valid even if one takes into account the
finite width of the ring by adding contributions from the transverse channels,
since disorder leads to a compensation between the channels cheu2 ; mont .
However, the measurements on an ensemble of $10^{7}$ Cu rings levy reported a
diamagnetic persistent current of average amplitude $3\times 10^{-3}$
$ev_{F}/L$ with half a flux-quantum periodicity. Such $\phi_{0}/2$
oscillations with diamagnetic response were also found in other persistent
current experiments consisting of ensemble of isolated rings deb ; reu .
Measurements on single isolated mesoscopic rings on the other hand detected
$\phi_{0}$-periodic persistent currents with amplitudes of the order of
$I_{0}\sim ev_{F}/L$, (closed to the value for an ordered ring). Theory and
experiment mailly seem to agree only when disorder is weak. In another recent
nice experiment Bluhm et al. blu have measured the magnetic response of $33$
individual cold mesoscopic gold rings, one ring at a time, using a scanning
SQUID technique. They have measured $h/e$ component and predicted that the
measured current amplitude agrees quite well with theory cheu1 in a single
ballistic ring mailly and an ensemble of $16$ nearly ballistic rings raba .
However, the amplitudes of the currents in single-isolated-diffusive gold
rings chand were two orders of magnitude larger than the theoretical
estimates. This discrepancy initiated intense theoretical activity, and it is
generally believed that the electron-electron correlation plays an important
role in the disordered diffusive rings abra ; bouz ; giam . An explanation
based on the perturbative calculation in presence of interaction and disorder
has been proposed and it seems to give a quantitative estimate closer to the
experimental results, but still it is less than the measured currents by an
order of magnitude, and the interaction parameter used in the theory is not
well understood physically. Most of these theoretical results have been
obtained based on a tight-binding framework within the nearest-neighbor
hopping (NNH) approximation. This is an important approximation and it has
been shown that within the NNH model electronic correlation provides a small
enhancement of current amplitude in disordered materials i.e., a weak
delocalizing effect is observed in presence of electron-electron (e-e)
interaction. As an attempt in the present work we modify the traditional NNH
model by incorporating the effects of higher order hopping integrals, at least
second-neighbor hopping (SNH), in addition to the NNH integral. It is also
quite physical since electrons have some finite probabilities to hop from one
site to other sites apart from nearest-neighbor with reduced strengths. We
will show that the inclusion of higher order hopping integrals gives
significant enhancement of current amplitude and it reaches quite closer to
the current amplitude of ordered systems.
The other important controversy comes for the determination of the sign of
low-field currents and still it is an unresolved issue between theoretical and
experimental results. In an experiment on persistent current Levy et al. levy
have shown diamagnetic nature for the measured currents at low-field limit.
While, in other experiment Chandrasekhar et al. chand have obtained
paramagnetic response near zero field limit. Jariwala et al. jari have
predicted diamagnetic persistent current in their experiment and similar
diamagnetic response in the vicinity of zero field limit were also supported
in an experiment done by Deblock deb et al. on Ag rings. Yu and Fowler yu
have shown both diamagnetic and paramagnetic responses in mesoscopic Hubbard
rings. Though in a theoretical work Cheung et al. cheu2 have predicted that
the direction of current is random depending on the total number of electrons
in the system and the specific realization of the random potentials. Hence,
prediction of the sign of low-field currents is still an open challenge and
further studies on persistent current in mesoscopic systems are needed to
remove the existing controversies.
In the present paper we address the behavior of persistent current in an
interacting mesoscopic ring with finite width threaded by an Aharonov-Bohm
flux $\phi$. A simple tight-binding Hamiltonian is used to illustrate the
system and all the calculations are performed within a mean field approach.
Using a generalized Hartree-Fock (HF) approximation kato ; kam , we compute
numerically persistent current ($I$) as functions of AB flux $\phi$, total
number of electrons $N_{e}$, e-e interaction strength $U$, second-neighbor
hopping integral, disorder strength $W$ and system size $N$. The main
motivation of the present work is to illustrate the effects of higher order
hopping integrals on the enhancement of persistent current in disordered
mesoscopic cylinders. Our results can be utilized to study magnetic response
in any interacting mesoscopic system.
In what follows, we present the results. In Section II, we describe the
geometric model and generalized Hartree-Fock theory to study magnetic response
in the model quantum system. Section III contains the numerical results.
Finally, in Section IV we draw our conclusions.
## II Model and synopsis of the theoretical formulation
Let us start by referring to Fig. 1, where a small metallic cylinder is
threaded by a magnetic flux $\phi$. The filled black circles correspond to the
positions of the atomic sites in the cylinder. To predict
Figure 1: (Color online). Schematic view of a $1$D mesoscopic cylinder
penetrated by a magnetic flux $\phi$. The red dashed line corresponds to the
second-neighbor hopping integral and the filled black circles represent the
positions of the atomic sites. A persistent current $I$ is established in the
cylinder.
the size of a cylinder we use two parameters $N$ and $M$, where the $1$st one
($N$) represents total number of atomic sites in each circular ring and the
other one ($M$) gives total number of identical circular rings. For the
description of our model quantum system we use a tight-binding (T-B) framework
and in order to incorporate the effect of higher order hopping integrals to
the Hamiltonian here we consider second-neighbor hopping (SNH) (shown by the
red dashed line in Fig. 1) in addition to the nearest-neighbor hopping (NNH)
of electrons. Considering both NNH and SNH integrals the T-B Hamiltonian for
the cylindrical system in Wannier basis looks in the form,
$\displaystyle H_{\mbox{c}}$ $\displaystyle=$
$\displaystyle\sum_{i,j,\sigma}\epsilon_{i,j,\sigma}c_{i,j,\sigma}^{\dagger}c_{i,j,\sigma}+\sum_{i,j,\sigma}t_{l}\left[e^{i\theta_{l}}c_{i,j,\sigma}^{\dagger}c_{i,j+1,\sigma}\right.$
(1) $\displaystyle+$
$\displaystyle\left.h.c.\right]+\sum_{i,j,\sigma}t_{d}\left[e^{i\theta_{d}}c_{i,j,\sigma}^{\dagger}c_{i+1,j+1,\sigma}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{ij}Uc_{i,j,\uparrow}^{\dagger}c_{i,j,\uparrow}c_{i,j,\downarrow}^{\dagger}c_{i,j,\downarrow}$
where, ($i,j$) represent the co-ordinate of a lattice site. The index $i$ runs
from $1$ to $M$, while the integer $j$ goes from $1$ to $N$.
$\epsilon_{i,j,\sigma}$ is the on-site energy of an electron at the site
($i,j$) of spin $\sigma$ ($\uparrow,\downarrow$). $t_{l}$ and $t_{d}$ are the
NNH and SNH integrals, respectively. Due to the presence of magnetic flux
$\phi$ (measured in unit of the elementary flux quantum $\phi_{0}=ch/e$), a
phase factor $\theta_{l}=2\pi\phi/N$ appears in the Hamiltonian when an
electron hops longitudinally from one site to its neighboring site, and
accordingly, a negative sign comes when the electron hops in the reverse
direction. $\theta_{d}$ is the associated phase factor for the diagonal motion
of an electron between two neighboring concentric rings. No phase factor
appears when an electron moves along the vertical direction which is set by
proper choice of the gauge for the vector potential $\vec{A}$ associated with
the magnetic field $\vec{B}$, and this choice makes the phase factors
($\theta_{l}$, $\theta_{d}$) identical to each other for the longitudinal and
diagonal motions. Since the magnetic field corresponding to the AB flux $\phi$
does not penetrate anywhere of the surface of the cylinder, we ignore Zeeman
term in the above tight-binding Hamiltonian (Eq. 1).
$c_{i,j,\sigma}^{\dagger}$ and $c_{i,j,\sigma}$ are the creation and
annihilation operators, respectively, of an electron at the site ($i,j$) with
spin $\sigma$. $U$ is the on-site Hubbard interaction term.
### II.1 Decoupling of the interacting Hamiltonian
To get the energy eigenvalues of the interacting model quantum system
described by the above tight-binding Hamiltonian given in Eq. 1, first we
decouple the interacting Hamiltonian using generalized Hartree-Fock approach,
the so-called mean field approximation. In this procedure, the full
Hamiltonian is completely decoupled into two parts. One is associated with the
up-spin electrons, while the other is related to the down-spin electrons with
their modified site energies. For up and down spin Hamiltonians, the modified
site energies are expressed in the form,
$\epsilon_{i,j,\uparrow}^{\prime}=\epsilon_{i,j,\uparrow}+U\langle
n_{i,j,\downarrow}\rangle$ (2)
$\epsilon_{i,j,\downarrow}^{\prime}=\epsilon_{i,j,\downarrow}+U\langle
n_{i,j,\uparrow}\rangle$ (3)
where, $n_{i,j,\sigma}=c_{i,j,\sigma}^{\dagger}c_{i,j,\sigma}$ is the number
operator. With these site energies, the full Hamiltonian (Eq. 1) can be
written in the decoupled form as,
$\displaystyle H_{\mbox{c}}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\epsilon_{i,j,\uparrow}^{\prime}n_{i,j,\uparrow}+\sum_{i,j}t_{l}\left[e^{i\theta_{l}}c_{i,j,\uparrow}^{\dagger}c_{i,j+1,\uparrow}+h.c.\right]$
(4) $\displaystyle+$
$\displaystyle\sum_{i,j}t_{d}\left[e^{i\theta_{d}}c_{i,j,\uparrow}^{\dagger}c_{i+1,j+1,\uparrow}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{i,j}\epsilon_{i,j,\downarrow}^{\prime}n_{i,j,\downarrow}+\sum_{i,j}t_{l}\left[e^{i\theta_{l}}c_{i,j,\downarrow}^{\dagger}c_{i,j+1,\downarrow}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{i,j}t_{d}\left[e^{i\theta_{d}}c_{i,j,\downarrow}^{\dagger}c_{i+1,j+1,\downarrow}+h.c.\right]$
$\displaystyle-$ $\displaystyle\sum_{i,j}U\langle
n_{i,j,\uparrow}\rangle\langle n_{i,j,\downarrow}\rangle$ $\displaystyle=$
$\displaystyle H_{\uparrow}+H_{\downarrow}-\sum_{i,j}U\langle
n_{i,j,\uparrow}\rangle\langle n_{i,j,\downarrow}\rangle$
where, $H_{\uparrow}$ and $H_{\downarrow}$ correspond to the effective tight-
binding Hamiltonians for the up and down spin electrons, respectively. The
last term is a constant term which provides an energy shift in the total
energy.
### II.2 Self consistent procedure
With these decoupled Hamiltonians ($H_{\uparrow}$ and $H_{\downarrow}$) of up
and down spin electrons, now we start our self consistent procedure
considering initial guess values of $\langle n_{i,j,\uparrow}\rangle$ and
$\langle n_{i,j,\downarrow}\rangle$. For these initial set of values of
$\langle n_{i,j,\uparrow}\rangle$ and $\langle n_{i,j,\downarrow}\rangle$, we
numerically diagonalize the up and down spin Hamiltonians. Then we calculate a
new set of values of $\langle n_{i,j,\uparrow}\rangle$ and $\langle
n_{i,j,\downarrow}\rangle$. These steps are repeated until a self consistent
solution is achieved.
### II.3 Calculation of ground state energy
After achieving the self consistent solution, the ground state energy $E_{0}$
for a particular filling at absolute zero temperature ($T=0$K) can be
determined by taking the sum of individual states up to Fermi energy ($E_{F}$)
for both up and down spins. Thus, we can write the final form of ground state
energy as,
$E_{0}=\sum_{p}E_{p,\uparrow}+\sum_{p}E_{p,\downarrow}-\sum_{i,j}U\langle
n_{i,j,\uparrow}\rangle\langle n_{i,j,\downarrow}\rangle$ (5)
where, the index $p$ runs for the states up to the Fermi level.
$E_{p,\uparrow}$ ($E_{p,\downarrow}$) is the single particle energy eigenvalue
for $p$-th eigenstate obtained by diagonalizing the Hamiltonian $H_{\uparrow}$
($H_{\downarrow}$).
### II.4 Calculation of persistent current
At absolute zero temperature, total persistent current of the system is
obtained from the expression,
$I(\phi)=-c\frac{\partial E_{0}(\phi)}{\partial\phi}$ (6)
where, $E_{0}(\phi)$ is the ground state energy for a particular filling.
In the present work we perform all the essential features of persistent
current at absolute zero temperature and use the units where $c=h=e=1$.
Throughout our numerical calculations we set the nearest-neighbor hopping
strength $t_{l}=-1$ and fix $M=2$ i.e., cylinders with two identical rings.
Energy scale is measured in unit of $t_{l}$.
## III Numerical results and discussion
Following the above theoretical prescription now we start to analyze our
numerical results. We describe the results in three different parts. In the
first part, we consider perfect cylinders with only nearest-neighbor hopping
integral. In the second part, disordered cylinders described with only NNH
integral are considered. Finally, in the third part we discuss the effect of
second-neighbor hopping (SNH) integral on the enhancement of persistent
current in disordered cylinders.
### III.1 Perfect cylinders with NNH integral
For perfect cylinders we choose
$\epsilon_{i,j,\uparrow}=\epsilon_{i,j,\downarrow}=0$ for all ($i,j$). Since
here we consider the cylinders described with NNH integral only, the second-
neighbor hopping strength $t_{d}$ is fixed to zero.
#### III.1.1 Energy-flux characteristics
As illustrative examples, in Fig. 2 we show the variation of ground state
energy levels as a function of magnetic flux $\phi$ for some typical
mesoscopic cylinders where $N$ is fixed at $5$ (odd $N$). In (a) the results
are given for the quarterly-filled ($N_{e}=5$) cylinders, while in (b) the
curves correspond to the results for the half-filled ($N_{e}=10$) cylinders.
The red, green and blue lines represent the ground state energy levels for
$U=0$, $0.5$ and $1$, respectively. It is observed that the ground state
energy shows oscillatory behavior as a function of $\phi$ and the energy
increases as the electronic correlation strength $U$ gets increased. Most
significantly we see that the ground state energy levels provide two
Figure 2: (Color online). Ground state energy levels as a function of flux
$\phi$ for some perfect cylinders with $N=5$ and $M=2$. The red, green and
blue curves correspond to $U=0$, $0.5$ and $1$, respectively. (a) Quarter-
filled case and (b) Half-filled case.
Figure 3: (Color online). Ground state energy levels as a function of flux
$\phi$ for some perfect cylinders considering $N=8$ and $M=2$. The red, green
and blue curves correspond to $U=0$, $0.5$ and $1$, respectively. (a) Quarter-
filled case and (b) Half-filled case.
different types of periodicities depending on the electron filling. At
quarter-filling, ground state energy level gives $\phi_{0}$ ($=1$, since
$c=e=h=1$ in our chosen unit system) flux-quantum periodicity. On the other
hand, at half-filling it shows $\phi_{0}/2$ flux-quantum periodicity. The
situation becomes quite different when the total number of atomic sites $N$ in
individual rings is even. For our illustrative purposes in Fig. 3 we plot the
lowest energy levels as a function of $\phi$ for some typical mesoscopic
cylinders considering $N=8$ (even $N$). The curves of different colors
correspond to the identical meaning as in Fig. 2. From the spectra given in
Figs. 3(a) (quarter-filled case) and (b) (half-filled case) it is clearly
observed that the ground state energy levels vary periodically with AB flux
$\phi$ exhibiting only $\phi_{0}$ flux-quantum periodicity. Thus it can be
emphasized that the appearance of half flux-quantum periodicity strongly
depends on the electron filling as well as on the oddness and evenness of the
total number of atomic sites $N$ in individual rings. Only for the half-filled
cylinders with odd $N$, the lowest energy level gets $\phi_{0}/2$ periodicity
with flux $\phi$. Now it is important to note that this half flux-quantum
periodicity does not depend on the width ($M$) of the cylinder and also it is
independent of the Hubbard correlation strength $U$. Hence, depending on the
system size and filling of electrons variable periodicities are observed in
the variation of lowest energy level. It may provide an important signature in
studying magnetic response in nano-scale loop geometries.
#### III.1.2 Current-flux characteristics
In Fig. 4 we display the current-flux characteristics for some impurity free
mesoscopic cylinders considering $M=2$. In (a) the
Figure 4: (Color online). Persistent current as a function of flux $\phi$ for
some ordered mesoscopic cylinders considering $M=2$. (a) Half-filled case with
$N=15$. The red, green and blue curves correspond to $U=0$, $1.5$ and $2$,
respectively. (b) Quarter-filled case with $N=20$. The red, green and blue
curves correspond to $U=0$, $2$ and $3$, respectively.
results are given for the half-filled case where we set $N=15$. The red line
corresponds to the current for the non-interacting ($U=0$) case, while the
green and blue lines represent the currents when $U=1.5$ and $2$,
respectively. From the curves we notice that the current amplitude gradually
decreases with the increase of electronic correlation strength $U$. The reason
is that at half-filling each site is occupied by at least one electron of up
spin or down spin, and the placing of a second electron of opposite spin needs
more energy due to the repulsive effect of $U$. Thus conduction becomes
difficult as it requires more energy when an electron hops from its own site
and situates at the neighboring site. Now both for the non-interacting and
interacting cases, current shows half flux-quantum periodicity as a function a
$\phi$ obeying the energy-flux characteristics since here we choose odd $N$
($N=15$). The behavior of the persistent currents for even $N$ is shown in (b)
where we set $N=20$. The currents are drawn for the quarter-filled case i.e.,
$N_{e}=20$, where the red, green and blue curves correspond to $U=0$, $2$ and
$3$, respectively. The reduction of current amplitude with the increase of
Hubbard interaction strength is also observed for this quarter-filled case,
similar to the case of half-filled as described earlier. But the point is that
at quarter-filling, the reduction of current amplitude is much smaller
compared to the half-filled situation. This is quite obvious in the sense that
at less than half-filling ‘empty’ lattice sites are available where electrons
can hop easily without any cost of extra energy and the conduction becomes
much easier than the half-filled situation. In this quarter-filled case,
persistent currents provide only $\phi_{0}$ flux-quantum periodicity following
the $E$-$\phi$ diagram. From these current-flux characteristics it can be
concluded that for ‘ordered’ cylinders current amplitude always decreases with
the enhancement in Hubbard correlation strength $U$.
### III.2 Disordered cylinders with NNH integral
In order to describe the effect of impurities on electron transport now we
focus our attention on the results of some typical disordered cylinders
described with NNH integral. Here we consider the diagonal disordered
cylinders i.e., impurities are introduced only at the site energies without
disturbing the hopping integrals. The site energies in each concentric ring
are chosen from a correlated distribution function which looks in the form,
$\epsilon_{j,\uparrow}=\epsilon_{j,\downarrow}=W\cos\left(j\lambda\pi\right)$
(7)
where, $W$ is the impurity strength. $\lambda$ is an irrational number and we
choose $\lambda=(1+\sqrt{5})/2$, for the sake of our illustration. Setting
$\lambda=0$, we get back the pure system with uniform site energy $W$. Now,
instead of considering site energies from a correlated distribution function,
as mentioned above in Eq. 7, we can also take them randomly from a “Box”
distribution function of width $W$. But in the later case we have to take the
average over a large number of disordered configurations (from the stand point
of statistical average) and since it is really a difficult task in the aspect
of numerical computation we select the other option. Not only that in the
averaging process several mesoscopic phenomena may disappear. Therefore, the
averaging process is an important issue in low-dimensional systems.
In presence of disorder, energy levels get modified significantly. For our
illustrative purposes in Fig. 5 we plot ground state energy levels as a
function of magnetic flux $\phi$ for some disordered mesoscopic cylinders when
they are half-filled. The Hubbard interaction strength $U$ is set at $1$ and
the impurity strength $W$ is fixed to $2$. In (a) the ground state energy
level is shown for a cylinder with $N=5$ (odd), while in (b) it is presented
for a cylinder taking $N=8$ (even). Quite interestingly we see that for the
cylinder with odd $N$, the half flux-quantum periodicity of the lowest energy
level disappears in the presence of impurity and it provides conventional
$\phi_{0}$ periodicity. Hence, for cylinders with odd $N$, $\phi_{0}/2$ flux-
quantum periodicity will be observed only when they are free from any
impurity. For the disordered cylinder with even $N$ ($N=8$), the lowest energy
level as usual provides $\phi_{0}$ periodicity similar to the impurity free
cylinders containing even $N$. Apart from this periodic nature, impurities
play another significant role in the determination of the slope of the
Figure 5: (Color online). Ground state energy level as a function of flux
$\phi$ for half-filled disordered mesoscopic cylinders ($M=2$) considering
$U=1$ and $W=2$. (a) $N=5$ and (b) $N=8$.
energy levels. The slope of the lowest energy level decreases significantly
compared to the perfect case, and therefore, a prominent change in current
amplitude also takes place.
To justify the above facts, in Fig. 6 we present the variations of persistent
currents with AB flux $\phi$ for a half-filled mesoscopic cylinder, described
in the framework of NNH model, considering $N=15$ and $M=2$. The red curve
represents the current for the ordered ($W=0$) non-interacting ($U=0$)
cylinder. It shows saw-tooth like nature with flux $\phi$ providing
$\phi_{0}/2$ flux-quantum periodicity. The situation becomes completely
different when impurities are introduced in the cylinder as seen by the other
two curves. The green curve represents the current for the case only when
impurities are considered but the effect of electronic correlation is not
taken into account. It shows a continuous like nature with $\phi_{0}$ flux-
quantum periodicity. The most important observation is that the current
amplitude gets reduced enormously, even an order of magnitude, compared to the
perfect cylinder. This is due to the localization of the energy eigenstates in
the presence of impurity, which is the so-called Anderson localization. Hence,
a large difference exists in the current amplitudes of an ordered and
disordered non-interacting cylinders and it was the main controversial issue
among the theoretical and experimental predictions. Experimental verifications
suggest that the measured current amplitude is quite comparable to the
theoretical current amplitude obtained in a perfect system. To remove this
controversy, as a first attempt, we include the effect of e-e correlation in
the disordered cylinder described by the NNH model. The result is shown by the
blue curve where $U$ is fixed at $1.5$. It is observed that the current
amplitude gets increased compared to the non-interacting disordered cylinder,
though the increment is too small. Not only that the enhancement can take
place only for small values of $U$, while for large enough $U$ the current
amplitude rather decreases. This phenomenon can be implemented as follows.
Figure 6: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder considering $N=15$ and $M=2$. The red line
corresponds to the ordered case when $U=0$, whereas the green and blue lines
correspond to the disordered case ($W=2$) when $U=0$ and $1.5$, respectively.
For the non-interacting disordered cylinder the probability of getting two
opposite spin electrons becomes higher at the atomic sites where the site
energies are lower than the other sites since the electrons get pinned at the
lower site energies to minimize the ground state energy, and this pinning of
electrons becomes increased with the rise of impurity strength $W$. As a
result the mobility of electrons and hence the current amplitude gets reduced
with the increase of impurity strength $W$. Now, if we introduce electronic
correlation in the system then it tries to depin two opposite spin electrons
those are situated together due to the Coulomb repulsion. Therefore, the
electronic mobility is enhanced which provides larger current amplitude. But,
for large enough interaction strength, no electron can able to hop from one
site to other at the half-filling since then each site is occupied either by
an up or down spin electron which does not allow other electron of opposite
spin due to the repulsive term $U$. Accordingly, the current amplitude
gradually decreases with $U$. On the other hand, at less than half-filling
though there is some finite probability to hop an electron from one site to
the other available ‘empty’ site but still it is very small. So, in brief, we
can say that within the nearest-neighbor hopping (NNH) approximation electron-
electron interaction does not provide any significant contribution to enhance
the current amplitude, and hence the controversy regarding the current
amplitude still persists.
### III.3 Disordered cylinders with NNH and SNH integrals
To overcome the existing situation regarding the current amplitude, in this
sub-section, finally we make an attempt by incorporating the effect of second-
neighbor hopping (SNH) integral in addition to the nearest-neighbor hopping
(NNH) integral.
A significant change in current amplitude takes place when we include the
contribution of second-neighbor hopping (SNH) integral in addition to the NNH
integral. As representative examples, in Fig. 7 we plot the current-flux
characteristics for a half-filled mesoscopic cylinder considering $N=15$ and
$M=2$. The black,
Figure 7: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder taking $N=15$ and $M=2$ in the presence of
NNH and SNH integrals. The black line corresponds to the ordered case when
$U=0$, whereas the magenta and gold lines correspond to the disordered case
($W=2$) when $U=0$ and $1.5$, respectively. Here SNH integral is fixed at
$-0.6$. The currents shown by the red, green and blue lines for the ring
described with NNH model (identical to Fig. 6) are re-plotted to judge the
effect of SNH integral over NNH model much clearly.
magenta and gold lines correspond to the results in the presence of SNH
integral, while the other three colored curves (red, green and blue) represent
the currents in the absence of SNH integral. Here we choose $t_{d}=-0.6$. The
black curve refers to the persistent current for the perfect ($W=0$) non-
interacting ($U=0$) cylinder and it achieves much higher amplitude compared to
the NNH model (red curve). This additional contribution comes from the SNH
integral since it allows electrons to hop further. In addition it is also
noticed that the current varies periodically with $\phi$ providing $\phi_{0}$
flux-quantum periodicity, instead of $\phi_{0}/2$ as in the case of NNH
integral model (red curve). Thus, it can be emphasized that $\phi_{0}/2$
periodicity will be observed only when the cylinder is (a) free from impurity,
(b) half-filled, (c) made with odd $N$, and (d) described by the nearest-
neighbor hopping model. The main focus of this sub-section is to interpret the
combined effect of SNH integral and electron-electron correlation on the
enhancement of persistent current amplitude in disordered cylinder. To do this
first we narrate the effect of SNH integral in disordered non-interacting
cylinder. The nature of the current for this particular case is shown by the
magenta curve of Fig. 7. It shows that the current amplitude gets reduced
compared to the perfect case (black line), which is expected, but the
reduction of the current amplitude is quite small than the NNH integral model.
This is due the fact that the SNH integral tries to delocalize the electronic
Figure 8: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder taking $N=15$ and $M=2$ in the presence of
NNH and SNH integrals. The black line corresponds to the ordered case when
$U=0$, whereas the magenta and gold lines correspond to the disordered case
($W=2$) when $U=0$ and $1.5$, respectively. Here SNH integral is fixed at
$-0.8$. The currents shown by the red, green and blue lines for the ring
described with NNH model (identical to Fig. 6) are re-plotted to judge the
effect of SNH integral over NNH model much clearly.
states, and therefore, the mobility of the electrons is enriched. The
situation becomes more interesting when we include the effect of Hubbard
interaction. The behavior of the current in the presence of interaction is
plotted by the gold curve of Fig. 7 where we fix $U=1.5$. Very interestingly
we see that the current amplitude is enhanced significantly and quite
comparable to that of the perfect cylinder.
For better clarity of the results discussed above, in Fig. 8 we also present
the similar feature of persistent current for other hopping strength of SNH
integral. Here we set $t_{d}=-0.8$. From these curves we see that the current
amplitude gets enhanced more as we increase the SNH strength.
Thus, it can be predicted that the presence of SNH integral and Hubbard
interaction can provide a persistent current which may be comparable to the
measured current amplitudes. In this presentation we consider the effect of
only SNH integral as a higher order hopping integral in addition to the NNH
model, and, illustrate how such a higher order hopping integral leads an
important role on the enhancement of current amplitude in presence of Hubbard
correlation for disordered cylinders. Instead of considering only the SNH
integral we can also take the contributions from all possible higher order
hopping integrals with reduced hopping strengths. Since the strengths of other
higher order hopping integrals are too small, the contributions from these
factors are reasonably small and they will not provide any significant change
in the current amplitude. Finally, we can say that further studies are needed
by incorporating all these factors.
## IV Closing remarks
To summarize, in the present work we have addressed the behavior of persistent
current in an interacting mesoscopic cylinder threaded by an Aharonov-Bohm
flux $\phi$. We have adopted a tight-binding Hamiltonian to describe the model
quantum system and all the numerical calculations have been done within a mean
field approximation. Using the generalized Hartree-Fock (HF) approximation, we
have computed persistent current as functions of SNH integral, impurity
strength $W$, AB flux $\phi$, electron filling $N_{e}/N$ and system size $N$.
Our numerical results have provided several interesting features and the
present study may be helpful in understanding magnetic response in nano-scale
loop geometries.
The essential features observed from our analysis are as follows.
(i) In the determination of the lowest energy level we see that the energy
level varies periodically with $\phi$ exhibiting both $\phi_{0}/2$ and
$\phi_{0}$ flux-quantum periodicities depending on the choices of the
parameters describing the tight-binding Hamiltonian.
(ii) In the NNH model, current amplitude gets significantly reduced when
impurities are introduced in the system. With the inclusion of Hubbard
interaction ($U$), the current amplitude can be enhanced, though the
enhancement becomes too small compared to the experimental verifications.
(iii) A significant change in the current amplitude takes place when the
effect of SNH integral is taken into account. The combined effect of SNH and
Hubbard interaction can provide the current which is quite comparable to the
experimental realizations. So in short we can say that the conventional NNH
model can be modified by incorporating the higher order hopping integrals.
Throughout the analysis we have kept the width of the cylinders at a fixed
value ($M=2$), for the sake of our illustration. All these results are also
valid for cylinders of larger widths. Here we have considered several
important approximations by ignoring the effects of temperature, electron-
phonon interaction, etc. Due to these factors, any scattering process that
appears in the cylinder would have influence on electronic phases. At the end,
we would like to say that we need further study in such systems by
incorporating all these effects.
ACKNOWLEDGMENTS
I acknowledge with deep sense of gratitude the illuminating comments and
suggestions I have received from Prof. S. N. Karmakar during the calculations.
## References
* (1) M. Büttiker, Y. Imry, and R. Landauer, Phys. Lett. 96A, 365 (1983).
* (2) L. P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. 64, 2074 (1990).
* (3) V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gallagher, and A. Kleinsasser, Phys. Rev. Lett. 67, 3578 (1991).
* (4) D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 (1993).
* (5) E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, and R. A. Webb, Phys. Rev. Lett. 86, 1594 (2001).
* (6) R. Deblock, R. Bel, B. Reulet, H. Bouchiat, and D. Mailly, Phys. Rev. Lett. 89, 206803 (2002).
* (7) B. Reulet, M. Ramin, H. Bouchiat, and D. Mailly, Phys. Rev. Lett. 75, 124 (1995).
* (8) H. F. Cheung, Y. Gefen, E. K. Riedel, and W. H. Shih, Phys. Rev. B 37, 6050 (1988).
* (9) H. F. Cheung and E. K. Riedel, Phys. Rev. Lett. 62, 587 (1989).
* (10) L. K. Castelano, G.-Q. Hai, B. Partoens, and F. M. Peeters, Phys. Rev. B 78, 195315 (2008).
* (11) M. Zarenia, M. J. Pereira, F. M. Peeters, and G. de Farias, Phys. Rev. B 81, 045431 (2010).
* (12) D. Y. Vodolazov, F. M. Peeters, T. T. Hongisto, and K. Yu. Arutyunov, Europhys. Lett. 75, 315 (2006).
* (13) G. Montambaux, H. Bouchiat, D. Sigeti, and R. Friesner, Phys. Rev. B 42, 7647 (1990).
* (14) H. Bouchiat and G. Montambaux, J. Phys. (Paris) 50, 2695 (1989).
* (15) B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. 66, 88 (1991).
* (16) F. von Oppen and E. K. Riedel, Phys. Rev. Lett. 66, 84 (1991).
* (17) A. Schmid, Phys. Rev. Lett. 66, 80 (1991).
* (18) V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65, 381 (1990).
* (19) M. Abraham and R. Berkovits, Phys. Rev. Lett. 70, 1509 (1993).
* (20) G. Bouzerar, D. Poilblanc, and G. Montambaux, Phys. Rev. B 49, 8258 (1994).
* (21) T. Giamarchi and B. S. Shastry, Phys. Rev. B 51, 10915 (1995).
* (22) N. Yu and M. Fowler, Phys. Rev. B 45, 11795 (1992).
* (23) S. Bellucci and P. Onorato, Physica E 41, 1393 (2009).
* (24) P. A. Orellana and M. Pacheco, Phys. Rev. B 71, 235330 (2005).
* (25) H.-M. Li and J.-L. Xiao, Physica B 396, 91 (2007).
* (26) H.-M. Li, Y.-L. Huang, and J.-L. Xiao, Int. J. Mod. Phys. B 22, 2255 (2008).
* (27) S. K. Maiti, Physica E 31, 117 (2006).
* (28) S. K. Maiti, Solid State Phenomena 155, 87 (2009).
* (29) S. K. Maiti, J. Chowdhury, and S. N. karmakar, J. Phys.: Condens. Matter 18, 5349 (2006).
* (30) H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, and K. A. Moler, Phys. Rev. Lett. 102, 136802 (2009).
* (31) W. Rabaud, L. Saminadayar, D. Mailly, K. Hasselbach, A. Benoit, and B. Etienne, Phys. Rev. Lett. 86, 3124 (2001).
* (32) H. Kato and D. Yoshioka, Phys. Rev. B 50, 4943 (1994).
* (33) A. Kambili, C. J. Lambert, and J. H. Jefferson, Phys. Rev. B 60, 7684 (1999).
|
arxiv-papers
| 2010-07-17T15:28:13 |
2024-09-04T02:49:11.680993
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1007.2935"
}
|
1007.2969
|
# Representation of Itô Integrals
by Lebesgue/Bochner Integrals
Qi Lü, Jiongmin Yong, and Xu Zhang School of Mathematics, Sichuan University,
Chengdu 610064, China. e-mail:luqi59@163.com. Department of Mathematics,
University of Central Florida, FL 32816, USA. This work was partially
supported by the NSF under grant DMS-1007514. e-mail:jyong@mail.ucf.edu. Key
Laboratory of Systems and Control, Academy of Mathematics and Systems
Sciences, Chinese Academy of Sciences, Beijing 100190, China; and Yangtze
Center of Mathematics, Sichuan University, Chengdu 610064, China. This work is
supported by the NSFC under grants 10831007, 60821091 and 60974035, and the
project MTM2008-03541 of the Spanish Ministry of Science and Innovation.
e-mail:xuzhang@amss.ac.cn.
###### Abstract
In [22], it was proved that as long as the integrand has certain properties,
the corresponding Itô integral can be written as a (parameterized) Lebesgue
integral (or a Bochner integral). In this paper, we show that such a question
can be answered in a more positive and refined way. To do this, we need to
characterize the dual of the Banach space of some vector-valued stochastic
processes having different integrability with respect to the time variable and
the probability measure. The later can be regarded as a variant of the
classical Riesz Representation Theorem, and therefore it will be useful in
studying other problems. Some remarkable consequences are presented as well,
including a reasonable definition of exact controllability for stochastic
differential equations and a condition which implies a Black-Scholes market to
be complete.
2010 Mathematics Subject Classification. Primary 60G05; Secondary 60H05,
60G07.
Key Words. Itô integral, Lesbegue integral, Bochner integral, range inclusion,
Riesz-type Representation Theorem.
## 1 Introduction
Let $(\Omega,{\cal F},{\mathbb{F}},{\mathbb{P}})$ be a complete filtered
probability space with ${\mathbb{F}}=\\{{\cal F}_{t}\\}_{t\geq 0}$, on which a
one-dimensional standard Brownian motion $\\{W(t)\\}_{t\geq 0}$ is defined so
that ${\mathbb{F}}$ is its natural filtration augmented by all the
${\mathbb{P}}$-null sets. Let $H$ be a Banach space with the norm
$|\cdot|_{H}$ and with the dual space $H^{*}$. For any $p\in[1,\infty)$, let
$L_{{\cal F}_{T}}^{p}(\Omega;H)$ be the set of all ${\cal F}_{T}$-measurable
($H$-valued) random variables $\xi:\Omega\to H$ such that
$\mathbb{E}|\xi|_{H}^{p}<\infty$. Next, for any $p,q\in[1,\infty)$, put
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;H))=\Big{\\{}\varphi:[0,T]\times\Omega\to
H\bigm{|}\varphi(\cdot)\hbox{ is ${\mathbb{F}}$-progressively measurable
and}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{\mathbb{E}}\Big{(}\int_{0}^{T}|\varphi(t)|_{H}^{q}dt\Big{)}^{\frac{p}{q}}<\infty\Big{\\}},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{q}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))=\Big{\\{}\varphi:[0,T]\times\Omega\to
H\bigm{|}\varphi(\cdot)\hbox{ is ${\mathbb{F}}$-progressively measurable
and}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\int_{0}^{T}\Big{(}{\mathbb{E}}|\varphi(t)|_{H}^{p}\Big{)}^{\frac{q}{p}}dt<\infty\Big{\\}}.\end{array}$
(1.1)
In an obvious way, we may also define (for $1\leq p,q<\infty$)
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle L^{\infty}_{\mathbb{F}}(\Omega;L^{q}(0,T;H)),\quad
L^{p}_{\mathbb{F}}(\Omega;L^{\infty}(0,T;H)),\quad
L^{\infty}_{\mathbb{F}}(\Omega;L^{\infty}(0,T;H)),\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle L^{\infty}_{\mathbb{F}}(0,T;L^{p}(\Omega;H)),\quad
L^{q}_{\mathbb{F}}(0,T;L^{\infty}(\Omega;H)),\quad
L^{\infty}_{\mathbb{F}}(0,T;L^{\infty}(\Omega;H)).\end{array}\right.$
It is clear that
$L^{p}_{\mathbb{F}}(\Omega;L^{p}(0,T;H))=L^{p}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))\equiv
L^{p}_{\mathbb{F}}(0,T;H),\qquad 1\leq p\leq\infty.$
Also, by Minkovski’s inequality, it holds that
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;H))\subseteq
L^{q}_{\mathbb{F}}(0,T;L^{p}(\Omega;H)),\qquad 1\leq p\leq q\leq\infty,\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{q}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))\subseteq
L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;H)),\qquad 1\leq q\leq
p\leq\infty.\end{array}\right.$ (1.2)
In particular,
$L^{1}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))\subseteq
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H)),\qquad 1\leq p\leq\infty.$ (1.3)
We now introduce two linear operators
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{I}}:L^{1}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))\to
L^{1}_{{\cal F}_{T}}(\Omega;H)\quad\hbox{(when }H\hbox{ is a Hilbert
space)},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{I}}\big{(}\zeta(\cdot)\big{)}=\int_{0}^{T}\zeta(t)dW(t),\qquad\forall\;\zeta(\cdot)\in
L^{1}_{\mathbb{F}}(\Omega;L^{2}(0,T;H)),\end{array}\right.$ (1.4)
and
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{L}}:L^{1}_{\mathbb{F}}(0,T;H)\to L^{1}_{{\cal
F}_{T}}(\Omega;H),\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{L}}\big{(}u(\cdot)\big{)}=\int_{0}^{T}u(t)dt,\qquad\forall\;u(\cdot)\in
L^{1}_{\mathbb{F}}(0,T;H).\end{array}\right.$ (1.5)
We call ${\mathbb{I}}$ and ${\mathbb{L}}$ the Itô integral operator and the
Lebesgue integral operator, respectively. It is clear that
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{I}}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))\Big{)}\subseteq
L^{p}_{{\cal F}_{T}}(\Omega;H),\qquad\forall\;p\in[1,\infty)\quad\hbox{(when
}H\hbox{ is a Hilbert space)},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{L}}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))\Big{)}\subseteq
L^{p}_{{\cal
F}_{T}}(\Omega;H),\qquad\forall\;p\in[1,\infty).\end{array}\right.$ (1.6)
The first inclusion in (1.6) can be refined (when $H$ is a Hilbert space).
Indeed, for any $\xi\in L^{p}_{{\cal F}_{T}}(\Omega;H)$ (with
$p\in[1,\infty)$), ${\mathbb{E}}[\xi\,|\,{\cal F}_{t}]$ is an $H$-valued
continuous $L^{p}$-martingale. Hence, by the Martingale Representation Theorem
([11]), there is a unique $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$ (called the Malliavin derivative
([17]) of $\xi$ and sometimes denoted by $D_{\cdot}\xi$) such that
${\mathbb{E}}[\xi\,|\,{\cal
F}_{t}]={\mathbb{E}}\xi+\int_{0}^{t}\zeta(s)dW(s),\qquad\forall\;t\in[0,T].$
(1.7)
In particular, by taking $t=T$ in the above, we see that
$\xi={\mathbb{E}}\xi+\int_{0}^{T}\zeta(s)dW(s).$ (1.8)
Therefore, in the case that $H$ is a Hilbert space, the first inclusion in
(1.6) can be refined to the following equality:
$L^{p}_{{\cal
F}_{T}}(\Omega;H)=H\oplus\Big{[}{\mathbb{I}}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))\Big{)}\Big{]},$
(1.9)
where “$\oplus$” stands for a direct sum. Now, for the second inclusion in
(1.6), we have the following simple result.
Proposition 1.1. Let $H$ be a Hilbert space and $p\in[1,\infty)$. Then
$\overline{{\mathbb{L}}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))\Big{)}}^{\,L^{p}_{{\cal
F}_{T}}(\Omega;H)}=L^{p}_{{\cal F}_{T}}(\Omega;H),$ (1.10)
where $\overline{G}^{\,L^{p}_{{\cal F}_{T}}(\Omega;H)}$ stands for the closure
of $G$ in $L^{p}_{{\cal F}_{T}}(\Omega;H)$.
Proof. For any $\zeta\in L^{p}_{{\cal F}_{T}}(\Omega;H)$, let
$\xi(t)={\mathbb{E}}[\zeta\,|\,{\cal F}_{t}],\qquad t\in[0,T].$
Then $\xi(\cdot)$ is an $H$-valued $L^{p}$-martingale. By the martingale
representation theorem and the Burkholder–Davis–Gundy’s inequality, we have
${\mathbb{E}}\big{|}\xi(t)-\zeta|^{p}_{H}\leq
C{\mathbb{E}}\Big{(}\int_{t}^{T}|D_{s}\zeta|^{2}ds\Big{)}^{p\over 2}\to
0,\qquad\hbox{as }t\to T.$
Now, for any $\delta>0$, let
$u_{\delta}(t)={\xi(T-\delta)\over\delta}I_{[T-\delta,T]}(t),\qquad
t\in[0,T].$
Then $u_{\delta}(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{\infty}(0,T;H))\bigcap
L^{\infty}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))\subseteq
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))$, and
${\mathbb{E}}\Big{|}\int_{0}^{T}u_{\delta}(t)dt-\zeta\Big{|}^{p}={\mathbb{E}}\big{|}\xi(T-\delta)-\zeta\big{|}^{p}\to
0,\qquad\hbox{as }\delta\to 0,$
proving the proposition.
Remark 1.2. From the proof of Proposition 1.1, it is easy to see that we have
proved the following stronger result than (1.10):
$\overline{{\mathbb{L}}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{\infty}(0,T;H))\Big{)}}^{\,L^{p}_{{\cal
F}_{T}}(\Omega;H)}=\overline{{\mathbb{L}}\Big{(}L^{\infty}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))\Big{)}}^{\,L^{p}_{{\cal
F}_{T}}(\Omega;H)}=L^{p}_{{\cal F}_{T}}(\Omega;H).$ (1.11)
From Proposition 1.1, it is seen that we do not expect to have a refinement
for the Lebesgue integral operator ${\mathbb{L}}$ similar to (1.9). Instead,
it is very natural for us to pose the following problem:
Problem (E) Whether the following is true:
${\mathbb{L}}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))\Big{)}=L^{p}_{{\cal
F}_{T}}(\Omega;H)\;?$ (1.12)
Note that the above is equivalent to the following: When the range of the
operator ${\mathbb{L}}:L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))\to L^{p}_{{\cal
F}_{T}}(\Omega;H)$ is closed. An interesting problem closely related to the
above, taking into account (1.9), reads as follows.
Problem (R) Under what additional conditions on $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$, there will be a $u(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))$ such that the following holds
$\int_{0}^{T}\zeta(t)dW(t)=\int_{0}^{T}u(t)dt\;\qquad\hbox{\rm a.s.{ }}?$
(1.13)
For convenience, any $u(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))$
satisfying (1.13) is called a representor of $\zeta(\cdot)$. Since the Itô
integral in the usual sense can only be defined on Hilbert spaces, we pose
Problem (R) for the case that $H$ is a Hilbert space. It is clear that when
$u(\cdot)$ is a representor of $\zeta(\cdot)$ so is $u(\cdot)+v(\cdot)$ as
long as $\displaystyle\int_{0}^{T}v(t)dt=0$, almost surely. Therefore, if
$\zeta(\cdot)$ admits one representor, it admits infinitely many representors.
Problem (R) with $H={\mathbb{R}}$ was posed and studied in [22]. Various
integrability conditions were imposed on $\zeta(\cdot)$ so that it admits a
representor. Let us now briefly recall several relevant results from [22],
which will give us some feelings about the representation (1.13). To this end,
we define
$u_{\alpha}(s)\equiv{1-\alpha\over{(T-s)^{\alpha}}}\int_{0}^{s}{{\zeta(t)}\over{(T-t)^{1-\alpha}}}dW(t),\qquad
s\in[0,T),$ (1.14)
for $\alpha\in[0,1)$. The following is a summary of the relevant results
presented in [22].
Theorem 1.3. (i) Let $p\geq 1$. For any $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;{\mathbb{R}}))$,
$u_{0}(\cdot)\equiv\int_{0}^{\cdot}{{\zeta(t)}\over{T-t}}dW(t)\in\bigcup_{\varepsilon>0}L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T\negthinspace\negthinspace-\varepsilon;{\mathbb{R}})),$
(1.15)
and $(\ref{R})$ holds with $u(\cdot)=u_{0}(\cdot)$ in the following sense:
$\lim_{\varepsilon\to
0}{\mathbb{E}}\left|\int_{0}^{T-\varepsilon}u_{0}(t)dt-\int_{0}^{T}\zeta(t)dW(t)\right|^{p}=0.$
(1.16)
(ii) Suppose $\zeta(\cdot)\in
L^{1}_{\mathbb{F}}(0,T;L^{2}(\Omega;{\mathbb{R}}))$ such that
$\int_{0}^{T}\left[\int_{0}^{s}{{{\mathbb{E}}|\zeta(t)|^{2}}\over(T-t)^{2}}dt\right]^{1\over
2}ds<\infty.$ (1.17)
Then
$u_{0}(\cdot)\equiv\int_{0}^{\cdot}{{\zeta(t)}\over{T-t}}dW(t)\in
L^{1}_{\mathbb{F}}(0,T;{\mathbb{R}}),$ (1.18)
and $(\ref{R})$ holds with $u(\cdot)=u_{0}(\cdot)$.
(iii) Suppose $\zeta(\cdot)\in L^{1}_{\mathbb{F}}(0,T;{\mathbb{R}})$ such that
for some $\delta>0$ the following holds:
$\int_{0}^{T}{{{\mathbb{E}}|\zeta(t)|^{2}}\over{(T-t)^{\delta}}}dt<\infty.$
(1.19)
Then
$u_{\alpha}(\cdot)\in
L^{2}_{\mathbb{F}}(\Omega;L^{q}(0,T;{\mathbb{R}})),\qquad\forall\;\alpha\in\hbox{$({1-\delta\over
2},{1\over q})$}\bigcap[0,1],\quad
q\in\hbox{$[1,{2\over{2-\min(\delta,1)}})$},$ (1.20)
and
$u_{\alpha}(\cdot)\in
L^{q}_{\mathbb{F}}(0,T;{\mathbb{R}}),\qquad\forall\;\alpha\in\hbox{$(1-{\delta\over
2}-{1\over q},{1\over q})$}\bigcap[0,1],\quad
q\in\hbox{$[1,{2\over{2-\min(\delta,1)}})$}.$ (1.21)
Moreover, $(\ref{R})$ holds with $u(\cdot)=u_{\alpha}(\cdot)$.
(iv) Suppose $\zeta(\cdot)\in L^{p}_{\mathbb{F}}(0,T;{\mathbb{R}})$ for some
$p>2$. Then
$u_{\alpha}(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;{\mathbb{R}})),\qquad\forall\;\alpha\in\hbox{$({1\over
2},{1\over 2}+{1\over p})$}\bigcap[0,1],\quad
q\in\hbox{$[1,{2p\over{p+2}})$}.$ (1.22)
Moreover, $(\ref{R})$ holds with $u(\cdot)=u_{\alpha}(\cdot)$.
The above shows that there are many $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;{\mathbb{R}}))$ such that one can find a
corresponding representor $u(\cdot)$.
Note that although Problem (R) is posed for the case $H$ is a Hilbert space,
Problem (E) can be posed for general Banach space since Itô’s integral is not
involved here. The main purpose of this paper is to give a positive answer to
Problem (E) when $H$ is a Banach space with $H^{*}$ having the Radon–Nikodým
property. Our result seems to be a little surprising in some sense, and it
refines the results of [22] on Problem (R). More precisely, when the answer to
Problem (E) is positive, any $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$ (when $H$ is a Hilbert space) admits
a representor $u(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;H))$, without
assuming further integrability conditions on $\zeta(\cdot)$. This means that
an Itô’s integral on a given (fixed) interval can be represented by a
(parameterized) Bochner integral on that interval. We should emphasize here
that any representor $u(\cdot)$ of $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$ depends on $T$, in general. In
another word, it will be more proper to write
$\int_{0}^{T}\zeta(t)dW(t)=\int_{0}^{T}u(t,T)dt,\qquad\hbox{\rm a.s.{ }}$
(1.23)
Hence, by allowing the upper limit to change, we should have
$\int_{0}^{s}\zeta(t)dW(t)=\int_{0}^{s}u(t,s)dt,\qquad\forall\;s\in[0,T],\quad\hbox{\rm
a.s.{ }}$ (1.24)
According to Theorem 1.3, when $\zeta(\cdot)$ satisfies certain (better)
integrability conditions, we can find a representor of the following form:
$u(t,s)={1-\alpha\over(s-t)^{\alpha}}\int_{0}^{t}{\zeta(r)\over(s-r)^{1-\alpha}}dW(r),\qquad
0\leq t<s\leq T,$ (1.25)
for some $\alpha\in[0,1)$. Clearly, such an $s\mapsto u(t,s)$ is smooth in
$s\in(t,T]$. Therefore it is natural to further ask the following question,
without assuming the better integrability conditions on $\zeta(\cdot)$.
Problem (C) For any $\zeta(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$,
whether it has a representor $u(t,s)$ which is continuous with respect to the
variable $s$?
We will also show that the answer to Problem (C) is positive. Note that, since
the Itô integral $\displaystyle s\mapsto\int_{0}^{s}\zeta(t)dW(t)$ is at most
Hölder continuous up to order ${1\over 2}$, generally, one cannot expect that
the differentiability of $s\mapsto u(t,s)$ (given in (1.24)). Nevertheless, it
is natural to expect that $s\mapsto u(t,s)$ is Hölder continuous up to order
${1\over 2}$. But, we do not have a proof for this yet.
Remark 1.4. The fact that $u(\cdot)$ in (1.23) depends on $T$ tells us that,
the positive answer to Problem (E) does not mean that Itô integrals can be
completely replaced by (parameterized) Bochner integrals.
The rest of this paper is organized as follows. In Section 2, as a preliminary
result, we establish a Riesz-type Representation Theorem for the dual of the
Banach space $L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))$ (see Subsection 2.1 for
its definition). An interesting byproduct in this section is the
characterization on the dual of $L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;H))$ and
$L^{q}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))$, which will be useful in some
problems appeared in stochastic distributed parameter control systems and/or
stochastic partial differential equations. Section 3 is addressed to giving
answers to Problems (E) and (R). Section 4 is devoted to answering Problem
(C), for which the key tool we employ is the continuous selection theorem in
[15]. In Section 5, we present two remarkable consequences of our positive
solution to Problem (E), one of which is related to the reasonable formulation
of exact controllability for stochastic differential equations, and the other
a condition to guarantee a Black-Scholes market to be complete.
## 2 The Dual of $L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))$
As a key preliminary to answer Problem (E), we need to characterize the dual
of $L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;H))$ and
$L^{q}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))$. We will go a little further by
considering the dual of $L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))$, which will be
be defined below. It seems to us that this result has its own interest.
### 2.1 Statement of the result
Let $(X_{1},{\cal M}_{1},\mu_{1})$ and $(X_{2},{\cal M}_{2},\mu_{2})$ be two
finite measure spaces. Let ${\cal M}$ be a sub-$\sigma$-field of ${\cal
M}_{1}\otimes{\cal M}_{2}$ (the $\sigma$-field generated by ${\cal
M}_{1}\times{\cal M}_{2}$), and for any $1\leq p,q<\infty$, let
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))=\Big{\\{}\varphi:X_{1}\times X_{2}\to
H\bigm{|}\varphi(\cdot)\hbox{ is ${\cal M}$-measurable and}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_{X_{1}}\left(\int_{X_{2}}|\varphi(x_{1},x_{2})|_{H}^{q}d\mu_{2}\right)^{p\over
q}d\mu_{1}<\infty\Big{\\}}.\end{array}$
Likewise, let
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{\infty}_{\cal M}(X_{1};L^{q}(X_{2};H))=\Big{\\{}\varphi:X_{1}\times
X_{2}\to H\bigm{|}\varphi(\cdot)\hbox{ is ${\cal M}$-measurable and}\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathop{\rm
esssup}_{x_{1}\in
X_{1}}\left(\int_{X_{2}}|\varphi(x_{1},x_{2})|_{H}^{q}d\mu_{2}\right)^{1\over
q}<\infty\Big{\\}},\end{array}$ $\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt
minus 2.0pt\cr\displaystyle L^{p}_{\cal
M}(X_{1};L^{\infty}(X_{2};H))=\Big{\\{}\varphi:X_{1}\times X_{2}\to
H\bigm{|}\varphi(\cdot)\hbox{ is ${\cal M}$-measurable and}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\int_{X_{1}}\left(\mathop{\rm
esssup}_{x_{2}\in
X_{2}}|\varphi(x_{1},x_{2})|_{H}^{p}\right)d\mu_{1}<\infty\Big{\\}},\end{array}$
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{\infty}_{\cal M}(X_{1};L^{\infty}(X_{2};H))=\Big{\\{}\varphi:X_{1}\times
X_{2}\to H\bigm{|}\varphi(\cdot)\hbox{ is ${\cal M}$-measurable and}\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathop{\rm
esssup}_{(x_{1},x_{2})\in X_{1}\times
X_{2}}|\varphi(x_{1},x_{2})|_{H}<\infty\Big{\\}}.\end{array}$
We denote
$L^{p}_{\cal M}(X_{1}\times X_{2};H)=L^{p}_{\cal
M}(X_{1};L^{p}(X_{2};H)),\qquad 1\leq p\leq\infty.$
Also, for any $f\in L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))$ ($1\leq
p,q\leq\infty$), we denote
$\|f\|_{p,q,H}\equiv\|f\|_{L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};H))}\mathop{\buildrel\Delta\over{=}}\left[\int_{X_{1}}\left(\int_{X_{2}}|f(x_{1},x_{2})|_{H}^{q}d\mu_{2}\right)^{p\over
q}d\mu_{1}\right]^{1\over p}.$ (2.1)
The definition of $\|f\|_{\infty,q,H}$, $\|f\|_{p,\infty,H}$ and
$\|f\|_{\infty,\infty,H}$ are obvious. Let
$\|f\|_{p,H}\equiv\|f\|_{p,p,H}\;,\qquad\qquad 1\leq p\leq\infty.$ (2.2)
The definition of $L^{q}_{\cal M}(X_{2};L^{p}(X_{1};H))$ ($1\leq
p,q\leq\infty$) is similar. By Hölder’s inequality and Minkovski’s inequality,
we have the following inclusions:
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))\subseteq L^{r}_{\cal
M}(X_{1};L^{s}(X_{2};H)),\qquad 1\leq r\leq p\leq\infty,\quad 1\leq s\leq
q\leq\infty,\end{array}$ (2.3)
and (comparing with (1.2)–(1.3)),
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))\subseteq
L^{q}_{\cal M}(X_{2};L^{p}(X_{1};H)),\qquad 1\leq p\leq q\leq\infty,\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};H))\supseteq L^{q}_{\cal M}(X_{2};L^{p}(X_{1};H)),\qquad
1\leq q\leq p\leq\infty.\end{array}\right.$ (2.4)
Next, for any $p\in[1,\infty]$, denote
$p^{\prime}=\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{p\over p-1},\qquad 1<p<\infty,\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\displaystyle 1,\qquad\qquad p=\infty,\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr\displaystyle\infty,\qquad\quad~{}p=1.\end{array}\right.$
The definition of $q^{\prime}\in[1,\infty]$ for $q\in[1,\infty]$ is similar.
We have the following result.
Lemma 2.1. Let $H$ be a Banach space, $(X_{1},{\cal M}_{1},\mu_{1})$ and
$(X_{2},{\cal M}_{2},\mu_{2})$ be two finite measure spaces, ${\cal M}$ be a
sub-$\sigma$-field of ${\cal M}_{1}\otimes{\cal M}_{2}$, and let $1\leq
p,q<\infty$. Then, $H^{*}$ has the Radon–Nikodým property with respect to
$(X_{1}\times X_{2},{\cal M},\mu_{1}\times\mu_{2})$ if and only if for any
$F\in L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))^{*}$, there exists a unique $g\in
L^{p^{\prime}}_{\cal M}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$ such that
$F(f)=\int_{X_{1}\times
X_{2}}(f(x_{1},x_{2}),g(x_{1},x_{2}))_{H,H^{*}}d\mu_{1}d\mu_{2},\qquad\forall\;f\in
L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H)),$ (2.5)
and
$\|F\|_{L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};H))^{*}}=\|g\|_{p^{\prime},q^{\prime},H^{*}}.$ (2.6)
Due to the above result, we make the following identification (for the case
that $H^{*}$ has the Radon–Nikodým property with respect to $(X_{1}\times
X_{2},{\cal M},\mu_{1}\times\mu_{2})$):
$L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))^{*}=L^{p^{\prime}}_{\cal
M}(X_{1};L^{q^{\prime}}(X_{2};H^{*})),\qquad 1\leq p,q<\infty.$ (2.7)
The above is a Riesz-type Representation Theorem for the dual of space
$L^{p}_{\cal M}(X_{1};L^{q}(X_{2};H))$. It seems to us that Lemma 2.1 should
be a known result but we have not found an exact reference. Therefore, for
reader’s convenience, we provide a detailed proof in the next three
subsections. As a corollary of Lemma 2.1, we will characterize the dual of
$L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,T;H))$ and
$L^{q}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))$ in the last subsection.
The main idea for the proof of Lemma 2.1 is similar to that of the relevant
result in [4, Appendix B, pp. 375–376] (see also [7, Theorem 1, Chapter IV,
pp. 98–99]). However, Lemma 2.1 does not follow from the main result in [4,
Appendix B] because the later considered only the special case that $p=q$ and
$H={\mathbb{R}}$, for which, by Fubini’s Theorem, one can reduce the problem
to the case with one measure on the product space. Also, Lemma 2.1 does not
seem to be a corollary of [7, Theorem 1, Chapter IV, pp. 98–99] because of the
very fact that our ${\cal M}$ is an “interconnecting” sub-$\sigma$-field of
the $\sigma$-field generated by ${\cal M}_{1}\times{\cal M}_{2}$.
### 2.2 Proof of the necessity in Lemma 2.1 for the case $H={\mathbb{R}}$
As a key step to prove Lemma 2.1, in this subsection we show first the “only
if” part of this lemma for the special case $H={\mathbb{R}}$.
For any $g\in L^{p^{\prime}}_{\cal
M}(X_{1};L^{q^{\prime}}(X_{2};{\mathbb{R}}))$, define $F_{g}:L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))\mapsto{\mathbb{R}}$ by
$F_{g}(f)=\int_{X_{1}\times
X_{2}}f(x_{1},x_{2})g(x_{1},x_{2})d\mu_{1}d\mu_{2},\qquad\forall\;f\in
L^{p}_{\cal M}(X_{1};L^{q}(X_{2};{\mathbb{R}})).$
By the linearity of the integral, $g\mapsto F_{g}$ is a linear map. It follows
from Hölder’s inequality that
$|F_{g}(f)|\leq\|f\|_{p,q,{\mathbb{R}}}\|g\|_{p^{\prime},q^{\prime},{\mathbb{R}}},\qquad\forall\;f\in
L^{p}_{\cal M}(X_{1};L^{q}(X_{2};{\mathbb{R}})).$
Hence $F_{g}\in L^{p}_{\cal M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}$ and
$\|F_{g}\|_{L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}}\leq\|g\|_{p^{\prime},q^{\prime},{\mathbb{R}}}.$
(2.8)
Therefore, $g\mapsto F_{g}$ is a linear non-expanding map. Now, we show that
this map is surjective and is an isometry.
To show the surjectivity of $g\mapsto F_{g}$, take any $F\in L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}$. Since for any $A\in{\cal M}$,
$I_{A}\in L^{p}_{\cal M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))$, we may define
$\nu(A)=F(I_{A}),\qquad\forall\;A\in{\cal M}.$
Then $\nu$ is a totally finite signed measure on $(X_{1}\times X_{2},{\cal
M})$, and $\nu<\negthinspace\negthinspace\negthinspace<\mu_{1}\times\mu_{2}$.
By the Radon-Nikodým Theorem, there is an ${\cal M}$-measurable map $g\in
L^{1}_{\cal M}(X_{1}\times X_{2};{\mathbb{R}})$ such that
$\nu(A)=\int_{A}gd\mu_{1}d\mu_{2},\qquad\forall\;A\in{\cal M},$
i.e.,
$F(I_{A})=\int_{X_{1}\times
X_{2}}gI_{A}d\mu_{1}d\mu_{2},\qquad\forall\;A\in{\cal M}.$
Consequently, for any ${\cal M}$-measurable simple functions $f$,
$F(f)=\int_{X_{1}\times X_{2}}f(x_{1},x_{2})g(x_{1},x_{2})d\mu_{1}d\mu_{2}.$
Select a sequence $\\{A_{n}\\}_{n=1}^{\infty}\subset{\cal M}$ such that
$A_{n}\subset A_{n+1},\quad
n=1,2,\cdots,\qquad(\mu_{1}\times\mu_{2})\left(\big{(}X_{1}\times
X_{2}\big{)}\setminus\bigcup_{n=1}^{\infty}A_{n}\right)=0,$ (2.9)
and $g$ is bounded on each $A_{n}$. For any $n\geq 1$, note that
$f\mapsto\int_{X_{1}\times
X_{2}}f(x_{1},x_{2})g(x_{1},x_{2})I_{A_{n}}(x_{1},x_{2})d\mu_{1}d\mu_{2}$
is a bounded linear functional on $L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))$ which agrees with $F$ on all ${\cal
M}$-measurable simple functions which vanishes off $A_{n}$. It follows that
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
F(fI_{A_{n}})=\int_{X_{1}\times
X_{2}}fgI_{A_{n}}d\mu_{1}d\mu_{2},\qquad\forall\;f\in L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}})).\end{array}$ (2.10)
Since $gI_{A_{n}}$ is bounded, one has $gI_{A_{n}}\in L^{p^{\prime}}_{\cal
M}(X_{1};L^{q^{\prime}}(X_{2};{\mathbb{R}}))$. We claim that $g\in
L^{p^{\prime}}_{\cal M}(X_{1};L^{q^{\prime}}(X_{2};{\mathbb{R}}))$, and
$\|g\|_{p^{\prime},q^{\prime};{\mathbb{R}}}\leq\|F\|_{L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}}.$ (2.11)
To show this, we distinguish four cases.
Case 1: $p,q\in(1,\infty)$. Choose
$f=\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle
a\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{{p^{\prime}\over
q^{\prime}}-1}|g|^{q^{\prime}-1}(\hbox{\rm sgn$\,$}g)I_{A_{n}},&\hbox{ if
}\displaystyle\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\neq 0,\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle 0,&\hbox{ if
}\displaystyle\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}=0,\end{array}\right.$
where
$a=\left[\int_{X_{1}}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{p^{\prime}\over
q^{\prime}}d\mu_{1}\right]^{{1\over p^{\prime}}-1}.$
Then
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|f\|_{p,q}=\left[\int_{X_{1}}\left(\int_{X_{2}}|f|^{q}d\mu_{2}\right)^{p\over
q}d\mu_{1}\right]^{1\over p}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle=\left\\{\int_{X_{1}}\left[\int_{X_{2}}a^{q}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{\Big{(}{p^{\prime}\over
q^{\prime}}-1\Big{)}q}|g|^{(q^{\prime}-1)q}I_{A_{n}}d\mu_{2}\right]^{p\over
q}d\mu_{1}\right\\}^{1\over p}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle=a\left\\{\int_{X_{1}}\left[\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{\Big{(}{p^{\prime}\over
q^{\prime}}-1\Big{)}p}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{p\over
q}\right]d\mu_{1}\right\\}^{1\over p}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle=a\left\\{\int_{X_{1}}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{p^{\prime}\over
q^{\prime}}d\mu_{1}\right\\}^{1\over p}=1.\end{array}$
Taking the above $f$ in (2.10), we find that
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
F(f)=\int_{X_{1}}\int_{X_{2}}fgI_{A_{n}}d\mu_{2}d\mu_{1}=a\int_{X_{1}}\left[\int_{X_{2}}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{{p^{\prime}\over
q^{\prime}}-1}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right]d\mu_{1}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad=a\int_{X_{1}}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{p^{\prime}\over
q^{\prime}}d\mu_{1}=\left[\int_{X_{1}}\left(\int_{X_{2}}|g|^{q^{\prime}}I_{A_{n}}d\mu_{2}\right)^{p^{\prime}\over
q^{\prime}}d\mu_{1}\right]^{1\over p^{\prime}}\\\ \vskip 6.0pt plus 2.0pt
minus
2.0pt\cr\displaystyle\qquad\qquad=\|gI_{A_{n}}\|_{p^{\prime},q^{\prime};{\mathbb{R}}},\end{array}$
which gives
$\|gI_{A_{n}}\|_{p^{\prime},q^{\prime};{\mathbb{R}}}\leq\|F\|_{L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}}.$
Letting $n\to\infty$, by making use of Fatou’s Lemma, one concludes (2.11).
Case 2: $p=1$, $1<q<\infty$. In this case, we first take $p\in(1,\infty)$, and
take $f$ as in Case 1. Then
$\|f\|_{1,q}=\int_{X_{1}}\left(\int_{X_{2}}|f|^{q}d\mu_{2}\right)^{1\over
q}d\mu_{1}\leq\Big{[}\int_{X_{1}}\left(\int_{X_{2}}|f|^{q}d\mu_{2}\right)^{p\over
q}d\mu_{1}\Big{]}^{1\over p}\mu_{1}(X_{1})^{1\over
p^{\prime}}=\mu_{1}(X_{1})^{1\over p^{\prime}}.$
Consequently,
$\|gI_{A_{n}}\|_{p^{\prime},q^{\prime};{\mathbb{R}}}=F(f)\leq\|F\|_{L^{1}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}}\|f\|_{1,q}\leq\|F\|_{L^{1}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}}\mu_{1}(X_{1})^{1\over p^{\prime}}.$
Letting $n\to\infty$ and then letting $p\to 1$ (which means
$p^{\prime}\to\infty$), we obtain
$\|g\|_{\infty,q^{\prime};{\mathbb{R}}}\leq\|F\|_{L^{1}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}},$ (2.12)
which is (2.11) for the case $p=1$.
Case 3: $1<p<\infty$, $q=1$. In this case, we first take $q\in(1,\infty)$, and
take $f$ as in Case 1. Then
$\|f\|_{p,1;{\mathbb{R}}}=\left[\int_{X_{1}}\left(\int_{X_{2}}|f|d\mu_{2}\right)^{p}d\mu_{1}\right]^{1\over
p}\leq\left[\int_{X_{1}}\left(\int_{X_{2}}|f|^{q}d\mu_{2}\right)^{p\over
q}d\mu_{1}\right]^{1\over p}\mu_{2}(X_{2})^{1\over
q^{\prime}}=\mu_{2}(X_{2})^{1\over q^{\prime}}.$
Hence,
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|gI_{A_{n}}\|_{p^{\prime},q^{\prime};{\mathbb{R}}}=F(f)\leq\|F\|_{L^{p}_{\cal
M}(X_{1};L^{1}(X_{2};{\mathbb{R}}))^{*}}\|f\|_{p,1;{\mathbb{R}}}\leq\|F\|_{L^{p}_{\cal
M}(X_{1};L^{1}(X_{2};{\mathbb{R}}))^{*}}\mu_{2}(X_{1})^{1\over
q^{\prime}}\end{array}$
Letting $n\to\infty$ and then letting $q\to 1$ (which means
$q^{\prime}\to\infty$), we obtain
$\|g\|_{p^{\prime},\infty;{\mathbb{R}}}\leq\|F\|_{L^{p}_{\cal
M}(X_{1};L^{1}(X_{2};{\mathbb{R}}))^{*}},$ (2.13)
which is the case of (2.11) for $q=1$.
Case 4: $p=q=1$. In this case, we still first let $p,q\in(1,\infty)$, and take
$f$ as in Case 1 with $q=r$. Then
$\begin{array}[]{ll}\displaystyle\|f\|_{1,1}=\int_{X_{1}}\int_{X_{2}}|f|d\mu_{2}d\mu_{1}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\leq\left[\int_{X_{1}}\left(\int_{X_{2}}|f|^{q}d\mu_{2}\right)^{p\over
q}d\mu_{1}\right]^{1\over p}\mu_{1}(X_{1})^{1\over
p^{\prime}}\mu_{2}(X_{2})^{1\over q^{\prime}}=\mu_{1}(X_{1})^{1\over
p^{\prime}}\mu_{2}(X_{2})^{1\over q^{\prime}}.\end{array}$
Consequently,
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|gI_{A_{n}}\|_{p^{\prime},q^{\prime};{\mathbb{R}}}=F(f)\leq\|F\|_{L^{1}_{\cal
M}(X_{1};L^{1}(X_{2};{\mathbb{R}}))^{*}}\|f\|_{1,1}\leq\|F\|_{L^{1}_{\cal
M}(X_{1};L^{1}(X_{2};{\mathbb{R}}))^{*}}\mu_{1}(X_{1})^{1\over
p^{\prime}}\mu_{2}(X_{1})^{1\over q^{\prime}}.\end{array}$
Letting $n\to\infty$ and then letting $p,q\to 1$ (which means
$p^{\prime},q^{\prime}\to\infty$), we obtain
$\|g\|_{\infty;{\mathbb{R}}}\leq\|F\|_{L^{1}_{\cal
M}(X_{1};L^{1}(X_{2};{\mathbb{R}}))^{*}},$ (2.14)
which is the case of (2.11) for $p,q=1$.
Finally, (2.8) means that $F_{g}\in(L^{p}_{\cal
M}(X_{1};L^{q}(X_{2};{\mathbb{R}})))^{*}$ and since $F$ and $F_{g}$ coincides
on a dense subset of $L^{p}_{\cal M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))$, one
has $F=F_{g}$. Also, (2.6) follows easily from (2.8) and (2.11).
### 2.3 Proof of the necessity in Lemma 2.1 for the general case
We are now in a position to prove the “only if” part of Lemma 2.1 for the
general case. The proof is divided into two steps.
Step 1. We show that $L_{{\cal
M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$ is isometrically
isomorphic to a subspace $\cal H$ of $L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}$.
For any given $g\in L_{{\cal
M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$, define a linear
functional $F_{g}$ on $L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))$ as follows:
$F_{g}(f)=\int_{X_{1}\times X_{2}}\langle
f(x_{1},x_{2}),g(x_{1},x_{2})\rangle_{H,H^{*}}d\mu_{1}d\mu_{2},\qquad\forall
f\in L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H)).$ (2.15)
Then, by means of the Hölder inequality and similar to (2.8), we conclude that
$F_{g}$ belongs to $L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}$, and
$\|F_{g}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\leq\|g\|_{p^{\prime}q^{\prime},H^{*}}.$
(2.16)
Therefore the norm of $F_{g}$ is not greater than
$\|g\|_{p^{\prime}q^{\prime},H^{*}}$. Define
${\cal H}\equiv\\{F_{g}\;|\;g\in L_{{\cal
M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))\\}.$
It remains to prove the reverse of inequality (2.16). Clearly, without loss of
generality, we may assume that $g\not=0$.
Suppose first that $\displaystyle g=\sum_{i=1}^{\infty}h_{i}^{*}I_{E_{i}}$
where $h_{i}^{*}$ is a sequence in $H^{*}$ and $\\{E_{i}\\}_{i=1}^{\infty}$ is
a countable partition of $X_{1}\times X_{2}$ by members of ${\cal M}$ with
$(\mu_{1}\times\mu_{2})(E_{i})>0$ for all $i$. Since we have shown that
$L^{p}_{\cal M}(X_{1};L^{q}(X_{2};{\mathbb{R}}))^{*}=L^{p^{\prime}}_{\cal
M}(X_{1};L^{q^{\prime}}(X_{2};{\mathbb{R}}))$ (in Subsection 2.2) and noting
that $0<|g|_{H^{*}}\in L^{p^{\prime}}_{\cal
M}(X_{1};L^{q^{\prime}}(X_{2};{\mathbb{R}}))$, for any $\varepsilon>0$, there
exists a nonnegative function $\varphi\in L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};\mathbb{R}))$ such that
$0<\|\varphi\|_{p,q}\leq
1,\qquad\|g\|_{p^{\prime}q^{\prime},H^{*}}-\varepsilon\leq\int_{X_{1}\times
X_{2}}|g|_{H^{*}}\varphi d\mu_{1}d\mu_{2}.$
Further, choose $h_{i}\in H$ with $|h_{i}|_{H}=1$ such that
$|h_{i}^{*}|_{H^{*}}-\frac{\varepsilon}{\|\varphi\|_{1,1}}\leq
h_{i}^{*}(h_{i}),$
and define
$f=\sum_{i=1}^{\infty}\varphi h_{i}I_{E_{i}}\in L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H)).$
Then we have that $\|f\|_{p,q,H}=\|\varphi\|_{p,q}\leq 1$, and we have that
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\int_{X_{1}\times
X_{2}}\mathop{\langle}f(x_{1},x_{2}),g(x_{1},x_{2})\rangle_{H,H^{*}}d\mu_{1}d\mu_{2}=\int_{X_{1}\times
X_{2}}\varphi\sum_{i=1}^{\infty}\mathop{\langle}h_{i},h_{i}^{*}\rangle_{H,H^{*}}\chi_{E_{i}}d\mu_{1}d\mu_{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\geq\int_{X_{1}\times
X_{2}}\varphi\sum_{i=1}^{\infty}\Big{(}|h_{i}^{*}|_{H^{*}}-\frac{\varepsilon}{\|\varphi\|_{1,1}}\Big{)}\chi_{E_{i}}d\mu_{1}d\mu_{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\geq\int_{X_{1}\times
X_{2}}|g|_{H^{*}}\varphi
d\mu_{1}d\mu_{2}-\frac{\varepsilon}{\|\varphi\|_{1,1}}\int_{X_{1}\times
X_{2}}\varphi
d\mu_{1}d\mu_{2}\geq\|g\|_{p^{\prime},q^{\prime},H^{*}}-2\varepsilon.\end{array}$
This gives
$\|F_{g}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\geq\|g\|_{p^{\prime}q^{\prime},H^{*}},$
and therefore
$\|F_{g}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}=\|g\|_{p^{\prime}q^{\prime},H^{*}},$
whenever $g\in L_{{\cal M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$
is countably valued.
For the general case, we choose a sequence $\\{g_{n}\\}_{n=1}^{\infty}\subset
L_{{\cal M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$ such that each
$g_{n}$ is countably valued and
$\lim_{n\to\infty}\|g_{n}-g\|_{p^{\prime},q^{\prime},H^{*}}=0.$ (2.17)
We have obtained that
$\|F_{g_{n}}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}=\|g_{n}\|_{p^{\prime},q^{\prime},H^{*}},$
and by virtue of (2.16),
$\|F_{g_{n}}-F_{g}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}=\|F_{g_{n}-g}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\leq\|g_{n}-g\|_{p^{\prime},q^{\prime},H^{*}}.$
Therefore, noting (2.17), we end up with
$\displaystyle\|F_{g}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}=\lim_{n\to\infty}\|F_{g_{n}}\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}=\lim_{n\to\infty}\|g_{n}\|_{p^{\prime}q^{\prime},H^{*}}=\|g\|_{p^{\prime}q^{\prime},H^{*}}.$
Hence we get that $L_{{\cal
M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$ is isometrically
isomorphic to $\cal H$.
Step 2. We show that the subspace $\cal H$ is equal to $L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}$.
To this end, for $F\in L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}$, we define
$G(E)(h)=F(hI_{E}),\qquad\forall\;E\in{\cal M},\ h\in H.$ (2.18)
By
$|F(hI_{E})|\leq\|F\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\|hI_{E}\|_{p,q,H}\leq\|F\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}|h|_{H}\|I_{E}\|_{p,q},$
we see that $G:{\cal M}\to H^{*}$ and it is countably additive. Let
$E_{1},\cdots,E_{n}$ ($n\in{\mathbb{N}}$) be a partition of $X_{1}\times
X_{2}$ by members of ${\cal M}$ with $(\mu_{1}\times\mu_{2})(E_{i})>0$ for all
$1\leq i\leq n$. Then $G(E_{i})\in H^{*}$. Define
$G_{E_{i}}^{1}(h)={\mathop{\rm Re}\,}G(E)(h),\quad
G_{E_{i}}^{2}(h)={\mathop{\rm Im}\,}G(E)(h),\qquad\forall h\in H.$
Clearly,
$|G(E_{i})|_{H^{*}}\leq|G_{E_{i}}^{1}|_{H^{*}}+|G_{E_{i}}^{2}|_{H^{*}}$.
Noting that both $G_{E_{i}}^{1}$ and $G_{E_{i}}^{2}$ are real functionals, we
see that, for any $\varepsilon>0$, one can find $h_{i}^{1}$ and $h_{i}^{2}$ in
the closed unit ball of $H$ such that
$|G_{E_{i}}^{1}|_{H^{*}}-\frac{\varepsilon}{2n}<{\mathop{\rm
Re}\,}G(E_{i})(h_{i}^{1}),\qquad|G_{E_{i}}^{2}|_{H^{*}}-\frac{\varepsilon}{2n}<{\mathop{\rm
Im}\,}G(E_{i})(h_{i}^{2}).$
It follows that
$\displaystyle\sum_{i=1}^{n}|G(E_{i})|_{H^{*}}-\varepsilon<{\mathop{\rm
Re}\,}\sum_{i=1}^{n}G(E_{i})(h_{i}^{1})+{\mathop{\rm
Im}\,}\sum_{i=1}^{n}G(E_{i})(h_{i}^{2})$
$\displaystyle\qquad\qquad\qquad\qquad={\mathop{\rm
Re}\,}F\Big{(}\sum_{i=1}^{n}h_{i}^{1}I_{E_{i}}\Big{)}+{\mathop{\rm
Im}\,}F\Big{(}\sum_{i=1}^{n}h_{i}^{2}I_{E_{i}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\leq\|F\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\left(\Big{\|}\sum_{i=1}^{n}h_{i}^{1}I_{E_{i}}\Big{\|}_{p,q,H}+\Big{\|}\sum_{i=1}^{n}h_{i}^{2}I_{E_{i}}\Big{\|}_{p,q,H}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\leq 2\|F\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\Big{\|}\sum_{i=1}^{n}I_{E_{i}}\Big{\|}_{p,q}$
$\displaystyle\qquad\qquad\qquad\qquad\leq 2\|F\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}\mu_{1}(X_{1})^{\frac{1}{p}}\mu_{2}(X_{2})^{\frac{1}{q}}.$
Hence $|G(X_{1}\times X_{2})|_{H^{*}}<\infty$ and $G$ is a
$(\mu_{1}\times\mu_{2})$-continuous vector-valued measure of bounded
variation. Since $H^{*}$ has the Radon-Nikodým property with respect to
$(X_{1}\times X_{2},{\cal M},\mu_{1}\times\mu_{2})$, there exists a Bochner
integrable $g:X_{1}\times X_{2}\to H^{*}$ such that
$G(E)=\int_{E}gd\mu_{1}d\mu_{2},\qquad\forall\;E\in{\cal M}.$ (2.19)
Clearly, if $f\in L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))$ is a simple
function, then
$F(f)=\int_{X_{1}\times X_{2}}\langle
f(x_{1},x_{2}),g(x_{1},x_{2})\rangle_{H,H^{*}}d\mu_{1}d\mu_{2}.$
Select an expanding sequence $\\{E_{n}\\}_{n=1}^{\infty}$ in ${\cal M}$ such
that $\displaystyle\bigcup_{n=1}^{\infty}E_{n}=X_{1}\times X_{2}$ and such
that $g$ is bounded on each $E_{n}$. Fixing arbitrarily an
$n_{0}\in{\mathbb{N}}$ and noting that
$\displaystyle\int_{E_{n_{0}}}\langle\cdot,g(x_{1},x_{2})\rangle_{H,H^{*}}d\mu_{1}d\mu_{2}$
is a bounded linear functional on $L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))$
which agrees with $F$ on all simple functions supported on $E_{n_{0}}$, it
follows that
$F(fI_{E_{n_{0}}})=\int_{X_{1}\times X_{2}}\langle
f(x_{1},x_{2}),g(x_{1},x_{2})I_{E_{n_{0}}}\rangle_{H,H^{*}}d\mu_{1}d\mu_{2},\quad\forall\;f\in
L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H)).$ (2.20)
Further, since $gI_{E_{n_{0}}}$ is bounded, one has $gI_{E_{n_{0}}}\in
L_{{\cal M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$ and
$\|gI_{E_{n_{0}}}\|_{p^{\prime},q^{\prime},H^{*}}\leq\|F\|_{L_{{\cal
M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}}.$ (2.21)
Since inequality (2.21) holds for each $n_{0}$, by the Monotone Convergence
Theorem, we conclude that $g\in L_{{\cal
M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$.
Finally, for any $f\in L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))$, it follows
from (2.20) that
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
F(f)=\lim_{n\to\infty}\int_{X_{1}\times X_{2}}\langle
f(x_{1},x_{2}),g(x_{1},x_{2})I_{E_{n}}\rangle_{H,H^{*}}d\mu_{1}d\mu_{2}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\qquad\;=\int_{X_{1}\times
X_{2}}\langle
f(x_{1},x_{2}),g(x_{1},x_{2})\rangle_{H,H^{*}}d\mu_{1}d\mu_{2}=F_{g}(f).\end{array}$
This means that $F=F_{g}$. Hence $L_{{\cal M}}^{p}(X_{1};L^{q}(X_{2};H))^{*}$
coincides with $L_{{\cal M}}^{p^{\prime}}(X_{1};L^{q^{\prime}}(X_{2};H^{*}))$.
### 2.4 Proof of the sufficiency in Lemma 2.1
In order to complete the proof of Lemma 2.1, it remains to prove its “if”
part, which is the main concern in this subsection.
Let $G:{\cal M}\to H^{*}$ be a $(\mu_{1}\times\mu_{2})$-continuous vector
measure of bounded variation. We want to show that there exists a
$\widetilde{g}\in L^{1}_{\cal M}(X_{1};L^{1}(X_{2};H^{*}))$ such that
$G(E)=\int_{E}\widetilde{g}d\mu_{1}d\mu_{2},\qquad\forall E\in{\cal M}.$
(2.22)
Firstly, we show that if $E_{0}\in{\cal M}$ has a positive
$(\mu_{1}\times\mu_{2})$-measure, then $G$ has a Bochner integrable
Radon–Nikodým derivative on an ${\cal M}$-measurable set $B$ satisfying
$B\subset E_{0}$ and $(\mu_{1}\times\mu_{2})(B)>0$.
Denote by $|G|$ the variation of $G$, which is a scalar measure (see [7,
Definition 4 and Proposition 9 of Chapter 1, pp.2–3]). It is easy to see that
$|G|$ is a $(\mu_{1}\times\mu_{2})$-continuous ${\mathbb{R}}^{+}$-valued
measure. Applying the Radon–Nikodým Theorem (to $|G|$ and
$\mu_{1}\times\mu_{2}$), one can find an ${\cal M}$-measurable subset $B$ of
$E_{0}$ and a positive integer $k$ such that $|G|(A)\leq
k(\mu_{1}\times\mu_{2})(A)$ for all $A\in{\cal M}$ with $A\subset B$. Define a
linear functional $\ell$ on the subspace ${\cal S}$ of simple functions in
$L_{{\cal M}}^{p}(X_{1},L^{q}(X_{2},H))$ as follows:
$\ell(f)=\sum_{i=1}^{n}G(E_{i}\cap B)(x_{i}),$
where
$f=\sum_{i=1}^{n}x_{i}I_{E_{i}},\qquad x_{i}\in H,\quad 1\leq i\leq n,$
with $\\{E_{i},\;1\leq i\leq n\\}$ being a partition of $X_{1}\times X_{2}$.
It follows that
$\begin{array}[]{ll}\displaystyle|\ell(f)|=\Big{|}\sum_{i=1}^{n}G(E_{i}\cap
B)(x_{i})\Big{|}=\Big{|}\sum_{i=1}^{n}\frac{G(E_{i}\cap
B)}{(\mu_{1}\times\mu_{2})(E_{i}\cap
B)}\Big{(}(\mu_{1}\times\mu_{2})(E_{i}\cap B)x_{i}\Big{)}\Big{|}\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\;\;\leq\sum_{i=1}^{n}k|(\mu_{1}\times\mu_{2})(E_{i}\cap
B)x_{i}|\leq k\|f\|_{L^{1}(X_{i}\times X_{2};H)}\leq
k\mu_{1}(X_{1})^{\frac{1}{p}}\mu_{2}(X_{2})^{\frac{1}{q}}\|f\|_{L^{p}_{{\cal
M}}(X_{1};L^{q}(X_{2};H))}.\end{array}$
Therefore $\ell$ is a bounded linear functional on ${\cal S}$. By the Hahn-
Banach Theorem, it has a bounded linear extension to $L_{{\cal
M}}^{p}(X_{1},L^{q}(X_{2},H))$ (The extension is still denoted by $\ell$).
Hence there exists a $g\in L_{{\cal
M}}^{p^{\prime}}(X_{1},L^{q^{\prime}}(X_{2},H^{*}))$ such that
$\ell(f)=\int_{X_{1}\times X_{2}}\langle
f,g\rangle_{H,H^{*}}d\mu_{1}d\mu_{2}\qquad\forall\;f\in L_{{\cal
M}}^{p}(X_{1},L^{q}(X_{2},H)).$
We have
$G(E\cap B)(x)=\ell(xI_{E})=\int_{E}\langle
x,g\rangle_{H,H^{*}}d\mu_{1}d\mu_{2},\qquad\forall x\in H,~{}E\in M.$
Since $g\in L_{{\cal M}}^{p^{\prime}}(X_{1},L^{q^{\prime}}(X_{2},H^{*}))$ is
Bochner integrable, we see that
$G(E\cap B)(x)=\Big{(}\int_{E}gd\mu_{1}d\mu_{2}\Big{)}(x),\qquad\forall x\in
H,\quad E\in M.$
Consequently,
$\displaystyle G(E\cap B)=\int_{E}gd\mu_{1}d\mu_{2},\qquad\forall\;E\in{\cal
M}.$ (2.23)
Noting that $B\in{\cal M}$, and therefore replacing $E$ in (2.23) by $E\cap
B$, we see that
$G(E\cap B)=\int_{E\cap B}gd\mu_{1}d\mu_{2},\qquad\forall\;E\in{\cal M}.$
Now by the Exhaustion Lemma ([7, page 70]), there exist a sequence
$\displaystyle\\{A_{n}\\}_{n=1}^{\infty}$ of disjoint members of ${\cal M}$
such that $\displaystyle\bigcup_{n=1}^{\infty}A_{n}=X_{1}\times X_{2}$ and a
sequence $\displaystyle\\{g_{n}\\}_{n=1}^{\infty}$ of Bochner integrable
functions on $X_{1}\times X_{2}$ such that
$G(E\cap A_{n})=\int_{E\cap A_{n}}g_{n}d\mu_{1}d\mu_{2},\qquad\forall
E\in{\cal M},\quad n\in{\mathbb{N}}.$
Define $\widetilde{g}:X_{1}\times X_{2}\to H^{*}$ by
$\widetilde{g}(x_{1},x_{2})=g_{n}(x_{1},x_{2})$ if $(x_{1},x_{2})\in A_{n}$.
It is obvious that $\widetilde{g}$ is $(\mu_{1}\times\mu_{2})$-measurable.
Moreover, for each $E\in{\cal M}$ and all $m\in{\mathbb{N}}$, it holds
$G\Big{(}E\bigcap\Big{(}\bigcup_{n=1}^{m}A_{n}\Big{)}\Big{)}=\int_{E}\widetilde{g}I_{\cup_{n=1}^{m}A_{n}}d\mu_{1}d\mu_{2}.$
Consequently,
$G(E)=\lim_{m\to\infty}\int_{E}\widetilde{g}I_{\cup_{n=1}^{m}A_{n}}d\mu_{1}d\mu_{2},\qquad\forall
E\in{\cal M}.$
For $h\in H^{**}$, the variation
$|G(h)|(X_{1}\times X_{2})\geq\lim_{m\to\infty}\int_{X_{1}\times
X_{2}}|\langle
h,\widetilde{g}\rangle_{H^{**},H^{*}}|I_{\cup_{n=1}^{m}A_{n}}d\mu_{1}d\mu_{2}.$
Hence by the Monotone Convergence Theorem, $\langle
h,\widetilde{g}\rangle_{H^{**},H^{*}}\in L^{1}_{{\cal
M}}(X_{1};L^{1}(X_{2};{\mathbb{R}}))$ for each $h\in H^{**}$. If $E\in{\cal
M}$ and $h\in H^{**}$, from the Dominate Convergence Theorem, we have
$\displaystyle\langle
h,G(E)\rangle_{H^{**},H^{*}}=\lim_{m\to\infty}\int_{X_{1}\times X_{2}}\langle
h,\widetilde{g}\rangle_{H^{**},H^{*}}I_{\cup_{n=1}^{m}A_{n}}d\mu_{1}d\mu_{2}$
$\displaystyle\qquad\qquad\qquad\quad\;=\int_{X_{1}\times X_{2}}\langle
h,\widetilde{g}\rangle_{H^{**},H^{*}}d\mu_{1}d\mu_{2}.$
Therefore $\widetilde{g}$ is Pettis integrable and its Pettis integration
P-$\displaystyle\int_{X_{1}\times X_{2}}\widetilde{g}d\mu_{1}d\mu_{2}=G(E)$
for each $E\in{\cal M}$. Since $|G|(X_{1}\times X_{2})$ is finite,
$\displaystyle\int_{X_{1}\times
X_{2}}|\widetilde{g}|_{H^{*}}I_{\cup_{n=1}^{m}A_{n}}d\mu_{1}d\mu_{2}\leq|G|(X_{1}\times
X_{2})$ for all $m\in{\mathbb{N}}$. By the Monotone Convergence Theorem,
$|\widetilde{g}|_{H^{*}}\in L^{1}_{{\cal
M}}(X_{1};L^{1}(X_{2};{\mathbb{R}}))$. Hence $\widetilde{g}$ is Bochner
integrable. Since the Pettis and Bochner integrals coincide whenever they
coexist, we obtain (2.22), proving the Radon-Nikodým property of $H^{*}$ with
respect to $(X_{1}\times X_{2},{\cal M},\mu_{1}\times\mu_{2})$.
### 2.5 A corollary of Lemma 2.1
We now look an interesting corollary of Lemma 2.1. We first state the
following.
Lemma 2.2. Let
${\cal M}=\Big{\\{}A\in{\cal B}[0,T]\otimes{\cal F}_{T}\bigm{|}t\mapsto
I_{A}(t,\cdot)\hbox{ is ${\mathbb{F}}$-progressively measurable }\Big{\\}}.$
(2.24)
Then ${\cal M}$ is a sub-$\sigma$-field of ${\cal B}[0,T]\otimes{\cal F}_{T}$.
Moreover, a process $\varphi:[0,T]\times\Omega\to H$ is
${\mathbb{F}}$-progressively measurable if and only if it is ${\cal
M}$-measurable.
Remark 2.3. It is easy to see that the same conclusion in Lemma 2.2 holds for
any given filtration ${\mathbb{F}}$ (i.e., it is not necessarily the natural
filtration generated by the Brownian motion $\\{W(t)\\}_{t\geq 0}$), and also
if one replaces the ${\mathbb{F}}$-progressive measurability by any other
measurability requirement, for examples, adapted, optional or predictable,
etc.
According to Lemmas 2.1 and 2.2, we have the following interesting corollary,
whose proof is straightforward.
Corollary 2.4. Let $0<s\leq T$ and $H^{*}$ have the Radon–Nikodým property
with respect to $([0,T]\times\Omega,{\cal M},m\times{\mathbb{P}})$ (where $m$
is the Lebesgue measure). Then the following identities hold:
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle
L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,s;H))^{*}=L^{p^{\prime}}_{\mathbb{F}}(\Omega;L^{q^{\prime}}(0,s;H^{*})),\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
L^{q}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))^{*}=L^{q^{\prime}}_{\mathbb{F}}(0,s;L^{p^{\prime}}(\Omega;H^{*})).\end{array}\right.\qquad
1\leq p,q<\infty.$ (2.25)
The above is a Riesz-type Representation Theorem for the dual of spaces
$L^{p}_{\mathbb{F}}(\Omega;L^{q}(0,s;H))$ and
$L^{q}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$, which will be very useful below.
We refer to [14] for an application of Corollary 2.4 in the study of null
controllability of forward stochastic heat equations with one control. We will
give more application of this result in our forthcoming papers;
## 3 Answers to Problems (E) and (R)
In this section, we return to our complete filtered probability space
$(\Omega,{\cal F},{\mathbb{F}},{\mathbb{P}})$ and give answers to Problems (E)
and (R).
For any $p\in[1,\infty)$ and $0<s\leq T$, define an operator
${\mathbb{L}}_{s}:L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H))\to L^{p}_{{\cal
F}_{s}}(\Omega;H)$ by
${\mathbb{L}}_{s}\big{(}u(\cdot)\big{)}=\int_{0}^{s}u(t)dt,\qquad\forall\;u(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H)).$
Concerning Problem (E), noting that
$L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\subseteq
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H))$, we give the following positive
answer (which is a little stronger than the desired (1.12)):
Theorem 3.1. If $H^{*}$ has the Radon-Nikodým property, then
${\mathbb{L}}_{s}\Big{(}L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\Big{)}=L^{p}_{{\cal
F}_{s}}(\Omega;H).$ (3.1)
Moreover, for each $\phi(\cdot,s)\in L^{p}_{{\cal F}_{s}}(\Omega;H)$, there is
a $\varsigma(\cdot,s)\in L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$ such that
$\left\\{\begin{array}[]{ll}\displaystyle{\mathbb{L}}_{s}\big{(}\varsigma(\cdot,s)\big{)}=\phi(\cdot,s),\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|\varsigma(\cdot,s)\|_{L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))}\leq\|\phi(\cdot,s)\|_{L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))}.\end{array}\right.$
(3.2)
(In general, the above $\varsigma(\cdot,s)$ is NOT unique.)
The result in Theorem 3.1 turns out to be sharp for $p\in(1,\infty)$. Indeed,
we have the following result of negative nature.
Theorem 3.2. For any $p\in(1,\infty)$ and any $r\in(1,\infty]$, it holds that
${\mathbb{L}}_{s}\Big{(}L^{r}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\Big{)}\subsetneq
L^{p}_{{\cal F}_{s}}(\Omega;H).$ (3.3)
Remark 3.3. 1) In [6, VI, 68, pp. 130–131] and [8], some Radon-Nikodým type
theorems were established for real-valued or vector-valued processes with
finite variation. However, it seems that none of these results could be
applied to prove Theorem 3.1.
2) Thanks to Remark 2.3, the conclusion in Theorem 3.1 holds for any given
filtration ${\mathbb{F}}$; and one may replace the ${\mathbb{F}}$-progressive
measurability by any other measurability requirement.
3) We believe that (3.1) is sharp in the sense that, for any $r\in(1,\infty]$
and any $p\in[1,\infty]$,
$\left\\{\begin{array}[]{ll}{\mathbb{L}}_{s}\Big{(}L^{r}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\Big{)}\subsetneq
L^{p}_{{\cal F}_{s}}(\Omega;H),\\\\[5.69054pt]
{\mathbb{L}}_{s}\Big{(}L^{p}_{\mathbb{F}}(\Omega;L^{r}(0,s;H))\Big{)}\subsetneq
L^{p}_{{\cal F}_{s}}(\Omega;H).\end{array}\right.$ (3.4)
Theorem 3.2 shows that the first conclusion in (3.4) is true for
$p\in(1,\infty)$, and that, noting (1.2), the second conclusion in (3.4) is
true for $p\in(1,r]\cap(1,\infty)$. The general case is under our
investigation. Note that the above can also be written as
${\mathbb{L}}_{s}\left(\bigcup_{q>1}L_{\mathbb{F}}^{p}(\Omega;L^{q}(0,s;H))\right)\subsetneq
L^{p}_{{\cal F}_{s}}(\Omega;H).$ (3.5)
As a consequence of Theorem 3.1 and the Martingale Representation Theorem, our
answer to Problem (R) is as follows:
Corollary 3.4. If $H$ is a Hilbert space, then for any $p\in[1,\infty)$, one
can find a constant $C>0$ such that for any $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$, there is a $u(\cdot)\in
L^{1}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))$ so that equality $(\ref{R})$ holds
and
$\|u(\cdot)\|_{L^{1}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))}\leq
C\|\zeta(\cdot)\|_{L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))}.$ (3.6)
Remark 3.5. By point 2) in Remark 3.3, it is easy to see that the conclusion
in Corollary 3.4 holds also for adapted or optional or predictable stochastic
processes.
Corollary 3.4 shows the existence for the representation of Itô integrals by
Lebesgue/Bochner integrals. The proof of Corollary 3.4 follows easily from
Theorem 3.1 by noting the well-known result that any Hilbert space has the
Radon-Nikodým property (e.g., [7]) and using also the Burkholder-Davis-Gundy
inequality for vector-valued stochastic processes (see [5, Theorem 5.4] and
[16, Corollary 3.11]). The rest of this section is devoted to proving Theorems
3.1–3.2.
In order to prove Theorems 3.1–3.2, besides Corollary 2.4, we need the
following result concerning range inclusion for operators, which can be found
in [19, Lemma 4.13, pp. 94–95 and Theorem 4.15, p. 97], for example.
Lemma 3.6. Suppose $B_{X}$ and $B_{Z}$ are the open unit balls in Banach
spaces $X$ and $Z$ , respectively. Let $L:\;X\to Z$ be a linear bounded
operator whose range is denoted by ${\cal R}(L)$, and whose adjoint operator
is denoted by $L^{*}:Z^{*}\to X^{*}$. Then, the following two conclusions hold
(i) If ${\cal R}(L)=Z$, then there is a constant $C>0$ such that
$\|z^{*}\|_{Z^{*}}\leq C\|L^{*}z^{*}\|_{X^{*}},\qquad\forall\;z^{*}\in Z^{*}.$
(3.7)
(ii) If $(\ref{123})$ holds for some constant $C>0$, then
$B_{Z}\subset CL(B_{X})\equiv\big{\\{}CLx\;\big{|}\;x\in B_{X}\big{\\}}.$
(3.8)
Remark 3.7. 1) Clearly, by Lemma 3.6, we see that ${\cal R}(L)=Z$ if and only
if (3.7) holds for some constant $C>0$. But this lemma goes a little further
than this. Indeed, the second conclusion of this lemma provides a
“quantitative” characterization $B_{Z}\subset CL(B_{X})$, which is more
delicate than ${\cal R}(L)=Z$. We shall use this result essentially when we
answer Problem (C) in the next section;
2) One should compare Lemma 3.6 with the following general range inclusion
result (e.g., [13, Lemma 2.4 in Chap. 7]): Let $X,Y$ and $Z$ be Banach spaces
with $X$ being reflexive, and both $F:Y\to Z$ and $G:X\to Z$ be linear bounded
operators. Then,
$\begin{array}[]{ll}|F^{*}z^{*}|_{Y^{*}}\leq
C|G^{*}z^{*}|_{X^{*}},\quad\forall z^{*}\in Z^{*},\hbox{ \rm for some constant
}C>0\\\ \iff{\cal R}(F)\subseteq{\cal R}(G).\end{array}$ (3.9)
As shown in [1], the equivalence (3.9) may fail whenever $X$ is not reflexive.
Nevertheless, when $F$ is surjective (in particular when $Y=Z$ and $F=I$, the
identity operator, the case considered in Lemma 3.6), this equivalence remains
to be true (even without the reflexivity assumption for $X$) (see [20, Theorem
1.2 and Remark 1.3]). We refer to [21] for further range inclusion results.
Further, we need the following property for Wiener integrals, a special case
of Itô integrals with deterministic integrands (e.g., [12, Theorem 2.3.4 in
Chapter 2, p. 11]).
Lemma 3.8. For each $0\leq a<b\leq T$ and $f\in L^{2}(a,b)$ (for which $f$ is
a deterministic function, i.e., it does not depend on $\omega\in\Omega$), the
Wiener integral $\int^{b}_{a}f(t)dW(t)$ is a Gaussian random variable with
mean $0$ and variance $\int^{b}_{a}\left|f(t)\right|^{2}dt$.
We are now in a position to prove Theorems 3.1–3.2.
Proof of Theorem 3.1. It suffices to show (3.2). Since
$L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\subseteq
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H))$ (algebraically and topologically),
the restriction of operator
${\mathbb{L}}_{s}:L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H))\to L^{p}_{{\cal
F}_{s}}(\Omega;H)$ to $L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$ is a bounded
linear operator from $L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$ to
$L^{p}_{{\cal F}_{s}}(\Omega;H)$ (For simplicity, we still denote it by
${\mathbb{L}}_{s}$). By Conclusion (ii) in Lemma 3.6 and Corollary 2.4, by a
simple scaling, we see that the desired result (3.2) is implied by the
following:
$\|{\mathbb{L}}_{s}^{*}\eta\|_{L^{\infty}_{\mathbb{F}}(0,s;L^{p^{\prime}}(\Omega;H^{*}))}\geq\|\eta\|_{L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})},\qquad\forall\;\eta\in L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*}).$ (3.10)
In order to prove (3.10), let us first find the adjoint operator
${\mathbb{L}}_{s}^{*}$ of ${\mathbb{L}}_{s}$. For any $u(\cdot)\in
L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$, and $\eta\in L^{p}_{{\cal
F}_{s}}(\Omega;H)^{*}=L^{p^{\prime}}_{{\cal F}_{s}}(\Omega;H^{*})$, we have
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\mathop{\langle}{\mathbb{L}}_{s}u,\eta\mathop{\rangle}={\mathbb{E}}\left(\int_{0}^{s}u(t)dt,\eta\right)_{H,H^{*}}=\int_{0}^{s}{\mathbb{E}}\Big{(}u(t),\eta\Big{)}_{H,H^{*}}dt\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad=\int_{0}^{s}{\mathbb{E}}\Big{(}u(t),{\mathbb{E}}[\eta\,|\,{\cal
F}_{t}]\Big{)}_{H,H^{*}}dt=\mathop{\langle}u,{\mathbb{L}}_{s}^{*}\eta\mathop{\rangle},\end{array}$
(3.11)
which leads to
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{L}}_{s}^{*}:L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})\to
L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))^{*}=L^{\infty}_{\mathbb{F}}(0,s;L^{p^{\prime}}(\Omega;H^{*})),\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle({\mathbb{L}}_{s}^{*}\eta)(t)={\mathbb{E}}[\eta\,|\,{\cal
F}_{t}],\qquad t\in[0,s],~{}\forall\;\eta\in L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*}).\end{array}\right.$ (3.12)
This gives a representation of the adjoint operator ${\mathbb{L}}_{s}^{*}$ of
${\mathbb{L}}_{s}$.
Now, we let $p>1$. Making use of (3.12), we find that
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|{\mathbb{L}}_{s}^{*}\eta\|_{L^{\infty}_{\mathbb{F}}(0,s;L^{p^{\prime}}(\Omega;H^{*}))}=\left[\sup_{t\in[0,s]}{\mathbb{E}}\Big{|}{\mathbb{E}}[\eta\,|\,{\cal
F}_{t}]\Big{|}_{H^{*}}^{p^{\prime}}\right]^{1\over p^{\prime}}\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr\displaystyle\geq\left[{\mathbb{E}}\Big{|}{\mathbb{E}}[\eta\,|\,{\cal
F}_{s}]\Big{|}_{H^{*}}^{p^{\prime}}\right]^{1\over
p^{\prime}}=\left[{\mathbb{E}}|\eta|^{p^{\prime}}\right]^{1\over
p^{\prime}}=\|\eta\|_{L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})}.\end{array}$ (3.13)
Therefore, (3.10) holds for $p>1$.
Next, for $p=1$, we have that
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|{\mathbb{L}}_{s}^{*}\eta\|_{L^{\infty}_{\mathbb{F}}(\Omega;L^{\infty}(0,s;H^{*}))}=\mathop{\rm
esssup}_{\omega\in\Omega}\left[\sup_{t\in[0,s]}|{\mathbb{E}}[\eta\,|\,{\cal
F}_{t}]|_{H^{*}}\right]\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\qquad\qquad\qquad\qquad\qquad\geq\mathop{\rm
esssup}_{\omega\in\Omega}\Big{[}|{\mathbb{E}}[\eta\,|\,{\cal
F}_{s}]|_{H^{*}}\Big{]}=\mathop{\rm
esssup}_{\omega\in\Omega}|\eta(\omega)|_{H^{*}}=\|\eta\|_{L^{\infty}_{{\cal
F}_{s}}(\Omega;H^{*})}.\end{array}$ (3.14)
This implies that our conclusion also holds for $p=1$.
Proof of Theorem 3.2. Noting (2.3), it suffices to prove Theorem 3.2 for
$r\in(1,\infty)$. We use the contradiction argument. Assume that
${\mathbb{L}}_{s}\Big{(}L^{r}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\Big{)}=L^{p}_{{\cal
F}_{s}}(\Omega;H),\quad\hbox{ for some }p,r\in(1,\infty).$ (3.15)
Since $L^{r}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\subseteq
L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))\subseteq
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H))$ (algebraically and topologically),
the restriction of operator
${\mathbb{L}}_{s}:L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,s;H))\to L^{p}_{{\cal
F}_{s}}(\Omega;H)$ to $L^{r}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$ is again a
bounded linear operator from $L^{r}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$ to
$L^{p}_{{\cal F}_{s}}(\Omega;H)$ (For simplicity, we still denote it by
${\mathbb{L}}_{s}$). Similar to (3.12), the representation of the adjoint
operator ${\mathbb{L}}_{s}^{*}$ of ${\mathbb{L}}_{s}$ is given as follows:
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\mathbb{L}}_{s}^{*}:\ L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})\to
L^{r^{\prime}}_{\mathbb{F}}(0,s;L^{p^{\prime}}(\Omega;H^{*})),\\\ \vskip 6.0pt
plus 2.0pt minus
2.0pt\cr\displaystyle({\mathbb{L}}_{s}^{*}\eta)(t)={\mathbb{E}}[\eta\,|\,{\cal
F}_{t}],\qquad t\in[0,s],~{}\forall\;\eta\in L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*}).\end{array}\right.$ (3.16)
By (3.15), using the first conclusion in Lemma 3.6 and noting Corollary 2.4,
we conclude that there exists a constant $C>0$ such that for any $\eta\in
L^{p^{\prime}}_{{\cal F}_{s}}(\Omega;H^{*})$, it holds that
$\|\eta\|_{L^{p^{\prime}}_{{\cal F}_{s}}(\Omega;H^{*})}\leq
C\|{\mathbb{L}}_{s}^{*}\eta\|_{L^{r^{\prime}}_{\mathbb{F}}(0,s;L^{p^{\prime}}(\Omega;H^{*}))},$
(3.17)
where $r^{\prime}=r/(r-1)$.
Fix any $x_{0}\in H^{*}$ satisfying $|x_{0}|_{H^{*}}=1$ (which is independent
of the time variable $t$ and the sample point $\omega$). Consider a sequence
of random variables $\\{\eta_{n}\\}_{n=1}^{\infty}$ defined by
$\eta_{n}=\int_{0}^{s}e^{nt}dW(t)x_{0},\qquad n\in\mathbb{N}.$
It is obvious that $\eta_{n}\in L^{p^{\prime}}_{{\cal F}_{s}}(\Omega;H^{*})$
for any $n\in\mathbb{N}$. By Lemma 3.8, the integral
$\displaystyle\int_{0}^{s}e^{nt}dW(t)$ is a Gaussian random variable with mean
$0$ and variance $\frac{e^{2ns}-1}{2n}$. Hence,
$\begin{array}[]{ll}\displaystyle\displaystyle\left[\mathbb{E}\left|\int_{0}^{s}e^{nt}dW(t)\right|^{p^{\prime}}\right]^{\frac{1}{p^{\prime}}}\negthinspace\negthinspace\negthinspace&\displaystyle=\left[\int_{-\infty}^{\infty}\frac{\sqrt{n}\left|x\right|^{p^{\prime}}}{\sqrt{(e^{2ns}-1)\pi}}e^{-\frac{nx^{2}}{e^{2ns}-1}}dx\right]^{\frac{1}{p^{\prime}}}\\\\[8.53581pt]
&\displaystyle=\left[\int_{-\infty}^{\infty}\left(\frac{e^{2ns}-1}{n}\right)^{p^{\prime}/2}\frac{\left|x\right|^{p^{\prime}}}{\sqrt{\pi}}e^{-x^{2}}dx\right]^{\frac{1}{p^{\prime}}}\\\\[8.53581pt]
&\displaystyle=\left(\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\left|x\right|^{p^{\prime}}e^{-x^{2}}dx\right)^{\frac{1}{p^{\prime}}}\sqrt{\frac{e^{2ns}-1}{n}}.\end{array}$
(3.18)
Now, by (3.18), it is easy to see that
$\begin{array}[]{ll}\displaystyle\|\eta_{n}\|_{L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})}\negthinspace\negthinspace\negthinspace&\displaystyle=\left[\mathbb{E}\left|\int_{0}^{s}e^{nt}dW(t)x_{0}\right|^{p^{\prime}}\right]^{\frac{1}{p^{\prime}}}=\left[\mathbb{E}\left|\int_{0}^{s}e^{nt}dW(t)\right|^{p^{\prime}}\right]^{\frac{1}{p^{\prime}}}\\\\[8.53581pt]
&\displaystyle=\left(\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\left|x\right|^{p^{\prime}}e^{-x^{2}}dx\right)^{\frac{1}{p^{\prime}}}\sqrt{\frac{e^{2ns}-1}{n}}.\end{array}$
(3.19)
Using (3.18) again, we have
$\begin{array}[]{ll}\displaystyle\big{\|}\mathbb{E}[\eta_{n}|{\cal
F}_{t}]\big{\|}_{L^{r^{\prime}}_{{\mathbb{F}}}(0,s;L^{p^{\prime}}(\Omega;H^{*}))}\negthinspace\negthinspace\negthinspace&\displaystyle=\left\\{\int_{0}^{s}\left[\mathbb{E}\left|\int_{0}^{t}e^{n\tau}dW(\tau)x_{0}\right|^{p^{\prime}}\right]^{\frac{r^{\prime}}{p^{\prime}}}dt\right\\}^{\frac{1}{r^{\prime}}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr&\displaystyle=\left\\{\int_{0}^{s}\left[\mathbb{E}\left|\int_{0}^{t}e^{n\tau}dW(\tau)\right|^{p^{\prime}}\right]^{\frac{r^{\prime}}{p^{\prime}}}dt\right\\}^{\frac{1}{r^{\prime}}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr&\displaystyle=\left\\{\int_{0}^{s}\left[\left(\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\left|x\right|^{p^{\prime}}e^{-x^{2}}dx\right)^{\frac{1}{p^{\prime}}}\sqrt{\frac{e^{2nt}-1}{n}}\right]^{r^{\prime}}dt\right\\}^{\frac{1}{r^{\prime}}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr&\displaystyle\leq\frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\left|x\right|^{p^{\prime}}e^{-x^{2}}dx\right)^{\frac{1}{p^{\prime}}}\left(\int_{0}^{s}e^{nr^{\prime}t}dt\right)^{\frac{1}{r^{\prime}}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr&\displaystyle\leq\frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\left|x\right|^{p^{\prime}}e^{-x^{2}}dx\right)^{\frac{1}{p^{\prime}}}\frac{e^{ns}}{\left(nr^{\prime}\right)^{\frac{1}{r^{\prime}}}}.\end{array}$
(3.20)
From (3.19) and (3.20), it follows that
$\lim_{n\to\infty}\frac{\big{\|}\mathbb{E}[\eta_{n}|{\cal
F}_{t}]\big{\|}_{L^{r^{\prime}}_{{\mathbb{F}}}(0,s;L^{p^{\prime}}(\Omega;H^{*}))}}{\|\eta_{n}\|_{L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})}}\leq\lim_{n\to\infty}\frac{e^{ns}}{\left(nr^{\prime}\right)^{\frac{1}{r^{\prime}}}\sqrt{e^{2ns}-1}}=0.$
This, combined with (3.16), gives
$\lim_{n\to\infty}\frac{\big{\|}{\mathbb{L}}_{s}^{*}\eta_{n}\big{\|}_{L^{r^{\prime}}_{{\mathbb{F}}}(0,s;L^{p^{\prime}}(\Omega;H^{*}))}}{\|\eta_{n}\|_{L^{p^{\prime}}_{{\cal
F}_{s}}(\Omega;H^{*})}}=0,$
which contradicts inequality (3.17). This completes the proof of Theorem 3.2.
## 4 Answer to Problem (C)
This section is addressed to give a positive answer to Problem (C).
Theorem 3.1 tells us that any Itô integral
$\displaystyle\int_{0}^{s}\zeta(t)dW(t)$ with $\zeta(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$ admits a (parameterized) Bochner
integral representation, i.e. we can find a representor $u(\cdot,s)\in
L^{1}_{\mathbb{F}}(0,s;L^{p}(\Omega;H))$ (which is of course NOT unique) such
that
$\int_{0}^{s}\zeta(t)dW(t)=\int_{0}^{s}u(t,s)dt,\qquad\forall\;s\in[0,T].$
(4.1)
Put $Z\equiv L^{1}_{\mathbb{F}}(0,T;L^{p}(\Omega;H))$. We now show that one
can choose a $u(\cdot,s)$, which is continuous in $Z$ with respect to $s$,
such that (4.1) holds. More precisely, we have the following result:
Theorem 4.1. For any given $\zeta(\cdot)\in
L_{\mathbb{F}}^{p}(\Omega;L^{2}(0,T;H))$, define a (set-valued) mapping
$F:[0,T]\to 2^{Z}$ by
$\displaystyle F(s)=\Big{\\{}\eta(\cdot,s)\in
Z\;\Big{|}\int_{0}^{s}\eta(t,s)dt=\int_{0}^{s}\zeta(t)dW(t),\hbox{ and
}\eta(t,s)=0,\,\forall\,t>s\Big{\\}},\;\;\forall\;s\in[0,T].$ (4.2)
Then $F$ has a continuous selection $f$.
Remark 4.2. If we choose $u(\cdot,s)$ to be the above $f(s)$, then
$u(\cdot,s)$ is the desired process (for (4.1)), which is continuous in $Z$
with respect to $s$.
Before proving Theorem 4.1, we recall the following useful preliminary
results.
Lemma 4.3. Let $X$ and $Y$ be two topological spaces. Then, for any set-valued
mapping $\phi:X\to 2^{Y}$, the following two statements are equivalent:
(i) The map $\phi$ is lower semi-continuous, i.e., for any open subset $V$ of
$Y$, the set $\Big{\\{}x\in X\;\Big{|}\;\phi(x)\cap V\neq\emptyset\Big{\\}}$
is open in $X$;
(ii) If $x\in X$, $y\in\phi(x)$, and $V$ is a neighborhood of $y$ in $Y$, then
there exists a neighborhood $U$ of $x$ in $X$ such that for every
$x^{\prime}\in U$, there exists a $y^{\prime}\in\phi(x^{\prime})\cap V$.
Lemma 4.4. ([15, Theorem 3.2$\,{}^{\prime\prime}$]) The following properties
of a $T_{1}$-space are equivalent:
(i) $X$ is paracompact (i.e., any open cover of $X$ admits a locally finite
open refinement, which is the case if $X$ is compact or is a metric space);
(ii) If $Y$ is a Banach space, then every lower semi-continuous mapping
$F:X\to 2^{Y}$ such that $F(x)$ is a non-empty, closed, convex subset of $Y$
for any $x\in X$, admits a continuous selection, i.e., there exists a
continuous mapping $f:X\to Y$ such that $f(x)\in F(x)$ for any $x\in X$.
We can now give a proof of Theorem 4.1.
Proof of Theorem 4.1. The main idea is to use Lemma 4.4. It is obviously that
$[0,T]$ is an $T_{1}$-space and is paracompact. Hence we need only to prove
that $F(s)$ is a non-empty, closed, convex subset of $Z$ for any $s\in[0,T]$
and $F$ is lower semi-continuous. By Theorem 3.1, we see that $F(s)$ is non-
empty. Also, it is very easy to check that $F(s)$ is a convex subset of $Z$
and is closed in $Z$.
It remains to show that $F$ is lower semi-continuous. Fix any $s\in[0,T]$, any
$\eta(\cdot,s)\in F(s)$, and any neighborhood $V$ of $\eta(\cdot,s)$ in $Z$.
Clearly, there exists a $\delta>0$ such that
$V_{1}=\big{\\{}z(\cdot)\in
Z\,\big{|}\,\|z(\cdot)-\eta(\cdot,s)\|_{Z}<\delta\big{\\}}\subset V.$
We claim that there exists an $\varepsilon>0$ such that for any $r$ satisfying
$|r-s|<\varepsilon$, it holds that
$F(r)\cap V_{1}\neq\emptyset.$ (4.3)
This claim will yield the lower semi-continuity of $F(\cdot)$. To prove out
claim, we first make use of the Burkholder-Davis-Gundy inequality for vector-
valued stochastic process (see [5, Theorem 5.4] and [16, Corollary 3.11]) to
get the following:
${\mathbb{E}}\Big{|}\int_{r}^{s}\zeta(t)dW(t)\Big{|}_{H}^{p}\leq{\mathbb{E}}\Big{[}\sup_{r\leq
h\leq s}\Big{|}\int_{r}^{h}\zeta(t)dW(t)\Big{|}_{H}^{p}\Big{]}\leq
C{\mathbb{E}}\Big{[}\int_{r}^{s}|\zeta(t)|_{H}^{2}dt\Big{]}^{\frac{p}{2}}.$
(4.4)
Choose an increasing sequence $\\{r_{k}\\}_{k=1}^{\infty}$ such that $0\leq
r_{1}\leq r_{2}\leq\cdots\leq r_{k}\leq r_{k+1}\leq\cdots\to s$. Since
$\zeta(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;H))$, by the Dominated
Convergence Theorem, we have
$\lim_{k\to\infty}{\mathbb{E}}\Big{[}\int_{r_{k}}^{s}|\zeta(t)|_{H}^{2}dt\Big{]}^{\frac{p}{2}}=\lim_{{k\to\infty}}{\mathbb{E}}\Big{[}\int_{0}^{T}\chi_{[{r_{k}},s]}|\zeta(t)|_{H}^{2}dt\Big{]}^{\frac{p}{2}}=0.$
Hence,
$\lim_{r\to
s}{\mathbb{E}}\Big{[}\int_{r}^{s}|\zeta(t)|_{H}^{2}dt\Big{]}^{\frac{p}{2}}\leq\lim_{k\to\infty}{\mathbb{E}}\Big{[}\int_{r_{k}}^{s}|\zeta(t)|_{H}^{2}dt\Big{]}^{\frac{p}{2}}=0.$
(4.5)
Therefore, it follows from (4.4) that there exists an $\varepsilon_{1}>0$ such
that for any $0\leq s-r<\varepsilon_{1}$, the following holds
$\Big{\|}\int_{r}^{s}\zeta(t)dW(t)\Big{\|}_{L_{{\cal
F}_{s}}^{p}(\Omega;H)}<\frac{\delta}{3}.$ (4.6)
On the other hand, by the Hölder inequality and using the Dominated
Convergence Theorem, similar to the proof of (4.5), we see that there exists
an $\varepsilon_{2}>0$ (may depend on $s$) such that for any $0\leq
s-r<\varepsilon_{2}$, it holds
$\Big{\|}\int_{r}^{s}\eta(t,s)dt\Big{\|}_{L_{{\cal
F}_{s}}^{p}(\Omega;H)}\leq\int_{r}^{s}\|\eta(t,s)\|_{L^{p}_{{\cal
F}_{s}}(\Omega;H)}dt=\int_{r}^{s}\Big{[}{\mathbb{E}}\big{|}\eta(t,s)\big{|}^{p}_{H}\Big{]}^{\frac{1}{p}}dt<\frac{\delta}{3}.$
(4.7)
Put $\varepsilon_{3}=\min\\{\varepsilon_{1},\varepsilon_{2}\\}$. From
(4.6)–(4.7) and noting that
$\displaystyle\int_{0}^{s}\eta(t,s)dt=\int_{0}^{s}\zeta(t)dW(t)$, we conclude
that for any $r$ satisfies $0\leq s-r<\varepsilon_{3}$, it holds that
$\begin{array}[]{ll}\displaystyle\Big{\|}\int_{0}^{r}\eta(t,s)dt-\int_{0}^{r}\zeta(t)dW(t)\Big{\|}_{L_{{\cal
F}_{s}}^{p}(\Omega;H)}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\leq\negthinspace\negthinspace\Big{\|}\int_{0}^{r}\eta(t,s)dt-\int_{0}^{s}\eta(t,s)dt\Big{\|}_{L_{{\cal
F}_{s}}^{p}(\Omega;H)}\negthinspace+\negthinspace\Big{\|}\int_{0}^{r}\zeta(t)dW(t)-\int_{0}^{s}\zeta(t)dW(t)\Big{\|}_{L_{{\cal
F}_{s}}^{p}(\Omega;H)}<\frac{2\delta}{3}.\end{array}$ (4.8)
By the second conclusion in Theorem 3.1 and noting (4.8), we see that there is
a $\displaystyle\phi(\cdot,r)\in L^{1}_{\mathbb{F}}(0,r;L^{p}(\Omega;H))$ such
that
$\displaystyle\|\phi(\cdot,r)\|_{L^{1}_{\mathbb{F}}(0,r;L^{p}(\Omega;H))}<\frac{2\delta}{3}$,
and
$\int_{0}^{r}\phi(t,r)dt=\int_{0}^{r}\zeta(t)dW(t)-\int_{0}^{r}\eta(t,s)dt.$
Put $\varrho(\cdot,r)=\chi_{[0,r]}\phi(\cdot,r)+\chi_{[0,r]}\eta(\cdot,s)$. It
is obvious that $\varrho(\cdot,r)\in F(r)$, and
$\Big{\|}\eta(\cdot,s)-\varrho(\cdot,r)\Big{\|}_{L_{{\mathbb{F}}}^{1}(0,s;L^{p}(\Omega,H))}\leq\int_{r}^{s}\Big{[}{\mathbb{E}}|\eta(t,s)|^{p}_{H}\Big{]}^{\frac{1}{p}}dt+\|\phi(\cdot,r)\|_{L^{1}_{\mathbb{F}}(0,r;L^{p}(\Omega;H))}<\delta.$
Therefore, for any $0\leq s-r<\varepsilon_{3}$, it holds that
$\varrho(\cdot,r)\in V_{1}$, which gives (4.3). By a similar argument, one can
show that there exists an $\varepsilon_{4}>0$ such that (4.3) holds for any
$0\leq r-s<\varepsilon_{4}$. Choosing
$\varepsilon=\min\\{\varepsilon_{3},\varepsilon_{4}\\}$, we see that (4.3)
holds for any $|r-s|<\varepsilon$. By Lemma 4.3, we know that $F:[0,T]\to Z$
is lower semi-continuous.
Finally, thanks to Lemma 4.4, we conclude that there exists a continuous
selection $f$ of $F$.
## 5 Two Illustrative Applications
In this section, we give two simple applications of our Theorems 3.1–3.2. More
interesting and sophisticated applications will be presented in our
forthcoming publications.
### 5.1 Application to the controllability problem
Consider a one-dimensional controlled stochastic differential equation:
$dx(t)=[bx(t)+u(t)]dt+\sigma dW(t),$ (5.1)
with $b$ and $\sigma$ being given constants. We say that system (5.1) is
exactly controllable if for any $x_{0}\in{\mathbb{R}}$ and $x_{T}\in
L^{p}_{{\cal F}_{T}}(\Omega;{\mathbb{R}})$, there exists a control
$u(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;{\mathbb{R}}))$ such that the
corresponding solution $x(\cdot)$ satisfies $x(0)=x_{0}$ and $x(T)=x_{T}$. By
variation of constant formula, we have
$x(T)=e^{bT}x_{0}+\int_{0}^{T}e^{b(T-t)}u(t)dt+\int_{0}^{T}e^{b(T-t)}\sigma
dW(t).$
Thus, exact controllability is equivalent to the following:
$x_{T}-e^{bT}x_{0}-\int_{0}^{T}e^{b(T-t)}\sigma
dW(t)=\int_{0}^{T}e^{b(T-t)}u(t)dt.$ (5.2)
Since $x_{T}\in L^{p}_{{\cal F}_{T}}(\Omega;{\mathbb{R}})$, there exists a
unique $\zeta(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;{\mathbb{R}}))$
such that
$x_{T}={\mathbb{E}}x_{T}+\int_{0}^{T}\zeta(t)dW(t).$
Hence, to ensure (5.2), it suffices to have
${\mathbb{E}}x_{T}-e^{bT}x_{0}+\int_{0}^{T}\big{[}\zeta(t)-e^{b(T-t)}\sigma\big{]}dW(t)=\int_{0}^{T}e^{b(T-t)}u(t)dt,$
which is guaranteed by Theorem 3.1. This means that (5.1) is exactly
controllable.
On the other hand, surprisingly, in virtue of [18, Theorem 3.1], it is clear
that system (5.1) is NOT exactly controllable if one restricts to use
admissible controls $u(\cdot)$ in
$L^{2}_{\mathbb{F}}(\Omega;L^{2}(0,T;{\mathbb{R}}))$! Moreover, by Theorem
3.2, we see that system (5.1) is NOT exactly controllable, either provided
that one uses admissible controls $u(\cdot)$ in
$L^{2}_{\mathbb{F}}(\Omega;L^{q}(0,T;{\mathbb{R}}))$ for any $q\in(1,\infty]$.
This leads to a corrected formulation for the exact controllability of
stochastic differential equations, as presented below.
We consider the following linear stochastic differential equation:
$\left\\{\begin{array}[]{ll}dy(t)=\big{[}Ay(t)+Bu(t)\big{]}dt+\big{[}Cy(t)+Du(t)\big{]}dW(t),\qquad
0\leq t\leq T,\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
y(0)=y_{0}\in{\mathbb{R}}^{n},\end{array}\right.$ (5.3)
where $A,C\in{\mathbb{R}}^{n\times n}$ and $B,D\in{\mathbb{R}}^{n\times m}$
($n,m\in{\mathbb{N}}$) are matrices. Various controllability issues for system
(5.3) were studied, say, in [2, 3, 10, 18] and the references cited therein.
Note however that, unlike the classical deterministic case, as far as we know,
there exist no universally accepted notions for controllability in the
stochastic setting so far.
Motivated by the above observation, we introduce the following:
Definition 5.1. System (5.3) is said to be exactly controllable if for any
$y_{0}\in{\mathbb{R}}^{n}$ and $y_{T}\in L^{p}_{{\cal
F}_{T}}(\Omega;{\mathbb{R}}^{n})$, there exists a control $u(\cdot)\in
L^{p}_{\mathbb{F}}(\Omega;L^{1}(0,T;{\mathbb{R}}^{m}))$ such that
$Du(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;{\mathbb{R}}^{n}))$ and the
corresponding solution $y(\cdot)$ of (5.3) satisfies $y(T)=y_{T}$.
We need $Du(\cdot)\in L^{p}_{\mathbb{F}}(\Omega;L^{2}(0,T;{\mathbb{R}}^{n}))$
in the above definition because it appears in the Itô integral
$\displaystyle\int_{0}^{T}\big{[}Cy(t)+Du(t)\big{]}dW(t)$. It is clear that,
for the controllability of deterministic linear (time-invariant) ordinal
differential equations, there is no difference between the controllability by
using $L^{1}$ (in time) control and that by using $L^{2}$ (or even analytic in
time) control. However, our analysis above indicates that things are
completely different in the stochastic setting. A detailed study of the
controllability for system (5.3) (in the sense of Definition 5.1) seems to
deviate the theme of this paper, and therefore we shall address this topic in
our forthcoming works.
### 5.2 Application to a Black-Scholes model
Consider a Black-Scholes market model
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle dX_{0}(t)=rX_{0}(t)dt,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle dX(t)(t)=bX(t)dt+\sigma X(t)dW(t),\end{array}\right.$
(5.4)
with $r,b,\sigma$ being constants. Under conditions of self-financing, and no
transaction costs, the investor’s wealth process $Y(\cdot)$ satisfies the
following equation:
$dY(t)=\big{[}rY(t)+(b-r)Z(t)\big{]}dt+\sigma Z(t)dW(t),$ (5.5)
where $Z(t)$ is the amount invested in the stock. For convenience, a European
contingent claim with payoff at the maturity $T$ being $\xi\in L^{p}_{{\cal
F}_{T}}(\Omega;{\mathbb{R}})$ is identified with $\xi$. Any such a $\xi$ is
said to be replicatable if there exists a trading strategy $Z(\cdot)$ such
that for some $Y_{0}$ (the price of the contingent claim at $t=0$), one has
$Y(0)=Y_{0},\qquad Y(T)=\xi.$
In another word, contingent claim $\xi$ is replicatable if and only if the
following backward stochastic differential equation (BSDE, for short) admits
an adapted solution $(Y(\cdot),Z(\cdot))$:
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle dY(t)=\big{[}rY(t)+(b-r)Z(t)\big{]}dt+\sigma
Z(t)dW(t),\quad t\in[0,T],\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle Y(T)=\xi.\end{array}\right.$ (5.6)
In this case, $Y(t)$ is a price of the contingent claim at time $t$. See [9]
and [23] for some relevant presentations. Now, let us look at an extreme case,
$b-r>0,\qquad\sigma=0.$ (5.7)
In this case, $\xi$ is replicatable if and only if the following BSDE admits
an adapted solution $(Y(\cdot),Z(\cdot))$:
$\left\\{\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle dY(t)=\big{[}rY(t)+(b-r)Z(t)\big{]}dt,\qquad
t\in[0,T],\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
Y(T)=\xi.\end{array}\right.$
Similar to the above subsection, we see that the above admits an adapted
solution $(Y(\cdot),Z(\cdot))$, which means that $\xi$ is replicatable.
Further, since $\xi$ is arbitrary, this also means that the market with
conditions (5.7) is complete! This is a little surprising since $\sigma=0$ in
the market model. Some further careful study along this line will be carried
out in our future investigations.
## References
* [1] R. Bouldin, A counterexample in the factorization of Banach space operators, Proc. Amer. Math. Soc., 68 (1978), 327.
* [2] R. Buckdahn, M. Quincampoix and G. Tessitore, A characterization of approximately controllable linear stochastic differential equations, Stochastic Partial Differential Equations and Applications—VII, Lect. Notes Pure Appl. Math., vol. 245, Chapman & Hall/CRC, Boca Raton, FL, 2006, 53–60.
* [3] S. Chen, X. Li, S. Peng and J. Yong, On Stochastic linear controlled systems, Unpublished manuscript.
* [4] J. B. Conway, A Course in Functional Analysis. Second Edition, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990.
* [5] S. Cox and M. Veraar, Vector-valued decoupling and the Burkholder-Davis-Gundy inequality, Preprint (see http://fa.its.tudelft.nl/$\sim$veraar/research/papers/Cox$\\_$Veraar$\\_$Decoupling.pdf).
* [6] C. Dellacherie and P. A. Meyer, Probabilities and Potential B, North-Holland, 1982.
* [7] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys, vol. 15, American Mathematical Society, Providence, R.I., 1977.
* [8] N. Dinculeanu, Vector-valued stochastic processes. II. A Radon-Nikodym theorem for vector-valued processes with finite variation, Proc. Amer. Math. Soc., 102 (1988), 393–401.
* [9] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1–71.
* [10] A. Goreac, A Kalman-type condition for stochastic approximate controllability, C. R. Math. Acad. Sci. Paris, 346 (2008), 183–188.
* [11] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 1988.
* [12] H.-H. Kuo, Introduction to Stochastic Integration, Springer, 2006.
* [13] X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995\.
* [14] Q. Lü, Controllability of forward stochastic heat equations with one control, J. Funct. Anal., In submission.
* [15] E. Michael, Continuous selections I, Ann. of Math., 63 (1956), 361–382.
* [16] J. M. A. M. van Neerven, M. C. Veraar and L. Weis, Stochastic integration in UMD Banach space, Ann. Probab., 35 (2007), 1438–1478.
* [17] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, 1995.
* [18] S. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci. (English Ed.), 4 (1994), 274–284.
* [19] W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.
* [20] A. Vieru, On null controllability of linear systems in Banach spaces, Systems Control Lett., 54 (2005), 331–337.
* [21] P. Wang and X. Zhang, Range inclusion of operators on non-archimedean Banach space, Sci. China Ser. A, In submission.
* [22] J. Yong, Some problems related to the Black-Scholes type security markets, Stochastic Processes and Applications to Mathematical Finance, Edited by J. Akahori, A. Ogawa, and S. Watanabe, World Scientific, Singapore, 2004, 369–400.
* [23] J. Yong, Completeness of security markets and solvability of linear backward stochastic differential equations, J. Math. Anal. Appl., 319 (2006), 333–356.
|
arxiv-papers
| 2010-07-18T03:19:28 |
2024-09-04T02:49:11.690021
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qi L\\\"u, Jiongmin Yong and Xu Zhang",
"submitter": "Xu Zhang",
"url": "https://arxiv.org/abs/1007.2969"
}
|
1007.3051
|
# Essence of Special Relativity, Reduced Dirac Equation and Antigravity
Guang-jiong Ni a,b pdx01018@pdx.edu a Department of Physics, Portland State
University, Portland, OR97207, U. S. A.
b Department of Physics, Fudan University, Shanghai, 200433, China Suqing
Chen b suqing_chen@yahoo.com b Department of Physics, Fudan University,
Shanghai, 200433, China Senyue Lou c,d sylou@sjtu.edu.cn c Department of
Physics, Shanghai Jiao Tong University, Shanghai, 200030, China
d Department of Physics, Ningbo University, Ningbo 315211, China Jianjun Xu b
xujj@fudan.edu.cn b Department of Physics, Fudan University, Shanghai, 200433,
China
###### Abstract
The essence of special relativity is hiding in the equal existence of particle
and antiparticle, which can be expressed by two discrete symmetries within one
inertial frame — the invariance under the (newly defined) space-time inversion
(${\bf x}\to-{\bf x},t\to-t$), or equivalently, the invariance under a mass
inversion ($m\to-m$). The problems discussed are: the evolution of the $CPT$
invariance into a basic postulate, an unique solution to the original puzzle
in Einstein-Podolsky-Rosen paradox, the reduced Dirac equation for
hydrogenlike atoms, and the negative mass paradox leading to the prediction of
antigravity between matter and antimatter.
Keywords: Special relativity, Reduced Dirac Equation, Antiparticle,
Antigravity
PACS: 03.30.+p; 03.65.Pm; 04.50.kd
## I Introduction
It’s time to rethink the theory of special relativity ($SR$) established by
Einstein in 1905. As is well-known, he put $SR$ on two basic postulates:
$A$. Principle of constancy of the light speed;
$B$. Principle of relativity.
Both $A$ and $B$ are ”relativistic principles”. Why we need both of them? Some
authors have been devoting to reduce these two principles into one, trying to
ignore $A$. They might think as follows: Once we already have Maxwell’s theory
of classical electrodynamics ($CED$) in one inertial frame $S$, the principle
B would lead to the conclusion that the $CED$ must hold in another inertial
frame $S^{\prime}$, i.e., principle $A$ must be valid too.
All attempts mentioned above were doomed to failure because they overlooked
one clue point: In order to transfer from one frame $S$ to another
$S^{\prime}$, we need a transformation bringing the coordinates from ${\bf
x},t$ to ${\bf x}^{\prime},t^{\prime}$. And in this Lorentz transformation, an
universal constant $c$ must be fixed in advance, otherwise the transformation
would be meaningless.
However, the above argument, so-called a ”logic cycle”, does hint that the
$SR$ based on $A$ and $B$ is by no means a final story. If we insist on
discussing physics in one frame, there should be essentially one ”relativistic
principle” only. Like everything else, the essence of $SR$ can only be
revealed gradually after 1905. But where is the breakthrough point?
## II The $CPT$ Invariance Turning into a Basic Postulate
There are three discrete transformations in quantum mechanics ($QM$)1 (1),
quantum field theory ($QFT$) and particle physics2 (2):
(a) Space-inversion ($P$):
The sign change of space coordinates (${\bf x}\to-{\bf x}$) in the wavefuction
($WF$) of $QM$ may lead to two eigenstates:
$\psi_{\pm}({\bf x},t)\to\psi_{\pm}(-{\bf x},t)=\pm\psi_{\pm}({\bf x},t)$ (1)
with eigenvalues $1$ or $-1$ being the even or odd parity.
(b) Time reversal ($T$):
The so-called $T$ transformation is actually not a ”time reversal” but a
”reversal of motion”3 (3, 4), which implies an equivalence relation between
two $WFs$:
$\psi({\bf x},t)\sim\psi^{*}({\bf x},-t)$ (2)
where the equivalence notation $\sim$ possibly means some matrices in front of
$WF$ being ignored.
(c) Charge conjugation transformation($C$):
The $C$ transformation brings a particle (with charge $q$) into its
antiparticle (with charge $-q$) and implies a complex conjugation on the $WF$:
$\psi({\bf x},t)\to\psi_{c}({\bf x},t)\sim\psi^{*}({\bf x},t)$ (3)
Note that the $WF$ $\psi_{c}$ implies a negative-energy particle. To explain
it being an antiparticle, one has to resort to so-called ”hole theory” for
electron — the vacuum is fully filled with infinite negative-energy electrons
and a ”hole” created in the ”sea” would correspond to a positron1 (1, 5). But
how could the ”hole theory” be applied to the boson particle? No one knows.
(See the discussion between S. Weinberg and Dirac6 (6)).
(d) $CPT$ combined transformation
If taking the product of $C$, $P$ and $T$ transformations together, the
complex conjugation contained in the $C$ and $T$ will cancel each other,
yielding2 (2, 5)
$\psi({\bf x},t)\to CPT\psi({\bf x},t)=\psi_{CPT}({\bf x},t)\sim\psi(-{\bf
x},-t)$ (4)
On the right-hand-side (RHS), the $WF$ should be understood as to describe an
antiparticle. But it differs from the original $WF$ only in the sign change of
${\bf x}$ and $t$. What does it mean?
The historical discovery of parity violation in 1956-19577 (7, 8) reveals that
both $P$ and $C$ symmetries are violated in weak interactions. Since 1964, it
is found that $CP$ symmetry is also violated whereas the $CPT$ invariance
remains valid 2 (2), which in turn implies the violation of $T$ reversal
symmetry, as summarized in the Review of particle Physics9 (9).
Therefore, the relation between a particle $|a\rangle$ and its antiparticle
$|\bar{a}\rangle$ is not $|\bar{a}\rangle=C|a\rangle$ but (as defined by Lee
and Wu10 (10))
$|\bar{a}\rangle=CPT|a\rangle$ (5)
which means exactly the Eq.(4). For example, for an electron in free motion,
its $WF$ reads
$\langle{\bf x},t|e^{-},{\bf p},E\rangle=\psi_{e^{-}}({\bf
x},t)\sim\exp\left[\frac{i}{\hbar}({\bf p}\cdot{\bf x}-Et)\right]$ (6)
while the $WF$ for a positron is given by Eq.(4) or Eq.(5) as
$\langle{\bf x},t|e^{+},{\bf p},E\rangle=\psi_{e^{+}}({\bf
x},t)\sim\exp\left[-\frac{i}{\hbar}({\bf p}\cdot{\bf x}-Et)\right]$ (7)
Note that the momentum ${\bf p}$ and energy $E\,(>0)$ are the same in Eqs.(6)
and (7) (see Eq.(16.51) in 2 (2)).
The above relation should be viewed as a new symmetry: The (newly defined)
space-time inversion (${\bf x}\to-{\bf x},t\to-t$) is equivalent to particle-
antiparticle transformation. The transformation of a particle to its
antiparticle (denoted by $\cal C$) is not something which can be defined
independently but a direct consequence of the (newly defined) space-time
inversion $\cal PT$ (${\bf x}\to-{\bf x},t\to-t$)11 (11, 12, 4):
${\cal PT}={\cal C}$ (8)
Note that there is an important difference between a ”theorem” and a ”law”.
Various quantities contained in a theorem must be defined clearly and
unambiguously in advance before the theorem can be proved. On the other hand,
a law can often (not always) accommodate a definition of a physical quantity
which can only be defined unambiguously after the law is verified by
experiments. Two examples are:
The definition of inertial mass $m$ is contained in the Newton’s dynamical
law:
${\bf F}=m{\bf a}$ (9)
The definition of electric (magnetic) field strength $\bf E\,(B)$ is contained
in the Lorentz-force formula:
${\bf F}=m\dfrac{d{\bf v}}{dt}=q({\bf E}+\frac{1}{c}{\bf v}\times{\bf B})$
(10)
Hence, we see from Eqs.(6) and (7) that the familiar operator relations in
$QM$:
$\hat{\bf p}=-i\hbar\nabla,\quad\hat{E}=i\hbar\frac{\partial}{\partial t}$
(11)
are only valid for particle, they must be supplemented by
$\hat{\bf p}_{c}=i\hbar\nabla,\quad\hat{E}_{c}=-i\hbar\frac{\partial}{\partial
t}$ (12)
for antiparticle, in conformity with the basic postulate, Eq.(8).
From the beginning, we have been believing that Eq.(8) or Eq.(11) versus
Eq.(12) is the essence of $SR$. We derived $SR$ from $QM$, based on this
symmetry12 (12). Among various arguments for this claim (13 (13), see 4 (4)
for detail), we will discuss the $EPR$ paradox in the next section before
turning to another new arguments in our recent studies.
## III The Original Puzzle in Einstein-Podolsky-Rosen Paradox
The famous paper titled ”Can quantum mechanical description of physical
reality be considered complete?” by Einstein, Podolsky and Rosen ($EPR$, 14
(14)) is not easy to read. Quite naturally, beginning from Bohm 15 (15) and
Bell 16 (16), physicists have been turning their attention to the entanglement
phenomena of photons and electrons with spin. To our knowledge, H. Guan
(1935-2007) first clearly pointed out that 17 (17) the original puzzle in
$EPR$’s paper is involving spinless particles and what being overlooked is as
follows.
Consider two particles in one dimensional space with positions
$x_{i},\,(i=1,2)$ and momentum operators
$\hat{p}_{i}=-i\hbar\dfrac{\partial}{\partial x_{i}}$. Then the commutation
relation
$[x_{1}-x_{2},\hat{p}_{1}+\hat{p}_{2}]=0$ (13)
implies that there may be a state with two commutative (compatible)
observables:
$p_{1}+p_{2}=0,\quad(p_{2}=-p_{1})\quad\text{and}\quad x_{1}-x_{2}=D$ (14)
How can such a quantum state be realized?
Guan’s observation led to discussions in Refs.18 (18), 19 (19) and 4 (4),
where another commutation relations like
$\displaystyle{[x_{1}+x_{2},\hat{p}_{1}-\hat{p}_{2}]}$ $\displaystyle=$
$\displaystyle 0$ (15) $\displaystyle{[t_{1}-t_{2},\hat{E}_{1}+\hat{E}_{2}]}$
$\displaystyle=$ $\displaystyle 0$ (16)
$\displaystyle{[t_{1}+t_{2},\hat{E}_{1}-\hat{E}_{2}]}$ $\displaystyle=$
$\displaystyle 0$ (17)
($\hat{E}_{i}=i\hbar\frac{\partial}{\partial t_{i}}$) are considered in
connection with a wonderful experiment (in 1998) on an entangled state of
$K^{0}-\bar{K}^{0}$ system 20 (20). Now let us discuss it further.
As in 20 (20), we focus on back-to-back events. However, the evolution of
wavefunctions ($WFs$) will be considered in three inertial frames: The center-
of-mass system $S$ is at rest with its origin $t=0$ located at detector’s
center. The space-time coordinates in Eqs.(13)-(17) refer to particles moving
to right ($x_{1}>0$) and left ($x_{2}<0$) respectively. Then we take an
inertial system $S^{\prime}$ with its origin located at particle $1$ (i.e.,
$x^{\prime}_{1}=0$). $S^{\prime}$ is moving in a uniform velocity $v$ with
respect to $S$. (For Kaon’s momentum of $800MeV/c,\beta=\frac{v}{c}=0.849$).
Another $S^{\prime\prime}$ system is chosen with its origin located at
particle $2$ ($x^{\prime\prime}_{2}=0$). $S^{\prime\prime}$ is moving in a
velocity $-v$ with respect to $S$. Thus we have Lorentz transformations among
their space-time coordinates as:
$\left\\{\begin{array}[]{ll}x^{\prime}=\dfrac{x-vt}{\sqrt{1-\beta^{2}}},&\\\
t^{\prime}=\dfrac{t-vx/c^{2}}{\sqrt{1-\beta^{2}}},&\end{array}\right.\qquad\left\\{\begin{array}[]{ll}x^{\prime\prime}=\dfrac{x+vt}{\sqrt{1-\beta^{2}}},&\\\
t^{\prime\prime}=\dfrac{t+vx/c^{2}}{\sqrt{1-\beta^{2}}},&\end{array}\right.$
(18)
Here $t^{\prime}_{1}$ and $t^{\prime\prime}_{2}$ correspond to the proper
times $t_{a}$ and $t_{b}$ in 20 (20) respectively. The common time origin
$t=t^{\prime}=t^{\prime\prime}=0$ is adopted.
Surprisingly, we see that Eqs.(13) and (14) are just realized by $K^{0}K^{0}$
events with $p_{2}=-p_{1}$ and $x_{1}-x_{2}=D$ being the distance between two
particles when they are detected. Note that Eq.(17) is also realized in this
case with $E_{1}=E_{2}$ and $t_{1}+t_{2}=D/v$.
More interestingly, Eqs.(15) and (16) can be realized exactly by
$K^{0}\bar{K}^{0}$ events. For example, if particle $1$ is $K^{0}$, then
particle $2$ must be a $\bar{K}^{0}$ with ${\hat{p}}_{2}=-{\hat{p}}_{2}^{c}$
and ${\hat{E}}_{2}=-{\hat{E}}_{2}^{c}$ (see Eq.(12)). So
$p_{1}=-p_{2}^{c}>0,E_{1}=E_{2}^{c}>0$ and $x_{1}+x_{2}=v(t_{1}-t_{2})=\Delta
l$ implies the flight-path difference measured in the laboratory 20 (20).
A remarkable merit of experiment in 20 (20) lies in the fact that its data
cover $t_{1}\neq t_{2}$ cases and so go beyond the EPR-type correlation
($t_{1}=t_{2}$). However, the concept of ”simultaneity” in time is relative
and frame dependent as can be seen from Eq.(18):
$\displaystyle t^{\prime}_{2}-t^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-\beta^{2}}}[(t_{2}-t_{1})+\beta^{2}(t_{1}+t_{2})]>0,\quad(t_{2}>t_{1})$
(19) $\displaystyle t^{\prime\prime}_{1}-t^{\prime\prime}_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-\beta^{2}}}[(t_{1}-t_{2})+\beta^{2}(t_{1}+t_{2})]>0,\quad(t_{1}>t_{2})$
(20)
Here $t^{\prime}_{1}$ or $t^{\prime\prime}_{2}$ is the proper time of particle
first observed in the $K^{0}-\bar{K}^{0}$ system. We see that the ”causality”
is preserved because even at the EPR limit ($t_{1}=t_{2}$), Eqs.(19) and (20)
remain positive.
Once a particle (say $2$) is first detected, a destruction process on the
coherence of entangled state is triggered. This process will be accomplished
right at the detection of second particle (say $1$). In order to better
understand why the coherence can be maintained within this interval, let us
stay at $S^{\prime\prime}$ system and compare two velocities. The particle $1$
has a velocity being
$v^{\prime\prime}_{1}=\frac{x^{\prime\prime}_{1}}{t^{\prime\prime}_{1}}=\frac{2v}{1+v^{2}/c^{2}}$
(21)
as expected. On the other hand, the correlation between particles $1$ and $2$
has been established since $t^{\prime\prime}=0$ until $t^{\prime\prime}_{2}$,
during which a ”decoherence signal” from particle $2$ is triggered and it
reaches particle $1$ at time $t^{\prime\prime}_{1}$. The signal’s propagation
velocity must be no less than
$w^{\prime\prime}=\frac{x^{\prime\prime}_{1}-x^{\prime\prime}_{2}}{t^{\prime\prime}_{1}-t^{\prime\prime}_{2}}=\frac{2v}{(1+\beta^{2})-(1-\beta^{2})t_{2}/t_{1}}\,\xrightarrow[t_{1}\to
t_{2}]{\,}\frac{c^{2}}{v}>c$ (22)
which is superluminal! However, we learn from $RQM$ that the wave’s phase
velocity $u_{p}=\frac{\omega}{k}=\frac{E}{p}$, ($E^{2}=p^{2}c^{2}+m^{2}c^{4}$)
is different from its group velocity $u_{g}$, i.e., the particle’s velocity
$v$ ($v=u_{g}=\frac{d\omega}{dk}=\frac{pc^{2}}{E}$) with their relation being:
$u_{p}u_{g}=c^{2},\quad u_{p}=c^{2}/v$ (23)
so $u^{\prime\prime}_{p}=c^{2}/v^{\prime\prime}|_{v^{\prime\prime}\to
0}\to\infty$ as measured from particle $2$. The inequality
$v^{\prime\prime}_{1}<w^{\prime\prime}<u^{\prime\prime}_{p}$ ensures the
quantum correlation surviving throughout the time interval
$t^{\prime\prime}_{2}<t^{\prime\prime}<t^{\prime\prime}_{1}$.
The phase velocity is by no means an observable speed of energy transfer, but
it does keep the wave coherence globally. By contrast, the destruction of
coherence is triggered and accomplished by detectors locally. In the
wavefunction ($WF$) of $K-K$ system, there are both $K^{0}$ and $\bar{K}^{0}$
(with their space-time coordinates) in the wave propagating to the right or
left side. Actually, there is neither $K^{0}$ nor $\bar{K}^{0}$ particle in
the wave but a wave with its interference until particles $1$ and $2$ are
detected eventually with only one particle ( either $K^{0}$ or $\bar{K}^{0}$ )
at each side 20 (20). In our understanding 4 (4), the invisible $WF$ is the
amplitude of a ”fictitious measurement” to show the relevant ”potential
possibility”, which turns into the ”real probability” at a concrete
measurement.
Hence $EPR$ were right: For better understanding the $QM$, one needs to study
the two-particle entangled state, a nonlocal coherent state evolved over long
distance not only in space but also in time. As shown by experiment 20 (20)
and Eqs.(13)-(17), being a quantum system with less uncertainty, it is easier
to be observed. This is because anything could be and should be recognized
only in relationships as emphasized by $SR$.
$EPR$ were quite right: $QM$ cannot be considered complete in describing the
physical reality unless (a). we take the antiparticle with relevant operator
relations, Eq.(12), into account: (b). the relativistic relation between phase
velocity and group velocity, Eq. (23), is also taken into account. 111If
constrained by nonrelativistic relation $E=p^{2}/(2m)$ which would lead to
$u_{p}=(1/2)u_{g}=(1/2)v$, we even cannot understand the forming process
toward the Bose-Einstein-condensation where any two particles obey
Eqs.(13)-(17).
## IV Invariance Under Mass Inversion and Dirac Equation
There is another symmetry equivalent to the invariance of space-time
inversion, Eq.(8), showing the equal existence of particle and antiparticle.
As noticed in 4 (4), the Lorentz force law, Eq.(10) for an electron should be
transformed into that for a positron by an inversion of $m\to-m$ as a
substitution of changing $q=-e\,(e>0)$ into $q=e$ in classical physics. This
is because the particle-antiparticle transformation has already been replaced
by Eq.(5) or Eq.(8). Moreover, we can see from Eqs.(6) and (7) that the space-
time inversion (${\bf x}\to-{\bf x},t\to-t$) is equivalent to changing the
sign of the $\bf p$ and $E$, i.e., changing $m\to-m$.
However, it was not until 2003 that the importance of mass inversion became
clearer as discussed in 21 (21) (see also Appendix 9C in the 2nd Edition of 4
(4)). Let us look at the Dirac equation for a free electron
$i\hbar\dfrac{\partial}{\partial t}\psi=H\psi=(-i\hbar
c{\boldsymbol{\alpha}}\cdot\nabla+\beta mc^{2})\psi$ (24)
with ${\boldsymbol{\alpha}}$ and $\beta$ being $4\times 4$ matrices, the $WF$
$\psi$ is a four-component spinor:
$\psi=\begin{pmatrix}\phi\\\ \chi\end{pmatrix}$ (25)
Usually, the two-component spinors $\phi$ and $\chi$ are called ”positive” and
”negative” energy components. In our point of view, they are the hiding
”particle” and ”antiparticle” fields in a particle (electron) (4 (4), see
below). Substitution of Eq.(25) into Eq.(24) leads to
$\left\\{\begin{array}[]{l}i\hbar\dfrac{\partial}{\partial t}\phi=-i\hbar
c{\boldsymbol{\sigma}}\cdot\nabla\chi+mc^{2}\phi,\\\
i\hbar\dfrac{\partial}{\partial t}\chi=-i\hbar
c{\boldsymbol{\sigma}}\cdot\nabla\phi-mc^{2}\chi\end{array}\right.$ (26)
(${\boldsymbol{\sigma}}$ are Pauli matrices). Eq.(26) is invariant under the
space-time inversion (${\bf x}\to-{\bf x},t\to-t$) with
$\phi(-{\bf x},-t)\to\chi({\bf x},t),\quad\chi(-{\bf x},-t)\to\phi({\bf x},t)$
(27)
Alternatively, it also remains invariant under a mass inversion as (see also
19 (19))
$m\to-m,\quad\phi({\bf x},t)\to\chi({\bf x},t),\quad\chi({\bf
x},t)\to\phi({\bf x},t)$ (28)
Note that the transformation $m\to-m$ by no means implies antiparticle having
”negative mass”. Both particle and antiparticle have positive mass as shown by
using Eqs.(11) and (12) respectively. This will be clearer later.
## V Reduced Dirac Equation for Hydrogenlike Atoms
In nonrelativistic $QM$, a hydrogenlike atom (shown in Fig.1) is treated by
Schrödinger equation in the center-of-mass coordinate system ($CMCS$) as
($\hbar=c=1$)
$\begin{array}[]{c}i\dfrac{\partial}{\partial t}\psi({\bf r}_{1},{\bf
r}_{2},t)=\left[\dfrac{1}{2m}\hat{\bf p}_{1}^{2}+\dfrac{1}{2m_{N}}\hat{\bf
p}_{2}^{2}-\dfrac{Z\alpha}{r}\right]\psi({\bf r}_{1},{\bf
r}_{2},t)\\\\[14.22636pt] =\left[\dfrac{1}{2\mu}\hat{\bf
p}^{2}-\dfrac{Z\alpha}{r}\right]\psi({\bf r},t)\end{array}$ (29)
Here $m=m_{e}$ and $m_{N}$ are the masses of electron and nucleus while
$\mu=\dfrac{mm_{N}}{m+m_{N}}\equiv\dfrac{mm_{N}}{M}$ (30)
is the reduced mass. Eq.(29) is formally written down in a relative motion
coordinate system ($RMCS$) with the ”point nucleus” being its center and ${\bf
r}={\bf r}_{1}-{\bf r}_{2}$ ($\hat{\bf p}=-i\hbar\nabla$). Thus a two-body
problem is reduced into a one-body problem in a noninertial frame like $RMCS$.
However, for relativistic $QM$ ($RQM$), we cannot bring two kinetic energy
terms in the $CMCS$ (an inertial frame) into one like that in Eq.(29) (See the
page note after Eq.(34)). When Dirac equation is used for a hydrogenlike atom,
one just put $V(r)=-\frac{Z\alpha}{r}$ directly into Eq.(26), yielding
$\left\\{\begin{array}[]{l}(i\hbar\dfrac{\partial}{\partial
t}-V(r)-mc^{2})\phi=-i\hbar
c{\boldsymbol{\sigma}}\cdot\nabla\chi,\\\\[11.38109pt]
(i\hbar\dfrac{\partial}{\partial t}-V(r)+mc^{2})\chi=-i\hbar
c{\boldsymbol{\sigma}}\cdot\nabla\phi\end{array}\right.$ (31)
Notice that here two approximations have been made implicitly:
(a) The nucleus mass $m_{N}\to\infty$, so $\mu=m=m_{e}$ in Eq.(30).
(b) The invariance of space-time inversion, Eq.(27) (${\bf x}\to{\bf r}$),
must be supplemented by
$V({\bf r},t)\to V(-{\bf r},-t)=-V({\bf r},t)$ (32)
even $V(r)$ doesn’t contain time $t$ explicitly. 333Previously, the $V$ in
Eq.(32) was called as a ”vector potential”, having the same property like that
of ”energy” in the Lorentz transformation. Here the physical meaning of $V$ is
the electron’s potential energy in an ”external field” of nucleus.
In our point of view, what Eq.(32) means is: Under the space-time inversion,
while the electron transforms into a positron, the nucleus remains unchanged
at all! Hence the nucleus is treated as an ”inert core” in Eq.(31) not only in
the sense of (a), but also in that of (b).
In a prominent paper 22 (22), by using Dirac’s method, Marsch rigorously
solved the hydrogen atom as a two Dirac particle system bound by Coulomb
force. His solutions are composed of positive and negative pairs,
corresponding respectively to hydrogen and antihydrogen as expected. However,
surprisingly, in the hydrogen spectrum, besides the normal type-1 solution
with reduced mass $\mu$, there is another anomalous type-2 solution with
energy levels:
$E^{\prime}_{n}=Mc^{2}-2\mu c^{2}+\dfrac{1}{2}\mu
c^{2}\left(\frac{\alpha}{n}\right)^{2}+\cdots,\quad(n=1,2,\ldots)$ (33)
And, ”strange enough, the type-2 ground state ($n=1$) does not have lowest
energy but the continuum ($n=\infty$)”22 (22).
In our opinion, these anomalous solutions just imply a positron moving in the
field of proton. So all discrete states with energy $E^{\prime}_{n}$ are
actually unbound, they should be and can be ruled out in physics by either the
”square integrable condition” or the ”orthogonality condition” acting on their
rigorous $WF$s (for one body Dirac equation, see 21 (21), also p.28-31, 50 of
24 (24)). On the other hand, all continuum states ($n=\infty$) with energies
lower than $Mc^{2}-2\mu c^{2}$ correspond to scattering $WF$s with negative
phase shifts, showing the repulsive force between positron and proton (25
(25), section 1.5 in 24 (24) or section 9.5 in 4 (4)). Marsch’s work precisely
validates our understanding: (a) The negative energy state of a particle just
describes its antiparticle state. (b) The Coulomb potential allows a complete
set of solutions comprising two symmetric sectors, hydrogen and antihydrogen.
In the hydrogen sector, the negative energy states mean that the proton
remains unchanged but the electron has already been transformed into a
positron under the Coulomb interaction.
However, the nucleus of a hydrogenlike atom maybe either a fermion or a boson
like deuteron $d$ (of a deuterium atom $D$) with angular momentum $I=1$. To
solve two-particle problem individually would be a daunting task, it couldn’t
be rigorous eventually too. We prefer to improve the one-body Dirac equation,
Eq.(31), at the least labor cost. It is possible, just let the reduced mass
$\mu$ replacing the $m$ in Eq.(31) and claim the invarince of ”mass inversion”
in a noninertial frame ($RMCS$), ignoring a small centripetal acceleration of
the nucleus in the $CMCS$:
$\mu\to-\mu,\quad\phi({\bf r},t)\to\chi({\bf r},t),\quad\chi({\bf
r},t)\to\phi({\bf r},t)$ (34)
Such a reduced Dirac equation ($RDE$) 26 (26) should be tested by experiments.
444Since the nucleus is assumed to be ”inert” in the sense of Eq.(32), when
$m\to-m,m_{N}\to m_{N}$, $\mu\to-\mu(1+\frac{2m}{M})$ (whereas $V(r)$ remains
unchanged under the mass inversion). So Eq.(34) has an inaccuracy up to
$\frac{2m}{M}$ ($<1.1\times 10^{-3}$ for $H$) in this claim. Thanks to
remarkable advances in high resolution laser spectroscopy and optical
frequency metrology, the $1S-2S$ two-photon transition in atomic hydrogen $H$
(or deuterium $D$) with its natural line width of only 1.3Hz has been measured
to a very high precision. In 1997, Udem et al.,determined the $1S-2S$ energy
interval of $H$ being (27 (27), see also 43 (43)):
$f_{H}^{exp}(1S-2S)=2466061413187.34(84)\,kHz$ (35)
In 1998, Huber et al.measured the isotopy-shift of the $1S-2S$ transition of
$H$ and $D$ (28 (28), see also 29 (29))
$f_{D}^{exp}(2S-1S)-f_{H}^{exp}(2S-1S)=670994334.64(15)\,kHz$ (36)
As expected, the theoretical values calculated from $RDE$ turn out to be 26
(26)
$\Delta E_{H}^{RDE}(2S-1S)=2.466067984\times 10^{15}\,Hz$ (37)
which is only a bit larger than the measured data, Eq.(35), by $3\times
10^{-6}$, and
$\Delta E_{D-H}^{RDE}(2S-1S)=6.7101527879\times 10^{11}\,Hz$ (38)
which is larger than that in Eq.(36) by $3\times 10^{-5}$ only.
Further theoretical modifications will bring the discrepancies down to one
order of magnitude respectively 26 (26). See Appendix $A$.
## VI Negative Mass Paradox and Antigravity
The well-known Newton’s gravitation law reads
$F(r)=-G\dfrac{m_{1}m_{2}}{r^{2}}$ (39)
where $m_{1}$ and $m_{2}$ are the gravitational masses of two particles or
macroscopic bodies with spherical symmetry. Then an acute problem arises: can
a body have a negative mass? If so, a bizarre phenomenon would occur as
discussed by Bondi 30 (30), Schiff31 (31) and Will32 (32) respectively.
Suppose such a body (with mass $m_{1}<0$) is brought close to a normal body
(with mass $m_{2}>0$) and assume the validity of Newton’s dynamical law,
Eq.(9), together with
$m_{inert}=m_{grav}$ (40)
Then according to Eqs.(9) and (39), the positive-mass body ($m_{2}$) would
attract the negative-mass body ($m_{1}$) whereas $m_{1}$ would repel $m_{2}$.
The pair (a ”gravitational dipole”) would accelerate itself without outside
propulsion. incredible!
The above problem was named as a ”negative mass paradox” in 21 (21). Although
a ”positive energy theorem” was proved since 1979, saying that ”the total
asymptotically determined mass of any isolated body in general relativity
($GR$) must be non-negative”, we believed21 (21) that a thorough solution to
this paradox will tell us much more. As we learn from $RQM$, the emergence of
negative energy is inevitable and is intimately related to the existence of
antiparticle. The previous discussion enables us to establish a working rule:
Any theory, either quantum or classical, being capable of reflecting the equal
existence of particle versus antiparticle, must be invariant under a mass
inversion ($m\to-m$).
So it is quite natural to generalize Eq.(39) into 21 (21):
$F(r)=\pm G\dfrac{m_{1}m_{2}}{r^{2}}$ (41)
where the minus sign holds for $m_{1}$ and $m_{2}$ (both positive) being both
matters or antimatters whereas the plus sign holds for one of them being
antimatter, meaning that matter and antimatter repel each other.
Note: The root cause of ”negative mass paradox” is stemming from an incorrect
notion that the distinction between $m$ and $-m$ is absolute. But actually, it
is merely relative, not absolute.
Now consider a positronium and an ordinary atom. There will be no
gravitational force between them, showing $m_{grav}=0$ for the positronium
relative to any matter. But its $m_{inert}\neq 0$ due to Einstein’s equation:
111Throughout this paper, mass $m$ refers to the ”rest mass” as discussed by
Okun 42 (42).
$E_{0}=m_{inert}c^{2}=mc^{2}$ (42)
with the rest energy $E_{0}$ being positive definite. Hence the $GR$’s
”equivalence principle” in the (weak) sense of Eq.(40) ceases to be valid in
the case of coexistence of matter and antimatter. Despite of its great
success, $GR$ needs some modification to meet the requirement of $SR$. Indeed,
let us look at the Einstein field equation ($EFE$) (see e.g., 33 (33)):
$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}$ (43)
On the $RHS$ of $EFE$, the energy-momentum tensor ($EMT$) $T_{\mu\nu}$ is
proportional to the mass $m$ of matter. Hence the mass inversion $m\to-m$ will
change the $RHS$ of $EFE$ but not its left-hand-side (containing no mass).
To keep $EFE$ invariant under the mass inversion, a generalization is proposed
in 21 (21) that
$T_{\mu\nu}\to T_{\mu\nu}^{eff}=T_{\mu\nu}-T_{\mu\nu}^{c}$ (44)
(as before, the superscript $c$ refers to antimatter). Notice that the form of
$EMT$ is the same for both matter and antimatter. So under the mass inversion,
$T_{\mu\nu}\to-T_{\mu\nu}^{c}$ and $T_{\mu\nu}^{c}\to-T_{\mu\nu}$. Thus
Eq.(41) can be derived from Eq.(43) with modification, Eq.(44), in a weak-
field approximation (as described in 33 (33)).
## VII Summary and Discussion
There are two invariants in the kinematics of $SR$:
$c^{2}(t_{1}-t_{2})^{2}-({\bf x}_{1}-{\bf
x}_{2})^{2}=c^{2}(t^{\prime}_{1}-t^{\prime}_{2})^{2}-({\bf
x}^{\prime}_{1}-{\bf x}^{\prime}_{2})^{2}=\text{const}\\\ $ (45)
$E^{2}-c^{2}{\bf p}^{2}=m^{2}c^{4}$ (46)
In hindsight, it seems quite clear that Eq.(45) is invariant under the space-
time inversion (${\bf x}\to-{\bf x},t\to-t$) and Eq.(46) remains invariant
under the mass inversion $m\to-m$. However, these two discrete symmetries are
deeply rooted at the dynamics of $SR$. Their implication is focused on one
common essence of nature: Everything is in contradiction, i.e., it contains
two sides in confrontation inside. For instance, an electron’s $WF$ Eq.(25),
has two components, $\phi$ and $\chi$:
$\psi_{e^{-}}({\bf x},t)\sim\phi\sim\chi\sim\exp\left[\frac{i}{\hbar}({\bf
p}\cdot{\bf x}-Et)\right],\qquad(|\phi|>|\chi|)$ (47)
where $\phi$ dominates $\chi$. Under a space-time inversion, Eq.(27), the
electron transforms into a positron with $WF$ being
$\psi_{e^{+}}({\bf
x},t)\sim\chi_{c}\sim\phi_{c}\sim\exp\left[-\frac{i}{\hbar}({\bf p}\cdot{\bf
x}-Et)\right],\qquad(|\chi_{c}|>|\phi_{c}|)$ (48)
where $\chi_{c}$ dominates $\phi_{c}$. Note that the observed momentum ${\bf
p}$ and energy $E\,(>0)$ are the same for Eqs.(47) and (48), using Eqs.(11)
and (12) respectively.555Historically, in 1953, Konopinski and Mahmaud 34 (34)
wrote down operator relation like Eq.(12) in a page note while Schwinger’s
argument in 195835 (35) also contains some insight relevant to that in this
paper. The variations of complex $WF$s of electron and positron at a fixed
point, say ${\bf x}=0$, as functions of time $t$ are depicted on Fig.2.
Comparing Eq.(3) to Eq.(47) with Eq.(48), we see that the original ”charge
conjugation transformation $C$” could be regarded as ”correct” in form but
”incorrect” in explanation. So the discovery of $C$ violation in weak
interactions could be viewed as a warning that a correct understanding in
physics needs rigorous language in mathematics, careful comparison with
experiments and a sound logic in linking them together. If a theory (e.g., the
”hole theory”) can only be explained in ordinary language, it would be likely
incorrect, or at least missing something important in the basic concept.
The symmetry between Eqs.(47) and (48) (or Eqs.(11) and (12)) can also be
ascribed to that of $i$ versus $-i$, showing again the beauty and power of
mathematics.
By providing Eq.(12) as a supplement to Eq.(11), the space-time inversion
exhibits itself as the essence of $SR$ to show the symmetry of particle versus
antiparticle. $SR$ is compatible or in conformity with $QM$ essentially. In
some senses, the reason why $RQM$, $QFT$ and particle physics are capable of
developing with vitality is because they have inherited their ”genes” ($DNA$)
half (Eq.(11)) from $QM$ and half (Eq.(12)) from $SR$.
Being different in form but equivalent in essence, the mass inversion seems
more convenient in use for implementing the particle-antiparticle symmetry,
especially for a classical theory. The prediction about the antigravity
between matter and antimatter, though interesting, remains open to scientific
verification. As now the antihydrogen atoms have already been made in
laboratories on Earth, it’s time to consider the universe being filled not
only with matter galaxies, but also with antimatter galaxies at remote
distances. A tentative model calculation is currently being studied 36 (36).
Recently, two experiments have been proposed at Fermilab 37 (37) and CERN 38
(38) respectively, aiming at directly measuring the free fall acceleration of
antihydrogen in the field of Earth (quoted from 39 (39)). We anticipate a big
surprise from such an experiment. We don’t believe in the claim that existing
experiments already place stringent bounds (say, $10^{-7}$ 39 (39)) on any
gravitational asymmetry between matter and antimatter. This is because an
electron in motion as shown by Eq.(47) does contain some ”hidden positron
field” described by $\chi$, which is by no means a real ”positron particle”
ingredient with opposite charge. Being subordinate to $\phi$ in an electron,
$\chi$ can only display itself by various $SR$ effects, including
$E_{0}=mc^{2}$, the time dilatation and the enhancement of electron’s charge
at high-energy collisions, etc. Similarly, in our opinion, the virtual
$e^{+}e^{-}$ pairs in the loops of vacuum polarization and self-energy of
$QFT$ (see Figs.1 and 2 in 39 (39)) contain no real antimatter content (of
$e^{+}$) too. While there are many arguments against ”antigravity” 41 (41),
only further experiments can judge.
## Acknowledgements
We thank E. Bodegom, Y. X. Chen, T. P. Cheng, X. X. Dai, Y. Q. Gu, F. Han, J.
Jiao, A. Khalil, R. Konenkamp, D. X. Kong, J. S. Leung, P.T. Leung, D. Lu, Z.
Q. Ma, E. J. Sanchez, Z. Y. Shen, Z. Q. Shi, P. Smejtek, X.T. Song, R. K. Su,
F. Wang, Z. M. Xu, J. Yan, R. H. Yu, Y. D. Zhang and W. M. Zhou for
encouragement, collaboration and helpful discussions.
## Appendix A: Comparison between Dirac Equation and Reduced Dirac Equation
for hydrogenlike atoms
| Dirac Equation | Reduced Dirac Equation
---|---|---
Approximation made | keep $m_{e}$, i.e., assume $m_{N}\to\infty$ (combine $CMCS$ and $RMCS$ into one frame) | keep $m_{N}$ finite but reduce $m_{e}$ into $\mu=\frac{m_{e}m_{N}}{m_{e}+m_{N}}=\frac{m_{e}m_{N}}{M}$ in the $RMCS$ (noninertial frame)
Invariance under space-time inversion | $\begin{array}[]{c}{\bf r}_{1}\to-{\bf r}_{1},t\to-t,\\\ V({\bf r}_{1},t)\to V(-{\bf r}_{1},-t)=-V({\bf r}_{1},t),\\\ \phi({\bf r}_{1},t)\to\phi(-{\bf r}_{1},-t)=\chi({\bf r}_{1},t),\\\ \chi({\bf r}_{1},t)\to\chi(-{\bf r}_{1},-t)=\phi({\bf r}_{1},t)\end{array}$ | $\begin{array}[]{c}{\bf r}={\bf r}_{1}-{\bf r}_{2},{\bf r}\to-{\bf r},t\to-t,\\\ V({\bf r},t)\to V(-{\bf r},-t)=-V({\bf r},t),\\\ \phi({\bf r},t)\to\phi(-{\bf r},-t)=\chi({\bf r},t),\\\ \chi({\bf r},t)\to\chi(-{\bf r},-t)=\phi({\bf r},t)\end{array}$
Invariance under mass inversion | $\begin{array}[]{c}m_{e}\to-m_{e},\\\ V({\bf r}_{1})\to V({\bf r}_{1}),\\\ \phi({\bf r}_{1},t)\to\chi({\bf r}_{1},t),\\\ \chi({\bf r}_{1},t)\to\phi({\bf r}_{1},t)\end{array}$ | $\begin{array}[]{c}m_{e}\to-m_{e},m_{N}\to m_{N},\\\ \mu\to-\mu(1+\frac{2m_{e}}{M})\sim-\mu,\\\ V({\bf r})\to V({\bf r}),\\\ \phi({\bf r},t)\to\chi({\bf r},t),\\\ \chi({\bf r},t)\to\phi({\bf r},t)\end{array}$
Physical implication | With the increase of nucleus charge number $Z$, the electron’s energy decreases with $|\chi|$ rising against $|\phi|$. But even when electron turns into a positron ($|\chi|>|\phi|$), the nucleus remains unchanged at all. | Same as the case of Dirac equation even the mass of nucleus is finite.
Theoretical prediction
(its discrepancy from the experimental data) | $\begin{array}[]{c}\Delta E^{Dirac}_{H}(2S-1S)\\\ =2.467411048\times 10^{15}Hz\\\ (5.5\times 10^{-4}),\\\ \Delta E^{Dirac}_{D-H}(2S-1S)=0\\\ (100\%)\end{array}$ | $\begin{array}[]{c}\Delta E^{RDE}_{H}(2S-1S)\\\ =2.466067984\times 10^{15}Hz\\\ (3\times 10^{-6}),\\\ \Delta E^{RDE}_{D-H}(2S-1S)\\\ =6.7101527879\times 10^{11}Hz\\\ (3\times 10^{-5})\end{array}$
## Appendix B: Hints from Philosophy
Fig.2 could be compared with Fig.3.
## References
* (1) J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley Publishing Company, 1978.
* (2) O. Nachtman, Elementary Particle Physics, Concepts and Phenomena, Springer-Verlag, 1990.
* (3) J. J. Sakurai, Modern Quantum Mechanics, NewYork, John Wiley & Sons, Inc. 1994.
* (4) G. J. Ni and S. Q. Chen, Advanced Quantum Mechanics, 2nd Edition (Fudan University Press, 2003); English Edition was published by Rinton Press, 2002.
* (5) J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill Book Company. 1964.
* (6) S. Weinberg, The Quantum Theory of Fields, Vol.I, Cambridge University Press, Cambridge, 1995
* (7) T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956).
* (8) C. S. Wu et al., Phys. Rev. 105, 1413 (1957).
* (9) C. Amsler et al.,, Phys. Lett. B 667, 1 (2008) or the Particle Physics Booklet extract from it.
* (10) T. D. Lee and C. S. Wu, Ann. Rev. Nucl. Sci., 15, 381-476, (1965).
* (11) G. J. Ni, Journal of Fudan University (Natural Science), 1974, No.3-4, 125 (In fact, this paper was in collaboration with Suqing Chen, but her name was erased according to editor’s advice for promoting the publication at that time).
* (12) G. J. Ni and S. Q. Chen, Journal of Fudan University (Natural Science) 35, 325 (1996); English version is in ”Photon and Poincaré Group”, edit. V. Dvoeglazov (NOVA Science Publisher, Inc. 1999), Chapter III, 145-169.
* (13) G. J. Ni, Progress in Physics (Nanjing, China) 23, 484-503 (2003), (In English).
* (14) A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).
* (15) D. Bohm, Quantum Theory, Prentice Hall, 1956.
* (16) J. S. Bell, Physics, Long Island City, NewYork, 1, 195-200 (1964).
* (17) H. Guan, Basic Concepts in Quantum Mechanics, (High Education Press, Beijing, 1990), Chapter 7 (in Chinese).
* (18) G. J. Ni, H. Guan, W. M. Zhou and J. Yan, Chinese Phys. Lett. 17, 393-395 (2000), quant-ph/0001016.
* (19) G. J. Ni, in ”Quantum Mechanics, Picking up the Morning Flower in the Evening, Articles on teaching and Learning”, edit. W. Z. Wang, S. C. Ke and Q. W. Liu, Science Press, Beijing, 2004, 143-167 (in Chinese).
* (20) A. Apostolakis et al.(CPLEAR Collaboration),, Phys. Lett. B 422, 339-348 (1998).
* (21) G. J. Ni, in Relativity, Gravitation, Cosmology, Edit by V. V. Dvoeglazov and A. A. Espinoza Garrido, NOVA Science Publisher, 2004, 123-136, physics/0308038.
* (22) E. Marsch, Ann. Phys. (Leipzig) 14, No.5, 324-343 (2005).
* (23) X. X. Dai, Journal of Fudan University (Natural Science), 1977, No.1, 100.
* (24) G. J. Ni and S. Q. Chen, Levinson Theorem, Anomaly and the Phase Transition of Vacuum, (Shanghai Scientific & Technical Publishers, 1995, in Chinese).
* (25) Z. Q. Ma and G. J. Ni, Phys. Rev. D 31, 1482 (1985).
* (26) G. J. Ni, J. J. Xu and S. Y. Lou, Preprint, quant-ph/0511197.
* (27) Th. Udem et al., Phys. Rev. Lett. 79, 2646 (1997).
* (28) A. Huber, Th. Udem, B. Gross et al.Phys. Rev. Lett. 80, 468 (1998).
* (29) F. Schmidt-Kaler et al., Phys. Rev. Lett. 70, 2261 (1993).
* (30) H. Bondi, Rev. Mod. Phys. 29, 423-428 (1957).
* (31) L. I. Schiff, Phys. Rev. Lett. 1, 254-255 (1958)); Proc. Nat. Acad. Sci.US, 45, 69-80 (1959).
* (32) C. Will, in The New Physics, Edit by P. Davis, Cambridge University Press, 1989, Chapter 2, p.31.
* (33) S. Weinberg, Gravitation and Cosmology, John Wiley. 1972.
* (34) E. J. Konopinski and H. M. Mahmaud, Phys. Rev. 92, 1045 (1953).
* (35) J. Schwinger, Proc. Nat. Acad. Sci. US, 44, 223-228 (1958).
* (36) G. J. Ni and S. Y. Lou, in preparation.
* (37) A. D. Cronin et al., ”Letter of Intent: Antimatter Gravity Experiment (AGE) in Fermilab”, (2009)
* (38) G. Dobrychev et al., ”Proposal for the AEGIS experiment at the CERN Antiproton Decelerator (Antimatter Experiment: Gravity, Interferometry, Spectroscopy).” (2009)
* (39) D. S. M. Alves, M. Jankowick and P. Saraswat, arXiv: 0907.4110v1[hep-ph]23, July 2009.
* (40) F. Capra. The Tao of Physics, Shambhala, 2000.
* (41) M. M. Nieto and T. Goldman, Phys. Rep. 205, 221-281 (1991).
* (42) L. B. Okun, Phys. Today, 42, 31-36 (June 1989); Discussions in 42, pp.13, 15, 115, 117 (May 1990)
* (43) M. Niering et al., Phys. Rev. Lett. 84, 5496-5499 (2000).
Figure 1: A hydrogenlike atom in quantum mechanical description. The nucleus
with mass $m_{N}$ occupies a small sphere with radius $r_{N}$ (greatly
exaggerated in the diagram) while the electron with mass $m$ spreads over a
larger sphere with radius $R_{e}$ (i.e.atomic radius). Their common center is
the atom’s center of mass (CM). The wavefunction $\psi({\bf r})e^{-iEt}$ with
${\bf r}={\bf r}_{1}-{\bf r}_{2}$ shows the electron’s amplitude under a
”fictitious measurement” 4 (4), during which the electron and nucleus shrink
into two ”fictitious point particles ” located at ${\bf r}_{1}$ and ${\bf
r}_{2}$ simultaneously. The Coulomb potential $V(r)=-\frac{Ze^{2}}{r}$ between
them is a static one. The probability to find the electron at ${\bf r}$ is
$|\psi({\bf r})|^{2}$ while that to find its momentum being ${\bf p}$ is
$|\phi({\bf p})|^{2}$ with $\phi({\bf p})$ being the Fourier transform of
$\psi({\bf r})$.
Figure 2: The $WFs$ of (a) electron: $\psi_{e^{-}}(0,t)=e^{-i\omega t}$ and
(b) positron: $\psi_{e^{+}}(0,t)=e^{i\omega t},\;(\omega=\frac{E}{\hbar})$.
Their momentum $p$ is perpendicular to the paper and both particles move
towards us ($p>0$). The points $\psi_{e^{-}}$ and $\psi_{e^{+}}$ rotate on the
unit circle clockwise and anticlockwise respectively. $Re\psi$ and $Im\psi$
transform into each other during the particle’s motion or a gauge (phase)
transformation. So the distinction between them is merely relative, not
absolute. (see Appendix B) 19 (19).
Figure 3: Tai-chi Tu (Diagram of the supreme ultimate 40 (40)). (a) The
original one and (b) its ”mirror image” are rotating clockwise and
anticlockwise respectively. Black and white colors refer to ”yin” and ”yang”.
Two small circles hiding inside imply that ”there is yang hiding inside yin
and vice versa”. And the motion is triggered by mutual interactions between
yin and yang inside, not due to a push from the outside. The yin and yang
could be corresponding to, of course not precisely, the $Re\psi$ and $Im\psi$
of a $WF$ $\psi$ (see Fig.2)19 (19).
|
arxiv-papers
| 2010-07-19T01:56:49 |
2024-09-04T02:49:11.702880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guang-jiong Ni, Suqing Chen, Senyue Lou and Jianjun Xu",
"submitter": "Jianjun Xu",
"url": "https://arxiv.org/abs/1007.3051"
}
|
1007.3054
|
# Puzzles of Divergence and Renormalization
in Quantum Field Theory
Guang-jiong Ni a,b pdx01018@pdx.edu a Department of Physics, Portland State
University, Portland, OR97207, U. S. A.
b Department of Physics, Fudan University, Shanghai, 200433, China Jianjun Xu
b xujj@fudan.edu.cn b Department of Physics, Fudan University, Shanghai,
200433, China Senyue Lou c,d sylou@sjtu.edu.cn c Department of Physics,
Shanghai Jiao Tong University, Shanghai, 200030, China
d Department of Physics, Ningbo University, Ningbo 315211, China
###### Abstract
A regularization renormalization method ($RRM$) in quantum field theory
($QFT$) is discussed with simple rules: Once a divergent integral $I$ is
encountered, we first take its derivative with respect to some mass parameter
enough times, rendering it just convergent. Then integrate it back into $I$
with some arbitrary constants appeared. Third, the renormalization is nothing
but a process of reconfirmation to fix relevant parameters (mass, charge,
etc.) by experimental data via suitable choices of these constants. Various
$QFT$ problems, including the Lamb shift, the running coupling constants in
$QED$ and $QCD$, the $\lambda\phi^{4}$ model as well as Higgs mass in the
standard model of particle physics, are discussed. Hence the calculation,
though still approximate and limited in accuracy, can be performed in an
unambiguous way with no explicit divergence, no counter term, no bare
parameter and no arbitrarily running mass scale (like the $\mu$ in $QCD$).
Keywords: renormalization, quantum field theory, Lamb shift, running coupling
constant
PACS: 03.70.+k; 11.10.-z; 11.10.Gh; 11.10.Hi
## I Introduction
Since the establishment of quantum mechanics ($QM$) and the quantization of
electromagnetic field, the quantum electrodynamics ($QED$) in particular and
the quantum field theory ($QFT$) in general have been developed for over 80
years. A common prominent feature of $QFT$ is the emergence of the divergence
in calculations beyond the tree level. To handle these divergences, various
regularization and renormalization methods ($RRMs$) have been proposed.
Despite the great success of $QFT$, the present status of $RRMs$ remains
ambiguous to some extent. For example, in the theory of Chromodynamics ($QCD$)
for describing the strong interactions of colored quarks and gluons, a
commonly used renormalization scheme ($RS$) is the modified minimal
substraction ($\bar{MS}$) scheme (see a summary in the Review of Particle
Physics in 2008, 1 , p.157) where an arbitrary renormalization mass scale
$\mu$ is introduced (see next section). Physicists believe the fundamental
theorem of $RS$ dependence: Physical quantities, such as the cross-section
calculated to all orders in perturbation theory, should not depend on the
$RS$. However, it follows that a truncated series does exhibit $RS$
dependence. In practice, $QCD$ cross-section are known to different orders,
depending on the choice of $RS$ (and $\mu$) in different sensitive ways. We
still don’t know what is the ”best” choice for $\mu$ within a given scheme
(usually $\bar{MS}$). There is no definite answer to this question yet.
## II What a divergence means?
Physicists often talk about different orders of a divergence, based on its
dimension with respect to mass (i.e., momentum, we use natural unit system
with $\hbar=c=1$). For example, if a Feynman diagram integral ($FDI$) in $QFT$
reads
$I=\int\dfrac{d^{4}K}{(2\pi)^{4}}\dfrac{K^{n}}{(K^{2}-M^{2})^{2}}$ (1)
where $K$ corresponds to the (4-dimensional)momentum of virtual particle (say,
virtual photon in $QED$) and $M$ is a mass parameter (maybe in a complex form)
characterizing the $QFT$ under consideration. Then
$n=\left\\{\begin{array}[]{l}0,\quad\text{logarithmic divergence}\\\
1,\quad\text{linear divergence}\\\ 2,\quad\text{quadratic
divergence}\end{array}\right.$ (2)
However, among these categories, only the first one is really meaningful in
mathematics. This is because the definition of a number sequence
$A_{i}\;(i=1,2,\ldots)$ having a limit being $\infty$ is as follows: Given
arbitrarily a large number $M$, one can always find such a number $N$ so that
$A_{i}>M$, when $i>N$. Here $A_{i}$ and $M$, let alone $i$ and $N$, are all
dimensionless numbers. A number $M\gg 1$ is called a large number, whereas
$\varepsilon\ll 1$ a small number.
On the other hand, the space-time coordinates $\bf x$ and $t$, mass $m$ and
momentum $p$ (or $k$) are physical quantities and each with certain dimension.
If treating them as dimensionless numbers, we will run into trouble
inevitably.
Example A: Assume that in $QFT$, a $FDI$ with linear divergence is
approximately expressed as $I\sim 10^{3}M$ with $M$ being a mass parameter in
the unit of $mg$. If $M=1\,mg$, we have $I\sim 10^{3}mg$ with $10^{3}\gg 1$
being a large number. But if changing the unit from $mg$ to $kg$, we will have
$I\sim 10^{-3}kg$ with $10^{-3}\ll 1$ being a small number. A mathematician
would ask: ”Could you still treat your $I$ as a divergent quantity?”Who could
answer his question?
Example B: In the $\bar{MS}$ scheme, the renormalization mass scale $\mu$ is
introduced as follows (see p.137 in 2 (2))
$\dfrac{\Gamma(2-d/2)}{(4\pi)^{d/2}(m^{2})^{2-d/2}}=\dfrac{1}{(4\pi)^{2}}\left[\dfrac{2}{\varepsilon}-\gamma+\ln(4\pi)-\ln(m^{2})\right]\to\dfrac{1}{(4\pi)^{2}}\left[-\ln\left(\dfrac{m^{2}}{\mu^{2}}\right)\right]$
(3)
where $m$ is some mass parameter containing in the model. Eq.(3) is derived
from the ”dimensional regularization” method. The 4-dimensional (Euclidean)
space has been analytically continued into $d$-dimensional one with
$\varepsilon\sim 4-d\sim 0$ and Gamma function
$\Gamma(2-d/2)=\Gamma(\varepsilon/2)=2/\varepsilon-\gamma$
($\gamma=0.4772\ldots$ is the Euler constant). Obviously, the left-handed-side
($LHS$) has a dimension of $m^{-\varepsilon}$, whereas in the right-handed-
side ($RHS$), the function $\ln\left(\frac{m^{2}}{\mu^{2}}\right)$ becomes
dimensionless after the $\mu$ is introduced. However, the mathematician would
feel quite uncomfortable because $\varepsilon\neq 0$. He will focus on the
middle of Eq.(3) and ask: ”Why the divergent number $2/\varepsilon$ disappears
at the $RHS$ and becomes finite? Where the term $\ln(m^{2})$ comes from? What
is its dimension?”
We physicists accept Eq.(3) since it could be derived from a ”mathematical
formula” like (see p.57 in 3 (3))
$(m^{2})^{d/2-2}=\exp\left[\left(\frac{d}{2}-2\right)\ln m^{2}\right]\simeq
1+\left(\frac{d}{2}-2\right)\ln m^{2}$ (4)
Then the mathematician would say: ”No! In the mathematical formula
$x^{y}=\exp[y\ln x]$ (5)
both $x$ and $y$ must be dimensionless numbers. So in the $LHS$ of Eq.(3), you
correctly write down:
$(4\pi)^{-d/2}=(4\pi)^{2-d/2-2}=\dfrac{1}{(4\pi)^{2}}\exp\left[(2-\frac{d}{2})\ln(4\pi)\right]\simeq\dfrac{1}{(4\pi)^{2}}\left[1+(2-\frac{d}{2})\ln(4\pi)\right]$
But Eq.(4) is wrong because $x=m^{2}$ is a physical quantity with dimension.
That’s why you got a strange result at the $RHS$ of Eq.(3)”.
Hence if insisting on mathematical rigor, we should admit that the
introduction of $\mu$ via Eq.(3) is groundless. Then a question arises: Why
the $\mu$ seems necessary in $QCD$ ?
To our understanding, the answer lies in the fact that in high energy $QCD$,
the quarks’ masses were often neglected. Therefore, in order to express the
running coupling constant ($RCC$) of strong interaction, $\alpha_{s}$, as a
function of $Q$, the 3-dimensional momentum transfer in collision, one needs
$\mu$ as shown by the solution of renormalization-group-equation ($RGE$) (see
p.532 of 2 (2) and Eq.(53) below):
$\alpha_{s}(Q)=\dfrac{\alpha_{s}(\mu)}{1+\alpha_{s}(\mu)\frac{\beta_{0}}{2\pi}\ln(Q/\mu)}$
(6)
It is interesting to solve Eq.(6) for $\alpha_{s}(\mu)$, yielding
$\alpha_{s}(\mu)=\dfrac{\alpha_{s}(Q)}{1+\alpha_{s}(Q)\frac{\beta_{0}}{2\pi}\ln(\mu/Q)}$
(7) 111see Appendix of 4 (4), where a typing error exists in the denominator
of Eq.(A.5), ”$+$” should be ”$-$”.
We see that Eqs.(6) and (7) are symmetrical with respect to mutual change of
$Q\leftrightarrow\mu$. $Q$ and $\mu$ are essentially equivalent. Why we need
both of them? The answer is: only $Q/\mu$ is capable of expressing a
dimensionless $\alpha_{s}$. However, as shown in Eq.(3), the existence of
$\mu$ is doubtful, even superfluous. Once we take the quarks’ masses into
account, there will be no need of $\mu$ at all (see section V below).
## III Self-Energy Correction of an Electron, Lamb Shift
As is well known (see e.g., Refs.4 (4, 5, 6)), the $FDI$ of a free electron’s
self-energy at one loop ($L=1$) level of $QED$ in covariant form reads
$-i\Sigma(p)=(ie)^{2}\int\dfrac{d^{4}k}{(2\pi)^{4}}\dfrac{g_{\mu\nu}}{ik^{2}}\gamma^{\mu}\dfrac{i}{\not\\!\\!{p}-\not\\!\\!{k}-m}\gamma^{\nu}$
(8)
where the Bjorken-Drell metric ($\not\\!\\!{p}=\gamma^{\mu}p_{\mu}$) and
rationalized Gaussian units are adopted with electron charge $-e\,(e>0)$ and
mass $m=m_{e}$. In Eq.(8), $p$ and $k$ are momenta of electron and (virtual)
photon. After introducing the Feynman parameter $x$ and making a shift in
momentum integration: $k\to K=k-xp$, Eq.(8) is recast into
$-i\Sigma(p)=-e^{2}\int_{0}^{1}dx[-2(1-x)\not\\!\\!{p}+4m]I$ (9)
with
$I=\int\dfrac{d^{4}K}{(2\pi)^{4}}\dfrac{1}{(K^{2}-M^{2})^{2}},\quad
M^{2}=p^{2}x^{2}+(m^{2}-p^{2})x$ (10)
being a logarithmically divergent integral, see Eq.(1).
Note that in Eq.(10) we can change the unit of $M$ (and $K$) at our disposal
without any change in the value of $I$, which is just a ”dimensionless”,
”large” but ”uncertain” number. However, in the past, we used to pay too much
attention to its feature of being ”large”, trying to curb the divergence by
means of some regularization method, which led to complicated renormalization
schemes ($RS$).
By contrast, now we believe the more important, even essential feature of a
divergence is hiding in its ”uncertainty”. To stress this cognition, we just
use a simple trick to regulate the $I$ in Eq.(10) as follows.
To render it convergent, we perform a differentiation with respect to the
mass-square parameter $M^{2}$, yielding
$\dfrac{\partial I}{\partial
M^{2}}=2\int\dfrac{d^{4}K}{(2\pi)^{4}}\dfrac{1}{(K^{2}-M^{2})^{3}}=\dfrac{-i}{(4\pi)^{2}}\dfrac{1}{M^{2}}$
(11)
Then we reintegrate Eq.(11) with respect to $M^{2}$ and arrive at
$I=\dfrac{-i}{(4\pi)^{2}}(\ln
M^{2}+C_{1})=\dfrac{-i}{(4\pi)^{2}}\ln\dfrac{M^{2}}{\mu_{2}^{2}}$ (12)
where an arbitrary constant $C_{1}=-\ln\mu_{2}^{2}$ (with $\mu_{2}$ a mass
scale to be fixed later) is introduced so that the ambiguity of dimension in
the $\ln M^{2}$ term can be eliminated.
Further integration of Eq.(9) with respect to $x$ leads to
($\alpha=\frac{e^{2}}{4\pi}$)
$\begin{array}[]{l}\Sigma(p)=A+B\not\\!\\!{p},\\\\[14.22636pt]
A=\dfrac{\alpha}{\pi}m\left[2-2\ln\dfrac{m}{\mu_{2}}+\dfrac{(m^{2}-p^{2})}{p^{2}}\ln\dfrac{(m^{2}-p^{2})}{m^{2}}\right],\\\\[14.22636pt]
B=\dfrac{\alpha}{4\pi}\left\\{2\ln\dfrac{m}{\mu_{2}}-3-\dfrac{(m^{2}-p^{2})}{p^{2}}\left[1+\dfrac{(m^{2}+p^{2})}{p^{2}}\ln\dfrac{(m^{2}-p^{2})}{m^{2}}\right]\right\\}\end{array}$
(13)
Using the chain approximation, we can derive the modification on the electron
propagator as
$\dfrac{i}{\not\\!\\!{p}-m}\to\dfrac{i}{\not\\!\\!{p}-m}\dfrac{1}{1-\frac{\Sigma(p)}{\not\\!\\!{p}-m}}=\dfrac{iZ_{2}}{\not\\!\\!{p}-m_{R}}$
(14)
where
$Z_{2}=\dfrac{1}{1-B}$ (15)
is the renormalization factor for electron’s wave function and
$m_{R}=\dfrac{m+A}{1-B}$ (16)
is the renormalized mass of $m$. The increment of mass reads
$\delta m=m_{R}-m=\dfrac{A+mB}{1-B}$ (17)
In the past, many physicists viewed $\delta m$ as some real contribution of
”radiation correction”. While $m_{R}$ should be identified with the observed
mass $m_{obs}$, or physical mass $m_{e}$, the original $m$ (usually denoted by
$m_{0}$ or $m_{B}$ in the expression of Lagrangian density) was thought to be
a ”bare mass”. Both $\delta m$ and $m_{0}$ were divergent quantities. (see,
e.g., p.220 in 2 (2)).
We don’t think so. Let us read carefully the seminal paper by Bethe in 19478
(8). The theory for the hydrogenlike atom begins with a Hamiltonian in the
center-of-mass frame
$H_{0}=\dfrac{{\bf p}^{2}}{2m}+\dfrac{{\bf
p}^{2}}{2m_{N}}-\dfrac{Z\alpha}{r}=\dfrac{{\bf
p}^{2}}{2\mu}-\dfrac{Z\alpha}{r}$ (18)
Bethe pointed out that the effect of electron’s interaction with the vector
potential $\bf A$ of radiation field
$H_{int}=\dfrac{e}{mc}{\bf A}\cdot{\bf p}$ (19)
should properly be regarded as already included in the $m_{obs}$, which is
denoted by $m$ in Eqs.(18) and (19).
In our understanding on Bethe’s claim, the ”self-interaction” of electron with
radiation field is indivisible from the free electron mass $m$. In other
words, in the covariant form of $QED$, certain contributions of $FDIs$ for
”self-energy” (with Eq.(8) being merely that at $L=1$ order) at all orders (up
to $L\to\infty$) are already contained in the value of $m$. To show this
cognition, we impose the mass-shell condition $p^{2}=m^{2}$ in Eq.(17)
together with
$\delta m|_{p^{2}=m^{2}}=\dfrac{\alpha m}{4\pi}(5-6\ln\frac{m}{\mu_{2}})=0$
(20)
which in turn fixes the arbitrary constant $\mu_{2}$ to be
$\mu_{2}=me^{-5/6}$ (21)
and thus
$Z_{2}|_{p^{2}=m^{2}}=\dfrac{1}{1+\frac{\alpha}{3\pi}}\simeq
1-\dfrac{\alpha}{3\pi}$ (22)
Note that $m_{R}=m=m_{obs}=m_{e}$ with no bare mass at all and $Z_{2}$ is
fixed and finite, in sharp contrast to that in previous theories.
Our reader may wonder: ”In this case, does the calculation on $FDI$ for the
self-energy become worthless ?” The answer is ”No” due to two reasons. First,
at the $QM$ level, the parameters $m$ and $e$ in Eqs.(18) and (19) can be
regarded as well-defined. But they are not so at the level of $QED$. As
discussed before Eq.(20), the new effect of radiative corrections of $FDIs$
for self-energy is inevitably confused with that in the mass, the dividing
line between them is blurred. In some sense, the appearance of divergence in
the $FDI$ is just a warning: the new effect you want to calculate has become
entangled with the mass $m$, rendering both of them uncertain. Hence the aim
of so-called mass renormalization is nothing but a reconfirmation of $m$ as we
did in Eqs.(20)-(22), where the mass $m$ is renormalized on the mass-shell
$p^{2}=m^{2}$ with $m=m_{e}$ being fixed by the experimental value and thus
well-defined. This is one important thing we must do and at most we can do on
the mass-shell for a free electron.
Second, the increment of mass, $\delta m$, ceases to be zero once when the
electron is moving off-mass-shell ($p^{2}\neq m^{2}$). Then Eq.(17) will
provide some information about the new effect of radiation corrections. For
example, for a bound electron in a hydrogenlike atom, in Ref.7 (7), we replace
the electron mass $m=m_{e}$ by reduced mass
$\mu=\frac{m_{e}m_{N}}{m_{e}+m_{N}}$ (not to be confused with the $\mu$ in
$QCD$) and write (see also 29 (29)):
$p^{2}=\mu^{2}(1-\zeta)$ (23)
Here a dimensionless parameter $\zeta\,(>0)$ is introduced to show (on
average) how large the extent of ”off-mass-shell” is. Substitution of Eq.(23)
into Eq.(17) yielding
$\delta\mu\simeq\dfrac{\alpha\mu}{4\pi}\dfrac{(-\zeta+2\zeta\ln\zeta)}{1+\frac{\alpha}{3\pi}}$
(24)
where some terms of the order of $\zeta^{2}$ or $\zeta^{2}\ln\zeta$ are
neglected since $\zeta\ll 1$.
As a perturbative calculation at $L=1$ order, we may ascribe $\delta\mu$ to
the (minus) binding energy $B$ of electron in the Bohr theory
$\delta\mu=\varepsilon_{n}=-B=-\dfrac{Z^{2}\alpha^{2}}{2n^{2}}\mu$ (25)
Combination of Eqs.(24) and (25) gives the value of $\zeta=\zeta^{<S>}$ with
the superscript $<S>$ referring to ”self-energy (at $L=1$ order)”.
Another ”nonperturbative” method to fix the $\zeta$ in Eq.(23) is to resort to
the Virial theorem: For an electron in the Coulomb potential
$V=-\frac{Z\alpha}{r}$, its kinetic energy $T=\frac{1}{2\mu}{\bf p}^{2}$ can
be evaluated on average as
$<{\bf p}^{2}>=2\mu<T>=2\mu[-B-<V>]=2\mu B$ (26) $<p^{2}>=<E^{2}-{\bf
p}^{2}>=<(\mu-B)^{2}-{\bf p}^{2}>\simeq\mu^{2}(1-\dfrac{4B}{\mu})$ (27)
Comparing Eq.(27) with Eq.(23), we obtain
$\zeta^{<V>}=\dfrac{4B}{\mu}=\dfrac{2Z^{2}\alpha^{2}}{n^{2}}$ (28)
where the superscript $<V>$ refers to ”Virial theorem”.
In Ref.7 (7), for explaining the Lamb shift of energy levels in hydrogenlike
atoms, we find the result being expressed in terms of $\zeta$. Throughout the
entire calculation, all ultraviolet divergences are handled like that in
Eqs.(9)-(12) while the infrared divergence disappears due to the introduction
of $\zeta$. However, the formulas are still approximate and either one of
$\zeta^{<S>}$ and $\zeta^{<V>}$ is not reliable. So in the following table
$I$, not only $\zeta^{<S>}$ and $\zeta^{<V>}$, but also two kinds of
”average”, $\zeta^{<S+V>}=\frac{1}{2}(\zeta^{<S>}+\zeta^{<V>})$ and
$\zeta^{<SV>}=\sqrt{(\zeta^{<S>}\zeta^{<V>}}$ are given.
Table I. Off-mass-shell parameter $\zeta$ and $\ln\zeta$ |
---|---
$\frac{Z^{2}}{n^{2}}$ | $\zeta^{<S>}\times 10^{4}$ | -$\ln\zeta^{<S>}$ | $\zeta^{<V>}\times 10^{6}$ | -$\ln\zeta^{<V>}$ | $\zeta^{<S+V>}\times 10^{5}$ | -$\ln\zeta^{<S+V>}$ | $\zeta^{<SV>}\times 10^{5}$ | $-\ln\zeta^{<SV>}$
$\frac{1}{16}$ | $1.546093458$ | $8.77461$ | $\frac{\alpha^{2}}{8}=6.6564192$ | 11.91992886 | $8.0632$ | 9.425609 | 3.2080284 | 10.34727
$\frac{1}{4}$ | $7.446539697$ | 7.20259 | $\frac{\alpha^{2}}{2}=26.6256771$ | 10.5336345 | $38.5639$ | 7.860609 | 14.0808 | 8.86816225
1 | $37.73719345$ | 5.57969 | $2\alpha^{2}=106.502$ | 9.147340142 | $194.011$ | 6.2450103 | 63.39626 | 7.36351521
There are $8$ cases discussed in 7 (7). The first one is the hydrogen atom’s
”classical Lamb shift” measured as:
$L_{H}^{exp}(2S-2P)=E_{H}(2S_{1/2})-E_{H}(2P_{1/2})=1057.845\;MHz$ (29)
Theoretically, the radiative correction (at $L=1$ order) makes the dominant
contribution, yielding:
$\begin{array}[]{l}\Delta E_{H}^{Rad<S>}(2S-2P)=1000.657\;MHz\\\ \Delta
E_{H}^{Rad<S+V>}(2S-2P)=1089.651\;MHz\\\ \Delta
E_{H}^{Rad<SV>}(2S-2P)=1226.087\;MHz\\\ \Delta
E_{H}^{Rad<V>}(2S-2P)=1451.791\;MHz\end{array}$ (30)
Taking the small contribution from the nuclear size effect into account, we
adopt the $<S+V>$ scheme to obtain
$L_{H}^{theor}(2S-2P)=1089.794\;MHz$ (31)
which is larger than the experimental value, Eq.(29), by $3\%$.
The most interesting case is the $1S-2S$ two-photon transition in hydrogen $H$
or deuterium $D$ because its natural width is so tiny ($1.3Hz$) and thus
allows precision measurement in recent years9 (9):
$\Delta E_{H}^{exp}(1S-2S)=2466061413187.34(84)\;kHz$ (32)
The isotope shift of $1S-2S$ transition between $H$ and $D$ had been measured
first by Schmidt-Kalar et al.10 and quoted in 11 (11) as:
$\Delta E_{D-H}^{exp}(2S-1S)=670994337(22)\;kHz$ (33)
Theoretically, the above accurate data cannot be explained by the original
Dirac equation with nucleus having mass $m_{N}\to\infty$. We propose a reduced
Dirac equation ($RDE$) with electron mass replaced by reduced mass for $H$ and
$D$ being respectively
$\mu_{H}=\dfrac{m_{e}m_{p}}{m_{e}+m_{p}},\quad\mu_{D}=\dfrac{m_{e}m_{d}}{m_{e}+m_{d}}$
(34)
Then theoretically, the $RDE$ predicts:
$\Delta E_{H}^{RDE}(2S-1S)=2.466067984\times 10^{15}\;Hz$ (35) $\Delta
E_{D-H}^{RDE}(2S-1S)=6.7101527879\times 10^{11}\;Hz$ (36)
which are larger than the experimental values by only $3\times 10^{-6}$ and
$3\times 10^{-5}$ respectively. Further radiative corrections on Eq.(35) will
be sensitive to the choice of schemes in Table $I$, the best one is $\Delta
E_{H}^{Theor<SV>}(2S-1S)$, deviating from the experimental data, Eq.(32), by
$-1\times 10^{-7}$ only. On the other hand, besides Eq.(36), the $\Delta
E_{D-H}^{Theor}(2S-1S)$ is influenced considerably by the nuclear size effect
and so less sensitive to the scheme choice of the smaller radiative
correction, bringing the discrepancy between theory and experimental data,
Eq.(33), from $3\times 10^{-5}$ down to $3\times 10^{-6}$ approximately.
## IV Renormalization Group Equation ($RGE$) for $QED$ and Its Solution
In $QED$, the $FDI$ for photon self-energy (i.e., vacuum polarization) can
also be evaluated 4 (4), bringing the charge $e$ into its renormalized one:
$e^{2}\to e^{2}_{R}=Z_{3}e^{2}$ (37)
$Z_{3}=1+\dfrac{\alpha}{3\pi}\left(\ln\dfrac{m^{2}}{\mu^{2}_{3}}-\dfrac{q^{2}}{5m^{2}}+\cdots\right)$
(38)
Here $m$ is the fermion (say, electron) mass, $q$ is the momentum of photon
and $\mu_{3}$ is an arbitrary constant emerging from the treatment on the
divergence like that in Eqs.(9)-(12). Although Eq.(37) looks like that in
previous theories, it is really a new one: $e$ is the observed (physical)
charge, not a ”bare charge”, and $Z_{3}$ remains finite.
The vertex function between two fermions’ momenta $p$ and $p^{\prime}$ with
$p^{\prime}-p=q$ will give another $Z_{1}$ 4 (4, 7). Adding all the $FDIs$, we
find the renormalized charge being:
$e_{R}=\dfrac{Z_{2}}{Z_{1}}Z_{3}^{1/2}e$ (39)
But the Ward-Takahashi identity ($WTI$) implies that6 (6)
$Z_{1}=Z_{2}$ (40)
Hence Eq.(37) remains valid and
$e_{R}(Q)=e\left\\{1+\dfrac{\alpha}{2\pi}\left[\dfrac{1}{3}\ln\dfrac{m^{2}}{\mu^{2}_{3}}+\dfrac{1}{15}\dfrac{Q^{2}}{m^{2}}+\cdots\right]\right\\}$
(41)
where $Q^{2}=-q^{2}>0$, with $Q$ being the 3-dimensional momentum transfer at
fermion collision. The observed charge should be defined at $Q\to 0$ (Thomson
scattering limit):
$e_{obs}=e_{R}|_{Q=0}=e$ (42)
which dictates that
$\mu_{3}=m$ (43)
As usual, the beta function is defined as
$\beta(\alpha,Q)\equiv Q\dfrac{\partial}{\partial Q}\alpha_{R}(Q)$ (44)
From Eq.(41), it is found in 4 (4) that
$\beta(\alpha,Q)=\dfrac{2\alpha^{2}}{3\pi}-\dfrac{4\alpha^{2}m^{2}}{\pi
Q^{2}}\left[1+\dfrac{2m^{2}}{\sqrt{Q^{4}+4m^{2}Q^{2}}}\ln\dfrac{\sqrt{Q^{4}+4m^{2}Q^{2}}-Q^{2}}{\sqrt{Q^{4}+4m^{2}Q^{2}}+Q^{2}}\right]$
(45)
$\beta(\alpha,Q)=\dfrac{2\alpha^{2}}{15\pi}\dfrac{Q^{2}}{m^{2}},\quad(\dfrac{Q^{2}}{m^{2}}\ll
1)$ (46)
Evidently, the $e_{R}(Q)$ will increase with $Q$, becoming a running coupling
constant ($RCC$). To calculate it, usually a renormalization-group-equation
($RGE$) was derived for $QED$ by setting $\alpha\to\alpha_{R}(Q)$ and
$Q\to\infty$ in $\beta(\alpha,Q)$ yielding:
$Q\dfrac{d}{dQ}\alpha_{R}=\dfrac{2\alpha_{R}^{2}}{3\pi}$ (47)
with its solution
$\alpha_{R}(Q)=\dfrac{\alpha}{1-\frac{2\alpha}{3\pi}\ln\frac{Q}{m}}$ (48)
Here, the renormalization was forced to be made at $Q^{2}=m^{2}$ so that
$\alpha_{R}|_{Q^{2}=m^{2}}=\alpha$ (49)
This is inconsistent with the physical condition, Eq.(42), a defect due to
ignoring the mass $m$, which plays a dominant role at low $Q$ region as shown
by the Eq.(46). As an improvement, a more practical $RGE$ is constructed in 4
(4) by changing $\alpha$ into $\alpha_{R}(Q)$ in Eq.(45) for the entire $Q$
region and adding up contributions from $9$ elementary charged fermions
($e,\mu,\tau,u,d,s,c,b,t$). Then $\alpha_{R}$ can be numerically calculated as
a function of $\ln(Q/m_{e})$, starting from
$\alpha_{R}(Q=0)=\alpha=(137.03599)^{-1}$ (50)
and passing through another experimental data point 12 (12)
$\alpha_{R}(Q=M_{Z}=91.1880\,GeV)=(128.89)^{-1}$ (51)
In this way, after adopting three heavy quarks’ masses as
$m_{c}=1.031\,GeV,\;m_{b}=4.326\,GeV$ (see section 9.5D in 13 (13)) and
$m_{t}=175\,GeV$, three light quarks masses
$m_{u}=8\,MeV,\;m_{d}=10\,MeV,\;m_{s}=200\,MeV$ (52)
(or averaged mass for $u,d,s$ being $92\ MeV$) can be fitted as shown in Fig.1
of Ref.4 (4).
## V $RGE$ for $QCD$, Threshold Energies of Quarks Hadronization
Different from the $RCC$ in $QED$, the $RCC$ in $QCD$, denoted by
$\alpha_{s}(Q)=\frac{1}{4\pi}g_{s}^{2}(Q)$, is much larger and decreases with
the increase of $Q$. Usually, the $RGE$ for $QCD$ reads (see p.551-552 in 2
(2))
$Q\dfrac{d}{dQ}\alpha_{s}(Q)=\beta(\alpha_{s}(Q))=-\dfrac{\beta_{0}}{2\pi}\alpha_{s}^{2}(Q)$
(53)
where
$\beta_{0}=11-\dfrac{2}{3}n_{f}$ (54)
with $n_{f}$ being the number of quarks’ flavor. Eq.(53) looks similar to
Eq.(47), but the negative sign in beta function implies the property of
asymptotic freedom in the strong interaction. Solution of Eq.(53) is give by
Eq.(6), also bears some resemblance to Eq.(48).
Let us make a comparison between Eqs.(6) and (48). Besides the difference in
sign (”$+$” versus ”$-$”) in the denominator, there is another one: The mass
scale $\mu$ remains arbitrary in $QCD$ whereas $m$ in $QED$ means the mass of
an observed charged fermion (usually electron).
As we already see, even for $QED$, one mass ($m_{e}$) is far from enough, let
alone in $QCD$, where quarks’ masses are much heavier. How can we ignore them?
Usually, to remove the arbitrary mass scale $\mu$, a parameter $\Lambda_{QCD}$
is often defined via equation
$\dfrac{\beta_{0}}{2\pi}\alpha_{s}(\mu)\ln\left(\dfrac{\mu}{\Lambda_{QCD}}\right)=1$
(55)
such that a simpler formula for $\alpha_{s}(Q)$ can be found as
$\alpha_{s}(Q)=\dfrac{2\pi}{\beta_{0}\ln(Q/\Lambda_{QCD})}$ (56)
Then the precision experimental data 1
$\alpha_{s}(M_{Z}=91.1876\,GeV)=0.1176$ (57)
serves as a substitute for $\alpha_{s}(\mu)$ in Eq.(55), yielding
$\Lambda_{QCD}=M_{Z}\exp\left[\dfrac{-2\pi}{\alpha_{s}(M_{Z})\beta_{0}}\right]=\left\\{\begin{array}[]{l}240\;MeV,\;(n_{f}=3)\\\
150\;MeV,\;(n_{f}=4)\\\ 85.8\;MeV,\;(n_{f}=5)\\\
44.2\;MeV,\;(n_{f}=6)\end{array}\right.$ (58)
It is evident from Eq.(56) that
$\alpha_{s}(\Lambda_{QCD})\to\infty$ (59)
which implies the ”infrared confinement” of quarks.
However, the value of $\Lambda_{QCD}$ sensitively depends on the flavor number
$n_{f}$ as shown by Eq.(58) but is independent of the concrete flavor of a
quark under consideration. Moreover, the divergence of $\alpha_{s}$ appears at
$\Lambda_{QCD}$. These features seem not so reasonable and are not consistent
with the experimental fact that the lighter a quark’s mass is, the lower its
”threshold energy for hadronization” will be.
The way out of above difficulties is clear. Just like in Eq.(45) for $QED$,
where rather than just the first term, all terms for $9$ fermions should be
added, now for $RGE$ in $QCD$, all masses of $6$ quarks should be preserved.
In this way, the $\alpha_{si}(Q)$ are numerically calculated for $i=u,d,s,c,b$
respectively in Ref.4 (4). Starting from $\alpha_{s}(M_{Z})=0.118$ (the common
renormalization point), their running curves (as shown by Figs.2 and 3 in 4
(4)) follow the trend of experimental data (as shown on p.158 of Ref.1 ) quite
well but separate at the low $Q$ region. Each of them rises to a maximum
$\alpha_{si}^{max}$ at $Q=\Lambda_{i}$ and then suddenly drops to zero at
$Q=0$.
For example, for $b$ quark, $\Lambda_{b}=7.04\,GeV,\;\alpha_{sb}^{max}=0.161$.
If tentatively explain $L_{b}\sim\hbar/\Lambda_{b}$ as some critical length
scale of $b\bar{b}$ pair, then $L_{b}\sim 0.02805\,fm$. In 4 (4), it is
further guessed that $E_{b}^{theor}\sim\alpha_{sb}^{max}/L_{b}\sim 1.133\,GeV$
being the order of excitation energy for breaking the binding $b\bar{b}$ pair,
i.e., the hadronization threshold energy of Upsilon $\Upsilon(b\bar{b})$
against its dissociation into two bosons. It is indeed the case found
experimentally1 :
$M(\Upsilon(4S))-M(\Upsilon)=1.12\,GeV,\quad M(\Upsilon(4S))\to
B^{+}B\;\text{or}\;B^{0}\bar{B^{0}}$ (60)
Similarly, we can estimate from Ref.4 (4) that $E_{c}^{theor}\sim 0.398\,GeV$
which also corresponds to experiments that1
$M(\psi(3770))-M(\psi(3097))=673\,MeV,\quad\psi(3770)\to
D^{+}D^{-}\;\text{or}\;D^{0}\bar{D^{0}}$ (61)
It seems that $E_{s}^{theor}\sim 90\,MeV$ and $E_{u,d}^{theor}\sim 0.4\,MeV$
are not so reliable but still reasonable.
## VI $\lambda\phi^{4}$ Model and Higgs Mass in the Standard Model
The Lagrangian density of $\lambda\phi^{4}$ ($\phi(x)$ is a real scalar field)
model is defined by
${\cal
L}=\dfrac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\dfrac{1}{2}\sigma\phi^{2}-\dfrac{\lambda}{4!}\phi^{4}$
(62)
The importance of this model lies in the ”wrong sign” of mass term
($\sigma=-m^{2}>0$) which leads to the spontaneous symmetry breaking ($SSB$)
at the tree level ($L=0$). The effective potential ($EP$) reads
$V_{0}(\phi)=-\dfrac{1}{2}\sigma\phi^{2}+\dfrac{\lambda}{4!}\phi^{4}$ (63)
(The subscript ”$0$” refers to $L=0$). Obviously, $V_{0}(\phi)$ has two
extremum, one is a maximum:
$\phi_{0}=0\quad\text{(symmetric phase)}$ (64)
while the other one is a minimum:
$\phi_{1}^{2}=\dfrac{6\sigma}{\lambda}\quad\text{(SSB phase)}$ (65)
At the $QFT$ level, the $EP$ evolves into
$V=V_{0}+V_{1}+\cdots$ (66)
The theory for $EP$ had been developed by various authors 14 (14, 15, 16, 17,
18), with $L=1$ contribution to $EP$ being evaluated as:
$V_{1}(\phi)=\dfrac{1}{2}\int\dfrac{d^{4}k_{E}}{(2\pi)^{4}}\ln(k_{E}^{2}-\sigma+\frac{1}{2}\lambda\phi^{2})$
(67)
This highly divergent integral (in 4-dimensional Euclidean momentum space) is
treated in Ref.19 (19) like that for Eqs.(10)-(12). First, three times of
differentiation with respect to $M^{2}=-\sigma+\frac{1}{2}\lambda\phi^{2}$ are
needed before it becomes just convergent.
$\dfrac{\partial^{3}V_{1}}{\partial(M^{2})^{3}}=\int\dfrac{d^{4}k_{E}}{(2\pi)^{4}}\dfrac{1}{(k_{E}^{2}+M^{2})^{3}}=\dfrac{1}{2(4\pi)^{2}M^{2}}$
(68)
Second, three times integration with respect to $M^{2}$ are performed,
yielding
$V_{1}(\phi)=\dfrac{1}{2(4\pi)^{2}}\left\\{\dfrac{M^{4}}{2}(\ln
M^{2}-\dfrac{1}{2})-\dfrac{1}{2}M^{4}+\dfrac{1}{2}C_{1}M^{4}+C_{2}M^{2}+C_{3}\right\\}$
(69)
As expected, three arbitrary constants $C_{1},C_{2},C_{3}$ appear. The
renormalization amounts to fix them at our disposal.
Third, like that in Eq.(3), for eliminating the ambiguity of dimension in the
first term involving $\ln M^{2}$, the only possible choice of
$C_{1}=-\ln\mu^{2}$ is fixed. Then the choice of $C_{2}=\mu^{2}=2\sigma$ and
$C_{3}=-\sigma^{2}+(4\pi)^{2}\frac{3\sigma^{2}}{\lambda}$ leads to
$V=V_{0}+V_{1}$ with its derivatives being given at the Table II 19 (19).
Table II. Effective potential of $\lambda\phi^{4}$ model with $SSB$ |
---|---
| $SSB$ phase | symmetric phase
$\phi$ | $\phi_{1}=\sqrt{\dfrac{6\sigma}{\lambda}}$ | $\phi_{0}=0$
$V$ | $0$ | $-\dfrac{\sigma^{2}}{2(4\pi)^{2}}\left[\dfrac{15}{4}+\dfrac{1}{2}\ln 2-i\dfrac{\pi}{2}\right]+\dfrac{3}{2}\dfrac{\sigma^{2}}{\lambda}$
$\dfrac{dV}{d\phi}$ | $0$ | $0$
$\dfrac{d^{2}V}{d\phi^{2}}$ | $2\sigma$ | $-\sigma\left[1-\dfrac{\lambda}{2(4\pi)^{2}}(3+\ln 2-i\pi)\right]$
$\dfrac{d^{3}V}{d\phi^{3}}$ | $\lambda\sqrt{\dfrac{6\sigma}{\lambda}}\left[1+\dfrac{3\lambda}{2(4\pi)^{2}}\right]$ | $0$
$\dfrac{d^{4}V}{d\phi^{4}}$ | $\lambda\left[1+\dfrac{9\lambda}{2(4\pi)^{2}}\right]$ | $\lambda\left[1-\dfrac{3\lambda}{2(4\pi)^{2}}(\ln 2-i\pi)\right]$
Note that, with the above assignment of $C_{i}\,(i=1,2,3)$, both the position
of the $SSB$ phase, $\phi_{1}$, and the mass $m_{\sigma}$ excited above it
take the same expression as that at the tree level
$m_{\sigma}^{2}=\dfrac{d^{2}V}{d\phi^{2}}|_{\phi=\phi_{1}}=2\sigma$ (70)
However, the renormalization coupling constant
$\lambda_{R}=\dfrac{d^{4}V}{d\phi^{4}}|_{\phi=\phi_{1}}=\lambda\left[1+\dfrac{9\lambda}{2(4\pi)^{2}}\right]$
(71)
does receive some quantum correction on its classical value $\lambda$. Hence
it is suddenly realized that the invariant meaning of $\lambda$ in the
Lagrangian, Eq.(62), is by no means a ”coupling constant”, but the ratio of
two mass scales 21 (21).
$\lambda=3\dfrac{m_{\sigma}^{2}}{\phi_{1}^{2}}$ (72)
Two parameters, $\sigma$ and $\lambda$, together with Eqs.(70) and (72),
should all be preserved through out high loop ($L$) evaluations in
perturbation theory until $L\to\infty$, i.e., in any nonperturbative
treatment.
On the other hand, the above assignment of $C_{i}$ renders the appearance of
imaginary part in $V$ and its derivatives at the symmetric phase
($\phi_{0}=0$). It means the instability of symmetric phase at the presence of
stable $SSB$ phase.
It is interesting to see that an alternative choice of
$C_{1}=-\ln(-\sigma),\;C_{2}=-\sigma,\;C_{3}=-\dfrac{1}{4}\sigma^{2}$ (73)
would leads to the survival of $\phi_{0}=0$ as a semistable state with
$V(0)=\dfrac{dV}{d\phi}|_{\phi=0}=\dfrac{d^{3}V}{d\phi^{3}}|_{\phi=0}=0$ (74)
$\dfrac{d^{2}V}{d\phi^{2}}|_{\phi=0}=-\sigma,\;\dfrac{d^{4}V}{d\phi^{4}}|_{\phi=0}=\lambda$
(75)
whereas no real $SSB$ solution exists. Hence we see that two different choices
of $C_{i}$ lead two separable sectors in the effective potential 19 (19).
In 1989, we had estimated the upper and lower bounds of Higgs mass $M_{H}$ in
the standard model of particle physics by using a nonperturbative approach in
$QFT$ — the Gaussian effective potential ($GEP$) method, yielding20 (20)
$76\,GeV<M_{H}<170\,GeV$ (76)
Like many authors, we were bothered a lot by divergences. After a deeper
understanding on $\lambda\phi^{4}$ model19 (19), this problem was restudied in
1998 by a combination of $GEP$ with our $RRM$, yielding
$M_{H}=138\,GeV$ (77)
which is based on the following input of experimental data:
$\begin{array}[]{l}M_{W}=80.359\,GeV,\;M_{Z}=91.1884\,GeV,\\\
\alpha^{-1}=\dfrac{4\pi}{g^{2}\sin^{2}\theta_{W}}=128.89,\;\sin^{2}\theta_{W}=0.2317\end{array}$
(78)
where $\theta_{W}$ is the weak mixing (Weinberg) angle. As now the search for
Higgs particle becomes so urgent experimentally while the theoretical
estimation about its mass still remains uncertain1 , we think our method 21
(21) with its prediction, Eq.(77), deserves to be reconsidered.
## VII Summary and Discussion
Our $RRM$ was first proposed by J. F. Yang in 199422 , then elaborated in a
series of papers since 1998 (19 (19, 21, 23, 24, 4, 7) etc.). What we have
been thinking about is: Why the divergence emerges inevitably in $QFT$? What
is the essential meaning of a regularization and renormalization procedure?
For instance, in a pioneering work to explain the Lamb shift, Welton (25 (25),
see also section 9.6B in Ref.13 (13)) encountered a divergent integral
$I=\int\frac{d\omega}{\omega}$ with $\omega$ being the (angular) frequency of
virtual photon (vacuum fluctuation). He simply set the lower and upper bounds
by $\omega_{min}\sim mZ\alpha=\frac{Z}{a}$ ($a$ is Bohr radius) and
$\omega_{max}\sim m$ respectively, arriving at
$I\simeq\ln(\frac{1}{Z\alpha})=4.92\,(Z=1)$ which leads to an estimation of
Lamb shift $L_{H}^{theor}(2S_{1/2}-2P_{1/2})\simeq 668\,MHz$. However, if
instead of Bohr radius, the lower cutoff is provided by the electron’s binding
energy, one would get $I\simeq\ln(Z\alpha)^{-2}$ and
$L_{H}^{theor}(2S_{1/2}-2P_{1/2})\simeq 1336\,MHz$ (see Eq.(30) in the Ref.26
(26)). We see the integral $I$ being a dimensionless number, not very large
($I\leq 10$), but uncertain indeed. The root cause of uncertainty lies in the
fact that a reconfirmation process of electron mass like Eq.(20) was missing.
For further clarity, a study on Lamb shift in the form of noncovariant $QED$
was performed in 24 (24) (see also Appendix 9A in 13 (13)). Beginning with
Eqs.(18) and (19) (with $m\to\mu=\frac{mm_{N}}{m+m_{N}}$), the perturbative
calculation of electron’s self-energy at second order in noncovariant form
(corresponding to the one-loop ($L=1$) order in covariant form) leads to an
energy increase of an electron with momentum $\bf p$, $\Delta E_{p}$, which
contains divergence and can be handled just like that in Eqs.(10) and (67),
yielding
$\begin{array}[]{l}\Delta E_{p}=b_{1}p^{2}+b_{2}p^{4}+\cdots\\\\[14.22636pt]
b_{1}=\dfrac{\alpha}{\pi\mu}(\dfrac{4}{3}\ln
2+\dfrac{4}{3}\ln\mu-\dfrac{4}{3}C_{1})\\\\[14.22636pt]
b_{2}=\dfrac{\alpha}{\pi\mu^{3}}(-\dfrac{2}{15})\end{array}$ (79)
The only choice of arbitrary constant $C_{1}$ is to make $b_{1}=0$ such that
the reduced mass $\mu$ in Eq.(18) can be reconfirmed. However, Eq.(19) must be
supplemented by the interaction between electron’s spin and the radiation
field
$H^{\prime}_{int}=\dfrac{ge\hbar}{4\mu
c}{\boldsymbol{\sigma}}\cdot\nabla\times{\bf A}$ (80)
where $g=2\times 1.0011596522$ is gyromagnetic ratio of electron. Similar
treatment leads to
$\begin{array}[]{l}\Delta
E^{\prime}_{p}=b^{\prime}_{0}+b^{\prime}_{1}p^{2}+b^{\prime}_{2}p^{4}+\cdots\\\\[11.38109pt]
b^{\prime}_{0}=\dfrac{g^{2}}{4}\dfrac{\alpha\mu}{\pi}[4(\ln
2+\ln\mu)-4C_{2}-\dfrac{2C_{3}}{\mu}-\dfrac{C_{4}}{\mu^{2}}]\\\\[11.38109pt]
b^{\prime}_{1}=\dfrac{g^{2}}{4}\dfrac{\alpha}{\pi\mu}(\dfrac{4}{3}\ln
2+2+\dfrac{4}{3}\ln\mu-\dfrac{4}{3}C_{2})\\\\[11.38109pt]
b^{\prime}_{2}=\dfrac{g^{2}}{4}\dfrac{\alpha}{\pi\mu^{3}}(-\dfrac{1}{15})\end{array}$
(81)
Because $\mu$ has already been reconfirmed (by $b_{1}=0$), the only choice of
arbitrary constant $C_{2}$ is to cancel the ambiguous term with
$\ln\mu,\;C_{2}=\ln\mu$, leaving a nonzero $b^{\prime}_{1}p^{2}$ and combining
with $\frac{1}{2\mu}{\bf p}^{2}$. Hence $\mu$ really acquires a modification
as
$\mu\to\mu_{obs}=\dfrac{\mu}{1+\beta},\qquad\beta=\dfrac{g^{2}\alpha}{2\pi}(\dfrac{4}{3}\ln
2+2)$ (82)
where $\mu_{obs}=\frac{m_{e}m_{N}}{m_{e}+m_{N}}$. Then constants $C_{3}$ and
$C_{4}$ must be chosen such that $b^{\prime}_{0}=0$.
Notice that, however, the spin induced interaction, Eq.(80), endows electron
with relativistic feature, creating a term ($-\frac{1}{8\mu^{3}}p^{4}$) in its
kinetic energy. Yet the modification on $\mu$ shown in Eq.(82) does induce a
corresponding change
$-\frac{1}{8}(\frac{1}{\mu_{obs}^{3}}-\frac{1}{\mu^{3}})p^{4}$, which should
be regarded as an invisible ”background” and subtracted from the $p^{4}$ term
induced by the radiative corrections. Hence the ”renormalized” $b_{2}$ should
be
$b_{2}^{R}=b_{2}+b^{\prime}_{2}+\frac{1}{8\mu^{3}}(3\beta+3\beta^{2}+\beta^{3})\simeq\dfrac{\alpha}{\pi\mu_{obs}^{3}}(1.99808)$
(83)
where only the lowest approximation is kept ($\mu\simeq\mu_{obs}$). Hence the
radiative correction on the energy level of a stationary state $|Z,n,l\rangle$
in hydrogenlike atom simply reads
$\Delta E^{rad}(Z.n,l)=\langle
Z,n,l|b_{2}^{R}p^{4}|Z,n,l\rangle=[\dfrac{8n}{3l+1}-3]\dfrac{b_{2}^{R}Z^{4}\alpha^{4}}{n^{4}}\mu_{obs}^{4}$
(84)
This contribution, together with that from the vacuum polarization (borrowed
from covariant theory) and nuclear size effect, gives a theoretical value for
the Lamb shift 24 (24)
$L_{H}^{theor}(2S_{1/2}-2P_{1/2})=1056.52\,MHz$ (85)
which is smaller than the experimental value, Eq.(29), by $0.13\%$ only.
Despite the approximation involved, the above method clearly shows that our
regularization is by noo means a trick to curb the divergence. Rather, it is a
natural way to transform a divergence into some arbitrary constants, revealing
that the essential meaning of divergence is just the ”uncertainty” in the
theory. Thus so-called renormalization turn out to be nothing but a process of
reconfirmation to fix these constants via experiments. We must reconfirm a
mass before it could be modified via radiative corrections. Either ”skipping
over the first step” or ”combining two steps into one” is not allowed.
In deeper understanding, our $RRM$ is based on a ”principle of relativity” in
epistemology27 (27): Every thing is moving and becomes recognizable only in
relationship with other things. What we can understand is either no mass scale
or two mass scales, but never one mass scale. This scenario is clearly
displayed in the Gross-Neveu model28 (28), also in the $\lambda\phi^{4}$ model
with $SSB$ as shown by Eqs.(65), (70) and (72). The vacuum expectation value
of field, $\phi_{1}$, just provides a ”mass unit” for the mass $m_{\sigma}$
excited on the vacuum with $SSB$.
Similarly, in perturbative $QFT$, we will be able to calculate various
radiative corrections on a particle only when its mass $m$ can be reconfirmed
again and again throughout any high loop ($L$) order of theory until
$L\to\infty$. Just like one has to reconfirm his plane ticket before his
departure from the airport, he must use the same name throughout his entire
jouney4 (4).
## Acknowledgements
We thank S. Q. Chen, Y. S. Duan, R. T. Fu, S. S. Feng, T. Huang, P. T. Leung,
W. F. Lu, X. T. Song, F. Wang, H. B. Wang, K. Wu, Y. L. Wu, J. Yan, G. H.
Yang, J. F. Yang and Z. X. Zhang for close collaborations and/or helpful
discussions.
## References
* (1) C. Amsler et al.,, Phys. Lett. B 667, 1 (2008) We refer to the Particle Physics Booklet extractd from it.
* (2) M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, (Addison-Wesley Publishing Company, 1995).
* (3) J. C. Collins, Renormalization, (Cambridge University Press, Cambridge, 1984)
* (4) G. J. Ni, G. H. Yang and R. T.Fu, Int. J. Mod. Phys. A16, 2873(2001)
* (5) J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley Publishing Company, 1967)
* (6) C. Itzykson and J-B. Zuber, Quantum Field Theory (McGraw-Hill Book Company, 1980)
* (7) G. J. Ni, J. J. Xu and S. Y. Lou, Submitted to Chinese Physics B, quant-ph/0511197.
* (8) H. A. Bethe, Phys. Rev. 72, 339 (1947).
* (9) Th. Udem et al., Phys. Rev. Lett. 79, 2646 (1997).
* (10) F. Schmidt-Kaler et al., Phys. Rev. Lett. 70, 2261 (1993)
* (11) M. Weitz et al., Phys. Rev. A 52, 2664 (1995).
* (12) H. Burkhardt and B. Pietrzyk, Phys. Lett. B 356, 398 (1995)
* (13) G. J. Ni and S. Q. Chen, Advanced Quantum Mechanics, 2nd Edition (Fudan University Press, 2003); English Edition was published by Rinton Press, 2002.
* (14) S. Coleman and E. Weinberg, Phys. Rev. D 7, 1883 (1973)
* (15) R. Jackiw, Phys. Rev. D 9, 1686 (1974)
* (16) C. W. Bernard, Phys. Rev. D 9, 3312 (1974)
* (17) L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974)
* (18) S. Weinberg, Phys. Rev. D 9, 3357 (1974)
* (19) G. J. Ni and S. Q. Chen, Acta Physica Sinica (Overseas Edition), 7, 401 (1998).
* (20) S. Y. Lou and G. J. Ni, Phys. Rev. D 40, 3040 (1989)
* (21) G. J. Ni, S. Y. Lou, W. F. Lu and J. F. Yang, Science in China (Series A), 41, 1206 (1998), hep-ph/9801264.
* (22) J. F. Yang, Thesis for PhD (Fudan University, 1994); hep-th/9708104;
J. F. Yang and G. J. Ni, Acta Physica Sinica (Overseas Edition), 4, 88 (1995).
* (23) S. S. Feng and G. J. Ni, Int. J. Mod. Phys. A, 14, 4259 (1999).
* (24) G. J. Ni, H. B. Wang, J. Yan and H. L. Li, High Energy Physics and Nuclear Physics, 24, 400 (2000).
* (25) T. A. Welton, Phys. Rev. 74, 1157 (1948)
* (26) M. I. Eides, H. Grotch and V. A. Shelyuto, Phys. Rep. 342, 63 (2001)
* (27) G. J. Ni, Principle of Relativity in Physics and in Epistemology, in Relativity, Gravitation, Cosmology: New development, (NOVA Science Publisher, to be published)
* (28) D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974)
* (29) G. J. Ni, S. Q. Chen, J. J. Xu and S. Y. Lou, Essence of special relativity, reduced Dirac equation and antigravity, Preprint.
|
arxiv-papers
| 2010-07-19T02:09:40 |
2024-09-04T02:49:11.710907
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guang-jiong Ni, Jianjun Xu and Senyue Lou",
"submitter": "Jianjun Xu",
"url": "https://arxiv.org/abs/1007.3054"
}
|
1007.3092
|
# Variant Supercurrents and Linearized Supergravity
Sibo Zheng1 and Jia-Hui Huang2
1 Department of Physics, Chongqing University, Chongqing 400030, P.R. China
2 Center of Mathematical Science, Zhejiang University, Hangzhou 310027, P.R.
China
Abstract
In this paper the variant supercurrents based on consistency and completion in
off-shell $\mathcal{N}=1$ supergravity are studied. We formulate the embedding
relations for supersymmetric current and energy tensor into supercurrent
multiplet. Corresponding linearized supergravity is obtained with appropriate
choice of Wess-Zumino gauge in each gravity supermultiplet.
July 2010
## 1 Introduction
According to the structure of supersymmetry algebra, the $R$ current
$j^{5}_{\mu}$, supersymmetric current $j_{\mu}$ and energy tensor $T_{\mu\nu}$
corresponding to the $R$ charge, supercharge and spacetime momentum
respectively can be embedded into a supermultiplet. This multiplet is known as
supercurrent [1]. The superfield form of a supercurrent and the constraint it
satisfies are found to be quite model dependent, although some general
considerations from symmetries can be taken into account [2, 3, 4, 5].
There is a standard scheme for analyzing the structure of supercurrent and its
corresponding linearized supergravity for a given physical system. The
procedure is as follows:
1. 1.
Begin with the physics systems studied, and the conservation conditions,
$\displaystyle\partial^{\mu}j_{\mu}=0,~{}~{}~{}~{}~{}~{}~{}\partial^{\mu}T_{\mu\nu}=0$
one has to find the embedding relations for supersymmetric current and energy-
momentum tensor into supercurrent. During this stage the supercurrent
multiplet and the constraint it satisfies are determined at meantime.
2. 2.
Through the constraint that supercurrent satisfies we obtain the constraints
on gauge transformation superfield $L$ of gravity supermultiplet, which tell
us the analogy of Wess-Zumino gauge in gravity supermultiplet.
3. 3.
Collect the embedding relations of gravity and gravitino into gravity
supermultiplet together, the action of linearized supergravity can be directly
read in components.
In this paper we study the structures of three new variant supercurrent [7]
using the results obtained earlier in [8, 9]. The existence of these variant
supercurrent is based on consistency and completion in $N=1$ off-shell
linearized supergravity . Other supercurrents deduced via this viewpoint
include the Ferrara-Zumino (FZ) multiplet, new minimal multiplet [10] and $S$
multiplets [13] all of which has completions of quantum field theories (see
also [11]). The varaint supercurrents are defined as follows.
$\displaystyle{}Case~{}I:~{}~{}~{}~{}~{}\bar{D}^{\dot{\alpha}}J^{I}_{\alpha\dot{\alpha}}=i\eta_{\alpha},~{}~{}~{}~{}~{}\bar{D}\eta=D^{\alpha}\eta_{\alpha}-\bar{D}^{\dot{\alpha}}\bar{\eta}_{\dot{\alpha}}=0$
(1.1)
which is a minimal off-shell supergravity. The second case is,
$\displaystyle{}Case~{}II:~{}~{}\bar{D}^{\dot{\alpha}}J^{II}_{\alpha\dot{\alpha}}=i\eta_{\alpha}+\hat{\chi}_{\alpha},$
(1.2)
with
$\bar{D}^{\dot{\alpha}}\eta_{\alpha}=D^{\alpha}\eta_{\alpha}-\bar{D}^{\dot{\alpha}}\bar{\eta}_{\dot{\alpha}}=0$
and
$\bar{D}^{\dot{\alpha}}\hat{\chi}_{\alpha}=D^{\alpha}\hat{\chi}_{\alpha}-\bar{D}^{\dot{\alpha}}\bar{\hat{\chi}}_{\dot{\alpha}}=0$.
The last case is,
$\displaystyle{}Case~{}III:~{}~{}~{}\bar{D}^{\dot{\alpha}}J^{III}_{\alpha\dot{\alpha}}=i\eta_{\alpha}+D_{\alpha}X$
(1.3)
with
$\bar{D}^{\dot{\alpha}}\eta_{\alpha}=D^{\alpha}\eta_{\alpha}-\bar{D}^{\dot{\alpha}}\bar{\eta}_{\dot{\alpha}}=0$
and $\bar{D}X=0$.
There are some common results in variant supercurrent. Firstly, the $R$
current is not conserved, which can be easily observed from the constraint of
eq(1.1) to eq(1.3) . Secondly, there exists some special constraints for
energy tensor $T_{\mu\nu}$ as shown below. These constraints exclude some
simple physical systems we are familiar with. Thus, they might serve as the
necessary conditions for existence of variant supercurrent.
The paper is organized as follows. In section 2, we discuss the minimal case
I. Section 3 are denoted to study non-minimal cases II and III. The solutions
to the constraint (1.1) to eq(1.3) are obtained, with comments on conditions
that energy-tensor has to satisfy. The actions of linearized supergravity are
obtained after the analogy of Wess-Zumino gauge in each case are discussed. In
section 4, we conclude and discuss the difference between variant
supercurrents and other supercurrents in the literature.
## 2 Minimal Case I
In this note we follow the conventions of Wess and Bagger [6]. The real vector
superfield $J_{\mu}$ is defined in bi-spinor representation as
$\displaystyle
J_{\alpha\dot{\alpha}}=\sigma^{\mu}_{\alpha\dot{\alpha}}J_{\mu},~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}J_{\mu}=-\frac{1}{2}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}J_{\alpha\dot{\alpha}}.$
(2.1)
The components are expressed as,
$\displaystyle{}S$ $\displaystyle=$ $\displaystyle
C^{S}+i\theta\chi^{S}-i\bar{\theta}\bar{\chi}^{S}+\frac{i}{2}\theta^{2}(M^{S}+iN^{S})-\frac{i}{2}\bar{\theta}^{2}(M^{S}-iN^{S})-\theta\sigma^{m}\bar{\theta}\upsilon^{S}_{m}$
$\displaystyle+$ $\displaystyle
i\theta^{2}\bar{\theta}\left(\bar{\lambda}^{S}+\frac{i}{2}\bar{\sigma}^{m}\partial_{m}\chi^{S}\right)-i\bar{\theta}^{2}\theta\left(\lambda^{S}+\frac{i}{2}\sigma^{m}\partial_{m}\bar{\chi}^{S}\right)+\frac{1}{2}\theta^{2}\bar{\theta}^{2}\left(D^{S}+\frac{1}{2}\Box~{}C^{S}\right)$
Note that the lowest component field $C^{J}$ in supercurrent superfield $J$ is
the $R$ current $j^{5}_{\mu}$.
We deduce a new constraint on supercurrent from the constraint eq(1.1),
$\displaystyle{}\bar{D}^{\dot{\beta}}\bar{D}^{\dot{\alpha}}J^{I}_{\alpha\dot{\alpha}}=0$
(2.3)
The first equation in constraint eq.(1.1) can be classified into its real and
imaginary parts, respectively111Similar methods are applied to the other two
cases we will discuss in this note.. Explicit expressions for these components
can be found in [14].
Solving eq(2.3) and eq(1.1) we obtain,
$\displaystyle{}J_{\mu}^{I}$ $\displaystyle=$ $\displaystyle
C_{\mu}+\theta\left(j_{\mu}+\frac{1}{3}\sigma_{\mu}\bar{\sigma}^{\nu}j_{\nu}\right)+\bar{\theta}\left(\bar{j}_{\mu}+\frac{1}{3}\bar{\sigma}_{\mu}\sigma^{\nu}\bar{j}_{\nu}\right)$
$\displaystyle+$
$\displaystyle(\theta\sigma^{\nu}\bar{\theta})\left(aT_{\nu\mu}+bT\eta_{\nu\mu}+\frac{1}{4}\epsilon_{\nu\mu\rho\lambda}\left(\partial^{\rho}C^{\lambda}-\partial^{\lambda}C^{\rho}\right)-\frac{1}{2}\Phi_{\nu\mu}\right)$
$\displaystyle+$
$\displaystyle\theta^{2}\bar{\theta}\left(\frac{i}{3}\bar{\sigma}^{\nu}\partial_{\mu}j_{\nu}\right)+\bar{\theta}^{2}\theta\left(\frac{i}{3}\sigma^{\nu}\partial_{\mu}\bar{j}_{\nu}\right)+\bar{\theta}^{2}\theta^{2}\left(-\frac{1}{2}\Box~{}C_{\mu}-\frac{1}{2}\partial_{\mu}\partial^{\nu}C_{\nu}\right)$
and
$\displaystyle\eta_{\alpha}$ $\displaystyle=$
$\displaystyle-i\Lambda_{\alpha}(y)+\left(\delta^{\beta}_{\alpha}\Delta-2i\bar{\sigma}^{\mu}\sigma^{\nu}\Phi_{\mu\nu}(y)\right)\theta_{\beta}+\theta^{2}(\sigma^{\mu}\partial_{\mu}\bar{\Lambda}(y))_{\alpha}$
(2.5)
where the coefficient $a,~{}b$ is introduced to define
$\hat{T}_{\mu}\mid_{s}=aT_{\mu\nu}+b\eta_{\mu\nu}T$. In this case, constant
$a$, $b$ are given by,
$\displaystyle{}a=-4b,~{}~{}~{}~{}2b\partial_{\nu}T=-\partial^{\mu}\Phi_{\mu\nu},~{}~{}~{}~{}~{}~{}\Box~{}T=0$
(2.6)
The lower indices $s,a$ in $\hat{T}_{\mu\nu}\mid$ refer to the symmetric and
anti-symmetric part respectively. $\Phi^{\rho\sigma}$ and $\Delta$ is the
tensor field and $D$-term in $\eta$ superfield respectively. The degrees of
freedom of $\hat{T}_{\mu\nu}\mid_{a}$ can be considered as totally provided by
$\Phi^{\rho\sigma}$ . Physical systems with energy tensor $T_{\mu\nu}$ that
satisfies these special constraints are extraordinary. The non-existence of
these conditions might serve as a proof that the first kind of constrained
supercurrent is not physical. This question will be investigated further.
The degrees of freedom in this case are described by
$(C_{\mu},\chi_{\mu},\hat{T}_{\mu\nu}\mid_{s},\hat{T}_{\mu\nu}\mid_{a})$,
which imply supersymmetric theories correspond to 12/12 off-shell
supergravity. Gauging the supercurrent $J^{I}$ in supergravity via coupling
$\displaystyle{}\int d^{4}x\int
d^{4}\theta~{}J^{I}_{\alpha\dot{\alpha}}H^{\alpha\dot{\alpha}}$ (2.7)
Gauge invariance of action eq(2.7) under transformation
$H_{\mu}\rightarrow~{}H_{\mu}+\triangle_{\mu}$, or equivalently via its bi-
spinor expression
$\displaystyle{}H_{\alpha\dot{\alpha}}\rightarrow~{}H_{\alpha\dot{\alpha}}+D_{\alpha}\bar{L}_{\dot{\alpha}}-\bar{D}_{\dot{\alpha}}L_{\alpha}$
(2.8)
leads to
$\displaystyle{}\bar{D}_{\dot{\alpha}}D^{2}\bar{L}^{\dot{\alpha}}+D_{\alpha}\bar{D}^{2}L^{\alpha}=0$
(2.9)
Here superfield $L$ is defined as,
$\displaystyle{}\triangle_{\mu}=-\frac{1}{2}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}\left(D_{\alpha}\bar{L}_{\dot{\alpha}}-\bar{D}_{\dot{\alpha}}L_{\alpha}\right)$
(2.10)
$\triangle_{\mu}$ is a general real superfield. Eq(2.10) suggests that the
relations of embedding graviton and gravitino into supergravity multiplet
$H_{u}$ follow those of [12, 13] 222Following the conventions we take, one can
see these embedding relations are independent of constraints on $L_{\alpha}$.
They are valid throughout this note.. $H_{\mu}\mid_{\theta\bar{\theta}}$ is
divided into the symmetric part $\upsilon^{H}_{\mu\nu}$ and anti-symmetric
part $B_{\mu\nu}$. The gauge transformations are as follows,
$\displaystyle\delta~{}h_{\mu\nu}=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu},~{}~{}~{}~{}~{}\delta~{}\Psi_{\mu\alpha}=\partial_{\mu}\omega_{\alpha}$
The constraint Eq(2.9) impose some equations in components in $L$, which
implies a set of constraint equations in components of $\triangle_{\mu}$ via
eq(2.10). These constraints determine the analog of the Wess-Zumino gauge for
supermultiplet $H_{\mu}$. Define
$\displaystyle L_{\alpha}=iD_{\alpha}V$ (2.11)
Eq(2.9) leads to the identification of $V$ as Wess-Zumino gauged vector
superfield. The constraints on components in $\triangle_{\mu}$ are,
$\displaystyle L_{\alpha}\mid=L_{\alpha}\mid_{\theta}$ $\displaystyle=$
$\displaystyle
L_{\alpha}\mid_{\theta^{2}}=L_{\alpha}\mid_{\theta^{2}\bar{\theta}}=0$ (2.12)
and
$\displaystyle\partial^{m}(L_{\alpha}\mid_{\theta\sigma^{m}\bar{\theta}})$
$\displaystyle=$
$\displaystyle-2(L_{\alpha}\mid_{\theta^{2}\bar{\theta}^{2}})$ (2.13)
which imply that $B_{\mu\nu}$ field in gravity supermultiplet can not be set
to zero.
One can see that the analogy of Wess-Zumino gauge is as follows,
$\displaystyle{}H_{\mu}\mid=H_{\mu}\mid_{\theta}=H_{\mu}\mid_{\bar{\theta}}=H_{\mu}\mid_{\theta^{2}}=H_{\mu}\mid_{\bar{\theta}^{2}}=0$
(2.14)
The residual degrees of freedom in gravity supermultiplet are represented by
$(h_{\mu\nu}$,$B_{\mu\nu}$,$\Psi_{\mu\alpha}$ and $D^{H}_{\mu})$, which
describe 12/12 minimal supergravity. They match with that of supercurrent.
Following notation eq.(2), we obtain the action in components,
$\displaystyle{}S=-\upsilon_{\mu\nu}^{H}\hat{T}^{\mu\nu}\mid_{s}-B_{\mu\nu}\hat{T}^{\mu\nu}\mid_{a}+\frac{1}{2}j^{5\mu}D^{H}_{\mu}+\left(\frac{i}{2}\chi_{\mu}^{(J)}\lambda^{(H)\mu}+c.c\right)$
(2.15)
The kinetic term of graviton can be constructed via appropriate derivative
operator [7]. Starting with the constraint on gauge transformation superfield,
the results in [7] can be reproduced. Similar results can be found in non-
minimal cases.
## 3 Reducible Cases
Now we discuss the non-minimal case II and case III. Their supercurrent
multiplets both include $16+16$ degrees of freedom (supermultiplets with
$16+16$ degrees of freedom are also discussed in [18, 19]), which are
manifested by their corresponding gravity supermultiplets. In comparison with
the minimal case I, the gauge transformation superfield $L_{\alpha}$ is more
constrained, which is the origin of more degrees of freedom in gravity
supermultiplets.
### 3.1 Reducible Cases II
The constraint eq(1.2) implies that,
$\displaystyle{}\bar{D}^{\dot{\beta}}\bar{D}^{\dot{\alpha}}J^{II}_{\alpha\dot{\alpha}}=0$
(3.1)
Solving eq(3.1) and eq(1.2) gives,
$\displaystyle{}J_{\mu}^{II}$ $\displaystyle=$ $\displaystyle
C_{\mu}+\theta\left(j_{\mu}+\frac{1}{3}\sigma_{\mu}\bar{\sigma}^{\nu}j_{\nu}+\frac{1}{3}\sigma_{\mu}\bar{\psi}\right)+\bar{\theta}\left(\bar{j}_{\mu}+\frac{1}{3}\bar{\sigma}_{\mu}\sigma^{\nu}\bar{j}_{\nu}-\frac{1}{3}\bar{\sigma}_{\mu}\psi\right)$
$\displaystyle+$
$\displaystyle(\theta\sigma^{\nu}\bar{\theta})\left(aT_{\nu\mu}-\frac{b}{a+4b}Z\eta_{\nu\mu}+\frac{1}{4}\epsilon_{\nu\mu\rho\lambda}\left(\partial^{\rho}C^{\lambda}-\partial^{\lambda}C^{\rho}+\Sigma^{\rho\lambda}\right)-\frac{1}{2}\Phi_{\nu\mu}\right)$
$\displaystyle+$
$\displaystyle\theta^{2}\bar{\theta}\left(-\frac{2i}{3}\partial_{\mu}\bar{\psi}+\frac{i}{3}\bar{\sigma}^{\nu}\partial_{\mu}j_{\nu}\right)+\bar{\theta}^{2}\theta\left(\frac{2i}{3}\partial_{\mu}\psi+\frac{i}{3}\sigma^{\nu}\partial_{\mu}\bar{j}_{\nu}\right)$
$\displaystyle+$
$\displaystyle\bar{\theta}^{2}\theta^{2}\left(\frac{1}{2}\partial_{\mu}Z-\frac{1}{2}\Box~{}C_{\mu}-\frac{1}{2}\partial_{\mu}\partial^{\nu}C_{\nu}+\frac{3}{2}\partial^{\nu}\Sigma_{\mu\nu}\right)$
and
$\displaystyle\eta_{\alpha}$ $\displaystyle=$
$\displaystyle-i\Lambda_{\alpha}(y)+\left(\delta^{\beta}_{\alpha}\Delta-2i\bar{\sigma}^{\mu}\sigma^{\nu}\Phi_{\mu\nu}(y)\right)\theta_{\beta}+\theta^{2}(\sigma^{\mu}\partial_{\mu}\bar{\Lambda}(y))_{\alpha}$
$\displaystyle\hat{\chi}_{\alpha}$ $\displaystyle=$
$\displaystyle-i\psi_{\alpha}(y)+\left(\delta^{\beta}_{\alpha}Z-2i\bar{\sigma}^{\mu}\sigma^{\nu}\Sigma_{\mu\nu}(y)\right)\theta_{\beta}+\theta^{2}(\sigma^{\mu}\partial_{\mu}\bar{\psi}(y))_{\alpha}$
(3.3)
The coefficient $a,~{}b$ satisfy
$\displaystyle{}\left(a+4b\right)T=-Z,~{}~{}~{}~{}2b\partial_{\nu}T=-\partial^{\mu}\Phi_{\mu\nu}$
(3.4)
As emphasized above, the existence of $a,~{}b$ is necessary for physical
systems described by the case II. The multiplet $J_{\mu}^{II}$ contain $12+12$
degrees of freedom, a Weyl spinor $\psi$, a closed two-form $\Sigma_{\mu\nu}$,
and a real scalar $Z$. Thus it describes $16+16$ supermultiplet.
Gauging the supercurrent $J^{II}$ in supergravity via coupling
$\displaystyle{}\int d^{4}x\int
d^{4}\theta~{}J^{II}_{\alpha\dot{\alpha}}H^{\alpha\dot{\alpha}}$ (3.5)
Gauge invariance of the action under transformation eq.(2.8) leads to
$\displaystyle{}\bar{D}^{\dot{\alpha}}D^{2}\bar{L}_{\dot{\alpha}}=D^{\alpha}\bar{D}^{2}L_{\alpha}=0$
(3.6)
The embedding relations of graviton and gravitino into $H_{\mu}$ superfield is
the same as in case I. Note that the equation of motion of a field strength
chiral superfield without FI term is exactly the same with eq(3.6). The
analogy of Wess-Zumino gauge is given by,
$\displaystyle{}H_{\mu}\mid=H_{\mu}\mid_{\theta}=H_{\mu}\mid_{\bar{\theta}}=H_{\mu}\mid_{\theta^{2}}=H_{\mu}\mid_{\bar{\theta}^{2}}=0$
(3.7)
The residual degrees of freedom in gravity supermultiplet are represented by
$(h_{\mu\nu}$,$B_{\mu\nu}$,$\Psi_{\mu\alpha}$ and $D^{H}_{\mu})$, which
describe 16/16 linearized supergravity. They match with that of supercurrent.
Corresponding action is in components with notation eq.(2),
$\displaystyle{}S=-\upsilon_{\mu\nu}^{H}\hat{T}^{\mu\nu}\mid_{s}-B_{\mu\nu}\hat{T}^{\mu\nu}\mid_{a}+\frac{1}{2}j^{5\mu}D^{H}_{\mu}+\left(\frac{i}{2}\chi_{\mu}^{(J)}\lambda^{(H)\mu}+c.c\right)$
(3.8)
### 3.2 Reducible Cases III
Finally we address the third possible constraint satisfied by supercurrent.
Solving equation eq.(1.3) we obtain $J_{\mu}^{III}$,
$\displaystyle{}J_{\mu}^{III}$ $\displaystyle=$ $\displaystyle
C_{\mu}+\theta\left(j_{\mu}+\frac{1}{3}\sigma_{\mu}\bar{\sigma}^{\nu}j_{\nu}\right)+\bar{\theta}\left(\bar{j}_{\mu}+\frac{1}{3}\bar{\sigma}_{\mu}\sigma^{\nu}\bar{j}_{\nu}\right)-i\theta^{2}\partial_{\mu}\phi+i\bar{\theta}^{2}\partial_{\mu}\phi^{*}$
$\displaystyle+$
$\displaystyle(\theta\sigma^{\nu}\bar{\theta})\left(aT_{\nu\mu}-2Re(F)\eta_{\nu\mu}+\frac{1}{2}\epsilon_{\nu\mu\rho\lambda}\partial^{\rho}C^{\lambda}-\frac{1}{2}\Phi_{\nu\mu}\right)$
$\displaystyle+$
$\displaystyle\theta^{2}\bar{\theta}\left(\frac{i}{3}\bar{\sigma}^{\rho}\partial_{\mu}j_{\rho}-\sqrt{2}\partial_{\mu}\bar{\psi}\right)+\bar{\theta}^{2}\theta\left(\frac{i}{3}\sigma^{\rho}\partial_{\mu}\bar{j}_{\rho}-\sqrt{2}\partial_{\mu}\psi\right)$
$\displaystyle+$
$\displaystyle\bar{\theta}^{2}\theta^{2}\left(-2\partial_{\mu}(Im(F))+\frac{1}{2}\Box~{}C_{\mu}-\frac{3}{2}\partial_{\mu}\partial^{\rho}C_{\rho}\right)$
and
$\displaystyle X$ $\displaystyle=$
$\displaystyle\phi(y)+\sqrt{2}\theta\psi(y)+\theta^{2}F$
$\displaystyle\eta_{\alpha}$ $\displaystyle=$
$\displaystyle-i\Lambda_{\alpha}(y)+\left(\delta^{\beta}_{\alpha}\Delta-2i\bar{\sigma}^{\mu}\sigma^{\nu}\Phi_{\mu\nu}(y)\right)\theta_{\beta}+\theta^{2}(\sigma^{\mu}\partial_{\mu}\bar{\Lambda}(y))_{\alpha}$
(3.10)
The components fields in $\eta_{\alpha}$ satisfy extra constraints,
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle-\partial^{\mu}C_{\mu}-2Im(F),$ $\displaystyle\Lambda_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{i}{3}(\sigma^{\mu}\bar{j}_{\mu})_{\alpha}-\sqrt{2}\psi_{\alpha}$
(3.11)
Here the coefficient $a$ is given by $aT=6Re(F)$, with $F=Re(F)+iIm(F)$. The
multiplet $J_{\mu}^{III}$ contains $12+12$ degrees of freedom, a Weyl spinor
$\psi$, a complex scalar $\phi$, and a complex scalar $F$ (or equivalently
$Re(F)$ and $\Delta$), which imply that it is actually $16+16$ supermultiplet.
Compared with the $S$-multiplet that is introduced to solve problem of FI term
in supergravity [13], the scalar $Re(F)$ is now replaced by $F$. The embedding
realtions are also very different.
Gauging the supercurrent $J^{III}$ in supergravity via coupling
$\displaystyle{}\int d^{4}x\int
d^{4}\theta~{}J^{III}_{\alpha\dot{\alpha}}H^{\alpha\dot{\alpha}}$ (3.12)
Gauge invariance of the action under transformation eq.(2.8) leads to
$\displaystyle{}\bar{D}^{2}D^{\alpha}L_{\alpha}=0,~{}~{}~{}\bar{D}_{\dot{\alpha}}D^{2}\bar{L}^{\dot{\alpha}}=D_{\alpha}\bar{D}^{2}L^{\alpha}$
(3.13)
As more constraints are imposed, less component fields in gravity
supermultiplet can be set to zero. The constraint eq(3.13) suggests that the
analog of Wess-Zumino guage is
$\displaystyle{}H_{\mu}\mid=H_{\mu}\mid_{\theta}=H_{\mu}\mid_{\bar{\theta}}=0$
(3.14)
The action can be read in components with notation eq.(2),
$\displaystyle{}S=-\upsilon_{\mu\nu}^{H}\hat{T}^{\mu\nu}\mid_{s}-B_{\mu\nu}\hat{T}^{\mu\nu}\mid_{a}+\frac{1}{2}j^{5\mu}D^{H}_{\mu}+\left[\frac{i}{2}\chi_{\mu}^{(J)}\lambda^{(H)\mu}+\frac{1}{4}\left(M^{J}+iN^{J}\right)\left(M^{H}-iN^{H}\right)+c.c\right]$
## 4 Conclusions
In this note we study a set of variant supercurrents that arise from
consistency and completion in $\mathcal{N}=1$ off-shell supergravity. We use
the component languages of superfield to obtain the embedding relations of
supersymmetric current and energy-momentum tensor into formalism of linear
supergravity. The analogy of Wess-Zumino gauge in each case is analyzed in
details.
Instead of the superfield formalisms used to describe variant supercurrents,
we find more physical results are uncovered in the component expressions.
First, the consistent conditions for energy-momentum tensor of supersymmetric
theories that can be described by variant supercurrent multiplets are
determined explicitly. Second, the component results help identifying
corresponding linearized supergravity.
Although supercurrents that include $S$-multiplet [13], FZ-multiplet and
minimal mutiplet have rich constructions of quantum field theories and
important applications333Recently, it is found in [13] that $S$-multiplet is
useful to embed the Fayet-Iliopoulos term for abelian gauge supermultiplet
into supergravity. , the consistent conditions eq.(2.6) and eq.(3.4) for
variant supercurrents studied in this paper imply $\Box~{}T=0$. It can be
verified that the variant supercurrents are not viable for simple
supersymmetric field theories including pure supersymmetric Yang-Mills
theories and SQCD-like theories as a result of $\Box~{}T\neq 0$. The main
reason for this difference between variant supercurrent and $S$-multiplet is
that the $i$ factor in front of the linear superfield in eq.(1.1) to eq.(1.3)
leads to the real and imaginary part of ${\theta^{2}\bar{\theta}}$ component
in these constraint equations exchanged. In the case of $S$-multiplet, the
energy momentum tensor depends on $D$-term of $\eta_{\alpha}$ superfield 444As
shown in [13], $\Box~{}T$ is proportional to $\Box~{}Re~{}F_{X}$ and
$\Box~{}\Delta$. Although the component expressions of $F$ and $D$ terms are
quite involved, it is expected that both $\Box~{}Re~{}F_{X}\neq 0$ and
$\Box~{}\Delta\neq 0$ in terms of equations of motion of relevant fields. In
other words, $\Box~{}T=0$ is forbidden in case of $S$ multiplet. , and anti-
symmetric tensor field $\Phi_{\mu\nu}$ is related to the anti-symmetric part
of $\hat{T}_{\mu\nu}$ . In the case of variant supercurrents, however,
exchanging the real and imaginary parts of component
${\theta^{2}\bar{\theta}}$ in eq.(1.1) to eq.(1.3) leads to that the
dependence on anti-symmetric tensor field $\Phi_{\mu\nu}$ is transferred to
the derivative of energy-momentum tensor trace $\partial_{\mu}T$ , which is
the origin of the severe constraint $\Box~{}T=0$ for variant supercurrents.
## Acknowledgement
This work is supported in part by the Fundamental Research Funds for the
Central Universities with project number CDJRC10300002.
## References
* [1] S. Ferrara and B. Zumino, “Transformation Properties Of The Supercurrent,” Nucl. Phys. B 87, 207 (1975).
* [2] V. Ogievetsky and E. Sokatchev “Supercurrent,” Sov. J. Nucl. Phys28, 423 (1978), Yad. Fiz28, 825 (1978).
* [3] V. Ogievetsky and E. Sokatchev, “On vector superfield generated by supercurrent,” Nucl. Phys. B 124, 309 (1977).
* [4] P. S. Howe, K. S. Stelle, P. K. Townsend, “Supercurrents,” Nucl. Phys. B192, 332 (1981).
* [5] T. E. Clark, O. Piguet and K. Sibold, “Supercurrents, Renormaliztion and Anomalies,” Nucl. Phys. B143, 445 (1978).
* [6] J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton Univ. Pr. (1992).
* [7] S. M. Kuzenko,“Variant supercurrent multiplets,” JHEP04, 022 (2010), [arXiv:1002.4932].
* [8] S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, “Superspace, or one thousand and one lessons in supersymmetry,” Front. Phys. 58, 1 (1983) [arXiv:hep-th/0108200].
* [9] S. J. Gates Jr., S. M. Kuzenko and J. Phillips, “The off-shell (3/2,2) supermultiplets revisited,” Phys. Lett. B 576, 97 (2003) [arXiv:hep-th/0306288].
* [10] K. S. Stelle and P. C. West, “Minimal Auxiliary Fields For Supergravity,” Phys. Lett. B74, 330 (1978).
* [11] M. Magro, I. Sachs and S. Wolf, “Superfield Noether procedure,” Annals Phys 298, 123 (2002), [arXiv: hep-th/0110131].
* [12] S. Weinberg,“The Quantum Theory of Fields,” Vol III, Chapter 31, Cambridge. Pr. (2000).
* [13] Z. Komargodski, N. Seiberg, “Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity,” JHEP 07, 017 (2010), [arXiv:1002.2228].
* [14] K. R. Dienes and B. Thomas,“On the Inconsistency of Fayet-Iliopoulos Terms in Supergravity Theories,” Phys. Rev. D81, 065023 (2010), [arXiv:0911.0677].
* [15] Z. Komargodski and N. Seiberg, “Comments on the Fayet-Iliopoulos Term in Field Theory and Supergravity,” JHEP06, 007 (2009) [arXiv:0904.1159].
* [16] T. E. Clark and S. T. Love, “ The Supercurrent in supersymmetric field theories,” Int. J. Mod. Phys. A11, 2807 (1996), [hep-th/9506145].
* [17] K. Yonekura, “Notes on Operator Equations of Supercurrent Multiplets and Anomaly Puzzle in Supersymmetric Field Theories,” [arXiv:1004.1296].
* [18] W. Lang, J. Louis and B. A. Ovrut, “ (16+16) Supergravity Coupled To Matter: The Low-Energy Limit Of The Superstring,” Phys. Lett. B 158, 40 (1985).
* [19] W. Siegel, “ 16/16 Supergravity,” Class. Quant. Grav. 3, 47 (1986).
|
arxiv-papers
| 2010-07-19T09:09:38 |
2024-09-04T02:49:11.719818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng and Jia-Hui Huang",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1007.3092"
}
|
1007.3186
|
# UCAC3 Proper Motion Survey. I.
DISCOVERY OF NEW PROPER MOTION STARS IN UCAC3
WITH 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 BETWEEN DECLINATIONS
$-$90$\arcdeg$ and $-$47$\arcdeg$
Charlie T. Finch, Norbert Zacharias finch@usno.navy.mil U.S. Naval
Observatory, Washington DC 20392–5420 Todd J. Henry Georgia State
University, Atlanta, GA 30302–4106
###### Abstract
Presented here are 442 new proper motion stellar systems in the southern sky
between declinations $-$90$\arcdeg$ and $-$47$\arcdeg$ with 0$\farcs$40 yr-1
$>$ $\mu$ $\geq$ 0$\farcs$18 yr-1. These systems constitute a 25.3% increase
in new systems for the same region of the sky covered by previous SuperCOSMOS
RECONS (SCR) searches that used Schmidt plates as the primary source of
discovery. Among the new systems are 25 multiples, plus an additional seven
new common proper motion companions found to previously known primaries. All
stars have been discovered using the third U.S. Naval Observatory (USNO) CCD
Astrograph Catalog (UCAC3). A comparison of the UCAC3 proper motions to those
from the Hipparcos, Tycho-2, Southern Proper Motion (SPM4), and SuperCOSMOS
efforts is presented, and shows that UCAC3 provides similar values and
precision to the first three surveys. The comparison between UCAC3 and
SuperCOSMOS indicates that proper motions in RA are systematically shifted in
the SuperCOSMOS data but are consistent in DEC data, while overall showing a
significantly higher scatter. Distance estimates are derived for stars having
SuperCOSMOS Sky Survey (SSS) $B_{J}$, $R_{59F}$, and $I_{IVN}$ plate
magnitudes and Two-Micron All Sky Survey (2MASS) infrared photometry. We find
15 systems estimated to be within 25 pc, including UPM 1710-5300 our closest
new discovery estimated at 13.5 pc. Such new discoveries suggest that more
nearby stars are yet to be found in these slower proper motion regimes,
indicating that more work is needed to develop a complete map of the solar
neighborhood.
solar neighborhood — stars: distances — stars: statistics — surveys —
astrometry
## 1 INTRODUCTION
Proper motion surveys were once in the forefront of astronomy research.
Providing a vast library of low mass stars, proper motion surveys provide a
wealth of information to astronomers studying stellar populations, and in
particular the stellar luminosity and mass functions that reveal how the
Galaxy’s stellar mass is divided among different types of stars. Proper motion
surveys of faint objects, such as red dwarfs, subdwarfs, and white dwarfs,
play a crucial role in identifying the Sun’s nearest neighbors.
Historically, proper motion studies have been carried out by blinking
photographic plates taken at different epochs to detect the changing positions
of stars. The pioneering surveys of the second half of the last century
include the Lowell Proper Motion Survey (Giclas et al., 1971, 1978), the
Luyten Half-Second catalog (LHS) (Luyten, 1979) and the New Luyten Two-Tenths
catalog (NLTT) (Luyten, 1980). With the aid of plate scanning machines and
high powered computers, the traditional techniques used for proper motion
studies are carried out in much the same way, only now using digitized images
of the photographic plates. Utilizing various techniques and plate sets, these
new computerized searches have revealed many new proper motion systems. Recent
surveys of the southern sky — the region targeted by the present effort —
include (Wroblewski & Costa, 1999), (Scholz et al., 2000, 2002), (Oppenheimer
et al., 2001), the Southern Infrared Proper Motion Survey (SIPS) (Deacon et
al., 2005; Deacon & Hambly, 2007), the IPHAS-POSS-I proper motion survey of
the Galactic plane (Deacon et al., 2009) and Lepine’s SUPERBLINK survey
(Lépine, 2005, 2008).
The Research Consortium On Nearby Stars (RECONS) group111www.recons.org has
also been systematically canvassing the southern sky for new proper motion
systems as part of their effort to understand the stellar population of the
solar neighborhood. To date, these discoveries have been reported in five of
the The Solar Neighborhood (TSN) series of papers (Hambly et al., 2004; Henry
et al., 2004; Subasavage et al. 2005a, ; Subasavage et al. 2005b, ; Finch et
al., 2007). The new systems are given SCR (SuperCOSMOS-RECONS) names because
they have been discovered using the SuperCOSMOS Sky Survey (SSS) data (Hambly
et al. 2001a, ). The RECONS group continues to operate a trigonometric
parallax program at the CTIO 0.9m telescope to confirm stars within 25 pc,
with a focus on stars within 10 pc. In Table 1, we summarize the number of new
stellar systems reported to be within 25 pc by RECONS and others via proper
motion surveys that have distance estimates derived from photographic
relations.
In this investigation we focus on stars in the newly released third U.S. Naval
Observatory (USNO) CCD Astrograph Catalog (UCAC3) (Zacharias et al., 2010)
found between declinations $-$90$\arcdeg$ and $-$47$\arcdeg$ that have
0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1, where $\mu$ is the proper
motion. The search region and proper motion range match that in The Solar
Neighborhood XVIII (hereafter, TSN18) (Finch et al., 2007), in which the lower
proper motion cutoff was chosen to match that of the NLTT catalog. In TSN18 we
presented 1606 new SCR systems, including 54 candidate common proper motion
multiples. By utilizing the UCAC3 catalog with proper motions determined
without the sole use of photographic plates, we can probe for new proper
motion stars and companions that have been overlooked during previous
searches. The new objects reported here have been dubbed UPM, for this new
USNO Proper Motion search.
## 2 Method
### 2.1 UCAC3
The USNO CCD Astrograph Catalog (UCAC) project has been producing astrometric
catalogs since October 2000, with the first release (UCAC1) (Zacharias et al.,
2000) covering only 80% of the southern sky. The second catalog in this series
(UCAC2) (Zacharias et al., 2004) was released in July 2003 with about the same
level of completeness as UCAC1, but with early epoch plates paired with the
Astrograph CCD data for improved proper motions. The UCAC3, released in August
2009, is the first from the series to have all sky coverage, and contains just
over 100 million entries with a limiting magnitude of $\sim$16 in the UCAC
bandpass (579-642 nm). UCAC3 also includes double star fitting, and has a
slightly deeper limiting magnitude than UCAC2 due to a complete re-reduction
of the pixel data (Zacharias, 2010). A detailed introduction to the UCAC3 can
be found in the release paper (Zacharias et al., 2010) and the README file of
the data distribution.
The UCAC3 has been used in the present survey to probe for proper motion stars
that have been overlooked during previous SCR and other searches. The Two-
Micron All Sky Survey (2MASS) was used to probe for and reduce systematic
errors in UCAC CCD observations, giving a greater number of reference stars to
stack up residuals as a function of many parameters, such as observing site
and exposure time. A detailed description of the astrometric reductions of
UCAC3 can be found in (Finch et al., 2010).
### 2.2 PROPER MOTIONS
Out of the roughly 100 million stars in the UCAC3 catalog, about 95 million
have calculated absolute proper motions. Most proper motions are derived using
the Astrograph CCD data combined with various earlier epoch catalogs in much
the same manner as UCAC2. All input catalogs are reduced to the International
Celestial Reference Frame (ICRF) by utilizing Hipparcos data or a similar,
denser catalog, such as Tycho-2. For each position, standard errors are
estimated. These errors are then used as weights to compute a UCAC3 mean
position and proper motion, utilizing a weighted, least-squares adjustment
procedure. For bright stars (R$\sim$8–12), UCAC Astrograph CCD data were
combined with ground-based photographic and transit circle catalogs, including
all catalogs used for the Tycho-2 project (Høg et al., 2000), and $\sim$1.2
million positions from about 1950 AGK2 plates derived using the StarScan
machine (Zacharias et al., 2008). For fainter stars (R$\sim$12.5–16.5), UCAC
Astrograph CCD data were combined with scans from the StarScan machine of
roughly 3200 plates from the Hamburg Zone Astrograph, the USNO Black Birch
Astrograph, and the Lick Astrograph, as well as a complete new reduction of
the Yale Southern Proper Motion (SPM4) survey (Girard et al., 2010), and data
from the SuperCOSMOS project (Hambly et al. 2001a, ). The SuperCOSMOS data
were used in place of the Northern Proper Motions (NPM) (Girard et al., ), in
preparation, which was not complete when the UCAC3 was generated, but which
will be included in the anticipated UCAC4 final release. An estimated error
floor has been added to all catalogs used for the proper motion calculation.
The largest root mean square (RMS) error contribution added was 100 mas for
the SuperCOSMOS data due to zonal (plate pattern) systematic errors in the
range of 50 to 200 mas, when compared to 2MASS data. For a detailed
description of the derived UCAC3 proper motions see (Zacharias et al., 2010).
An effort was made to tag previously known High Proper Motion (HPM) stars in
the UCAC3 catalog using the VizieR on-line data tool, along with published
literature. The list includes roughly 51000 known proper motion stars covering
the entire sky with $\mu$ $>$ $\sim$0$\farcs$18 yr-1. In the North we used the
LSPM-North catalog (Lépine, 2005) containing 61977 new and previously found
stars having proper motions greater than 0$\farcs$15 yr-1. In the South we
used many smaller surveys along with the Revised NLTT Catalog (Salim & Gould,
2003), which produced 17730 stars with proper motions greater than 0$\farcs$15
yr-1. For a full list of catalogs used see the UCAC3 README file. This list is
not comprehensive due to not having a complete list of proper motion surveys
and the dificulty in matching some catalogs like the NLTT which do not have
reliable positions. The proper motion values given in UCAC3 for these
previously known stars come from the individual catalogs themselves and are
not derived in the same manner as the UCAC3 proper motions mentioned above.
These previously known proper motion stars are flagged in the UCAC3 data with
a Mean Position (MPOS) running star number greater than 140 million.
The errors in proper motions reported in the UCAC3 release for stars brighter
than mag 12 are only $\sim$1–3 mas/yr because of the large epoch spread,
oftentimes as long as 100 years. For fainter stars found in SPM4, the errors
are $\sim$2–3 mas/yr, while proper motions incorporating SuperCOSMOS data
result in errors of $\sim$6–8 mas/yr.
### 2.3 SEARCH CRITERIA
The initial sample of 177231 proper motion candidates for this search included
all UCAC3 stars in the southern sky between declinations $-$90$\arcdeg$ and
$-$47$\arcdeg$ with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1.
Winnowing of the sample was accomplished by examining previously known proper
motion stars meeting the survey criteria to find a combination of UCAC3 flags
with values indicative of real proper motion objects. To verify the set of
flags adopted for final target selection, visual inspections were done of
targets in selected sky regions to confirm true proper motion. In addition to
meeting the declination and proper motion survey limits, all stars (1) must be
in the 2MASS catalog with an e2mpho (2MASS photometry error) less than or
equal to 0.05 magnitudes in all three bands, (2) have a UCAC fit model
magnitude between 7 and 17 mag, (3) have a double star flag (dsf) equal to 0,
1, 5 or 6, meaning a single star or fitted double, (4) have an object flag
(objt) between $-$2 and 2 to exclude positions that used all overexposed
images in the fit, (5) have an MPOS number less than 140 million to exclude
already known high proper motion stars, and (6) have a LEDA galaxy flag of
zero, meaning that the source is not in the LEDA galaxy catalog. After
implementing these cuts, 9248 candidates remained.
These candidates were then cross-checked via VizieR, the published literature,
and SIMBAD to determine if they were previously known. VizieR was used to
cross-check various proper motion catalogs, such as NLTT, Hipparcos, and
Tycho-2. If a survey was not available on VizieR, data were obtained from the
published literature (as in the case for stars found in TSN18, which has only
recently been added to the VizieR database.) A final search was done by
checking the remaining candidates against the SIMBAD database. Cross-checks of
the various compendia were performed using a 90$\arcsec$ search radius, with
one exception (the NLTT catalog, see below). If a candidate was matched to a
known proper motion star having roughly the same proper motion and magnitude,
then it was labeled as previously known, and is not included in the sample
reported here.
A larger search radius of 180$\arcsec$ was used when comparing UPM candidates
to the NLTT catalog, which is known to have inaccurate positions. As shown in
the Figure 2 histogram of (Bakos et al., 2002), the distances between their
measured positions and Luyten’s listed positions of LHS stars can be quite
large. The number of objects with a given position offset levels off around
90$\arcsec$, beyond which fewer than 10 objects per 1$\arcsec$ bin are found.
Thus, UCAC3 proper motion candidates with positions differing from Luyten’s by
less than 90$\arcsec$ are considered known, those differing by 90–180$\arcsec$
are considered new discoveries but are noted as possible NLTT stars in the
tables, and those differing by more than 180$\arcsec$ are considered new
discoveries. It is not a goal of this paper to revise the NLTT catalog and
assign proper identifications and accurate positions to NLTT entries; rather,
the goal is to identify new high proper motion stars.
The various cross-checks for previously known stars reduced the number of new
candidates to a list of 4425. Each was visually inspected to verify proper
motion by blinking the $B_{J}$ and $R_{59F}$ SuperCOSMOS digitized plate
images. Objects without verifiable proper motions were then discarded, leaving
474 new proper motion discoveries. Of these new discoveries 32 were found to
be part of a Common Proper Motion (CPM) system, including seven new
discoveries having CPM to previously known primaries, leaving a total of 442
new systems.
A lower successful hit rate (5297 real proper motion objects / 9248 total
“good” candidates extracted) of 57.3% was found for this search than the 78.1%
successful hit rate obtained in TSN18. As in TSN18, the hit rate takes into
account new, known, and phantom proper motion objects (phantoms are identified
as moving objects but are not). The lower hit rate in the present effort is
the result of at least three factors. First, some real objects were discarded
early in the search due to the rigorous sifting mentioned above to obtain a
more manageable sample for investigating, i.e. some of the selection criteria,
particularly involving 2mass, were “too tight.” Second, many phantom proper
motion objects in the UCAC3 made the sample cuts because of incorrect matches
between catalogs during the proper motion calculation. This is particularly
common in the fainter stars for which the proper motion calculations rely on
only two catalog positions. Third, other misidentifications arise from blended
images, where two single star detections in the UCAC3 can be matched up to a
single image in an earlier epoch catalog.
## 3 RESULTS
The 442 new UCAC3 proper motion systems are listed in Table 2. In Table 3 we
highlight the 15 systems for which we estimate distances to be less than 25
pc. In both Tables we list names, coordinates, proper motions, 1$\sigma$
errors in the proper motions, plate magnitudes from SuperCOSMOS, near-IR
photometry from 2MASS, the computed $R_{59F}-J$ color, distance estimate, and
notes.
### 3.1 Positions and Proper Motions
All positions on the ICRF system, proper motions, and errors are taken
directly from UCAC3. For a few stars found visually, e.g. companions, no data
could be obtained from UCAC3, so information was obtained from other sources
(additional objects, see below). The average positional errors reported in the
UCAC3 catalog for this sample are 52 mas in RA and 53 mas in Dec. The average
proper motion errors are 8.5 mas/yr in $\mu_{\alpha}\cos\delta$ and 8.4 mas/yr
in $\mu_{\delta}$.
### 3.2 Photometry
In Tables 2 and 3, we give photographic magnitudes from SuperCOSMOS for three
plate emulsions: $B_{J}$, $R_{59F}$, and $I_{IVN}$. For sources fainter than
mag $\sim$15, plate errors are typically less than 0.3 mag, but errors
increase for brighter sources. Plate color errors are smaller, at roughly 0.07
mag (Hambly et al. 2001b, ). 2MASS $JHK_{s}$ infrared photometry is given,
with errors typically 0.05 mag or less due to the search criteria and because
the stars are usually brighter than 16 in UCAC3 and are red, making them
relatively bright, mag $\sim$ 10–14, in 2MASS. Companions found during visual
inspection may be fainter and have consequently larger photometric errors. The
optical (SuperCOSMOS) and infrared (2MASS) datasets are combined in a computed
$R_{59F}-J$ color to provide an indicator of the star’s color. In some cases,
SuperCOSMOS magnitudes may not be given due to blending, no source detection
or other problems where no magnitude is reported in the SuperCOSMOS data.
2MASS magnitudes are given for all but five objects that are not present in
the 2MASS catalog.
### 3.3 Distances
Distance estimates are computed using 11 colors generated from the six-band
photometry using the relations given in (Hambly et al., 2004). This method
assumes that all objects are main sequence stars. The accuracy reported for
this technique is roughly 26%, which is determined from the mean of the
absolute values of the differences between distances for stars with
trigonometric parallaxes and distances estimated via the relations. No
distance estimate is given for stars that are too blue for the relations. For
objects with incomplete photometry, the distance estimate will be less
reliable. While only one relation is needed to produce a distance estimate,
six are needed to be considered “reliable.” This is half of the 11 total
posibilities. Stars having fewer than six relations have been identified in
the notes. For any star expected to have an erroneous distance (white dwarf,
evolved star, subdwarf), the distance is given in brackets.
### 3.4 Additional Objects
During the visual inspection of the candidates, 27 additional proper motion
objects were found, listed in Table 4. These objects generally are CPM
companion candidates that either have a fainter limiting magnitude than
implemented for this search, were eliminated from the candidate list by the
search criteria, or have a UCAC3 proper motion less than 0$\farcs$18 yr-1.
Proper motions from the UCAC3 data less than the 0$\farcs$18 yr-1 cutoff of
this paper are considered suspect from a visual inspection that compared the
proper motion of the companion candidate. All objects detected during the
visual inspection were investigated using SIMBAD and VizieR for previous
identifications. If none were found, their proper motions were obtained from
UCAC3, SPM4, or SuperCOSMOS, in that order of priority. Magnitudes were then
obtained from SuperCOSMOS and 2MASS to compute distance estimates. Only four
of the 27 objects found visually did not have a proper motion reported in any
catalog. For stars that were not found in the UCAC3 data, positions were
computed using the epoch, coordinates, and proper motion obtained from the
corresponding catalog.
## 4 ANALYSIS
### 4.1 Color-Magnitude Diagram
In Figure 1 we show a color-magnitude diagram of the 465 proper motion objects
reported in this sample having a $R_{59F}-J$ color. New proper motion objects
are represented by closed circles while known objects (companions to new
objects) are represented by open circles. Data points that fall below
$R_{59F}\sim 17$ are CPM companion candidates noticed during visual
inspection. The brightest new object, UPM 1542-5041, has $R_{59F}$ = 9.924 and
is estimated to be at a distance of 33.1 pc. The reddest object found in this
search is UPM 1703-4934B with $R_{59F}-J$ = 6.30, $R_{59F}$ = 17.31, and
estimated distance of 40.7 pc.
The subdwarf population is not as well defined in this paper as in TSN18
because there are far fewer new objects. Nonetheless, a separation can be seen
below the concentration of main sequence stars. Finally, a single known white
dwarf, WD 0607-530B, can be seen in the lower left of Figure 1.
### 4.2 Reduced Proper Motion Diagram
We show in Figure 2 the Reduced Proper Motion (RPM) diagram for all 465
objects in this sample having a $R_{59F}-J$ color. New proper motion objects
are represented by closed circles while known objects (companions to new
objects) are represented by open circles. A reduced proper motion diagram
takes advantage of the assumption that objects with larger distances tend to
have smaller proper motions. While this is not always valid it can be used as
a good method to separate white dwarfs and subdwarfs from main-sequence stars.
We determine HR as in TSN18 using a modified distance modulus equation, in
which $\mu$ is substituted for distance.
$H_{R}=R_{59F}+5+5\log\mu.$
The dashed line in Figure 2 is the same empirical line used in TSN18 to
separate white dwarfs from subdwarfs. This separation line has been shown from
past SCR searches to be reliable in identifying white dwarf candidates. From
the present survey, only one known white dwarf WD 0607-530B, a CPM companion
candidate to UPM 0608-5301A, is seen clearly below the subdwarf region.
Subdwarf candidates have been selected using the same method as in TSN18 —
stars with $R_{59F}-J>$ 1.0 and within 4.0 mag in $H_{R}$ of the the empirical
line separating the white dwarfs are considered subdwarfs. From this survey
there are 31 subdwarf candidates, all with distance estimates greater than 147
pc. Large distance estimates can be used to identify both subdwarf and white
dwarf candidates, which are subluminous compared to main sequence stars and
yield large distance estimates because they are intrinsically fainter than the
main-sequence stars used to generate the photometric distance relations. The
presumably erroneous distances for these stars are given in brackets in Tables
2 and 3. Follow up spectroscopic observations will be needed to confirm all
subdwarf candidates.
### 4.3 New Common Proper Motion Systems
In this search we found 32 common proper motion systems (31 binaries and one
triple), including 25 entirely new systems and seven hybrid systems containing
both new and known objects. The triple system is a previously known system
discovered as part of the automated search to have a newly discovered third
component. The data for these systems are given in Table 4, where we list the
primaries and companions, their proper motions, and the companions’
separations and position angles relative to the primaries (defined to be the
brightest star in each system). The distance estimates were used in
conjunction with the proper motions and visual inspections to determine
whether or not a pair of stars is physically associated. Because these objects
were found during visual inspections, the proper motion and/or SuperCOSMOS
magnitudes may be missing or suspect; in such cases, identifications as CPM
systems are more tentative and identified in the notes.
In Figure 3 we compare the proper motions per coordinate for the 29 CPM
systems for which both components have proper motions. CPM candidates that
have proper motions from UCAC3 are represented by closed circles while those
with proper motions from other sources are represented by open circles. Proper
motions for the latter candidates were extracted manually from either SPM4 or
SuperCOSMOS.
### 4.4 Notes on Specific Stars
UPM 0608-5301A is an M dwarf at an estimated distance of 37.1 pc with a known
white dwarf as a possible companion. The B component (known white dwarf) is at
a separation of 21.5$\arcsec$ at position angle 120.7∘ from the primary. We
estimate a distance of 34.2 pc for the white dwarf with an error of 20% using
the relation of (Oppenheimer et al., 2001). See Table 4 for more details.
UPM 0835-6018C is in a possible triple system with NLTT 19906 and NLTT 19907.
The A and B components are separated by 5.1$\arcsec$. The C component has a
separation of 113.0$\arcsec$ at a position angle of 49.3∘ from the primary.
See Table 4 for more details.
UPM 1230-5736AB The A component is estimated to be at 22.2 pc, and has
$R_{59F}=$ 12.04 and proper motion per coordinate
($\mu_{\alpha}\cos\delta$,$\mu_{\delta}$) = (-227.6,-66.5) mas/yr. NLTT 30961
is found 1.61$\arcmin$ away, and NLTT lists a red photographic magnitude of
13.1 and proper motion per coordinate
($\mu_{\alpha}\cos\delta$,$\mu_{\delta}$) = (-216.7, -38.2) mas/yr.
The B component is estimated to be at 19.8 pc, and has $R_{59F}=$ 12.83 and
proper motion per coordinate ($\mu_{\alpha}\cos\delta$,$\mu_{\delta}$) =
(-243.0, -29.3) mas/yr. NLTT 30938 is found 1.68$\arcmin$ away, and NLTT lists
a red photographic magnitude of 12.6 and proper motion identical to NLTT
30961\. Thus, this nearby UPM double is likely the same as the NLTT double,
but the relatively large offset from the NLTT makes the identification
ambiguous.
UPM 1542-5041 is the brightest new HPM discovery from this effort. It has a
distance estimate of 33.1 pc, $R_{59F}=$ 9.92 and proper motion per coordinate
($\mu_{\alpha}\cos\delta$,$\mu_{\delta}$) = (182.4, -16.3) mas/yr. NLTT 40903
is found 2.46$\arcmin$ away, and NLTT lists a red photographic magnitude of
12.8 and proper motion per coordinate
($\mu_{\alpha}\cos\delta$,$\mu_{\delta}$) = (-244.2,-190.8) mas/yr. The
discordant magnitudes and proper motions indicate that UPM 1542-5041 is not
NLTT 40903.
UPM 1710-5300 has an estimated distance of only 13.5 pc, making it the nearest
candidate in the sample.
### 4.5 COMPARISON TO PREVIOUS PROPER-MOTION SURVEYS
Most previously known HPM stars have been tagged in UCAC3 and their listed
proper motions in UCAC3 were taken from their respective catalogs. Because no
UCAC3 proper motions were determined, comparisons to other catalogs/surveys
are therefore difficult. Nonetheless, within the sky coverage and proper
motion regime of this paper, 66 stars have been found in both the Hipparcos
and Tycho-2 catalogs that are not tagged as HPM stars in the UCAC3 catalog.
This constitutes a small but ample number of stars that can be used to compare
the bright end of UCAC3 proper motions to those in the Hipparcos and Tycho-2
catalogs. In Figure 4, we show the comparison between UCAC3 proper motions in
RA and DEC to the Hipparcos (top) and Tycho-2 (middle) catalogs. For
comparison we also plot the proper motion differences between Hipparcos and
Tycho-2 in the bottom panel of Figure 4. This plot implies that for both the
Hipparcos and Tycho-2 catalogs the UCAC3 proper motions show only small
differences per coordinate at all declinations in the present search, with no
significant systematics. The RMS differences of UCAC3 proper motions per
coordinate ($\Delta\mu_{\alpha}\cos\delta$, $\Delta\mu_{\delta}$) when
compared to Hipparcos are 7.5 and 6.6 mas/yr, respectively. When compared to
Tycho-2 proper motion coordinates we find 5.5 and 5.9 mas/yr, respectively. A
slightly lower RMS difference of 3.8 mas/yr in both coordinates is seen when
comparing the Hipparcos and Tycho-2 proper motions for these stars.
To investigate fainter stars in UCAC3, we compare UCAC3 proper motions to SPM4
and SuperCOSMOS results. We compare proper motions using the SuperCOSMOS
proper motions to bring the positions to the UCAC3 epoch with a 1.5 arcsec
match radius. A 1.5 arcsec radius was also used to match UCAC3 to SPM4 with no
need to correct for proper motions because both catalogs are on the same
system and set at the same epoch.
A total of 137 objects meeting the proper motion and declination limits of
this paper were found in all three catalogs. In Figure 5, we compare UCAC3
proper motions in the same manner as above with the SPM4 (top) and SuperCOSMOS
(middle) catalogs. Again for comparison, we also include a plot showing the
differences between SPM4 and SuperCOSMOS in the bottom panel of Figure 5. The
RMS differences between UCAC3 and SPM4 per coordinate
($\Delta\mu_{\alpha}\cos\delta$, $\Delta\mu_{\delta}$) are 6.6 and 4.1 mas/yr
respectively. Much higher RMS differences of 19.3 and 18.9 mas/yr are seen
when comparing UCAC3 to the SuperCOSMOS proper motions per coordinate
($\Delta\mu_{\alpha}\cos\delta$, $\Delta\mu_{\delta}$). This comparison also
indicates that proper motions in RA are systematically shifted in the
SuperCOSMOS data, but are consistent in DEC. This high RMS including the
systematic shift in RA is also seen when comparing the SPM4 to SuperCOSMOS
proper motions per coordinate, yielding RMS differences of 19.9 and 17.5
mas/yr in $\Delta\mu_{\alpha}\cos\delta$ and $\Delta\mu_{\delta}$,
respectively. The higher RMS differences for the SuperCOSMOS proper motions
are in agreement with the findings of TSN18 where SCR proper motions were
found to have an average deviation of 23 mas/yr total proper motion when
compared to the NLTT and Hipparcos proper motions.
In TSN18 a total of 1662 objects were reported, of which 1615 match the proper
motion and declination limits of this paper. During this UCAC3 search, 1298 of
the 1615 objects reported in TSN18 were recovered, or a 80.4% succesful
recovery rate. Objects missed in this UCAC3 survey are primarily those at the
faint end, as the TSN18 survey reached to $R_{59F}=$ 16.5.
The Hipparcos catalog contains 118218 total objects of which 722 meet the
proper motion and declination limits of this paper. Tycho-2 contains 2539913
total objects in the main catalog with 1273 of those objects matching the
proper motion and declination limits of this paper. We recover 646 Hipparcos
stars and 973 Tycho-2 stars using the search criteria of this paper, yielding
recovery rates of 89.5% and 76.4% respectively. Objects missed in this UCAC3
survey is primarily due to UCAC3 lacking a source detection for $\sim$15% of
the Tycho-2 objects. The relatively high recovery rates of UCAC3 when compared
to these three efforts implies UCAC3 can be used as a reliable source to
search for new proper motion stars with $\mu$ = 0.18–0.40 arcsec yr-1 for
other portions of the sky.
## 5 DISCUSSION
We have found 442 new proper motion systems including 474 objects with
0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 between declinations
$-$90$\arcdeg$ and $-$47$\arcdeg$. In Figure 6, we show the sky distribution
for the entire sample reported in this paper. The 474 new discoveries
represent a 25.3% increase in new systems for the same region of the sky
covered by previous (SCR) searches that used Schmidt plates as the primary
source of discovery. While many of these new UPM discoveries are found along
the Galactic plane, a region avoided by the SCR survey, additional new systems
were found far from the plane. Areas in Figure 6 with a lower density of new
discoveries have been heavily searched by previous proper motion surveys as
seen when comparing to a similar sky plot presented in TSN18.
As shown in Figure 4 and 5, the proper motions obtained from UCAC3 compare
well to the Hipparcos, Tycho-2, and SPM4 catalogs. However, we find that the
SuperCOSMOS proper motions have a significantly higher scatter when compared
to these catalogs, which confirms our similar result in TSN18.
We find 25 new CPM candidate systems, as well as 31 new subdwarf candidates
that will need future spectroscopic efforts to be confirmed. In Figure 7, we
show a histogram of the number of proper motion discoveries in 0$\farcs$01
yr-1 bins for the present sample, highlighting the number of those having
distance estimates within 50 pc. The increase in nearby systems at the lowest
proper motions sampled here implies that more nearby stars are likely to be
found at even slower proper motion regimes.
Finally, we have found 16 objects in 15 systems with distances estimated to be
within 25 pc, and an additional 109 objects in 107 systems between 25 and 50
pc. The discoveries include UPM 1542-5041, which at $R_{59F}=$ 9.92 is a
surprisingly bright new proper motion discovery with an estimated distance of
33.1 pc. UPM 1710-5300, which is our nearest new candidate with an estimated
distance at 13.5 pc. We anticipate that further exploration of the UCAC3 for
new proper motion discoveries will result in more nearby star candidates,
perhaps some even within 10 pc, where new discoveries are still being made
(Henry et al., 2006)
We thank the entire UCAC team for making this proper motion survey possible.
Special thanks go out to the RECONS team at Georgia State University for their
support, John Subasavage in particular for assistance with the SCR searches,
and Nigel Hambly for his work with the SuperCOSMOS Sky Survey. We would also
like to thank all USNO summer students who helped in this survey. This work
has made use of the SIMBAD, VizieR, and Aladin databases operated at the CDS
in Strasbourg, France. We have also made use of data from the Two-Micron All
Sky Survey, SuperCOSMOS Science Archive and the Southern Proper Motion
catalog.
## References
* Bakos et al. (2002) Bakos, G. Á., Sahu, K. C., & Németh, P. 2002, ApJS, 141, 187
* Deacon et al. (2005) Deacon, N. R., Hambly, N. C., & Cooke, J. A. 2005, A&A, 435, 363
* Deacon & Hambly (2007) Deacon, N. R., & Hambly, N. C. 2007, A&A, 468, 163
* Deacon et al. (2009) Deacon, N. R., et al. 2009, MNRAS, 397, 1685
* Finch et al. (2007) Finch, C. T., Henry, T. J., Subasavage, J. P., Jao, W.-C., & Hambly, N. C. 2007, AJ, 133, 2898
* Finch et al. (2010) Finch, C., Zacharias, N.,Wycoff, G. 2010, in prep. for AJ.
* Giclas et al. (1971) Giclas, H. L., Burnham, R., & Thomas, N. G. 1971, Flagstaff, Arizona: Lowell Observatory, 1971
* Giclas et al. (1978) Giclas, H. L., Burnham, R., & Thomas, N. G. 1978, Lowell Observatory Bulletin, 8, 89
* (9) Girard, T. et al., paper about NPM (in preparation)
* Girard et al. (2010) Girard, T. et al. 2010, paper about SPM4 (in preparation)
* (11) Hambly, N. C., MacGillivray, H. T., Read, M. A., Tritton, S. B., Thomson, E. B., Kelly, B. D., Morgan, D. H., Smith, R. E., Driver, S. P., Williamson, J., Parker, Q. A., Hawkins, M. R. S., Williams, P. M., Lawrence, A. 2001, MNRAS, 326, 1279
* (12) Hambly, N. C., Irwin, M. J., & MacGillivray, H. T. 2001, MNRAS, 326, 1295
* Hambly et al. (2004) Hambly, N. C., Henry, T. J., Subasavage, J. P., Brown, M. A., & Jao, W. 2004, AJ, 128, 437
* Henry et al. (2004) Henry, T. J., Subasavage, J. P., Brown, M. A., Beaulieu, T. D., Jao, W.-C., & Hambly, N. C. 2004, AJ, 128, 2460
* Henry et al. (2006) Henry, T. J., Jao, W.-C., Subasavage, J. P., Beaulieu, T. D., Ianna, P. A., Costa, E., & Méndez, R. A. 2006, AJ, 132, 2360
* Høg et al. (2000) Høg, E., et al. 2000, A&A, 355, L27
* Lépine (2005) Lépine, S. 2005, AJ, 130, 1247
* Lépine (2008) Lépine, S. 2008, AJ, 135, 2177
* Luyten (1979) Luyten, W. J. 1979, LHS Catalogue (Minneapolis: Univ. of Minnesota Press)
* Luyten (1980) Luyten, W. J. 1980, Proper Motion Survey with the 48-inch Telescope, Univ. Minnesota, 55, 1 (1980), 55, 1
* Oppenheimer et al. (2001) Oppenheimer, B. R., Hambly, N. C., Digby, A. P., Hodgkin, S. T., & Saumon, D. 2001, Science, 292, 698
* Salim & Gould (2003) Salim, S., & Gould, A. 2003, ApJ, 582, 1011
* Scholz et al. (2000) Scholz, R.-D., Irwin, M., Ibata, R., Jahreiß, H., & Malkov, O. Y. 2000, A&A, 353, 958
* Scholz et al. (2002) Scholz, R.-D., Szokoly, G. P., Andersen, M., Ibata, R., & Irwin, M. J. 2002, ApJ, 565, 539
* (25) Subasavage, J. P., Henry, T. J., Hambly, N. C., Brown, M. A., & Jao, W. C. 2005, AJ, 129, 413
* (26) Subasavage, J. P., Henry, T. J., Hambly, N. C., Brown, M. A., Jao, W. C., & Finch, C. T. 2005, AJ, 130, 1658
* Wroblewski & Costa (1999) Wroblewski, H., & Costa, E. 1999, A&AS, 139, 25
* Zacharias et al. (2000) Zacharias, N., Zacharias, M. I., & Rafferty, T. J. 2000, AJ, 118, 2503 (UCAC1 paper)
* Zacharias et al. (2004) Zacharias, N., Urban, S. E., Zacharias, M. I., Wycoff, G. L., Hall, D. M., Monet, D. G., & Rafferty, T. J. 2004, AJ, 127, 3043 (UCAC2 paper)
* Zacharias et al. (2008) Zacharias, N., Winter, L, Holdenried, E.R., De Cuyper,J.-P., Rafferty, T.J., Wycoff, G.L., 2008, PASP, 120, 644 (astro-ph 0806.0256)
* Zacharias et al. (2010) Zacharias, N. et al. submitted to AJ (UCAC3 release paper)
* Zacharias (2010) Zacharias, N. 2010, in preparation for AJ (UCAC3 pixel red.paper)
Figure 1: Color-apparent magnitude diagram for all proper motion systems in
the sample having an $R_{59F}-J$ color. New proper motion objects are
represented by closed circles while known objects (CPM companions to new
objects) are represented with open circles. Data points below $R_{59F}=$ 17
are CPM candidates noticed during the visual inspection. Figure 2: RPM diagram
for all proper motion systems in this sample having an $R_{59F}-J$ color. New
proper motion objects are represented by closed circles while known objects
(CPM companions to new objects) are represented with open circles. The
empirical line separates the white dwarf candidate from the subdwarf
candidates, which lie above the white dwarf stars and just below the
concentration of main sequence stars. Figure 3: Comparison of proper motions
per coordinate, $\mu_{\alpha}\cos\delta$ (top) and $\mu_{\delta}$ (bottom),
for components in CPM systems. Proper motions from the UCAC3 catalog are
represented by closed circles while proper motions manually obtained through
other means are denoted by open circles. The solid line indicates perfect
agreement.
Figure 4: Comparison of UCAC3, Hipparcos and Tycho-2 proper motions per
coordinate, $\Delta\mu_{\alpha}\cos\delta$ (left column) and
$\Delta\mu_{\delta}$ (right column). Figure 5: Comparison of UCAC3,
SuperCOSMOS and SPM4 proper motions per coordinate,
$\Delta\mu_{\alpha}\cos\delta$ (left column) and $\Delta\mu_{\delta}$ (right
column).
Figure 6: Sky distribution of all UCAC3 proper motion objects reported in this
sample, i.e. those between declinations $-$90$\arcdeg$ and $-$47$\arcdeg$
having 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1. The curve
represents the Galactic plane. Figure 7: Histogram showing the number of
proper motion objects in 0$\farcs$01 yr-1 bins for the entire sample (empty
bars) and the number of those objects having distance estimates within 50 pc
(filled bars).
Table 1: New Proper Motion Systems within 25 Parsecs Based on Distances Using SuperCOSMOS Photographic Magnitudes Paper | Systems | References
---|---|---
| $\leq$ 25 pc |
TSN08 | 3 | (Hambly et al., 2004)
TSN10 | 4 | (Henry et al., 2004)
TSN12 | 18 | (Subasavage et al. 2005a, )
TSN15 | 25 | (Subasavage et al. 2005b, )
TSN18 | 31 | (Finch et al., 2007)
SIPS1 | 1 | (Deacon et al., 2005)
SIPS2 | 12 | (Deacon & Hambly, 2007)
this paper | 15 | (Finch et al. 2010)
Table 2: New UCAC3 High Proper Motion Systems between Declinations $-$90$\arcdeg$ and $-$47$\arcdeg$ with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 Name | RA | DEC | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | sig$\mu_{\alpha}$ | sig$\mu_{\delta}$ | $B_{J}$ | $R_{59F}$ | $I_{IVN}$ | $J$ | $H$ | $K_{s}$ | $R_{59F}$ $-$ $J$ | Est Dist | Notes
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (mas) | (mas) | (mas/yr) | (mas/yr) | (mas/yr) | (mas/yr) | | | | | | | | (pc) |
UPM 0000-4843 | 480848 | 148604159 | 181.7 | 29.9 | 12.3 | 8.5 | 15.579 | 13.333 | 11.337 | 10.824 | 10.208 | 10.018 | 2.509 | 47.2 |
UPM 0016-7327 | 14557586 | 59575305 | 184.8 | -29.6 | 1.9 | 2.1 | 16.498 | 15.410 | 13.572 | 11.351 | 10.754 | 10.611 | 4.059 | 40.0 |
UPM 0028-5501 | 25583309 | 125920383 | 190.0 | -10.2 | 2.6 | 3.0 | 17.382 | 15.533 | 14.688 | 11.863 | 11.233 | 11.054 | 3.670 | 50.1 |
UPM 0029-6724 | 26938223 | 81322056 | 184.7 | 43.5 | 1.8 | 1.4 | 15.228 | 13.197 | 12.132 | 11.065 | 10.412 | 10.244 | 2.132 | 60.7 |
UPM 0032-6719 | 29092633 | 81607463 | 183.1 | 90.8 | 8.9 | 5.5 | 17.735 | 15.518 | 14.025 | 12.776 | 12.160 | 11.956 | 2.742 | 100.6 |
UPM 0037-6725 | 33311860 | 81260513 | 297.2 | -98.6 | 11.2 | 11.8 | 17.318 | 15.043 | 13.494 | 11.529 | 11.025 | 10.739 | 3.514 | 40.5 |
UPM 0037-6844 | 33623179 | 76513673 | 249.0 | 44.0 | 7.8 | 7.6 | 17.173 | 15.016 | 13.408 | 11.571 | 10.983 | 10.701 | 3.445 | 41.1 |
UPM 0040-6834 | 36433102 | 77158520 | 170.0 | 78.5 | 7.4 | 7.1 | 15.386 | 13.502 | 12.456 | 11.054 | 10.405 | 10.257 | 2.448 | 55.1 |
UPM 0049-7516 | 44833770 | 52993310 | 141.1 | 221.6 | 7.7 | 7.5 | 16.132 | 13.622 | 11.855 | 10.558 | 10.008 | 9.724 | 3.064 | 29.8 |
UPM 0050-8406 | 45673415 | 21181545 | 184.7 | 28.1 | 4.2 | 3.5 | $\cdots$ | $\cdots$ | $\cdots$ | 12.460 | 11.941 | 11.665 | $\cdots$ | $\cdots$ |
UPM 0052-7124 | 47330607 | 66954257 | 226.8 | 54.8 | 9.0 | 9.1 | 16.185 | 14.764 | 13.558 | 12.199 | 11.642 | 11.360 | 2.565 | 95.3 |
UPM 0055-8649 | 50082601 | 11435834 | 190.0 | 3.6 | 6.0 | 4.6 | 16.074 | 13.915 | 12.193 | 11.164 | 10.594 | 10.335 | 2.751 | 49.1 |
UPM 0106-6857 | 60014307 | 75757311 | 208.2 | -231.6 | 8.2 | 8.1 | 16.704 | 14.988 | 14.288 | 13.229 | 12.673 | 12.515 | 1.759 | [181.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0109-7523 | 62854828 | 52560670 | 163.3 | -80.0 | 3.6 | 6.6 | 17.846 | 15.536 | 13.984 | 12.662 | 12.070 | 11.892 | 2.874 | 91.2 |
UPM 0111-7655B | 64567429 | 47053087 | 182.5 | 35.6 | 4.9 | 5.0 | 18.581 | 16.585 | 15.107 | 13.369 | 12.772 | 12.531 | 3.216 | 109.9 | ccCommon proper motion companion; see Table 4
UPM 0111-7655A | 64749501 | 47075281 | 130.0 | 31.4 | 1.6 | 1.6 | 13.499 | 12.205 | 11.777 | 10.781 | 10.269 | 10.237 | 1.424 | 64.4 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 0116-7214 | 69103288 | 63920551 | 247.5 | -77.0 | 8.7 | 8.3 | 16.481 | 14.783 | 13.992 | 13.300 | 12.745 | 12.651 | 1.483 | [219.0] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0118-7606 | 70334783 | 49987168 | 177.9 | -34.1 | 2.4 | 2.8 | 17.350 | 15.149 | 13.999 | 13.075 | 12.395 | 12.278 | 2.074 | 154.5 |
UPM 0119-7347 | 71653364 | 58371207 | 222.9 | 28.3 | 8.7 | 8.7 | 17.291 | 15.254 | 13.531 | 12.177 | 11.624 | 11.376 | 3.077 | 70.0 |
UPM 0128-4754 | 79755558 | 151547143 | 120.9 | -188.5 | 7.3 | 7.0 | 15.375 | 13.287 | 11.314 | 10.210 | 9.640 | 9.384 | 3.077 | 28.0 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0133-7503 | 83912414 | 53819904 | 101.6 | -180.7 | 13.0 | 7.2 | 18.504 | 15.996 | 14.065 | 12.476 | 11.886 | 11.623 | 3.520 | 57.3 |
UPM 0135-6359 | 86122021 | 93629302 | 183.4 | -20.8 | 4.4 | 10.5 | 17.836 | 16.443 | 15.139 | 12.260 | 11.647 | 11.423 | 4.183 | 52.8 |
UPM 0152-8815 | 101325332 | 6293882 | 193.7 | -8.5 | 12.3 | 10.5 | 14.706 | 13.795 | 13.226 | 12.837 | 12.447 | 12.357 | 0.958 | 173.6 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0157-7324 | 106019256 | 59755191 | 183.6 | -39.7 | 2.7 | 2.7 | 16.972 | 15.144 | 14.468 | 13.599 | 12.931 | 12.849 | 1.545 | [230.6] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0204-5012 | 112153869 | 143253073 | 164.2 | 97.5 | 7.5 | 4.9 | 16.042 | 15.107 | 13.986 | 11.073 | 10.552 | 10.474 | 4.034 | 35.7 |
UPM 0216-5328 | 123095616 | 131482914 | 179.4 | 102.1 | 7.5 | 8.6 | 13.818 | 11.862 | 11.263 | 10.721 | 10.127 | 9.951 | 1.141 | 61.6 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0233-5122 | 137727579 | 139047453 | 259.3 | 117.5 | 11.4 | 11.2 | 16.907 | 15.392 | 14.597 | 13.798 | 13.198 | 12.963 | 1.594 | [234.4] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0245-8833A | 148994471 | 5191128 | 174.6 | 59.1 | 5.7 | 5.8 | 17.617 | 16.231 | $\cdots$ | 11.828 | 11.188 | 10.958 | 4.403 | 34.5 | ccCommon proper motion companion; see Table 4
UPM 0245-8833B | 149064071 | 5197674 | 175.7 | 89.7 | 6.5 | 6.7 | $\cdots$ | $\cdots$ | $\cdots$ | 11.726 | 11.071 | 10.870 | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 0320-4847B | 180208650 | 148355200 | 208.7 | 93.3 | 13.1 | 12.5 | 18.877 | 16.817 | 15.849 | 14.740 | 14.276 | 14.179 | 2.077 | 396.9 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 0320-4847A | 180270958 | 148347787 | 230.8 | 147.8 | 2.4 | 1.8 | 15.184 | 13.008 | 11.487 | 10.402 | 9.856 | 9.624 | 2.606 | 37.7 | ccCommon proper motion companion; see Table 4,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0342-5044 | 200622342 | 141302005 | 165.5 | -95.8 | 16.9 | 11.6 | $\cdots$ | 12.651 | 10.850 | 10.259 | 9.607 | 9.376 | 2.392 | 36.5 |
UPM 0350-5947 | 207350403 | 108760589 | 159.8 | -161.7 | 8.5 | 15.4 | 18.166 | 15.461 | 14.618 | 11.923 | 11.380 | 11.139 | 3.538 | 46.8 |
UPM 0352-7409 | 209621476 | 57009023 | 149.8 | 117.9 | 4.0 | 2.8 | 15.983 | 13.991 | 12.806 | 11.522 | 10.892 | 10.664 | 2.469 | 64.5 |
UPM 0356-7246 | 212775775 | 62017326 | 108.8 | 143.5 | 4.5 | 6.3 | 16.730 | 14.913 | 13.075 | 11.703 | 11.059 | 10.844 | 3.210 | 53.1 |
UPM 0418-6900 | 232848877 | 75566269 | -227.1 | -181.1 | 7.4 | 6.7 | 13.503 | 11.465 | 10.434 | 10.362 | 9.721 | 9.505 | 1.103 | 52.4 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0422-5114 | 236677358 | 139530080 | -162.2 | -140.7 | 8.8 | 8.3 | 16.833 | 14.644 | 12.724 | 11.870 | 11.292 | 11.042 | 2.774 | 67.8 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0424-6956 | 238195953 | 72180172 | 163.8 | 167.3 | 9.5 | 9.3 | 17.246 | 15.106 | 13.055 | 12.101 | 11.530 | 11.247 | 3.005 | 67.9 |
UPM 0426-7849 | 239769264 | 40214583 | 76.8 | 163.5 | 3.7 | 3.8 | 16.767 | 15.569 | 14.214 | 11.418 | 10.732 | 10.572 | 4.151 | 37.0 |
UPM 0435-8142 | 248223169 | 29825780 | 187.1 | 31.6 | 2.9 | 2.7 | 13.762 | 11.941 | 10.621 | 11.028 | 10.406 | 10.195 | 0.913 | 66.4 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0437-7232 | 249681505 | 62832156 | 104.8 | -167.1 | 2.8 | 2.8 | 18.041 | 14.183 | 16.621 | 12.016 | 11.365 | 11.131 | 2.167 | 63.8 |
UPM 0443-5236 | 254948887 | 134592463 | 105.7 | 176.7 | 5.5 | 8.2 | 15.377 | 13.074 | 11.399 | 10.479 | 9.914 | 9.640 | 2.595 | 37.2 |
UPM 0445-6544 | 257096346 | 87304149 | 44.6 | 180.0 | 10.0 | 8.2 | 14.420 | 12.835 | 11.824 | 11.819 | 11.202 | 11.025 | 1.016 | 107.8 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0447-6445 | 259144878 | 90863676 | 116.5 | 159.1 | 4.0 | 4.0 | 15.524 | 13.123 | 12.514 | 11.705 | 11.063 | 10.906 | 1.418 | 94.7 |
UPM 0450-4940 | 261002744 | 145179168 | 163.5 | -171.0 | 6.5 | 7.2 | 14.913 | 12.696 | 10.703 | 10.194 | 9.593 | 9.368 | 2.502 | 35.2 |
UPM 0457-4802 | 267813979 | 151048953 | 17.4 | 183.0 | 2.2 | 2.6 | 16.158 | 14.093 | 12.076 | 11.516 | 10.905 | 10.682 | 2.577 | 64.4 |
UPM 0458-6741 | 268276278 | 80291179 | 83.7 | 171.0 | 2.2 | 2.2 | 13.851 | 12.481 | 11.923 | 10.275 | 9.692 | 9.502 | 2.206 | 37.7 |
UPM 0459-6824 | 269511253 | 77743694 | 133.3 | -126.8 | 9.4 | 9.6 | 16.915 | 14.914 | 13.406 | 12.577 | 11.963 | 11.699 | 2.337 | 111.7 |
UPM 0510-4952 | 279040730 | 144454421 | 106.3 | 168.2 | 7.4 | 7.3 | 16.569 | 14.203 | 11.922 | 11.062 | 10.487 | 10.257 | 3.141 | 39.8 |
UPM 0514-7456 | 283007084 | 54221478 | -145.8 | -151.0 | 3.7 | 3.3 | 16.945 | 15.205 | 14.128 | 12.800 | 12.226 | 12.059 | 2.405 | 134.2 |
UPM 0514-4902 | 283498019 | 147437611 | -86.8 | -199.5 | 13.2 | 13.8 | 18.404 | 16.620 | 15.186 | 12.623 | 12.056 | 11.782 | 3.997 | 60.3 |
UPM 0518-4934 | 286357991 | 145552843 | 15.9 | -229.9 | 10.3 | 10.2 | 17.480 | 15.481 | 13.494 | 12.173 | 11.596 | 11.361 | 3.308 | 63.4 |
UPM 0521-5227 | 289705231 | 135166103 | 118.5 | 188.5 | 9.2 | 8.8 | 16.471 | 14.343 | 12.654 | 11.384 | 10.771 | 10.520 | 2.959 | 47.9 |
UPM 0530-5423 | 297516602 | 128202518 | 162.1 | -84.6 | 2.9 | 2.9 | 16.901 | 14.944 | 14.172 | 13.686 | 13.098 | 13.000 | 1.258 | [264.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0533-5210A | 300341870 | 136155536 | -80.5 | 180.7 | 3.4 | 3.3 | 16.348 | 13.641 | 12.161 | 11.350 | 10.762 | 10.512 | 2.291 | 58.2 | ccCommon proper motion companion; see Table 4
UPM 0533-5210B | 300345900 | 136148200 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 13.885 | 13.311 | 13.020 | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 0537-4924 | 303746224 | 146154602 | 196.2 | 42.2 | 9.1 | 9.0 | 16.734 | 14.567 | 12.263 | 11.299 | 10.787 | 10.488 | 3.268 | 43.7 |
UPM 0538-5313 | 304573280 | 132361506 | 244.6 | 90.0 | 7.4 | 7.2 | 16.220 | 14.231 | 12.810 | 12.202 | 11.640 | 11.450 | 2.029 | 114.2 |
UPM 0546-8356 | 311935231 | 21813762 | 39.8 | 185.9 | 12.9 | 12.9 | 16.962 | 15.380 | 13.325 | 12.060 | 11.498 | 11.219 | 3.320 | 65.3 |
UPM 0546-7124 | 312294377 | 66931727 | -55.1 | -176.9 | 6.1 | 6.2 | 16.395 | 15.073 | 13.380 | 11.792 | 11.285 | 11.011 | 3.281 | 63.8 |
UPM 0550-4839 | 315022816 | 148809418 | 55.3 | 283.7 | 13.2 | 11.2 | 14.508 | 12.328 | 10.676 | 10.576 | 10.015 | 9.769 | 1.752 | 54.7 |
UPM 0550-5557 | 315113926 | 122546698 | -14.8 | 208.1 | 16.0 | 13.7 | 18.420 | 16.668 | 15.148 | 12.467 | 11.897 | 11.602 | 4.201 | 51.7 |
UPM 0552-5648 | 317640102 | 119472712 | 4.3 | 203.5 | 12.6 | 13.3 | 18.140 | 16.090 | 14.761 | 13.133 | 12.574 | 12.362 | 2.957 | 114.7 |
UPM 0553-5014 | 317828071 | 143105615 | -52.7 | -180.0 | 10.9 | 10.5 | 16.940 | 14.790 | 12.751 | 11.813 | 11.335 | 11.017 | 2.977 | 63.3 |
UPM 0554-5009 | 319442059 | 143452591 | 45.4 | 214.7 | 10.6 | 9.7 | 16.264 | 14.063 | 11.271 | 10.784 | 10.199 | 9.890 | 3.279 | 33.9 |
UPM 0555-4932 | 319998070 | 145629559 | -112.4 | 213.2 | 12.6 | 11.6 | 17.018 | 14.941 | 12.272 | 11.646 | 11.058 | 10.703 | 3.295 | 49.1 |
UPM 0556-7215 | 321269817 | 63842657 | 58.9 | 176.9 | 1.4 | 1.3 | 12.742 | 10.698 | 9.700 | 9.268 | 8.590 | 8.380 | 1.430 | 30.5 |
UPM 0558-5358 | 322864429 | 129701481 | 110.3 | -200.4 | 10.9 | 10.4 | 16.171 | 14.204 | 12.863 | 11.834 | 11.328 | 11.103 | 2.370 | 86.4 |
UPM 0559-5225 | 323598169 | 135295015 | 120.7 | 152.6 | 4.5 | 4.4 | 14.532 | 12.000 | 10.287 | 9.628 | 8.951 | 8.703 | 2.372 | 24.9 |
UPM 0600-4707 | 324124205 | 154347926 | 3.0 | 223.7 | 8.9 | 8.7 | 16.447 | 14.358 | 12.375 | 11.194 | 10.635 | 10.374 | 3.164 | 42.5 |
UPM 0602-6209 | 326452097 | 100243766 | -57.4 | 171.1 | 1.9 | 1.9 | 16.507 | 14.476 | 12.695 | 11.805 | 11.198 | 10.993 | 2.671 | 71.0 |
UPM 0604-5054 | 327760875 | 140727784 | 179.0 | 32.7 | 2.6 | 2.6 | 14.600 | 12.440 | 10.549 | 10.265 | 9.659 | 9.444 | 2.175 | 41.6 |
UPM 0606-5342 | 329823432 | 130660944 | 141.7 | 125.4 | 13.6 | 11.9 | 17.659 | 15.758 | 14.102 | 13.026 | 12.523 | 12.269 | 2.732 | 129.6 |
UPM 0606-6524 | 329873509 | 88555612 | -27.1 | 182.5 | 7.3 | 7.4 | 17.434 | 15.078 | 13.889 | 12.288 | 11.747 | 11.522 | 2.790 | 78.8 |
UPM 0606-4907 | 330128370 | 147129405 | -85.7 | 165.2 | 16.3 | 15.9 | 17.839 | 15.654 | 13.470 | 12.532 | 11.980 | 11.747 | 3.122 | 82.3 |
UPM 0607-5751 | 330761393 | 115683690 | -96.8 | 198.5 | 11.2 | 11.2 | 17.343 | 15.150 | 13.871 | 13.063 | 12.444 | 12.293 | 2.087 | 157.6 |
UPM 0608-5301A | 331826617 | 133116470 | -131.1 | 208.6 | 12.2 | 10.8 | 14.341 | 12.234 | 10.865 | 10.046 | 9.431 | 9.217 | 2.188 | 37.1 | ccCommon proper motion companion; see Table 4
UPM 0612-5326 | 335637420 | 131623962 | -167.4 | -72.7 | 3.4 | 2.2 | 16.044 | 13.982 | 12.129 | 11.044 | 10.486 | 10.207 | 2.938 | 43.6 |
UPM 0619-4901 | 341823528 | 147490465 | 109.8 | 221.9 | 7.2 | 7.0 | 14.297 | 12.153 | 10.008 | 9.639 | 9.010 | 8.813 | 2.514 | 27.5 |
UPM 0621-6111 | 343586117 | 103726637 | 11.4 | 204.6 | 5.8 | 5.3 | 16.108 | 14.256 | 12.028 | 10.427 | 9.788 | 9.534 | 3.829 | 21.8 |
UPM 0632-6656 | 353475538 | 82985482 | -29.6 | 183.3 | 2.2 | 2.2 | 14.698 | 12.495 | 10.749 | 10.165 | 9.563 | 9.315 | 2.330 | 36.6 |
UPM 0636-6639 | 356884852 | 84035552 | -4.4 | 183.9 | 3.4 | 3.3 | 16.255 | 14.246 | 11.945 | 11.030 | 10.411 | 10.135 | 3.216 | 38.2 |
UPM 0636-7412 | 357254645 | 56842839 | -163.8 | 92.9 | 5.9 | 3.9 | 16.355 | 14.629 | 13.026 | 12.345 | 11.740 | 11.537 | 2.284 | 112.5 |
UPM 0638-5827 | 358555223 | 113559044 | 200.5 | -149.5 | 10.9 | 11.0 | 16.676 | 15.164 | 14.461 | 13.562 | 13.096 | 12.873 | 1.602 | [228.9] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0639-6849 | 359685140 | 76237418 | -40.9 | 182.9 | 8.3 | 8.1 | 16.409 | 14.289 | 12.750 | 11.592 | 11.026 | 10.812 | 2.697 | 63.0 |
UPM 0642-5012 | 362591504 | 143221516 | -20.5 | 239.7 | 7.3 | 7.2 | 15.084 | 13.703 | 13.051 | 12.763 | 12.317 | 12.141 | 0.940 | 159.9 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0644-7317 | 363638069 | 60127410 | -10.7 | 183.8 | 3.7 | 4.4 | 17.367 | 15.430 | 13.466 | 11.932 | 11.319 | 11.023 | 3.498 | 49.0 |
UPM 0646-5452 | 365990094 | 126421377 | 10.4 | -216.8 | 22.1 | 49.6 | 15.711 | 13.616 | 11.583 | 10.954 | 10.325 | 10.078 | 2.662 | 46.4 |
UPM 0649-5354 | 368443271 | 129948539 | 69.8 | 187.0 | 10.0 | 14.3 | 15.012 | 12.792 | 11.410 | 10.767 | 10.160 | 9.939 | 2.025 | 54.2 |
UPM 0659-7648 | 377784601 | 47493923 | 33.0 | 185.0 | 6.8 | 6.8 | 17.668 | 15.759 | 13.416 | 11.619 | 10.982 | 10.658 | 4.140 | 31.2 |
UPM 0711-5513 | 387967786 | 125160489 | -96.5 | 152.0 | 3.8 | 4.1 | 15.809 | 13.727 | 11.903 | 11.249 | 10.650 | 10.391 | 2.478 | 57.9 |
UPM 0713-5836 | 390581091 | 113012021 | -35.9 | 179.8 | 2.8 | 2.8 | 16.176 | 14.017 | 12.046 | 11.214 | 10.605 | 10.354 | 2.803 | 48.8 |
UPM 0720-5237 | 396383525 | 134568885 | 113.9 | -148.2 | 4.8 | 4.3 | 17.373 | 15.353 | 14.468 | 13.853 | 13.216 | 13.056 | 1.500 | [261.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0730-7501 | 405560351 | 53890475 | -49.5 | 197.9 | 27.9 | 11.2 | 17.706 | 16.172 | 14.101 | 12.401 | 11.822 | 11.539 | 3.771 | 61.3 |
UPM 0731-6642 | 406499377 | 83824278 | -130.0 | 127.8 | 2.5 | 2.4 | 12.675 | 10.861 | 10.080 | 9.856 | 9.195 | 9.011 | 1.005 | 41.9 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0731-7942 | 406797553 | 37071878 | -85.3 | 164.1 | 7.2 | 9.1 | 14.558 | 12.750 | 12.113 | 11.162 | 10.523 | 10.366 | 1.588 | 64.9 |
UPM 0734-7509 | 409044755 | 53455762 | -54.3 | 192.3 | 13.2 | 14.5 | 17.118 | 14.957 | 13.044 | 12.194 | 11.614 | 11.305 | 2.763 | 76.6 |
UPM 0739-8840 | 413322282 | 4758933 | -114.4 | 254.0 | 10.6 | 10.0 | 16.884 | 14.766 | $\cdots$ | 11.015 | 10.452 | 10.159 | 3.751 | 26.7 |
UPM 0740-5408 | 414078074 | 129084313 | -168.2 | -76.0 | 8.3 | 8.5 | 13.856 | 12.626 | 11.970 | 11.557 | 11.021 | 10.927 | 1.069 | 89.1 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0740-5207A | 414137868 | 136369560 | -37.7 | -221.5 | 10.4 | 9.6 | 17.906 | 16.477 | 14.997 | 11.898 | 11.292 | 11.058 | 4.579 | 38.9 | ccCommon proper motion companion; see Table 4
UPM 0740-5207B | 414151948 | 136382302 | 6.6 | -240.0 | 13.8 | 13.8 | 18.475 | 18.193 | 17.723 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,ggSource not in 2MASS
UPM 0747-6428 | 421129780 | 91870083 | 29.4 | 178.6 | 2.5 | 1.5 | 13.799 | 12.892 | 12.476 | 11.735 | 11.361 | 11.290 | 1.157 | 104.9 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0759-6658 | 431215147 | 82902895 | 5.2 | 191.4 | 11.6 | 12.5 | 18.734 | 16.654 | 15.144 | 12.797 | 12.223 | 11.967 | 3.857 | 64.1 |
UPM 0800-7205 | 432654941 | 64470732 | -24.8 | 248.1 | 9.0 | 9.0 | 16.397 | 14.513 | 12.843 | 11.533 | 10.958 | 10.703 | 2.980 | 55.0 |
UPM 0802-4926 | 434306031 | 146007969 | -196.5 | -35.5 | 10.6 | 11.4 | 15.982 | 14.237 | 12.336 | 11.304 | 10.766 | 10.531 | 2.933 | 55.3 |
UPM 0803-4955 | 435307518 | 144246351 | -118.6 | 166.7 | 3.1 | 3.1 | 14.957 | 14.873 | 12.586 | 12.535 | 11.880 | 11.774 | 2.338 | 113.3 |
UPM 0806-5903 | 437482203 | 111417097 | -85.0 | 163.4 | 5.8 | 5.7 | 16.791 | 14.877 | 13.182 | 11.872 | 11.273 | 11.018 | 3.005 | 62.1 |
UPM 0806-5409 | 437729739 | 129055037 | -221.3 | 255.4 | 9.8 | 12.6 | 17.265 | 15.305 | 13.290 | 12.030 | 11.507 | 11.211 | 3.275 | 61.4 |
UPM 0807-6913 | 438748524 | 74797001 | 157.4 | 107.6 | 7.9 | 6.0 | 13.404 | 11.430 | 10.211 | 9.541 | 8.878 | 8.692 | 1.889 | 32.8 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0811-6952 | 442531801 | 72439872 | 19.1 | -197.6 | 10.7 | 9.7 | 15.543 | 13.944 | 12.225 | 11.181 | 10.610 | 10.349 | 2.763 | 55.2 |
UPM 0815-4941 | 445774778 | 145093080 | -149.8 | 129.7 | 1.3 | 1.3 | 13.306 | 11.518 | 10.737 | 9.978 | 9.406 | 9.237 | 1.540 | 44.5 |
UPM 0823-4700 | 453122494 | 154795726 | 184.3 | -18.9 | 4.3 | 4.3 | 13.960 | 11.819 | 10.807 | 10.855 | 10.213 | 10.058 | 0.964 | 68.2 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0834-4832 | 463092742 | 149223005 | -40.8 | 178.8 | 1.9 | 1.8 | 15.441 | 13.279 | 11.683 | 11.452 | 10.788 | 10.558 | 1.827 | 76.6 |
UPM 0835-8332 | 463882202 | 23233252 | -74.5 | 169.3 | 2.9 | 6.9 | 14.641 | 12.652 | 11.481 | 10.870 | 10.329 | 10.130 | 1.782 | 66.7 |
UPM 0835-6716 | 463888892 | 81802666 | -146.1 | 194.8 | 7.9 | 7.5 | 18.175 | 16.354 | 14.711 | 11.834 | 11.286 | 11.017 | 4.520 | 35.1 |
UPM 0835-7247 | 464095507 | 61975334 | -61.3 | 170.1 | 7.2 | 2.2 | 17.097 | 15.168 | 13.609 | 12.364 | 11.820 | 11.586 | 2.804 | 89.5 |
UPM 0835-6418 | 464128846 | 92464726 | -91.0 | 165.1 | 7.9 | 8.0 | 16.860 | 15.270 | 14.489 | 13.616 | 12.988 | 12.838 | 1.654 | [217.3] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0835-6018C | 464201386 | 106885496 | -184.6 | -17.7 | 8.9 | 11.2 | $\cdots$ | 15.671 | 13.904 | 12.162 | 11.608 | 11.318 | 3.509 | 50.3 | ccCommon proper motion companion; see Table 4,eeProper motions suspect
UPM 0837-6435B | 465804750 | 91491299 | -29.6 | 158.4 | 5.8 | 6.0 | 18.701 | 16.753 | 14.850 | 13.323 | 12.829 | 12.561 | 3.430 | 106.6 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 0837-6435A | 465826276 | 91489399 | -46.7 | 175.2 | 4.2 | 3.1 | 18.204 | 16.568 | 14.644 | 12.203 | 11.658 | 11.455 | 4.365 | 46.0 | ccCommon proper motion companion; see Table 4
UPM 0838-4935 | 466900627 | 145482276 | -191.0 | -39.4 | 5.9 | 6.0 | 15.071 | 13.044 | 11.663 | 11.255 | 10.658 | 10.491 | 1.789 | 77.9 |
UPM 0846-7345B | 473694134 | 58464369 | -125.4 | 112.9 | 3.0 | 3.3 | 15.708 | 13.930 | 12.802 | 11.789 | 11.205 | 10.965 | 2.141 | 88.4 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 0846-7345A | 473701026 | 58488516 | -129.3 | 127.6 | 2.9 | 3.1 | 15.702 | 13.617 | 12.348 | 11.330 | 10.710 | 10.430 | 2.287 | 61.6 | ccCommon proper motion companion; see Table 4
UPM 0847-5952 | 474714326 | 108445130 | -76.0 | 164.0 | 4.4 | 3.2 | 17.254 | 15.190 | 13.486 | 12.638 | 12.063 | 11.818 | 2.552 | 108.7 |
UPM 0847-6114 | 475060646 | 103508456 | -139.5 | 113.8 | 6.7 | 6.4 | 17.554 | 15.670 | 13.833 | 13.069 | 12.548 | 12.297 | 2.601 | 140.6 |
UPM 0849-5624 | 476802766 | 120928221 | 183.9 | -34.2 | 17.1 | 5.4 | 17.324 | 15.253 | 13.668 | 12.312 | 11.730 | 11.476 | 2.941 | 76.2 |
UPM 0850-5609 | 477752730 | 121836946 | -89.3 | 161.7 | 2.2 | 8.1 | 17.705 | 16.336 | 14.428 | 11.673 | 11.054 | 10.899 | 4.663 | 34.1 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0853-7257 | 480561006 | 61351874 | -163.1 | 85.0 | 7.7 | 6.9 | 16.464 | 14.521 | 12.993 | 11.935 | 11.315 | 11.091 | 2.586 | 76.8 |
UPM 0855-7628 | 481940284 | 48709869 | -169.7 | 64.5 | 5.7 | 7.8 | 17.305 | 15.711 | 14.140 | 12.948 | 12.418 | 12.193 | 2.763 | 130.1 |
UPM 0856-7737 | 483109280 | 44538257 | -80.3 | 233.0 | 3.4 | 3.4 | 16.221 | 15.091 | 13.332 | 11.754 | 11.121 | 10.873 | 3.337 | 59.1 |
UPM 0857-5644 | 484130611 | 119756593 | -62.0 | -187.2 | 3.5 | 3.5 | 15.015 | 13.200 | 12.269 | 11.960 | 11.492 | 11.223 | 1.240 | 117.2 |
UPM 0900-7548 | 486663656 | 51069120 | -269.7 | 63.8 | 16.1 | 10.5 | 16.230 | 15.592 | 13.725 | 12.571 | 12.029 | 11.820 | 3.021 | 116.2 |
UPM 0901-6526 | 487375530 | 88399668 | -68.4 | 196.4 | 6.7 | 6.7 | 16.269 | 14.219 | 11.471 | 10.140 | 9.589 | 9.282 | 4.079 | 18.0 |
UPM 0904-5040 | 489923010 | 141595163 | -148.7 | 151.6 | 11.4 | 16.0 | 18.151 | 16.823 | 14.978 | 13.008 | 12.436 | 12.204 | 3.815 | 86.3 |
UPM 0904-7300 | 490461535 | 61175505 | -139.5 | 115.7 | 2.0 | 2.0 | 16.181 | 14.199 | 12.448 | 11.242 | 10.623 | 10.367 | 2.957 | 46.2 |
UPM 0907-7337 | 492965065 | 58937692 | -128.1 | 139.1 | 7.1 | 7.0 | 14.938 | 12.962 | 11.817 | 11.052 | 10.509 | 10.264 | 1.910 | 68.2 |
UPM 0908-5735 | 493590843 | 116658425 | -128.8 | 178.3 | 14.0 | 8.7 | 15.670 | 15.333 | 13.381 | 12.791 | 12.208 | 11.987 | 2.542 | 118.8 |
UPM 0912-5501 | 497648346 | 125899556 | -181.5 | 88.6 | 4.0 | 4.2 | 15.179 | 14.259 | 11.439 | 11.944 | 11.395 | 11.177 | 2.315 | 77.1 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0913-6058 | 497794860 | 104510459 | 23.8 | -189.1 | 2.6 | 2.3 | 13.487 | 12.365 | 10.861 | 11.696 | 11.171 | 10.908 | 0.669 | 94.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0913-6333 | 497821252 | 95208293 | -176.3 | -65.2 | 7.4 | 7.1 | 16.840 | 15.043 | 14.283 | 13.665 | 13.160 | 12.915 | 1.378 | [251.0] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0913-5405 | 497914131 | 129246075 | 129.3 | -163.0 | 9.6 | 9.6 | 16.199 | 15.920 | 12.810 | 12.957 | 12.365 | 12.123 | 2.963 | 98.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0915-5930 | 500081336 | 109741135 | 101.1 | 195.8 | 10.8 | 10.7 | $\cdots$ | 15.704 | $\cdots$ | 14.063 | 13.562 | 13.349 | 1.641 | [313.3] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0915-5515 | 500370580 | 125053511 | 107.8 | -182.2 | 6.5 | 6.8 | 15.996 | 14.231 | 11.880 | 11.645 | 10.957 | 10.736 | 2.586 | 68.9 |
UPM 0917-4707 | 501408590 | 154348882 | -175.8 | -171.2 | 3.6 | 3.0 | 15.396 | 13.501 | 11.295 | 10.304 | 9.806 | 9.544 | 3.197 | 31.0 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0919-6205 | 503552308 | 100495615 | 204.0 | 52.6 | 7.1 | 7.0 | 15.583 | 13.720 | 11.948 | 11.703 | 11.070 | 10.854 | 2.017 | 87.2 |
UPM 0924-7319 | 507837850 | 60043020 | -173.0 | 76.6 | 5.2 | 5.3 | 18.285 | 16.869 | 15.047 | 12.394 | 11.906 | 11.604 | 4.475 | 50.5 |
UPM 0927-4925 | 510592049 | 146043500 | -226.1 | 68.0 | 7.0 | 7.0 | 19.157 | 19.607 | 20.120 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ddNot detected during automated search but noticed by eye during the blinking process,ggSource not in 2MASS
UPM 0928-5442A | 511346324 | 127068710 | -129.4 | 137.3 | 6.3 | 6.1 | 14.539 | 12.199 | 10.397 | 10.702 | 10.072 | 9.849 | 1.497 | 57.1 | ccCommon proper motion companion; see Table 4
UPM 0928-5442B | 511392300 | 127069700 | -91.0 | 147.9 | 7.5 | 7.8 | 20.410 | 18.880 | 16.134 | 14.150 | 13.607 | 13.295 | 4.730 | 93.0 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect,hhSuperCOSMOS plate magnitudes suspect
UPM 0928-4736 | 511977850 | 152615832 | -151.2 | 132.9 | 4.0 | 6.7 | 18.129 | 16.786 | 14.890 | 12.509 | 11.889 | 11.664 | 4.277 | 55.1 |
UPM 0931-5214 | 514422292 | 135948112 | -182.9 | -46.6 | 3.4 | 4.3 | 14.343 | 12.375 | 11.129 | 11.335 | 10.735 | 10.499 | 1.040 | 77.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0934-8226 | 516828159 | 27228505 | -134.3 | 124.9 | 3.5 | 3.4 | 17.326 | 15.196 | 13.416 | 12.173 | 11.634 | 11.357 | 3.023 | 70.1 |
UPM 0946-5721 | 527727964 | 117517068 | 194.4 | -6.3 | 12.3 | 7.3 | 15.794 | 13.902 | 12.640 | 12.623 | 11.989 | 11.810 | 1.279 | [150.9] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 0948-5807 | 529927201 | 114732425 | -249.0 | 161.6 | 12.8 | 12.8 | 16.700 | 14.743 | 12.547 | 12.154 | 11.639 | 11.406 | 2.589 | 93.8 |
UPM 0948-4840 | 530079096 | 148753388 | -172.0 | 55.8 | 2.9 | 5.1 | 14.042 | 12.832 | 12.555 | 12.466 | 12.030 | 11.933 | 0.366 | 150.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 0952-5004 | 532988430 | 143714640 | 144.1 | 122.4 | 4.0 | 4.0 | 16.780 | 13.537 | 13.458 | 12.811 | 12.163 | 11.957 | 0.726 | 144.6 |
UPM 0952-5404 | 533479000 | 129344116 | 33.5 | -262.1 | 9.7 | 9.5 | 14.935 | 12.574 | 10.116 | 9.993 | 9.324 | 9.098 | 2.581 | 29.0 |
UPM 0959-6511 | 539778440 | 89339446 | -229.7 | 47.3 | 9.2 | 9.3 | 16.281 | 15.042 | 13.103 | 11.857 | 11.317 | 11.117 | 3.185 | 72.6 |
UPM 1001-7450 | 540996018 | 54585231 | -49.3 | 175.2 | 3.7 | 4.2 | 17.005 | 15.147 | 12.911 | 11.181 | 10.596 | 10.296 | 3.966 | 29.3 |
UPM 1002-5346 | 542138816 | 130406875 | -211.8 | 85.4 | 14.4 | 15.3 | 17.471 | 15.616 | 13.425 | 12.588 | 12.082 | 11.881 | 3.028 | 100.0 |
UPM 1004-7143 | 543907730 | 65807409 | -154.7 | 107.0 | 1.5 | 3.0 | 13.698 | 11.777 | 10.860 | 9.849 | 9.238 | 9.066 | 1.928 | 38.7 |
UPM 1007-5654 | 546325232 | 119102081 | -178.0 | 66.5 | 4.5 | 4.6 | 17.079 | 15.025 | 13.362 | 12.746 | 12.177 | 11.925 | 2.279 | 128.2 |
UPM 1015-6848 | 554027327 | 76286632 | -183.1 | 40.0 | 7.4 | 7.6 | 16.784 | 14.899 | 13.950 | 12.921 | 12.309 | 12.085 | 1.978 | 152.7 |
UPM 1016-5354 | 554981277 | 129915776 | 153.2 | -120.3 | 4.5 | 4.4 | 16.038 | 14.239 | 13.999 | 13.325 | 12.707 | 12.543 | 0.914 | 165.2 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1016-8230 | 555178676 | 26962130 | -242.7 | 49.6 | 12.4 | 10.6 | 17.666 | 15.813 | 14.179 | 12.622 | 12.006 | 11.790 | 3.191 | 81.9 |
UPM 1018-5540 | 556901250 | 123585233 | -231.3 | 106.5 | 7.2 | 6.9 | 15.108 | 12.714 | 10.596 | 10.142 | 9.605 | 9.313 | 2.572 | 32.7 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1020-5039 | 558143896 | 141657641 | -273.1 | 156.0 | 5.5 | 5.5 | 15.783 | 13.462 | 11.592 | 10.903 | 10.321 | 10.093 | 2.559 | 47.0 |
UPM 1024-5014 | 561747625 | 143139523 | -254.2 | 35.8 | 8.2 | 7.6 | 17.315 | 15.265 | 13.224 | 12.364 | 11.820 | 11.647 | 2.901 | 89.7 |
UPM 1025-6853 | 563154280 | 76012205 | -232.9 | -54.0 | 8.2 | 7.7 | 16.338 | 14.232 | 12.254 | 10.835 | 10.243 | 9.967 | 3.397 | 30.8 |
UPM 1026-5220 | 563859528 | 135555177 | -142.7 | 145.1 | 18.7 | 18.7 | 15.783 | 13.573 | 11.631 | 10.719 | 10.224 | 9.928 | 2.854 | 39.7 |
UPM 1029-4717 | 566272965 | 153747533 | -191.1 | 78.1 | 31.5 | 9.0 | 18.142 | 16.158 | 14.562 | 13.301 | 12.824 | 12.561 | 2.857 | 137.8 |
UPM 1029-5833 | 566322949 | 113212773 | -140.3 | 145.3 | 4.5 | 4.5 | 15.877 | 13.186 | 11.789 | 11.224 | 10.638 | 10.395 | 1.962 | 63.3 |
UPM 1030-5522 | 567848867 | 124674624 | -170.2 | 72.1 | 13.8 | 12.0 | 16.693 | 14.689 | 12.425 | 11.484 | 10.849 | 10.609 | 3.205 | 47.6 |
UPM 1033-5703 | 570262042 | 118605552 | -199.0 | -93.6 | 11.3 | 11.5 | 17.406 | 15.678 | 13.149 | 12.258 | 11.753 | 11.469 | 3.420 | 72.1 |
UPM 1034-5524 | 571158773 | 124522000 | -197.0 | -21.6 | 6.0 | 6.0 | 14.270 | 12.104 | 10.735 | 10.789 | 10.171 | 10.007 | 1.315 | 64.3 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1037-7107 | 574194223 | 67973139 | -209.1 | -.1 | 8.2 | 8.4 | 17.147 | 15.246 | 13.842 | 12.789 | 12.274 | 12.049 | 2.457 | 130.5 |
UPM 1039-6147 | 575499606 | 101537935 | -197.4 | 64.0 | 7.1 | 7.2 | 16.967 | 14.621 | 12.133 | 11.414 | 10.764 | 10.531 | 3.207 | 44.0 |
UPM 1039-4757 | 575999373 | 151362747 | -172.1 | 67.6 | 6.3 | 4.7 | 18.025 | 15.870 | 13.756 | 12.701 | 12.107 | 11.823 | 3.169 | 81.3 |
UPM 1040-5621 | 576632314 | 121134540 | -173.7 | 61.3 | 13.3 | 13.3 | 16.539 | 15.734 | 14.579 | 13.989 | 13.397 | 13.205 | 1.745 | [259.3] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,hhSuperCOSMOS plate magnitudes suspect
UPM 1040-5728A | 576899309 | 117072931 | -151.4 | 131.1 | 5.1 | 5.9 | 17.195 | 15.395 | 13.254 | 12.448 | 11.828 | 11.607 | 2.947 | 88.8 | ccCommon proper motion companion; see Table 4
UPM 1040-5728B | 576912600 | 117071399 | -224.2 | 138.9 | 5.2 | 5.6 | 18.382 | 16.178 | 13.855 | 13.298 | 12.732 | 12.435 | 2.880 | 164.7 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,hhSuperCOSMOS plate magnitudes suspect
UPM 1041-5743 | 576937649 | 116163103 | -162.2 | 140.3 | 6.8 | 6.9 | 16.863 | 14.599 | 12.833 | 11.986 | 11.468 | 11.254 | 2.613 | 80.4 |
UPM 1044-7053A | 580394596 | 68773915 | -217.5 | 17.6 | 7.3 | 7.4 | 16.376 | 14.437 | 12.659 | 11.564 | 10.952 | 10.724 | 2.873 | 57.8 | ccCommon proper motion companion; see Table 4
UPM 1044-7053B | 580396649 | 68739099 | -217.5 | 36.5 | 8.4 | 8.5 | 20.085 | 17.841 | 15.418 | 13.843 | 13.250 | 12.957 | 3.998 | 92.3 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 1049-5024 | 584145300 | 142551809 | -177.3 | -56.6 | 4.8 | 4.4 | 17.250 | 15.168 | 13.358 | 11.933 | 11.301 | 11.067 | 3.235 | 54.6 |
UPM 1051-6453 | 586032816 | 90413577 | -182.4 | 85.1 | 7.4 | 6.8 | 16.646 | 15.014 | 13.501 | 12.117 | 11.566 | 11.333 | 2.897 | 80.9 |
UPM 1053-6848 | 587914658 | 76268649 | 100.5 | 167.3 | 5.2 | 5.3 | 15.903 | 14.550 | 13.447 | 13.255 | 12.751 | 12.637 | 1.295 | [218.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1054-5945 | 588842452 | 108899458 | -244.9 | 52.1 | 8.8 | 9.4 | 17.515 | 15.478 | 13.363 | 12.059 | 11.462 | 11.212 | 3.419 | 55.7 |
UPM 1055-5934 | 590266470 | 109511211 | -217.8 | 99.6 | 7.3 | 7.3 | 17.227 | 14.861 | 12.711 | 11.615 | 10.974 | 10.715 | 3.246 | 44.7 |
UPM 1056-5750 | 590521836 | 115757473 | -194.4 | 121.9 | 6.9 | 7.4 | 17.175 | 14.966 | 12.873 | 11.339 | 10.727 | 10.450 | 3.627 | 33.7 |
UPM 1057-5048 | 591643375 | 141061116 | -116.6 | 164.1 | 3.8 | 3.8 | 16.024 | 12.974 | 11.947 | 11.375 | 10.766 | 10.567 | 1.599 | 72.8 |
UPM 1058-5516 | 592209732 | 125011151 | -57.8 | 204.2 | 4.5 | 4.5 | $\cdots$ | $\cdots$ | $\cdots$ | 13.132 | 12.615 | 12.357 | $\cdots$ | $\cdots$ |
UPM 1100-6615 | 594036873 | 85493226 | 181.0 | -6.1 | 40.0 | 7.8 | 19.612 | 14.988 | 17.336 | 13.231 | 12.665 | 12.383 | 1.757 | 97.0 |
UPM 1101-6112 | 595586097 | 103635531 | -184.4 | 62.3 | 6.7 | 7.2 | 16.499 | 14.183 | 12.383 | 11.977 | 11.450 | 11.232 | 2.206 | 92.8 |
UPM 1101-6918 | 595737805 | 74472734 | -170.1 | 78.2 | 9.4 | 5.4 | 15.700 | 15.057 | 13.623 | 13.080 | 12.398 | 12.229 | 1.977 | 153.7 |
UPM 1104-6232 | 598106935 | 98845495 | -207.8 | -58.5 | 6.0 | 6.2 | 16.423 | 14.086 | 11.484 | 10.256 | 9.677 | 9.357 | 3.830 | 19.4 |
UPM 1104-7107A | 598372077 | 67942248 | -197.3 | -83.8 | 7.7 | 7.7 | 17.740 | 16.102 | 14.741 | 12.287 | 11.770 | 11.545 | 3.815 | 60.9 | ccCommon proper motion companion; see Table 4
UPM 1104-7107B | 598388051 | 67951710 | 20.3 | 23.1 | 3.5 | 5.9 | 15.524 | 13.595 | 15.033 | 12.855 | 12.444 | 12.183 | 0.740 | 44.6 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 1105-5825A | 598956034 | 113681441 | -176.1 | 51.5 | 4.2 | 4.2 | 16.916 | $\cdots$ | 13.713 | 10.298 | 9.711 | 9.496 | $\cdots$ | 15.2 | ccCommon proper motion companion; see Table 4,hhSuperCOSMOS plate magnitudes suspect
UPM 1105-5825B | 599009180 | 113697850 | -141.8 | 56.9 | 12.4 | 6.6 | 17.721 | 15.259 | 13.109 | 11.615 | 11.070 | 10.796 | 3.644 | 38.4 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 1107-7032 | 600625006 | 70069523 | -192.8 | 134.6 | 4.8 | 3.5 | 15.169 | 13.238 | 11.793 | 11.129 | 10.513 | 10.258 | 2.109 | 63.5 |
UPM 1109-8518 | 602953600 | 16907251 | -189.7 | 35.3 | 3.1 | 3.1 | 16.926 | 14.899 | 13.318 | 11.652 | 11.113 | 10.801 | 3.247 | 49.0 |
UPM 1111-6653 | 604096236 | 83206908 | -182.5 | 13.6 | 7.1 | 6.9 | 17.073 | 15.007 | 13.596 | 12.187 | 11.629 | 11.392 | 2.820 | 77.9 |
UPM 1111-6014 | 604347021 | 107105218 | -195.7 | 52.2 | 5.3 | 5.3 | 17.103 | 14.488 | 11.285 | 12.775 | 12.171 | 11.959 | 1.713 | 121.4 |
UPM 1111-5903 | 604421714 | 111381636 | -144.2 | 109.1 | 3.9 | 16.3 | 14.777 | 12.424 | 10.606 | 11.144 | 10.541 | 10.293 | 1.280 | 72.3 |
UPM 1112-4834 | 605254455 | 149144350 | -245.3 | 106.6 | 10.1 | 10.0 | 18.765 | 16.634 | 14.154 | 11.953 | 11.352 | 11.035 | 4.681 | 28.8 |
UPM 1112-5246 | 605337027 | 134015761 | -33.2 | -181.6 | 3.7 | 3.3 | 17.313 | 14.887 | 13.078 | 12.705 | 12.109 | 11.844 | 2.182 | 119.1 |
UPM 1112-5551 | 605392402 | 122918291 | -212.5 | 67.0 | 7.9 | 6.0 | 14.114 | 11.879 | 10.866 | 11.008 | 10.356 | 10.182 | 0.871 | 71.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1113-4908 | 605758353 | 147100509 | -196.7 | 32.0 | 6.8 | 6.8 | 15.399 | 14.112 | 13.662 | 12.671 | 12.164 | 12.048 | 1.441 | [147.3] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1113-5546 | 605999181 | 123199403 | -174.7 | 56.3 | 3.2 | 3.1 | 15.809 | 13.571 | 11.484 | 11.180 | 10.589 | 10.304 | 2.391 | 56.2 |
UPM 1120-7753 | 612430190 | 43574189 | -178.5 | 38.5 | 3.0 | 3.4 | 16.632 | 14.920 | 13.758 | 12.063 | 11.388 | 11.134 | 2.857 | 70.1 |
UPM 1122-4916 | 613897272 | 146609505 | -184.2 | 34.4 | 6.1 | 9.6 | 17.581 | 15.504 | 13.135 | 12.255 | 11.714 | 11.483 | 3.249 | 71.7 |
UPM 1122-7946 | 614301816 | 36791042 | -181.4 | 35.5 | 2.8 | 3.0 | 15.309 | 13.424 | 12.158 | 11.101 | 10.526 | 10.224 | 2.323 | 57.7 |
UPM 1123-4839 | 615142692 | 148852167 | -181.3 | 83.0 | 3.7 | 3.7 | $\cdots$ | $\cdots$ | $\cdots$ | 11.281 | 10.698 | 10.362 | $\cdots$ | $\cdots$ |
UPM 1125-5127 | 616862846 | 138732194 | 170.6 | 147.4 | 4.5 | 4.5 | 17.136 | 13.432 | 13.395 | 12.728 | 12.159 | 11.930 | 0.704 | 129.5 |
UPM 1134-6455 | 625214057 | 90272065 | -186.6 | 2.3 | 4.5 | 10.5 | 14.941 | 12.946 | 11.645 | 10.527 | 9.953 | 9.681 | 2.419 | 42.6 |
UPM 1135-5554 | 625712836 | 122753701 | -200.8 | -116.8 | 18.5 | 9.4 | 15.127 | 13.022 | 11.497 | 11.248 | 10.698 | 10.497 | 1.774 | 77.7 |
UPM 1136-5358A | 626495784 | 129716980 | -215.3 | -74.1 | 6.9 | 6.7 | 14.772 | 12.473 | 10.510 | 10.206 | 9.574 | 9.302 | 2.267 | 36.3 | ccCommon proper motion companion; see Table 4,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1136-5358B | 626496300 | 129732900 | -247.7 | -69.6 | 6.6 | 6.4 | 19.322 | 18.182 | 17.904 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,ggSource not in 2MASS,hhSuperCOSMOS plate magnitudes suspect
UPM 1142-6440 | 632248846 | 91165839 | -183.4 | -205.1 | 32.2 | 16.8 | 16.067 | 14.081 | 11.418 | 10.393 | 9.846 | 9.481 | 3.688 | 23.7 |
UPM 1142-6344 | 632535830 | 94528424 | -259.0 | 40.0 | 7.5 | 7.3 | 15.965 | 13.490 | 10.971 | 11.583 | 10.985 | 10.800 | 1.907 | 72.0 |
UPM 1143-5324 | 632832624 | 131734543 | -188.4 | 120.7 | 2.9 | 18.4 | 13.722 | 11.666 | 10.146 | 10.058 | 9.463 | 9.242 | 1.608 | 45.1 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1143-4836 | 633486209 | 149024797 | -119.3 | 165.6 | 3.1 | 3.1 | 15.959 | 13.889 | 12.573 | 12.733 | 12.228 | 12.000 | 1.156 | [155.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1144-4923 | 633657394 | 146191433 | -277.6 | -16.1 | 8.4 | 8.1 | 15.894 | 13.774 | 12.050 | 11.847 | 11.229 | 11.032 | 1.927 | 93.9 |
UPM 1144-5557 | 633896464 | 122551085 | -245.2 | 37.0 | 7.1 | 7.1 | 17.383 | 15.306 | 13.222 | 12.097 | 11.538 | 11.266 | 3.209 | 63.3 |
UPM 1147-4733 | 636701757 | 152783368 | -206.3 | 126.8 | 11.3 | 11.3 | 17.985 | 15.817 | 13.616 | 12.055 | 11.537 | 11.254 | 3.762 | 48.0 |
UPM 1149-7948 | 638766897 | 36660917 | -203.2 | 38.5 | 10.4 | 10.4 | 16.103 | 15.183 | 14.614 | 13.205 | 12.729 | 12.637 | 1.978 | 159.8 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1152-4906 | 641000199 | 147194325 | 190.9 | 10.2 | 8.4 | 8.3 | $\cdots$ | $\cdots$ | $\cdots$ | 11.818 | 11.247 | 10.996 | $\cdots$ | $\cdots$ |
UPM 1157-4902 | 645382627 | 147472855 | -182.0 | -5.2 | 5.0 | 6.4 | 17.658 | 15.389 | 13.655 | 12.831 | 12.271 | 12.048 | 2.558 | 116.8 |
UPM 1158-4740 | 646653103 | 152372902 | -188.8 | 35.4 | 6.9 | 7.9 | 17.845 | 16.206 | 14.692 | 12.675 | 12.074 | 11.872 | 3.531 | 77.2 |
UPM 1200-6048 | 648207506 | 105083449 | -179.9 | 32.5 | 3.6 | 15.0 | 14.999 | 12.193 | 10.693 | 11.014 | 10.469 | 10.258 | 1.179 | 70.1 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1201-6030 | 648904878 | 106182272 | -184.4 | -14.8 | 6.3 | 6.3 | 17.271 | 15.315 | 12.937 | 12.251 | 11.654 | 11.423 | 3.064 | 76.7 |
UPM 1203-4910B | 651090352 | 146962333 | -177.1 | -40.5 | 4.2 | 4.2 | 15.446 | 13.202 | 11.735 | 11.692 | 11.124 | 10.914 | 1.510 | 97.5 | ccCommon proper motion companion; see Table 4
UPM 1204-7506 | 651612980 | 53600108 | -265.4 | 48.9 | 6.7 | 6.6 | 15.549 | 13.867 | 12.272 | 10.779 | 10.221 | 9.999 | 3.088 | 39.8 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1208-6352 | 655915161 | 94051538 | -180.8 | 11.6 | 3.9 | 3.9 | 12.899 | 10.962 | 10.047 | 11.126 | 10.510 | 10.283 | -0.164 | 81.3 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,hhSuperCOSMOS plate magnitudes suspect
UPM 1210-5144 | 657807554 | 137758940 | -175.3 | 64.4 | 7.0 | 10.8 | 12.323 | 10.599 | 9.736 | 10.203 | 9.530 | 9.365 | 0.396 | 55.3 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1211-5740 | 658107210 | 116377531 | -72.8 | 167.5 | 10.9 | 8.5 | 17.816 | 16.003 | 14.320 | 13.537 | 12.989 | 12.756 | 2.466 | 183.9 |
UPM 1211-4738 | 658674798 | 152461987 | -186.6 | 18.1 | 2.8 | 2.8 | 17.353 | 15.654 | 14.772 | 13.855 | 13.243 | 13.044 | 1.799 | [237.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1218-5642 | 664279147 | 119833488 | -181.2 | -13.0 | 8.0 | 7.9 | 15.224 | 13.229 | 11.622 | 10.975 | 10.407 | 10.178 | 2.254 | 58.6 |
UPM 1219-5935 | 665312081 | 109493550 | -182.5 | 139.5 | 4.2 | 4.1 | 16.132 | 15.777 | 13.703 | 12.731 | 12.174 | 11.938 | 3.046 | 93.9 |
UPM 1219-5639 | 665642490 | 120010019 | -214.6 | 8.3 | 7.1 | 7.0 | 17.522 | 15.398 | 14.005 | 11.404 | 10.748 | 10.520 | 3.994 | 31.0 |
UPM 1221-5118 | 667250601 | 139300531 | -239.0 | -102.7 | 8.5 | 9.0 | 16.943 | 15.688 | 13.803 | 12.740 | 12.240 | 12.037 | 2.948 | 123.3 |
UPM 1221-5305 | 667280397 | 132879517 | 181.8 | -132.5 | 3.6 | 3.6 | 14.393 | 12.853 | 11.881 | 12.836 | 12.249 | 12.091 | 0.017 | 188.7 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,hhSuperCOSMOS plate magnitudes suspect
UPM 1224-5429 | 669796204 | 127850432 | -181.6 | 7.1 | 4.1 | 4.1 | 16.525 | 14.619 | 13.884 | 12.922 | 12.232 | 12.096 | 1.697 | 164.0 |
UPM 1224-5909 | 669916109 | 111047246 | -194.8 | -52.8 | 3.4 | 3.4 | 14.920 | 13.632 | 11.792 | 11.702 | 11.073 | 10.887 | 1.930 | 87.2 |
UPM 1227-6342 | 672319589 | 94676890 | 174.5 | -44.9 | 4.6 | 12.5 | 16.333 | 14.224 | 12.635 | 12.742 | 12.210 | 12.039 | 1.482 | 161.6 |
UPM 1228-5146 | 673673311 | 137589485 | -190.6 | -17.4 | 8.3 | 5.3 | 17.393 | 15.609 | 13.787 | 12.722 | 12.165 | 11.925 | 2.887 | 105.2 |
UPM 1229-5139 | 674128500 | 138032073 | -239.8 | -122.1 | 7.4 | 7.3 | 13.836 | 13.582 | 13.083 | 12.724 | 12.372 | 12.235 | 0.858 | 165.5 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1229-6033 | 674141823 | 105973539 | -183.9 | 2.6 | 7.0 | 7.1 | 16.974 | 14.994 | 13.107 | 12.440 | 11.968 | 11.709 | 2.554 | 108.9 |
UPM 1230-5736B | 675249073 | 116560461 | -243.0 | -29.3 | 6.9 | 6.5 | 14.928 | 12.828 | 10.998 | 9.694 | 9.033 | 8.785 | 3.134 | 19.8 | ccCommon proper motion companion; see Table 4,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1230-5736A | 675783245 | 116623954 | -227.6 | -66.5 | 6.8 | 6.5 | 13.794 | 12.035 | 10.550 | 9.348 | 8.712 | 8.445 | 2.687 | 22.2 | ccCommon proper motion companion; see Table 4,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1233-5438 | 678078870 | 127263281 | -67.0 | 173.0 | 3.8 | 3.6 | 15.403 | 13.583 | 12.441 | 11.565 | 10.990 | 10.792 | 2.018 | 85.3 |
UPM 1239-7327 | 683401087 | 59573302 | -180.1 | -29.1 | 3.0 | 3.0 | 16.017 | $\cdots$ | 10.602 | 10.963 | 10.328 | 10.089 | $\cdots$ | 34.6 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,hhSuperCOSMOS plate magnitudes suspect
UPM 1239-6228 | 683531021 | 99084175 | 212.3 | -155.5 | 8.5 | 9.1 | 15.513 | 15.123 | 12.290 | 12.025 | 11.419 | 11.185 | 3.098 | 78.1 |
UPM 1240-5637 | 684685297 | 120146483 | -245.8 | -139.0 | 6.9 | 6.6 | 14.348 | 13.818 | 12.542 | 11.489 | 10.932 | 10.712 | 2.329 | 66.0 |
UPM 1243-6904 | 687517359 | 75354839 | -198.3 | -76.6 | 10.6 | 12.6 | 17.311 | 15.316 | 13.542 | 12.819 | 12.303 | 12.038 | 2.497 | 127.2 |
UPM 1248-6018 | 691205255 | 106888292 | 187.3 | 4.1 | 20.9 | 13.0 | 15.923 | 13.444 | 11.840 | 11.311 | 10.782 | 10.503 | 2.133 | 65.6 |
UPM 1248-6825 | 691899930 | 77658523 | -185.0 | -39.5 | 12.1 | 13.9 | 14.313 | 12.883 | 13.164 | 11.350 | 10.656 | 10.528 | 1.533 | 55.7 |
UPM 1253-5924 | 696374392 | 110140798 | -177.5 | 34.4 | 7.9 | 3.1 | 16.438 | 14.707 | 12.458 | 11.271 | 10.701 | 10.479 | 3.436 | 43.4 |
UPM 1255-6915 | 697926133 | 74668638 | -169.2 | -109.2 | 14.1 | 5.8 | 15.934 | 13.804 | 11.864 | 10.907 | 10.350 | 10.098 | 2.897 | 42.2 |
UPM 1255-5654 | 698007911 | 119128301 | 268.5 | -2.5 | 13.5 | 13.7 | $\cdots$ | 16.078 | 14.497 | 13.044 | 12.512 | 12.296 | 3.034 | 100.8 |
UPM 1255-6817 | 698247493 | 78148828 | -189.8 | -34.4 | 8.6 | 7.9 | 16.149 | 14.011 | 12.529 | 11.674 | 11.108 | 10.880 | 2.337 | 75.9 |
UPM 1257-5107 | 699318268 | 139935108 | -167.8 | -69.7 | 7.7 | 3.3 | 18.006 | 16.101 | 13.909 | 12.484 | 11.923 | 11.683 | 3.617 | 66.0 |
UPM 1259-5144 | 701235466 | 137713698 | -185.2 | 1.9 | 3.2 | 3.9 | 17.695 | 15.716 | 13.851 | 12.237 | 11.649 | 11.383 | 3.479 | 58.2 |
UPM 1308-5535 | 709469852 | 123890378 | -213.0 | -9.5 | 7.5 | 7.9 | 16.138 | 14.372 | 12.622 | 11.536 | 10.960 | 10.704 | 2.836 | 60.8 |
UPM 1308-7437 | 710077137 | 55354493 | -131.6 | -161.3 | 9.3 | 7.5 | 16.086 | 14.421 | 12.251 | 11.514 | 10.905 | 10.627 | 2.907 | 59.1 |
UPM 1309-5339 | 710289427 | 130807194 | -187.7 | -9.1 | 5.2 | 7.3 | 17.244 | 15.525 | 13.484 | 12.040 | 11.427 | 11.126 | 3.485 | 54.7 |
UPM 1312-5933 | 712889351 | 109595597 | -223.2 | -40.8 | 7.1 | 7.4 | 15.749 | 14.358 | 11.945 | 11.061 | 10.462 | 10.186 | 3.297 | 43.9 |
UPM 1317-5100 | 717471129 | 140352770 | -137.6 | 117.9 | 4.3 | 4.2 | 17.484 | 16.767 | 15.525 | 13.578 | 13.062 | 12.767 | 3.189 | 149.2 |
UPM 1318-6705 | 718242602 | 82452844 | -196.7 | -57.2 | 8.6 | 8.7 | 17.162 | 15.557 | 14.570 | 13.880 | 13.363 | 13.041 | 1.677 | [250.1] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1322-5730 | 721872886 | 116945756 | -214.6 | -67.6 | 6.5 | 6.3 | 15.574 | 13.935 | 12.350 | 11.266 | 10.699 | 10.455 | 2.669 | 59.6 |
UPM 1324-7123 | 724317780 | 66983446 | -188.8 | 14.7 | 5.9 | 3.4 | 16.176 | 14.167 | 12.888 | 11.975 | 11.342 | 11.166 | 2.192 | 92.2 |
UPM 1325-8321 | 724631438 | 23929798 | -105.5 | 159.4 | 2.4 | 2.8 | 15.209 | 13.416 | 11.780 | 10.794 | 10.149 | 9.925 | 2.622 | 45.5 |
UPM 1330-5844 | 729060131 | 112556039 | -190.1 | 22.8 | 6.3 | 6.1 | 15.543 | 13.370 | 12.336 | 11.118 | 10.468 | 10.270 | 2.252 | 56.7 |
UPM 1332-7421 | 731048730 | 56335255 | 175.9 | 50.5 | 47.8 | 45.3 | 14.290 | 12.439 | 10.758 | 10.057 | 9.483 | 9.229 | 2.382 | 36.9 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1335-5708 | 734150978 | 118269665 | -225.3 | 29.0 | 13.6 | 12.7 | 18.565 | 17.027 | 15.200 | 12.487 | 11.866 | 11.592 | 4.540 | 46.6 |
UPM 1339-7507 | 737590334 | 53557535 | -177.7 | -48.5 | 11.2 | 14.3 | 18.288 | 16.752 | 14.841 | 12.737 | 12.205 | 11.930 | 4.015 | 66.7 |
UPM 1340-6431 | 738068289 | 91682046 | -179.2 | -44.8 | 10.0 | 10.0 | 14.464 | 13.006 | 12.265 | 10.686 | 10.098 | 9.936 | 2.320 | 50.7 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position,hhSuperCOSMOS plate magnitudes suspect
UPM 1344-6829 | 741720593 | 77453474 | -256.9 | -180.2 | 6.8 | 7.6 | 17.635 | 16.274 | 14.268 | 11.780 | 11.185 | 10.941 | 4.494 | 36.3 |
UPM 1346-6135 | 743664758 | 102274414 | -199.8 | -61.5 | 8.2 | 7.1 | 16.859 | 14.674 | 13.054 | 11.999 | 11.423 | 11.173 | 2.675 | 73.9 |
UPM 1354-7121 | 751408498 | 67092222 | -165.0 | -132.7 | 6.5 | 6.8 | 12.598 | 10.665 | 8.750 | 8.549 | 7.920 | 7.672 | 2.116 | 19.3 |
UPM 1401-6837 | 757371795 | 76964116 | -174.5 | -56.7 | 8.8 | 13.0 | 16.977 | 15.882 | 13.535 | 12.571 | 11.991 | 11.744 | 3.311 | 96.0 |
UPM 1401-6405B | 757530750 | 93309800 | -185.2 | -149.3 | 5.8 | 5.7 | 19.027 | 17.453 | 15.687 | 13.273 | 12.699 | 12.398 | 4.180 | 76.3 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 1401-6405A | 757555362 | 93281664 | -172.2 | -140.9 | 7.1 | 7.0 | 15.536 | 13.909 | 12.787 | 10.296 | 9.649 | 9.407 | 3.613 | 24.1 | ccCommon proper motion companion; see Table 4
UPM 1403-6140 | 758976520 | 101999830 | 193.6 | 149.2 | 10.6 | 10.6 | $\cdots$ | 13.252 | $\cdots$ | 10.991 | 10.366 | 10.174 | 2.261 | 55.8 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1406-5850 | 761880385 | 112189733 | -175.4 | -64.0 | 4.2 | 6.6 | 16.746 | 15.180 | 13.568 | 12.214 | 11.621 | 11.380 | 2.966 | 80.6 |
UPM 1407-5749 | 762455713 | 115832007 | 253.6 | -52.6 | 10.0 | 10.1 | 17.890 | 16.185 | 14.303 | 13.311 | 12.795 | 12.541 | 2.874 | 145.1 |
UPM 1408-6315 | 763992896 | 96269737 | -167.5 | -72.4 | 5.1 | 5.3 | 17.577 | 16.267 | 14.705 | 12.362 | 11.777 | 11.517 | 3.905 | 60.7 |
UPM 1412-6959 | 766992114 | 72040016 | -178.2 | -39.7 | 4.5 | 4.6 | 16.912 | 14.912 | 14.256 | 12.205 | 11.665 | 11.385 | 2.707 | 74.4 |
UPM 1414-6023A | 768913676 | 106583792 | -200.8 | -87.5 | 6.4 | 6.5 | 16.109 | 14.192 | 12.328 | 11.739 | 11.151 | 10.937 | 2.453 | 78.4 | ccCommon proper motion companion; see Table 4
UPM 1414-6023B | 768922049 | 106594300 | -189.1 | -61.3 | 4.8 | 4.7 | 16.568 | 17.150 | 17.718 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,ggSource not in 2MASS
UPM 1415-6311 | 769792263 | 96488756 | -246.7 | -29.1 | 10.4 | 7.5 | 17.414 | 15.880 | 14.282 | 11.599 | 11.016 | 10.763 | 4.281 | 35.7 |
UPM 1415-5844 | 770247340 | 112534141 | -142.0 | -240.6 | 7.6 | 7.7 | 17.067 | 15.128 | 13.915 | 12.666 | 12.023 | 11.801 | 2.462 | 110.1 |
UPM 1416-6547 | 770760647 | 87154578 | -154.5 | 97.2 | 13.3 | 18.2 | 16.250 | 14.702 | 13.371 | 11.036 | 10.461 | 10.144 | 3.666 | 33.6 |
UPM 1416-6846 | 770849386 | 76397755 | -182.9 | -131.7 | 5.2 | 4.0 | 16.417 | 15.759 | 14.404 | 13.367 | 12.835 | 12.591 | 2.392 | 154.5 |
UPM 1418-7605 | 772902679 | 50072692 | -43.9 | -181.7 | 7.5 | 9.5 | 14.643 | 12.745 | 11.734 | 10.339 | 9.734 | 9.482 | 2.406 | 39.0 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1419-5109 | 773453770 | 139843693 | -125.6 | -152.6 | 13.5 | 5.2 | 16.238 | 14.670 | 13.099 | 11.961 | 11.399 | 11.181 | 2.709 | 83.3 |
UPM 1426-6340 | 779857214 | 94764404 | -183.5 | 67.4 | 6.8 | 18.1 | 15.798 | 14.747 | 14.011 | 13.550 | 13.079 | 12.941 | 1.197 | [250.9] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1427-6705 | 780541416 | 82481944 | -170.3 | -110.3 | 7.2 | 7.0 | 16.896 | 14.891 | 13.557 | 12.078 | 11.543 | 11.368 | 2.813 | 79.4 |
UPM 1430-7722B | 783295650 | 45468699 | -67.0 | -44.4 | 7.3 | 7.1 | 20.009 | 18.111 | 16.593 | 15.116 | 14.488 | 14.296 | 2.995 | 280.6 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 1430-5448 | 783845768 | 126695594 | -195.4 | -111.9 | 6.9 | 6.7 | 16.262 | 15.582 | 13.144 | 12.323 | 11.716 | 11.538 | 3.259 | 97.6 |
UPM 1432-6213 | 785364757 | 99996107 | -149.4 | -144.0 | 6.2 | 6.3 | 16.798 | 15.410 | 13.136 | 12.517 | 11.844 | 11.615 | 2.893 | 98.9 |
UPM 1433-6149 | 786022560 | 101424123 | -198.2 | -12.9 | 8.1 | 6.3 | 16.961 | 14.678 | 13.386 | 12.504 | 11.919 | 11.717 | 2.174 | 115.2 |
UPM 1435-6728 | 787903652 | 81063756 | -277.5 | 12.8 | 7.3 | 7.1 | 16.652 | 14.492 | 12.999 | 12.098 | 11.571 | 11.368 | 2.394 | 93.7 |
UPM 1436-6144 | 788585214 | 101740867 | -97.4 | -233.3 | 7.9 | 8.2 | 16.174 | 14.344 | 13.071 | 11.855 | 11.299 | 11.091 | 2.489 | 82.5 |
UPM 1437-6116 | 789964428 | 103381853 | 123.3 | -155.2 | 9.0 | 22.6 | 15.313 | 13.941 | 12.227 | 11.541 | 10.855 | 10.665 | 2.400 | 75.0 |
UPM 1439-6433 | 791205139 | 91561306 | -198.3 | -43.0 | 7.1 | 7.0 | 15.121 | 14.112 | 13.604 | 13.110 | 12.518 | 12.422 | 1.002 | [171.5] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1442-5937 | 793810374 | 109363585 | -166.2 | -73.5 | 5.5 | 9.3 | 17.286 | 15.442 | 12.698 | 11.310 | 10.702 | 10.454 | 4.132 | 30.9 |
UPM 1442-6702 | 794095735 | 82640171 | -131.0 | -128.7 | 4.8 | 5.7 | 16.827 | 14.652 | 12.674 | 11.716 | 11.181 | 10.862 | 2.936 | 58.4 |
UPM 1442-5915 | 794124307 | 110687177 | -243.6 | -222.5 | 10.8 | 10.8 | 17.846 | 16.231 | 14.848 | 13.288 | 12.736 | 12.574 | 2.943 | 141.4 |
UPM 1443-6554 | 795028077 | 86711555 | -183.2 | -182.4 | 7.8 | 8.6 | 16.370 | 14.644 | 12.681 | 11.880 | 11.296 | 11.000 | 2.764 | 73.2 |
UPM 1448-6357 | 799625574 | 93760146 | -168.0 | -197.1 | 4.5 | 4.5 | $\cdots$ | $\cdots$ | $\cdots$ | 11.091 | 10.420 | 10.275 | $\cdots$ | $\cdots$ |
UPM 1449-5926 | 800302397 | 109989509 | -171.8 | -75.5 | 7.9 | 45.0 | 15.760 | 14.604 | $\cdots$ | 12.384 | 11.701 | 11.583 | 2.220 | 108.1 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1449-6056 | 800518005 | 104607864 | 187.8 | 58.2 | 28.3 | 20.1 | 14.203 | 11.948 | 10.818 | 10.224 | 9.608 | 9.357 | 1.724 | 45.2 |
UPM 1449-5130 | 800558946 | 138542514 | -207.9 | -161.2 | 7.8 | 9.0 | 16.622 | 15.191 | 13.019 | 11.750 | 11.123 | 10.926 | 3.441 | 55.9 |
UPM 1452-5459 | 803454332 | 126035105 | -250.3 | -116.9 | 10.1 | 10.5 | 17.452 | 16.162 | 14.856 | 11.426 | 10.817 | 10.561 | 4.736 | 31.2 |
UPM 1453-5010 | 803753281 | 143395678 | -173.6 | -91.2 | 11.2 | 11.0 | 18.047 | 16.525 | 14.412 | 13.280 | 12.689 | 12.436 | 3.245 | 120.3 |
UPM 1453-7305 | 803890316 | 60868008 | -21.5 | 211.7 | 15.2 | 15.2 | $\cdots$ | 14.823 | $\cdots$ | 11.052 | 10.501 | 10.261 | 3.771 | 27.9 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1454-6013 | 804616852 | 107210805 | -253.1 | -198.5 | 10.3 | 8.0 | 16.823 | 15.696 | 13.935 | 11.618 | 11.103 | 10.858 | 4.078 | 44.5 |
UPM 1454-6322 | 804798590 | 95867997 | -110.5 | -151.9 | 4.7 | 10.5 | 14.814 | 13.233 | 12.794 | 11.063 | 10.436 | 10.270 | 2.170 | 55.3 |
UPM 1454-5809 | 805350729 | 114653491 | -36.4 | 188.0 | 13.1 | 13.1 | 16.770 | 14.890 | 13.200 | 10.635 | 10.063 | 9.776 | 4.255 | 20.9 |
UPM 1455-4904 | 806346721 | 147309534 | -156.8 | -89.2 | 6.6 | 6.5 | 14.785 | 13.811 | 11.602 | 11.580 | 10.949 | 10.737 | 2.231 | 72.9 |
UPM 1503-5007B | 813027719 | 143532327 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 11.727 | 11.174 | 10.921 | $\cdots$ | $\cdots$ | ddNot detected during automated search but noticed by eye during the blinking process,ccCommon proper motion companion; see Table 4
UPM 1503-5007A | 813028013 | 143524824 | -238.2 | -193.2 | 7.5 | 7.0 | 14.234 | 12.109 | 10.235 | 10.655 | 10.109 | 9.835 | 1.454 | 57.6 | ccCommon proper motion companion; see Table 4
UPM 1511-5319 | 820599480 | 132025375 | 10.6 | -179.8 | 3.5 | 3.5 | 16.131 | 14.364 | 13.682 | 11.998 | 11.430 | 11.202 | 2.366 | 85.4 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1513-5602 | 822002567 | 122233112 | -190.6 | -97.3 | 4.3 | 3.5 | 15.978 | 14.268 | 12.471 | 11.162 | 10.617 | 10.319 | 3.106 | 45.7 |
UPM 1515-6258 | 823574156 | 97313936 | -190.1 | -110.5 | 9.1 | 8.2 | 16.900 | 16.131 | 14.793 | 11.735 | 11.130 | 10.852 | 4.396 | 35.6 |
UPM 1515-7325 | 823782967 | 59696719 | 174.0 | -189.4 | 10.1 | 15.5 | 17.600 | 15.781 | 13.903 | 12.626 | 12.141 | 11.913 | 3.155 | 94.0 |
UPM 1517-5807 | 826033939 | 114763609 | -188.8 | -17.3 | 7.0 | 6.9 | 15.883 | 14.016 | 14.209 | 11.903 | 11.226 | 11.088 | 2.113 | 73.0 |
UPM 1519-7255 | 827454427 | 61484348 | -194.5 | -216.5 | 7.7 | 7.5 | 16.426 | 14.266 | 11.940 | 10.898 | 10.297 | 9.993 | 3.368 | 32.3 |
UPM 1521-4835 | 829428454 | 149044795 | -234.3 | -146.1 | 11.3 | 11.3 | 17.385 | 15.988 | 13.830 | 12.255 | 11.683 | 11.428 | 3.733 | 62.1 |
UPM 1523-4913 | 831258441 | 146791866 | -94.7 | -180.7 | 8.7 | 7.9 | 16.331 | 14.663 | 13.027 | 11.475 | 10.810 | 10.588 | 3.188 | 48.5 |
UPM 1523-5454 | 831355688 | 126325591 | -60.6 | 200.9 | 14.9 | 14.9 | 17.770 | 11.791 | 9.697 | 9.767 | 9.194 | 8.871 | 2.024 | 20.4 |
UPM 1524-5439 | 831659121 | 127212227 | -187.7 | -39.2 | 7.7 | 7.9 | 15.966 | 14.134 | 12.376 | 11.510 | 10.966 | 10.751 | 2.624 | 68.6 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1526-4930 | 833429461 | 145750882 | -154.9 | -115.6 | 8.9 | 7.9 | 13.599 | 11.582 | 10.794 | 9.983 | 9.350 | 9.177 | 1.599 | 43.9 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1528-5839B | 835757400 | 112820999 | 5.9 | -197.8 | 6.5 | 6.5 | 18.311 | 18.503 | 18.776 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,ggSource not in 2MASS
UPM 1528-5839A | 835780889 | 112825365 | 2.1 | -197.2 | 4.2 | 4.1 | 15.182 | 13.233 | 11.679 | 10.416 | 9.777 | 9.528 | 2.817 | 33.3 | ccCommon proper motion companion; see Table 4
UPM 1530-5511 | 837643179 | 125327924 | -95.1 | -153.8 | 6.9 | 7.0 | 15.742 | 13.925 | 12.143 | 11.511 | 10.875 | 10.684 | 2.414 | 71.3 |
UPM 1533-6556 | 840377734 | 86586399 | -148.6 | -137.4 | 7.3 | 7.4 | 16.368 | 14.167 | 12.510 | 11.836 | 11.297 | 11.026 | 2.331 | 81.2 |
UPM 1537-5610 | 843606725 | 121747354 | -133.2 | -133.6 | 3.5 | 3.5 | 15.275 | 13.265 | 11.568 | 10.839 | 10.207 | 10.004 | 2.426 | 49.8 |
UPM 1538-5456 | 844492903 | 126230428 | -102.9 | -207.1 | 14.3 | 14.3 | 17.152 | 16.178 | 13.882 | 12.213 | 11.678 | 11.392 | 3.965 | 62.1 |
UPM 1540-5023 | 846454757 | 142615035 | -229.8 | -95.6 | 8.3 | 8.0 | 15.638 | 14.696 | 14.199 | 13.166 | 12.711 | 12.579 | 1.530 | [186.4] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1540-5550 | 846546026 | 122995128 | -173.8 | -262.2 | 7.9 | 7.9 | 15.877 | 14.542 | 12.672 | 11.654 | 11.053 | 10.762 | 2.888 | 66.5 |
UPM 1540-6447 | 846621188 | 90731820 | -153.0 | -199.7 | 8.4 | 8.4 | 14.958 | 15.441 | 12.219 | 12.503 | 11.872 | 11.671 | 2.938 | 85.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1542-5659 | 848144097 | 118835541 | -171.0 | -207.7 | 2.8 | 2.9 | 14.702 | 14.200 | 12.327 | 12.050 | 11.401 | 11.247 | 2.150 | 96.0 |
UPM 1542-5041 | 848609832 | 141523338 | 182.4 | -16.3 | 3.6 | 3.6 | $\cdots$ | 9.924 | 9.592 | 9.440 | 8.867 | 8.690 | 0.484 | 33.1 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1543-5449 | 848811762 | 126606709 | -149.9 | -135.8 | 10.4 | 8.2 | 16.944 | 15.432 | 13.822 | 12.092 | 11.625 | 11.359 | 3.340 | 70.5 |
UPM 1543-5812 | 848814656 | 114427930 | -204.9 | -101.7 | 6.9 | 6.6 | 15.176 | 13.101 | 11.994 | 10.727 | 10.080 | 9.847 | 2.374 | 45.0 |
UPM 1543-7306 | 848944114 | 60833296 | -136.4 | -124.6 | 6.3 | 5.6 | 14.182 | 13.352 | 12.361 | 11.797 | 11.119 | 11.016 | 1.555 | 98.1 |
UPM 1544-5208 | 849748641 | 136318898 | -57.1 | -180.3 | 7.6 | 7.5 | 17.691 | 16.196 | 14.821 | 12.951 | 12.442 | 12.262 | 3.245 | 111.1 |
UPM 1544-5341 | 850150522 | 130717165 | -259.8 | -95.8 | 6.8 | 7.1 | 17.198 | 15.609 | 14.004 | 12.663 | 12.117 | 11.920 | 2.946 | 106.0 |
UPM 1545-5259 | 850605613 | 133228438 | -182.9 | -168.4 | 5.4 | 3.6 | 17.864 | 16.312 | 14.084 | 12.045 | 11.455 | 11.147 | 4.267 | 40.8 |
UPM 1545-5018 | 851366897 | 142904161 | -252.2 | -81.7 | 9.0 | 8.3 | 13.765 | 12.692 | 11.167 | 10.694 | 10.093 | 9.841 | 1.998 | 53.9 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1548-5045A | 853752961 | 141274236 | -92.4 | -175.3 | 6.9 | 6.5 | 15.788 | 13.638 | 11.611 | 10.526 | 9.912 | 9.660 | 3.112 | 30.6 | ccCommon proper motion companion; see Table 4
UPM 1548-5045B | 853766317 | 141285738 | -57.1 | -155.8 | 8.5 | 8.5 | 18.117 | 16.525 | 14.265 | 12.184 | 11.580 | 11.295 | 4.341 | 41.9 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect
UPM 1549-8434 | 854968181 | 19516753 | -82.3 | -227.6 | 3.6 | 3.1 | 14.195 | 13.148 | 12.361 | 13.187 | 12.606 | 12.344 | -0.039 | 227.6 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,hhSuperCOSMOS plate magnitudes suspect
UPM 1555-5939 | 859698048 | 109256622 | -148.1 | -139.3 | 7.1 | 9.2 | 16.489 | 15.353 | 14.277 | 13.814 | 13.177 | 12.962 | 1.539 | [242.2] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1556-4823 | 861083858 | 149761425 | -177.9 | -187.5 | 7.2 | 20.4 | 14.810 | 14.203 | 12.467 | 11.234 | 10.637 | 10.404 | 2.969 | 56.9 |
UPM 1557-4852 | 861748184 | 148032598 | -155.8 | -90.3 | 12.7 | 8.6 | 16.301 | 14.588 | 12.855 | 11.408 | 10.922 | 10.709 | 3.180 | 54.2 |
UPM 1557-8159 | 862016282 | 28823775 | -195.9 | -195.8 | 9.4 | 9.9 | 16.861 | 14.747 | 12.909 | 11.655 | 11.101 | 10.850 | 3.092 | 54.0 |
UPM 1600-8109 | 864215257 | 31836852 | -211.6 | -5.7 | 11.3 | 11.4 | 17.252 | 15.740 | 13.875 | 12.369 | 11.846 | 11.606 | 3.371 | 77.8 |
UPM 1600-7402 | 864883254 | 57469688 | -150.2 | -108.1 | 5.6 | 8.1 | 18.249 | 16.461 | 15.124 | 12.875 | 12.309 | 12.080 | 3.586 | 81.2 |
UPM 1602-5650 | 866299723 | 119342424 | -140.2 | -115.0 | 4.5 | 4.5 | 15.480 | $\cdots$ | 12.298 | 10.461 | 9.865 | 9.617 | $\cdots$ | 29.3 |
UPM 1603-4710 | 867405600 | 154185744 | -127.6 | -132.7 | 8.3 | 16.6 | 17.317 | 15.898 | 14.368 | 11.992 | 11.456 | 11.183 | 3.906 | 51.4 |
UPM 1606-5801 | 869542767 | 115123561 | -163.3 | -152.5 | 27.1 | 27.1 | 15.164 | 13.617 | 11.964 | 11.369 | 10.723 | 10.479 | 2.248 | 70.9 |
UPM 1606-5502 | 870063998 | 125864104 | 170.0 | -97.6 | 16.5 | 16.5 | 17.273 | 14.092 | 13.190 | 12.050 | 11.416 | 11.188 | 2.042 | 78.5 | hhSuperCOSMOS plate magnitudes suspect
UPM 1606-6424 | 870071528 | 92152496 | -87.6 | -180.2 | 4.7 | 13.2 | 16.111 | 14.240 | 12.086 | 11.667 | 11.033 | 10.848 | 2.573 | 72.5 |
UPM 1612-6211 | 875631410 | 100096982 | -224.6 | -196.4 | 9.4 | 10.0 | 15.267 | 13.734 | 11.920 | 11.881 | 11.321 | 10.990 | 1.853 | 90.8 |
UPM 1614-5644 | 877400692 | 119748242 | -172.4 | 132.6 | 36.6 | 6.6 | 18.660 | 15.727 | 16.251 | 12.346 | 11.702 | 11.535 | 3.381 | 39.8 |
UPM 1616-5634 | 879009940 | 120353677 | -70.2 | 178.7 | 9.7 | 9.7 | $\cdots$ | $\cdots$ | $\cdots$ | 11.411 | 10.838 | 10.628 | $\cdots$ | $\cdots$ |
UPM 1617-6126 | 879365812 | 102805939 | 179.1 | 42.7 | 17.7 | 4.8 | 13.051 | 12.228 | 11.139 | 11.461 | 10.892 | 10.685 | 0.767 | 88.4 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1618-5500 | 880614502 | 125984834 | -157.1 | -140.8 | 11.6 | 8.8 | 16.144 | 14.059 | 12.493 | 11.928 | 11.333 | 11.116 | 2.131 | 92.5 |
UPM 1622-5139 | 884266236 | 138014836 | -175.0 | -71.2 | 4.6 | 4.7 | $\cdots$ | $\cdots$ | $\cdots$ | 12.950 | 12.303 | 12.138 | $\cdots$ | $\cdots$ |
UPM 1626-5519 | 887625825 | 124812716 | 59.6 | 184.7 | 13.0 | 12.5 | $\cdots$ | $\cdots$ | $\cdots$ | 12.012 | 11.358 | 11.126 | $\cdots$ | $\cdots$ |
UPM 1628-4844 | 890083966 | 148558590 | -75.0 | -172.4 | 7.0 | 7.5 | 16.048 | 13.841 | 11.418 | 10.613 | 10.014 | 9.704 | 3.228 | 30.4 |
UPM 1629-6116 | 890347726 | 103423813 | -82.8 | -214.7 | 13.2 | 13.7 | 15.763 | 15.487 | 12.728 | 12.748 | 12.150 | 11.907 | 2.739 | 101.2 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1632-5634 | 893446159 | 120340545 | -107.2 | -165.7 | 9.5 | 27.5 | 17.638 | 16.345 | 14.770 | 12.654 | 12.149 | 11.885 | 3.691 | 80.2 |
UPM 1635-5202 | 896058321 | 136679193 | -185.5 | 94.4 | 12.3 | 9.3 | 16.727 | 14.731 | 12.217 | 10.408 | 9.795 | 9.559 | 4.323 | 17.6 |
UPM 1636-8117 | 896834696 | 31361390 | 164.7 | -165.3 | 11.3 | 10.8 | 16.856 | 14.775 | 12.984 | 11.730 | 11.194 | 10.924 | 3.045 | 57.6 |
UPM 1637-5341 | 897823748 | 130696230 | -145.2 | -177.6 | 14.8 | 14.8 | 17.660 | 14.602 | 15.098 | 13.321 | 12.569 | 12.422 | 1.281 | [155.0] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1640-5303 | 900133048 | 132983329 | -29.3 | -180.0 | 10.9 | 6.1 | 16.624 | 14.572 | 12.596 | 12.182 | 11.541 | 11.324 | 2.390 | 92.7 |
UPM 1641-4957 | 901066844 | 144129914 | -176.7 | -65.6 | 8.8 | 8.8 | $\cdots$ | 15.196 | 13.389 | 11.012 | 10.478 | 10.202 | 4.184 | 22.0 |
UPM 1641-4822 | 901220912 | 149877925 | -184.8 | -111.7 | 6.9 | 6.9 | 16.645 | 15.221 | 13.345 | 11.625 | 11.119 | 10.836 | 3.596 | 50.1 |
UPM 1641-5351 | 901600603 | 130095538 | -123.7 | -159.0 | 8.5 | 8.5 | 14.447 | 12.496 | 10.954 | 11.853 | 11.255 | 11.023 | 0.643 | 84.5 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1643-6144 | 903514948 | 101747311 | -130.5 | -125.0 | 8.3 | 8.4 | 17.004 | 14.684 | 13.184 | 12.734 | 12.159 | 11.972 | 1.950 | 141.3 |
UPM 1644-4757 | 904185355 | 151357617 | 130.5 | -146.0 | 9.9 | 9.6 | 17.355 | 15.071 | 13.129 | 11.733 | 11.181 | 10.903 | 3.338 | 47.5 |
UPM 1645-4853 | 905208118 | 148011805 | -79.3 | -185.3 | 12.0 | 12.3 | 18.232 | 16.791 | 15.248 | 13.113 | 12.607 | 12.331 | 3.678 | 95.5 |
UPM 1651-5406 | 910203559 | 129209633 | -25.3 | -190.5 | 8.7 | 5.2 | 14.705 | 12.587 | 10.957 | 9.955 | 9.302 | 9.103 | 2.632 | 29.0 |
UPM 1658-5934 | 916311580 | 109519753 | 129.4 | 164.2 | 11.3 | 11.2 | 16.306 | 14.415 | 12.710 | 11.745 | 11.086 | 10.882 | 2.670 | 68.1 |
UPM 1658-5311A | 917041848 | 132497063 | -163.5 | -229.3 | 8.2 | 7.8 | 15.892 | 13.661 | 11.399 | 10.490 | 9.914 | 9.560 | 3.171 | 28.6 | ccCommon proper motion companion; see Table 4
UPM 1658-5311B | 917159205 | 132450388 | -11.6 | -134.1 | 9.5 | 9.4 | 16.346 | 14.562 | 11.906 | 11.001 | 10.400 | 10.014 | 3.561 | 33.0 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect,hhSuperCOSMOS plate magnitudes suspect
UPM 1703-5441 | 921435004 | 127082284 | -171.6 | -150.6 | 5.3 | 5.4 | 16.594 | 15.442 | 13.273 | 12.643 | 12.124 | 11.935 | 2.799 | 128.5 |
UPM 1703-4934B | 921462541 | 145546027 | -28.6 | 72.6 | 31.3 | 31.3 | 15.243 | 17.305 | 16.288 | 11.005 | 10.397 | 10.132 | 6.300 | 40.7 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process,eeProper motions suspect,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1704-7751 | 922050168 | 43712233 | -162.3 | -196.3 | 11.2 | 10.9 | 18.261 | 16.287 | 14.432 | 13.197 | 12.657 | 12.449 | 3.090 | 118.0 |
UPM 1704-8055 | 922365146 | 32647192 | -109.9 | -229.3 | 15.6 | 12.3 | 18.327 | 16.572 | 15.085 | 13.459 | 12.921 | 12.666 | 3.113 | 131.7 |
UPM 1706-5449 | 924280382 | 126610913 | -84.1 | -215.3 | 48.8 | 42.1 | 15.393 | 14.004 | 12.177 | 11.549 | 10.972 | 10.756 | 2.455 | 78.7 |
UPM 1709-5644 | 926355078 | 119715597 | -82.3 | -160.7 | 3.6 | 3.8 | 15.744 | 14.072 | 12.643 | 11.914 | 11.329 | 11.127 | 2.158 | 97.7 |
UPM 1710-5300 | 927664633 | 133174754 | -34.3 | -179.9 | 2.5 | 1.5 | 12.469 | 10.032 | 7.989 | 8.001 | 7.407 | 7.163 | 2.031 | 13.5 |
UPM 1713-7336 | 930246842 | 59036684 | -66.7 | -182.7 | 7.9 | 18.1 | 17.420 | 15.351 | 13.715 | 12.637 | 12.030 | 11.795 | 2.714 | 98.4 |
UPM 1719-6255B | 935394345 | 97476797 | -94.0 | -153.7 | 6.8 | 7.2 | 17.187 | 15.079 | 13.376 | 12.554 | 12.016 | 11.735 | 2.525 | 105.6 | ccCommon proper motion companion; see Table 4
UPM 1720-4836 | 936087252 | 149033584 | -78.1 | -167.1 | 7.8 | 28.8 | 16.962 | 15.144 | 13.163 | 12.115 | 11.508 | 11.290 | 3.029 | 72.9 |
UPM 1725-4709 | 941154374 | 154211346 | -57.6 | -191.5 | 6.9 | 6.6 | 15.987 | 13.942 | 12.770 | 12.865 | 12.232 | 12.013 | 1.077 | [168.1] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1727-6239 | 942304521 | 98410597 | -230.9 | -48.7 | 9.7 | 9.3 | 16.500 | 13.266 | 11.233 | 11.034 | 10.434 | 10.182 | 2.232 | 47.7 |
UPM 1728-5128 | 943526046 | 138700980 | -29.6 | -178.3 | 7.3 | 6.7 | 17.366 | 14.772 | 13.268 | 12.348 | 11.809 | 11.558 | 2.424 | 91.9 |
UPM 1732-4700 | 947055769 | 154749702 | -45.8 | -174.9 | 3.0 | 3.0 | 16.079 | 14.033 | 11.310 | 11.327 | 10.752 | 10.551 | 2.706 | 50.7 |
UPM 1732-4736 | 947129106 | 152581933 | -205.7 | -203.9 | 6.9 | 6.7 | 14.880 | 12.769 | 10.996 | 9.586 | 8.977 | 8.682 | 3.183 | 18.5 |
UPM 1744-4922 | 958370713 | 146277461 | -139.9 | -127.0 | 46.4 | 6.9 | 17.069 | 15.168 | 14.370 | 13.521 | 12.964 | 12.794 | 1.647 | [227.4] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1746-5251 | 959779165 | 133698582 | -160.1 | 191.7 | 5.4 | 5.7 | 17.552 | 15.253 | 13.867 | 13.050 | 12.447 | 12.282 | 2.203 | 147.8 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1748-7247 | 961476350 | 61937161 | -195.5 | -246.2 | 11.9 | 10.2 | 17.436 | 15.452 | 13.687 | 12.481 | 11.941 | 11.722 | 2.971 | 88.5 |
UPM 1749-7530 | 962401296 | 52159553 | 159.0 | -113.5 | 9.0 | 9.0 | 14.809 | 13.643 | 12.653 | 12.925 | 12.363 | 12.117 | 0.718 | 181.6 |
UPM 1754-7701 | 967289539 | 46733484 | -1.3 | -208.7 | 10.4 | 10.6 | 14.829 | 13.029 | 12.189 | 11.369 | 10.680 | 10.578 | 1.660 | 78.9 |
UPM 1756-7406A | 969047217 | 57217628 | -78.4 | -201.7 | 15.6 | 15.6 | 18.282 | 16.576 | 14.247 | 12.051 | 11.470 | 11.236 | 4.525 | 37.3 | ccCommon proper motion companion; see Table 4
UPM 1756-7406B | 969067800 | 57217099 | -279.1 | -203.5 | 9.3 | 10.8 | $\cdots$ | 17.937 | $\cdots$ | 13.836 | 13.318 | 12.960 | 4.101 | 83.4 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 1756-6126 | 969289853 | 102830753 | -50.2 | -173.6 | 6.9 | 17.6 | 16.331 | 13.869 | 12.214 | 11.073 | 10.526 | 10.282 | 2.796 | 44.3 |
UPM 1758-4705 | 970965439 | 154485573 | 156.1 | -93.2 | 7.8 | 7.9 | 16.856 | 14.924 | 13.319 | 12.708 | 12.175 | 11.932 | 2.216 | 135.6 |
UPM 1812-6433 | 983319061 | 91560079 | -13.6 | -181.7 | 10.5 | 9.9 | 16.634 | 13.901 | 12.936 | 11.822 | 11.244 | 11.008 | 2.079 | 78.3 |
UPM 1815-6406 | 985813582 | 93223937 | -178.9 | -178.5 | 5.2 | 4.5 | 14.412 | 12.436 | 10.579 | 9.748 | 9.219 | 8.965 | 2.688 | 28.6 |
UPM 1817-8649 | 987374698 | 11424241 | -189.2 | 237.9 | 10.3 | 11.0 | 16.396 | 14.483 | $\cdots$ | 11.927 | 11.369 | 11.161 | 2.556 | 85.2 |
UPM 1824-4850 | 994028323 | 148141997 | 163.0 | -243.1 | 9.0 | 9.7 | 16.821 | 14.645 | 13.265 | 12.647 | 12.069 | 11.859 | 1.998 | 134.5 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1833-7136 | 1002337631 | 66220145 | -6.7 | -188.9 | 14.9 | 13.8 | 18.476 | 16.563 | 14.288 | 12.973 | 12.340 | 12.102 | 3.590 | 79.7 |
UPM 1834-6047 | 1003145785 | 105173051 | 14.7 | -189.0 | 10.2 | 9.8 | 14.884 | 13.363 | 11.637 | 10.850 | 10.274 | 10.038 | 2.513 | 54.0 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1836-5826 | 1004981205 | 113593259 | 61.4 | -177.6 | 5.2 | 6.1 | 16.118 | 14.741 | 13.216 | 12.081 | 11.409 | 11.135 | 2.660 | 82.7 |
UPM 1852-6027 | 1018955088 | 106373799 | -218.6 | -204.3 | 7.7 | 7.8 | 16.450 | 14.503 | 12.940 | 11.684 | 11.116 | 10.900 | 2.819 | 64.3 |
UPM 1858-6804 | 1024609765 | 78939442 | -129.6 | -125.5 | 5.5 | 3.8 | 16.990 | 15.315 | 14.638 | 13.659 | 13.213 | 13.034 | 1.656 | [243.8] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 1900-7029 | 1026744100 | 70203767 | -54.0 | -173.7 | 8.3 | 8.4 | 17.506 | 15.443 | 13.346 | 11.990 | 11.408 | 11.145 | 3.453 | 52.9 |
UPM 1909-4927 | 1034415320 | 145978861 | -127.5 | -222.8 | 13.5 | 10.6 | 17.147 | 15.145 | 13.945 | 13.029 | 12.383 | 12.195 | 2.116 | 151.4 |
UPM 1912-4942 | 1037156800 | 145077070 | 24.7 | -189.1 | 9.0 | 9.2 | 16.544 | 14.410 | 12.745 | 11.732 | 11.107 | 10.890 | 2.678 | 65.0 |
UPM 1917-6941 | 1041816777 | 73138163 | 35.6 | -208.4 | 11.7 | 10.0 | 16.928 | 15.030 | 12.970 | 11.763 | 11.149 | 10.882 | 3.267 | 52.6 | ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1923-5050 | 1047094221 | 140965899 | -152.7 | -96.9 | 14.5 | 10.5 | 15.979 | 14.159 | 13.200 | 12.438 | 11.863 | 11.722 | 1.721 | 137.5 |
UPM 1924-5328 | 1048004792 | 131493209 | -120.3 | 137.8 | 2.8 | 2.7 | 13.888 | 11.697 | 10.744 | 10.919 | 10.301 | 10.073 | 0.778 | 70.0 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable
UPM 1924-6646 | 1048323630 | 83623834 | 49.2 | -202.2 | 8.1 | 8.3 | 16.538 | 14.551 | 12.610 | 11.645 | 11.073 | 10.878 | 2.906 | 62.3 |
UPM 1926-5838 | 1050092012 | 112868363 | -262.3 | -182.4 | 8.7 | 8.9 | 16.783 | 14.705 | 12.843 | 11.047 | 10.570 | 10.298 | 3.658 | 32.8 |
UPM 1927-6334 | 1050328043 | 95124273 | 201.0 | -144.0 | 35.7 | 15.1 | 17.580 | 15.245 | 14.151 | 12.991 | 12.462 | 12.259 | 2.254 | 142.4 |
UPM 1930-7112 | 1053133323 | 67662282 | -153.4 | -109.2 | 9.1 | 9.3 | 16.626 | 14.906 | 12.993 | 11.722 | 11.111 | 10.833 | 3.184 | 55.0 |
UPM 1932-6707 | 1055210338 | 82373484 | 39.4 | -188.9 | 2.5 | 3.4 | 14.962 | 13.038 | 12.098 | 11.033 | 10.422 | 10.257 | 2.005 | 65.3 |
UPM 1935-5936 | 1057892042 | 109409490 | 45.0 | -194.9 | 9.4 | 9.6 | 16.515 | 14.637 | 12.770 | 11.755 | 11.128 | 10.893 | 2.882 | 63.1 |
UPM 1941-4928 | 1063289596 | 145911664 | -180.0 | -30.6 | 9.0 | 9.0 | 16.876 | 12.514 | 11.181 | 11.367 | 10.808 | 10.514 | 1.147 | 58.9 |
UPM 1943-6049 | 1065224052 | 105007320 | -61.9 | -216.5 | 9.7 | 9.8 | 17.207 | 15.251 | 13.636 | 12.565 | 11.991 | 11.785 | 2.686 | 103.0 |
UPM 1950-6013B | 1071297914 | 107145608 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 13.978 | 13.500 | 13.202 | $\cdots$ | $\cdots$ | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 1950-6013A | 1071312857 | 107171254 | -31.2 | -268.7 | 10.1 | 10.3 | 15.298 | 13.502 | 12.186 | 11.538 | 11.009 | 10.771 | 1.964 | 87.1 | ccCommon proper motion companion; see Table 4
UPM 1952-5539 | 1073529066 | 123644245 | 18.1 | -188.5 | 11.8 | 12.1 | 14.853 | 13.632 | 12.898 | 12.297 | 11.862 | 11.801 | 1.335 | 148.1 |
UPM 1956-6116 | 1076858244 | 103394149 | -151.0 | -167.7 | 9.7 | 10.4 | 18.048 | 16.396 | 15.079 | 13.653 | 13.080 | 12.919 | 2.743 | 178.1 |
UPM 2002-4735 | 1082313871 | 152672097 | 16.5 | -199.3 | 10.8 | 11.1 | 16.823 | 15.193 | 14.310 | 13.745 | 13.137 | 12.931 | 1.448 | [248.6] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 2008-6305B | 1088024100 | 96844499 | 77.4 | -181.2 | 13.3 | 14.7 | 13.811 | 12.005 | 10.679 | 12.395 | 11.150 | 11.582 | -0.390 | 125.7 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 2010-5353 | 1089659723 | 129992428 | -3.9 | -218.7 | 8.7 | 9.1 | 17.183 | 14.982 | 12.701 | 11.399 | 10.732 | 10.446 | 3.583 | 34.5 |
UPM 2013-5751 | 1092393887 | 115706680 | 37.1 | -257.5 | 11.5 | 12.4 | 15.821 | 14.400 | 13.682 | 12.930 | 12.373 | 12.298 | 1.470 | [181.8] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 2021-5858 | 1099009333 | 111686918 | -150.9 | -181.6 | 17.4 | 16.7 | 18.041 | 16.064 | 14.328 | 13.016 | 12.519 | 12.238 | 3.048 | 108.5 |
UPM 2024-4824 | 1102094399 | 149701952 | -180.5 | -46.0 | 3.9 | 3.9 | 17.838 | 15.911 | 14.287 | 13.138 | 12.638 | 12.406 | 2.773 | 135.0 |
UPM 2036-5548A | 1112699891 | 123087548 | 45.1 | -174.6 | 2.3 | 2.1 | 16.152 | 13.565 | 12.298 | 12.103 | 11.511 | 11.324 | 1.462 | 114.8 | ccCommon proper motion companion; see Table 4,ffPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 2036-5548B | 1112708550 | 123089700 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 19.025 | $\cdots$ | $\cdots$ | 14.229 | 13.732 | 13.493 | $\cdots$ | 233.8 | bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable,ccCommon proper motion companion; see Table 4 ,ddNot detected during automated search but noticed by eye during the blinking process
UPM 2038-4938 | 1114739048 | 145270905 | 70.2 | -204.3 | 9.4 | 10.0 | 15.787 | 13.595 | 11.507 | 10.754 | 10.089 | 9.838 | 2.841 | 37.2 |
UPM 2042-5552 | 1117911466 | 122822515 | -237.8 | -87.2 | 13.6 | 10.9 | 17.157 | 15.252 | 14.108 | 12.839 | 12.320 | 12.073 | 2.413 | 132.0 |
UPM 2043-5751 | 1119153753 | 115707535 | 249.5 | -125.3 | 13.5 | 12.6 | 18.103 | 16.133 | 14.452 | 12.891 | 12.309 | 12.068 | 3.242 | 89.2 |
UPM 2049-5415B | 1124176381 | 128699125 | 197.2 | 44.3 | 7.0 | 11.8 | 18.253 | 16.119 | 14.134 | 12.625 | 12.006 | 11.717 | 3.494 | 64.7 | ccCommon proper motion companion; see Table 4,ddNot detected during automated search but noticed by eye during the blinking process
UPM 2049-5415A | 1124200971 | 128684871 | 180.4 | 53.2 | 11.4 | 11.1 | 16.068 | 13.982 | 12.284 | 11.104 | 10.480 | 10.244 | 2.878 | 44.3 | ccCommon proper motion companion; see Table 4
UPM 2050-5044 | 1125736631 | 141342485 | 209.4 | -17.7 | 8.9 | 8.9 | 17.608 | 15.385 | 13.619 | 12.420 | 11.864 | 11.620 | 2.965 | 79.7 |
UPM 2051-5733 | 1126543636 | 116760305 | 124.5 | -130.1 | 8.0 | 6.8 | 17.840 | 15.995 | 14.387 | 12.742 | 12.147 | 11.891 | 3.253 | 83.5 |
UPM 2055-6059 | 1130089106 | 104450175 | 91.2 | -164.8 | 11.3 | 11.3 | 16.081 | 14.762 | 13.173 | 11.509 | 10.955 | 10.684 | 3.253 | 54.6 |
UPM 2057-5839 | 1132155449 | 112852294 | 185.8 | -261.0 | 10.0 | 10.3 | 16.739 | 14.688 | 13.046 | 11.610 | 10.990 | 10.770 | 3.078 | 51.6 |
UPM 2059-5711 | 1133803361 | 118094897 | 194.7 | -136.3 | 18.2 | 18.2 | 18.607 | 16.418 | 14.608 | 13.039 | 12.442 | 12.169 | 3.379 | 83.1 |
UPM 2106-5706B | 1139440736 | 118437233 | 162.6 | -178.8 | 16.3 | 15.1 | 18.226 | 16.111 | 14.254 | 12.687 | 12.186 | 11.927 | 3.424 | 76.6 | ccCommon proper motion companion; see Table 4
UPM 2109-5305 | 1142615989 | 132880256 | 33.0 | -207.9 | 14.4 | 14.5 | 18.133 | 15.963 | 15.182 | 14.271 | 13.625 | 13.314 | 1.692 | [279.0] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 2112-5121 | 1145134513 | 139091864 | 153.6 | -137.4 | 7.5 | 8.5 | 16.504 | 14.213 | 11.974 | 11.122 | 10.491 | 10.221 | 3.091 | 39.5 |
UPM 2112-5954 | 1145241065 | 108340688 | -177.4 | -51.5 | 11.5 | 11.5 | 15.928 | 14.047 | 12.755 | 11.392 | 10.773 | 10.583 | 2.655 | 59.2 |
UPM 2112-6215 | 1145389750 | 99864555 | -76.1 | 175.3 | 5.0 | 4.2 | $\cdots$ | $\cdots$ | $\cdots$ | 13.961 | 13.358 | 13.204 | $\cdots$ | $\cdots$ |
UPM 2113-5307 | 1146511831 | 132730907 | 211.6 | -172.0 | 12.5 | 11.4 | 17.439 | 15.535 | 14.916 | 13.617 | 12.999 | 12.869 | 1.918 | [211.7] | aaSubdwarf candidate picked from RPM diagram; plate distance [in bracket] is incorrect
UPM 2114-5545 | 1147399145 | 123271869 | -147.6 | -113.1 | 2.3 | 2.3 | 13.292 | 11.856 | 11.127 | 10.571 | 10.047 | 9.949 | 1.285 | 63.1 |
UPM 2119-5856 | 1151370742 | 111836222 | -142.4 | -117.5 | 10.4 | 3.5 | 15.891 | 14.298 | 12.323 | 11.383 | 10.758 | 10.492 | 2.915 | 55.2 |
UPM 2119-8445 | 1151637008 | 18871742 | -47.9 | -174.6 | 4.2 | 4.2 | $\cdots$ | $\cdots$ | $\cdots$ | 11.413 | 10.817 | 10.575 | $\cdots$ | $\cdots$ |
UPM 2129-7843 | 1160418602 | 40599605 | 14.9 | -179.8 | 11.7 | 3.9 | 15.628 | 14.579 | 14.055 | 12.986 | 12.402 | 12.264 | 1.593 | 153.4 |
UPM 2148-5404 | 1177649631 | 129334900 | -135.2 | -119.2 | 4.3 | 2.5 | 17.123 | 15.118 | 13.384 | 11.960 | 11.344 | 11.101 | 3.158 | 58.6 |
UPM 2247-5707 | 1230566367 | 118338374 | 179.7 | -120.1 | 12.6 | 12.7 | 17.245 | 15.259 | 13.878 | 12.656 | 12.167 | 11.879 | 2.603 | 111.0 |
UPM 2316-6309 | 1256781310 | 96640937 | 248.8 | 41.7 | 10.4 | 10.3 | 17.413 | 15.371 | 13.843 | 12.257 | 11.699 | 11.494 | 3.114 | 72.1 |
UPM 2353-7426 | 1289770038 | 55991827 | 24.1 | -179.1 | 4.2 | 3.8 | 16.614 | 14.526 | 13.240 | 12.605 | 12.032 | 11.884 | 1.921 | 142.2 |
Table 3: New UCAC3 High Proper Motion Systems estimated to be within 25 pc between Declinations $-$90$\arcdeg$ and $-$47$\arcdeg$ with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 Name | RA | DEC | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | sig $\mu_{\alpha}$ | sig $\mu_{\delta}$ | $B_{J}$ | $R_{59F}$ | $I_{IVN}$ | $J$ | $H$ | $K_{s}$ | $R_{59F}$ $-$ $J$ | Est Dist | Notes
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (mas) | (mas) | (mas/yr) | (mas/yr) | (mas/yr) | (mas/yr) | | | | | | | | (pc) |
UPM 0559-5225 | 323598169 | 135295015 | 120.7 | 152.6 | 4.5 | 4.4 | 14.532 | 12.000 | 10.287 | 9.628 | 8.951 | 8.703 | 2.372 | 24.9 |
UPM 0621-6111 | 343586117 | 103726637 | 11.4 | 204.6 | 5.8 | 5.3 | 16.108 | 14.256 | 12.028 | 10.427 | 9.788 | 9.534 | 3.829 | 21.8 |
UPM 0901-6526 | 487375530 | 88399668 | -68.4 | 196.4 | 6.7 | 6.7 | 16.269 | 14.219 | 11.471 | 10.140 | 9.589 | 9.282 | 4.079 | 18.0 |
UPM 1104-6232 | 598106935 | 98845495 | -207.8 | -58.5 | 6.0 | 6.2 | 16.423 | 14.086 | 11.484 | 10.256 | 9.677 | 9.357 | 3.830 | 19.4 |
UPM 1105-5825A | 598956034 | 113681441 | -176.1 | 51.5 | 4.2 | 4.2 | 16.916 | $\cdots$ | 13.713 | 10.298 | 9.711 | 9.496 | $\cdots$ | 15.2 | aaCommon proper motion companion; see Table 4,bbSuperCOSMOS plate magnitudes suspect
UPM 1142-6440 | 632248846 | 91165839 | -183.4 | -205.1 | 32.2 | 16.8 | 16.067 | 14.081 | 11.418 | 10.393 | 9.846 | 9.481 | 3.688 | 23.7 |
UPM 1230-5736B | 675249073 | 116560461 | -243.0 | -29.3 | 6.9 | 6.5 | 14.928 | 12.828 | 10.998 | 9.694 | 9.033 | 8.785 | 3.134 | 19.8 | aaCommon proper motion companion; see Table 4,ccPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1230-5736A | 675783245 | 116623954 | -227.6 | -66.5 | 6.8 | 6.5 | 13.794 | 12.035 | 10.550 | 9.348 | 8.712 | 8.445 | 2.687 | 22.2 | aaCommon proper motion companion; see Table 4,ccPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1354-7121 | 751408498 | 67092222 | -165.0 | -132.7 | 6.5 | 6.8 | 12.598 | 10.665 | 8.750 | 8.549 | 7.920 | 7.672 | 2.116 | 19.3 |
UPM 1401-6405A | 757555362 | 93281664 | -172.2 | -140.9 | 7.1 | 7.0 | 15.536 | 13.909 | 12.787 | 10.296 | 9.649 | 9.407 | 3.613 | 24.1 | aaCommon proper motion companion; see Table 4
UPM 1454-5809 | 805350729 | 114653491 | -36.4 | 188.0 | 13.1 | 13.1 | 16.770 | 14.890 | 13.200 | 10.635 | 10.063 | 9.776 | 4.255 | 20.9 |
UPM 1523-5454 | 831355688 | 126325591 | -60.6 | 200.9 | 14.9 | 14.9 | 17.770 | 11.791 | 9.697 | 9.767 | 9.194 | 8.871 | 2.024 | 20.4 |
UPM 1635-5202 | 896058321 | 136679193 | -185.5 | 94.4 | 12.3 | 9.3 | 16.727 | 14.731 | 12.217 | 10.408 | 9.795 | 9.559 | 4.323 | 17.6 |
UPM 1641-4957 | 901066844 | 144129914 | -176.7 | -65.6 | 8.8 | 8.8 | $\cdots$ | 15.196 | 13.389 | 11.012 | 10.478 | 10.202 | 4.184 | 22.0 |
UPM 1710-5300 | 927664633 | 133174754 | -34.3 | -179.9 | 2.5 | 1.5 | 12.469 | 10.032 | 7.989 | 8.001 | 7.407 | 7.163 | 2.031 | 13.5 |
UPM 1732-4736 | 947129106 | 152581933 | -205.7 | -203.9 | 6.9 | 6.7 | 14.880 | 12.769 | 10.996 | 9.586 | 8.977 | 8.682 | 3.183 | 18.5 |
Table 4: Common Proper Motion Candidate Systems Primary | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | Distance | Secondary/Tertiary | $\mu_{\alpha}\cos\delta$ | $\mu_{\delta}$ | Distance | Separation | $\theta$ | notes
---|---|---|---|---|---|---|---|---|---|---
| (mas/yr) | (mas/yr) | (pc) | | (mas/yr) | (mas/yr) | (pc) | ($\arcsec$) | ($\arcdeg$) |
UPM 0111-7655A | 130.0 | 31.4 | 64.4 | UPM 0111-7655B | 182.5 | 35.6 | 109.9 | 46.8 | 241.7 | aaNot detected during automated search but noticed by eye during the blinking process,ffProper motions suspect
UPM 0245-8833A | 174.6 | 59.1 | 34.5 | UPM 0245-8833B | 175.7 | 89.7 | $\cdots$ | 6.8 | 15.0 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 0320-4847A | 230.8 | 147.8 | 37.7 | UPM 0320-4847B | 208.7 | 93.3 | 396.9 | 41.7 | 280.2 | aaNot detected during automated search but noticed by eye during the blinking process,bbPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 0533-5210A | -80.5 | 180.7 | 58.2 | UPM 0533-5210B | $\cdots$ | $\cdots$ | $\cdots$ | 7.7 | 161.4 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 0608-5301A | -131.1 | 208.6 | 37.1 | WD 0607-530B | -115.5 | 172.1 | [98.8] | 21.5 | 120.7 | aaNot detected during automated search but noticed by eye during the blinking process,ccWhite dwarf, plate distance [in bracket] is incorrect with a more accurate distance in the notes,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable, 34.2pc
UPM 0740-5207A | -37.7 | -221.5 | 38.9 | UPM 0740-5207B | 6.6 | -240.0 | $\cdots$ | 15.4 | 34.2 | aaNot detected during automated search but noticed by eye during the blinking process,eeSource not in 2MASS
NLTT 19906A | -150.5 | 338.0 | $\cdots$ | NLTT 19907B | -150.5 | 338.0 | $\cdots$ | 5.1 | 71.5 | ffProper motions suspect
| | | | UPM 0835-6018C | -184.6 | -17.7 | 50.3 | 113.0 | 49.3 | ffProper motions suspect
UPM 0837-6435A | -46.7 | 175.2 | 46.0 | UPM 0837-6435B | -29.6 | 158.4 | 106.6 | 9.4 | 281.6 | aaNot detected during automated search but noticed by eye during the blinking process,ffProper motions suspect
UPM 0846-7345A | -129.3 | 127.6 | 61.6 | UPM 0846-7345B | -125.4 | 112.9 | 88.4 | 24.2 | 184.6 | aaNot detected during automated search but noticed by eye during the blinking process,ffProper motions suspect
UPM 0928-5442A | -129.4 | 137.3 | 57.1 | UPM 0928-5442B | -91.0 | 147.9 | 93.0 | 26.6 | 87.9 | aaNot detected during automated search but noticed by eye during the blinking process,ggSuperCOSMOS plate magnitudes suspect,ffProper motions suspect
UPM 1040-5728A | -151.4 | 131.1 | 88.8 | UPM 1040-5728B | -224.2 | 138.9 | 164.7 | 7.3 | 102.1 | aaNot detected during automated search but noticed by eye during the blinking process,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable,ggSuperCOSMOS plate magnitudes suspect
UPM 1044-7053A | -217.5 | 17.6 | 57.8 | UPM 1044-7053B | -217.5 | 36.5 | 92.3 | 34.8 | 178.9 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 1104-7107A | -197.3 | -83.8 | 60.9 | UPM 1104-7107B | 20.3 | 23.1 | 44.6 | 10.8 | 28.6 | aaNot detected during automated search but noticed by eye during the blinking process,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable,ffProper motions suspect,ggSuperCOSMOS plate magnitudes suspect
UPM 1105-5825A | -176.1 | 51.5 | 15.2 | UPM 1105-5825B | -141.8 | 56.9 | 38.4 | 32.3 | 59.5 | aaNot detected during automated search but noticed by eye during the blinking process,ffProper motions suspect,ggSuperCOSMOS plate magnitudes suspect
UPM 1136-5358A | -215.3 | -74.1 | 36.3 | UPM 1136-5358B | -247.7 | -69.6 | $\cdots$ | 15.9 | 1.1 | aaNot detected during automated search but noticed by eye during the blinking process,bbPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position,eeSource not in 2MASS,ggSuperCOSMOS plate magnitudes suspect
NLTT 29430A | -175.4 | -42.5 | $\cdots$ | UPM 1203-4910B | -177.1 | -40.5 | 97.5 | 41.4 | 270.2 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 1230-5736A | -227.6 | -66.5 | 22.2 | UPM 1230-5736B | -243.0 | -29.3 | 19.8 | 293.1 | 257.5 | bbPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position
UPM 1401-6405A | -172.2 | -140.9 | 24.1 | UPM 1401-6405B | -185.2 | -149.3 | 76.3 | 30.1 | 339.1 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 1414-6023A | -200.8 | -87.5 | 78.4 | UPM 1414-6023B | -189.1 | -61.3 | $\cdots$ | 11.3 | 21.5 | aaNot detected during automated search but noticed by eye during the blinking process,eeSource not in 2MASS
NLTT 37372A | -180.8 | -79.8 | 80.4 | UPM 1430-7722B | -67.0 | -44.4 | 280.6 | 18.8 | 37.5 | aaNot detected during automated search but noticed by eye during the blinking process,ffProper motions suspect
UPM 1503-5007A | -238.2 | -193.2 | 57.6 | UPM 1503-5007B | $\cdots$ | $\cdots$ | $\cdots$ | 7.5 | 358.6 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 1528-5839A | 2.1 | -197.2 | 33.3 | UPM 1528-5839B | 5.9 | -197.8 | $\cdots$ | 13.0 | 250.3 | aaNot detected during automated search but noticed by eye during the blinking process,eeSource not in 2MASS
UPM 1548-5045A | -92.4 | -175.3 | 30.6 | UPM 1548-5045B | -57.1 | -155.8 | 41.9 | 14.3 | 36.3 | aaNot detected during automated search but noticed by eye during the blinking process,ffProper motions suspect
UPM 1658-5311A | -163.5 | -229.3 | 28.6 | UPM 1658-5311B | -11.6 | -134.1 | 33.0 | 84.4 | 123.6 | aaNot detected during automated search but noticed by eye during the blinking process,ggSuperCOSMOS plate magnitudes suspect,ffProper motions suspect
NLTT 44051A | -189.2 | 19.2 | 38.5 | UPM 1703-4934B | -28.6 | 72.6 | 40.7 | 8.1 | 315.1 | aaNot detected during automated search but noticed by eye during the blinking process,bbPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position,ggSuperCOSMOS plate magnitudes suspect,ffProper motions suspect
WT 549A | -94.4 | -155.3 | 65.5 | UPM 1719-6255B | -94.0 | -153.7 | 105.6 | 84.0 | 185.0 | aaNot detected during automated search but noticed by eye during the blinking process
UPM 1756-7406A | -78.4 | -201.7 | 37.3 | UPM 1756-7406B | -279.1 | -203.5 | 83.4 | 5.7 | 95.4 | aaNot detected during automated search but noticed by eye during the blinking process,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable
UPM 1950-6013A | -31.2 | -268.7 | 87.1 | UPM 1950-6013B | $\cdots$ | $\cdots$ | $\cdots$ | 26.7 | 196.1 | aaNot detected during automated search but noticed by eye during the blinking process
NLTT 48738A | 52.5 | -173.0 | 43.4 | UPM 2008-6305B | 77.4 | -181.2 | 125.7 | 5.7 | 67.2 | aaNot detected during automated search but noticed by eye during the blinking process,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable
UPM 2036-5548A | 45.1 | -174.6 | 114.8 | UPM 2036-5548B | $\cdots$ | $\cdots$ | 233.8 | 5.3 | 66.1 | aaNot detected during automated search but noticed by eye during the blinking process,bbPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable
UPM 2049-5415A | 180.4 | 53.2 | 44.3 | UPM 2049-5415B | 197.2 | 44.3 | 64.7 | 20.2 | 314.8 | aaNot detected during automated search but noticed by eye during the blinking process
NLTT 50475A | 133.9 | -198.0 | 12.5 | UPM 2106-5706B | 162.6 | -178.8 | 76.6 | 166.4 | 356.4 | aaNot detected during automated search but noticed by eye during the blinking process,ddNumber of relations used for distance estimate $<$ 6; plate distance less reliable
|
arxiv-papers
| 2010-07-19T15:42:40 |
2024-09-04T02:49:11.730515
|
{
"license": "Public Domain",
"authors": "Charlie T. Finch, Norbert Zacharias, Todd J. Henry",
"submitter": "Charlie Finch",
"url": "https://arxiv.org/abs/1007.3186"
}
|
1007.3294
|
# Testing quantum adiabaticity with quench echo
H. T. Quan and W. H. Zurek whzurek@gmail.com http://public.lanl.gov/whz/
Theoretical Division, MS B213, Los Alamos National Laboratory, Los Alamos, NM,
87545, U.S.A.
###### Abstract
Adiabaticity of quantum evolution is important in many settings; one example
is the adiabatic quantum computation (AQC). Nevertheless, to date, there is no
effective method available for testing the adiabaticity of the evolution for
the case where the eigenenergies of the driven Hamiltonian are not known. We
propose a simple method for checking adiabaticity of a quantum process for an
arbitrary quantum system. We further propose an operational method for finding
more efficient protocols that approximate adiabaticity, and suggest a
“uniformly adiabatic” quench scheme based on the Kibble-Zurek mechanism for
the case where the initial and the final Hamiltonians are given. This method
should help in implementing AQC and other tasks where preserving the system in
the ground state of a time-dependent Hamiltonian is desired.
###### pacs:
03.65.Aa, 03.65.Vf, 03.67.-a, 05.30.Rt
## I Introduction
In a quantum quench process, when the Hamiltonian of a quantum system is
driven from $H_{0}$ to $H_{1}$, interstate excitations of the system usually
occur, owing to the non-commutativity of the Hamiltonians at different
moments. However, when the quench process is slow enough, the interstate
excitations will be suppressed. According to the quantum adiabatic theorem
adiabatictheorem , when the condition for quantum adiabatic approximation
$\left\langle\Phi_{\mathrm{ground}}(t)|\frac{dH(t)}{dt}|\Phi_{\mathrm{excited}}(t)\right\rangle\ll\Delta(t)^{2}$
is satisfied, the system will remain in the ground state – its evolution will
be adiabatic – except for some special situations clarification . Here $H(t)$
is the changing Hamiltonian, and $\Delta(t)$ is the minimal energy gap between
the ground state $\left|\Phi_{\mathrm{ground}}(t)\right\rangle$ and the first
excited state $\left|\Phi_{\mathrm{excited}}(t)\right\rangle$ of $H(t)$.
In order to ensure that a quantum system evolves adiabatically, one usually
needs to find the energy spectrum (or at least the smallest energy gap
$\Delta$) of the driven Hamiltonian. One can then use the quantum adiabatic
theorem to choose a proper time scale, so that the conditions for quantum
adiabatic approximation are satisfied and the evolution remains adiabatic.
Nevertheless, in practice, neither eigenenergies nor eigenstates of a complex
quantum many-body system are easy to obtain. This is often the case in
implementing the adiabatic quantum computation (AQC) qac as well as quantum
annealing nishimori98 ; anarb2008 . Hence, one does not have the ingredients
to use quantum adiabatic theorem. One cannot count on the direct comparison
between the final state and the instantaneous ground state of the final
Hamiltonian, either. Thus, it would be useful to find a reliable method for
evaluating the adiabaticity of an evolution under an arbitrary Hamiltonian,
especially when one has no idea about the eigenstates and/or eigenenergies of
the system (except at the initial moment).
The quench echo method we propose here is one solution to the above problem.
It will allow one to evaluate unambiguously the adiabaticity of a process.
What is more, it can help one find the efficient “uniformly adiabatic” quench
path in the parameter space of the Hamiltonian. Such ideas may have
applications in the implementation of AQC qac . This paper is organized as
follows: In section II, we introduce the quench echo method and briefly
explain its underlying physics. In section III, we use a simple model to
demonstrate main ideas of the general theory. In Section IV, we propose a
uniformly adiabatic scheme that is based on the application of the Kibble-
Zurek mechanism (KZM) to quantum phase transitions. In Section V we give
discussions and conclusions.
## II Quench echo
Consider a system described by the Hamiltonian $H(g(t))$, where $g(t)$ is a
time-dependent parameter. The system is initially prepared in the ground state
of $H(g(t=0))$. The system evolves under the influence of the driven
Hamiltonian, which changes from $H(g(t=0))$ to $H(g(t=T))$. Our aim is to test
the adiabaticity of this evolution, but we know nothing about the
eigenenergies and eigenstates of the time-dependent Hamiltonian except at
$t=0$. Hence we cannot count on the comparison between the evolving state and
the instantaneous ground state. Neither can we use the adiabatic theorem.
Nevertheless, we can apply a backward “echo” quench following the initial
quench (from $t=0$ to $t=T$). That is, one extends the evolution from $t=T$ to
$t=2T$ footnote :
$H(g(t))=\left\\{\begin{array}[]{c}\begin{split}&H(g(t)),(0<t<T)\\\
&H(g(2T-t)),(T<t<2T)\end{split}\end{array}\right..$ (1)
The final Hamiltonian is identical to the initial Hamiltonian
$H(g(t=2T))\equiv H(g(t=0))$. Hence, we can use the fidelity of the initial
state, e.g., the ground state of $H(g(t=0))$ and the final evolving state as a
criterion for the adiabaticity of the evolution.
$F=\left|\left\langle
GS\right|\hat{T}e^{-i\int_{T}^{2T}H(g(2T-t))dt}\hat{T}e^{-i\int_{0}^{T}H(g(t))dt}\left|GS\right\rangle\right|^{2},$
(2)
where $\hat{T}$ is the time-ordered operator and $\left|GS\right\rangle$ is
the ground state of $H(g(t=0))$. When the fidelity $F$ is greater than a
threshold value close to unity, (e.g., $0.999$, the error tolerance is
$0.001$), the whole process $(0<t<2T)$ is adiabatic. This implies that the
forward quench process $(0<t<T)$ is adiabatic. The underlying mechanism for
this “quench echo” method is straightforward: Except for the eigenstate of the
Hamiltonian at the initial moment $H(g(t=0))$, we do not have any information
about its eigenstates at other moments. Hence, we can only quench the
Hamiltonian back, so that it goes back to its initial $H(g(t=0))$, and we can
measure the final state (and compare it with the initial state). The quench
echo protocol (Eq. (1)) ensures that the excitation probabilities in the
forward quench process and those in the backward quench process are similar
(but not identical; see Refs. bogdan06 ; Mosseri08 ). As a result, when the
forward process is adiabatic (no excitations), so is the backward quench
process. Otherwise both the forward and the backward processes are
nonadiabatic, and the phase accumulated between transitions (known as the
Stückelberg phase) may result in constructive or destructive interference
sahel ; nori . Usually the excitations in the forward and the backward
processes cannot cancel each other out (but see Ref. nori ). Hence, through
this quench echo method, without knowing about the eigenenergies and
eigenstates, one is able to evaluate the adiabaticity of an arbitrary
evolution in most cases. Nevertheless, in some special cases (e. g., impulse
evolution, which is so fast that the state of the system is frozen) the final
fidelity is equal to unity, but the process may not be adiabatic. A solution
to this problem is to let the system evolve freely for some time before the
backward quench. We will discuss this in detail in the next section. By
utilizing quench echo one can even find a uniformly adiabatic quench protocol
for a given Hamiltonian by repeating the above process with different quench
time scales.
## III A case study: Ising chain in a transverse magnetic field
Figure 1: Three regimes of the quench dynamics: (a) nearly adiabatic regime,
$\tau_{Q}=150$; (b) the intermediate regime $\tau_{Q}=35$; and (c) nearly
impulse regime $\tau_{Q}=0.004$. The horizontal axis represents the parameter
$g(t)$, which varies between 0 and 10, and the vertical axis represents the
probability of the system being in the instantaneous ground state during the
evolution. There is a quantum phase transition at $g_{c}=1$. The red solid
line represents the forward quench (from $g_{0}=10$ to $g_{T}=0$) and the
green dashed line represents the quench echo (from $g_{T}=0$ to $g_{0}=10$).
In both the nearly adiabatic and the nearly impulse regimes, the fidelity at
final moment is close to unity. In the first row (a)-(c), there is no time
delay at the turnaround point. In the second row (a′)-(c′), the delay time at
$g_{T}=0$ is $\Delta t=0.1$, $0.2$, $0.3$, $0.4$. In the third row
(a′′)-(c′′), the delay time at $g_{T}=0$ is $\Delta t=10$, $20$, $30$, $40$.
The number of the spins in the Ising chain is $N=50$.
We now use a simple model to demonstrate our central ideas: Consider the
quench dynamics of an Ising model in an transverse magnetic field sachdev .
The time-dependent Hamiltonian is
$H(t)=-J\sum_{i=1}^{N}\left[\sigma_{i}^{x}\sigma_{i+1}^{x}+g(t)\sigma_{i}^{z}\right],$
(3)
where $J$ indicates the energy scale; $\sigma_{i}^{\alpha}$, $\alpha=x,y,z$ is
the Pauli matrix on the $i$ th lattice site; and $g(t)$ is the reduced
strength of the magnetic field, which varies with time. It is known that for
this model there is a finite energy gap $\Delta=2J\frac{\pi}{N}$ at
$g=\cos(\pi/N)$ when the size of the system $N$ is finite. For simplicity, we
consider a linear quench protocol
$g(t)=\left\\{\begin{array}[]{c}\begin{split}&g_{0}-\frac{t}{\tau_{Q}},(0<t<(g_{0}-g_{T})\tau_{Q}),\\\
2g_{T}-&g_{0}+\frac{t}{\tau_{Q}},((g_{0}-g_{T})\tau_{Q}<t<2(g_{0}-g_{T})\tau_{Q}),\end{split}\end{array}\right.$
(4)
where $\tau_{Q}$ is the time scale of the quench. The larger the $\tau_{Q}$,
the slower the quench. In the forward quench the strength of the magnetic
field is ramped from $g=g_{0}$ to $g=g_{T}$ continuously, and in the quench
echo, it is ramped back from $g=g_{T}$ to $g=g_{0}$, where $g_{T}$ is the
turnaround point. Initially the system is prepared in the ground state of
$H(g=g_{0})$. When one quenches the system at different rates (by choosing
different $\tau_{Q}$), the fidelity (2) will be different.
The Hamiltonian of the Ising model (3) can be decoupled into $N$ independent
fermionic modes sachdev .
$H(t)=\sum_{k}\Lambda_{k}(g(t))\left[\left|+(t)\right\rangle_{k}\left\langle+(t)\right|_{k}-\left|-(t)\right\rangle_{k}\left\langle-(t)\right|_{k}\right],$
(5)
where $\left|+(t)\right\rangle_{k}$ and $\left|-(t)\right\rangle_{k}$ are the
two instantaneous eigenstates of the $k$ mode. Their corresponding
eigenenergies are $\pm\Lambda_{k}(g(t))$, and
$\Lambda(g(t))=J\sqrt{g(t)^{2}-2g(t)\cos{k}+1}$. Here $k=(2s+1)\pi/N$,
$s=0,1,2,\cdots,N/2-1$ is the wave vector, and the number of spins $N$ is
even.
We write the Schrödinger equation $i\hbar\frac{\partial}{\partial
t}\left|\Phi(t)\right\rangle=H(t)\left|\Phi(t)\right\rangle$ in the
instantaneous eigenbases $\left|+(t)\right\rangle_{k}$ and
$\left|-(t)\right\rangle_{k}$ of $H(t)$, where
$\left|\Phi(t)\right\rangle=\prod_{k}\alpha_{k}(t)\left|+(t)\right\rangle_{k}+\beta_{k}(t)\left|-(t)\right\rangle_{k}$.
For simplicity we choose $\hbar=1$ hereafter. For both the forward quench
($0<t<(g_{0}-g_{T})\tau_{Q}$) and the backward
($(g_{0}-g_{T})\tau_{Q}<t<2(g_{0}-g_{T})\tau_{Q}$) process , the Schrödinger
equation can be written as
$\imath\frac{d}{dt}\left[\begin{array}[]{c}\alpha_{k}(t)\\\
\beta_{k}(t)\end{array}\right]=\left[\begin{array}[]{c
c}\begin{split}2\Lambda_{k}(g(t)),\frac{-iJ^{3}\sin{k}}{2\Lambda_{k}^{2}(g(t))}\frac{dg(t)}{dt}\\\
\frac{iJ^{3}\sin{k}}{2\Lambda_{k}^{2}(g(t))}\frac{dg(t)}{dt},-2\Lambda_{k}(g(t))\end{split}\end{array}\right]\left[\begin{array}[]{c}\alpha_{k}(t)\\\
\beta_{k}(t)\end{array}\right],$ (6)
where the initial condition for Eq. (6) is $\alpha_{k}(t=0)=0$,
$\beta_{k}(t=0)=1$. The modulus square of the overlap between the final state
of Eq. (6) and the instantaneous ground state at $g=g_{0}$ gives the fidelity
(2)
$F=P_{GS}(2(g_{0}-g_{T})\tau_{Q})=\prod_{k>0}\left|\beta_{k}(2(g_{0}-g_{T})\tau_{Q})\right|^{2}.$
(7)
In the following, we will focus on the solution of the Eq. (7). We will
consider both the numerical and the analytical results.
### III.1 Kibble-Zurek mechanism and three regimes
Before the quantitative study of the fidelity and its relation with the time
scales of the quench, we describe the Kibble-Zurek mechanism (KZM) kibble ;
zurek of second-order phase transitions, which provides a quantitative
understanding of the quench process. The KZM describes e.g., the relation
between the density of topological defects, which are generated during
quenching across a phase transition, and the time scale of the quench (see
Ref. jacek09review for a good review). The KZM was first introduced in the
classical phase transitions kibble ; zurek , and later generalized to quantum
phase transitions zurek05 . In our study, however, we will not focus on the
density of topological defects, but on the adiabaticity of the evolution of
the system.
A quantum phase transition is characterized by a vanishing excitation gap
$\Delta(g(t))\approx\Delta_{0}|g(t)-g_{c}|^{z\nu}$ and a divergent correlation
length $\xi\approx\xi_{0}/|g(t)-g_{c}|^{\nu}$, where $z$ and $\nu$ are the
critical exponents, and $\Delta_{0}$ and $\xi_{0}$ are constants sachdev . We
define a dimensionless distance from the critical point $g_{c}$ by
$\epsilon(t)=\frac{g(t)-g_{c}}{g_{c}}.$ (8)
A generic $\epsilon(t)$ can be linearized near the critical point
$\epsilon(t)=0$ as explain :
$\epsilon(t)\approx-\frac{t}{\tau_{Q}}.$ (9)
There are two interlinked time scales during a quench: the system reaction
time given by the inverse of the gap
$\tau(\epsilon(t))=1/\Delta_{0}|g(t)-g_{c}|^{z\nu}$ and the time scale of
transition given by $|g(t)-g_{c}|^{z\nu}/\frac{d}{dt}|g(t)-g_{c}|^{z\nu}$.
Away from the critical point the reaction time is small in comparison with the
time scale of transition and the evolution is adiabatic. Near the critical
point, however, the opposite situation occurs and the evolution is
approximately impulse (the state of the system is frozen out). The boundary
$\hat{t}$ between the two regions is determined by the relation
$\tau(\epsilon(t))=\epsilon/\dot{\epsilon}|_{\hat{t}}$, or
$\frac{1}{|g(\hat{t})-g_{c}|^{z\nu}}\sim\frac{|g(\hat{t})-g_{c}|^{z\nu}}{\frac{d}{dt}|g(\hat{t})-g_{c}|^{z\nu}}.$
(10)
That is, $\hat{t}\sim(\frac{\tau_{Q}}{\hat{t}})^{z\nu}$, which gives
$\hat{t}\sim\tau_{Q}^{\frac{z\nu}{1+z\nu}}$ zurek ; zurek05 . For the Ising
model, we have $z=\nu=1$ resulting in $\hat{t}\sim\tau_{Q}^{\frac{1}{2}}$
zurek05 . According to KZM when $t\in(-\hat{t},\hat{t})$, the system will not
evolve (the wavefunction will be frozen). Outside this time interval the
system will evolve approximately adiabatically.
For an infinitely large system, the energy gap is vanishingly small at the
critical point, and one can always find a $\hat{t}$. According to the KZM this
implies that, no matter how slow one quenches the Hamiltonian in an infinite
system, the evolution across the critical point can never be adiabatic. For a
finite-size system, however, there is a finite energy gap even at the critical
point. When one quenches the system sufficiently slowly (large $\tau_{Q}$),
$\hat{t}$ approaches very near the critical point where – for a finite systems
– scalings no longer hold. As a consequence, the KZM does not lead to simple
scaling, as $\tau(\epsilon(t))=\epsilon/\dot{\epsilon}|_{\hat{t}}$ leads to a
more difficult equation which has to be solved to obtain $\hat{t}$. zurek2000
. Indeed – in accord with the adiabatic theorem – the KZM predicts that when
$\tau_{Q}$ is larger than the inverse of the gap the transition will remain
adiabatic throughout. Thus, a finite energy gap allows an adiabatic evolution
across the critical point when the Hamiltonian is driven sufficiently slowly.
This is the adiabatic quench regime. By contrast, when one quenches the
Hamiltonian very fast (small $\tau_{Q}$), there is a big $\hat{t}$ and there
is an approximately impulse regime for $t\in(-\hat{t},\hat{t})$ when the
quench is essentially instantaneous. In this time interval the system will
approximately cease to evolve – its wavefunction will be frozen. This is the
so-called impulse regime zurek05 ; bogdan05 . When one chooses a time scale of
quench $\tau_{Q}$ between the above two limiting cases, the system will evolve
adiabatically when either $t<-\hat{t}$ or $t>\hat{t}$, and will be frozen when
$t\in(-\hat{t},\hat{t})$. We call this regime the intermediate regime. We can
summarize the quench behavior as follows: For a finite-size system, when
$\tau_{Q}$ is large enough, the evolution will be adiabatic; When $\tau_{Q}$
is extremely small, the state of the system will be frozen; When $\tau_{Q}$ is
in between these two limiting cases, the process is in the intermediate
regime.
### III.2 Numerical and analytical results
Having obtained the qualitative understanding of the quench dynamics from the
above KZM arguments, in the following we will study the Ising model
quantitatively, and compare the results with the estimates obtained above. We
consider a spin chain with a finite size $N=50$, and start evolving it at
$g_{0}=10$ and let it turn around at $g_{T}=0$. There is a finite energy gap
for this system at the quantum critical point $g_{c}=1$. We choose three
different quench time scales $\tau_{Q}=150$, $\tau_{Q}=35$, and
$\tau_{Q}=0.004$ which correspond to the adiabatic, intermediate, and impulse
regimes. The system evolves under the time-dependent Schrödinger equation. We
plot the probability $P_{GS}(g)$ in the instantaneous ground state as a
function of the controlling parameter $g$ during the quench process in Fig.
1a-1c.
Figure 2: The fidelity as a function of the time scale of quench $\tau_{Q}$.
Panels (a)-(d) represent different free evolution time at turnaround point
$g_{T}=0$ before the quench echo. Here $N=50$, and start point at $g_{0}=10$,
and the times of free evolutions are chosen to be $\Delta t=0$, $\Delta
t=0.1$, $\Delta t=0.3$, $\Delta t=0.7$. It can be seen that in the adiabatic
regime the fidelity is always equal to unity, but in the impulse regime it is
less than unity and varies with the time of free evolution $\Delta t$.
From Fig. 1a it can be seen that when the time scale of the quench is
relatively large, the system evolves almost adiabatically in the whole range
of the parameter $g_{T}=0<g<g_{0}=10$, $P_{GS}(g)$ is always close to unity
(except a tiny decay and partial revival at the critical points. So is the
fidelity of the quench echo (see Fig. 1a).
When the time scale of the quench is reduced to $\tau_{Q}=35$ (see Fig. 1b),
the quench dynamics enters the intermediate regime. It can be seen that away
from the critical point, the evolution is adiabatic. But near the critical
point, the probability in the instantaneous ground state $P_{GS}(g)$ decays
sharply and oscillates rapidly. This is due to the interstate transitions at
the anti-cross point. Soon after passing through the quantum critical point
the adiabatic evolution resumes.
When the time scale of the quench is further reduced to $\tau_{Q}=0.004$
(almost instantaneous quench), the wave function of the system is nearly
frozen. Hence, the probability of being in the instantaneous ground state is
simply equal to the overlap of the initial state and the instantaneous ground
state. In the backward quench, the same situation arises. Because in both the
forward and the backward quench, the wave functions of the system are frozen,
and hence are almost identical, the curves of $P_{GS}(g)$ of the forward and
the backward quench almost collapse onto the same curve (see Fig. 1c), and the
fidelity at $t=2(g_{0}-g_{T})\tau_{Q}$ is close to unity.
We also plot the fidelity as a function of the quench time scale $\tau_{Q}$
(see Fig. 2a). It can be seen that in both the impulse regime
$(\tau_{Q}<\thicksim 10^{-3})$ and the adiabatic regime $(\tau_{Q}>\thicksim
10^{2.5})$, the fidelity is equal to unity. This agrees with our intuition.
Meanwhile, in the intermediate regime, $\thicksim 10^{-3}<\tau_{Q}<\thicksim
10^{2.5}$, the fidelity oscillates rapidly (see Fig. 2a). When we plot the
fidelity as a function of the quench time $\tau_{Q}$, instead of
$\ln{\tau_{Q}}$, we found that there is a regular quasi-periodic oscillation
(see Fig. 3). We obtain an accurate expression of fidelity in the intermediate
regime,
$\begin{split}F\approx\prod_{k>0}^{\pi/2}&\left|e^{-2\pi\tau_{Q}\sin^{2}{k}}e^{i\phi_{k}}+\frac{2\pi\tau_{Q}\sin^{2}{k}}{\Gamma^{2}(1-i\tau_{Q}\sin^{2}{k})}e^{-\pi\tau_{Q}\sin^{2}{k}}e^{-i\phi_{k}}\right|^{2},\end{split}$
(11)
where
$\phi_{k}=2\tau_{Q}[(-g_{T}+\cos{k})^{2}+\sin^{2}{k}\ln\sqrt{4\tau_{Q}(-g_{T}+\cos{k})^{2}}]$,
and $\Gamma(1-i\tau_{Q}\sin^{2}{k})$ is the Gamma function (see Appendix A for
details of the derivation). From Fig. 3 it can be seen that the analytical
results agree with the numerical simulations, and that the fidelity oscillates
quasi-periodically with the increase of $\tau_{Q}$ as expected. This
oscillation can actually also be observed in Fig. 1 (see Fig. 1b’ and Fig.
1c’).
Numerical simulations agree with the results obtained from the KZM very well,
i.e., they account for three regimes that correspond to different $\tau_{Q}$.
We are especially interested in the first regime – the adiabatic regime. From
Fig. 1a and Fig. 1c, it can be seen that in both the adiabatic regime and the
impulse regime, the fidelity is close to unity. In the next subsection, we
will introduce a method to eliminate the “degeneracy” of the adiabatic regime
and the impulse regime.
Figure 3: Fidelity as a function of the time scale $\tau_{Q}$. Here the black
dashed line represents the numerical results whereas the red solid line
represents the analytical results (Eq. (11)). It can be seen that the
analytical results (Eq. (11)) agree well with the numerical results except for
the case $\tau_{Q}\to 0$, where the quench process enters the impulse regime.
The length of the spin chain is $N=50$, and the delay time at the turnaround
point is $\Delta t=0$.
### III.3 Free evolution and decay of fidelity
To distinguish the adiabatic and the impulse regime using quench echo, one can
let the system evolve freely for some time at the turnaround point before
quenching back. A study of the Landau-Zener problem with waiting at the
minimum gap has been reported in Ref. sen10 . It was observed that the waiting
influences the excitation probability. Similarly in our study the free
evolution at the turnaround point leads to a decay in the fidelity in the
impulse regime, but makes no difference in the adiabatic regime (see Figs.
1a′-1c′ and Figs. 1a′′-1c′′). The reason is straightforward. Let us first
consider the adiabatic regime. Because the system is always in its
instantaneous ground state, the effect of the free evolution is simply a
global phase factor, which does not affect the fidelity (see Figs. 1a′, 1a′′,
and Figs. 2b-2d). In the impulse regime, the wavefunction before the free
evolution is the ground state of the initial Hamiltonian $H(g_{0}=10)$, and
alternatively, a superposition of the excited and the ground states of
$H(g_{T}=0)$. The excited and ground states acquire different phase factors
during the free evolution. Thus the wave function acquires relative phase
factors in its components and is no longer the ground state of $H(g_{T}=0)$,
but a superposition of its ground and excited states. Hence, in the impulse
regime when one quenches the system back to the initial Hamiltonian
$H(g_{0}=10)$, the system will no longer be in its ground state, but in a
superposition of the ground state and the excited states. As a result, the
fidelity is less than unity (see Fig. 1c′, Fig. 1c′′, and Figs. 2b-2d). The
length of time of the free evolution $\Delta t$ also influences the fidelity.
One can analytically calculate the fidelity as a function of the time of free
evolution $\Delta t$:
$\left|-(g=+\infty)\right\rangle_{k}=\sin{\left[\frac{\theta_{k}}{2}\right]}\left|+(g_{T})\right\rangle_{k}+\cos{\left[\frac{\theta_{k}}{2}\right]}\left|-(g_{T})\right\rangle_{k},$
where $\theta_{k}=\arctan(\frac{-\sin k}{\cos k-g_{T}})$. After free evolution
for $\Delta t$, the wave function becomes
$\sin{\left[\frac{\theta_{k}}{2}\right]}e^{-i\Lambda_{k}(g_{T})\Delta
t}\left|+(g_{T})\right\rangle_{k}+\cos{\left[\frac{\theta_{k}}{2}\right]}e^{i\Lambda_{k}(g_{T})\Delta
t}\left|-(g_{T})\right\rangle_{k}.$
The fidelity can then be calculated as
$F=\prod_{k>0}\left(1-\frac{\sin^{2}{k}\sin^{2}{[\Lambda_{k}(g_{T})\Delta
t]}}{1-2g_{T}\cos{k}+g^{2}_{T}}\right).$ (12)
Note that for a fixed chain size, the value of $\Delta t$ needed to scramble
all the relevant phases is relevant to the range of the spectrum of the system
or the size of gap $\Lambda_{k}(g_{T})$ of different $k$ at the turnaround
point $g_{T}$. When $\Lambda_{k}(g_{T})$ is very small, i.e., the energy
spectrum of the system is concentrated within a very small energy range, one
needs to wait for a long time in order to scramble all the relevant phases:
$\Delta t$ is inversely proportional to the energy scale $J$ of
$\Lambda_{k}(g_{T})$. For a spin chain of $N=50$, when the time of free
evolution is very short, e.g., $\Delta t=0.1$, there is a pronounced decay in
the fidelity in the impulse regime (see Fig. 2b). The analytical result gives
$F\approx 0.882$, which agrees with the numerical result. The fidelity
decreases with the increase of time of the free evolution. The fidelity decays
to $0.002$ when $\Delta t=0.7$ (see Fig. 2d). Hence the quench echo with a
free evolution at the turnaround point can distinguish the adiabatic and the
impulse regime. Our numerical results confirm our theoretical predictions.
## IV Beyond the linear quench
In the above discussion, we focused on the linear quench. One may repeat the
above process with different $\tau_{Q}$ until one finds the smallest
$\tau_{Q}^{c}$, under which the process is sufficiently adiabatic, for example
$F\geq 0.9$. Nevertheless, the linear quench with $\tau_{Q}^{c}$ obtained
above may waste a lot of time. The reason is obvious: in different regions of
the parameter $g$, the energy gaps are different. According to KZM, different
energy gaps correspond to different relaxation time $\tau$. For a linear
quench protocol, we are treating the whole range of the parameter uniformly,
and the relaxation time is determined by the global minimal energy gap. Thus,
we waste a lot of time. Usually we want to ensure that the process not only
nearly adiabatic but also as fast as possible. In the following we will
consider nonlinear quench.
### IV.1 Adjusting quench rate to the instantaneous gap
An improved scheme is to divide the whole range of the parameter into many,
e.g., $M$, parts with equal length $(g_{0}-g_{T})/M$, and then apply the above
linear quench protocol to these ranges separately to find the uniformly
adiabatic quench for each range $\tau_{Q}^{ci}$, $i=1,2,\cdots,M$. We can also
use the KZM to find a uniformly adiabatic quench protocol. From the discussion
in Section III.A we know that the transition time scale is given by the
absolute value of $\Delta(g(t))/\frac{d}{dt}\Delta(g(t))$. Meanwhile, the
relaxation time scale is given by $1/\Delta(g(t))$. When the former is many
times larger than the latter, the process should be uniformly adiabatic. That
is, when the parameter $g(t)$ satisfies the relation
$\left|\frac{\Delta(g(t))}{\frac{d}{dt}\Delta(g(t))}\right|=\frac{\gamma}{\Delta(g(t))}.$
(13)
where $\gamma$ is a constant many times larger than unity, e.g., $\gamma=10$,
the process is uniformly adiabatic in the sense that the ratio of two time
scales remains a constant. Such a quench scheme is better than the linear
quench. The solution to the above ordinary differential equation is
$\Delta(g(t))=\frac{1}{\mp\frac{1}{\gamma}t+c},$ (14)
where $\mp$ corresponds to the sign of $\Delta/\dot{\Delta}$ on the left-hand-
side of Eq. (13) being positive or negative, and $c$ is a constant of
integration. For simplicity $c$ can be chosen such that at $t=0$ $\Delta$ in
Eq. (14) is the minimal gap. Now, we know exactly the energy gap as a function
of the controlling parameter (see Fig. 4a) gap
$\begin{split}\Delta(g(t))=&2J\sqrt{1-2g(t)\cos(\frac{\pi}{N})+g^{2}(t)}.\end{split}$
(15)
Therefore, $c$ can be determined by $g(t=0)=\cos{(\pi/N)}$. Combining Eqs.
(14) and (15), we find the following uniformly adiabatic quench protocol (see
Fig. 4b)
$g_{\mathrm{KZ}}(t)=\left\\{\begin{array}[]{c}\begin{split}&\cos(\frac{\pi}{N})-\sqrt{-\left(\sin(\frac{\pi}{N})\right)^{2}+\frac{(\gamma)^{2}}{4J^{2}\left(t+\frac{\gamma}{2J\sin{(\pi/N)}}\right)^{2}}},(-\frac{\gamma}{2J\sin{(\pi/N)}}<t<0)\\\
&\cos(\frac{\pi}{N})+\sqrt{-\left(\sin(\frac{\pi}{N})\right)^{2}+\frac{(\gamma)^{2}}{4J^{2}\left(-t+\frac{\gamma}{2J\sin{(\pi/N)}}\right)^{2}}},(0<t<\frac{\gamma}{2J\sin{(\pi/N)}})\end{split}\end{array}.\right.$
(16)
It can be seen that the time required for the whole process (quenching the
controlling parameter from $g=0$ to $g=\infty$) is given by
$\Delta t_{\mathrm{KZ}}=\frac{\gamma}{J\sin{(\pi/N)}}\approx\frac{\gamma
N}{J\pi},$ (17)
or $\Delta t_{\mathrm{KZ}}=\frac{2\gamma}{\Delta_{\min}}$, which is
proportional to the chain size $N$ and the ratio $\gamma$, and inversely
proportional to the minimum energy gap
$\Delta_{\min}=2J\sin\frac{\pi}{N}\approx 2J\frac{\pi}{N}$. In linear quench
the minimal time required for the adiabatic evolution grows with the system
size like $N^{2}$ jacek . The quench scheme of Eq. (16) is obviously better.
This agrees with previous studies that “non-linear” quench can improve the
adiabaticity (minimize excitation) jacek09review ; nonlinear1 ; nonlinear2 .
The energy gap and the protocols for uniformly adiabatic quench (Eq. (16)) are
shown in Fig. 4.
Figure 4: (a) Energy gap as a function of the controlling parameter $g$ for a
finite size chain. Here the solid line indicates the spin chain of $N=50$ and
the dashed line indicates the gap of an infinite chain. (b) The uniformly
adiabatic quench protocol associated with the KZM criterion (solid) (16) and
associated with RC criterion (dashed) (19). Here the ratio $\gamma=2$, and the
energy scale $J=1/2$. We have chosen the condition $g(t=0)=\cos{(\pi/N)}$.
Note that the criterion for uniformly adiabatic evolution derived from the KZM
(Eq. (13)) is similar, but not identical, to the criterion proposed by Roland
and Cerf that was derived from the quantum adiabatic theorem (See Eq. (17) of
reference cerf ). In the RC model the energy gap is inversely proportional to
$\sqrt{N}$, and the minimum time required is proportional to $\sqrt{N}$. But
in the Ising chain, the energy gap is inversely proportional to $N$, and the
minimum time required is proportional to $N$. It can be proved that if one
uses Roland and Cerf’s criterion to evaluate the minimum time required for the
uniformly adiabatic evolution, the minimum time is also proportional to $N$.
It is interesting to compare the two criteria for uniformly adiabatic
evolution in the Ising chain. In the following we will first solve the
equation of the quench protocol for uniformly adiabatic evolution
$g_{\mathrm{RC}}(t)$ associated with the Roland and Cerf’s criterion and then
simulate the dynamic evolution of the Ising chain with both
$g_{\mathrm{RC}}(t)$ and $g_{\mathrm{KZ}}(t)$. We will fix the time of quench
process, and compare the fidelity of the two protocols. The Roland and Cerf’s
criterion (see Eq. (17) of Ref. cerf ) is
$\left|\frac{d}{dt}\left|g(t)-g_{c}\right|\right|=\frac{1}{\gamma^{\prime}}\Delta^{2}(g(t)),$
(18)
where $\gamma^{\prime}$ is the ratio between the two time scale. Obviously,
when $N\to\infty$, $\Delta(g)=\left|g(t)-g_{c}\right|$ is valid for arbitrary
$g$. In this respect, the two criteria, Eq. (13) and Eq. (18), are equivalent.
Nevertheless, when $N$ is finite, the two criteria differ slightly because the
gap $\Delta(g)$ deviates from $\left|g(t)-g_{c}\right|$ near the critical
point (see Eq. (15) and Fig. 4a). As a result there is a small discrepancy in
the quench protocols $g_{\mathrm{RC}}(t)$ and $g_{\mathrm{KZ}}(t)$ associated
with two criteria, especially when $g(t)$ is close to $g_{c}$.
By substituting Eq. (15) into Eq. (18), we obtain the quench protocol:
$g_{\mathrm{RC}}(t)=\cos(\frac{\pi}{N})+\sin(\frac{\pi}{N})\tan{\left(\frac{2J\sin(\frac{\pi}{N})}{\gamma^{\prime}}t\right)},(-\frac{\gamma^{\prime}N}{4J}<t<\frac{\gamma^{\prime}N}{4J}).$
(19)
We plot the solution $g_{\mathrm{RC}}(t)$ along with $g_{\mathrm{KZ}}(t)$ in
Fig. 4b. There is a “kink” at the anti-cross point of the energy levels in
$g_{\mathrm{KZ}}(t)$ associated with the KZM criterion, but there is none in
$g_{\mathrm{RC}}(t)$ associated with the RC criterion (see Fig. 4b and the
inset). Although there is a singularity (divergent time derivative of g) in
$g_{\mathrm{KZ}}(t)$ at $t=0$, the time interval of this region is vanishingly
small. As a result the total change in $g$ in this singular region is very
small (See Eq. 4b), and the eigenstates of H(t) do not change significantly
within it. This is in the same spirit as quantum fidelity zanardi06 , where
fidelity susceptibility diverges at quantum critical point, but the fidelity
is nonzero indicating that the ground state does not changes significantly.
Hence, the ’kink’ at $t=0$ will not lead to a lot of excitations. Our
simulation verifies this point. Similar to Eq. (17), we obtain the time
required for the uniformly adiabatic evolution
$\Delta t_{\mathrm{RC}}=\frac{\gamma^{\prime}N}{2J}.$ (20)
Comparing Eq. (17) and Eq. (20), we find that when
$\gamma^{\prime}=\frac{2}{\pi}\gamma$, the time required for two criteria are
equal. In the following we will simulate the dynamics of the Ising chain under
the two quench protocols: Eq. (16) and Eq. (19). Substituting Eq. (16) and Eq.
(19) into Eq. (6), one obtains the instantaneous fidelity as a function of the
time $F=\left|\beta(t)\right|^{2}$ associated with two criteria. We plot the
fidelity as a function of the time in Fig. 5. When one chooses a different
initial condition, the fidelity as a function of the time differs a lot. In
the left panel of Fig. 5, we plot the fidelity as a function of the time
quenching from $t=-\frac{\gamma}{2J\sin{(\pi/N)}}$ ($g=-\infty$). The fidelity
associated with the Roland and Cerf criterion decays when the system is near
the anti-crossing point, and then revives. But the fidelity associated with
the KZM does not change much, and remains close to unity all the time. At
$t=\frac{\gamma}{2J\sin{(\pi/N)}}$ ($g=\infty$), the fidelity associated with
the Roland and Cerf’s criterion is a bit higher than that associated with the
KZM. In the right panel of Fig. 5, we plot the fidelity as a function of the
time starting from $t=-\frac{1}{3}\frac{\gamma}{2J\sin{(\pi/N)}}$. In this
case the fidelity associated with the Roland and Cerf’s criterion oscillates
rapidly and finally reaches a stable value around $0.85$. By contrast the
fidelity associated with the the KZM does not oscillate and remains very close
to unity. From the above facts, we conclude that in some cases, the Roland and
Cerf’s criterion is better than the KZM criterion, but in some other cases, it
is worse. Hence, we cannot say which criterion is definitely better, but the
KZM provides new insights into the conditions for uniformly adiabatic
evolution.
One is tempted at this point to undertake a variational study in search of
optimal quenches. While such a study is beyond the scope of this paper, we
note that in practical applications (e.g., adiabatic quantum computing)
optimization would involve not just varying rate, but (as it was done in Fig.
5) also the starting and final points of the quenches can be brought closer to
the “critical point”. Resulting errors can be detected and the correct result
can be ascertained by repeating the computation many times.
Figure 5: Comparison of the Roland and Cerf criterion and the KZM criterion.
The horizontal axis indicates the time, and the vertical axis depicts the
instantaneous fidelity. The solid (dashed) line describes the fidelity as a
function of the time $t$ associated with the Roland and Cerf (KZM) criterion.
Left panel: The evolution starts from the ground state of $t=-\frac{\gamma
N}{2J\pi}$ ($g=-\infty$). Right panel: The evolution starts from the ground
state of $t=-\frac{1}{3}\frac{\gamma N}{2J\pi}$.
### IV.2 Gauging the distance from the adiabatic quench
Fig. 2 indicates that when the time scale of the quench $\tau_{Q}$ is in the
range $\tau_{Q}\in(10^{-1},1)$, the fidelity is almost equal to zero. However,
this does not reveal how far the quench is from the adiabaticity. For example
when one out of many ($N=50$ in our numerical simulation) modes get excited,
the fidelity will decay to nearly zero due to the orthogonality of one mode.
But, in a sense, the system is still close to the ground state, as all but one
excited state are empty. In this sense, the fidelity is not a good criterion
for measuring how far away the quench is from the adiabatic evolution.
A better gauge of the distance of a quench process from the adiabaticity may
be obtained using other variables, such as the magnetization per site along
the direction of the external magnetic field
$m=\frac{1}{N}\sum_{i=1}^{N}{\left\langle\sigma_{i}^{z}\right\rangle}=\frac{1}{N}\sum_{k=1}^{N}2\left|\beta_{k}((g_{0}-2g_{T})\tau_{Q})\right|^{2}-1.$
(21)
When the magnetic field is large, the ground state corresponds to $m=1$. We
plot the final magnetization as a function of $\tau_{Q}$ in FIg. 6. In the
range of $\tau_{Q}\in(10^{-1},1)$, the fidelity is vanishingly small, but the
magnetization per site is still large. This indicates that the system is not
very far away from the instantaneous ground state. Moreover, when one delays
for some time at the turnaround point, the magnetization per site of the
impulse regime will decrease, but that of the adiabatic regime will not (see
Fig. 6). This is similar to the fidelity and agrees with our intuition. Last
but not least, the magnetization is experimentally easier to the measure than
the fidelity, and it has been used as a tool to study the adiabaticity of
quantum dynamics in Ref. anarb .
One can also use the kink density
$\frac{1}{2}\sum_{i}(1-\left\langle\sigma_{i}\sigma_{i+1}\right\rangle)$ jacek
as a measure of the distance of the system from the adiabaticity. The relation
between the fidelity and the density of defects has been studied in Ref.
jacek09prb . Other variables, such as the residual energy fazio , can be also
used to gauge the distance from the adiabatic quench. Such obvious measures of
how far the quench is from the adiabaticity work well in the one-dimensional
Ising model, but finding their useful analogues in other situations (e.g.,
adiabatic quantum computing) may not be easy.
Figure 6: Final magnetization of the Ising chain as a function of the time
scale of the quench $\tau_{Q}$. All the parameters are the same as those in
Fig. 2(a). One can see that the spin chain is not very far away from
equilibrium except when the time scale of the quench is in the range
$\tau_{Q}\in(10^{-1},1)$. Left panel: delay time $\Delta t=0$; Right panel:
delay time $\Delta t=0.7$.
## V Summary and conclusion
We have proposed a strategy to test the adiabaticity without knowing either
the eigenstates or eigenenergies of the Hamiltonian. Instead of having to find
the gap of the Hamiltonian, and then using the quantum adiabatic approximation
to evaluate the adiabaticity, one can use a quench echo to evaluate the
adiabaticity of an evolution. The underlying mechanism is that when the time
scale of the quench is large in comparison with the inverse of the energy gap,
both the forward and the backward evolutions are adiabatic. As a result, the
fidelity of the initial state and the final state is close to unity.
Otherwise, the evolution is not adiabatic, and the fidelity is less than
unity. The method for testing the adiabaticity of an evolution presented in
this paper is universally valid. It does not depend the model or the validity
of conditions for adiabatic approximation. We further proposed a method for
finding the uniformly adiabatic quench protocol based on the KZM, and
discussed the problem of gauging how non-adiabatic is a quench. Given the
importance of the adiabaticity in various applications, we believe that our
results will be broadly applicable, and may be useful in experimental
applications.
###### Acknowledgements.
This work is supported by U.S. Department of Energy through the LANL/LDRD
Program. We thank Bogdan Damski, Anarb Das, and Rishi Sharma for helpful
discussions.
## Appendix A fidelity in the intermediate regime
From Refs. jacek ; bogdan , we know that in the wave vector $k$
representation, the Schrödinger equation for the forward quench can be
rewritten as Landau-Zener type equations (see Eq. (6) for a comparison):
$i\frac{d}{dt^{\prime}}\left[\begin{array}[]{c}v_{k}(t^{\prime})\\\
u_{k}(t^{\prime})\end{array}\right]=\frac{1}{2}\left[\begin{array}[]{lr}\frac{t^{\prime}}{\tau_{Q}^{\prime}}&1\\\
1&-\frac{t^{\prime}}{\tau_{Q}^{\prime}}\end{array}\right]\left[\begin{array}[]{c}v_{k}(t^{\prime})\\\
u_{k}(t^{\prime})\end{array}\right],$ (1)
where $t^{\prime}=4\tau_{Q}\sin k[-g(t)+\cos k]$ and
$\tau_{Q}^{\prime}=4\tau_{Q}\sin^{2}k$. This equation can be solved in terms
of Weber functions. The initial conditions are $v_{k}(t^{\prime}=-\infty)=0$
and $u_{k}(t^{\prime}=-\infty)=1$. The solution for this equation is bogdan
$\begin{split}u_{k}(t)\approx&e^{-\pi\tau_{Q}\sin^{2}{k}}e^{i\tau_{Q}[(-g+\cos{k})^{2}+\sin^{2}{k}\ln\sqrt{4\tau_{Q}(-g+\cos{k})^{2}}]},\\\
v_{k}(t)\approx&\frac{\sqrt{2\pi\tau_{Q}\sin^{2}{k}}}{\Gamma(1-i\tau_{Q}\sin^{2}{k})}e^{-\frac{\pi\tau_{Q}\sin^{2}{k}}{2}}e^{-i\tau_{Q}[(-g+\cos{k})^{2}+\sin^{2}{k}\ln\sqrt{4\tau_{Q}(-g+\cos{k})^{2}}]}.\end{split}$
(2)
Similarly, we can obtain the solution for the quench echo. Combining the
forward and the quench echo process, we find the solution of the fidelity
(11). It is worth pointing out that the above solution is only good for the
turnaround point far away from the critical point $g_{T}\ll 1$. For example
$g_{T}=0.5$.
## References
* (1) A. Messiah Quantum Mechanics , (Dover Publications, Mineola, New York, 1999).
* (2) In recent years the conditions for the adiabatic approximation have become a subject of controversy controversy ; anarb . The standard criterion is found to be neither sufficient nor necessary jianda . Additional assumptions are required to ensure the stated conditions to be rigorous. See Ref. nori for a clarification of weakness of adiabaticity conditions.
* (3) K.-P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 95, 110407 (2005); S. Boixo and R. D. Somma, Phys. Rev. A 81, 032308 (2010).
* (4) Jian-da Wu, Mei-sheng Zhao, Jian-lan Chen, Yong-de Zhang, Phys. Rev. A 77, 062114 (2008); Jiangfeng Du, Lingzhi Hu, Ya Wang, Jianda Wu, Meisheng Zhao, and Dieter Suter, Phys. Rev. Lett. 101, 060403 (2008).
* (5) S. N. Shevchenko, S. Ashhab, and Franco Nori, Physics Reports 492, 1 (2010).
* (6) E. Farhi, et al, Science, 292, 472 (2001).
* (7) T. Kadowaki, and H. Nishimori, Phys. Rev. E 58, 5355 (1998).
* (8) A. Das, and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061 (2008).
* (9) Reference sen08 studied the density of topological defects under such quench echo. But our emphasis is different.
* (10) V. Mukherjee, A. Dutta, and D. Sen, Phys. Rev. B 77, 214427 (2008); U. Divakaran and A. Dutta, Phys. Rev. B 79, 214408 (2009).
* (11) B. Damski, and W. H. Zurek, Phys. Rev. A 73, 063405 (2006).
* (12) P. Solinas, P. Ribeiro, and R. Mosseri, Phys. Rev. A 78, 052329 (2008).
* (13) S. Ashhab, J. R. Johansson, A. M. Zagoskin, and Franco Nori, Phys. Rev. A 75, 063414 (2008).
* (14) S. Sachdev Quantum Phase Transitions, (Cambridge University Press, Cambridge, 2000).
* (15) T. W. B. Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep. 67, 183 (1980);
* (16) W. H. Zurek, Nature 317, 505 (1985); Acta Physica Polonica B 24, 1301 (1993); Phys. Rep. 276, 177 (1996).
* (17) Kibble-Zurek mechanism with nonlinear quench schemes is also discussed recently, see Refs. jacek09review ; nonlinear1 ; nonlinear2 .
* (18) J. Dziarmaga, arXiv: 0912.4034, Adv. Phys. in press.
* (19) D. Sen, K. Sengupta, and S. Mondal, Phys. Rev. Lett. 101, 016806 (2008); Phys. Rev. B 79, 045128 (2009).
* (20) R. Barankov and A. Polkovnikov, Phys. Rev. Lett. 101, 076801 (2008).
* (21) W.H. Zurek, U. Dorner and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005).
* (22) W. H. Zurek, L. M. A. Bettencourt, J. Dziarmaga, and N. D. Antunes, Acta. Phys. Polon. B 31, 2937 (2000).
* (23) B. Damski, Phys. Rev. Lett. 95, 035701 (2005); L. C. Wang, X. L. Huang, X. X. Yi, Eur. Phys. J. D 46, 345 (2008).
* (24) J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
* (25) U. Divakaran, A. Dutta, and D. Sen, Phys. Rev. B 81, 054306 (2010).
* (26) The energy gap of other modes is much larger than the considered mode $k=\pi/N$. So one usually consider only the excitations (if there is any) in the smallest mode $k=\pi/N$, and ignore the excitations in the other modes
* (27) J. Roland and N. J. Cerf, Phys. Rev. A 65, 042308 (2002).
* (28) B. Damski, H. T. Quan, and W. H. Zurek, arXiv: 0911.5729.
* (29) A. Das arXiv: 0904.2172.
* (30) Paolo Zanardi, and Nikola Paunkovi, Phys. Rev. E 74, 031123 (2006).
* (31) L. Cincio, J.Dziarmaga, J. Meisner, and M. M. Rams, Phys. Rev. B 79, 094421 (2009).
* (32) F. Pellegrini, S. Montangero, G. E. Santoro, and R. Fazio, Phys. Rev. B 77, 140404 (2008).
|
arxiv-papers
| 2010-07-19T21:58:54 |
2024-09-04T02:49:11.768872
|
{
"license": "Public Domain",
"authors": "H. T. Quan and W. H. Zurek",
"submitter": "Haitao Quan",
"url": "https://arxiv.org/abs/1007.3294"
}
|
1007.3335
|
# Pseudorandom selective excitation in NMR
Jamie D. Walls jwalls@miami.edu Department of Chemistry, University of Miami,
Coral Gables, FL 33124 Alexandra Coomes
###### Abstract
In this work, average Hamiltonian theory is used to study selective excitation
in a spin-1/2 system evolving under a series of small flip-angle
$\theta-$pulses $(\theta\ll 1)$ that are applied either periodically [which
corresponds to the DANTE pulse sequence] or aperiodically. First, an average
Hamiltonian description of the DANTE pulse sequence is developed; such a
description is determined to be valid either at or very far from the DANTE
resonance frequencies, which are simply integer multiples of the inverse of
the interpulse delay. For aperiodic excitation schemes where the interpulse
delays are chosen pseudorandomly, a single resonance can be selectively
excited if the $\theta$-pulses’ phases are modulated in concert with the time
delays. Such a selective pulse is termed a pseudorandom-DANTE or p-DANTE
sequence, and the conditions in which an average Hamiltonian description of
p-DANTE is found to be similar to that found for the DANTE sequence. It is
also shown that averaging over different p-DANTE sequences that are selective
for the same resonance can help reduce excitations at frequencies away from
the resonance frequency, thereby improving the apparent selectivity of the
p-DANTE sequences. Finally, experimental demonstrations of p-DANTE sequences
and comparisons with theory are presented.
## I Introduction
Of the multitude of radiofrequency (RF) schemes used for exciting and
controlling spin dynamics in NMR, most can be placed into one of two
categories: aperiodic RF pulse sequences or periodic RF pulse sequences. For
many aperiodic sequences, the RF phases, amplitudes and pulse delays are often
chosen randomly or in a pseudorandom manner. Such sequences have been used to
generate white noise or broadband excitation in NMR noise spectroscopyErnst
(1970); Kaiser (1970, 1974); Bartholdi _et al._ (1976), while sequences that
generate colored noise have been used in early spin decoupling schemes, such
as in noise decouplingErnst (1966). Theoretical models of a spin system’s
response to pseudorandom pulse sequences typically use a Volterra or
perturbation series in the randomly applied RF pulsesB.Blumich (1987). Since
many pseudorandom sequences are designed by considering only the first term in
the Volterra series, pseudorandom sequences are typically low power and result
in small, linear spin excitations.
Unlike aperiodic sequences, periodic RF pulse sequences are commonly used in a
variety of NMR experiments and are often found to be superior to their
pseudorandom counterparts; for example, two periodic sequences, MLEVLevitt and
Freeman (1981) and WALTZ-16Shaka _et al._ (1983), provide better
heteronuclear decoupling over noise decoupling under most conditions. Many
periodic RF pulse sequences are designed using average Hamiltonian theory
(AHT)Haeberlen and Waugh (1968), where the necessary RF pulse sequence that
generates a desired average Hamiltonian $\overline{H}_{avg}$ over a time
$\tau_{c}$ must be determined ($\tau_{c}$ is the length of the pulse
sequence). Repeated application of the pulse sequence introduces frequencies
into the dynamics that are integer multiples of $\frac{1}{\tau_{c}}$, which
may result in higher-order contributions to $\overline{H}_{avg}$ that degrade
the sequence’s performance. It has been previously noted that random or
asynchronous pulse imperfections placed into pulse sequences can often improve
their performanceBosman _et al._ (2004). Recently, Uhrig dynamical decoupling
(UDD) sequencesUhrig (2009) which utilize unequally spaced $\pi-$pulses, were
shown to be superior in preserving spin coherence to the standard Carr-
Purcell-Meiboom-Gill (CPMG) sequenceCarr and Purcell (1954), which uses
equally spaced $\pi-$pulses.
Selective pulsesFreeman (1991) are one class of pulses that do not fall neatly
into either category. The design of most commonly used selective pulses, such
as the gaussian and the sinc pulse shapes, is guided by the fact that a spin
system’s response to an applied pulse as a function of frequency/offset is
proportional to the Fourier transformation of the applied pulseTomlinson and
Hill (1973). Using linear response to design selective pulses has been used to
develop colored noise sequences for selective excitation in imaging
applicationsOrdige (1987). While sequences designed from the linear response
are valid for small flip-angles, these pulses fail as the degree of excitation
increases. As such, most methods for designing selective pulses of arbitrary
flip-angles use the linear response pulse shapes as starting points in
numerical searchesVeshtort and Griffin (2005). However, pulse shapes generated
by these numerical techniques often do not lend themselves to an easy physical
interpretation behind their selectivity.
One of the earliest and most easily understood periodic selective pulses that
is rigorously valid for all flip-angles is the DANTE sequenceBodenhausen _et
al._ (1976), which is shown in Fig. 1(A). The DANTE sequence consists of a
series of $N$ small-tip, broadband $\theta$-pulses that selectively rotate
those spins resonating at integer multiples of the interpulse delay by
$\Theta=N\theta$ about an axis in the transverse plane. The DANTE sequence’s
periodicity is responsible for this frequency response, which can be
calculated analyticallyCanet _et al._ (1995). To excite a single frequency,
however, the periodicity of the DANTE sequence must be violated. Breaking
DANTE’s symmetry for removing excitation at other frequencies has been
previously accomplished by modulating the phasesKacynski _et al._ (1992),
amplitudes, and delays of the $\theta-$pulsesRoumestand and Canet (2000).
However, these excitation sculpting modifications of the DANTE sequence are
still based on the assumptions of linear response.
In the following paper, we use AHT to provide insight into the selective
excitation of a spin-1/2 system by a series of periodically and aperiodically
small-flip $\theta$-pulses. First, the conditions where an AHT description of
the DANTE pulse sequence is valid is determined. Next, an AHT description for
a series of aperiodically spaced and phase-modulated $\theta-$pulses is
developed. Such sequences are referred to as pseudorandom-DANTE or p-DANTE
selective pulses [Fig. 1(B)]. Finally, experiments performed in acetone and in
an acetone/DMSO/water solution are used to demonstrate and validate the
selectivity of the p-DANTE sequences.
## II General Theory
Both the DANTE [Fig. 1(A)] and the p-DANTE [Fig. 1(B)] pulse sequences involve
the application of a series of $N$ small flip-angle $\theta-$pulses that
selectively rotate spins about an axis lying in the transverse plane by an
angle $\Theta=N\theta$. For the DANTE sequence, spins resonating at
$\nu_{Z}=\frac{n}{\tau}$ are selectively rotated by $\Theta$ (where $n$ is an
integer), whereas for the p-DANTE sequence, only those spins resonating at
$\nu_{Z}=\nu_{0}$ are rotated by $\Theta$.
Figure 1: Pulse sequence for (A) DANTEBodenhausen _et al._ (1976) and (B)
pseudorandom-DANTE or p-DANTE selective excitation. (A) The DANTE sequence
consists of a series of $N$ small-flip, $\theta$-pulses, equally spaced
between periods of free evolution of time $\tau$. The periodicity of the
sequence results in a rotation of $\Theta=N\theta$ about an axis perpendicular
in the transverse plane for those spins resonating (in the rotating frame) at
a frequency $\nu_{Z}=\frac{n}{\tau}$ where $n$ is an integer. (B) The p-DANTE
sequence consists of a series of $N$ unequally spaced small-flip,
$\theta$-pulses where $\tau_{k}$ is the time delay between the $k^{th}$ and
$(k+1)^{th}$, and $\phi_{k}=2\pi\sum_{k=1}^{k-1}\nu_{0}\tau_{k}$ is the phase
of the $k^{th}$ pulse with $\phi_{1}=0$. In the p-DANTE sequence, only those
spins resonating (in the rotating frame) at a frequency given by $\nu_{0}$ are
rotated by an angle $\Theta=N\theta$.
To understand the selectivity of both the DANTE and p-DANTE sequences within
the framework of AHT, it is useful to revisit the dynamics of a spin-1/2
system under a non-resonant RF irradiation. The Hamiltonian during the
application of an RF pulse is given by:
$\frac{\widehat{H}}{\hbar}=\omega_{z}\widehat{I}_{Z}+\omega_{RF}\left(\widehat{I}_{X}\cos(\phi)+\widehat{I}_{Y}\sin(\phi)\right)$,
where $\phi$ and $\omega_{RF}$ are the phase and amplitude of the RF pulse,
and $\omega_{Z}=2\pi\nu_{Z}$ is the resonance offset that the spin experiences
in the rotating frame. The propagator for an RF pulse applied for a time
$T_{p}$ can be written as:
$\displaystyle\widehat{P}^{\text{exact}}_{\phi}(T_{p})$ $\displaystyle=$
$\displaystyle\exp\left(-i\frac{\widehat{H}}{\hbar}T_{p}\right)=\exp\left(-i\left[\omega_{Z}\widehat{I}_{Z}+\omega_{RF}\left(\widehat{I}_{X}\cos(\phi)+\widehat{I}_{Y}\sin(\phi)\right)\right]T_{p}\right)$
$\displaystyle=$
$\displaystyle\cos\left(T_{p}\frac{\sqrt{\omega_{Z}^{2}+\omega_{RF}^{2}}}{2}\right)\widehat{1}-i\frac{2\sin\left(T_{p}\frac{\sqrt{\omega_{Z}^{2}+\omega_{RF}^{2}}}{2}\right)}{\sqrt{\omega_{Z}^{2}+\omega_{RF}^{2}}}\left(\omega_{Z}\widehat{I}_{Z}+\omega_{RF}\left(\cos(\phi)\widehat{I}_{X}+\sin(\phi)\widehat{I}_{Y}\right)\right)$
Alternatively, the propagator in Eq. (LABEL:eq:pulexact) can be transformed
into an interaction frame defined by $\omega_{Z}\widehat{I}_{Z}$ and is given
by:
$\displaystyle\widehat{P}^{\text{exact}}_{\phi}(T_{p})$ $\displaystyle=$
$\displaystyle\exp\left(-i\omega_{Z}T_{p}\widehat{I}_{Z}\right)\widehat{T}\exp\left(-i\int^{T_{p}}_{0}\text{d}t^{\prime}\frac{\omega_{RF}}{2}\left[\widehat{I}_{+}e^{i\left(\omega_{Z}t^{\prime}-\phi\right)}+\widehat{I}_{-}e^{-i\left(\omega_{Z}t^{\prime}-\phi\right)}\right]\right]$
(2) $\displaystyle=$
$\displaystyle\widehat{U}_{\text{free}}(\omega_{Z}T_{p})\widehat{T}\exp\left(-i\int^{T_{p}}_{0}\text{d}t^{\prime}\frac{\widehat{H}_{INT}(t^{\prime})}{\hbar}\right)$
where $\widehat{T}$ is the Dyson time-ordering operator,
$\widehat{U}_{\text{free}}(\omega_{Z}T_{p})=\exp\left(-i\omega_{Z}T_{p}\widehat{I}_{Z}\right)$,
and $\widehat{H}_{INT}(t^{\prime})$, the Hamiltonian in the interaction frame,
represents a purely phase-modulated RF pulse. The time-dependent propagator in
Eq. (2) can be approximated by:
$\displaystyle\widehat{T}\exp\left(-\frac{i}{\hbar}\int^{T_{p}}_{0}\text{d}t^{\prime}\widehat{H}_{INT}(t^{\prime})\right)=\exp\left(-\frac{iT_{p}}{\hbar}\overline{H}_{p,\phi}\right)$
(3)
In Eq. (3), $\overline{H}_{p,\phi}$ is the average HamiltonianHaeberlen and
Waugh (1968) and is given by
$\overline{H}_{p,\phi}=\sum_{n=1}^{\infty}\overline{H}_{p,\phi}^{(n)}$ where
the first two terms in the series are:
$\displaystyle\frac{\overline{H}_{p,\phi}^{(1)}}{\hbar}$ $\displaystyle=$
$\displaystyle\frac{1}{T_{p}}\int^{T_{p}}_{0}\text{d}t^{\prime}\widehat{H}_{INT,\phi}(t^{\prime})$
$\displaystyle=$
$\displaystyle\frac{\omega_{RF}}{2}\text{sinc}\left(\frac{\omega_{Z}T_{p}}{2}\right)\left(\widehat{I}_{+}e^{i\left(\frac{\omega_{Z}T_{p}}{2}-\phi\right)}+I_{-}e^{-i\left(\frac{\omega_{Z}T_{p}}{2}-\phi\right)}\right)$
$\displaystyle\frac{\overline{H}_{p,\phi}^{(2)}}{\hbar}$ $\displaystyle=$
$\displaystyle\frac{1}{2iT_{p}}\int^{T_{p}}_{0}\text{d}t^{\prime}\int^{t^{\prime}}_{0}\text{d}t^{\prime\prime}\left[\widehat{H}_{INT,\phi}(t^{\prime}),\widehat{H}_{INT,\phi}(t^{\prime\prime})\right]$
(4) $\displaystyle=$
$\displaystyle\frac{\omega_{RF}^{2}}{4it_{p}}\widehat{I}_{Z}\int^{T_{p}}_{0}\text{d}t^{\prime}\int^{t^{\prime}}_{0}\text{d}t^{\prime\prime}\left(e^{i\omega_{Z}(t^{\prime}-t^{\prime\prime})}-e^{-i\omega_{Z}(t^{\prime}-t^{\prime\prime})}\right)$
$\displaystyle=$
$\displaystyle\frac{\omega_{RF}^{2}}{2\omega_{Z}}\left(1-\text{sinc}(\omega_{Z}T_{p})\right)\widehat{I}_{Z}$
$\widehat{P}^{\text{exact}}_{\phi}(T_{p})$ in Eq. (2) can be approximated as:
$\displaystyle\widehat{P}_{\phi}^{\text{exact}}(T_{p})$ $\displaystyle\approx$
$\displaystyle\widehat{P}_{\phi}(T_{p})\approx\exp\left(-iT_{p}\omega_{Z}\widehat{I}_{Z}\right)\exp\left(-\frac{iT_{p}}{\hbar}\left(\overline{H}_{p,\phi}^{(1)}+\overline{H}_{p,\phi}^{(2)}\right)\right)$
(5)
For $\omega_{RF}T_{p}\leq\frac{2\pi}{9}$,
$||\widehat{P}_{\phi}^{\text{exact}}(T_{p})-\widehat{P}_{\phi}(T_{p})||\leq
10^{-3}$ for all $\omega_{Z}$, where $||A||=\sqrt{\text{Tr}[A^{\dagger}A]}$
represents the Frobenius matrix norm (if $A$ represents the difference of two
unitary matrices, then the maximum value of $||A||$ is $2\sqrt{n}$ where $n$
is the matrix dimension). Since the DANTE and p-DANTE sequences both consist
of a series of small flip-angle $\theta-$pulses with $\theta<\frac{2\pi}{9}$,
the approximation
$\widehat{P}^{\text{exact}}_{\phi}(T_{p})\approx\widehat{P}_{\phi}(T_{p})$ in
Eq. (5) will be used in the rest of this paper.
For future comparison of the propagator in Eq. (2) with the propagator for a
spin-1/2 evolving under either the DANTE or the p-DANTE sequences in Fig. 1,
it is useful to consider an alternative description of
$\widehat{P}_{\phi}^{\text{exact}}$ in the interaction frame by dividing
$\widehat{T}\exp\left(-\frac{i}{\hbar}\int^{T_{p}}_{0}\widehat{H}_{\text{INT}}(t^{\prime})\text{d}t^{\prime}\right)$
in Eq. (2) into $N\gg 1$ smaller propagators, which is illustrated in Figure
2(A). In this case, $P_{\phi}^{\text{exact}}(T_{p})$ can be rewritten as:
$\displaystyle\widehat{P}_{\phi}^{\text{exact}}(T_{p})$ $\displaystyle=$
$\displaystyle\widehat{U}_{\text{free}}(\omega_{Z}T_{p})\widehat{T}\exp\left(-i\int^{T_{p}}_{0}\text{d}t^{\prime}\frac{\widehat{H}_{INT}(t^{\prime})}{\hbar}\right)$
(6) $\displaystyle\approx$
$\displaystyle\widehat{U}_{\text{free}}(\omega_{Z}T_{p})\widehat{T}\prod_{j=1}^{N}\exp\left(-i\int^{j\frac{T_{p}}{N}}_{(j-1)\frac{T_{p}}{N}}\text{d}t^{\prime}\frac{\widehat{H}_{INT}(t^{\prime})}{\hbar}\right)$
$\displaystyle\approx$
$\displaystyle\widehat{U}_{\text{free}}(\omega_{Z}T_{p})\widehat{T}\prod_{j=1}^{N}\exp\left(-i\frac{T_{p}}{N}\omega_{RF}\text{sinc}\left(\frac{\omega_{Z}T_{p}}{2N}\right)\left(\widehat{I}_{X}\cos\left(\phi_{j}^{*}\right)-I_{Y}\sin\left(\phi_{j}^{*}\right)\right)\right)$
where $\phi_{j}^{*}=\frac{\omega_{Z}(j-1)T_{p}}{N}+\frac{T_{p}}{2N}-\phi$.
With respect to Eq. (6) and Fig. 2(A), the total propagator for an RF pulse of
strength $\omega_{RF}$ applied off-resonance by $\omega_{Z}$ for a time
$T_{p}$ is equivalent to the application of $N$ continuous, small-flip
$\theta=\frac{T_{p}}{N}\text{sinc}\left(\frac{\omega_{Z}T_{p}}{2N}\right)$,
phase-modulated RF pulses where the phase of the $j^{th}$ pulse is
$-\phi_{j}^{*}$, followed by a rotation about the $\widehat{z}-$axis by
$\omega_{Z}T_{p}$.
### II.1 DANTE Pulse Sequence
The DANTE sequenceBodenhausen _et al._ (1976) consists of a series of $N$,
equally spaced small-tip, $\theta$-pulses of constant phase and duration
$t_{p}$ where $\omega_{RF}t_{p}=\theta\ll 1$ [Figure 1(A)]. The full
propagator for the DANTE pulse sequence can be written as:
$\displaystyle\widehat{U}_{\text{exact}}(T_{\text{tot}})$ $\displaystyle=$
$\displaystyle\widehat{P}_{0}(t_{p})\left(\widehat{U}_{\text{free}}(\omega_{Z}\tau)\widehat{P}_{0}(t_{p})\right)^{N-1}=\left(\widehat{P}_{0}(t_{p})\widehat{U}_{\text{free}}(\omega_{Z}\tau)\right)^{N-1}\widehat{P}_{0}(t_{p})$
(7) $\displaystyle=$
$\displaystyle\widehat{U}_{\text{free}}(\omega_{Z}\left((N-1)\tau_{t}+t_{p}\right))\widehat{T}\prod_{k=0}^{N-1}\widehat{P}_{-k\omega_{Z}\tau_{t}}(t_{p})$
where $\tau$ is the time delay between pulses, $\tau_{t}=\tau+t_{p}$ and
$\displaystyle\widehat{P}_{-k\omega_{Z}\tau_{t}}(t_{p})$ $\displaystyle=$
$\displaystyle\widehat{U}^{\dagger}_{\text{free}}(\omega_{Z}k\tau_{t})\widehat{P}_{0}(t_{p})\widehat{U}_{\text{free}}(\omega_{Z}k\tau_{t})\equiv\exp\left(-i\widehat{H}_{k}t_{p}\right)$
(8) $\displaystyle\widehat{H}_{k}$ $\displaystyle=$
$\displaystyle\overline{H}_{p,-k\omega_{Z}\tau_{t}}$ $\displaystyle\approx$
$\displaystyle\frac{\omega_{RF}}{2}\text{sinc}\left(\frac{\omega_{Z}t_{p}}{2}\right)\left(\widehat{I}_{+}e^{i\omega_{Z}\frac{2k\tau_{t}+t_{p}}{2}}+\widehat{I}_{-}e^{-i\omega_{Z}\frac{2k\tau_{t}+t_{p}}{2}}\right)+\frac{\omega_{RF}^{2}}{2\omega_{Z}}\left(1-\text{sinc}(\omega_{Z}t_{p})\right)\widehat{I}_{Z}$
As has been previously notedShinnar and Leigh (1987); Shinnar _et al._ (1989,
1989), the propagator for the DANTE sequence in Eq. (7) is the same as the
propagator for a continuous series of $N$, phase modulated small-flip pulses,
where the phase modulation depends upon the spin’s chemical shift,
$\omega_{Z}$, followed by a rotation about the $\widehat{z}-$ axis by
$\omega_{Z}T_{\text{tot}}$. This is illustrated in Figure 2(B). If
$\text{mod}\left[\omega_{Z}\tau_{t},2\pi\right]\approx 0$, then all $N$ pulses
are effectively applied along the same direction since
$\phi_{k}\approx\phi_{j}$ for all $k$ and $j$. The small rotations are
therefore additive and lead to an overall rotation of $\Theta\approx N\theta$
about an axis in the transverse plane is generated. When
$\text{mod}\left[\omega_{Z}\tau_{t},2\pi\right]\neq 0$, the pulses are
effectively applied about different directions ($\phi_{k}\neq\phi_{j}$ for
$k\neq j$ in general) thereby reducing the overall spin rotation. Comparing
Fig. 2(B) and Eqs. (7) and (LABEL:eq:Hk) with Fig. 2(A) and Eq. (6), the
propagator for the DANTE sequence is similar to that of an off resonant, RF
pulse of duration $Nt_{p}=T_{p}$ followed by a rotation about the
$\widehat{z}$-axis. That is,
$\displaystyle\widehat{U}_{\text{exact}}(T_{\text{tot}})$
$\displaystyle\approx$
$\displaystyle\widehat{U}_{\text{free}}(\omega_{Z}T_{\text{tot}}-\omega_{Z}^{\prime}Nt_{p})\exp\left[-iT_{p}\left(\omega_{Z}^{\prime}\widehat{I}_{Z}+\omega_{RF}^{\prime}\left(\widehat{I}_{X}\cos(\phi^{\prime})+\widehat{I}_{Y}\sin(\phi^{\prime})\right)\right)\right]$
where
$\omega_{Z}^{\prime}=\frac{2\pi}{t_{p}}\text{mod}\left[\omega_{Z}\tau_{t},2\pi\right]$,
$\omega_{RF}^{\prime}=\frac{\text{sinc}\left(\frac{\omega_{Z}t_{p}}{2}\right)}{\text{sinc}\left(\frac{\omega_{Z}^{\prime}t_{p}}{2}\right)}\omega_{RF}$,
and $\phi^{\prime}=\frac{t_{p}}{2}(\omega_{Z}^{\prime}-\omega_{Z})$. As
mentioned above, when $\text{mod}\left[\omega_{Z}\tau_{t},2\pi\right]=0$, then
$\omega_{Z}^{\prime}=0$, and the effective pulse is applied on resonance and
rotates the spin by $\Theta$. When
$\text{mod}\left[\omega_{Z}\tau_{t},2\pi\right]\neq 0$, then
$\omega_{Z}^{\prime}$ can be quite large since $\frac{2\pi}{t_{p}}\gg 1$ for
short pulses ($t_{p}\ll 1$). In this case, the pulse appears to be applied
very far off resonance when $\omega_{Z}^{\prime}\gg\omega_{RF}^{\prime}$.
Figure 2: The connection between the DANTE pulse sequence and the application
of an off-resonant RF pulse. (A) An RF pulse applied off resonance by
$\omega_{Z}$ and with strength $\omega_{RF}$ for a time $T_{p}$ is equivalent
to applying $N\gg 1$ phase-modulated, small flip-angle
$\theta=\omega_{RF}\frac{T_{p}}{N}\text{sinc}\left(\frac{\omega_{Z}T_{p}}{2N}\right)$
pulses followed by a rotation about the $\widehat{z}$-axis by an angle
$\omega_{Z}T_{p}$. The phase of the $j^{th}$ pulse is given by
$-\phi_{j}^{*}=(j-1)\omega_{Z}\frac{T_{p}}{N}+\omega_{Z}\frac{T_{p}}{2N}-\phi$
(in Fig. 2(A), $\phi=0$). In (B), a DANTE pulse sequence is equivalent to a
series of $N$ phase-modulated small-flip pulses followed by a rotation about
the $\widehat{z}-$axis by an angle $N\omega_{Z}\tau$ [Eq. (7)]. The phase of
the $k^{th}$ pulse is
$\phi_{k}=\omega_{Z}\left((k-1)(\tau_{t})+\frac{t_{p}}{2}\right)$ where
$\tau_{t}=\tau+t_{p}$, $t_{p}$ is the length of the small-flip pulses, and
$T_{\text{tot}}=(N-1)\tau_{t}+t_{p}$.
In order to make the above arguments more quantitative, AHT can be used to
rewrite $\widehat{U}(T_{\text{tot}})$ in Eq. (7) as
$\widehat{U}(T_{\text{tot}})\approx\widehat{U}_{\text{AHT}}(T_{\text{tot}})$
where
$\widehat{U}_{\text{AHT}}(T_{\text{tot}})=\widehat{U}_{\text{free}}(\omega_{Z}T_{\text{tot}})\exp\left(-iNt_{p}\overline{H}_{avg}\right)$,
where the first two terms in the average Hamiltonian,
$\overline{H}_{avg}=\sum_{n=1}^{\infty}\overline{H}_{avg}^{(n)}$, are given by
[setting $a=\omega_{RF}\text{sinc}\left(\frac{\omega_{Z}t_{p}}{2}\right)$ and
$b=\frac{\omega_{RF}^{2}}{2\omega_{Z}}\left(1-\text{sinc}\left(\omega_{Z}t_{p}\right)\right)$]:
$\displaystyle\overline{H}_{avg}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{Nt_{p}}\sum_{k=1}^{N-1}t_{p}\widehat{H}_{k}$
$\displaystyle=$ $\displaystyle
a\frac{\text{sinc}\left(\frac{N\omega_{Z}\tau_{t}}{2}\right)}{\text{sinc}\left(\frac{\omega_{Z}\tau_{t}}{2}\right)}\widehat{I}_{T}(\omega_{Z},\tau_{t},t_{p},N)+b\widehat{I}_{Z}$
$\displaystyle\overline{H}_{avg}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2iNt_{p}}\sum_{k>j}\left[\widehat{H}_{k}t_{p},\widehat{H}_{j}t_{p}\right]=-\frac{a^{2}t_{p}}{2\omega_{Z}\tau_{t}}\frac{\text{sinc}\left(N\omega_{Z}\tau_{t}\right)-\text{sinc}(\omega_{Z}\tau_{t})}{\left(\text{sinc}\left(\frac{\omega_{Z}\tau_{t}}{2}\right)\right)^{2}}\widehat{I}_{Z}$
$\displaystyle+$
$\displaystyle\frac{abt_{p}(N^{2}-1)}{4N\omega_{Z}\tau_{t}}\frac{\text{sinc}\left(\frac{(N+1)\omega_{Z}\tau_{t}}{2}\right)-\text{sinc}\left(\frac{(N-1)\omega_{Z}\tau_{t}}{2}\right)}{\text{sinc}^{2}\left(\frac{\omega_{Z}\tau_{t}}{2}\right)}\widehat{I}_{T}(\omega_{Z},\tau_{t},t_{p},N)$
where
$\widehat{I}_{T}(\omega_{Z},\tau_{t},t_{p},N)=\widehat{I}_{X}\cos\left(\frac{\omega_{Z}T_{\text{tot}}}{2}\right)-\widehat{I}_{Y}\sin\left(\frac{\omega_{Z}T_{\text{tot}}}{2}\right)$.
Although the form of $\overline{H}_{avg}$ in Eq. (LABEL:eq:Havg) is somewhat
complicated, the physical picture behind $\overline{H}_{avg}$ in Eq.
(LABEL:eq:Havg) can be seen in Fig. 2(B). When $\omega_{Z}\tau_{t}=2\pi n$ for
integer $n$, $\phi_{j}=\phi_{k}$ and therefore
$\left[\widehat{H}_{k},\widehat{H}_{j}\right]=0$ for all $k\neq j$, so
$\overline{H}_{avg}=\overline{H}_{avg}^{(1)}$ exactly. In this case, the
propagator represents a rotation about an axis in the transverse plane of
phase $(-1)^{n}\omega_{Z}t_{p}/2$ by a total angle of
$\Theta=N\omega_{RF}t_{p}\text{sinc}\left(\omega_{Z}t_{p}/2\right)\approx
N\omega_{RF}t_{p}=N\theta$ for $\omega_{Z}t_{p}\ll 1$. For
$\omega_{Z}\tau_{t}\neq 2\pi n$ for integer $n$, the various $\widehat{H}_{k}$
are pointing in different directions, so that the average transverse field in
$\overline{H}_{avg}^{(1)}$ is lessened. Furthermore, since the effective
rotation directions no longer commute with one another, i.e.,
$[\widehat{H}_{k},\widehat{H}_{j}]\neq 0$ for $k\neq j$, there is a
contribution to $\overline{H}_{avg}$ at second-order,
$\overline{H}_{avg}^{(2)}$, of an effective field along the $\widehat{z}$
direction. When $|\overline{H}_{avg}^{(2)}|\gg|\overline{H}_{avg}^{(1)}|$, the
effective field lies mostly about the $\widehat{z}-$axis and the spins are
minimally excited (this argument is similar to the concept of second-
averagingDybowski and Vaughan (1975)).
In order to see under what conditions AHT can be used in calculating the DANTE
pulse sequence, Figure 3 shows
$||\widehat{U}_{\text{exact}}(T_{\text{tot}})-\widehat{U}_{\text{AHT}}(T_{\text{tot}})||$
using
$\overline{H}_{\text{avg}}\approx\overline{H}_{\text{avg}}^{(1)}+\overline{H}_{\text{avg}}^{(2)}$,
as a function of $N$ and $\omega_{Z}/\omega_{RF}$. Two calculations are shown
in Fig. 3, one for a total pulse rotation for an on-resonant RF pulse of
$\Theta=\pi/2$ [Fig. 3(A)] and one for $\Theta=2\pi$ [Fig. 3(B)]. For both
calculations, $\frac{\tau}{t_{p}}=1000$. As mentioned above,
$\widehat{U}_{\text{AHT}}(T_{\text{tot}})\approx\widehat{U}_{\text{exact}}(T_{\text{tot}})$
when $2\pi
n=\omega_{Z}\tau_{t}=\frac{\omega_{Z}}{\omega_{RF}}\omega_{RF}t_{p}\left(\frac{\tau}{t_{p}}+1\right)=\frac{\omega_{Z}}{\omega_{RF}}\frac{\Theta}{N}\left(\frac{\tau}{t_{p}}+1\right)$,
where $n$ is an integer. Therefore, at a resonance condition for $n\neq 0$,
there exists a linear relationship between $N$ and
$\frac{\omega_{Z}}{\omega_{RF}}$ that is given by
$\displaystyle N$ $\displaystyle=$
$\displaystyle\text{INT}\left[\frac{\omega_{Z}}{\omega_{RF}}\frac{\Theta}{2\pi
n}\left(\frac{\tau}{t_{p}}+1\right)\right]$ (12)
where $\text{INT}[x]$ gives the nearest integer near $x$. From Fig. 3, for
$\theta\leq\frac{\pi}{60}$ [$N\geq 30$ in Fig. 3(A) and $N\geq 120$ in Fig.
3(B)], $\widehat{U}_{\text{AHT}}(T_{\text{tot}})$ is a good approximation to
$\widehat{U}_{\text{exact}}(T_{\text{tot}})$ for all
$\frac{\omega_{Z}}{\omega_{RF}}$ except near the resonance conditions in Eq.
(12). The deviations of $\widehat{U}_{\text{AHT}}(T_{\text{Tot}})$ from
$\widehat{U}_{\text{exact}}(T_{\text{tot}})$ occur when $\omega_{Z}$ is
slightly away from the resonance condition, $\omega_{Z}=\frac{2\pi
n}{\tau_{t}}$ for integer $n$. From numerical calculations, the range of
frequencies in which $\widehat{U}_{\text{AHT}}(T_{\text{Tot}})$ is a good
approximation to $\widehat{U}_{\text{exact}}(T_{\text{tot}})$ is found to be
approximately given by $\delta\omega_{Z}\gg\frac{6\theta\pi}{5\tau_{t}\Theta}$
or $\delta\omega_{Z}\ll\frac{6\theta\pi}{5\tau_{t}\Theta}$, where
$\delta\omega_{Z}=\text{min}\left[|\omega_{Z}-\frac{2\pi n}{\tau_{t}}|\right]$
is the smallest frequency difference between $\omega_{Z}$ and the nearest
integer multiple of the DANTE resonance frequency, $\frac{2\pi}{\tau_{t}}$.
Although the range of $\omega_{Z}$ where
$\widehat{U}_{\text{AHT}}(T_{\text{tot}})$ is a good approximation increases
with decreasing $\theta$ (increasing $N$), the maximum of
$||\widehat{U}_{\text{AHT}}(T_{tot})-\widehat{U}_{\text{exact}}(T_{tot})||$
mainly depends upon the overall rotation angle, $\Theta$. For
$\Theta<\frac{5\pi}{9}$,
$\text{max}\left[||\widehat{U}_{\text{AHT}}(T_{tot})-\widehat{U}_{\text{exact}}(T_{tot})||\right]\leq
0.1$ for all $N$ and $\omega_{Z}$. Physically, this can be understood as
follows: for $\delta\omega_{Z}t_{p}\gg\frac{6\theta\pi}{5\tau_{t}\Theta}$, the
effective phases of the pulses [see Fig. 2(B)] are modulated faster than the
effective tip of the pulse, $\omega_{RF}t_{p}=\theta$, so that
$\omega_{Z}\gg\omega_{RF}$. In this case, AHT works well, just as in using the
typical rotating wave approximation. For
$\omega_{Z}\ll\frac{6\pi\theta}{5\tau_{t}\Theta}$, the phases of the pulses in
Fig. 2(B) are relatively unchanged during the course of the sequence; in this
case, the various Hamiltonians, $\widehat{H}_{k}$ in Eq. (LABEL:eq:Hk) commute
with one another, so $\overline{H}_{avg}\approx\overline{H}_{avg}^{(1)}$ and
$\widehat{U}_{\text{exact}}(T_{tot})\approx\widehat{U}_{\text{AHT}}(T_{tot})$.
Figure 3: The Frobenius norm of
$||\widehat{U}_{\text{exact}}(T_{\text{tot}})-\widehat{U}_{\text{AHT}}(T_{\text{tot}})||$
as a function of the number of small tip $\theta$-pulses, $N$, with
$\theta=\frac{\Theta}{N}$, and $\frac{\omega_{Z}}{\omega_{RF}}$. In Fig. 3,
two calculations are shown for total maximum flip angle of (A)
$\Theta=\frac{\pi}{2}$ and (B) $\Theta=2\pi$.
$\widehat{U}_{\text{exact}}(T_{\text{tot}})$ [Eq. (7)] is the exact propagator
for the DANTE sequence, and $\widehat{U}_{\text{AHT}}(T_{\text{tot}})$ is the
propagator calculated using the average Hamiltonian,
$\overline{H}_{avg}\approx\overline{H}_{avg}^{(1)}+\overline{H}_{avg}^{(2)}$
in Eq. (LABEL:eq:Havg). In both calculations, $\frac{\tau}{t_{p}}=1000$. The
greatest deviations between $\widehat{U}_{\text{exact}}(T_{\text{tot}})$ and
$\widehat{U}_{\text{AHT}}(T_{\text{tot}})$ occur about the resonance
conditions in Eq. (12), and the maxima in
$||\widehat{U}_{\text{exact}}(T_{\text{tot}})-\widehat{U}_{\text{AHT}}(T_{\text{tot}})||$
are approximately described by two parallel lines given by
$N=\text{INT}\left[\frac{\omega_{Z}}{\omega_{RF}}\frac{\Theta}{2\pi
n}\left(\frac{\tau}{t_{p}}+1\right)\pm\frac{3}{5n}\right]$ for integer $n\neq
0$. The agreement between $\widehat{U}_{\text{AHT}}(T_{\text{tot}})$ and
$\widehat{U}_{\text{exact}}(T_{\text{tot}})$ improves with decreasing
$\Theta$.
### II.2 pseudorandom DANTE (p-DANTE)
In the DANTE sequence, a natural frequency of $\frac{1}{\tau_{t}}$ is
introduced into the dynamics due to the periodicity of the pulse sequence;
this leads to efficient excitation at frequencies
$\nu^{\text{DANTE}}_{n}=\frac{n}{\tau_{t}}$ for integer $n$. However, suppose
that one was interested in using a DANTE-like sequence to efficiently excite
only one particular frequency, say at $\nu^{\text{DANTE}}_{0}=0$ Hz. One way
to accomplish this using a DANTE sequence would be to make $\tau_{t}$ small
enough such that all $\nu^{\text{DANTE}}_{n\neq 0}$ lie outside the relevant
spectral width. If the spectral width for the system of interest is large,
however, this would necessitate using small $\tau_{t}$, where the smallest
time possible $\tau_{t}$ is $\tau_{t}\approx t_{p}$ (i.e., when $\tau=0$). The
selectivity or width of the excitation spectrum about $\nu^{\text{DANTE}}_{n}$
is approximately given by $\frac{1}{N\tau_{t}}$. For $\tau_{t}\approx t_{p}$,
this means the selectivity is roughly proportional to
$\frac{1}{Nt_{p}}=\frac{2\pi\nu_{RF}}{\Theta}$. In this limit, the effect of
the DANTE sequence is similar to evolution under continuous RF irradiation,
which leads to a very broad excitation profile unless $\nu_{RF}$ is weak or
$\Theta\gg 2\pi$. Under these conditions, the DANTE sequence would be
equivalent to applying a long, low-amplitude RF pulse.
An alternative way to excite only a single resonance using a DANTE-like
sequence would be to violate the periodicity of the DANTE sequence. This could
be accomplished in a variety of ways, such as using aperiodic delays,
modulating the pulse amplitudes and delays, etc. One example of such an
aperiodic DANTE sequence is illustrated in Figure 1(B) where $N$ small-flip
$\theta$-pulses are applied with a delay of $\tau_{k}$ between the separation
between the $k^{th}$ and $(k+1)^{th}$ pulse, where in general,
$\tau_{k}\neq\tau_{j}$. For such pulse sequences to selectively and
efficiently excite spins at a single resonance frequency $\nu_{0}$, the phases
of the pulses, $\phi_{k}$, must be modulated. In this case, the phase of the
$(k+1)^{th}$ pulse is given by
$\phi_{k+1}=-2\pi\nu_{0}\left(kt_{p}+\sum_{j=1}^{k}\tau_{j}\right)=-2\pi\nu_{0}T_{k}$
with $T_{k}=kt_{p}+\sum_{j=1}^{k}\tau_{j}$, and $\phi_{1}=0$ and $T_{0}=0$.
Such a set of aperiodic DANTE sequences are referred to as pseudorandom-DANTE
or p-DANTE sequences.
As in the DANTE case, the propagator for the p-DANTE sequence can be written
as
$\widehat{U}(T_{\text{tot}})=\widehat{U}_{\text{free}}(\omega_{Z}(T_{\text{tot}}))\exp\left(-\frac{i}{\hbar}Nt_{p}\overline{H}_{avg}\right)$,
where $T_{\text{tot}}=T_{N-1}+t_{p}$, and $\overline{H}_{avg}$ is the average
Hamiltonian for the p-DANTE sequence, with the first two terms given by:
$\displaystyle\overline{H}^{(1)}_{\text{avg}}$ $\displaystyle=$
$\displaystyle\frac{a}{N}\sum_{k=1}^{N}\left(\widehat{I}_{+}e^{i\left(\Delta\omega
T_{k}+\frac{\omega_{Z}t_{p}}{2}\right)}+I_{-}e^{-i\left(\Delta\omega
T_{k}+\frac{\omega_{Z}t_{p}}{2}\right)}\right)+b\widehat{I}_{Z}$ (13)
$\displaystyle\overline{H}^{(2)}_{\text{avg}}$ $\displaystyle=$
$\displaystyle\frac{abt_{p}}{2N}\left[\sum_{j<k}\sin\left(\frac{\Delta\omega(T_{j}-T_{k})}{2}\right)\left(\widehat{I}_{+}e^{i\frac{\Delta\omega(T_{j}+T_{k})+\omega_{Z}t_{p}}{2}}+\widehat{I}_{-}e^{-i\frac{\Delta\omega(T_{j}+T_{k})+\omega_{Z}t_{p}}{2}}\right)\right]$
(14) $\displaystyle+$
$\displaystyle\frac{a^{2}t_{p}}{2N}\widehat{I}_{Z}\sum_{j<k}\sin\left(\Delta\omega(T_{k}-T_{j})\right)$
where $\Delta\omega=2\pi\left(\nu_{Z}-\nu_{0}\right)$, and $a$ and $b$ were
previously defined before Eq. (LABEL:eq:Havg). Unlike in the DANTE case, the
average delay between pulses fluctuates within the p-DANTE sequence, i.e.,
$\frac{T_{k}}{k}\neq\frac{T_{j}}{j}$ for $k\neq j$. For $\nu_{Z}\neq\nu_{0}$,
a p-DANTE sequence is effectively equivalent to applying an RF field with a
fluctuating offset [Fig. 2(A)], the result of which is a seemingly random
excitation profile for those spins with $\Delta\omega_{Z}\neq 0$. Spins with
$\Delta\omega_{Z}=0$ are rotated by $N\theta=\Theta$. This is illustrated in
Figures 4(A) and 4(B), which show the excitation and
$\widehat{z}-$magnetization profiles under a p-DANTE sequence respectively. In
Figs. 4(A) and 4(B), the various $N-1$ $\tau_{k}$’s were chosen randomly but
were scaled to ensure that $\sum_{k=1}^{N-1}\tau_{k}=46.4$ ms, and one hundred
different sets of randomly generated p-DANTE sequences were generated.
Consider the excitation and $\widehat{z}-$magnetization profile for a single
p-DANTE sequence [red curve ($N_{avg}=1$)] shown in Figs. 4(A) and 4(B). A
maximum rotation by $\Theta=\frac{\pi}{2}$ occurs at $\delta\nu=0$, where
$-\langle\widehat{I}_{Y}\rangle=1$ and $\langle\widehat{I}_{Z}\rangle=0$. Away
from $\delta\nu=0$, the excitation and $\widehat{z}-$magnetization profiles
are quite noisy, but $|\langle\widehat{I}_{Y}\rangle|<1$ and
$\langle\widehat{I}_{Z}\rangle>0$. Note that the $\widehat{z}-$magnetization
profile is less noisy, since rotations away from the $\widehat{z}$ direction
go as $\cos(\theta)\approx 1-\frac{\theta^{2}}{2}$ whereas excitations go as
$\sin(\theta)\approx\theta$ for $\theta\ll 1$.
If the excitation and $\widehat{z}-$magnetization profiles are averaged over
different p-DANTE sequences that possess the same total pulse length and are
all selective for $\nu_{0}$, then the fluctuations in both the excitation
[Fig. 4(B)] and $\widehat{z}-$magnetization [Fig. 4(B)] profiles for
$\delta\nu\neq 0$ decrease relative to the excitation at $\delta\nu=0$ roughly
as $\frac{1}{\sqrt{N_{avg}}}$. However, even when $N_{avg}\gg 1$, there still
exists a ”baseline” excitation at $\delta\nu\neq 0$ which is nonzero
(averaging simply decreases the fluctuations about the baseline excitation).
The average ”baseline” excitation is approximately given by
$\overline{|\langle\widehat{I}_{Y}\rangle|}\approx\frac{\Theta}{N+2}$ and
$\langle\overline{\widehat{I}_{Z}\rangle}=1-\frac{(N+1)\theta^{2}}{2}$. Thus
decreasing $\theta$ and $N$ will decrease the ”baseline” excitation as
$\theta\propto\frac{1}{N}$. Similar schemes averaging over random sequences
have been previously used for stochastic dipolar recouplingR.Tycko (2007);
Tycko (2008).
Figure 4: Theoretically calculated "averaged" excitation [(B) and (D)] and
$\widehat{z}-$magnetization profiles [(A) and (C)] for a series of p-DANTE
sequences [Fig. 1(B)]. In all p-DANTE sequences, $N=30$,
$\theta=\frac{\pi}{60}$, and $\phi_{k}=0$ for all $k$ so that only spins
resonating at $\delta\nu=0$ Hz are rotated about the $\widehat{x}$-axis by
$\Theta=\frac{\pi}{2}$. In (A) and (B), one hundred different p-DANTE
sequences with randomly chosen delays were generated such that the average
pulse delay, $\frac{1}{29}\sum_{k=1}^{29}\tau_{k}=1.6$ ms, was the same for
all p-DANTE sequences. The excitation and $\widehat{z}$-magnetization profiles
were averaged over $N_{avg}$ p-DANTE sequences with $N_{avg}=1$ (red curve),
$N_{avg}=25$ (blue curve), and $N_{avg}=100$ (green curve(. In (C) and (D),
one hundred different p-DANTE sequences were chosen such that the $k^{th}$
delay for the $p^{th}$ experiment was
$\tau^{p}_{k}=\tau_{p}\left[1+\frac{\delta\tau}{\tau}\cos\left(\frac{2\pi
k}{f_{p}}\right)\right]$ where $\frac{\delta\tau}{\tau}=\frac{1}{\sqrt{2}}$
and $f_{p}$ was inversely proportional to the square root of the $p^{th}$
prime number (e.g., $f_{1}=1/\sqrt{2}$, $f_{2}=1/\sqrt{3}$,
$f_{100}=1/\sqrt{541}$). $\tau_{p}$ was chosen to ensure that the averaged
delay was $\frac{1}{29}\sum_{k=1}^{29}\tau^{p}_{k}=1.6$ ms for all $p$. The
averaged excitation and $\widehat{z}$-magnetization profiles for $N_{avg}=1$
(red curve with $f_{1}=1/\sqrt{2}$), $N_{avg}=25$ (blue curve, averaging from
$f_{1}=1/\sqrt{2}$ to $f_{25}=1/\sqrt{97}$), and $N_{avg}=100$ (green curve,
averaging from $f_{1}=1/\sqrt{2}$ to $f_{100}=1/sqrt{541}$) are shown. As
$N_{avg}$ increases, the "fluctuations" in both $\langle I_{Z}\rangle$ and
$-\langle I_{Y}\rangle$ decrease, and the results are similar for p-DANTE
sequences using randomly chosen delays [(A) and (B)] and those using
periodically modulated delays [(C) and (D)].
Besides randomly chosen delays, averaging over different sets of delays that
are periodically modulated can also lead to selective excitation. Consider a
series of delays where the $k^{th}$ delay is given by
$\tau_{k}=\tau+\delta\tau\cos\left(\frac{2\pi k}{f}\right)$, where $f$ is a
real number, and $\tau\geq\delta\tau$ so that $\tau_{k}\geq 0$. For such a
sequence to selectively excite spins resonating at $\nu_{0}$, the phase of the
$k^{th}$ pulse must be given by $\phi_{k}=-2\pi\nu_{0}T_{k-1}$ with $T_{0}=0$
and :
$\displaystyle T_{k}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k}\tau_{k}=k\tau-\frac{\delta\tau}{2}\left(1-\csc\left(\frac{\pi}{f}\right)\sin\left(\frac{2k+1}{f}\pi\right)\right)$
(15)
where the total time of the sequence is given by
$T_{\text{tot}}=Nt_{p}+T_{N-1}=Nt_{p}+(N-1)\tau-\frac{\delta\tau}{2}+\frac{\delta\tau}{2}\csc\left(\frac{\pi}{f}\right)\sin\left(\frac{\pi(2N-1)}{f}\right)$.
Using the values of $T_{k}$ in Eq. (15), $\overline{H}_{avg}^{(1)}$ in Eq.
(13) can be evaluated and is given by:
$\displaystyle\overline{H}^{(1)}_{avg}$ $\displaystyle=$
$\displaystyle\frac{a}{2N}\sum_{k=1}^{N}\left(\widehat{I}_{+}e^{i\left(\Delta\omega
T_{k}+\frac{\omega_{Z}t_{p}}{2}\right)}+I_{-}e^{-i\left(\Delta\omega
T_{k}+\frac{\omega_{Z}t_{p}}{2}\right)}\right)+b\widehat{I}_{Z}$ (16)
$\displaystyle=$
$\displaystyle\frac{a}{2}\sum_{n=-\infty}^{\infty}J_{n}\left(\frac{\Delta\omega\delta\tau}{2}\csc\left(\frac{\pi}{f}\right)\right)\frac{\text{sinc}\left(\frac{N(\Delta\omega\tau+\frac{2n\pi}{f})}{2}\right)}{\text{sinc}\left(\frac{\Delta\omega\tau+\frac{2n\pi}{f}}{2}\right)}\left(I_{+}e^{i\chi_{n}}+I_{-}e^{-i\chi_{n}}\right)+b\widehat{I}_{Z}$
where $\Delta\omega=\omega_{Z}-\omega_{0}$, $J_{n}$ is a bessel function of
order $n$, and
$\chi_{n}=\Delta\omega\frac{(N-1)\tau-\delta\tau}{2}+\frac{Nn\pi}{f}+\frac{\omega_{Z}t_{p}}{2}$.
From Eq. (16), $\overline{H}^{(1)}_{\text{avg}}$ is maximal at the conditions
$2\pi\Delta\nu\tau+\frac{2n\pi}{f}=2m\pi$ or at
$\Delta\nu=\frac{m}{\tau}-\frac{n}{f\tau}$ where $m$ and $n$ are integers.
These define the resonance conditions for this type of p-DANTE sequence.
However, $\overline{H}_{avg}^{(1)}$ is scaled by
$J_{n}\left(\left(m-\frac{n}{f}\right)\frac{\pi\delta\tau}{\tau}\csc\left(\frac{\pi}{f}\right)\right)$,
which is greatest when $m=n=0$. For $m\neq 0$ and $n\neq 0$, this scaling is
less than one, which results in a smaller total rotation.
Figure 4(C) and (D) show the numerically averaged $\widehat{z}$-magnetization
and excitation profiles respectively, averaged for up to one hundred different
p-DANTE sequences using periodically modulated delays. In the simulations,
$N=30$ and $\frac{\delta\tau}{\tau}=\frac{1}{\sqrt{2}}$. For the $p^{th}$
p-DANTE sequence, $f_{p}$ was set to be equal to the inverse of the square
root of the $p^{th}$ prime number, i.e., $f_{1}=1/\sqrt{2}$,
$f_{2}=1/\sqrt{3}$, $f_{100}=1/\sqrt{541}$. In order to better compare these
results to the results for the p-DANTE sequences using random delays [Figs.
4(A) and 4(B)], $\tau_{p}$ for the $p^{th}$ experiment was chosen so that
$\frac{1}{29}\sum_{j=1}^{29}\tau_{j}^{p}=1.6$ ms. First consider the
$N_{avg}=1$ case (red curve) in which $f=\frac{1}{\sqrt{2}}$ and
$\frac{1}{\tau}=625.13$ Hz. Unlike the case of using random delays [red curves
in Fig. 4(A) and 4(B)] where the resulting excitations appear randomly
distributed throughout the spectral range, the excitation profile using
periodically modulated delays occur at discrete $\delta\nu$ given by the
resonance condition $\delta\nu=625.13\left(m-\sqrt{2}n\right)$ Hz [Eq. (16)].
Note that while the resonance at $\delta\nu=0$ ($m=0$ and $n=0$) is maximally
excited ($\langle\widehat{I}_{Z}\rangle=0$ and
$-\langle\widehat{I}_{Y}\rangle=1)$, the degree of excitation at other
resonance conditions is less. In particular, the resonances at $\delta\nu=\pm
625.13$ Hz ($m=\pm 1$, $n=0$) are not observed in the calculated profile,
since at these conditions, $\overline{H}_{avg}^{(1)}$ is scaled by
$J_{0}(-2.3046)=0.053$, whereas the resonances at $\delta\nu=\pm 366.2$ Hz
($n=\mp 1$ and $m=\pm 2$) are clearly observed $(|J_{1}(1.35)|=0.5325$). As
was the case for p-DANTE sequences using randomly chosen delays, averaging
over different sets of periodically modulated p-DANTE sequences reduces the
excitation for all resonances except at $\delta\nu=0$, which is a common
resonance for all p-DANTE sequences. From Fig. 4, the
$\widehat{z}$-magnetization and excitation profiles using periodically
modulated delays [Figs. 4(C) and 4(D)] become similar to those using the
randomly chosen delays [Figs. 4(A) and 4(D)] as $N_{avg}$ increases.
Finally, it should be noted that the conditions under which the average
Hamiltonian in Eq. (16) provides a valid description ofthe p-DANTE sequence
are approximately the same as those found for the DANTE sequence [Fig. 3].
Figure 5 shows the difference in the excitation and
$\widehat{z}$-magnetization profiles calculated using either the exact
propagator or the propagator calculated using the average Hamiltonian up to
second-order [Eq. 14 and Eq. (16)] for the p-DANTE sequences used in Figs.
4(C) and 4(D). AHT works well for all $\delta\nu$ away from resonance
conditions which is evident from Fig. 5 for the $N_{avg}=1$ curve. The
magnitude of the error in this approximation is the same as that found for a
DANTE sequence with $\Theta=\pi/2$. The agreement of the AHT calculations with
the exact calculations appears to improve upon averaging over different
p-DANTE sequences, except for the $\delta\nu=0$ resonance. This is due to the
fact that only the $\delta\nu=0$ resonance is the same for all p-DANTE
sequence used in Fig. 5.
Figure 5: Difference in the calculated (A) $\widehat{z}$-magnetization and (B)
excitation profiles for the p-DANTE sequences used in Fig. 4(C) and 4(D)
between the profiles calculated using the propagator from AHT,
$\widehat{U}_{\text{AHT}}(T_{\text{tot}})=e^{-iT_{\text{Tot}}\omega_{Z}\widehat{I}_{Z}}e^{-iNt_{p}\left(\overline{H}^{(1)}_{\text{avg}}+\overline{H}^{(2)}_{\text{avg}}\right)}$
[Eq. (16) and Eq. (14)] and the profiles calculated using exact propagator,
$\widehat{U}_{\text{exact}}(T_{\text{tot}})$. The agreement is relatively good
over a wide frequency range except near the $\delta\nu=0$ Hz resonance
condition, $\delta\nu\approx\pm\frac{3\theta}{5\Theta\tau_{t}}=\pm 13$ Hz. For
all $|\delta\nu|>>\frac{3\theta}{5\Theta\tau_{t}}$ and
$|\delta\nu|\ll\frac{3\theta}{5\Theta\tau_{t}}$, the agreement between AHT and
the exact calculation improves with averaging over different p-DANTE
sequences.
## III Experimental
All experiments were performed on a 300 MHz Avance Bruker spectrometer (static
magnetic field of 7 T and an operating frequency for ${}^{1}H$ of 300.13 MHz),
using a 5-mm Bruker BBO probe. A 2M solution of acetone in acetone$-d_{6}$ was
used to experimentally determine the excitation and
$\widehat{z}$-magnetization profiles as a function of frequency offset from
the acetone resonance for both the DANTE and two pseudorandom pulse sequences.
The carrier frequency was incremented between $-580$ Hz below to $580$ Hz
above the acetone resonance in intervals of 10 Hz in order to experimentally
determine the excitation and $\widehat{z}$-magnetization profiles (a total of
1161 measurements), and the integral of the acetone peak was measured. In
order to measure the $\widehat{z}$-magnetization, a $\frac{\pi}{2}$ pulse
(Rabi frequency of 21.4 kHz) was applied after the DANTE and p-DANTE
sequences, which was phase cycled in concert with the receiver phase so that
only the $\widehat{z}-$magnetization prior to the last $\frac{\pi}{2}$ pulse
was measured. A delay of 40 seconds was used between scans in all experiments
in order to ensure that the system had relaxed back to equilibrium which was
necessary to avoid any distortions in the observed profiles.
In order to demonstrate the improved selectivity in the excitation and
$\widehat{z}-$magnetization profiles by signal averaging over different
p-DANTE sequences (as shown in Fig. 4), experiments using different p-DANTE
sequences were performed on a solution of acetone, dimethyl sulfoxide
(DMSO),and water diluted in $D_{2}O$, such that
$\frac{[\text{Acetone}]}{[\text{DMSO}]}=0.822$ and
$\frac{[\text{H}_{2}\text{O}]}{[\text{DMSO}]}=1.4355$. All chemicals were
obtained from Sigma-Aldrich.
## IV Results and Discussion
The experimentally determined excitation and $\widehat{z}-$magnetization
profiles under the DANTE and two different p-DANTE sequences obtained using a
2M acetone solution in acetone-$d_{6}$ are shown in Figure 6, where the blue
and red curves correspond to the theoretical and experimentally observed
profiles respectively. In these experiments, $N=30$, $\theta=\frac{\pi}{60}$,
and $t_{p}=720$ ns were used with a maximum rotation of
$\Theta=N\theta=\frac{\pi}{2}$. For the DANTE sequence, $\tau=2$ ms. Over the
spectral range shown in Fig. 6(A) and 6(D), excitations at frequencies
$\delta\nu=\frac{\pm 1}{\tau}=\pm 500$ Hz and at $\delta\nu=0$ Hz were
observed. For the p-DANTE sequences, the $k^{th}$ delay was given by either
[Figs. 6(C) and 6(F)]
$\tau_{k}=2.063\left[1-\cos\left(\frac{k\pi}{29+1}\right)\right]$ms=$4.126\sin^{2}\left(\frac{k\pi}{2(29+1)}\right)$
(which is a similar set of delays used in the UDD sequencesUhrig (2009)) or
[Figs. 6(B) and 6(E)]
$\tau_{k}=2.096\left[1+\frac{1}{3}\cos\left(\frac{2k\pi}{23}\right)\right]$ms.
In both cases, the average delay between pulses was equal to
$\frac{1}{29}\sum_{k=1}^{29}\tau_{k}=2$ ms in order to allow for better
comparison with the DANTE sequence used in Figs. 6(A) and 6(D). Both p-DANTE
sequences generated a maximum excitation at $\delta\nu=0$ Hz, and smaller
excitations for $\delta\nu\neq 0$ Hz were also observed, as expected. Note
that for the UDD-like p-DANTE sequence [Fig. 6(C) and 6(F)], the excitation
and $\widehat{z}$-magnetization profiles look similar to that of a p-DANTE
sequence using randomly chosen delays [the red curve in Figs. 4(A) and 4(B)],
whereas excitations using the other p-DANTE sequence [Figs. 6(B) and 6(E)]
appear to be concentrated within a smaller frequency range.
Figure 6: The experimental (red) and theoretical (blue) excitation [(A)-(C)]
and $\widehat{z}$-magnetization [(D)-(F)] profiles under application of the
DANTE [(A),(D)] and p-DANTE sequences [(B),(E),(C),(F)] as a function of the
applied RF’s offset away from the acetone resonance, $\delta\nu$. In all
experiments, $N=30$ and $\theta=\frac{\pi}{60}$. The profiles were generated
using a 2M acetone solution in acetone-d6 by changing the RF carrier frequency
from $-580$ Hz to $580$ Hz in 10 Hz increments, and the resulting acetone
resonance was integrated. For the DANTE sequence, $\tau=2$ ms; with these
parameters, the acetone resonance was maximally excited at
$\delta\nu\approx\pm\frac{1}{\tau}=\pm 500$ Hz and at $\delta\nu=0$ Hz over
the spectral range $[-580$ Hz, $580$ Hz]. For the p-DANTE sequences, the
$k^{th}$ delay was either given by
$\tau_{k}=2.096\left[1+\frac{1}{3}\cos\left(\frac{2k\pi}{23}\right)\right]$ms
[Figs. 6(B) and 6(E)] or by the UDDUhrig (2009)-like delay
$\tau_{k}=2.063\left[1-\cos\left(\frac{k\pi}{29+1}\right)\right]$ms=$4.126\sin^{2}\left(\frac{k\pi}{2(29+1)}\right)$
[Figs. 6(C) and 6(F)]. In both cases, the average delay between pulses was
$\frac{1}{29}\sum_{k=1}^{29}\tau_{k}=2$ ms in order to enable comparison with
the results from the DANTE sequence in Figs. 6(A) and 6(D). For both p-DANTE
sequences, there is minimal excitation at $\delta\nu=\pm 500$ Hz, although the
UDD-like sequence [Figs. 6(C) and 6(F)] appears to generate smaller
excitations over a wider frequency range than the other p-DANTE sequence
[Figs. 6(B) and 6(E)]. In all cases, there is good agreement between theory
(blue) and experiment (red).
In order to examine the effects of averaging over different p-DANTE sequences,
experiments were performed on a DMSO-acetone-water solution in $D_{2}O$. The
spectrum of the solution after a simple $\frac{\pi}{2}$-acquire sequence is
shown in Fig. 7(A), where the RF was applied on resonance with respect to the
water resonance [$\delta\nu_{\text{acetone,water}}=-768$ Hz and
$\delta\nu_{\text{DMSO,water}}=-620.2$ Hz]. The experimental excitation and
$\widehat{z}-$magnetization weighted spectra after the application of a DANTE
sequence with $N=30$, $\theta=\frac{\pi}{60}$ ($t_{p}=630$ ns), and
$\tau\approx\frac{1}{|\delta\nu_{\text{DMSO,water}}|}=1.6$ ms are shown in
Figure 7(B) and Figure 7(C) respectively. With this choice of $\tau$, the
DANTE sequence efficiently excites both the water
$\left(\delta\nu=\frac{0}{\tau}\right)$ and DMSO
$\left(\delta\nu=-\frac{1}{\tau}\right)$ resonances [Fig. 7(B)] and leaves the
acetone magnetization mostly about the $\widehat{z}-$axis [Fig. 7(C)].
The averaged excitation [Fig. 8(B)] and $\widehat{z}-$magnetization weighted
spectra [Fig. 8(A)] for the DMSO/acetone/water solution was obtained using up
to one hundred different p-DANTE sequences, and the results are shown for
$N_{avg}=1$ (red curve), $N_{avg}=25$ (blue curve), and $N_{avg}=100$ (green
curve) in Figure 8. The p-DANTE sequences used in Fig. 8 were the same used in
the theoretical calculations shown in Figs. 4(C) and 4(D), where the $k^{th}$
delay used in $p^{th}$ experiment was given by:
$\displaystyle\tau^{p}_{k}=\frac{46.77\text{ms}}{29-\frac{1}{2\sqrt{2}}\left(1-\csc\left(\frac{\pi}{f_{p}}\right)\sin\left(\frac{59\pi}{f_{p}}\right)\right)}\left(1+\frac{1}{\sqrt{2}}\cos\left(\frac{2\pi
k}{f_{p}}\right)\right)$ (17)
which ensured that the average delay,
$\frac{1}{29}\sum_{k=1}^{29}\tau_{k}^{p}\approx\frac{1}{|\delta\nu_{\text{DMSO,water}}|}=1.6$
ms, was the same as the delay used in the DANTE sequence shown in Figs. 7(B)
and 7(C). As in Figs. 4(C) and 4(D), the water ($\delta\nu=0$ Hz) was
maximally excited whereas the averaged excitation at the acetone and DMSO
resonances decreased upon averaging over different p-DANTE sequences.
Similarly, the $\widehat{z}$-magnetization weighted spectra indicated that the
acetone and DMSO magnetization remained mostly about the $\widehat{z}$-axis
after application of the p-DANTE sequence, whereas there was little
$\widehat{z}$-magnetization at the water resonance.
Figure 7: (A) The spectrum after a $\frac{\pi}{2}$-acquire experiment for a
DMSO, acetone, and water solution. The spectrum is centered on the water
resonance $(\delta\nu=0)$. (B) The spectrum after application of a DANTE pulse
sequence with $N=30$, $\theta=\frac{\pi}{60}$, $\Theta=N\theta=\frac{\pi}{2}$,
and $\tau=\frac{1}{|\delta\nu_{\text{DMSO,water}}|}\approx 1.6$ ms. In this
case, both the water and DMSO resonances are excited whereas very little
excitation occurs at the acetone resonance
($\delta\nu_{\text{acetone,water}}=-768$ Hz). (C) The
$\widehat{z}-$magnetization weighted spectrum after application of the DANTE
pulse sequence. As expected from (B), there is substantial
$\widehat{z}$-magnetization for the acetone resonance and little
$\widehat{z}$-magnetization for both the water and DMSO resonances after
application of the DANTE sequence. Figure 8: Experimental stacked plots of the
(A) $\widehat{z}$-magnetization weighted spectra and the (B) excitation
spectra after averaging over $N_{avg}=1$ (red), $N_{avg}=25$ (blue), and
$N_{avg}=100$ (green) p-DANTE sequences [same sequences used in Figs. 4(C) and
4(D)] applied to the acetone, DMSO, and water solution used in Fig. 7. From
Figs. 4(C) and 4(D), only the water resonance at $\delta\nu=0$ Hz should be
efficiently excited. From Fig. 6(B) the water resonance $(\delta\nu=0)$ is
efficiently excited, and the amount of excitation at the acetone and DMSO
resonances decreases with averaging over more p-DANTE sequences. In 8(A), the
$\widehat{z}-$ magnetization weighted spectra are shown, illustrating that
both the acetone and DMSO magnetization lie mostly along the
$\widehat{z}$-direction after application of the p-DANTE sequences.
## V Conclusions
In this work, average Hamiltonian theory (AHT) was used to calculate the
effective propagators for the both the DANTE [Fig. 1(A)] and pseudorandom-
DANTE or p-DANTE [Fig. 1(B)] sequences. It was found that an AHT description
the DANTE sequence is valid when $\theta=\frac{\pi}{60}$ and for total pulse
flip-angles of $\Theta=N\theta\leq\frac{5\pi}{9}$ over all frequencies [Fig.
3]. The validity of the AHT description was also found to depend upon the
spin’s resonance frequency, $\nu$, and an AHT description of DANTE works well
for frequencies in the range $\delta\nu\gg\frac{2\theta}{5\tau_{t}\Theta}$ and
$\delta\nu\ll\frac{2\theta}{5\tau_{t}\Theta}$ where
$\delta\nu=\text{min}\left[\left|\nu-\frac{n}{\tau_{t}}\right|\right]$ is the
smallest frequency difference between $\nu$ and the nearest resonance of the
DANTE sequence, $\frac{n}{\tau_{t}}$ where $n$ is an integer. Understanding
the limitations of an AHT description for DANTE enabled us to develop an AHT
description of the p-DANTE sequences [Fig. 1(B)] where the delays and phases
of the pulses are modulated in concert throughout the sequence. This
modulation of delays and phases breaks the periodicity of the DANTE sequence
and enables the p-DANTE sequence to excite spins at a single frequency,
$\nu_{0}$. The ability to use an AHT description for the DANTE and p-DANTE
sequences might also provide additional insights into other selective pulse
sequences, since any shaped pulses can be cast into a DANTE-like
descriptionShinnar and Leigh (1987); Shinnar _et al._ (1989, 1989). While the
excitation and $\widehat{z}-$magnetization profiles for a single p-DANTE
sequence are not particularly clean, i.e., small excitations exist at many
frequencies, averaging over different p-DANTE sequences helps to ”clean-up"
the excitation profiles so that only a baseline excitation exists everywhere
except at $\nu_{0}$, which is excited. Experimental demonstrations [Fig. 6 and
Fig. 8] of the p-DANTE sequences were found to be in good agreement with
theoretical predictions.
For future work, determining the existence of an optimal set of p-DANTE
sequences that generate the "cleanest" excitation profiles using the smallest
number of p-DANTE sequences will be investigated. Since the frequency
selection in p-DANTE sequences is determined by correlating the pulse phases
with the delays, the p-DANTE sequences could also be incorporated into
ultrafast NMRFrydman _et al._ (2002, 2003) techniques to selectively excite
certain resonances in different parts of the sample volume. Furthermore,
extending the AHT results obtained in this paper to coupled spin systems is
currently underway, whereby a DANTE-like or p-DANTE-like sequences can be used
to selectively excite a particular multiple-quantum spin transition. The
conditions under which such an AHT description can be applied in these systems
are approximately the same as those found in this paper, since any subspace of
two transitions can be describedFeynman _et al._ (1957) as an effective
spin-1/2. Coupling these techniques with ultra-fast NMR should enable the
quick determination of all spin transitions in a given molecular system.
Acknowledgments We would like to thank Alex Burum for a careful reading of
this manuscript. This work was supported by a Camille and Henry Dreyfus New
Faculty award, a Provost Research award and startup funds from the University
of Miami.
## References
* Ernst (1970) R. Ernst, J. Mag. Res., 3, 10 (1970).
* Kaiser (1970) R. Kaiser, J. Mag. Res., 3, 28 (1970).
* Kaiser (1974) R. Kaiser, J. Mag. Res., 15, 44 (1974).
* Bartholdi _et al._ (1976) E. Bartholdi, A. Wokaun, and R. Ernst, Chem. Phys., 18, 57 (1976).
* Ernst (1966) R. Ernst, J. Chem. Phys., 45, 3845 (1966).
* B.Blumich (1987) B.Blumich, Prog. Nucl. Magn. Res., 19, 331 (1987).
* Levitt and Freeman (1981) M. Levitt and R. Freeman, J. Mag. Res., 43, 502 (1981).
* Shaka _et al._ (1983) A. Shaka, J. Keeler, T. Frenkiel, and R. Freeman, J. Magn. Res., 52, 335 (1983).
* Haeberlen and Waugh (1968) U. Haeberlen and J. Waugh, Phys. Rev., 175, 453 (1968).
* Bosman _et al._ (2004) L. Bosman, P. Madhu, S. Vega, and E. Vinogradov, J. Mag. Res., 169, 39 (2004).
* Uhrig (2009) G. Uhrig, Phys. Rev. Lett., 102, 120502 (2009).
* Carr and Purcell (1954) H. Carr and E. Purcell, Physical Review, 94, 630 (1954).
* Freeman (1991) R. Freeman, Chem. Rev., 91, 1397 (1991).
* Tomlinson and Hill (1973) B. Tomlinson and H. Hill, J. Chem. Phys., 59, 1775 (1973).
* Veshtort and Griffin (2005) M. Veshtort and R. Griffin, Chem.Phys.Chem., 5, 834 (2005).
* Ordige (1987) R. Ordige, Mag. Res. Med., 5, 93 (1987).
* Bodenhausen _et al._ (1976) G. Bodenhausen, R. Freeman, and G. Morris, J. Magn. Reson., 23, 171 (1976).
* Canet _et al._ (1995) D. Canet, J. Brondeau, and C. Roumestand, J. Mag. Res. Ser. A, 117, 103 (1995).
* Kacynski _et al._ (1992) J. Kacynski, N. Dodd, and B. Wood, J. Mag. Res., 100, 453 (1992).
* Roumestand and Canet (2000) C. Roumestand and D. Canet, J. Mag. Res., 147, 331 (2000).
* Shinnar and Leigh (1987) M. Shinnar and J. Leigh, J. Magn. Res., 75, 502 (1987).
* Shinnar _et al._ (1989) M. Shinnar, S. Eleff, H. Subramanian, and J. Leigh, Magn. Res. Med., 12, 74 (1989a).
* Shinnar _et al._ (1989) M. Shinnar, L. Bolinger, and J. Leigh, Mag. Res. Med., 12, 88 (1989b).
* Dybowski and Vaughan (1975) C. Dybowski and R. Vaughan, Macromolecules, 8, 50 (1975).
* R.Tycko (2007) R.Tycko, Phys. Rev. Lett., 99, Art. no. 187601 (2007).
* Tycko (2008) R. Tycko, J. Phys. Chem. B, 112, 6114 (2008).
* Frydman _et al._ (2002) L. Frydman, T. Scherf, and A. Lupulescu, Proc. Nat. Acad. Sci, 99, 15858 (2002).
* Frydman _et al._ (2003) L. Frydman, A. Lupulescu, and T. Scherf, J. Am. Chem. Soc., 125, 9204 (2003).
* Feynman _et al._ (1957) R. Feynman, F. Vernon, and R. Hellwarth, J. Appl. Phys., 28, 49 (1957).
|
arxiv-papers
| 2010-07-20T04:55:07 |
2024-09-04T02:49:11.776553
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jamie D. Walls and Alexandra Coomes",
"submitter": "Jamie Walls",
"url": "https://arxiv.org/abs/1007.3335"
}
|
1007.3727
|
arxiv-papers
| 2010-07-19T15:27:44 |
2024-09-04T02:49:11.789523
|
{
"license": "Public Domain",
"authors": "Giuseppe Iurato",
"submitter": "Giuseppe Iurato",
"url": "https://arxiv.org/abs/1007.3727"
}
|
|
1007.3728
|
# A possible quantic motivation of the structure of quantum group
Giuseppe Iurato
> Abstract. Following a suggestion of A. Connes (see [Co], § I.1), we build up
> a (first) simple natural structure of a no finitely generated braided non-
> commutative Hopf Algebra, suggested by elementary quantum mechanics.
### 1\. The notion of EBB-groupoid
The idea of quantum group has been introduced, in a pure mathematical context,
independently by V.G. Drinfeld111Drinfeld put, as basic examples of a quantum
group structure, some group algebras dually related with certain function
algebras, starting from simple quantum mechanics considerations (see [Dr2], §
1). ([Dr1], [Dr2]) and M. Jimbo ([Ji1], [Ji2]), who used the adjective
’quantum’ for the fact that a such structure is obtained quantizing (according
to geometric quantization) a Poisson symplectic structure introduced on the
algebra $\mathfrak{F}(G,\mathbb{C})$ of the differentiable
$\mathbb{C}$-functions defined on a Lie group $G$. This structure of quantum
group is the result of a non- commutative Hopf algebra achieved as non-
commutative (non-trivial) deformations ([BFFLS]) of the initial Hopf algebra
$\mathfrak{F}(G,\mathbb{C})$, or of the universal enveloping Hopf algebras of
an arbitrary semisimple Lie algebras. A little later, Yu. Manin (see [Mn]) and
S.L. Woronowicz ([Wo]) independently constructed non-commutative deformations
of the algebra of functions on $SL_{2}(\mathbb{C})$ and $SU_{2}$,
respectively.
On the other hand, the study of quantum group was inspired by the works of
physicists on integrable XYZ models with highest spin. Moreover, L.D. Faddeev
and coworkers ([Fa], [FST], [FT]), had already used similar structures in
mathematical physics, for the algebra of quantum inverse scattering transform
of the theory of integrable models, as well as P.P. Kulish, N.Yu. Reshetikhin
([KR]) and E.K. Sklyanin ([KuS]).
However, from elementary quantum mechanics, it is possible constructs a simple
model of a braided non-commutative Hopf algebra, through an algebraically
modern re-examination of few elementary notions of the original papers of W.
Heisenberg, M. Born e P. Jordan ([33]) on matrix quantum mechanics.
The basic algebraic structure for our model is that of groupoid.
Independently, A. Baer ([Ba]), and W. Brandt ([Br]) in some of his researches
on quadratic forms on $\mathbb{Z}$, introduced a well-defined new algebraic
structure, that we will call Ehresmann-Baer-Brandt groupoid (or EBB-groupoid)
(see [Co3], [CW], [La], [Hi], [Mak]). It is a partial algebraic structure
because is an algebraic system provided by a partial (or no total) composition
law, that is, not defined everywhere.
Historically, S. Eilenberg and N. Steenrod ([ES]) formulated the notion of
abstract category on the basis of the previous notion introduced by S.
Eilenberg and S. MacLane ([EM]), considering a groupoid as a special category
in which every morphism has an inverse (see also [MacL]).
A little later, the notions of abstract category and groupoid appeared too in
the work of C. Ehresmann ([Eh]).
However, retrospectively, the common substructure to all these notions of
groupoid, is that of multiplicative system (or Eilenberg-Steenrod groupoid, or
ES-groupoid), say $(G,\mathcal{D}_{\diamond},\diamond)$, where $\diamond$ is a
partial composition law over $G$, defined on a domain
$\mathcal{D}_{\diamond}\subseteq G\times G$. It was introduced by [ES].
The further categorial (over) structure of a ES-groupoid, leads to the notion
of Eilenberg-MacLane groupoid (or EM-groupoid), introduced in [EM], and to the
equivalent222See [Wa] for a proof of this equivalence. notion of Ehresmann
groupoid (or E-groupoid), introduced in [Eh]. We will use the notion of
E-groupoid.
An E-groupoid is an algebraic system of the type $(G,G^{(0)},r,s,\star)$,
where $G,G^{(0)}$ are non-void sets such that $G^{(0)}\subseteq G$,
$r,s:G\rightarrow G^{(0)}$ and $\star:G^{(2)}\rightarrow G$ with
$G^{(2)}=\\{(g_{1},g_{2})\in G\times G,s(g_{1})=r(g_{2})\\}$, satisfying the
following conditions:
* •
$\\!\\!\\!{}_{1}$ $s(g_{1}\star g_{2})=s(g_{2})$, $r(g_{1}\star
g_{2})=r(g_{1})$ $\forall(g_{1},g_{2})\in G^{(2)}$;
* •
$\\!\\!\\!{}_{2}$ $s(g)=r(g)=g\ \ \forall g\in G^{(0)}$;
* •
$\\!\\!\\!{}_{3}$ $g\star\alpha(s(g))=\alpha(r(g))\star g=g\ \ \forall g\in
G$;
* •
$\\!\\!\\!{}_{4}$ $(g_{1}\star g_{2})\star g_{3}=g_{1}\star(g_{2}\star g_{3})\
\ \forall g_{1},g_{2},g_{3}\in G$;
* •
$\\!\\!\\!{}_{5}$ $\forall g\in G,\ \exists g^{-1}\in G:\ g\star
g^{-1}=\alpha(r(g)),\ g^{-1}\star g=\alpha(s(g))$,
where $\alpha:G^{(0)}\hookrightarrow G$ is the immersion of $G^{(0)}$ into
$G$. The maps $r,s$ are called, respectively, range (or target) and source,
$G$ the support and $G^{(0)}$ the set of units of the groupoid. $g^{-1}$ is
said to be the (bilateral) inverse of $g$, so that we have an inversion map of
the type $i_{G}:g\rightarrow g^{-1}$, defined on the whole of $G$. Instead,
the map $\star$ is a partial map defined on $G^{(2)}\subseteq G\times G$, and
not on the whole of $G\times G$. However, from now on, for simplicity, we will
suppresses the symbol $\alpha$ in 3. and 5., writing only $r(g),s(g)$ instead
of $\alpha(r(g)),\alpha(s(g))$.
However, a E-groupoid is more general than the first notion of groupoid used
by W. Brandt and A. Baer, that we call Baer-Brandt groupoid (or BB-groupoid);
indeed, a BB-groupoid may be defined as a E-groupoid satisfying the further
condition
* •
$\\!\\!\\!{}_{6}$ $\forall g,g^{\prime\prime}\in G,\ \ \exists g^{\prime}\in
G\ \mbox{such\ that}\ (g,g^{\prime})\in G^{(2)},\
(g^{\prime},g^{\prime\prime})\in G^{(2)}$,
so that, we will call an Ehresmann-Baer-Brandt groupoid(or EBB-groupoid333A.
Nijenhuis ([Ni]) sets the structures of groupoid in the general context of the
group theory, whereas a topological characterizations of such structures is
due to S. Golab ([Go]).), an algebraic system satisfying $\bullet_{i}\ \
i=1,...,6$.
In view of the applications to quantum mechanics, it is important the
following particular examples of EBB-groupoid (see [Co3], § II.5).
If $X$ is an abstract (non-empty) set, let $G=X^{2}$,
$G^{(0)}=\Delta(X^{2})=\\{(x,x);\ x\in X\\}$, $r=pr_{1}:(x,y)\rightarrow x,\
s=pr_{2}:(x,y)\rightarrow y\ \ \forall x,y\in X$ and $(x,y)\star(y,z)=(x,z)$.
Then, it is easy verify that $(G,G^{(0)},r,s,\star)$ is a EBB-groupoid, with
$(x,y)^{-1}_{d}=(x,y)^{-1}_{s}=(y,x)$. It is called444Or pair EBB-groupoid, or
coarse EBB-groupoid. the natural EBB-groupoid on $X$, and denoted by ${\cal
G}_{EBB}(X)$.
On the other hand, there exists E-groupoids that are not EBB-groupoids. For
instance, if $\Gamma$ is a group that acts on a set $X$ by the action
$\psi:\Gamma\times X\rightarrow X$, if $e$ is the identity of $\Gamma$, and if
we put555In the definition of $r$ and $s$, it is necessary to consider the
bijection $x\rightarrow(e,x)\ \ \forall x\in X$, that identifies $X$ with
$\\{e\\}\times X$. $G=\Gamma\times X,\ G^{(0)}=\\{e\\}\times X,\ r(g,x)=x,\
s(g,x)=\psi(g,x)$ and $(g_{1},x)\star(g_{2},y)=(g_{1}g_{2},x)$ if and only
if666Indeed, $G^{(2)}=\\{((g_{1},x),(g_{2},y))\in G^{2};\
s(g_{1},x)=r(g_{2},y)\\}$ and $s(g_{1},x)=\psi(g_{1},x)=g_{1}x=y=r(g_{2},y)$,
that is $y=g_{1}x$. $y=\psi(g_{1},x)$, then we obtain a E-groupoid, with
$(g,x)^{-1}=(g^{-1},\psi(g,x))$, called the semi-direct product E-groupoid of
$\Gamma$ by $X$ and denoted by $\Gamma\ltimes_{E}X$.
Nevertheless, such a structure cannot be a EBB-groupoid if, for example, the
action $\psi$ is not transitive. In fact, for every $(g_{1},x),(g_{3},z)\in
G$, there exists $(g_{2},y)\in G$ such that $(g_{1},x)\star(g_{2},y)\in
G^{(2)}$ and $(g_{2},y)\star(g_{3},z)\in G^{(2)}$ if and only if
$y=\psi(g_{1},x)$ and $z=\psi(g_{2},y)$, that is, if and only if there exists
$g_{2}\in G$ such that $z=\psi(g_{2},\psi(g_{1},x))$ for given
$z,\psi(g_{1},x)\in X$. Therefore, if $\psi$ is a no transitive action,
follows that $\Gamma\ltimes_{E}X$ is a E-groupoid but not a EBB-groupoid.
The following result puts in relation the two structures of E-groupoid and
EBB-groupoid (see [Wa]).
I. A multiplicative system $(G,\mathcal{D}_{\diamond},\diamond)$ is a
E-groupoid if and only if there exists a unique partition $\mathcal{P}$ of $G$
such that $\mathcal{D}_{\diamond}\subseteq\bigcup_{A\in\mathcal{P}}A$ and that
the induced multiplicative system $(A,\diamond_{A})$ be a EBB-groupoid for
every $A\in\mathcal{P}$.
For instance, if the given multiplicative system is a E-groupoid, namely
$(G,G^{(0)},r,s,\star)$ (with $\star=\diamond$), then we consider the relation
$\sim\ \doteq\\{(g,g^{\prime\prime})\in G\times G;\ \exists g^{\prime}\in G,\
(g,g^{\prime})\in G^{(2)},\ (g^{\prime},g^{\prime\prime})\in G^{(2)}\\}.$
Hence, it is possible to prove that $\sim$ is an equivalence relation, so that
$\mathcal{P}=G/\sim$ is a partition of $G$. If we set $A^{(2)}=G^{(2)}\cap A$,
$A^{(0)}=G^{(0)}\cap A$, $r_{A}=r_{|_{A}},s_{A}=s_{|A}$ and
$\star_{A}=\star_{|_{A^{(2)}}}$, for every $A\in\mathcal{P}$, then
$(A,A^{(0)},r_{A},s_{A},\star_{A})$ is a EBB-groupoid.
### 2\. The notion of convolution structure777See [Gr], [No], [AM], [Vr].
Let $A$ be an unitary commutative ring, $J$ a set of indices and $X=\\{x_{j};\
j\in J\\}=(x_{j})_{j\in J}$ a family of abstract symbols. Let
$\langle X\rangle=\langle(x_{j})_{j\in J}\rangle=\\{\sum_{j\in J}a_{j}x_{j};\
a_{j}\in A,\ x_{j}\in X\\}$
be the set of the formal linear combinations of the elements of $(x_{j})_{j\in
J}$ with coefficients on $A$: $a_{j}x_{j}$ must be understood as the value of
a map of the type $A\times X\rightarrow\langle X\rangle$, whereas $\sum_{j\in
J}a_{j}x_{j}$ have all coefficients $a_{j}$ zero except a finite number;
$\langle X\rangle$ is said to be the free set generated by $X$. With the
operations
$None$ $\big{(}\sum_{j\in J}a_{j}x_{j}\big{)}+\big{(}\sum_{j\in
J}b_{j}x_{j}\big{)}=\sum_{j\in J}(a_{j}+b_{j})x_{j},$ $None$
$a\cdot\big{(}\sum_{j\in J}a_{j}x_{j}\big{)}=\sum_{j\in J}(aa_{j})x_{j},\ \
a\in A,$
$\langle X\rangle$ is a $A$-module, said to be the free $A$-module on $X$,
with base X. In this context, it is possible to prove that its elements admits
a unique decompositions of the type $\sum_{j\in J}a_{j}x_{j}$. We will denote
this free $A$-module by ${\cal M}_{A}(X)=(\langle X\rangle,+,\cdot)$, the
operations $+,\cdot$ being (5), (6). Furthermore, it is possible to prove
that: a $A$-module $(M,+,\cdot)$ is free on $(\emptyset\neq)X\subseteq M$ if
and only if, for any $A$-module $(M^{\prime},+,\cdot)$ and for any map
$\varphi:X\rightarrow M^{\prime}$, there exists a unique $A$-homomorphism of
$A$-modules $\psi:M\rightarrow M^{\prime}$ such that
$\varphi=\psi{\big{|}}_{X}$ (in such a case, $\psi$ is said to be an
$A$-extension of $\varphi$, whereas $X$ is a base of $M$, that is $M=\langle
X\rangle$). A base identifies a unique free $A$-module in the sense that, if
$M,M^{\prime}$ are two free $A$-modules with respective bases $X,X^{\prime}$
and $f:X\rightarrow X^{\prime}$ is a bijection, then $M,M^{\prime}$ are
$A$-isomorphic.
These last results are important in the processes of linear extension from a
base.
On the free $A$-module ${\cal M}_{A}(X)=(\langle X\rangle,+,\cdot)$ it is
possible to establish a structure of unitary ring (in general, non-
commutative) by a precise888A priori, the formal choice of this convolution
product may be arbitrary. Nevertheless, in the mathematical physics context,
often there are cases where such choice is forced by some ’ad hoc’ physical
reasons (as, for example, causality $-$ see [Vr]). product $\ast$ (of
convolution) of elements of $\langle X\rangle$. The resulting algebraic
structure999The unitary ring (in general, non-commutative) of this structure
$\mathcal{C}_{A}(X)$, is $(\langle X\rangle,+,\ast)$, whereas the convolution
product $\ast$ and the $A$-module product $\cdot$, are linked by the
compatibility relation $a\cdot(x\ast y)=(a\cdot x)\ast y=x\ast(a\cdot y)\ \
\forall a\in A,\ \forall x,y\in\langle X\rangle$; as usual, the product
$\cdot$ is implicit. ${\cal C}_{A}(X)=(\langle X\rangle,+,\cdot,\ast)$, is
said to be a convolution structure associated to the free $A$-module ${\cal
M}_{A}(X)$.
It is important to point out that such structure is strictly related to the
choice of the product $\ast$, and, to this purpose, the following remarks are
meaningful. When the set $X$ is already endowed with a given algebraic
structure of the type $(X,\diamond)$ (for example, that of EBB-groupoid),
formal coherence principles imposes that such (convolution) product must be
’predetermined’ by such preexistent structure. For instance, it is usually
required such product to be the result of the linear $A$-extension of the
operation $\diamond$, eventually taking into account further informal
requirements (of physical nature).
In particular, in view of the next arguments, $A$ will be a scalar field
$\mathbb{K}$, so that the $A$-module $(\langle X\rangle,+,\cdot)$ is a
$\mathbb{K}$-linear space that will be extended to a linear
$\mathbb{K}$-algebra (often said the convolution $\mathbb{K}$-algebra of the
given $A$-module), by means a convolution product.
Historically, the first structures of convolution $\mathbb{K}$-algebras were
the so-called group algebra, convolution structures made on a given group.
Group algebras were introduced (for finite groups) by T.Molien and G.
Frobenius, for investigations of representations of these groups.
Subsequently, at the beginning of the 20th-century, I. Schur ([Sh]) and H.
Weyl ([We]) used systematically the group algebras for investigations of
compact groups and commutative locally compact groups.
For applications of group algebra structures to quantum mechanics, see, for
example, [Lo].
### 3\. The notion of EBB-groupoid algebra
With any EBB-groupoid, a unique, well defined, natural structure of linear
$\mathbb{K}$-algebra is associated by means a precise convolution product
$\ast$.
If $\mathbb{K}$ is a field, let us consider the free $\mathbb{K}$-module
generated by the support of the given EBB-groupoid, hence we will consider the
related convolution structure provided by an adapted convolution product. Such
a group algebra is more appropriately called the EBB-groupoid algebra
associated to the given EBB-groupoid (although it is nothing else that a group
algebra on the support of a EBB-groupoid).
If ${\cal G}=(G,G^{(0)},r,s,\star)$ is a EBB-groupoid and $\mathbb{K}$ is a
scalar field, we put $X=G$, and thus $\langle G\rangle=\\{\sum_{g\in
G}\lambda(g)g;\ \lambda(g)\in\mathbb{K},\ \forall g\in G\\}$ is the set of the
formal combinations (with coefficients in $\mathbb{K}$) of elements of $G$,
equipped with the operations
$None$ $\big{(}\sum_{g\in G}\lambda(g)g\big{)}+\big{(}\sum_{g\in
G}\mu(g)g\big{)}=\sum_{g\in G}(\lambda(g)+\mu(g))g,$ $None$
$\mu\cdot\big{(}\sum_{g\in G}\lambda(g)g\big{)}=\sum_{g\in G}(\mu\lambda(g))g\
\ \ \forall\mu\in\mathbb{K}.$
${\cal M}_{\mathbb{K}}(G)=(\langle G\rangle,+,\cdot)$ is a free
$\mathbb{K}$-module (that is a $\mathbb{K}$-linear space) with base $G$. We
define the convolution product
$None$ $\big{(}\sum_{g\in G}\lambda(g)g\big{)}\ast\big{(}\sum_{g\in
G}\mu(g)g\big{)}=\sum_{g\in G}\xi(g)g$
where (with Cauchy)
$None$ $\xi(g)=\sum_{g_{1}\star g_{2}=g}\lambda(g_{1})\mu(g_{2})\ \ \ \forall
g\in G.$
Therefore, it is immediate to verify that ${\cal C}_{\mathbb{K}}(G)=(\langle
G\rangle,+,\cdot,\ast)$, as convolution structure associated to ${\cal M}(G)$,
is a linear $\mathbb{K}$-algebra, that we will call the EBB-groupoid algebra
(over the field $\mathbb{K}$) associated to the EBB-groupoid ${\cal
G}=(G,G^{(0)},r,s,\star)$. We will denote it by ${\cal A}_{\mathbb{K}}({\cal
G})$. As regards what has been said at the end of § 2, it is possible a
(unique) linear extension of the algebraic relations of the EBB-groupoid $\cal
G$ to ${\cal A}_{\mathbb{K}}(\cal G)$.
### 4\. Brief outlines of atomic spectroscopy101010See [Em], Chapt. 7, and
[An], [Ec], [Fe], [Gi], [Hu], [Lu].
The physical phenomenology leading to the first theoretical formulations of
quantum mechanics, were the atomic spectroscopy of emission and absorption of
electromagnetic waves. Nevertheless, the classical physics failed in the
interpretation of the structure of the observed atomic spectra of the various
chemical elements. An atomic spectrum of emission [absorption ] of an
arbitrary chemical element, is characterized by a well defined sequence of
spectral lines. Each spectroscopic line is related to the emission
[absorption] of an electromagnetic radiation of well precise frequency, from
the atom of the given chemical element under examination. Hence, each spectral
line is identified by that well-determined frequency $\nu$ of the e.m.
radiation corresponding to it. Moreover, such lines appears organized into
groups called (spectral) series (of the given spectrum).
From these experimental investigations, born the atomic spectroscopy of the
beginning of the 20th-century. It were, mainly, a coherent set of qualitative
and descriptive rules concerning the symmetry and regularity properties of
these spectral lines. The new quantum theory was built on the basis of it.
The first semi-quantitative spectroscopic rules has been formulated with the
quantum theory of N.H. Bohr, A. Sommerfeld and W. Wilson in the years
1913-1914, on the basis of the pioneering works of M. Planck, A. Einstein, E.
Rutherford, and others.
Nevertheless, in experimental spectroscopy, the works of J.R. Rydberg (see
[38]) and W. Ritz (see [38]) got a prominent rule, and led to the formulation
of their homonymous principle.
From the analysis of the (spectral) series of lines of several single atomic
spectra, Rydberg established the following (Rydberg) combination principle:
$\bullet_{1}$ each spectral lines, of any series, can be described in terms of
suitable spectral terms $T_{(s)}(n)\ s,n\in\mathbb{N}$;
$\bullet_{2}$ the frequency (or the wave number) of a line is given by a
relation of the type
$None$
$\nu_{(n,n^{\prime})}^{(s,s^{\prime})}=T_{(s^{\prime})}(n^{\prime})-T_{(s)}(n)$
where $s,s^{\prime}$ are indices of spectral series (of lines), $n,n^{\prime}$
indices of lines (of spectral series), and
$\lim_{n\rightarrow\infty}T_{(s)}(n)=0$ for any fixed $s$;
$\bullet_{3}$ there are precise selection rules that are imposed
instructions111111For instance, a simple selection rule is $|s-s^{\prime}|=1$.
In any case, according to these rules, not every possible combination of
values of $s,s^{\prime}$, corresponds to an effectively observed spectral
lines, but only those that satisfy well defined algebraic relations. on the
possible values $s,s^{\prime}$ of $(\spadesuit)$ in order that the frequency
predicted by this formula represents observed spectral lines.
Successively, Ritz stated that:
1) on the basis of the Rydberg’s formula, found for the hydrogenoid atoms,
according to
$T_{(s)}(n)={R_{H}}/{(n-a_{s})^{2}}$
(where $R_{H}$ is a universal constant (of Rydberg) and
$a_{0},a_{1},a_{2},...$ are real constants, typical of each hydrogenoid atom,
and that characterizes the different spectral series of its spectrum), it is
possible to formulate, for the spectral terms, the following more general
expression
$T_{(s)}(n)=\Phi_{(s)}(n)(K/n^{2})\qquad\mbox{\rm(\it{Ritz-Rydberg
formula}\rm)}$
where $K$ is a universal constant, and
$\lim_{n\rightarrow\infty}\Phi_{(s)}(n)=1$ for each fixed $s$;
2) the frequencies of the lines of each spectral series (that is, for each
fixed $s^{\prime}=s$), verify the following Ritz’s combination (or
composition) law $\nu_{(i,j)}^{(s)}+\nu_{(j,k)}^{(s)}=\nu_{(i,k)}^{(s)}$.
Subsequently, Ritz verified the validity of law 2) also for the lines of
different series, reaching to a more general Ritz-Rydberg composition
law121212Independently by eventual selection rules on $s,s^{\prime}$, as,
instead, required by the Rydberg’s combination principle (see $\bullet_{3}$).
Moreover, $(\clubsuit)$ is verified by $(\spadesuit)$, under the suitable
selection rules.:
$None$
$\nu_{(i,j)}^{(s,s^{\prime})}+\nu_{(j,k)}^{(s,s^{\prime})}=\nu_{(i,k)}^{(s,s^{\prime})}\
\ \ \ \ \forall s,s^{\prime},\ \ \ \ \forall i,j,k.$
The principle achieved by the above conditions 1) and 2), with the last
extension due to Ritz, it is usually called the Ritz-Rydberg composition
principle (as extension of the Rydberg’s composition principle). The main
interesting part of this principle is the Ritz-Rydberg composition law
131313Neglecting the series indices $(s,s^{\prime})$.
$\nu_{(i,j)}+\nu_{(j,k)}=\nu_{(i,k)}$.
The principle has been confirmed by a large class of spectroscopic phenomena,
ranging from atomic spectra to the molecular ones, from those optical to the X
ray spectra, and so on.
The class of the fundamental Franck-Hertz experiments ([FH]; [CCP], Chap. IV,
§ 10; [Her]), experimentally confirmed the existence of discrete energy levels
in an atom (predicted by N.H. Bohr), each characterized by a certain frequency
$\nu$ ($=E/h$, $E$ being the typical energy value of the given level). Hence,
the set of the energy levels of an atom is characterized by a well determined
set of frequencies ${\cal F}_{I}=\\{\nu_{i};\nu_{i}\in\mathbb{R}^{+},\ i\in
I\subseteq\mathbb{N}\\}\subseteq\mathbb{R}^{+}$, that the Franck-Hertz
experiences proved to be finite or countable, hence labelled by a subset $I$
of $\mathbb{N}$. Since $I$ is a non-empty subset of $(\mathbb{N};\leq)$ (with
the usual order), ${\cal F}_{I}$ can be ordered according to the increasing
indices of $I$, so that we can assume that the map
$\rho:i\rightarrow\nu_{i}\in{\cal F}_{I}\ \ \forall i\in I$ is monotonically
increasing and thus $I\cong{\cal F}_{I}$. Moreover, at each energy level
correspond the energy $E_{i}=h\nu_{i}$, where $h$ is the Planck’s constant.
Furthermore, we recall another basic principle of the spectroscopy, namely the
principle of A. Conway (see [Mo], Chapt. 13, § 11), according to any spectral
line is produced, once at a time, by a unique (perinuclear) electron, with
$\nu_{ij}\neq 0$ whenever $i\neq j$.
A spectral line correspond to the electromagnetic radiation involved in the
transition of an electron from a given level to another. If, throughout such
transition $i\rightarrow j$, the initial energy level is $\nu_{i}$ and the
final is $\nu_{j}$, with $i\neq j$, then the only physical observable
(according to P.A.M. Dirac $-$ see [Di2]) is the electromagnetic radiation of
emission (if $i>j$), or of absorption (if $i<j$), with frequency
$\nu_{i\rightarrow j}=\nu_{j}-\nu_{i}=(E_{j}-E_{i})/h$. In the emission it is
$\nu_{i\rightarrow j}<0$, whereas in the absorption it is $\nu_{i\rightarrow
j}>0$. Let we put141414With $\nu_{(i,j)}=\nu_{ij}$. $\nu_{i\rightarrow
j}=\nu_{ij}$, so that, if $\nu_{ij}=\nu_{j}-\nu_{i}<0$, then we have an
emission line, whereas, if $\nu_{ji}=-\nu_{ij}>0$, then we have an absorption
line. It follows the existence of a simple (opposite) symmetry, called the
Kirchhoff-Bunsen (inversion) symmetry (or KB-symmetry), between the emission
spectrum and the absorption spectrum of the same atom, given by
$\nu_{ij}=-\nu_{ji}$ (and that follows from the Ritz-Rydberg composition law).
Therefore, the lines of the atomic spectrum (of emission/absorption) of a
chemical element, are represented (in $\mathbb{R}$) by the elements of the set
$\Delta{\cal F}_{I}=\\{\nu_{ij}=\nu_{j}-\nu_{i};\ i,j\in
I\subseteq\mathbb{N}\\}\subseteq\mathbb{R}$, symmetric respect to the origin,
whose positive part represents the absorption spectrum, whereas that negative
represents the emission spectrum.
Finally, other spectroscopic principles prescribes symmetries and regularities
of this set, from which it is possible infer further algebraic properties of
it. However, as seen, between these principles, the Ritz-Rydberg combination
principle has a prominent rule in justifies the intrinsic non-commutativity of
the formal quantum theory.
### 5\. The Heisenberg-Born-Jordan EBB-groupoid
In general, $(\Delta{\cal F}_{I};+)$ is not a subgroup of the commutative
group $(\mathbb{R};+)$ because, if $\nu_{ij},\nu_{lk}\in\Delta{\cal F}_{I}$,
not even $\nu_{ij}\pm\nu_{lk}$ correspond to an observed spectral line, that
is may be $\nu_{ij}\pm\nu_{lk}\notin\Delta{\cal F}_{I}$. The elements of
$\Delta{\cal F}_{I}$ combines by means the Ritz-Rydberg combination principle,
according to $\nu_{ij}+\nu_{lk}\in\Delta{\cal F}_{I}$ if and only if151515This
last condition suggests the formal presence of a EBB-groupoid structure.
Indeed, this structure is endowed with a partial binary operation defined on a
domain $G^{(2)}=\\{(g_{1},g_{2})\in G\times G;s(g_{1})=r(g_{2})\\}$, where the
condition $s(g_{1})=r(g_{2})$ correspond to $j=l$, as we shall see. $j=l$,
whence $\nu_{ij}+\nu_{lk}=\nu_{ik}$. It is just this last principle that make
$+$ a partial law161616That is, do not defined for all possible pairs
$(\nu_{ij},\nu_{lk})$. in $\Delta{\cal F}_{I}$, so that $(\Delta{\cal
F}_{I};+)$ not only is not a subgroup, but neither a groupoid in the sense of
Universal Algebra (see [Co], or171717This Author introduces a structure that
he call magma, corresponding to the notion of groupoid of Universal Algebra.
[Bou]).
We will prove, instead, that it is a EBB-groupoid. Indeed, if $G=\Delta{\cal
F}_{I},$ we have $G^{(0)}={\cal F}_{I},$ $r(\nu_{ij})=\nu_{i},$ and
$s(\nu_{ij})=\nu_{j}$, with $\nu_{ij}\star\nu_{lk}$ defined on
$G^{(2)}=\\{(\nu_{ij},\nu_{lk})\in\Delta{\cal F}_{I}\times\Delta{\cal F}_{I};\
s(\nu_{ij})=r(\nu_{lk})\\}$. Therefore, since
$s(\nu_{ij})=\nu_{j}=\nu_{l}=r(\nu_{lk})$, we have $\nu_{j}=\nu_{l}$, from
which (by the bijectivity of $\rho:i\rightarrow\nu_{i}$ of § 4) follows that
$j=l$, hence $\nu_{ij}\star\nu_{lk}=\nu_{ij}\star\nu_{jk}$. Thus, if
$\star=\tilde{+}$, we have181818$\tilde{+}$ denotes the usual addition of
$(\mathbb{R};+)$, only partially defined on $G^{(2)}\subset\mathbb{R}^{2}$,
and that, substantially, represents the Ritz-Rydberg composition law.
$\nu_{ij}\star\nu_{jk}=\nu_{ij}\tilde{+}\nu_{jk}=\nu_{ik}$ by the Ritz-Rydberg
composition law, and hence it is immediate to verify that ${\cal
G}_{HBJ}({\cal F}_{I})=(\Delta{\cal F}_{I},{\cal F}_{I},r,s,\tilde{+})$ is a
EBB-groupoid, said to be191919The motivation for this terminology will be
given at § 6. the Heisenberg-Born-Jordan EBB-groupoid (or HBJ EBB-groupoid).
There are many different representation (or equivalent models) of ${\cal
G}_{HBJ}({\cal F}_{I})$. We will consider a first representation in
$\mathbb{N}^{2}$. As regard what has been said above, the map
$\xi:\nu_{ij}\rightarrow(i,j)\in I^{2}(\subseteq\mathbb{N}^{2})\ \
\forall\nu_{ij}\in\Delta{\cal F}_{I}$ is bijective, so that $\Delta{\cal
F}_{I}\cong I^{2}$, that is $I^{2}$ is a Cartesian representation of
$\Delta{\cal F}_{I}$. Considering the Cartesian lattice $\mathbb{N}^{2}$, if
$\Delta(\mathbb{N}^{2})=\\{(i,i);\ i\in\mathbb{N}\\}$, then the points
$(i,j)\in I^{2}$ with $i<j\ [i>j]$ represents the lines of the absorption
[emission] spectrum, whereas the points with $i=j$ represents the energy
levels of the atom because, being $\rho^{-1}:\nu_{i}\rightarrow i\in I\ \
\forall\nu_{i}\in{\cal F}_{I}$ bijective and $\alpha:i\rightarrow(i,i)$
bijection of $\mathbb{N}$ in $\Delta(\mathbb{N}^{2})$, we have that
$(\alpha\circ\rho^{-1})({\cal F}_{I})=\\{(i,i);\ i\in
I)\\}=\Delta(I^{2})(\subseteq\Delta(\mathbb{N}^{2}))$ represents ${\cal
F}_{I}$ in $\mathbb{N}^{2}$.
Thus, we have a Cartesian representation (in $\mathbb{N}^{2}$) of the atomic
spectrum $\Delta{\cal F}_{I}$: the upper half plane respect to
$\Delta(\mathbb{N}^{2})$ represents the absorption spectrum, whereas the lower
half plane represents the emission spectrum, and finally $\Delta(I^{2})$
represents the set of energy levels of the atom. The points of the two
emission and absorption half planes, are pairwise correlated by the
(inversion) Kirchhoff-Bunsen symmetry, that it is a simple reflexive
symmetry202020From this symmetry it is possible to construct a second
equivalent representation of the given EBB-groupoid. respect to
$\Delta(\mathbb{N}^{2})$.
The map $\xi^{-1}:(i,j)(\in I^{2})\rightarrow\nu_{ij}$ provides the
kinematical time evolution of a dynamic system through the coordinate
$\mathfrak{q}$ and the momentum $\mathfrak{p}$ given by the (Hermitian)
matrices
$None$ $\mathfrak{q}=[q_{ij}\ e^{2\pi i\nu_{ij}t}],\ \ \mathfrak{p}=[p_{ij}\
e^{2\pi i\nu_{ij}t}]\qquad\mbox{\rm(\it Heisenberg's\ representation\rm)},$
from which follows that any other physical observable
$\mathfrak{g}(\mathfrak{q},\mathfrak{p})$ can always be written in the form
$\mathfrak{g}=[{g_{ij}}\ e^{2\pi i\nu_{ij}t}]$. The celebrated canonical
commutation relations between $\mathfrak{q}$ and $\mathfrak{p}$, does follows
from the Kuhn-Thomas relation (interpreted according to Perturbation Theory;
see [BJ]).
Finally, we have the identifications $\Delta{\cal F}_{I}\cong I^{2}\cong{\cal
F}_{I}^{2}$, that give a representation (in $\mathbb{N}^{2}$) of the algebraic
system ${\cal G}_{HBJ}({\cal F}_{I})$. In fact, if we put $X=I$, it is
immediate to verify that ${\cal G}_{HBJ}({\cal F}_{I})$ is identifiable (since
isomorphic to it) with the natural EBB-groupoid ${\cal G}_{Br}(X)$ (of the
last part of § 1) with $X=I$, $G=I^{2},\ G^{(0)}=\Delta(I^{2}),\
r:(i,j)\rightarrow i,\ s:(i,j)\rightarrow j,\ (i,j)\star(j,k)=(i,k)$; ${\cal
G}_{Br}(I)$ is said to be the Heisenberg-Born-Jordan natural EBB-groupoid.
Two EBB-groupoids ${\cal G}_{1}=(G_{1},G^{(0)}_{1},r_{1},s_{1},\star_{1}),\
{\cal G}_{2}=(G_{2},G^{(0)}_{2},r_{2},s_{2},\star_{2})$ are said212121This
definition of isomorphism must be intended in the sense of category theory.
isomorphic if there exists $\psi:G_{1}\rightarrow G_{2},\
\psi_{0}:G^{(0)}_{1}\rightarrow G^{(0)}_{2}$ both bijective, such that
$r_{2}\circ\psi=\psi_{0}\circ r_{1},\ s_{2}\circ\psi=s_{1}\circ\psi_{0}$ and
$\psi(g\star_{1}g^{\prime})=\psi(g)\star_{2}\psi(g^{\prime})\ \ \forall
g,g^{\prime}\in G_{1}$. In this case, we will write ${\cal G}_{1}\cong{\cal
G}_{2}$.
If ${\cal G}_{1}={\cal G}_{HBJ}({\cal F}_{I}),\ {\cal G}_{2}={\cal
G}_{Br}(I)$, the maps $\psi:\nu_{ij}\rightarrow(i,j)\in I^{2}\
\forall\nu_{ij}\in\Delta{\cal F}_{I}$,
$\psi_{0}:\nu_{i}\rightarrow(i,i)\in\Delta(I^{2})\ \ \forall\nu_{i}\in{\cal
F}_{I}$, are bijective and make isomorphic the two given B-groupoid222222For
instance, we have
$\psi(\nu_{ij}\tilde{+}\nu_{jk})=\psi(\nu_{ik})=(i,k)=(i,j)\star(j,k)=\psi(\nu_{ij})\star\psi(\nu_{jk})$,
the remaining conditions being easily verified., that is ${\cal G}_{HBJ}({\cal
F}_{I})\cong{\cal G}_{Br}(I)$, hence they are identifiable.
### 6\. The Heisenberg-Born-Jordan EBB-groupoid algebra
Let us prove that the EBB-groupoid algebra of the Heisenberg-Born-Jordan EBB-
groupoid, in general, is a non-commutative linear $\mathbb{K}$-algebra,
isomorphic, in the finite dimensional case, to a well defined matrix
$\mathbb{K}$-algebra.
First, from ${\cal G}_{Br}(I)\cong{\cal G}_{HBJ}({\cal F}_{I})$, follows
that232323Indeed, the base of the free $\mathbb{K}$-modules constructed on
such EBB-groupoid, that is to say their supports, are respectively $I^{2}$ and
$\Delta{\cal F}_{I}$, and since $I^{2}\cong\Delta{\cal F}_{I}$, follows that
the respective $\mathbb{K}$-modules are $\mathbb{K}$-isomorphic (see § 3),
whence ${\cal A}_{\mathbb{K}}({\cal G}_{Br}(I))\cong{\cal
A}_{\mathbb{K}}({\cal G}_{HBJ}({\cal F}_{I}))$. ${\cal A}_{\mathbb{K}}({\cal
G}_{Br}(I))\cong{\cal A}_{\mathbb{K}}({\cal G}_{HBJ}({\cal F}_{I}))$. These
two isomorphic structures represent the so called242424See § 7. Heisenberg-
Born-Jordan EBB-groupoid algebra (or HBJ EBB-algebra).
It is ${\cal A}_{\mathbb{K}}({\cal G}_{HBJ}({\cal F}_{I}))=(\langle\Delta{\cal
F}_{I}\rangle,+,\cdot,\ast)$ with
$\langle\Delta{\cal F}_{I}\rangle=\\{\sum_{g\in\Delta{\cal
F}_{I}}\lambda(g)g;\ \lambda(g)\in\mathbb{K},\ g\in\Delta{\cal F}_{I}\\}=$
$=\\{\sum_{\nu_{ij}\in\Delta{\cal F}_{I}}\lambda(\nu_{ij})\nu_{ij};\
\lambda(\nu_{ij})\in\mathbb{K},\ \nu_{ij}\in\Delta{\cal F}_{I}\\},$
the operations $+,\cdot$ being definite as in (1), (2), whereas the
convolution product $\ast$ is defined as follow. If we consider two arbitrary
elements of $\langle\Delta{\cal F}_{I}\rangle$, their convolution product must
be an element of this set, and hence it is expressible as unique linear
combination of the base elements of $\Delta{\cal F}_{I}$. Hence, in general,
we have
$\big{(}\sum_{\nu_{ij}\in\Delta{\cal
F}_{I}}\lambda(\nu_{ij})\nu_{ij}\big{)}\ast\big{(}\sum_{\nu_{lk}\in\Delta{\cal
F}_{I}}\mu(\nu_{lk})\nu_{lk}\big{)}=\sum_{\nu_{ps}\in\Delta{\cal
F}_{I}}\xi(\nu_{ps})\nu_{ps},$
where $\xi(\nu_{ps})\in\mathbb{K}$ are uniquely determined by:
$\xi(\nu_{ps})\doteq\sum_{\nu_{ij}\tilde{+}\nu_{lk}=\nu_{ps}}\lambda(\nu_{ij})\mu(\nu_{lk}),$
and, since $\nu_{ij}\tilde{+}\nu_{lk}$ is defined if and only if
$s(\nu_{ij})=r(\nu_{lk})$, that is $j=l$, we have
$\nu_{ij}\tilde{+}\nu_{lk}=\nu_{ij}\tilde{+}\nu_{jk}=\nu_{ik}$, and thus
$\nu_{ps}=\nu_{ik}$, that is $p=i,s=k$, so that
$None$
$\xi(\nu_{ik})=\sum_{\nu_{ij}\tilde{+}\nu_{jk}=\nu_{ik}}\lambda(\nu_{ij})\mu(\nu_{jk}).$
In the last sum, the indexes $i,k$ are saturated while $j(\in I)$ is free, so
that the sum is extended only to last index, that is we can write252525All
these convolution structures are based on the Rytz-Rydberg composition law,
and, the latter, has been at the foundations of the works of W. Heisenberg, M.
Born and P. Jordan on matrix mechanics (see [An], [BJ], [Di1], [Di2], [Ec],
[Fe], [Gi], [HBJ], [He1,2], [Hu], [Lu], [Sch]). For modern quantum theories
see [St], [Em], whereas for the geometric developments of the non-
commutativity, and its physical applications, see [Co1], [Co2].
$None$ $\xi(\nu_{ik})=\sum_{j\in I}\lambda(\nu_{ij})\mu(\nu_{jk})$
and this is the expression of the generic element of the product262626Rows by
columns. matrix of the two formal matrices272727Of order $card\ I$.
$[\lambda(\nu_{ij})],[\mu(\nu_{lk})]$. Then, in general, the Heisenberg-Born-
Jordan EBB-groupoid algebra ${\cal A}_{\mathbb{K}}({\cal G}_{HBJ}({\cal
F}_{I}))$, is non-commutative, and, if card $I<\infty$, then it is isomorphic
to a matrix $\mathbb{K}$-algebra (see [Pi], Chap. 5, Theor. 3-2, or [Bou],
[Ma], [Mac]).
Nevertheless, since, in general, $I$ is not finite, it follows that
$\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ is not a
finitely generated $\mathbb{K}$-algebra.
### 7\. The notion of Hopf algebra282828We follow [Ka], [KS], [CP], [Md], and
[Mn], [As].
Let $V_{\mathbb{K}}$ be a $\mathbb{K}$-linear space and id its identity map. A
$\mathbb{K}$-algebra is an algebraic system ${\cal A}=(V_{\mathbb{K}},m,\eta)$
with $m:V\otimes V\rightarrow V$ (product) and $\eta:\mathbb{K}\rightarrow V$
(unit) $\mathbb{K}$-linear maps such that: 1) (associativity)
$m\circ(m\otimes\rm id)=\it m\circ(\rm id\otimes\it m)$; 2)
$m\circ(\eta\otimes\rm id)=\it m\circ(\rm id\otimes\eta)=id$. We have
$m(a\otimes b)=ab$.
A $\mathbb{K}$-coalgebra is an algebraic system $c{\cal
A}=(V_{\mathbb{K}},\Delta,\varepsilon)$ with $\Delta:V\rightarrow V\otimes V$
(coproduct) and $\varepsilon:V\rightarrow\mathbb{K}$ (counit)
$\mathbb{K}$-linear maps such that: 1’) (coassociativity) $(\Delta\otimes\rm
id)\circ\Delta=(id\otimes\Delta)\circ\Delta$; 2’) $(\varepsilon\otimes\rm
id)\circ\Delta=(id\otimes\varepsilon)\circ\Delta=id$. A $\mathbb{K}$-bialgebra
is an algebraic system $b{\cal A}=(V_{\mathbb{K}},\Delta,\varepsilon,m,\eta)$
such that $(V_{\mathbb{K}},\Delta,\varepsilon)$ is a $\mathbb{K}$-coalgebra,
$(V_{\mathbb{K}},m,\eta)$ a $\mathbb{K}$-algebra and $\Delta,\varepsilon$
$\mathbb{K}$-algebras homomorphism (see [Ka], Chapt. III, Theor. III.2.1). Let
$f\in End(V_{\mathbb{K}})$, and $\\{v_{i};i\in I\\}$ a base of
$V_{\mathbb{K}}$ with dim${}_{\mathbb{K}}V_{\mathbb{K}}=card\ I$; then
$\\{v_{i}\otimes v_{j};(i,j)\in I^{2}\\}$ is a base of $V\otimes V$, so that
$v=\sum_{(i,j)\in I_{v}}\lambda_{ij}(v)v_{i}\otimes v_{j}$ for certain
$\lambda_{ij}(v)\in\mathbb{K}$ and $I_{v}\subseteq I^{2}$, for any $v\in
V\otimes V$. Then, we can use the following (Sweedler) sigma notation
according to we will write simply $v=\sum_{(v)}v^{\prime}\otimes
v^{\prime\prime}$ for given $v^{\prime},v^{\prime\prime}\in V_{\mathbb{K}}$;
in particular, since $\Delta(v)\in V\otimes V\ \ \forall v\in V_{\mathbb{K}}$,
we will have $\Delta(v)=\sum_{(v)}v^{\prime}\otimes v^{\prime\prime}$. If
$(V_{\mathbb{K}},\Delta,\varepsilon)$ is a $\mathbb{K}$-coalgebra and
$(W_{\mathbb{K}},m,\eta)$ a $\mathbb{K}$-algebra, for each $f,g\in Hom(V,W)$
we have $f\otimes g\in Hom(V\otimes V,W\otimes W)$, so that, by exploiting the
relations
$V\stackrel{{\scriptstyle\Delta}}{{\longrightarrow}}V\otimes
V\stackrel{{\scriptstyle f\otimes g}}{{\longrightarrow}}W\otimes
W\stackrel{{\scriptstyle m}}{{\longrightarrow}}W,$
it is possible to consider the composition map $m\circ(f\otimes
g)\circ\Delta\in Hom(V,W)$, denoted by $f\ \hat{\ast}\ g$ and given by
$(f\ \hat{\ast}\ g)(v)=(m\circ(f\otimes g)\circ\Delta)(v)=(m\circ(f\otimes
g))(\Delta(x))=$ $=(m\circ(f\otimes g))(\sum_{(v)}v^{\prime}\otimes
v^{\prime\prime})=m(\sum_{(v)}(f\otimes g)(v^{\prime}\otimes
v^{\prime\prime}))=$ $=m(\sum_{(v)}f(v^{\prime})\otimes
g(v^{\prime\prime}))=\sum_{(v)}f(v^{\prime})g(v^{\prime\prime})\ \ \forall
v\in V,$
defining the (internal) binary operation (also called convolution)
$\hat{\ast}:Hom(V,W)\times Hom(V,W)\rightarrow Hom(V,W)$
$(f,g)\rightsquigarrow f\ \hat{\ast}\ g\ \ \ \forall f,g\in Hom(V,W).$
If $b{\cal A}=(V_{\mathbb{K}},\Delta,\varepsilon,m,\eta)$ is a
$\mathbb{K}$-bialgebra, what said above also subsists if we put $V=W$; an
element $a\in End(V)$ is said to be an antipode of $b\cal A$ if $a\
\hat{\ast}\ \rm id=id\ \hat{\ast}\ \it a=\eta\circ\varepsilon$, and the
algebraic system $(b{\cal A},a)$ is called a Hopf $\mathbb{K}$-algebra
292929In a $\mathbb{K}$-bialgebra, may exist, at most, only one antipode (see
[Ka], Def. III.3.2., p. 51). of support $V_{K}$.
The commutativity (or not) [cocommutativity (or not)] of such a structure,
follows from the commutativity (or not) [cocommutativity (or not)] of the
linear $\mathbb{K}$-algebra $(V_{\mathbb{K}},m,\eta)$ [$\mathbb{K}$-coalgebra
$(V_{\mathbb{K}},\Delta,\varepsilon)$].
A $\mathbb{K}$-bialgebra
$b\mathcal{A}=(V_{\mathbb{K}},\Delta,\varepsilon,m,\eta)$ is said to be quasi-
cocommutative if there exists an invertible element $R$ (called an universal
R-matrix) of $V_{\mathbb{K}}\otimes V_{\mathbb{K}}$ such that
$\Delta^{op}(x)=R\Delta(x)R^{-1}\ \ \forall x\in V_{\mathbb{K}}$, where
$\Delta^{op}=\tau_{V_{\mathbb{K}},V_{\mathbb{K}}}\circ\Delta$ is the opposite
coproduct on $V_{\mathbb{K}}$ and $\tau_{V_{\mathbb{K}},V_{\mathbb{K}}}$ is
the flip switching the factors. We will denote such a quasi-cocommutative
$\mathbb{K}$-bialgebra with $(V_{\mathbb{K}},\Delta,\varepsilon,m,\eta,R)$.
Any cocommutative $\mathbb{K}$-bialgebra is also quasi-cocommutative with
universal R-matrix $R=1_{V_{\mathbb{K}}}\otimes 1_{V_{\mathbb{K}}}$. A Hopf
$\mathbb{K}$-algebra whose underling $\mathbb{K}$-bialgebra has a universal
R-matrix, is said to be a quasi-cocommutative Hopf $\mathbb{K}$-algebra.
For any $R\in V_{\mathbb{K}}\otimes V_{\mathbb{K}}$, we set $R_{12}=R\otimes
1_{V_{\mathbb{K}}}\in V_{\mathbb{K}}\otimes V_{\mathbb{K}}\otimes
V_{\mathbb{K}}=\bigotimes^{3}V_{\mathbb{K}}$,
$R_{23}=1_{V_{\mathbb{K}}}\otimes R\in\bigotimes^{3}V_{\mathbb{K}}$,
$R_{13}=(\mbox{\rm
id}\otimes\tau_{V_{\mathbb{K}},V_{\mathbb{K}}})(R_{12})=(\tau_{V_{\mathbb{K}},V_{\mathbb{K}}}\otimes\mbox{\rm
id})(R_{23})\in\bigotimes^{3}V_{\mathbb{K}}$. A quasi-cocommutative
$\mathbb{K}$-bialgebra [Hopf $\mathbb{K}$-algebra]
$b\mathcal{A}=(V_{\mathbb{K}},\Delta,\varepsilon,m,\eta,R)$
[($b\mathcal{A},a)$] is braided (or quasi-triangular) if the universal
R-matrix $R$ satisfies the relations $(\Delta\otimes\mbox{\rm
id}_{V_{\mathbb{K}}})(R)=R_{13}R_{23}$ and $(\mbox{\rm
id}_{V_{\mathbb{K}}}\otimes\Delta)(R)=R_{13}R_{12}$. All cocommutative
bialgebras are braided with universal R-matrix $R=1_{V_{\mathbb{K}}}\otimes
1_{V_{\mathbb{K}}}$.
Often, a non-cocommutative braided Hopf $\mathbb{K}$-algebra is called a
quantum group.
### 8\. The HBJ EBBH-algebra
It is possible to associate a natural structure of (non-commutative) braided
Hopf $\mathbb{K}$-algebra to the HBJ EBB-algebra. So, we will get an explicit
example of non-commutative Hopf $\mathbb{K}$-algebra, that can be taken as
basic structure of a (particular) quantum group.
On ${\cal A}_{\mathbb{K}}({\cal G}_{HBJ}({\cal F}_{I}))=(\langle\Delta{\cal
F}_{I}\rangle,+,\cdot,\ast)$, let us consider the unique (see § 2) linear
extension $\tilde{i}$ of the inversion map $i_{G}:g\rightarrow g^{-1}$ (with
$G=\Delta{\cal F}_{I}$) to $\langle\Delta{\cal F}_{I}\rangle$ and set303030In
the notations of § 7. (if necessary, in the tensor product algebra ${\cal
A}_{\mathbb{K}}\otimes{\cal A}_{\mathbb{K}}$):
$V_{\mathbb{K}}=(\langle\Delta{\cal F}_{I}\rangle,+,\cdot)$, $m=\ast,\
\eta=\eta(1)(=1\ \mbox{the\ unit\ of}\ {\cal A}_{\mathbb{K}}({\cal
G}_{HBJ}({\cal F}_{I}))),\ \Delta(x)=x\otimes x\ \mbox{(group-like\
elements)},\ \varepsilon(x)=1\ \ \forall x\in\langle\Delta{\cal F}_{I}\rangle$
and $a=\tilde{i}\in End(\langle\Delta{\cal F}_{I}\rangle)$. Hence
$\Delta(m(x,y))=\Delta(x\ast y)=(x\ast y)\otimes(x\ast y)=$ $=(x\otimes
y)\ast(x\otimes y)=\Delta(x)\ast\Delta(y)=m(\Delta(x),\Delta(y))\ \ \forall
x,y\in\langle\Delta({\cal F}_{I})\rangle,$
$\varepsilon(m(x,y))=\varepsilon(x\ast y)=\varepsilon(z)=1=1\ast 1=$
$=\varepsilon(x)\ast\varepsilon(y)=m(\varepsilon(x),\varepsilon(y))\ \ \forall
x,y\in\langle\Delta({\cal F}_{I})\rangle$
(where $z=m(x,y)$, and $\varepsilon(x)=1\ \forall x\in\Delta({\cal F}_{I})$),
so that $\Delta,\varepsilon$ are homomorphism of $\mathbb{K}$-algebras.
Therefore, it is immediate to prove that $b{\cal A}_{\mathbb{K}}({\cal
G}_{HBJ}({\cal F}_{I}))=(\langle\Delta{\cal
F}_{I}\rangle,\Delta,\varepsilon,m,\eta)$ is a (non-commutative)
$\mathbb{K}$-bialgebra (see, also, [KS], Chapt. I, § 1.2.6., Example 7). In
fact, we have
$(a\ \hat{\ast}\
\mbox{id})(x)=(m\circ(\tilde{i}\otimes\mbox{id})\circ\Delta)(x)=m(\tilde{i}\otimes\mbox{id})(x\otimes
x)==m(\tilde{i}(x)\otimes x)=$ $=m(x^{-1}\otimes x)=\ x^{-1}\ast
x=1=(\eta\circ\varepsilon)(x)\big{(}=(\mbox{id}\ \hat{\ast}\ a)(x)\big{)}$
for each $x\in\Delta({\cal F}_{I})$. Hence $a\ \hat{\ast}\
\mbox{id}=\mbox{id}\ \hat{\ast}\ a=\eta\circ\varepsilon$ and therefore $a$ is
an antipode, that is $(b{\cal A}_{HBJ}({\cal F}_{I}),a)$ is a non-commutative
Hopf $\mathbb{K}$-algebra333333Furthermore, it is also a ${\ast}$-Hopf algebra
with the matrix transposition, when $\mathbb{K}=\mathbb{C}$ and card
$I<\infty$., naturally braided by $R=1\otimes 1$. We will denote it by ${\cal
H}_{\mathbb{K}}({\cal F}_{I})$, and will be said the343434HBJ EBBH-algebra
stands for Heisenberg-Born-Jordan Ehresmann-Baer-Brandt-Hopf algebra.
Heisenberg-Born-Jordan EBBH-algebra (or HBJ EBBH-algebra).
As already said, this last structure may be regarded as the basic structure of
a quantum group that also got, a posteriori, a more physical motivation in its
name. In conclusion, in the HBJ EBBH-algebra may be recognized the eventually
quantic origins of the basilar structure of quantum group, although it has
been obtained endowing the basic EBB-groupoid algebra with a trivial structure
of braided Hopf $\mathbb{K}$-algebra. There are further, well-known, (over)
structures and properties on such EBB-groupoid algebra when $I$ is finite
(since we shall get a finitely generated algebra), but very few when $I$ is
infinite (that is, when the algebra is not finitely generated).
In this first paper, we have considered the only possible (although trivial)
structure of braided non-commutative Hopf algebra on such EBB-groupoid
algebra, when $I$ is infinite (no finitely generation).
In a further paper, we shall try to find other non-trivial structure on it, in
spite of its (interesting) no finitely (algebraic) generation.
## References
[An] S. Antoci, Quando la Fisica parlava Tedesco, Quaderni del Gruppo
Nazionale di Fisica Matematica (GNFM) del CNR, Firenze, 2004.
[As] P. Aschieri, On the Geometry of Inhomogeneous Quantum Groups, Tesi di
Perfezionamento, Scuola Normale Superiore (SNS), Pisa, 1999.
[AM] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra,
Addison-Wesley Publishing Comp., Reading (Massachusetts), 1969 (tr. it.
Introduzione all’Algebra Commutativa, Feltrinelli, Milano, 1981).
[Ba] A. Baer, Zur Einführung des Scharbegriffs, Journal für die Reine und
Angewandte Mathematik, 160 (1929) 199-207.
[BFFLS] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer,
Deformation Theory and Quantization I, II, Annals of Physics, 111 (1978)
61-110, 111-151.
[BJ] M. Born, P. Jordan, Zeitschrift für Physik, 34 (1925) 858.
[Bou] N. Bourbaki, Algèbre I, Hermann, Paris, 1970.
[Br] W. Brandt, Über eine verallgemeineurung des gruppenbegriffes,
Mathematischen Annalen, 96 (1926) 360-366.
[CW] A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative
Algebras, University of California at Berkeley, Berkeley, 1998.
[CCP] P. Caldirola, R. Cirelli, G.M. Prosperi, Introduzione alla Fisica
Teorica, UTET, Torino, 1982.
[CP] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University
Press, Cambridge, 1995.
[Co] P.M. Cohn, Universal Algebra, Harper and Row, New York, 1965 (tr. it.
Algebra Universale, Feltrinelli, Milano, 1971).
[Co1] A. Connes, Géométrie non commutative, InterEditions, Paris, 1990\.
[Co2] A. Connes, Noncommutative Geometry, in: Symposium on the Current State
and Prospects of Mathematics, Barcelona, June, 1991, Lecture Notes in
Mathematics, Springer-Verlag, Berlin, 1992.
[Co3] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.
[Di1] P.A.M. Dirac, Proceedings of the Royal Society of London, A109 (1925)
642.
[Di2] P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press,
London, 1931.
[Dr1] V.G. Drinfeld, Hopf algebras and quantum Yang-Baxter equation, Soviet
Math. Dokl., 32 (1985) 254-258.
[Dr2] V.G. Drinfeld, Quantum groups, in: Proceedings of the International
Congress of Mathematicians, Berkeley, 1986, pp. 798-820, American Mathematical
Society, Providence, Rod Island, 1987.
[Ec] C. Eckart, The Physical Review, 28 (1926) 711.
[Eh] C. Ehresmann, Gattungen von lokalen Strukturen, Jahresbericht der
Deutschen Math. Vereinigung, 60 (1957) 49-77.
[EM] S. Eilenberg, S. MacLane, General theory of natural equivalence, Trans.
Amer. Math. Soc., 58 (1945) 231-294.
[ES] S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, Princeton
University Press, Princeton, 1952.
[Em] G.G.Emch, Mathematical and Conceptual Foundations of 20th-Century
Physics, North-Holland, Amsterdam, 1984.
[Fa] L.D. Faddeev, Integrable models in (1+1)-dimensional QFT, Les Houches
1982, Elsevier Science Publishers, 1984.
[FST] L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan, The quantum inverse
problem, Theor. Math. Phys., 40 (1979) 194-220.
[FT] L. Faddeev, L.A. Takhtajan, Hamiltonian Approach to Solitons Theory,
Springer-Verlag, Berlin-New York, 1987.
[Fe] B. Ferretti, Le Radici Classiche della Meccanica Quantica, Boringhieri,
Torino, 1980\.
[FH] J. Franck, G. Hertz, Verh. D. Phys. Ges., 16 (1914) 457, 512.
[Gi] E. Giordano, Origini e Sviluppi della Meccanica Ondulatoria e della
Meccanica delle Matrici, Istituto di Fisica dell’Università di Parma, Parma,
1974.
[Go] S. Golab, Über den Begriff der Pseudogrouppe von Transformationen, Math.
Annalen, 116 (1939) 768-780.
[Gr] K. Gruenberg, Una Introduzione all’Algebra Omologica, Pitagora Editrice,
Bologna, 2002.
[HBJ] W. Heisenberg, M. Born, P. Jordan, Zeitschrift für Physik, 35 (1926)
557\.
[He1] W. Heisenberg, Zeitschrift für Physik, 33 (1925) 879;
[He2] W. Heisenberg, Die Physikalischen Prinzipien der Quantentheorie, S.
Hirzel Verlag, Lipsia, 1930 (tr. it. I Principi Fisici della Teoria dei
Quanti, Boringhieri, Torino, 1976).
[Her] G. Herzberg, Atomspektren und Atomstruktur, Th. Steinkopff, Dresda, 1936
(tr. it. Spettri Atomici e Struttura Atomica, Boringhieri, Torino, 1961).
[Hi] P.J. Higgins, Categories and Groupoids, D. Van Nostrand Publ. Comp.,
Amsterdam, 1971.
[Hu] F. Hund, Geschichte der Quantentheorie, Bibliographishes Institut AG,
Zurigo, 1975 (tr. it. Storia della Teoria dei Quanti, Boringhieri, Torino,
1980).
[Ji1] M. Jimbo, A q-difference analogue of $U(\mathfrak{g})$ and the Yang-
Baxter equation, Lett. Math. Phys., 10 (1985) 63-69.
[Ji2] M. Jimbo, A q-analogue of $U(\mathfrak{gl}(N+1))$, Hecke algebra and the
Yang-Baxter equation, Lett. Math. Phys., 11 (1986) 247-252.
[Ka] C. Kassel, Quantum Groups, Springer-Verlag, Berlin, 1995.
[KS] A. Klimyk, K. Schmudgen, Quantum Groups and Their Representations,
Springer-Verlag, Berlin, 1997.
[KR] P.P. Kulish, N.Y. Reshetikhin, The quantum linear problem for the sine-
Gordon equation and highest representations, J. Soviet Math., 23 (1983)
2435-2441.
[KuS] P.P. Kulish, E.K. Sklyanin, Solutions of the Yang-Baxter equation, J.
Soviet Math., 19 (1982) 1596-1620.
[La] N.P. Landsman, Mathematical Topics Between Classical and Quantum
Mechanics, Spinger, New York 1998.
[Lo] P.O. Löwdin, Group Algebra, Convolution Algebra, and applications to
Quantum Mechanics, Reviews of Modern Physics, 39 (2) (1967) 259-287.
[Lu] G. Ludwig, Wave Mechanics, Pergamon Press, London, 1968.
[Mac] C.C. Macduffee, The Theory of Matrices, Chelsea Publ. Company, New York,
1946.
[Mak] K. Mackenzie, Lie groupoids and Lie algebroids in Differential Geometry,
Cambridge University Press, Cambridge, 1987.
[MacL] S. MacLane, Categories for the Working Mathematician, Springer-Verlag,
New York, 1971.
[Ma] I.J. Maddox, Elements of Functional Analysis, Cambridge University Press,
Cambridge, 1970.
[Md] S. Majid, Foundations of Quantum Group Theory, Cambridge University
Press, Cambridge, 1995.
[Mn] Y.I. Manin, Quantum Groups and Non-Commutative Geometry, Publications du
CRM de l’Universite de Montreal, Montreal, 1988.
[Mo] W.J. Moore, Physical Chemistry, tr. it. Chimica Fisica, Piccin Editore,
Padova, 1980.
[Ni] A. Nijenhuis, Theory of the geometric object, Amsterdam, 1952.
[No] D.G. Northcott, Ideal Theory, Cambridge University Press, Cambridge,
1953.
[Pi] B. Pini, Primo Corso di Algebra, Editrice CLUEB, Bologna, 1967.
[Sch] E. Schrödinger, Annalen der Physik, 79 (1926) 361, 489, 734; 80 (1926)
437; 81 (1926) 109.
[Sh] I. Schur, Sitzungsberichte der Preuss. Akademie der Wissenschaften,
Berlin (1905) 406-432.
[St] F. Strocchi, An Introduction to the Mathematical Structure of Quantum
Mechanics, lecture notes given at the Scuola Normale Superiore, Pisa, 1996,
or: F. Strocchi, An Introduction to the Mathematical Structure of Quantum
Mechanics, World Scientific Publishing Company, Singapore, 2005.
[Vr] A. Vretblad, Fourier Analysis and its Applications, GTM 223, Springer,
Berlin, 2003.
[Wa] W. Waliszewki, Categories, groupoids, pseudogroups and analytic
structures, Rozprawy Matematyczne, Instytut Matematyczny Polskiej Akademii
Nauk, XLV, Warszawa, 1965.
[We] H. Weyl, The Classical Groups, Princeton University Press, Princeton,
1946\.
[Wo] S.L. Woronowicz, Compact matrix pseudo-groups, Comm. Math. Phys., 111
(1987) 613-665.
|
arxiv-papers
| 2010-06-01T14:34:19 |
2024-09-04T02:49:11.793146
|
{
"license": "Public Domain",
"authors": "Giuseppe Iurato",
"submitter": "Giuseppe Iurato",
"url": "https://arxiv.org/abs/1007.3728"
}
|
1007.3812
|
11institutetext: Graduate School of Science and Technology, Hirosaki
University, Hirosaki, 036-8561, Japan 22institutetext: Department of Science,
School of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502,
Japan 33institutetext: Department of Physics, Moscow State University, Moscow,
119992, Russia 44institutetext: Department of Physics, Saitama University,
Saitama, 338-8570, Japan 55institutetext: Kyowa Interface Science Co.,Ltd.,
Saitama, 351-0033, Japan 66institutetext: Inovative Research Organization,
Saitama University, Saitama, 338-8570, Japan 77institutetext: Research
Institute for Science and Engineering, Waseda University, Tokyo, 169-0092,
Japan
77email: konish@si.hirosaki-u.ac.jp
# On the Sensitivity of L/E Analysis of Super-Kamiokande Atmospheric Neutrino
Data to Neutrino Oscillation Part 1
— The Effect of Quasi-Elastic Scattering over the Direction of the Emitted
Lepton in the Neutrino Events inside the Super-Kamiokande Detector —
E. Konishi 11 Y. Minorikawa 22 V.I. Galkin 33 M. Ishiwata 44 I. Nakamura 44 N.
Takahashi 11 M. Kato 55 A. Misaki 6677
###### Abstract
It is said that the finding of the maximum oscillation in neutrino oscillation
by Super-Kamiokande is one of the major achievements of the SK. In present
paper, we examine the assumption made by Super-Kamiokande Collaboration that
the direction of the incident neutrino is approximately the same as that of
the produced lepton, which is the cornerstone in their $L/E$ analysis and we
find this approximation does not hold even approximately. In the Part 2 of the
subsequent paper, we apply the results from Figures 12, 13 and 14 to $L/E$
analysis and conclude that one cannot obtain the maximum oscillation in $L/E$
analysis which shows strongly the oscillation pattern from the neutrino
oscillation.
###### pacs:
13.15.+g, 14.60.-z
## 1 Introduction
According to the results obtained from the Super-Kamio- kande Experiments on
atmospheric neutrinos, it is said that oscillation phenomena have been found
between two neutrinos, $\nu_{\mu}$and $\nu_{\tau}$. Published reports on the
confirmation to the oscillation between the neutrinos, $\nu_{\mu}$and
$\nu_{\tau}$, and the history foregoing these experiments will be critically
reviewed and details are in the following:
* (1)
During 1980’s Kamiokande and IMB observed smaller atmospheric neutrino flux
ratio $\nu_{\mu}/\nu_{e}$ than the predicted value Hirata .
* (2)
Kamiokande found anomaly in the zenith angle distribution Hatakeyama .
* (3)
Super-Kamiokande found $\nu_{\mu}$-$\nu_{\tau}$ oscillation Kajita2 and
Soudan2 and MACRO confirmed the Super-Kamiokande result Mann .
* (4)
K2K, the first accelerator-based long baseline experiment, confirmed
atmospheric neutrino oscillationK2K .
* (5)
MINOS’s precision measurement gives the consistent results with Super-
Kamiokande onesMINOS .
It is well known that Super-Kamiokande Collaboration examined all possible
types of the neutrino events, such as, say, Sub-GeV e-like, Multi-GeV e-like,
Sub-GeV $\mu$-like, Multi-GeV $\mu$-like, Multi-ring Sub-GeV $\mu$-like,
Multi-ring Multi-GeV $\mu$-like, PC, Upward Stopping Muon Events and Upward
Through Going Muon Events. In other words, all possible interactions by
neutrinos, such as, elastic and quasielastic scattering, single-meson
production and deep scattering are considered in their analyses. Furthermore,
all topologically different types of neutrino events lead to the unified
numerical oscillation parameters, say, $\Delta m^{2}=2.4\times
10^{-3}\rm{eV^{2}}$ and $sin^{2}2\theta=1.0$ Ashie2 .
However, these parameters are obtained from the analysis of the zenith angle
distribution of the incident neutrinos which are based on the survival
probability of a given flavor( see Eq.(7)). The most important result among
the achievements on neutrino oscillation made by Super-
Kamiokande Collaboration is the finding of the maximum oscillation in neutrino
oscillation, because it is the ultimate result in the sense that they observe
the oscillation pattern itself directly in neutrino oscillation.
Taking account of all factors mentioned above, it is natural that the majority
believes the finding of the $\mu-\tau$ neutrino oscillation by Super-
Kamiokande Collaboration.
However, it should be emphasized strongly that Super-Kamiokande Collaboration
put the fundamental assumption in all possible analyses of the atmospheric
neutrino oscillation which is never self-evident and should be carefully
examined. This assumption is that the directions of the incident neutrinos are
approximately the same as those of emitted leptons.
In order to avoid any misunderstanding toward the SK assumption on the
direction, we reproduce this assumption from the original SK papers and their
related papers in italic.
[A] Kajita and Totsuka Kajita1 state 111see page 101 of the paper concerned.:
> ”However, the direction of the neutrino must be estimated from the
> reconstructed direction of the products of the neutrino interaction. In
> water Cheren-kov detectors, the direction of an observed lepton is assumed
> to be the direction of the neutrino. Fig.11 and Fig.12 show the estimated
> correlation angle between neutrinos and leptons as a function of lepton
> momentum. At energies below 400 MeV/c, the lepton direction has little
> correlation with the neutrino direction. The correlation angle becomes
> smaller with increasing lepton momentum. Therefore, the zenith angle
> dependence of the flux as a consequence of neutrino oscillation is largely
> washed out below 400 MeV/c lepton momentum. With increasing momentum, the
> effect can be seen more clearly.”
[B] Ishitsuka Ishitsuka states222see page 138 of the paper concerned.:
> ” 8.4 Reconstruction of $L_{\nu}$
>
> Flight length of neutrino is determined from the neutrino incident zenith
> angle, although the energy and the flavor are also involved. First, the
> direction of neutrino is estimated for each sample by a different way. Then,
> the neutrino flight lenght is calclulated from the zenith angle of the
> reconstructed direction.
>
> 8.4.1 Reconstruction of Neutrino Direction
>
> FC Single-ring Sample
>
> The direction of neutrino for FC single-ring sample is simply assumed to be
> the same as the reconstructed direction of muon. Zenith angle of neutrino is
> reconstructed as follows:
>
> $\hskip 56.9055pt\cos\Theta^{rec}_{\nu}=\cos\Theta_{\mu}\hskip
> 56.9055pt(8.17)$
>
> ,where $\cos\Theta^{rec}_{\nu}$ and $\cos\Theta_{\mu}$ are cosine of the
> reconstructed zenith angle of neutrino and muon, respectively.”
[C] Jung, Kajita et al. Jung state 333see page 453 of the paper concerned.:
> ”At neutrino energies of more than a few hundred MeV, the direction of the
> reconstructed lepton approximately represents the direction of the original
> neutrino. Hence, for detectors near the surface of the Earth, the neutrino
> flight distance is a function of the zenith direction of the lepton. Any
> effects, such as neutrino oscillations, that are a function of the neutrino
> flight distance will be manifest in the lepton zenith angle distributions.”
Hereafter, we call the fundamental assumption by the Super-Kamiokande
Experiment simply as the SK assumption on the direction.
Among various types of the neutrino events analyzed by Super-Kamiokande, the
most important events are single ring muon(electron) events which are
generated in the detector and terminate in it, because these events give more
essential physical quantities for clear interpretation on neutrino
oscillation, namely, the kinds of the neutrino, the transferred energies and
the directions of the produced leptons. These single ring muon events are
generated from the quasi elastic scattering (QEL). If the existence of
neutrino oscillation is verified surely in the analyses of single ring muon
events among Fully Contained Events under the SK assumption on the direction,
then we can expect the corroboration of the oscillation in the analyses of
other types of the interaction with less accuracy. Therefore, the SK
assumption on the direction should be carefully examined in the analysis of
single ring events due to QEL among Fully Contained Events.
Our paper is organized as follows. In section 2, we treat the differential
cross section for QEL in the stochastic manner as exactly as possible and
obtain the zenith angle distributions of the emitted leptons,
$\cos\theta_{\mu}$, for given neutrinos with definite zenith angles, taking
account of the azimuthal angles of the emitted leptons in QEL. As a result of
it, we show that the SK assumption on the direction does not hold any more for
the incident neutrinos with smaller energies. In section 3, we examine the SK
assumption on the direction in the light of $L_{\nu}$ and $L_{\mu}$ and reach
the same conclusion obtained in the section 2, as it must be. In section 4, we
examine all possible choices of combination of $L$ and $E$, namely,
$L_{\nu}/E_{\nu}$, $L_{\nu}/E_{\mu}$, $L_{\mu}/E_{\nu}$, $L_{\mu}/E_{\mu}$, in
$L/E$ analysis and show that the maximum oscillation can be realized in the
combination of $L_{\nu}/E_{\nu}$, as it must be. In section 5, we compare our
results obtained from the numerical experiments with real observation obtained
by Super-Kamiokande Collaboration. In section 6, we conclude that SK cannot
observe the maximum oscillation in their $L_{\mu}/E_{\nu}$ analysis.
Here, we designate the minimum extrema in $L/E$ distribution for neutrino
oscillation as the maximum oscillation by the terminology already utilized in
the Super-Kamiokande Collaboration Ashie1 .
## 2 Single Ring Events among Fully Contained Events which are Produced by
Quasi Elastic Scsattering.
### 2.1 Differential cross section of quasi elastic scattering and influence
over various quantities concerned
As stated in Introduction, the finding of the observation of the maximum
oscillation in the $L/E$ analysis is the ultimate verification of the finding
of the neutrino oscillation by Super-Kamiokande. For the examination of the
Super-Kamiokande’s assertion, we analyze the $L/E$ distribution of the single
ring events among Fully Contained Events.
In order to examine the validity of the SK assumption on the direction, we
consider the single ring events due to the following quasi elastic
scattering(QEL):
$\displaystyle\nu_{e}+n\longrightarrow p+e^{-}$
$\displaystyle\nu_{\mu}+n\longrightarrow p+\mu^{-}$
$\displaystyle\bar{\nu}_{e}+p\longrightarrow n+e^{+}$ (1)
$\displaystyle\bar{\nu}_{\mu}+p\longrightarrow n+\mu^{+},$
The differential cross section for QEL is given as follows r4 .
$\displaystyle\frac{{\rm d}\sigma_{\ell(\bar{\ell})}(E_{\nu(\bar{\nu})})}{{\rm
d}Q^{2}}=\frac{G_{F}^{2}{\rm cos}^{2}\theta_{C}}{8\pi
E_{\nu(\bar{\nu})}^{2}}\Biggl{\\{}A(Q^{2})\pm
B(Q^{2})\biggl{[}\frac{s-u}{M^{2}}\biggr{]}+$ $\displaystyle
C(Q^{2})\biggl{[}\frac{s-u}{M^{2}}\biggr{]}^{2}\Biggr{\\}},$ (2)
where
$\displaystyle A(Q^{2})$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{4}\Biggl{[}f^{2}_{1}\biggl{(}\frac{Q^{2}}{M^{2}}-4\biggr{)}+f_{1}f_{2}\frac{4Q^{2}}{M^{2}}$
$\displaystyle+f_{2}^{2}\biggl{(}\frac{Q^{2}}{M^{2}}-\frac{Q^{4}}{4M^{4}}\biggr{)}+g_{1}^{2}\biggl{(}4+\frac{Q^{2}}{M^{2}}\biggl{)}\Biggr{]},$
$\displaystyle B(Q^{2})$ $\displaystyle=$
$\displaystyle(f_{1}+f_{2})g_{1}Q^{2},$ $\displaystyle C(Q^{2})$
$\displaystyle=$
$\displaystyle\frac{M^{2}}{4}\biggl{(}f^{2}_{1}+f^{2}_{2}\frac{Q^{2}}{4M^{2}}+g_{1}^{2}\biggr{)}.$
The signs $+$ and $-$ refer to $\nu_{\mu(e)}$ and $\bar{\nu}_{\mu(e)}$ for
charged current (c.c.) interactions, respectively. The $Q^{2}$ denotes the
four momentum transfer between the incident neutrino and the charged lepton.
Details of other symbols are given in r4 .
The relation among $Q^{2}$, $E_{\nu(\bar{\nu})}$, the energy of the incident
neutrino, $E_{\ell}$, the energy of the emitted charged lepton (muon or
electron or their anti-particles) and $\theta_{\rm s}$, the scattering angle
of the emitted lepton, is given as
$Q^{2}=2E_{\nu(\bar{\nu})}E_{\ell}(1-{\rm cos}\theta_{\rm s}).$ (3)
Also, the energy of the emitted lepton is given by
$E_{\ell}=E_{\nu(\bar{\nu})}-\frac{Q^{2}}{2M}.$ (4)
Now, let us examine the magnitude of the scattering angle of the emitted
lepton in a quantitative way, as this plays a decisive role in determining the
accuracy of the direction of the incident neutrino, which is directly related
to the reliability of the zenith angle distribution of single ring muon
(electron) events in the Super-Kamiokande Experiment.
By using Eqs. (2) to (4), we obtain the distribution function for the
scattering angle of the emitted leptons and the related quantities by a Monte
Carlo method. The procedure for determining the scattering angle for a given
energy of the incident neutrino is described in Appendix A. Figure 1 shows
this relation for muon, from which we can easily understand that the
scattering angle $\theta_{\rm s}$ of the emitted lepton ( muon here ) cannot
be neglected. For a quantitative examination of the scattering angle, we
construct the distribution function for ${\theta_{\rm s}}$ of the emitted
lepton from Eqs. (2) to (4) by using the Monte Carlo method.
Figure 2 gives the distribution function for $\theta_{\rm s}$ of the muon
produced in the muon neutrino interaction. It can be seen that the muons
produced from lower energy neutrinos are scattered over wider angles and that
a considerable part of them are scattered even in backward directions. Similar
results are obtained for anti-muon neutrinos, electron neutrinos and anti-
electron neutrinos.
Figure 1: Relation between the energy of the muon and its scattering angle for
different incident muon neutrino energies, 0.5, 1, 2, 5, 10 and 100 GeV.
Figure 2: Distribution functions for the scattering angle of the muon for
muon-neutrino with incident energies, 0.5 , 1.0 and 2 GeV. Each curve is
obtained by the Monte Carlo method (one million sampling per each curve).
Also, in a similar manner, we obtain not only the distribution function for
the scattering angle of the charged leptons, but also their average values
$<\theta_{\rm s}>$ and their standard deviations $\sigma_{\rm s}$. Table 1
shows them for muon neutrinos, anti-muon neutrinos, electron neutrinos and
anti-electron neutrinos. From Table 1, it seems to be clear that the
scattering angles could not be neglected, taking account of the fact that the
frequency of the neutrino events with smaller energies is far larger than that
of the neutrino events with larger energies due to high steep of the neutrino
energy spectrum. However, Super-Kamiokande Collaboration assume that the
direction of the neutrino is approximately the same as that of the emitted
lepton even for the neutrino events with smaller energies, as cited in the
three passages mentioned above Kajita1 , Ishitsuka ,Jung . However, it has
never been verified by Super-Kamiokande Collaboration.
Table 1: The average values $<\theta_{\rm s}>$ for scattering angle of the emitted charged leptons and their standard deviations $\sigma_{\rm s}$ for various primary neutrino energies $E_{\nu(\bar{\nu})}$. $E_{\nu(\bar{\nu})}$ (GeV) | angle | $\nu_{\mu(\bar{\mu})}$ | $\bar{\nu}_{\mu(\bar{\mu})}$ | $\nu_{e}$ | $\bar{\nu_{e}}$
---|---|---|---|---|---
| (degree) | | | |
0.2 | $<\theta_{\mathrm{s}}>$ | 89.86 | 67.29 | 89.74 | 67.47
| $\sigma_{\mathrm{s}}$ | 38.63 | 36.39 | 38.65 | 36.45
0.5 | $<\theta_{\mathrm{s}}>$ | 72.17 | 50.71 | 72.12 | 50.78
| $\sigma_{\mathrm{s}}$ | 37.08 | 32.79 | 37.08 | 32.82
1 | $<\theta_{\mathrm{s}}>$ | 48.44 | 36.00 | 48.42 | 36.01
| $\sigma_{\mathrm{s}}$ | 32.07 | 27.05 | 32.06 | 27.05
2 | $<\theta_{\mathrm{s}}>$ | 25.84 | 20.20 | 25.84 | 20.20
| $\sigma_{\mathrm{s}}$ | 21.40 | 17.04 | 21.40 | 17.04
5 | $<\theta_{\mathrm{s}}>$ | 8.84 | 7.87 | 8.84 | 7.87
| $\sigma_{\mathrm{s}}$ | 8.01 | 7.33 | 8.01 | 7.33
10 | $<\theta_{\mathrm{s}}>$ | 4.14 | 3.82 | 4.14 | 3.82
| $\sigma_{\mathrm{s}}$ | 3.71 | 3.22 | 3.71 | 3.22
100 | $<\theta_{\mathrm{s}}>$ | 0.38 | 0.39 | 0.38 | 0.39
| $\sigma_{\mathrm{s}}$ | 0.23 | 0.24 | 0.23 | 0.24
### 2.2 Influence of azimuthal angle in QEL over the zenith angle of single
ring events
In the present subsection, we examine the effect of the azimuthal angles,
$\phi$, of the emitted leptons over their own zenith angles,
$\theta_{\mu(\bar{\mu}))}$, for given zenith angles of the incident neutrinos,
$\theta_{\nu(\bar{\nu}))}$ in QEL, which was not be considered in the detector
simulation carried by Super-Kamiokande Collaboration 444Throughout this paper,
we measure the zenith angles of the emitted leptons from the upward vertical
direction of the incident neutrino. Consequently, notice that the sign of our
direction is opposite to that of the Super-Kamiokande Experiment ( our
$\cos\theta_{\nu(\bar{\nu})}$ = - $\cos\theta_{\nu(\bar{\nu})}$ in SK). The
influence of this effect over the zenith angle cannot be neglected
particularly in horizontal-like neutrino events.
For three typical cases (vertical, horizontal and diagonal), Figure 3 gives a
schematic representation of the relationship between,
$\theta_{\nu(\bar{\nu})}$, the zenith angle of the incident neutrino, and (
$\theta_{\rm s}$, $\phi$), a pair of scattering angle of the emitted lepton
and its azimuthal angle.
From Figure 3-a, it can been seen that the zenith angle
$\theta_{\mu(\bar{\mu})}$ of the emitted lepton is not influenced by its
$\phi$ in the vertical incidence of the neutrinos
$(\theta_{\nu(\bar{\nu})}=0^{\rm o})$, as it must be. From Figure 3-b,
however, it is obvious that the influence of $\phi$ of the emitted leptons on
their own zenith angle is the strongest in the case of horizontal incidence of
the neutrino $(\theta_{\nu(\bar{\nu})}=90^{\rm o})$. Namely, one half of the
emitted leptons are recognized as upward going, while the other half is
classified as downward going ones. The diagonal case (
$\theta_{\nu(\bar{\nu})}=43^{\rm o}$) is intermediate between the vertical and
the horizontal. In the following, we examine the cases for vertical,
horizontal and diagonal incidence of the neutrinos with different energies,
say, $E_{\nu(\bar{\nu})}=0.5$ GeV, $E_{\nu(\bar{\nu})}=1$ GeV and
$E_{\nu(\bar{\nu})}=5$ GeV, as the typical cases.
Figure 3: Schematic view of the zenith angles of the charged muons for
different zenith angles of the incident neutrinos, focusing on their azimuthal
angles.
### 2.3 Dependence of the spreads of the zenith angle for the emitted leptons
on the energies of the emitted leptons for different incident directions of
the neutrinos with different energies
The detailed procedure for the Monte Carlo simulation is described in Appendix
A. We give the scatter plots between the fractional energies of the emitted
muons and their zenith angle for a definite zenith angle of the incident
neutrino with different energies in Figures 6 to 6. In Figure 6, we give the
scatter plots for vertically incident neutrinos with different energies 0.5, 1
and 5 GeV. In this case, the relations between the emitted energies of the
muons and their zenith angles are unique, which comes from the definition of
the zenith angle of the emitted lepton. However, the densities (frequencies of
the event number) along each curves are different in position to position and
depend on the energies of the incident neutrinos. Generally speaking,
densities along the curves become greater toward
$\cos\theta_{\mu(\bar{\mu})}=1$. In this case, $\cos\theta_{\mu(\bar{\mu})}$
is never influenced by the azimuthal angel in the scattering by the definition
555The zenith angles of the particles concerned are measured from the vertical
direction..
On the contrast, it is shown in Figure 6 that the horizontally incident
neutrinos give the widest zenith angle distributions for the emitted muons
with the same fractional energies due to the effect of the azimuthal angles.
The lower the energies of the incident neutrinos are, the wider the spreads of
the scattering angles of emitted muons $\theta_{\mu}$ become, which leads to
wider zenith angle distributions for the emitted muons. As easily understood
from Figure 6, the diagonally incident neutrinos give the intermediate zenith
angle distributions for the emitted muons between those for vertically
incident neutrinos and those for horizontally incident neutrinos.
(a) (b) (c)
Figure 4: The scatter plots between the fractional energies of the produced
muons and their zenith angles for vertically incident muon neutrinos with 0.5
GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
(a) (b) (c)
Figure 5: The scatter plots between the fractional energies of the produced
muons and their zenith angles for horizontally incident muon neutrinos with
0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each
case.
(a) (b) (c)
Figure 6: The scatter plots between the fractional energies of the produced
muons and their zenith angles for diagonally incident muon neutrinos with 0.5
GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
(a) (b) (c)
Figure 7: Zenith angle distribution of the muon for the vertically incident
muon neutrino with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number
is 10000 for each case. SK stands for the corresponding ones under the SK
assumption.
(a) (b) (c)
Figure 8: Zenith angle distribution of the muon for the horizontally incident
muon neutrino with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number
is 10000 for each case. SK stands for the corresponding ones under the SK
assumption.
(a) (b) (c)
Figure 9: Zenith angle distribution of the muon for the diagonally incident
muon neutrino with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number
is 10000 for each case. SK stands for the corresponding ones under the SK
assumption.
In Figures 9 to 9, we express Figures 6 to 6 in a different way. We sum up
muon events with different emitted energies for given zenith angles. As the
result of it, we obtain frequency distribution of the neutrino events as a
function of $cos\theta_{\mu}$ for different incident directions and different
incident energies of neutrinos.
In Figures 9(a) to 9(c), we give the zenith angle distributions of the emitted
muons for the case of vertically incident neutrinos with different energies,
say, $E_{\nu}=$ 0.5, 1 and 5 GeV.
Comparing the case for 0.5 GeV with that for 5 GeV, we understand the big
contrast between them as for the zenith angle distribution. The scattering
angle of the emitted muon for 5 GeV neutrino is relatively small (See, Table
1), so that the emitted muons keep roughly the same direction as their
original neutrinos. In this case, the effect of their azimuthal angle on the
zenith angle is also smaller. However, in the case for 0.5 GeV which is the
dominant energy for single ring muon events in the Super-Kamiokande, there is
even a possibility for the emitted muon to be emitted in the backward
direction due to the larger angle scattering, the effect of which is enhanced
by their azimuthal angle.
The most frequent occurrence in the backward direction of the emitted muon
appears in the horizontally incident neutrino as shown in Figs. 8(a) to 8(c).
In this case, the zenith angle distribution of the emitted muon should be
symmetrical with regard to the horizontal direction. Comparing the case for 5
GeV with those both for $\sim$0.5 GeV and for $\sim$1 GeV, even 1 GeV incident
neutrinos lose almost the original sense of the incidence if we measure it by
the zenith angle of the emitted muon. Figures 9(a) to 9(c) for the diagonally
incident neutrinos tell us that the situation for diagonal case lies between
the case for the vertically incident neutrinos and that for horizontally
incident ones. SK in the figures denotes the SK assumption on the direction of
incident neutrinos. From the Figures 9(a) to 9(c), it is clear that the
scattering angles of emitted muons influence their zenith angles, which is
enhanced by their azimuthal angles, particularly for more inclined directions
of the incident neutrinos.
## 3 Super-Kamiokande Assumption on the Direction in the Light of $L_{\nu}$
and $L_{\mu}$
In the previous section, we show that the SK assumption on the direction does
not hold as for scattering angles of the leptons even if statistically. This
assumption is logically equivalent to the statement that $L_{\nu}$ is
approximately the same as $L_{\mu}$ in $L/E$ analysis , where $L_{\nu}$
denotes the distance on the incident neutrino from the interaction point of
the neutrino events to the intersection of the Earth surface toward its
arriving direction and $L_{\mu}$ denotes the corresponding distance on the
emitted muon. Consequently, if our indication on the invalidity of the SK
assumption on the direction is correct, the same conclusion should be expected
in the relation between $L_{\nu}$ and $L_{\mu}$. In the present section and
subsequent sections, we examine directly the validity of the implicit SK
assumption that $L_{\nu}$ is approximated by $L_{\mu}$, taking into
consideration the neutrino energy spectrum at the Super-Kamiokande site.
Figure 10: Schematic view of relations among $L_{\nu}$, $L_{\mu}$,
$\theta_{s}$ and $\phi_{s}$ .
Figure 11: The procedure for our numerical experiment for obtaining $L_{\mu}$
from a given $L_{\nu}$.
In section 3.1 we give the procedure how to obtain $L_{\mu}$ from a neutrino
event with given $L_{\nu}$ in the stochastic manner. In section 3.2 we give
the correlations between $L_{\nu}$ and $L_{\mu}$, taking account of the effect
of the backscattering as well as the effect of the azimuthal angle in the QEL
in stochastic manner. As the result of it, we show that $L_{\nu}\approx
L_{\mu}$, namely the SK assumption on the direction, does not hold even if
statistically in both the absence and the presence of neutrino oscillation
(Figure 13 and Figure 13). Also, we treat the correlation between $E_{\nu}$
and $E_{\mu}$, in the stochastic manner. We show that the approximation of
$E_{\nu}$ with $E_{\mu}$ by Super-Kamiokande Collaboration does not make so
serious error compared with the approximation of $L_{\nu}$ by $L_{\mu}$,
although their treatment is theoretically unsuitable (Figure 6).
In section 4, we show that $L_{\nu}/E_{\nu}$ distribution can reproduce the
minimum extrema for neutrino oscillation which SK’s neutrino oscillation
parameters demand and ,furthermore, it may give the differnt mimimum extrema
in the neutrino oscillation under the different neutrino oscillation
parameters from SK’s. We show $L_{\nu}/E_{\nu}$ distribution can reproduce the
minimum extrema for neutrino oscillation (the maximum oscillation) which
Super-Kamiokande Collaboration demand, by using their neutrino oscillation
parameters ($\Delta m^{2}=2.4\times 10^{-3}\rm{eV^{2}}$ and
$sin^{2}2\theta=1.0$). Furthermore, it may give the different minimum extrema
in the neutrino oscillation under the different neutrino oscillation
parameters from the Super-Kamiokande Collaboration. This fact denotes that our
numerical computer experiment is done in a correct manner.
### 3.1 Derivation of $L_{\mu}$ from a given $L_{\nu}$ in the single ring
muon event among Fully Contained Events
In our numerical computer experiment, we obtain single ring muon events among
the Fully Contained Events resulting from QEL in the virtual SK detector, the
details of which are described in Appendix A. For the neutrino event with a
definite neutrino energy thus generated, we simulate its interaction point
inside the detector and the emitted energy of the muon concerned which gives
its scattering angle uniquely. The determination of the neutrino energy, the
emitted energy of the muon and its scattering angle are described in Appendix
A. The muon thus generated is pursued in the stochastic manner by using GEANT
3 and finally we judge whether the muon concerned stops inside the detector
(the Fully Contained Event) or passes through the detector (the Partially
Contained Event). For Fully Contained Events thus obtained, we know the
directions of the incident neutrinos, the generation points and termination
points of the events generated inside the detector, the emitted muon energies,
their scattering angles and their azimuthal angles in QEL which give their
zenith angles, $L_{\nu}$ and $L_{\mu}$ 666 The azimuthal angle is but that in
QEL, not that with regard to the Earth here. finally.
Figure 12: Correlation diagram for $L_{\nu}$ and $L_{\mu}$ without oscillation
for 1489.2 live days. The blue points and orange points denote neutrino events
and ani-neutrino events, respectively.
Figure 13: Correlation diagram for $L_{\nu}$ and $L_{\mu}$ with oscillation
for 1489.2 live days. The blue points and orange points denote neutrino events
and ani-neutrino events, respectively.
In Figure 11, we show the relation between $L_{\nu}$ and $L_{\mu}$
schematically. Figure 11 shows the procedure for obtaining $L_{\mu}$ from
$L_{\nu}$ which is equivalent to the corresponding procedure for obtaining
$cos\theta_{\mu}$ from $cos\theta_{\nu}$.
The relation between direction cosine of the incident neutrino,
$(\ell_{\nu(\bar{\nu})},m_{\nu(\bar{\nu})},n_{\nu(\bar{\nu})})$, and that of
the corresponding emitted lepton, $(\ell_{\rm r},m_{\rm r},n_{\rm r})$, for a
given scattering angle, $\theta_{\rm s}$, and its azimuthal angle, $\phi$,
resulting from QEL is given in Appendix A.
$L_{\nu}$ and $L_{\mu}$ are functions of the direction cosine of the incident
neutrino, $cos\theta_{\nu}$, and that of emitted muon, $cos\theta_{\mu}$,
respectively and they are given as,
$L_{\nu}=R_{g}\times(r_{SK}cos\theta_{\nu}+\sqrt{r_{SK}^{2}cos^{2}\theta_{\nu}+1-r_{SK}^{2}})\,\,\,\,(5-1)$
$L_{\mu}=R_{g}\times(r_{SK}cos\theta_{\mu}+\sqrt{r_{SK}^{2}cos^{2}\theta_{\mu}+1-r_{SK}^{2}})\,\,\,\,(5-2)$
where $R_{g}$ is the radius of the Earth and $r_{SK}=1-D_{SK}/R_{g}$, with the
depth, $D_{SK}$, of the Super-Kamiokande Experiment detector from the surface
of the Earth. It should be noticed that the $L_{\nu}$ and $L_{\mu}$ are
regulated by both the energy spectrum of the incident neutrino and the
production spectrum of the muon (QEL in the present case). Consequently, their
mutual relation is influenced by either the absence of the oscillation or the
presence of the oscillation which depend on the combination of the oscillation
parameters.
Figure 14: The correlation diagram between $E_{\nu}$ and $E_{\mu}$ for
oscillation for 1489.2 live days. The continous line denotes the polynomial
expression by Super-Kamiokande Collabolation.
### 3.2 The correlation between $L_{\nu}$ and $L_{\mu}$
In Figure 13, we give the correlation diagram between $L_{\nu}$ and $L_{\mu}$
for single ring muon events among Fully Contained Events for the 1489.2 live
days in the absence of neutrino oscillation which corresponds to the actual
Super-Kamiokande ExperimentAshie2 . In Figure 13, blue points denote neutrino
events while orange points denote anti-
neutrino events. Throughout all correlation diagrams in the present paper,
blue points and orange ones have the same meaning shown in Figure 13. The
aggregates of the (anti-) neutrino events which correspond to a definite
combination of $L_{\nu}$ and $L_{\mu}$ are essentially classified into four
groups in the following:
Group A is defined as the aggregate for neutrino events in which both
$L_{\nu}$ and $L_{\mu}$ are rather small. It denotes that the downward
neutrinos produce the downward muons with smaller scattering angles. In this
case, the energies of the produced muons are near the energies of the incident
neutrinos due to smaller scattering angles.
Group B is defined as the aggregate for neutrino events in which both
$L_{\nu}$ and $L_{\mu}$ are rather large. It denotes that the upward neutrinos
produce upward muons with smaller scattering angles. In this case, the energy
relation between the incident neutrinos and the produced muons is essentially
the same as in Group A, because the flux of the upward neutrino events is
symmetrical to that of the downward neutrino events in the absence of neutrino
oscillation.
Group C is defined as the aggregate for neutrino events in which $L_{\nu}$ are
rather small and $L_{\mu}$ are rather large. It denotes that the downward
neutrinos produce the upward muons by the possible effect reusulting from both
backscattering and azimuthal angle in QEL. In this case, the energies of the
produced muons are smaller than those of the energies of the incident
neutrinos due to larger scattering angles.
Group D is defined as the aggregate for the neutrino events in which $L_{\nu}$
are rather large and $L_{\mu}$ are rather small. It denotes that the upward
neutrinos produce the downward muons. The energy relation between the incident
neutrinos and the produced muons is essentially the same as in Group C in the
absence of neutrino oscillation.
It is clear from Figure 13 that there exist the symmetries between Group A and
Group B, and also between Group C and Group D, which reflect the symmetry
between the upward neutrino flux and the downward neutrino one for null
oscillation.
In Figure 13, we give the correlation between $L_{\nu}$ and $L_{\mu}$ under
their neutrino oscillation parameters, say, $\Delta m^{2}=2.4\times
10^{-3}\rm{eV^{2}}$ and $sin^{2}2\theta=1.0$ Ashie2 . In the presence of
neutrino oscillation, the property of the symmetry which holds in the absence
of neutrino oscillation (see $\langle$Group A and Group B$\rangle$ and/or
$\langle$Group C and Group D$\rangle$ in Figure 13) is lost due to the
different incident neutrino fluxes in the upward direction and downward one.
If we compare Group A with Group B, the event number in Group B (upward $\nu$
$\rightarrow$ upward $\mu$) is smaller than that in group A (downward $\nu$
$\rightarrow$ downward $\mu$), which comes from smaller flux of the upward
neutrinos. The similar relation between Group C (downward $\nu$ $\rightarrow$
upward $\mu$ ) and Group D (upward $\nu$ $\rightarrow$ downward $\mu$) is held
in Figure 13.
Summarizing the characteristics among Groups A to D in the Figures 13 and 13,
we could conclude that $\langle$Group A and Group B$\rangle$ and
$\langle$Group C and Group D$\rangle$ are in symmetrical situations in the
absence of neutrino oscillation, while such a symmetrical situation is lost in
the presence of neutrino oscillation. Also, it is clear from Figures 13 and 13
that $L_{\nu}\approx L_{\mu}$, namely the SK assumption on the direction, does
not hold both in the absence of neutrino oscillation and in the presence of
neutrino oscillation even if statistically.
Here, it should be noticed that the approximation of $L_{\nu}\approx L_{\mu}$
does not hold completely in the region C and region D. The event numbers in
Group C and Group D could not be neglected among the total event number
concerned. In these regions, neutrino events consist of those with
backscattering and/or neutrino events in which the neutrino directions are
horizontally downward (upward), but their produced muons turn upward
(downward) resulting from the effect of azimuthal angles in QEL.
### 3.3 The correlation between $E_{\nu}$ and $E_{\mu}$
Super-Kamiokande Collaboration estimate $E_{\nu}$ from $E_{\mu}$, the visible
energy of the muon, from their Monte Carlo simulation, by the following
equationIshitsuka (see page 135 of the paper concerned) :
$E_{\nu,SK}=E_{\mu}\times(a+b\times x+c\times x^{2}+d\times x^{3}),\,\,\,(6)$
where $x=log_{10}(E_{\mu})$.
The idea that $E_{\nu}$ could be approximated as the polynomial means that
there is unique relation between $E_{\nu}$ and $E_{\mu}$. However, in the
light of stochastic characters inherent in both the incident neutrino energy
spectrum and the production spectrum of the muon, such a treatment is not
suitable theoretically, which may kill a real correlation effect between the
incident neutrino energy and the emitted muon energy. In Figure 14, we give
the correlation between $E_{\nu}$ and $E_{\mu}$ together with that obtained
from the polynomial expression by Super-Kamiokande Collaboration under their
neutrino oscillation parameters and their incident neutrino energy
spectrumHonda . It is clear from the figure that the part of the lower energy
incident neutrino deviates largely from the approximated formula, which
reflects explicitly the stochastic character of QEL. We can give the similar
relation for null oscillation, the shape of which may be different from that
with oscillation due to the difference in the incident neutrino energy
spectrum.
Thus, we could choose four combinations, namely $L_{\nu}/E_{\nu}$,
$L_{\mu}/E_{\mu}$, $L_{\mu}/E_{\nu}$ and $L_{\nu}/E_{\mu}$ for the examination
of maximum oscillations due to neutrino oscillation in $L/E$ analysis.
However, only the combination of $L_{\mu}/E_{\mu}$ out of these four
combinations can be physically measurable.
## 4 Summary
Since one cannot measure $L_{\nu}$ and $E_{\nu}$, so one is forced to utilize
$L_{\mu}$ and $E_{\mu}$ in the $L/E$ analysis in place of them. Then, Super-
Kamiokande Collaboration assume that the direction of the incident neutrino is
the same as that of the emitted lepton the SK assumption on the direction and
$E_{\nu}$ can be estimated from the some polynomial formula of the variable
$E_{\mu}$ in $L/E$ analysis. However, it is clear from Figures 12 and 13 that
the SK assumption on the direction does not hold even approximately and the
transformation of $E_{\mu}$ into $E_{\nu}$ is not uniquely.
In the Part 2 of the subsequent paper, we apply the results from Figures 12,
13 and 14 to $L/E$ analysis and conclude that one cannot obtain the maximum
oscillation in $L/E$ analysis which shows strongly the oscillation pattern
from the neutrino oscillation.
APPENDIX
## Appendix A Monte Carlo Procedure for the Decision of Emitted Energies of
the Leptons and Their Direction Cosines
Here, we give the Monte Carlo Simulation procedure for obtaining the energy
and its direction cosines, $(l_{r},m_{r},n_{r})$, of the emitted lepton in QEL
for a given energy and its direction cosines, $(l,m,n)$, of the incident
neutrino.
The relation among $Q^{2}$, $E_{\nu(\bar{\nu})}$, the energy of the incident
neutrino, $E_{\ell(\bar{\ell})}$, the energy of the emitted lepton (muon or
electron or their anti-particles) and $\theta_{\rm s}$, the scattering angle
of the emitted lepton, is given as
$Q^{2}=2E_{\nu(\bar{\nu})}E_{\ell(\bar{\ell})}(1-{\rm cos}\theta_{\rm s}).$
(A·1)
Also, the energy of the emitted lepton is given by
$E_{\ell(\bar{\ell})}=E_{\nu(\bar{\nu})}-\frac{Q^{2}}{2M}.$ (A·2)
Procedure 1
We decide $Q^{2}$ from the probability function for the differential cross
section with a given $E_{\nu(\bar{\nu})}$ (Eq. (2) in the text) by using the
uniform random number, ${\xi}$, between (0,1) in the following
$\xi=\int_{Q_{\rm
min}^{2}}^{Q^{2}}P_{\ell(\bar{\ell})}(E_{\nu(\bar{\nu})},Q^{2}){\rm d}Q^{2},$
(A·3)
where
$\displaystyle P_{\ell(\bar{\ell})}(E_{\nu(\bar{\nu})},Q^{2})=$
$\displaystyle\frac{{\rm
d}\sigma_{\ell(\bar{\ell})}(E_{\nu(\bar{\nu})},Q^{2})}{{\rm
d}Q^{2}}\Bigg{/}\\!\\!\\!\\!\int_{Q_{\rm min}^{2}}^{Q_{\rm max}^{2}}\frac{{\rm
d}\sigma_{\ell(\bar{\ell})}(E_{\nu(\bar{\nu})},Q^{2})}{{\rm d}Q^{2}}{\rm
d}Q^{2}.$
From Eq. (A$\cdot$1), we obtain $Q^{2}$ in histograms together with the
corresponding theoretical curve in Figure 15. The agreement between the
sampling data and the theoretical curve is excellent, which shows the validity
of the utlized procedure in Eq. (A$\cdot$3) is right.
Figure 15: The reappearance of the probability function for QEL cross
section. Histograms are sampling results, while the curves concerned are
theoretical ones for given incident energies.
Figure 16: The relation between the direction cosine of the incident neutrino
and that of the emitted charged lepton.
Procedure 2
We obtain $E_{\ell(\bar{\ell})}$ from Eq. (A$\cdot$2) for the given
$E_{\nu(\bar{\nu})}$ and $Q^{2}$ thus decided in the Procedure 1.
Procedure 3
We obtain $\cos{\theta_{\rm s}}$, cosine of the the scattering angle of the
emitted lepton, for $E_{\ell(\bar{\ell})}$ thus decided in the Procedure 2
from Eq. (A$\cdot$1) .
Procedure 4
We decide $\phi$, the azimuthal angle of the scattering lepton, which is
obtained from
$\phi=2\pi\xi.$ (A·5)
Here, $\xi$ is a uniform random number between (0, 1).
As explained schematically in the text(see Figure 3 in the text), we must take
account of the effect due to the azimuthal angle $\phi$ in the QEL to obtain
the zenith angle distribution both for Fully Contained Events and Partially
Contained Events correctly.
Procedure 5
The relation between direction cosines of the incident neutrinos,
$(\ell_{\nu(\bar{\nu})},m_{\nu(\bar{\nu})},n_{\nu(\bar{\nu})})$, and those of
the corresponding emitted lepton, $(\ell_{\rm r},m_{\rm r},n_{\rm r})$, for a
certain $\theta_{\rm s}$ and $\phi$ is given as
$\left(\begin{array}[]{c}\ell_{\rm r}\\\ m_{\rm r}\\\ n_{\rm
r}\end{array}\right)=\left(\begin{array}[]{ccc}\displaystyle\frac{\ell
n}{\sqrt{\ell^{2}+m^{2}}}&-\displaystyle\frac{m}{\sqrt{\ell^{2}+m^{2}}}&\ell_{\nu(\bar{\nu})}\\\
\displaystyle\frac{mn}{\sqrt{\ell^{2}+m^{2}}}&\displaystyle\frac{\ell}{\sqrt{\ell^{2}+m^{2}}}&m_{\nu(\bar{\nu})}\\\
-\sqrt{\ell^{2}+m^{2}}&0&n_{\nu(\bar{\nu})}\end{array}\right)\left(\begin{array}[]{c}{\rm
sin}\theta_{\rm s}{\rm cos}\phi\\\ {\rm sin}\theta_{\rm s}{\rm sin}\phi\\\
{\rm cos}\theta_{\rm s}\end{array}\right),$ (A·6)
where $n_{\nu(\bar{\nu})}={\rm cos}\theta_{\nu(\bar{\nu})}$, and $n_{\rm
r}={\rm cos}\theta_{\ell}$. Here, $\theta_{\ell}$ is the zenith angle of the
emitted lepton.
The Monte Carlo procedure for the determination of $\theta_{\ell}$ of the
emitted lepton for the parent (anti-)neutrino with given
$\theta_{\nu(\bar{\nu})}$ and $E_{\nu(\bar{\nu})}$ involves the following
steps:
We obtain $(\ell_{r},m_{r},n_{r})$ by using Eq. (A·6). The $n_{r}$ is the
cosine of the zenith angle of the emitted lepton which should be contrasted to
$n_{\nu}$, that of the incident neutrino.
Repeating the procedures 1 to 5 just mentioned above, we obtain the zenith
angle distribution of the emitted leptons for a given zenth angle of the
incident neutrino with a definite energy.
In the SK analysis, instead of Eq. (A·6), they assume
$n_{r}=n_{\nu(\bar{\nu})}$ uniquely for ${E_{\mu(\bar{\mu})}}\geq$ 400 MeV.
## References
* (1) Hirata, KS et al., Phys.Lett.B205(1988)416.
Hirata, KS et al., Phys.Lett.B280(1992)146.
Casper, D et al., Phys.Rev.Lett.66(1991)2561.
Becker-Szendy, R et al., Phys.Rev. D 46(1992)3720.
* (2) Hatakeyama, S et al., Phys.Rev.Lett.81(1998)2010.
* (3) Kajita, T, Neutrino 98, Takayama, Japan, June 4-9 19982.
Fukuda, Y, Phys.Rev.Lett.81(1998)1562.
* (4) Mann, WA, Nucl.Phys.Proc.Supple Vol.91(2000)134.
Ambrosio, Met al., Phys.Lett.B478(2000)3.
* (5) K2K, Phys.Rev. D74(2006)72003.
* (6) Michael, DG et al., Phys.Rev.Lett.97(2006) 191801\.
* (7) Ashie,Y. et al., Phys. Rev. D 71 (2005) 112005.
* (8) Kajita, T. and Totsuka, Y. Rev. Mod. Phys., 73 (2001)85. See p.101.
* (9) Ishitsuka, M, Ph.D thesis, University of Tokyo (2004). See p. 138.
* (10) Jung, CK, Kajita, T, Mann, T and McGrew, C, Anual. Rev. Mod. Sci. vol.15 (2005) 431.
* (11) Renton, P., Electro-weak Interaction, Cambridge University Press (1990). See p. 405.
* (12) Honda, M., et al., Phys. Rev. D 52 (1996) 4985.
Honda, M., et al., Phys. Rev. D 70 (2004)043008-1.
* (13) Ashie,Y et al., Phys.Rev.Lett.93 (2004)101801-1.
|
arxiv-papers
| 2010-07-22T07:32:34 |
2024-09-04T02:49:11.802079
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Konishi (1), Y. Minorikawa (2), V.I. Galkin (3), M. Ishiwata (4),\n I. Nakamura (4), N. Takahashi (1), M. Kato (5) and A. Misaki (6) ((1)\n Graduate School of Science and Technology, Hirosaki University, Hirosaki,\n Japan, (2) Department of Science, School of Science and Engineering, Kinki\n University, Higashi-Osaka, Japan, (3) Department of Physics, Moscow State\n University, Moscow, Russia, (4) Department of Physics, Saitama University,\n Saitama, Japan, (5) Kyowa Interface Science Co., Ltd., Saitama, Japan, (6)\n Research Institute for Science and Engineering, Waseda University, Tokyo,\n Japan )",
"submitter": "Eiichi Konishi",
"url": "https://arxiv.org/abs/1007.3812"
}
|
1007.3819
|
# FO(FD): Extending Classical Logic with Rule-Based Fixpoint Definitions
PING HOU BROES DE CAT and MARC DENECKER
Department of Computer Science K.U.Leuven Belgium
{ping.hou, broes.decat, marc.denecker}@cs.kuleuven.be
###### Abstract
We introduce fixpoint definitions, a rule-based reformulation of fixpoint
constructs. The logic FO(FD), an extension of classical logic with fixpoint
definitions, is defined. We illustrate the relation between FO(FD) and FO(ID),
which is developed as an integration of two knowledge representation
paradigms. The satisfiability problem for FO(FD) is investigated by first
reducing FO(FD) to difference logic and then using solvers for difference
logic. These reductions are evaluated in the computation of models for FO(FD)
theories representing fairness conditions and we provide potential
applications of FO(FD).
## 1 Introduction
Two mainstream knowledge representation paradigms of the moment are on the one
hand, classical logic-based approaches such as description logics [Baader et
al. (2003)], and on the other hand, rule-based approaches from logic
programming and extensions such as Answer Set Programming and Abductive Logic
Programming [Baral (2003), Kakas et al. (1992)]. The latter disciplines are
rooted in the discipline of Non-Monotonic Reasoning [McCarthy (1986)]. FO(ID)
[Denecker and Ternovska (2008)] integrates both paradigms in a tight,
conceptually clean manner. The key to integrate “rules” into classical logic
(FO) is the observation that natural language, or more precisely, the informal
language of mathematicians, has an informal rule-based construct: the
construct of inductive/recursive definitions (IDs). FO(ID) extends FO not only
with an inductive definition construct but also with an expressive and precise
non-monotonic reasoning principle. It is an extension of FO with inductive
definitions and an integration of FO and LP. It integrates monotonic and non-
monotonic logics. The inductive definition construct of FO(ID) formally
generalizes Datalog [Abiteboul et al. (1995)]. FO(ID) is also strongly related
to fixpoint logics. Monotone definitions in FO(ID) are a different rule-based
syntactic sugar of the fixpoint formulas of Least Fixpoint Logic (LFP) [Park
(1969)]. Last but not least, FO(ID), being a clear, well-founded integration
of rules into classical logic, might play a unifying role in the current
attempts of extending FO-based description logics with rules [Vennekens and
Denecker (2009)]. It thus appears that FO(ID) occupies quite a central
position in the spectrum of computational and knowledge representation logics.
The work in this paper is inspired by work on FO(ID) to integrate LP-style
rules into fixpoint constructs. The resulting constructs are called fixpoint
definitions (FDs). Fixpoint definitions use the rule-based format which will
enable us to more easily link fixpoint constructs with the rule-based
knowledge representation paradigm and the FO(ID) formalism. We define the
logic FO(FD), which is an extension of classical logic with fixpoint
definitions. In FO(FD), almost all kinds of inductions can be expressed in a
natural way. The study of FO(FD) contributes to the understanding of rule-
based systems and thus, to the study of the relation between non-monotonic
inductive definitions and fixpoint definitions, to the study of the
correspondence between well-founded and fixpoint semantics and to the
integration of classical logic-based and rule-based approaches for knowledge
representation.
We investigate the connection between FO(FD) and FO(ID) by presenting
equivalence preserving transformations from FO(ID) to FO(FD). It turns out
that all kinds of inductive definitions in FO(ID) can be expressed in FO(FD).
Meanwhile, due to the allowance of the nesting of least and greatest fixpoint
constructs in FO(FD), the nesting of induction and coinduction can be
represented in FO(FD). Thus, some concepts, e.g., infinite structures and the
nesting of recursion and corecursion [Barwise and Moss (1996)], which can not
be defined in FO(ID) in a well-founded way, can be handled naturally in
FO(FD). We show that in general, FO(FD) is strictly more expressive than
FO(ID).
On the computational level, the satisfiability problem for FO(FD), deciding
the satisfiability of FO(FD) theories, is a major research topic. One research
direction is towards developing solvers for extensions of propositional logic,
e.g., SMT. Difference logic [Nieuwenhuis and Oliveras (2005)] can be seen as
an instance of an SMT framework where propositional logic is extended with
simple linear constraints. Efficient implementation techniques for difference
logic are emerging in the SMT domain [Nieuwenhuis and Oliveras (2005), Cotton
and Maler (2006)], which makes it a good choice as base technology. In this
paper, we develop translations from FO(FD) to difference logic, based on
similar reductions of logic programs presented in [Janhunen et al. (2009),
Niemelä (2008)]. The translations reduce the satisfiability check of FO(FD)
theories to finding satisfying interpretations of difference logic theories.
This provides a novel approach to model expansion for FO(FD). We also present
experimental results.
The paper is organized as follows. In Section 2, we introduce fixpoint
definitions and the logic FO(FD). FO(ID) and the relationship between FO(FD)
and FO(ID) are presented in Section 3. We investigate the satisfiability
problem for FO(FD) by providing the reductions from FO(FD) to difference logic
in Section 4. The reductions are evaluated experimentally in Section 5. In
Section 6, we present some potential applications of FO(FD) and a conclusion
follows in Section 7.
## 2 FO(FD): A logic of fixpoint definitions
In this section, we extend first-order logic (FO) with an alternative rule-
based fixpoint construct: the construct of fixpoint definitions (FDs), to
formalize a new logic FO(FD), which can be viewed as an extension of first-
order logic with mixed induction and coinduction.
### 2.1 Syntax
We assume familiarity with classical logic. A vocabulary $\Sigma$ consists of
a set of predicate and function symbols. Terms and FO formulae are defined as
usual, and are built inductively from variables, constant and function
symbols, logical connectives ($\neg$, $\land$, $\lor$) and quantifiers
($\forall$, $\exists$). Note that predicate symbols occurring in a fixpoint
definition are viewed as predicate constants but not predicate variables.
A rule over a vocabulary $\Sigma$ is an expression of the form
$\forall\overline{x}(P(\overline{x})\leftarrow\varphi[\overline{x}])$, where
$P$ is a predicate symbol of $\Sigma$ and $\varphi[\overline{x}]$ is an
arbitrary first-order formula over $\Sigma$. Atomic formula $P(\overline{x})$
is known as the head of the rule and $\varphi[\overline{x}]$ is known as the
body of the rule. The defined predicate of the rule is $P$. The connective
$\leftarrow$ is called definitional implication and is to be distinguished
from material implication $\supset$, an abbreviation for $\lnot body\lor
head$. We say that a predicate symbol occurs positively (negatively) in a
formula if it occurs in the scope of an even (odd) number of negations. A rule
is positive in a set of predicate symbols if these symbols occur only
positively in $\varphi$.
For a set $\mathcal{R}$ of rules, we denote $\mbox{def}(\mathcal{R})$ as the
set of defined predicates of its rules, and we denote
$\mbox{open}(\mathcal{R})$ as the set of all other symbols occurring in
$\mathcal{R}$.
Without loss of generality, we assume from now on, that rule sets contain for
each of its defined predicates exactly one rule of the form
$\forall\overline{x}(P(\overline{x})\leftarrow\varphi_{P}[\overline{x}])$.
Indeed, any set of rules
$\\{\forall\bar{x}(P(\bar{x})\leftarrow\varphi_{1}[\overline{x}]),\ldots,\forall\bar{x}(P(\bar{x})\leftarrow\varphi_{n}[\overline{x}])\\}$
can be transformed into a single rule
$\forall\bar{x}(P(\bar{x})\leftarrow\varphi_{1}[\overline{x}]\lor\ldots\lor\varphi_{n}[\overline{x}])$.
###### Definition 2.1
We define a least fixpoint definition (LFD), respectively greatest fixpoint
definition (GFD) over vocabulary $\Sigma$ by simultaneous induction, as a
finite expression $\mathcal{D}$ of the form
$\left\lfloor\begin{array}[]{l}\mathcal{R},\Delta_{1},\dots,\Delta_{m},\nabla_{1},\dots,\nabla_{n}\end{array}\right\rfloor\mbox{,
respectively}\left\lceil\begin{array}[]{l}\mathcal{R},\Delta_{1},\dots,\Delta_{m},\nabla_{1},\dots,\nabla_{n}\end{array}\right\rceil$
with $0\leq n,m$ such that:
1. 1.
$\mathcal{R}$ is a set of rules over $\Sigma$.
2. 2.
Each $\Delta_{i}$ is a least fixpoint definition and each $\nabla_{j}$ is a
greatest fixpoint definition.
To express the remaining conditions, we need some auxiliary concepts and
notations. For such an expression $\mathcal{D}$, we say that a rule $r$ is
locally defined in $\mathcal{D}$ if $r\in\mathcal{R}$, and that a predicate
$P$ is locally defined in $\mathcal{D}$ if $P\in\mbox{def}(\mathcal{R})$, and
that $P$ is defined in $\mathcal{D}$ if $P$ is locally defined in
$\mathcal{D}$ or defined in any of its subdefinitions
$\Delta_{1},\dots,\nabla_{n}$. The set of defined predicates of $\mathcal{D}$
is denoted $\mbox{def}(\mathcal{D})$. A symbol is open in $\mathcal{D}$ if it
occurs in $\mathcal{D}$ and is not defined in it. The set of open symbols of
$\mathcal{D}$ is denoted $\mbox{open}(\mathcal{D})$.
1. 3.
Every defined symbol of $\mathcal{D}$ has only positive occurrences in the
bodies of rules in $\mathcal{D}$.
2. 4.
Each symbol $P\in\mbox{def}(\mathcal{D})$ has exactly one local definition in
$\mathcal{D}$. Formally,
$\\{\mbox{def}(\mathcal{R}),\mbox{def}(\Delta_{1}),\dots,\mbox{def}(\nabla_{n})\\}$
is a partition of $\mbox{def}(\mathcal{D}).$
3. 5.
For every subdefinition $\mathcal{D}^{\prime}$ of $\mathcal{D}$,
$\mbox{open}(\mathcal{D}^{\prime})\subseteq\mbox{open}(\mathcal{D})\cup\mbox{def}(\mathcal{R})$.
In particular, a symbol defined in another subdefinition
$\mathcal{D}^{\prime\prime}\neq\mathcal{D}^{\prime}$, does not occur in
$\mathcal{D}^{\prime}$.
A fixpoint definition is either a least fixpoint definition or a greatest
fixpoint definition. We allow arbitrary nesting of least and greatest fixpoint
definitions.
An FO(FD) formula is either an FO formula or a fixpoint definition. An FO(FD)
theory is a set of fixpoint definitions and FO sentences.
###### Example 2.2
Assume a binary predicate $T$ denoting a transition graph on a set of
vertices, representing the states. Assume a property on states $R$, i.e., a
unary predicate on vertices. The set of states $P$ that have an (infinite)
path passing an infinite number of times through a state satisfying $R$, is
defined by:
$\left\lceil\begin{array}[]{l}\forall x\ (P(x)\leftarrow Q(x))\\\
\left\lfloor\begin{array}[]{l}\forall x\ (Q(x)\leftarrow R(x)\land\exists
y(T(x,y)\land P(y)))\\\ \forall x\ (Q(x)\leftarrow\exists y(T(x,y)\land
Q(y)))\end{array}\right\rfloor\end{array}\right\rceil$
### 2.2 Semantics
The semantics of FO(FD) is an integration of standard FO semantics with
fixpoint semantics of definitions. We start by defining the fixpoint
semantics.
Given two disjoint first-order vocabularies $\Sigma$ and $\Sigma^{\prime}$, a
$\Sigma$-interpretation $I$ and a $\Sigma^{\prime}$-interpretation
$I^{\prime}$, the $\Sigma\cup\Sigma^{\prime}$-interpretation mapping each
element $e$ of $\Sigma$ to $e^{I}$ and each $e\in\Sigma^{\prime}$ to
$e^{I^{\prime}}$ is denoted by $I+I^{\prime}$. When
$\Sigma^{\prime}\subseteq\Sigma$, we denote the restriction of a
$\Sigma$-interpretation $I$ to the symbols of $\Sigma^{\prime}$ by
${{I}\rvert_{\Sigma^{\prime}}}$. For a $\Sigma$-interpretation $I$ and a tuple
of domain elements $\overline{d}$, we denote by $I[\overline{x}/\overline{d}]$
the interpretation that has the same domain as $I$, interprets
$\overline{x}=(x_{1},\ldots,x_{n})$ by $\overline{d}=(d_{1},\ldots,d_{n})$,
and coincides with $I$ on all other symbols.
With a set $\mathcal{R}$ of rules over $\Sigma$ and a (partial) two-valued
$\Sigma$-interpretation $I$ interpreting at least all open symbols and no
defined symbols, i.e., $\Sigma\cap\mbox{def}(\mathcal{R})=\emptyset$ and
$\mbox{open}(\mathcal{R})\subseteq\Sigma$, there is a standard way of
associating an operator $\Gamma_{I}^{\mathcal{R}}$ on the set of
$\mbox{def}(\mathcal{R})$-interpretations with the domain of $I$. For two such
interpretations $J,K$, we define $\Gamma_{I}^{\mathcal{R}}(J)=K$ if for every
$\forall\overline{x}(P(\overline{x})\leftarrow\varphi_{P}[\overline{x}])\in\mathcal{R}$,
$P^{K}=\\{\overline{d}|(I+J)[\overline{x}/\overline{d}]\models\varphi_{P}[\overline{x}]\\}$.
If each defined symbol in $\mbox{def}(\mathcal{R})$ has only positive
occurrences in the body of a rule in $\mathcal{R}$, the operator
$\Gamma_{I}^{\mathcal{R}}$ is monotone with respect to the standard truth
order on interpretations and hence, it has least and greatest fixpoints in
this set denoted $\mbox{lfp}(\Gamma_{I}^{\mathcal{R}})$, respectively
$\mbox{gfp}(\Gamma_{I}^{\mathcal{R}})$. Importantly, if $P^{I}\leq
P^{I^{\prime}}$ for every symbol $P\in\mbox{open}(\mathcal{R})$ with only
positive occurrences in rule bodies of $\mathcal{R}$, then
$\mbox{lfp}(\Gamma_{I}^{\mathcal{R}})\leq\mbox{lfp}(\Gamma_{I^{\prime}}^{\mathcal{R}})$
and
$\mbox{gfp}(\Gamma_{I}^{\mathcal{R}})\leq\mbox{gfp}(\Gamma_{I^{\prime}}^{\mathcal{R}})$.
Given an expression $\mathcal{D}$ which might be an LFD or a GFD, and an
$\mbox{open}(\mathcal{D})$-interpretation $I$ interpreting at least all open
symbols of $\mathcal{D}$ and no defined ones. We define an operator
$\Gamma_{I}^{\mathcal{D}}$ on the set of
$\mbox{def}(\mathcal{D})$-interpretations with domain $dom(I)$. This operator
is monotone with respect to the standard truth order on interpretations and
hence, it has least and greatest fixpoints in this set. We define
$\Gamma_{I}^{\mathcal{D}}(J)$ inductively as the interpretation $K+K^{\prime}$
where
* •
$K$ is the
$(\mbox{def}(\mathcal{D})\setminus\mbox{def}(\mathcal{R}))$-interpretation
such that, for $J^{\prime}=I+J|_{\mbox{def}(\mathcal{R})}$:
* –
$K|_{\mbox{def}(\Delta_{i})}=\mbox{lfp}(\Gamma_{J^{\prime}}^{\Delta_{i}})$ for
all $i=1,\ldots,m$.
* –
$K|_{\mbox{def}(\nabla_{j})}=\mbox{gfp}(\Gamma_{J^{\prime}}^{\nabla_{j}})$ for
all $j=1,\ldots,n$.
Observe that $J^{\prime}$ interprets all open symbols in every subdefinition
of $\mathcal{D}$.
* •
$K^{\prime}$ is the $\mbox{def}(\mathcal{R})$-interpretation
$\Gamma_{I+K}^{\mathcal{R}}(J|_{\mbox{def}(\mathcal{R})})$.
###### Definition 2.3 (Model of $\mathcal{D}$)
Let $\mathcal{D}$ be a fixpoint definition and $I$ a two-valued
$\Sigma$-interpretation such that $\Sigma$ contains all symbols in
$\mathcal{D}$. If $\mathcal{D}$ is an LFD, then $I$ satisfies $\mathcal{D}$,
or $I$ is a model of $\mathcal{D}$, iff
$I|_{\mbox{def}(\mathcal{D})}=\mbox{lfp}(\Gamma_{I|_{\mbox{open}(\mathcal{D})}}^{\mathcal{D}})$.
If $\mathcal{D}$ is a GFD, then $I$ satisfies $\mathcal{D}$, or $I$ is a model
of $\mathcal{D}$, iff
$I|_{\mbox{def}(\mathcal{D})}=\mbox{gfp}(\Gamma_{I|_{\mbox{open}(\mathcal{D})}}^{\mathcal{D}})$.
As usual, this is denoted $I\models\mathcal{D}$.
###### Example 2.4 (Continued 2.2)
Semantically, the fixpoint definition in Example 2.2 has the following
meaning: the relationship $P$ is the result of iteratively computing a least
(for $P$) and a greatest fixpoint (for $Q$). In the $n$-th iteration of the
outer fixpoint, $P$ will contain a vertex iff it has a (finite) path that goes
through at least $n$ times through vertices with property $R$. At fixpoint,
$P$ (and $Q$) will contain a vertex iff it has a path that infinitely often
reaches a vertex with property $R$.
###### Definition 2.5 (Model of an FO(FD) theory)
Let $T$ be an FO(FD) theory over $\Sigma$ and $I$ a two-valued
$\Sigma$-interpretation. Then $I$ is a model of $T$, denoted by $I\models T$,
iff $I\models\varphi$ for every $\varphi\in T$.
###### Definition 2.6 (Equivalence)
A theory $T_{1}$ with vocabulary $\Sigma_{1}$ is equivalent to a theory
$T_{2}$ with vocabulary $\Sigma_{2}$ iff each model $M_{1}$ of $T_{1}$
restricted to $\Sigma_{2}$ can be extended to a model $M_{2}$ of $T_{2}$ and
vice versa.
### 2.3 PC(FD)
In this section, we introduce PC(FD), the propositional fragment of FO(FD). We
assume familiarity with propositional logic.
A propositional vocabulary $\Sigma$ is a set of propositional atoms. A literal
is an atom $p$ or its negation $\neg p$. An atom $p$ is called a positive
literal, $\neg p$ a negative one. For a literal $l$, we identify $\neg\neg l$
with $l$.
A propositional fixpoint definition is a fixpoint definition such that all
symbols occurring in it are propositional symbols.
###### Example 2.7
Consider the propositional fixpoint definition
$\mathcal{D}=\left\lfloor\begin{array}[]{l}p\leftarrow q\lor r\\\ q\leftarrow
p\\\ \left\lceil\begin{array}[]{l}r\leftarrow p\\\ s\leftarrow t\lor a\\\
t\leftarrow s\end{array}\right\rceil\end{array}\right\rfloor$
It is obvious that $a$ is the only open atom in this fixpoint definition.
There are only two interpretations satisfying $\mathcal{D}$, namely,
$I_{1}=\\{a\mapsto\mbox{f},p\mapsto\mbox{f},q\mapsto\mbox{f},r\mapsto\mbox{f},s\mapsto\mbox{t},t\mapsto\mbox{t}\\}$
and
$I_{2}=\\{a\mapsto\mbox{t},p\mapsto\mbox{f},q\mapsto\mbox{f},r\mapsto\mbox{f},s\mapsto\mbox{t},t\mapsto\mbox{t}\\}$.
The construction of $I_{1}$ is illustrated as follows:
$I_{1}^{1}=\\{a\mapsto\mbox{f},p\mapsto\mbox{f},q\mapsto\mbox{f},r\mapsto\mbox{t},s\mapsto\mbox{t},t\mapsto\mbox{t}\\}$
and, because the body of the only rule for $r$ is false,
$I_{1}^{2}=\\{a\mapsto\mbox{f},p\mapsto\mbox{f},q\mapsto\mbox{f},r\mapsto\mbox{f},s\mapsto\mbox{t},t\mapsto\mbox{t}\\}$,
which is the limit of the iterations and thus, $I_{1}$ = $I_{1}^{2}$.
A propositional fixpoint definition $\mathcal{D}$ is in definitional normal
form (DefNF) if for any $p\in\Sigma$, the fixpoint definition contains at most
one rule $p\leftarrow\varphi_{p}$, and either $\varphi_{p}=\bigvee B_{p}$ or
$\varphi_{p}=\bigwedge B_{p}$, where $B_{p}$ is a set of literals called the
body literals. Any propositional fixpoint definition can be transformed into
DefNF in polynomial time using Tseitin transformation [Tseitin (1968)]. Hence
without loss of generality, we can from now on assume that propositional
fixpoint definitions are in DefNF.
A PC(FD) theory is a set of propositional formulas and propositional fixpoint
definitions. An interpretation $I$ satisfies a PC(FD) theory if it satisfies
every formula and every definition of the theory.
## 3 A comparison of FO(FD) and FO(ID)
FO(ID) is an extension of first-order logic with a new construct, namely
generalized inductive definitions, for representing definitions that occur
often in mathematics, but in general cannot be expressed in first-order logic.
It was originally introduced in [Denecker (2000)], and further developed in
[Denecker and Ternovska (2008)]. In this section, we compare FO(FD) to FO(ID)
by providing transformations from generalized inductive definitions to
alternating fixpoint definitions and showing that in general, the FO(FD)
formalism is strictly more expressive than the FO(ID).
###### Definition 3.1
Let $\Sigma$ be a vocabulary. A (generalized) inductive definition (GID) $D$
over $\Sigma$ is a finite set of rules over $\Sigma$. Its sets of defined
symbols $\mbox{def}(D)$, respectively open symbols $\mbox{open}(D)$ are
defined as usual.
We do not insist on defined predicates to occur positively in rule bodies in a
generalized inductive definition, but allow non-monotone inductive
definitions.
An FO(ID) formula is a Boolean combination of FO formulas and generalized
inductive definitions. An FO(ID) theory is a set of generalized inductive
definitions and FO sentences. A model of a generalized inductive definition is
a two-valued well-founded model [Denecker and Ternovska (2008)]. The semantics
of FO(ID) is an integration of standard two-valued FO semantics with the well-
founded semantics of generalized inductive definitions.
###### Example 3.2
Consider the following non-monotone inductive definition of even and odd
numbers over the structure of the natural numbers with zero and the successor
function:
$\left\\{\begin{array}[]{l}\forall x(Even(x)\leftarrow x=0\lor\exists
y(x=s(y)\land\neg Even(y)))\\\ \forall x(Odd(x)\leftarrow\exists y(x=s(y)\land
Even(y)))\end{array}\right\\}$
We begin our comparison of FO(FD) and FO(ID) by presenting equivalence
preserving transformations from generalized inductive definitions to
alternating fixpoint definitions. New symbols may be introduced to the
original vocabulary $\Sigma$.
###### Definition 3.3
Let $D$ be a generalized inductive definition. For each defined predicate $P$
of $D$, we introduce a new predicate symbol $P^{\neg}$ of the same arity of
$P$. For each formula $\varphi$, let $\overline{\varphi}$ denote the formula
obtained by substituting each negative occurrence $P(\bar{t})$ of a defined
predicate $P$ in $\varphi$ by $\lnot P^{\neg}(\bar{t})$. We define two sets of
rules:
$\mathcal{R}_{D}=\\{\forall\bar{x}(P(\bar{x})\leftarrow\overline{\varphi_{P}[\bar{x}]})\mid
P\in\mbox{def}(D)\\}$ and
$\mathcal{R}^{\neg}_{D}=\\{\forall\bar{x}(P^{\neg}(\bar{x})\leftarrow\overline{\neg\varphi_{P}[\bar{x}]})\mid
P\in\mbox{def}(D)\\}$. Now define $\Delta_{D}$ as
$\left\lfloor\begin{array}[]{l}\mathcal{R}_{D},\left\lceil\begin{array}[]{l}\mathcal{R}^{\neg}_{D}\end{array}\right\rceil\end{array}\right\rfloor$.
Let $D$ be a generalized inductive definition over $\Sigma$. Then $\Delta_{D}$
is a least fixpoint definition over $\Sigma^{\prime}=\Sigma\cup\\{P^{\neg}\mid
P\in\mbox{def}(D)\\}$. Note that $\mbox{open}(D)=\mbox{open}(\Delta_{D})$.
###### Example 3.4 (Continued 3.2)
Translating the previous FO(ID) formula into FO(FD) leads to
$\left\lfloor\begin{array}[]{l}\forall x(Even(x)\leftarrow x=0\lor\exists
y(x=s(y)\land{Even}^{\neg}(y)))\\\ \forall x(Odd(x)\leftarrow\exists
y(x=s(y)\land Even(y)))\\\ \left\lceil\begin{array}[]{l}\forall
x({Even}^{\neg}(x)\leftarrow x\not=0\land\forall y(x=s(y)\supset Even(y)))\\\
\forall x({Odd}^{\neg}(x)\leftarrow\forall
y(x=s(y)\supset{Even}^{\neg}(y)))\end{array}\right\rceil\end{array}\right\rfloor$
###### Theorem 3.5
Let $D$ be a generalized inductive definition over $\Sigma$. Then there exists
a one-to-one mapping between the $\Sigma$-models $I$ of $D$ and the
$\Sigma^{\prime}$-models $I^{\prime}$ of $\Delta_{D}$ such that the domain of
$I$ is the same as that of $I^{\prime}$, $I^{\prime}|_{\Sigma}=I$ and
$(P^{\neg})^{I^{\prime}}$ is the (relative) complement of $P^{I}$ for each
$P\in\mbox{def}(D)$.
In the following we show that in general, FO(FD) and FO(ID) do not have the
same expressive power.
Theorem 4.4 in [Schlipf (1995)], for the well-founded semantics, states that a
relation is definable in the well-founded semantics iff it is inductively
($\Pi_{1}^{1}$) definable over the natural numbers. However, on the other
hand, Theorem 10 in [Bradfield (1996)] presents that the FO(FD) alternation
hierarchy, the hierarchy of alternating LFD and GFD expressions (ordered along
the number of alternations) in any fixpoint definitions, is strict. A
consequence is the following result.
###### Corollary 3.6
FO(ID) is strictly less expressive than FO(FD) on infinite structures.
## 4 Satisfiability of FO(FD)
The second part of this paper presents an approach to _finite model expansion_
for FO(FD), the inference task consisting of, given a theory $T$, generating a
model for the theory. As a declarative problem solving technique, model
generation for FO(FD) allow to represent e.g. temporal properties in an
application, increasing its general applicability to among others program
verification.
Finite model expansion is equivalent to checking the satisfiability of a
Boolean formula, the satisfiability problem, solved by _SAT solvers_. One
approach to check the satisfiability of FO theories, taken by many state-of-
the-art solvers, is by reducing the theory to propositional logic (a
transformation called _grounding_) and using a SAT solver afterwards.
Grounding generally consists of replacing all variables in a formula by all
possible substitutions, but intelligent techniques exist that greatly reduce
the size of such a grounding, see e.g. [Wittocx et al. (2008)].
Satisfiability checking of FO(FD) theories can be done in a similar way. First
the FO(FD) theory is grounded to a PC(FD) theory. Afterwards, the PC(FD)
theory is reduced to _difference logic_ [Nieuwenhuis and Oliveras (2005)],
propositional logic extended with linear constraints, and a difference logic
solver is used to check the satisfiability of the resulting theory. In the
domain of SMT, efficient difference logic solvers have been developed, see
e.g. [Cotton and Maler (2006)].
_Difference logic_ , denoted PC(DL), is the extension of propositional logic
with linear difference constraints of the form $x+c<y$, where $x,~{}y$ and $c$
are integer variables, of which $c$ is known. Syntactically, a linear
constraint can occur in the same positions as an atom. An interpretation of a
difference logic theory assigns truth values to atoms and integer values to
variables.
We first introduce the grounding of FO(FD) to a variable free form. Then, we
address the reductions of PC(FD) to difference logic.
Without loss of generality, we only consider theories in function free FO(FD)
for the rest of the paper (any FO(FD) theory can be transformed into a
function free theory in polynomial time).
### 4.1 Grounding FO(FD)
The reduction of an FO(FD) theory $T$ to a PC(FD) theory is defined by:
###### Definition 4.1
Given an FO(FD) theory $T$ and a finite domain $\mathfrak{D}$. To allow
grounding of quantified formulas, we introduce a new constant $c_{d}$ for each
domain element $d\in\mathfrak{D}$, which maps to $d$ in every interpretation
$I$. The grounding of $T$ according to domain $\mathfrak{D}$, denoted $G(T)$,
consists of all $G(\varphi)$ where $\varphi\in T$ and $\varphi$ is either an
FO sentence or a fixpoint definition, and $G(\varphi)$ is defined as:
$G(\varphi)=\begin{cases}\bigwedge_{d\in\mathfrak{D}}G(\psi[x/c_{d}])&\text{if
}\varphi:=\forall x\ \psi[x]\\\
\bigvee_{d\in\mathfrak{D}}G(\psi[x/c_{d}])&\text{if }\varphi:=\exists x\
\psi[x]\\\ G(\psi_{1})\wedge G(\psi_{2})&\text{if
}\varphi:=\psi_{1}\wedge\psi_{2}\\\ G(\psi_{1})\vee G(\psi_{2})&\text{if
}\varphi:=\psi_{1}\vee\psi_{2}\\\
\left\lfloor\begin{array}[]{l}G(\psi)\end{array}\right\rfloor&\text{if
}\varphi:=\left\lfloor\begin{array}[]{l}\psi\end{array}\right\rfloor\\\ \neg
G(\psi)&\text{if }\varphi:=\neg\psi\\\
\left\lceil\begin{array}[]{l}G(\psi)\end{array}\right\rceil&\text{if
}\varphi:=\left\lceil\begin{array}[]{l}\psi\end{array}\right\rceil\\\
p\leftarrow G(\psi)&\text{if }\varphi:=p\leftarrow\psi\text{ and $p$ is an
atom}\\\ \psi&\text{if }\psi\text{ is an atom}\\\ \end{cases}$
###### Proposition 4.2
An interpretation $I$ is a model of an FO(FD) theory $T$ iff it is a model of
$G(T)$.
### 4.2 Reduction to difference logic
The aim is to reduce a PC(FD) theory $G(T)$ to an _equivalent_ theory $DL(T)$
in difference logic. The reduction of FO sentences to a PC(DL) theory
coincides with their grounding, so for each FO sentence $\varphi\in T$,
$DL(T)$ contains a sentence $G(\varphi)$. The reduction of fixpoint
definitions consists of the _completion_ and _level mapping_ constraints.
#### 4.2.1 Completion
The _completion_ , introduced by [Clark (1978)] for logical rules, expresses
in FO the consistency between the truth value of the head and the body of a
rule.
The completion of a propositional rule $r=p\leftarrow\varphi_{p}$, denoted
$Comp(r)$, is given by the formula $p\equiv\varphi_{p}$. The completion of a
propositional fixpoint definition $\mathcal{D}$, denoted by
$Comp(\mathcal{D})$, is $\bigcup_{r\in\mathcal{D}}Comp(r)$.
An important property is that $I\models\mathcal{D}$ implies $I\models
Comp(\mathcal{D})$. The converse is not true, $\mathcal{D}$ generally has
fewer models than $Comp(\mathcal{D})$.
###### Example 4.3
Consider the propositional fixpoint definition
$\mathcal{D}=\left\lfloor\begin{array}[]{l}p\leftarrow p\lor a\\\
\left\lceil\begin{array}[]{l}q\leftarrow q\land
p\end{array}\right\rceil\end{array}\right\rfloor$
Then $Comp(\mathcal{D})=(p\equiv p\lor a)\land(q\equiv q\land p)$.
$\mathcal{D}$ has two models:
$\\{a\mapsto\mbox{f},p\mapsto\mbox{f},q\mapsto\mbox{f}\\}$ and
$\\{a\mapsto\mbox{t},p\mapsto\mbox{t},q\mapsto\mbox{t}\\}$;
$Comp(\mathcal{D})$ has the same two models, and the additional three models:
$\\{a\mapsto\mbox{f},p\mapsto\mbox{t},q\mapsto\mbox{t}\\}$,
$\\{a\mapsto\mbox{f},p\mapsto\mbox{t},q\mapsto\mbox{f}\\}$ and
$\\{a\mapsto\mbox{t},p\mapsto\mbox{t},q\mapsto\mbox{f}\\}$.
#### 4.2.2 Level mappings
To obtain equivalence of $T$ and $DL(T)$, it is necessary to ensure that only
interpretations consistent with the operator $\Gamma_{I}^{\mathcal{D}}$ are
models of $DL(T)$. We take a _level mapping_ approach to characterize the
models of the fixpoint operator. This is an extension of the technique
presented in [Janhunen et al. (2009), Niemelä (2008)], where stable model
generation of logic programs is obtained by reduction to difference logic.
###### Definition 4.4 (level mapping)
Given a fixpoint definition $\mathcal{D}$, define a function
$l_{\mathcal{D}}:\mbox{def}(\mathcal{D})\rightarrow\mathbb{N}$, with
$\mbox{def}(\mathcal{D})$ the set of all defined atoms in $\mathcal{D}$.
Function $l$ is then the _level mapping_ function and $l_{\mathcal{D}}(p)$ is
the _level_ of defined atom $p$ for fixpoint definition $\mathcal{D}$.
A level mapping function $l_{\mathcal{D}}$ is introduced for each (nested)
fixpoint definition $\mathcal{D}$ in $G(T)$. In ground form, for each fixpoint
definition $\mathcal{D}$ and for each defined atom $p$ in $\mathcal{D}$, we
introduce an integer variable, denoted $l_{\mathcal{D}}^{p}$.
The level mapping should ensure that the truth of a least fixpoint relation or
the falsity of a greatest fixpoint relation can always be _finitely justified_
in terms of locally defined atoms or open ones.
#### 4.2.3 Level mapping constraints
We introduce PC(DL) formulas which, as part of $DL(T)$, act as constraints on
the relation between the levels of different defined atoms within one fixpoint
definition. Theory $DL(T)$ will be satisfiable iff such a finite justification
exists.
As mentioned earlier, all rules are considered to be in DefNF. For a given
rule $r$ in fixpoint definition $\mathcal{D}$, $h$ denotes the head and
$body(r)$ is the set of all literals occurring in the body of $r$. The sets
$BL_{def}(\mathcal{D},r)$ and $BL_{open}(\mathcal{D},r)$ denote the set of
defined, respectively open body literals(BL)
$\displaystyle BL_{def}(\mathcal{D},r)$
$\displaystyle=\\{d|d\in\mbox{def}(\mathcal{D})\cup\lnot\mbox{def}(\mathcal{D})\
\text{and}\ d\in body(r)\\}$ (1) $\displaystyle BL_{open}(\mathcal{D},r)$
$\displaystyle=\\{o|o\in\mbox{open}(\mathcal{D})\cup\lnot\mbox{open}(\mathcal{D})\
\text{and}\ o\in body(r)\\}$ (2)
We now introduce the constraints.
No justification is necessary for an atom defined in a GFD if it is true, nor
for an atom defined in an LFD which is false. Formally represented by the
constraints:
if $\mathcal{D}$ is a GFD: $\displaystyle\qquad a\supset
l_{\mathcal{D}}^{h}=0$ (3) if $\mathcal{D}$ is an LFD:
$\displaystyle\quad\lnot a\supset l_{\mathcal{D}}^{h}=0$ (4)
When an atom defined in a GFD is not true or an atom defined in an LFD is not
false, a _justification_ is necessary. A justification is a set of body
literals of a rule sufficient to derive the head in a given interpretation.
Although looping is allowed over literals defined in lower fixpoints, it has
to be possible to construct a justification which does not loop over literals
in the same level.
Deriving that the head of a rule with a disjunctive body in an LFD is true
requires only one body atom to be true. If it were a rule with a conjunctive
body, all body literals would be necessary as justification. This also holds
for the relation between their levels: in the disjunctive rule, the minimal
level of all true body literals can act as the level of the justification. In
the conjunctive case, the level is the maximum level of all body literals.
These ideas can be generalized and formalized as constraints. For clarity, the
constraints are not in PC(DL), but we introduce $min\\{\\}$ and $max\\{\\}$
notation to represent respectively the minimum and maximum of a set of levels.
Assume an interpretation $I$ to further simplify the aggregate notation. All
aggregates can be translated out easily, independent of $I$ (see further).
Also assume a fixpoint definition $\mathcal{D}$ with a locally defined atom
$h$ in a rule $r$.
1. 1.
If $\mathcal{D}$ is an LFD and $r$ has a conjunctive body, the translation of
$r$ is:
$h\supset l_{\mathcal{D}}^{h}>max\\{l_{\mathcal{D}}^{d}|d\in
BL_{def}(\mathcal{D},r)\ \text{and}\ I(d)=\mbox{t}\\}$ (5)
2. 2.
If $\mathcal{D}$ is an LFD and $r$ has a disjunctive body, the translation of
$r$ is:
$\begin{split}h\supset&(l_{\mathcal{D}}^{h}>min\\{l_{\mathcal{D}}^{d}|d\in
BL_{def}(\mathcal{D},r)\ \text{and}\ I(d)=\mbox{t}\\}\\\ &\lor\bigvee_{d\in
BL_{def}(\mathcal{D},r)}d\ \lor\bigvee_{o\in
BL_{open}(\mathcal{D},r)}o)\end{split}$ (6)
3. 3.
If $\mathcal{D}$ is a GFD and $r$ has a disjunctive body, the translation of
$r$ is:
$\lnot h\supset l_{\mathcal{D}}^{h}>max\\{l_{\mathcal{D}}^{d}|d\in
BL_{def}(\mathcal{D},r)\ \text{and}\ I(d)=\mbox{f}\\}$ (7)
4. 4.
If $\mathcal{D}$ is a GFD and $r$ has a conjunctive body, the translation of
$r$ is:
$\begin{split}\lnot
h\supset&(l_{\mathcal{D}}^{h}>min\\{l_{\mathcal{D}}^{d}|d\in
BL_{def}(\mathcal{D},r)\ \text{and}\ I(d)=\mbox{f}\\}\\\ &\lor\bigvee_{d\in
BL_{def}(\mathcal{D},r)}\lnot d\ \lor\bigvee_{o\in
BL_{open}(\mathcal{D},r)}\lnot o)\end{split}$ (8)
Similar constraints apply for the level of the head $h$ of rules defined in a
subdefinition of $\mathcal{D}$, but the inequality
$l_{\mathcal{D}}^{h}>\ldots$ is relaxed to $l_{\mathcal{D}}^{h}\geq\ldots$
###### Proposition 4.5
The truth value of a higher defined atom can only be justified by finite
looping over literals in the same definition or infinite looping over literals
in lower definitions. This is expressed by using similar constraints for
locally defined rules and for rules defined in subdefinitions, but dropping
the strict order requirement on the second, effectively allowing infinite
looping over literals defined in subdefinitions.
###### Example 4.6
In the following fixpoint definition, using only strict ordering would lead to
a contradiction, although a model exists.
$\left\lfloor\begin{array}[]{l}a\leftarrow c\\\
\left\lceil\begin{array}[]{l}c\leftarrow d\\\ d\leftarrow
c\end{array}\right\rceil\end{array}\right\rfloor$
###### Theorem 4.7
If an FO(FD) theory is transformed using the presented reduction to PC(DL) via
PC(FD), the resulting PC(DL) theory will be satisfiable iff the FO(FD) theory
is satisfiable. Any model of the PC(DL) theory can be transformed into a model
of the FO(FD) theory.
#### 4.2.4 Aggregate reduction
To obtain PC(DL) constraints, the aggregates $min$ and $max$ have to be
transformed into difference constraints, which can be done in the following
fashion:
$\begin{array}[]{ll}\text{Replace
}&l_{\mathcal{D}}^{h}>max(\\{l_{\mathcal{D}}^{d}|d\in BL_{def}(\mathcal{D},r)\
\text{and}\ I(d)=\mbox{t}\\})\\\ \text{by }&\bigwedge_{d\in
BL_{def}(\mathcal{D},r)}(l_{\mathcal{D}}^{h}>l_{\mathcal{D}}^{d}\lor\lnot
d)\\\ &\\\ \text{Replace }&l_{\mathcal{D}}^{h}>min(\\{l_{\mathcal{D}}^{d}|d\in
BL_{def}(\mathcal{D},r)\ \text{and}\ I(d)=\mbox{t}\\})\\\ \text{by
}&\bigvee_{d\in
BL_{def}(\mathcal{D},r)}(l_{\mathcal{D}}^{h}>l_{\mathcal{D}}^{d}\land
d)\end{array}$
For a condition $I(d)=\mbox{f}$ instead of $I(d)=\mbox{t}$, the literal $d$ is
replaced with $\lnot d$.
#### 4.2.5 Optimization: partial level mapping
Level mappings constraints are used to enforece dependencies between defined
atoms. Often, a preprocessing step (before PC(DL) reduction) allows to deduce
that certain atoms will never depend on each other. In that case, less mapping
constraints are necessary. A simple example are non-cyclic dependencies, for
which no level mapping constraints are necessary
($Comp(\mathcal{D})\models\mathcal{D}$). These dependencies can be obtained by
calculating the _strongly connected components_ [Tarjan (1972)] on the
_dependency graph_ of the fixpoint definition, a general technique used among
others in stable model generation [Syrjänen and Niemelä (2001)].
The _dependency graph_ consists of all edges $h\leadsto b$, for each rule $r$
in $\mathcal{D}$ with head $h$ and for each body literal $b$ of $r$ that is
defined in $\mathcal{D}$ or in a parent of $\mathcal{D}$. A _strongly
connected component_ of a directed graph is a maximal subset in which a path
exists between any two nodes in the set.
###### Proposition 4.8
Only defined atoms that are in a strongly connected component with
$\|nodes\|\geq 2$ or have recursion over themselves (e.g. $h\leadsto h$) need
a level mapping. Body atoms that are not in the same strongly connected
component as the head can be treated as open instead of defined atoms.
To implement this idea, the set of open body literals
$BL_{open}(\mathcal{D},r)$ is redefined: for a rule $r$, a body literal of $r$
is considered open if it is not defined, defined in an ancestor of the
definition of $r$ or if it is not in the same strongly connected component as
the head of $r$. The set $BL_{def}(\mathcal{D},r)$ contains all remaining body
literals.
#### 4.2.6 Optimization: stronger constraints
The presented constraints are _weak_ : infinitely many models of the PC(DL)
reduction exist that are equivalent (modulo shared vocabulary) to one model of
the FO(FD) theory. Exact one-to-one mapping is not possible because
expressions of the form $x=c$, where $c$ is a known integer constant, cannot
be expressed in difference logic. By expressing all constraints in terms of
one integer variable, which acts as a floating ground, we can greatly reduce
the number of redundant models.
The presented constraints can be adapted to obtain such _stronger_ constraints
by enforcing that the level of the head of a rule is the minimum allowed by
its associated constraint, adapted from in [Janhunen et al. (2009), Niemelä
(2008)]. For example for a rule with a conjunctive body in a least fixpoint,
which is subject to the constraint expressed by equation 5, a second
constraint is added of the form:
$h\supset\bigvee_{d\in
BL_{def}(\mathcal{D},r)}(l_{\mathcal{D}}^{h}=l_{\mathcal{D}}^{d}+1\land d)$
(9)
## 5 Implementation and experiments
In this section, we report our first experiments, on model checking of
fairness conditions, with a prototype implementation of the reductions from
FO(FD) to difference logic. We used the $\mu$-calculus fairness expression
presented in [Liu et al. (1998)]:
$\nu X.\mu Y.[-](\langle a\rangle X)\lor Y$ (10)
It expresses that a state in the transition system is fair if on all possible
paths, an $a$-labeled edge is infinitely often taken. Translated into an
FO(FD) theory:
$\left\lceil\begin{array}[]{l}\forall x\ (P(x)\leftarrow Q(x))\\\
\left\lfloor\begin{array}[]{l}\forall x\ (Q(x)\leftarrow\forall y\
(Edge(x,y)\supset(L(y,a)\land P(y))\lor
Q(y)))\end{array}\right\rfloor\end{array}\right\rceil$
where the relations $P$ and $Q$ contain states from which infinitely often a
state labelled $a$ will be reached. The predicate $L$ is the labelling
relation, expressing that a state has a certain label. The predicate $Edge$ is
the transition relation.
The task consists of doing model expansion, where the transitions and
labellings are known, to decide which nodes are fair. Both weak and strong
constraints were tested. The experiments were done on the graph depicted in
Figure 1. The results of these experiments are as shown in Table 1, grounding
times are included. The machine used is a dual-core 2.4 GHz with 4 Gb RAM,
with Ubuntu 8.04 OS. Yices2 was used as difference logic solver.
Figure 1: A transition graph $\|nodes\|$ | weak(sec) | strong(sec)
---|---|---
503 | 0.011 | 0.004
1503 | 0.21 | 0.09
2503 | 20.51 | 14.19
| |
Table 1: Model checking results
From these preliminary results, we conclude that fairness conditions can be
evaluated efficiently using our reduction to difference logic. Strong
constraints are significantly faster due to their fewer degrees of freedom,
which presumably allow more propagation and pruning of the search space. In
[Keinänen and Niemelä (2004)], similar results were obtained with the same
experiment.
## 6 Applications
Many applications can be found on the use of fixpoint expressions. Most of
them focus on inductive and coinductive definitions (which have nesting depth
1), used e.g. for expressing transitive closure (reachability), bisimulation
and situation calculus. One important application domain for nested fixpoint
definitions is the verification of automata. Temporal logics like CTL* allow
to express time-variant properties of automata, e.g. fairness. The
$\mu$-calculus, a superset logic of those temporal logics bound on fixpoint
expressions, can be transformed into fixpoint definitions. So any application
of model checking or model generation of temporal logics can be expressed in
FO(FD). Another application domain are so-called _parity games_ , which are
infinite games played on a graph with priority-annotated nodes. For more
information we refer to [Friedmann and Lange (2009)]. Parity games can be
expressed in fixpoint logic, the nesting increasing polynomially with the
number of priorities.
## 7 Conclusions and related work
In this paper, we have introduced fixpoint definitions, an alternative rule-
based expression of fixpoint constructs, and the logic FO(FD), which is an
extension of classical logic with fixpoint definitions. We have compared
FO(FD) and FO(ID) by providing equivalence preserving transformations of non-
monotone inductive definitions to alternating fixpoint definitions and showed
that FO(FD) is strictly more expressive than FO(ID) on infinite structures. We
have investigated the satisfiability problem for FO(FD) by developing
reductions from FO(FD) to difference logic. Hence, SMT solvers supporting
difference logic can be used for computing fixpoint models of FO(FD) theories
without any modifications. We have implemented these reductions and evaluated
the resulting solver in the computation of models of FO(FD) theories. In
general, our transformation to difference logic is exponential in the nesting
depth of a fixpoint definition, but for most practical applications they prove
compact and efficient.
$\mu\text{MALL}^{=}$, which is the logic obtained by extending MALL
(multiplicative, additive linear logic) with equality, quantification (via
$\forall$ and $\exists$) and mixed least and greatest fixpoint constructors,
was introduced in [Baelde and Miller (2007)]. It seems that
$\mu\text{MALL}^{=}$ has the same expressive power as FO(FD). However,
$\mu\text{MALL}^{=}$ is developed from a proof theory standpoint whereas
FO(FD) is developed from a model theory point of view.
Gupta et al. in [Gupta et al. (2007)] introduced coinduction, corresponding to
the greatest fixpoint constructor, into logic programming to obtain
coinductive logic programming. Discussed applications are verification, model
checking, non-monotonic reasoning, etc. However, in coinductive logic
programming, naively mixing coinduction and induction leads to contradictions
while arbitrary cyclical nesting of least and greatest fixpoint constructs is
allowed in FO(FD). Another difference is on the computational level. The main
computational task for FO(FD) is model generation in the context of a finite
domain. However, model generation in coinductive logic programming is applied
to constructs of an infinite Herbrand model based on an infinite Herbrand
universe.
Niemelä, Janhunen et al. in [Janhunen et al. (2009), Niemelä (2008)]
introduced stable model generation of general logic programs via reductions to
difference logic. They also used stable model generation to find solutions to
Boolean equation systems [Keinänen and Niemelä (2004)]. This is a related
fixpoint formalism, in which among others $\mu$-calculus can be expressed.
There are several other solvers for solving the satisfiability and validity
problems for fixpoint logics, e.g., [Friedmann and Lange (2009)]. Our
reduction is based on SMT solver technology, whereas referenced works are
based on characterizations of satisfiability through infinite (cyclic)
tableaux. Well-foundedness for unfoldings of least fixpoints is then checked
using deterministic parity automata.
## References
* Abiteboul et al. (1995) Abiteboul, S., Hull, R., and Vianu, V. 1995\. Foundations of Databases. Addison-Wesley.
* Baader et al. (2003) Baader, F., Calvanese, D., McGuinness, D. L., Nardi, D., and Patel-Schneider, P. F., Eds. 2003. The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press.
* Baelde and Miller (2007) Baelde, D. and Miller, D. 2007\. Least and greatest fixed points in linear logic. In LPAR, N. Derschowitz and A. Voronkov, Eds. LNCS, vol. 4790\. Springer, 92–106.
* Baral (2003) Baral, C. 2003\. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press.
* Barwise and Moss (1996) Barwise, J. and Moss, L. S., Eds. 1996. Vicious Circles: On the mathematics of Non-Wellfounded Phenomena. University of Chicago Press.
* Bradfield (1996) Bradfield, J. C. 1996\. The modal mu-calculus alternation hierarchy is strict. In CONCUR, U. Montanari and V. Sassone, Eds. LNCS, vol. 1119\. Springer, 233–246.
* Clark (1978) Clark, K. L. 1978\. Negation as failure. In Logic and Data Bases. Plenum Press, 293–322.
* Cotton and Maler (2006) Cotton, S. and Maler, O. 2006\. Fast and flexible difference constraint propagation for DPLL(T). In SAT, A. Biere and C. P. Gomes, Eds. LNCS, vol. 4121. Springer, 170–183.
* Denecker (2000) Denecker, M. 2000\. Extending classical logic with inductive definitions. In CL, J. W. Lloyd, V. Dahl, U. Furbach, M. Kerber, K.-K. Lau, C. Palamidessi, L. M. Pereira, Y. Sagiv, and P. J. Stuckey, Eds. LNCS, vol. 1861. Springer, 703–717.
* Denecker and Ternovska (2008) Denecker, M. and Ternovska, E. 2008\. A logic of nonmonotone inductive definitions. ACM Transactions on Computational Logic (TOCL) 9, 2, Article 14.
* Friedmann and Lange (2009) Friedmann, O. and Lange, M. 2009\. Solving parity games in practice. In ATVA, Z. Liu and A. P. Ravn, Eds. LNCS, vol. 5799. Springer, 182–196.
* Gupta et al. (2007) Gupta, G., Bansal, A., Min, R., Simon, L., and Mallya, A. 2007\. Coinductive logic programming and its applications. In ICLP, V. Dahl and I. Niemelä, Eds. LNCS, vol. 4670\. Springer, 27–44.
* Janhunen et al. (2009) Janhunen, T., Niemelä, I., and Sevalnev, M. 2009\. Computing stable models via reductions to difference logic. In LPNMR, E. Erdem, F. Lin, and T. Schaub, Eds. LNCS, vol. 5753. Springer, 142–154.
* Kakas et al. (1992) Kakas, A. C., Kowalski, R. A., and Toni, F. 1992\. Abductive logic programming. J. Log. Comput. 2, 6, 719–770.
* Keinänen and Niemelä (2004) Keinänen, M. and Niemelä, I. 2004\. Solving alternating boolean equation systems in answer set programming. In INAP/WLP, D. Seipel, M. Hanus, U. Geske, and O. Bartenstein, Eds. LNCS, vol. 3392. Springer, 134–148.
* Liu et al. (1998) Liu, X., Ramakrishnan, C. R., and Smolka, S. A. 1998\. Fully local and efficient evaluation of alternating fixed points (extended abstract). In TACAS, B. Steffen, Ed. LNCS, vol. 1384. Springer, 5–19.
* McCarthy (1986) McCarthy, J. 1986\. Applications of circumscription to formalizing common-sense knowledge. Artificial Intelligence 28, 1, 89–116.
* Niemelä (2008) Niemelä, I. 2008\. Stable models and difference logic. Ann. Math. Artif. Intell. 53, 1-4, 313–329.
* Nieuwenhuis and Oliveras (2005) Nieuwenhuis, R. and Oliveras, A. 2005\. DPLL(T) with exhaustive theory propagation and its application to difference logic. In CAV, K. Etessami and S. K. Rajamani, Eds. LNCS, vol. 3576\. Springer, 321–334.
* Park (1969) Park, D. 1969\. Fixpoint induction and proofs of program properties. Machine Intelligence 5, 59–78.
* Schlipf (1995) Schlipf, J. S. 1995\. The expressive powers of the logic programming semantics. J. Comput. Syst. Sci. 51, 1, 64–86.
* Syrjänen and Niemelä (2001) Syrjänen, T. and Niemelä, I. 2001\. The smodels system. In LPNMR, T. Eiter, W. Faber, and M. Truszczyński, Eds. LNCS, vol. 2173. Springer, 434–438.
* Tarjan (1972) Tarjan, R. E. 1972\. Depth-first search and linear graph algorithms. SIAM Journal on Computing 1, 2, 146–160.
* Tseitin (1968) Tseitin, G. S. 1968\. On the complexity of derivation in propositional calculus. In Studies in Constructive Mathematics and Mathematical Logic II, A. O. Slisenko, Ed. Consultants Bureau, N.Y., 115–125.
* Vennekens and Denecker (2009) Vennekens, J. and Denecker, M. 2009\. FO(ID) as an extension of DL with rules. In ESWC, L. Aroyo, P. Traverso, F. Ciravegna, P. Cimiano, T. Heath, E. Hyvönen, R. Mizoguchi, E. Oren, M. Sabou, and E. P. B. Simperl, Eds. LNCS, vol. 5554. Springer, 384–398.
* Wittocx et al. (2008) Wittocx, J., Mariën, M., and Denecker, M. 2008\. Grounding with bounds. In AAAI, D. Fox and C. P. Gomes, Eds. AAAI Press, 572–577.
|
arxiv-papers
| 2010-07-22T08:05:56 |
2024-09-04T02:49:11.809782
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hou Ping and Broes De Cat and Marc Denecker",
"submitter": "Broes De Cat",
"url": "https://arxiv.org/abs/1007.3819"
}
|
1007.3820
|
# Reply to Sergiu I. Vacaru’s “Critical remarks on Finsler modifications of
gravity and cosmology by Zhe Chang and Xin Li”
Xin Li1,3 lixin@itp.ac.cn Zhe Chang2,3 changz@ihep.ac.cn 1Institute of
Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China
2Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
3Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
###### Abstract
This is our reply to ”Critical remarks on Finslerian modifications of gravity
and cosmology by Zhe Chang and Xin Li”, Sergiu I. Vacaru, Phys. Lett. B 690
(2010) 224. It is pointed out that the Finslerian modifications of gravity and
cosmology (Zhe Chang and Xin Li, Phys. Lett. B 676 (2009) 173; ibid 668 (2008)
453) is a suggestion on the generalization of Einstein’s gravity and
cosmology, but not a proof for theorems in geometry. False or true of the
theory should be tested by experiments or observations. We show that the
arguments of Sergiu I. Vacaru were based a wrong logic. A personal claim can
not be used to prove any other theory be wrong. To get the claim: “we may
construct more “standard” physical Finsler classical/quantum gravity theories
for metric compatible connections like the Cartan d-connection” , Sergiu I.
Vacaru should complete a consistent presentation at least. We suggest Sergiu
I. Vacaru to make some predictions on gravity and cosmoligy using his
“standard” physical Finsler classical/quantum gravity theories as we did, and
compare them with astronomical observations. By the way, we should say that it
is still really far from a theory of quantum gravity.
Recently, Sergiu I. Vacaru published critical remarksVacaru on our work
Finslerian modifications of gravity and cosmology Finsler DE ; Finsler DM .
First of all, we thank Vacaru for paying attention to our researches. We are
happy to read any kind of criticisms and comments on the papers. The
Finslerian modifications of gravity and cosmology are really not complete and
in the course of development. However, we found that the comments of Sergiu I.
Vacaru was based a wrong logic. The Finslerian modifications of gravity and
cosmology is a suggestion on the generalization of Einstein’s gravity and
cosmology, but not a proof for theorems in geometry. False or true of the
theory should be tested by experiments or observations. A personal claim can
not be used to prove any other researches to be wrong. To get the claim: “we
may construct more “standard” physical Finsler classical/quantum gravity
theories for metric compatible connections like the Cartan d-connection” ,
Sergiu I. Vacaru should complete a consistent presentation at least. We
suggest Sergiu I. Vacaru to make some predictions on gravity and cosmoligy
using his “standard” physical Finsler classical/quantum gravity theories as we
did, and compare them with astronomical observations.
In the following, we will reply to some key points in the comments of Sergiu
I. Vacaru.
Question1: “The Chern connection is not metric compatible and not the unique
connection in Finsler geometry”.
Reply: It is correct. The Chern connection is not metric compatible. The
statement we gave in Finsler DM should be replaced by a more clear
presentation. In fact, in the second paper of the seriesFinsler DE , just
before the formula (2), we have already altered the statement as: “In Finsler
manifold, there exists a unique linear connection-the Chern connection. It is
torsion freeness and almost metric compatibility”. Here the word “unique” just
means that the Chern connection is determined by the conditions(or structural
equation) of torsion freeness and almost metric compatibility. It should not
be read as that the Chern connection is the unique connection in Finsler
geometry.
Question2: “The metric incompatibility make more difficult the definition of
spinors and conservation laws in Finsler gravity and does not allow “simple”
(super) string and noncommutative generalizations like we proposed.”
Reply: Our paper just presented a classical modification of gravity and
cosmology and did not concern any aspect of quantum theory of gravity.
Therefore, we do not think it is a comment on our papers. It is strange that a
claim on quantum gravity can be used to criticize a classical theory of
gravity. We should say that here the logic of Sergiu I. Vacaru is wrong. Even
though, we still would like to point out that the comment made a strong
conclusion without any proof. It is a pity that a proof of the no go theorem
can not be found in the commentsVacaru .
Question3: “The Ricci tensor introduced by H. Akbar-Zadeh Akbar is not
correct for all Finsler geometry/gravity models. There were considered various
types of Ricci type tensors in Finsler geometries.”
Reply: We do not know any mathematician has presented the theorem. Sergiu I.
Vacaru did not give any sound proof about his assertion either. In fact, these
various types of Ricci tensors in Finsler geometry and the so-called
“gravitational field equations” constructed by them depend on the chosen
connection. It implies that different gravitational field equations could be
obtained while one uses different connections to calculate it, even all the
connections are metric compatible. This brings up the problem that which
gravitational field equation is the physical one. On the contrary, the Ricci
tensor that introduced by Akbar-Zadeh Akbar does not face such a problem. It
is given as
$Ric_{\mu\nu}=\frac{\partial^{2}\left(\frac{1}{2}F^{2}Ric\right)}{\partial
y^{\mu}\partial y^{\nu}},$ (1)
where the Finsler metric is defined as
$g_{\mu\nu}\equiv\frac{\partial^{2}\left(\frac{1}{2}F^{2}\right)}{\partial
y^{\mu}\partial y^{\nu}}$, and the Ricci scalar “$Ric$” is the trace of the
predecessor of the flag curvature. The flag curvature Book by Bao in Finsler
geometry is the counterpart of the sectional curvature in Riemannian geometry.
It is a geometrical invariant. Furthermore, the same flag curvature is
obtained for any connection that chosen in Finsler space. Thus, the same Ricci
tensor is obtained for any connection that chosen in Finsler space Therefore,
the Ricci tensor introduced by Akbar-Zadeh Akbar is a reasonable and well-
defined one.
Question4: “Sergiu I. Vacaru claimed that he may construct more “standard”
physical Finsler classical/quantum gravity theories for metric compatible
connections like the Cartan d-connection”.
Reply: To become a theory, a claim should be complete and consistent at least.
The most important thing for a physical theory is that it makes predictions
that can be tested through experiments and observations. To our point of view,
Sergiu I. Vacaru’s Finsler gravity theories are still premature. Newton’s
theory of gravity can not be used to kill Einstein’s general relativity.
Einstein’s equations of gravitational field can not be used to kill the
hypothesis of dark energy and dark matter. Another reason for the
prematureness is that the Einstein’s tensor $E(\hat{D})$ in Sergiu I. Vacaru’s
Finsler gravity theories is not a conserved quantity Vacaru1 (in the sense of
covariant differentiation). This is also pointed out by Sergiu I. Vacaru
himself in his comments (see the footnote 3 and the formula (3) in Vacaru ).
It is well-known that in general relativity the Einstein tensor is a conserved
quantity. The Einstein’s gravitational field equation constructed in such a
form due to the requirement that the energy-momentum tensor must conserve. And
this conservation law of energy-momentum tensor is of vital importance and has
been extensively used and embedded in different branches of modern physics.
Any theory that does not subject to this rule can not be recognized as a
physical one. Neither can Sergiu I. Vacaru’s Finsler gravity theories. At
least, Sergiu I. Vacaru should give a conserved quantity which is the
counterpart of energy-momentum tensor in the framework of Finsler geometry.
Sergiu I. Vacaru has claimed that the quantum gravity theory is “almost sure”
of generalized Finsler type. We wish him publish his claim in an isolated
paper. To discuss details on this topic is out of range of our reply.
Acknowledgements
We would like to thank Prof. Z. Shen and Dr. M. Li for useful discussions. The
work was supported by the NSF of China under Grant No. 10525522 and 10875129.
## References
* (1) S. Vacaru, Phys. Lett. B 690 (2010) 224, arXiv:1003.0044v2 [gr-qc].
* (2) Z. Chang and X. Li, Phys. Lett. B 676 (2009) 173.
* (3) Z. Chang and X. Li, Phys. Lett. B 668 (2008) 453.
* (4) H. Akbar-Zadeh, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988) 281.
* (5) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics 200, Springer, New York, 2000.
* (6) S. Vacaru, P. Stavrinos, E. Gaburov, D. Gonta, arXiv:gr-qc/ 0508023\.
|
arxiv-papers
| 2010-07-22T08:08:32 |
2024-09-04T02:49:11.816922
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and Zhe Chang",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1007.3820"
}
|
1007.3827
|
# A realization theorem for modules of constant Jordan type and vector bundles
Dave Benson and Julia Pevtsova∗
###### Abstract.
Let $E$ be an elementary abelian $p$-group of rank $r$ and let $k$ be a field
of characteristic $p$. We introduce functors $\mathcal{F}_{i}$ from finitely
generated $kE$-modules of constant Jordan type to vector bundles over
projective space $\mathbb{P}^{r-1}$. The fibers of the functors
$\mathcal{F}_{i}$ encode complete information about the Jordan type of the
module.
We prove that given any vector bundle $\mathcal{F}$ of rank $s$ on
$\mathbb{P}^{r-1}$, there is a $kE$-module $M$ of stable constant Jordan type
$[1]^{s}$ such that $\mathcal{F}_{1}(M)\cong\mathcal{F}$ if $p=2$, and such
that $\mathcal{F}_{1}(M)\cong F^{*}(\mathcal{F})$ if $p$ is odd. Here,
$F\colon\mathbb{P}^{r-1}\to\mathbb{P}^{r-1}$ is the Frobenius map. We prove
that the theorem cannot be improved if $p$ is odd, because if $M$ is any
module of stable constant Jordan type $[1]^{s}$ then the Chern numbers
$c_{1},\dots,c_{p-2}$ of $\mathcal{F}_{1}(M)$ are divisible by $p$.
∗ partially supported by the NSF
## 1\. Introduction
The class of modules of constant Jordan type was introduced by Carlson,
Friedlander and the second author [5], and then consequently studied in [1, 2,
4, 6, 7, 8]. The connection between modules of constant Jordan type and
algebraic vector bundles on projective varieties was first observed and
developed by Friedlander and the second author in [7] in the general setting
of an arbitrary infinitesimal group scheme. In the present paper, we study
this connection for an elementary abelian $p$-group.
Let $k$ be a field of characteristic $p$ and let $E$ be an elementary abelian
$p$-group of rank $r$. We define functors $\mathcal{F}_{i}$ ($1\leq i\leq p$)
from finitely generated $kE$-modules of constant Jordan type to vector bundles
on projective space $\mathbb{P}^{r-1}$, capturing the sum of the socles of the
length $i$ Jordan blocks. The following is the main theorem of this paper.
###### Theorem 1.1.
Given any vector bundle $\mathcal{F}$ of rank $s$ on $\mathbb{P}^{r-1}$, there
exists a finitely generated $kE$-module $M$ of stable constant Jordan type
$[1]^{s}$ such that
* (i)
if $p=2$, then $\mathcal{F}_{1}(M)\cong\mathcal{F}$.
* (ii)
if $p$ is odd, then $\mathcal{F}_{1}(M)\cong F^{*}(\mathcal{F})$, the pullback
of $\mathcal{F}$ along the Frobenius morphism
$F\colon\mathbb{P}^{r-1}\to\mathbb{P}^{r-1}$.
The $kE$-modules produced this way are usually large. For example, in [1], the
first author showed how to produce a finitely generated $kE$-module $M$ of
constant Jordan type such that $\mathcal{F}_{2}(M)$ is isomorphic to the rank
two Horrocks–Mumford bundle on $\mathbb{P}^{4}$. In this case, the
construction used to prove our main theorem produces a module $M$ of dimension
many hundred times $p^{5}$ plus two such that $\mathcal{F}_{1}(M)\cong
F^{*}(\mathcal{F}_{\mathsf{HM}})$, whereas the construction in [1] produces a
module of dimension $30p^{5}$ of stable constant Jordan type
$[p-1]^{30}[2]^{2}[1]^{26}$ such that applying $\mathcal{F}_{2}$ gives
$\mathcal{F}_{\mathsf{HM}}(-2)$.
The theorem for $p=2$ may be thought of as a version of the
Bernstein–Gelfand–Gelfand correspondence [3], since the group algebra of an
elementary abelian $2$-group in characteristic two is isomorphic to an
exterior algebra. But for $p$ odd it says something new and interesting. In
particular, it is striking that the $p$ odd case of Theorem 1.1 cannot be
strengthened to say that $\mathcal{F}_{1}(M)\cong\mathcal{F}$. The following
theorem, which is proved in Section 5, gives limitations on the vector bundles
appearing as $\mathcal{F}_{1}(M)$ with $M$ of stable constant Jordan type
$[1]^{s}$.
###### Theorem 1.2.
Suppose that $M$ has stable constant Jordan type $[1]^{s}$. Then $p$ divides
the Chern numbers $c_{m}(\mathcal{F}_{1}(M))$ for $1\leq m\leq p-2$.
The paper is organized as follows. In Section 2 we give basic definitions of
the functors $\mathcal{F}_{i}$ and show that applied to modules of constant
Jordan type, they produce algebraic vector bundles. Section 3 analyzes
behavior of the functors $\mathcal{F}_{i}$ with respect to Heller shifts and
duals. This analysis plays a key role in the proof of our main theorem.
Theorems 1.1 and 1.2 are proved in Sections 4 and 5 respectively.
## 2\. Definition of the functors $\mathcal{F}_{i}$
Let $k$ be a perfect field of characteristic $p$. Let $E=\langle
g_{1},\dots,g_{r}\rangle$ be an elementary abelian $p$-group of rank $r$, and
set $X_{i}=g_{i}-1\in kE$ for $1\leq i\leq r$. Let $J(kE)=\langle
X_{1},\ldots,X_{r}\rangle$ be the augmentation ideal of $kE$. The images of
$X_{1},\dots,X_{r}$ form a basis for $J(kE)/J^{2}(kE)$, which we think of as
affine space $\mathbb{A}_{k}^{r}$ over $k$. Let $K/k$ be a field extension. If
$0\neq\alpha=(\lambda_{1},\dots,\lambda_{r})\in\mathbb{A}_{K}^{r}$, we define
$X_{\alpha}=\lambda_{1}X_{1}+\dots+\lambda_{r}X_{r}\in KE.$
This is an element of $J(KE)$ satisfying $X_{\alpha}^{p}=0$. If $M$ is a
finitely generated $kE$-module then $X_{\alpha}$ acts nilpotently on
$M_{K}=M\otimes K$, and we can decompose $M_{K}$ into Jordan blocks. They all
have eigenvalue zero, and length between $1$ and $p$. We say that $M$ has
constant Jordan type $[p]^{a_{p}}\dots[1]^{a_{1}}$ if there are $a_{p}$ Jordan
blocks of length $p$, …, $a_{1}$ blocks of length $1$, independently of choice
of $\alpha$. Since $a_{p}$ is determined by $a_{p-1},\dots,a_{1}$ and the
dimension of $M$, we also say that $M$ has stable constant Jordan type
$[p-1]^{a_{p-1}}\dots[1]^{a_{1}}$. Note that the property of having constant
Jordan type and the type itself do not depend on the choice of generators
$\langle g_{1},\dots,g_{r}\rangle$ (see [5]).
We write $k[Y_{1},\dots,Y_{r}]$ for the coordinate ring $k[\mathbb{A}^{r}]$,
where the $Y_{i}$ are the linear functions defined by
$Y_{i}(X_{j})=\delta_{ij}$ (Kronecker delta). We write $\mathbb{P}^{r-1}$ for
the corresponding projective space. Let $\mathcal{O}$ be the structure sheaf
on $\mathbb{P}^{r-1}$. If $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules
and $j\in\mathbb{Z}$, we write $\mathcal{F}(j)$ for the $j$th Serre twist
$\mathcal{F}\otimes_{\mathcal{O}}\mathcal{O}(j)$. If $M$ is a finitely
generated $kE$-module, we write $\widetilde{M}$ for the trivial vector bundle
$M\otimes_{k}\mathcal{O}$, so that
$\widetilde{M}(j)=M\otimes_{k}\mathcal{O}(j)$. Friedlander and the second
author [7, §4] define a map of vector bundles
$\theta_{M}\colon\widetilde{M}\to\widetilde{M}(1)$ by the formula
$\theta_{M}(m\otimes f)=\sum_{i=1}^{r}X_{i}(m)\otimes Y_{i}f.$
By abuse of notation we also write $\theta_{M}$ for the twist
$\theta_{M}(j)\colon\widetilde{M}(j)\to\widetilde{M}(j+1)$. With this
convention we have $\theta_{M}^{p}=0$.
We define functors $\mathcal{F}_{i,j}$ for $0\leq j<i\leq p$ from finitely
generated $kE$-modules to coherent sheaves on $\mathbb{P}^{r-1}$ by taking the
following subquotients of $\widetilde{M}$:
$\mathcal{F}_{i,j}(M)=\frac{\mathsf{Ker}\,\theta_{M}^{j+1}\cap\mathsf{Im}\,\theta_{M}^{i-j-1}}{(\mathsf{Ker}\,\theta_{M}^{j+1}\cap\mathsf{Im}\,\theta_{M}^{i-j})+(\mathsf{Ker}\,\theta_{M}^{j}\cap\mathsf{Im}\,\theta_{M}^{i-j-1})}$
We then define
$\mathcal{F}_{i}(M)=\mathcal{F}_{i,0}(M)=\frac{\mathsf{Ker}\,\theta_{M}\cap\mathsf{Im}\,\theta_{M}^{i-1}}{\mathsf{Ker}\,\theta_{M}\cap\mathsf{Im}\,\theta_{M}^{i}}$
For a point $0\not=\alpha\in\mathbb{A}^{r}$ and the corresponding operator
$X_{\alpha}\colon M\to M$, we also define
$\mathcal{F}_{i,\alpha}(M)=\frac{\mathsf{Ker}\,X_{\alpha}\cap\mathsf{Im}\,X_{\alpha}^{i-1}}{\mathsf{Ker}\,X_{\alpha}\cap\mathsf{Im}\,X_{\alpha}^{i}}$
Note that $\mathcal{F}_{i,\alpha}(M)$ is evidently well-defined for
$\bar{\alpha}\in\mathbb{P}^{r-1}$.
In the next Proposition we show that functors $\mathcal{F}_{i}$ take modules
of constant Jordan type to algebraic vector bundles (equivalently, locally
free sheaves), and that they commute with specialization.
###### Proposition 2.1.
1. (1)
Let $M$ be a $kE$-module of constant Jordan type
$[p]^{a_{p}}\dots[1]^{a_{1}}$. Then the sheaf $\mathcal{F}_{i}(M)$ is locally
free of rank $a_{i}$.
2. (2)
Let $f\colon M\to N$ be a map of modules of constant Jordan type. For any
point $\bar{\alpha}=[\lambda_{1}:\dots:\lambda_{r}]\in\mathbb{P}^{r-1}$ with
residue field $k(\bar{\alpha})$ we have a commutative diagram
$\textstyle{\mathcal{F}_{i}(M)\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{F}_{i}(f)}$$\textstyle{\mathcal{F}_{i}(N)\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{F}_{i,\alpha}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i,\alpha}(N)}$
###### Proof.
(1). Since the module $M$ is fixed throughout the proof, we shall use $\theta$
to denote $\theta_{M}$.
Note that
$\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i}=\mathsf{Ker}\,\\{\theta\colon\mathsf{Im}\,\theta^{i}\to\mathsf{Im}\,\theta^{i+1}\\}$.
Hence, we have a short exact sequence
(2.1.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{Im}\,\theta^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\mathsf{Im}\,\theta^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
Since $M$ has constant Jordan type, $\mathsf{Im}\,\theta^{i}$ is locally free
by [7, 4.13]. Therefore, specialization of the sequence (2.1.1) at any point
$\bar{\alpha}=[\lambda_{1}:\dots:\lambda_{r}]$ of $\mathbb{P}^{r-1}$ yields a
short exact sequence of vector spaces
(2.1.2)
$0\to(\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i})\otimes_{\mathcal{O}}k(\bar{\alpha})\to\mathsf{Im}\,\theta^{i}\otimes_{\mathcal{O}}k(\bar{\alpha})\to\mathsf{Im}\,\theta^{i+1}\otimes_{\mathcal{O}}k(\bar{\alpha})\to
0.$
By [7, 4.13],
$\mathsf{Im}\,\theta^{i}\otimes_{\mathcal{O}}k(\bar{\alpha})\simeq\mathsf{Im}\,\\{X^{i}_{\alpha}\colon
M\to M\\}$. In particular, the dimension of fibers of
$\mathsf{Im}\,\theta^{i}$ is constant and equals
$\sum\limits_{j=i+1}^{p}a_{j}(j-i)$. We can rewrite the sequence (2.1.2) as
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i})\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
151.55563pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
151.55563pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{Im}\,X_{\alpha}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
188.36478pt\raise 5.82222pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9611pt\hbox{$\scriptstyle{X_{\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 207.81848pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
207.81848pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{Im}\,X_{\alpha}^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
264.08133pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
264.08133pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
Hence the fiber of $\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i}$ at a
point $\bar{\alpha}$ equals
$\mathsf{Ker}\,X_{\alpha}\cap\mathsf{Im}\,X_{\alpha}^{i}$. In particular,
$\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i}$ has fibers of constant
dimension, equal to
$\sum\limits_{j=i+1}^{p}a_{j}(j-i)-\sum\limits_{j=i+2}^{p}a_{j}(j-i-1)=\sum\limits_{j=i+1}^{p}a_{j}.$
Applying [7, 4.10] (see also [10, V. ex. 5.8]), we conclude that
$\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i}$ is locally free of rank
$\sum\limits_{j=i+1}^{p}a_{j}$.
Consider the short exact sequence that defines $\mathcal{F}_{i}(M)$:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
Specializing at $\bar{\alpha}$, we get
$\textstyle{(\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i})\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathsf{Ker}\,\theta\cap\mathsf{Im}\,\theta^{i-1})\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}(M)\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathsf{Ker}\,X_{\alpha}\cap\mathsf{Im}\,X_{\alpha}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{Ker}\,X_{\alpha}\cap\mathsf{Im}\,X_{\alpha}^{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}(M)\otimes_{\mathcal{O}}k(\bar{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
The first arrow of the bottom row is clearly an injection. Hence,
$\dim(\mathcal{F}_{i}(M)\otimes_{\mathcal{O}}k(\bar{\alpha}))=\sum\limits_{j=i}^{p}a_{j}-\sum\limits_{j=i+1}^{p}a_{j}=a_{i}$
for any point $\alpha\in\mathbb{P}^{r-1}$. Applying [7, 4.10] again, we
conclude that $\mathcal{F}_{i}(M)$ is locally free (of rank $a_{i}$).
Statement (2) follows immediately by applying the last diagram to both $M$ and
$N$. ∎
###### Lemma 2.2.
$\widetilde{M}$ has a filtration in which the filtered quotients are
isomorphic to $\mathcal{F}_{i,j}(M)$ for $0\leq j<i\leq p$.
###### Proof.
We consider two filtrations on $\widetilde{M}$, the “kernel filtration” and
the “image filtration”:
$\displaystyle
0\subset\mathsf{Ker}\,\theta_{M}\subset\ldots\subset\mathsf{Ker}\,\theta_{M}^{p-1}\subset\widetilde{M}$
$\displaystyle
0=\mathsf{Im}\,\theta_{M}^{p}\subset\mathsf{Im}\,\theta_{M}^{p-1}\subset\ldots\subset\mathsf{Im}\,\theta_{M}\subset\mathsf{Im}\,\theta_{M}^{0}=\widetilde{M}$
To simplify notation, we set $\mathcal{K}_{j}=\mathsf{Ker}\,\theta^{i}_{M}$
and $\mathcal{I}_{i}=\mathsf{Im}\,\theta_{M}^{p-i}$. Using the standard
refinement procedure, we refine the kernel filtration by the image filtration:
$\mathcal{K}_{j}\subset(\mathcal{K}_{j+1}\cap\mathcal{I}_{1})+\mathcal{K}_{j}\subset\ldots\subset(\mathcal{K}_{j+1}\cap\mathcal{I}_{\ell})+\mathcal{K}_{j}\subset(\mathcal{K}_{j+1}\cap\mathcal{I}_{\ell+1})+\mathcal{K}_{j}\subset\ldots\subset\mathcal{K}_{j+1}$
For any three sheaves $A,B,C$ with $B\subset A$, the second isomorphism
theorem and the modular law imply that
$\frac{A+C}{B+C}\simeq\frac{A+(B+C)}{B+C}\simeq\frac{A}{A\cap(B+C)}\simeq\frac{A}{B+(A\cap
C)}.$
Hence, we can identify the subquotients of the refined kernel filtration above
as
$\frac{(\mathcal{K}_{j+1}\cap\mathcal{I}_{\ell+1})+\mathcal{K}_{j}}{(\mathcal{K}_{j+1}\cap\mathcal{I}_{\ell})+\mathcal{K}_{j}}\simeq\frac{\mathcal{K}_{j+1}\cap\mathcal{I}_{\ell+1}}{(\mathcal{K}_{j+1}\cap\mathcal{I}_{\ell})+(\mathcal{K}_{j}\cap\mathcal{I}_{\ell+1})}$
Setting $i=p-\ell+j$, we get that the latter quotient is precisely
$\mathcal{F}_{i,j}(M)$ (note that when $j>\ell$, the corresponding subquotient
is trivial). ∎
###### Lemma 2.3.
For $0\leq j<i$, we have a natural isomorphism
$\mathcal{F}_{i,j}(M)\cong\mathcal{F}_{i}(M)(j)$.
###### Proof.
For $0<j<i$, the map $\theta_{M}\colon\widetilde{M}\to\widetilde{M}(1)$
induces a natural isomorphism
$\mathcal{F}_{i,j}(M)\to\mathcal{F}_{i,j-1}(M)(1)$. Since
$\mathcal{F}_{i,0}=\mathcal{F}_{i}$, the result follows by induction on $j$. ∎
###### Remark 2.4.
It follows from the proof of Proposition 2.1 that the subquotient functors
$\mathcal{F}_{i,j}$ are linked as follows:
|
---|---
$\textstyle{\mathcal{F}_{p,p-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p-1,p-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p,p-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p-2,p-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p-1,p-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\dots\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots}$$\textstyle{\
\ \mathcal{F}_{1,0}\ \
}$$\textstyle{\mathcal{F}_{p-1,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\dots\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p-2,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p-1,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{p,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
We finish this section with an example.
###### Example 2.5.
Let $M=kE/J^{2}(kE)$. Then $M$ has constant Jordan type $[2][1]^{r-1}$. In the
short exact sequence of vector bundles
$0\to\widetilde{M/\mathsf{Rad}M}\xrightarrow{\theta}\widetilde{\mathsf{Rad}M}(1)\to\mathcal{F}_{1}(M)(1)\to
0$
the map $\theta$ (induced by $\theta_{M}$) is equal to the map defining the
tangent bundle (or sheaf of derivations) $\mathcal{T}$ of $\mathbb{P}^{r-1}$:
$0\to\mathcal{O}\to\mathcal{O}(1)^{r}\to\mathcal{T}\to 0.$
It follows that $\mathcal{F}_{1}(M)\cong\mathcal{T}(-1)$. On the other hand we
have $\mathcal{F}_{2,1}(M)\cong\mathcal{O}$, and hence
$\mathcal{F}_{2}(M)\cong\mathcal{O}(-1)$.
## 3\. Twists and syzygies
We need a general lemma whose proof we provide for completeness.
###### Lemma 3.1.
Let $X$ be a Noetherian scheme over $k$, and let $M,N$ be locally free
$\mathcal{O}_{X}$-modules. Let $f\colon M\to N$ be a morphism of
$\mathcal{O}_{X}$-modules such that
$f\otimes_{\mathcal{O}_{X}}k(x)\colon M\otimes_{\mathcal{O}_{X}}k(x)\to
N\otimes_{\mathcal{O}_{X}}k(x)$
is an isomorphism for any $x\in X$. Then $f$ is an isomorphism.
###### Proof.
It suffices to show that $f$ induces an isomorphism on stalks. Hence, we may
assume that $X=\operatorname{Spec}\nolimits R$, where $R$ is a local ring with
the maximal ideal $\mathfrak{m}$, and $M$, $N$ are free modules. Since
specialization is right exact, $f$ is surjective by Nakayama’s lemma. Hence,
we have an exact sequence of $R$-modules:
$0\to\ker f\to M\to N\to 0.$
Since $N$ is free, $\operatorname{Tor}\nolimits_{1}^{R}(N,R/\mathfrak{m})$
vanishes, and hence $\ker f\otimes_{R}R/\mathfrak{m}=0$. By Nakayama’s lemma,
$\ker f=0$; therefore, $f$ is injective. ∎
###### Theorem 3.2.
Let $M$ be a finite dimensional $kE$-module and let $1\leq i\leq p-1$. Then
there is a natural isomorphism
$\mathcal{F}_{i}(M)(-p+i)\cong\mathcal{F}_{p-i}(\Omega M).$
###### Proof.
Consider the diagram
(3.2.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\Omega
M}}$$\textstyle{\widetilde{P_{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{P_{M}}}$$\textstyle{\widetilde{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{M}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
M}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P_{M}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
where $P_{M}$ is a projective cover of $M$. Let
$\delta\colon\mathsf{Ker}\,\theta_{M}\to\mathsf{Coker}\,\theta_{\Omega M}$
be the switchback map. A simple diagram chase in conjunction with the fact
that $\theta_{P_{M}}^{p}=0$ yields that the restriction of $\delta$ to
$\mathsf{Ker}\,\theta_{M}\cap\mathsf{Im}\,\theta^{i-1}_{M}$ lands in
$\frac{\mathsf{Ker}\,\theta^{p-i}_{\Omega
M}}{\mathsf{Ker}\,\theta^{p-i}_{\Omega
M}\cap\mathsf{Im}\,\theta_{\Omega(M)}}(1).$
Projecting the latter onto
$\mathcal{F}_{p-i,p-i-1}(\Omega M)(1)=\frac{\mathsf{Ker}\,\theta^{p-i}_{\Omega
M}}{\mathsf{Ker}\,\theta^{p-i-1}_{\Omega M}+\mathsf{Ker}\,\theta^{p-i}_{\Omega
M}\cap\mathsf{Im}\,\theta_{\Omega(M)}}(1),$
we get a map of bundles:
$\delta\colon\mathsf{Ker}\,\theta_{M}\cap\mathsf{Im}\,\theta^{i-1}\to\mathcal{F}_{p-i,p-i-1}(\Omega
M)(1).$
Since $\delta$ evidently kills
$\mathsf{Ker}\,\theta_{M}\cap\mathsf{Im}\,\theta^{i}_{M}$, we conclude that
$\delta$ factors through $\mathcal{F}_{i}(M)$. Hence, we have an induced map
$\delta\colon\mathcal{F}_{i}(M)\to\mathcal{F}_{p-i,p-i-1}(\Omega M)(1).$
A simple block count shows that this is an isomorphism at each fiber. Hence,
by Lemma 3.1, this is an isomorphism of bundles. Thus using Lemma 2.3 (i.e.,
applying $\theta_{\Omega M}$ a further $p-i-1$ times), we have
$\mathcal{F}_{i}(M)\cong\mathcal{F}_{p-i,p-i-1}(\Omega
M)(1)\cong\mathcal{F}_{p-i}(\Omega M)(p-i).$
Twisting by $\mathcal{O}(-p+i)$, we get the desired isomorphism.
Let $f\colon M\to N$ be a map of $kE$-modules. The naturality of the
isomorphism $\mathcal{F}_{i}(M)(-p+i)\cong\mathcal{F}_{p-i}(\Omega M)$ is
equivalent to the commutativity of the diagram
(3.2.2)
$\textstyle{\mathcal{F}_{i}(M)(-p+i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{F}_{i}(f)(-p+i)}$$\textstyle{\mathcal{F}_{i}(N)(-p+i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{F}_{p-i}(\Omega
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{p-i}(\Omega
f)}$$\textstyle{\mathcal{F}_{p-i}(\Omega N).}$
The commutativity follows from the construction of the map $\delta$ and
naturality of the “shifting isomorphism” of Lemma 2.3. ∎
###### Corollary 3.3.
Let $M$ be a finite dimensional $kE$-module and let $1\leq i\leq p-1$. Then
$\mathcal{F}_{i}(\Omega^{2}M)\cong\mathcal{F}_{i}(M)(-p)$.
###### Proof.
Apply the theorem twice. ∎
###### Corollary 3.4.
We have $\mathcal{F}_{1}(\Omega^{2n}k)\cong\mathcal{O}(-np)$, and
$\mathcal{F}_{p-1}(\Omega^{2n-1}k)\cong\mathcal{O}(1-np)$.
###### Proof.
This follows from the theorem and the corollary, using the isomorphism
$\mathcal{F}_{1}(k)\cong\mathcal{O}$. ∎
###### Remark 3.5.
If $p=2$ then Theorem 3.2 and Corollary 3.4 reduce to the statements that
$\mathcal{F}_{1}(\Omega M)\cong\mathcal{F}_{1}(M)(-1)$ and
$\mathcal{F}_{1}(\Omega^{n}k)\cong\mathcal{O}(-n)$.
For a coherent sheaf $\mathcal{E}$, we denote by
$\mathcal{E}^{\vee}=\mathcal{H}om_{\mathcal{O}}(\mathcal{E},\mathcal{O})$ the
dual sheaf.
###### Theorem 3.6.
Let $M^{*}$ be the $k$-linear dual of $M$, as a $kE$-module. Then
$\mathcal{F}_{i}(M^{*})\cong\mathcal{F}_{i}(M)^{\vee}(-i+1).$
###### Proof.
This follows from the more obvious isomorphism
$\mathcal{F}_{i,i-1}(M^{*})\cong\mathcal{F}_{i,0}(M)^{\vee}$ together with
Lemma 2.3. ∎
We finish this section with exactness properties of the functors
$\mathcal{F}_{i}$ which will be essential in the proof of the main theorem.
Let $\mathcal{C}(kE)$ be the exact category of modules of constant Jordan type
as introduced in [4]. This is an exact category in the sense of Quillen: the
objects are finite dimensional $kE$-modules of constant Jordan type, and the
admissible morphisms are morphisms which can be completed to a locally split
short exact sequence. We call a sequence of $kE$-modules
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
locally split if it is split upon restriction to
$k[X_{\alpha}]/X_{\alpha}^{p}$ for any $0\not=\alpha\in\mathbb{A}^{r}$.
###### Proposition 3.7.
The functor
$\mathcal{F}_{i}\colon\mathcal{C}(kE)\to\operatorname{Coh}\nolimits(\mathbb{P}_{k}^{r-1})$
is exact for $1\leq i\leq p-1$.
###### Proof.
Let
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
be a locally split short exact sequence of modules of constant Jordan type.
Then the Jordan type of the middle term is the sum of Jordan types of the end
terms. Hence,
$\operatorname{rk}\nolimits\mathcal{F}_{i}(M_{2})=\operatorname{rk}\nolimits\mathcal{F}_{i}(M_{1})+\operatorname{rk}\nolimits\mathcal{F}_{i}(M_{3})$
for any $i$. Consider the map
$\mathcal{F}_{i}(M_{2})\to\mathcal{F}_{i}(M_{3})$. By Proposition 2.1, the
specialization
$\mathcal{F}_{i}(M_{2})\otimes_{\mathcal{O}}k(\bar{\alpha})\to\mathcal{F}_{i}(M_{3})\otimes_{\mathcal{O}}k(\bar{\alpha})$
is surjective at any point $\bar{\alpha}\in\mathbb{P}^{r-1}$. Arguing as in
Lemma 3.1, we conclude that $\mathcal{F}_{i}(M_{2})\to\mathcal{F}_{i}(M_{3})$
is surjective. Similarly, we show that
$\mathcal{F}_{i}(M_{1})\to\mathcal{F}_{i}(M_{2})$ is injective. Finally, the
equality
$\operatorname{rk}\nolimits\mathcal{F}_{i}(M_{2})=\operatorname{rk}\nolimits\mathcal{F}_{i}(M_{1})+\operatorname{rk}\nolimits\mathcal{F}_{i}(M_{3})$
implies exactness in the middle term. ∎
## 4\. The construction
Since $H^{1}(E,k)$ is the vector space dual of $J(kE)/J^{2}(kE)$, there are
elements $y_{1},\dots,y_{r}$ forming a vector space basis for $H^{1}(E,k)$ and
corresponding to the linear functions $Y_{1},\dots,Y_{r}$ on $J(kE)/J^{2}(kE)$
introduced in Section 2. Because of the difference in structure of the
cohomology ring, we divide the discussion into two cases, according as $p=2$
or $p$ is odd.
Case 1: $p=2$. In this case the cohomology ring $H^{*}(E,k)$ is the polynomial
algebra $k[y_{1},\dots,y_{r}]$. We define a $k$-algebra homomorphism
$\rho\colon H^{*}(E,k)=k[y_{1},\dots,y_{r}]\to k[Y_{1},\dots,Y_{r}]$
by $\rho(y_{i})=Y_{i}$. Recall that we have an isomorphism
$\mathcal{O}(-n)=\mathcal{F}_{1}(k)(-n)\simeq\mathcal{F}_{1}(\Omega^{n}k)$ by
Remark 3.5.
###### Lemma 4.1.
If $\zeta\in H^{n}(E,k)$ is represented by a cocycle
$\hat{\zeta}\colon\Omega^{n+j}k\to\Omega^{j}k$ (with $j\in\mathbb{Z}$) then
the diagram
$\textstyle{\mathcal{O}(-n-j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho(\zeta)}$$\textstyle{\mathcal{O}(-j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{F}_{1}(\Omega^{n+j}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{1}(\hat{\zeta})}$$\textstyle{\mathcal{F}_{1}(\Omega^{j}k)}$
commutes.
###### Proof.
Consider $\hat{\zeta}\colon\Omega^{n}k\to k$. The commutative diagram 3.2.2
applied to $\hat{\zeta}$ and iterated $j$ times becomes
$\textstyle{\mathcal{F}_{1}(\Omega^{n}k)(-j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{F}_{1}(\hat{\zeta})(-j)}$$\textstyle{\mathcal{F}_{1}(k)(-j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{F}_{1}(\Omega^{n+j}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{1}(\Omega^{j}\hat{\zeta})}$$\textstyle{\mathcal{F}_{1}(\Omega^{j}k).}$
Hence, it suffices to assume that $j=0$. Additivity of the functor
$\mathcal{F}_{1}$ allows us to assume that $\zeta$ is a monomial on generators
$y_{1},\ldots,y_{r}$. Finally, since multiplication in cohomology corresponds
to composition of the corresponding maps on Heller shifts of $k$, it suffices
to prove our statement for a degree one generator $\zeta=y_{i}$.
In the case $j=0$, $\zeta=y_{i}$, we need to show that the following diagram
commutes (this is the diagram above twisted by $\mathcal{O}(1)$)
(4.1.1)
$\textstyle{\mathcal{F}_{1}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{Y_{i}}$$\textstyle{\mathcal{F}_{1}(k)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{1}(\Omega
k)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{1}(\hat{y}_{i})}$$\textstyle{\mathcal{F}_{1}(k)(1).}$
Let $E_{i}$ be the subgroup of index two in $E$ such that $y_{i}$ is inflated
from $E/E_{i}$ to $E$, namely the subgroup generated by all of
$g_{1},\dots,g_{r}$ except $g_{i}$. Then $y_{i}$ represents the class of the
extension
$0\to k\to M_{i}\to k\to 0$
where $M_{i}$ is the permutation module on the cosets of $E_{i}$. This is a
length two module on which $X_{1},\dots,X_{r}$ act as zero except for $X_{i}$,
which acts as a Jordan block of length two. We have a commutative diagram of
$kE$-modules
(4.1.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega
k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{i}}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
The left vertical isomorphism
$\delta\colon\mathcal{F}_{1}(k)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}_{1}(\Omega
k)(1)$ of the diagram 4.1.1 is given by the switchback map for the short exact
sequence
$0\to\Omega k\to P_{0}\to k\to 0$
as in diagram (3.2.1). Applying $\theta$ to the commutative diagram on free
$\mathcal{O}$-modules induced by the module diagram (4.1.2), we get a
commutative diagram
| |
---|---|---
| |
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{y}_{i}}$$\textstyle{\widetilde{P_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
k}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{y}_{i}(1)}$$\textstyle{\widetilde{P_{0}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M_{i}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
where all horizontal arrows going back to front are given by the operator
$\theta$ on the corresponding module. The map
$\widetilde{y}_{i}\colon\widetilde{\Omega k}\to\mathcal{O}$ is induced by
$y_{i}\colon\Omega k\to k$. To compute the composite
$\mathcal{F}_{1}(\hat{y}_{i})\circ\delta$ we first do the switchback map of
the top layer and then push the result down via $\tilde{y}_{i}(1)$. Since the
diagram is commutative, we can first push down via the identity map of the
right vertical back arrow and then do the switchback of the bottom layer.
Hence, the composite $\mathcal{F}_{1}(\hat{y}_{i})\circ\delta$ is given by the
switchback map of the bottom layer; that is, of the diagram
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{k}}$$\textstyle{\widetilde{M}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{M_{i}}}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{k}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M}_{i}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
The left and right hand vertical maps here are zero. Hence, the switchback map
$\delta\colon\mathcal{O}\to\mathcal{O}(1)$ is given by multiplication by
$\theta_{M_{i}}$, which is given by multiplication by $Y_{i}$ in this
situation. ∎
Case 2: $p$ is odd. We write $\beta\colon H^{1}(E,k)\to H^{2}(E,k)$ for the
Bockstein map, and we set $x_{i}=\beta(y_{i})$. In terms of Massey products,
this is given by $x_{i}=-(y_{i},y_{i},\dots,y_{i})$ ($p$ terms).
The cohomology ring is a tensor product of an exterior algebra with a
polynomial algebra:
$H^{*}(E,k)\cong\Lambda(y_{1},\dots,y_{r})\otimes_{k}k[x_{1},\dots,x_{r}].$
We define a $k$-algebra homomorphism
$\rho\colon k[x_{1},\dots,x_{r}]\to k[Y_{1},\dots,Y_{r}]$
by $\rho(x_{i})=Y_{i}^{p}$.
###### Lemma 4.2.
Let $\zeta$ be a degree $n$ polynomial in $k[x_{1},\dots,x_{r}]$, regarded as
an element of $H^{2n}(E,k)$. If $\zeta$ is represented by a cocycle
$\hat{\zeta}\colon\Omega^{2(n+j)}k\to\Omega^{2j}k$ then the diagram
$\textstyle{\mathcal{O}(-p(n+j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho(\zeta)}$$\scriptstyle{\cong}$$\textstyle{\mathcal{O}(-pj)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{F}_{1}(\Omega^{2(n+j)}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{1}(\hat{\zeta})}$$\textstyle{\mathcal{F}_{1}(\Omega^{2j}k)}$
commutes.
###### Proof.
The proof is similar to the proof in the case $p=2$, but more complicated.
Again it suffices to treat the case where $\zeta=x_{i}$ and $j=0$. In other
words, we need to compute the composite $\mathcal{F}_{1}(\hat{x}_{i})\circ f$
in the diagram
$\textstyle{\mathcal{F}_{1}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{f}$$\scriptstyle{Y^{p}_{i}}$$\textstyle{\mathcal{F}_{1}(k)(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{1}(\Omega^{2}k)(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}_{1}(\hat{x}_{i})}$$\textstyle{\mathcal{F}_{1}(k)(p).}$
where $f\colon\mathcal{F}_{1}(k)\to\mathcal{F}_{1}(\Omega^{2}k)(p)$ is the
isomorphism of Corollary 3.3. Let
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{2}k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
be a truncated projective resolution of $k$. Tracing through the proof of
Theorem 3.2, we see that
$f\colon\mathcal{F}_{1}(k)\to\mathcal{F}_{1}(\Omega^{2}k)(p)$ is a composite
of three maps:
1. (1)
the switchback of the top two rows of the diagram 4.2.1 below which gives the
isomorphism $\mathcal{F}(k)\to\mathcal{F}_{p-1,p-2}(\Omega k)(1)$,
2. (2)
followed by the isomorphism $\theta^{p-2}_{\Omega
k}\colon\mathcal{F}_{p-1,p-2}(\Omega
k)(1)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}_{p-1}(\Omega k)(p-1)$ of
Lemma 2.3;
3. (3)
followed by another switchback map, now for the bottom two rows of diagram
4.2.1, which gives the isomorphism $\mathcal{F}_{p-1}(\Omega
k)(p-1)\simeq\mathcal{F}_{1}(\Omega^{2}k)(p)$.
(4.2.1)
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
k}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\Omega
k}^{p-2}}$$\textstyle{\widetilde{P}_{0}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega^{2}k}(p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P_{1}}(p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
k}(p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega^{2}k}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P_{1}}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Omega
k}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
Let $E_{i}$ be the subgroup of index $p$ such that $x_{i}$ is inflated from
$E/E_{i}$, namely the subgroup generated by all of $g_{1},\dots,g_{r}$ except
for $g_{i}$. We let $M_{i}$ be the permutation module on the cosets of
$E_{i}$. This is a length $p$ module on which $X_{1},\dots,X_{r}$ act as zero
except for $X_{i}$, which acts as a Jordan block of length $p$. Then $x_{i}$
represents the class of the 2-fold extension
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
where the middle map is multiplication by $X_{i}$. We construct a diagram
analogous to (4.2.1) for this extension:
(4.2.2)
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{N_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{N_{i}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{N_{i}}^{p-2}}$$\textstyle{\widetilde{M_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M_{i}}(p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{N_{i}}(p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M_{i}}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
Here, $N_{i}=\mathsf{Im}\,\\{X_{i}\colon M_{i}\to M_{i}\\}$. Just as in the
proof of Lemma 4.2, the module diagram
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{2}k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{i}}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
induces a commutative diagram of vector bundles with (4.2.1) on top and
(4.2.2) at the bottom. Arguing as in the proof of Lemma 4.1, we compute the
composite $\mathcal{F}_{1}(\hat{x}_{i})\circ f$ by first mapping the rightmost
$\mathcal{O}$ of diagram 4.2.1 identically to the rightmost $\mathcal{O}$ of
diagram 4.2.2, and then applying our composite of a switchback, followed by
$\theta_{N_{i}}^{p-2}$, followed by another switchback in the diagram 4.2.2.
The maps $\theta_{M_{i}}\colon\widetilde{M_{i}}\to\widetilde{M_{i}}(1)$ and
$\theta_{N_{i}}\colon\widetilde{N_{i}}\to\widetilde{N_{i}}(1)$ are simply
multiplication by $Y_{i}$. Since the leftmost and rightmost vertical arrows in
(4.2.2) are zero, to compute the composite of the three maps involved in
diagram 4.2.2, we have to multiply first by $Y_{i}$, then by $Y_{i}^{p-2}$,
then by $Y_{i}$ again. Hence, $\mathcal{F}_{1}(\hat{x}_{i})\circ f=Y_{i}^{p}$.
∎
We are ready to prove the main theorem.
###### Theorem 4.3.
Given any vector bundle $\mathcal{F}$ of rank $s$ on $\mathbb{P}^{r-1}$, there
exists a finitely generated $kE$-module $M$ of stable constant Jordan type
$[1]^{s}$ such that
* (i)
if $p=2$, then $\mathcal{F}_{1}(M)\cong\mathcal{F}$.
* (ii)
if $p$ is odd, then $\mathcal{F}_{1}(M)\cong F^{*}(\mathcal{F})$, the pullback
of $\mathcal{F}$ along the Frobenius morphism
$F\colon\mathbb{P}^{r-1}\to\mathbb{P}^{r-1}$.
###### Proof.
Given a vector bundle $\mathcal{F}$ on $\mathbb{P}^{r-1}$, using the Hilbert
syzygy theorem we can form a resolution by sums of twists of the structure
sheaf:
$0\to\bigoplus_{j=1}^{m_{r}}\mathcal{O}(a_{r,j})\to\cdots\to\bigoplus_{j=1}^{m_{1}}\mathcal{O}(a_{1,j})\to\bigoplus_{j=1}^{m_{0}}\mathcal{O}(a_{0,j})\to\mathcal{F}\to
0.$
Each of the maps in this resolution is a matrix whose entries are homogeneous
polynomials in $Y_{1},\dots,Y_{r}$. Replacing each $Y_{i}$ with $x_{i}$ gives
matrices of cohomology elements which we may use to form a sequence of modules
and homomorphisms, which takes the form
$0\to\bigoplus\Omega^{-\varepsilon
a_{r,j}}(k)\to\dots\to\bigoplus\Omega^{-\varepsilon
a_{1,j}}(k)\to\bigoplus\Omega^{-\varepsilon a_{0,j}}(k)$
where $\varepsilon=1$ if $p=2$ and $\varepsilon=2$ if $p$ is odd. This
sequence is a complex in the stable module category $\mathsf{stmod}(kE)$. We
complete the first map to a triangle whose third object we call $M_{r-1}$:
$\bigoplus\Omega^{-\varepsilon a_{r,j}}(k)\to\bigoplus\Omega^{-\varepsilon
a_{r-1,j}}(k)\to M_{r-1}.$
Since the first two entries in the triangle are modules of trivial stable
constant Jordan type, the same must be true for $M_{r-1}$. Moreover, the short
exact sequence corresponding to this triangle must be locally split.
Continuing by downwards induction on $i$ from $i=r-2$ to $i=0$ we complete
triangles
$M_{i+1}\to\bigoplus\Omega^{-\varepsilon a_{i,j}}(k)\to M_{i}$
and then finally we set $M=M_{0}$. By construction, $M_{0}$ is a module of
trivial stable constant Jordan type, and all intermediate triangles correspond
to locally split sequences of modules of constant Jordan type. Applying
$\mathcal{F}_{1}$ to this construction, we obtain an exact sequence of vector
bundles by Proposition 3.7. If $p=2$ it is isomorphic to the original
resolution by Lemma 4.1 and so we have $\mathcal{F}_{1}(M)\cong\mathcal{F}$.
On the other hand, if $p$ is odd, each of the original matrices has been
altered by replacing the variables $Y_{i}$ by their $p$th powers by Lemma 4.2.
This is the pullback of the original resolution along the Frobenius map
$F\colon\mathbb{P}^{r-1}\to\mathbb{P}^{r-1}$, and so it is a resolution of
$F^{*}(\mathcal{F})$. So in this case we have $\mathcal{F}_{1}(M)\cong
F^{*}(\mathcal{F})$. ∎
###### Remark 4.4.
The construction given above is not functorial, despite appearances. The
problem is that given a commutative square in $\mathsf{stmod}(kG)$ in which
the vertical maps are isomorphisms, it may be completed to an isomorphism of
triangles, but the third arrow is not unique.
## 5\. Chern numbers
Recall that the Chow ring of $\mathbb{P}^{r-1}$ is
$A^{*}(\mathbb{P}^{r-1})\cong\mathbb{Z}[h]/(h^{r}).$
If $\mathcal{F}$ is a vector bundle on $\mathbb{P}^{r-1}$, we write
$c(\mathcal{F},h)=\sum_{j\geq 0}c_{j}(\mathcal{F})h^{j}\in
A^{*}(\mathbb{P}^{r-1})$
for the Chern polynomial, where $c_{0}(\mathcal{F})=1$ and the
$c_{i}(\mathcal{F})\in\mathbb{Z}$ ($1\leq i\leq r-1$) are the Chern numbers of
$\mathcal{F}$.
If $0\to\mathcal{F}\to\mathcal{F}^{\prime}\to\mathcal{F}^{\prime\prime}\to 0$
is a short exact sequence of vector bundles then we have the Whitney sum
formula
$c(\mathcal{F}^{\prime},h)=c(\mathcal{F},h)c(\mathcal{F}^{\prime\prime},h).$
###### Lemma 5.1.
The formula for Chern numbers of twists of a rank $s$ vector bundle is
$c_{m}(\mathcal{F}(i))=\sum_{j=0}^{m}i^{j}\binom{s-m+j}{j}c_{m-j}(\mathcal{F}).$
Equivalently, the total Chern class of the twists is given by
(5.1.1)
$c(\mathcal{F}(i),h)=\sum_{n=0}^{s}c_{n}(\mathcal{F})h^{n}(1+ih)^{s-n}$
###### Proof.
See Fulton [9, Example 3.2.2]. ∎
More explicitly,
$\displaystyle c_{1}(\mathcal{F}(i))$ $\displaystyle=c_{1}(\mathcal{F})+is$
$\displaystyle c_{2}(\mathcal{F}(i))$
$\displaystyle=c_{2}(\mathcal{F})+i(s-1)c_{1}(\mathcal{F})+i^{2}\binom{s}{2}$
$\displaystyle c_{3}(\mathcal{F}(i))$
$\displaystyle=c_{3}(\mathcal{F})+i(s-2)c_{2}(\mathcal{F})+i^{2}\binom{s-1}{2}c_{1}(\mathcal{F})+i^{3}\binom{s}{3}$
and so on.
###### Lemma 5.2.
For a vector bundle $\mathcal{F}$ of rank $s$ on $\mathbb{P}^{r-1}$ we have
$c(\mathcal{F},h)c(\mathcal{F}(1),h)\cdots c(\mathcal{F}(p-1),h)\equiv
1-sh^{p-1}\pmod{(p,h^{p})}.$
###### Proof.
We write
$c(\mathcal{F})=\prod_{j=1}^{s}(1+\alpha_{j}h),$
where the $\alpha_{j}$ are the Chern roots. Then the formula (5.1.1) is
equivalent to
$c(\mathcal{F}(i))=\prod_{j=1}^{s}(1+(\alpha_{j}+i)h).$
Thus we have
$c(\mathcal{F})c(\mathcal{F}(1))\cdots
c(\mathcal{F}(p-1))=\prod_{j=1}^{s}(1+\alpha_{j}h)(1+(\alpha_{j}+1)h)\cdots(1+(\alpha_{j}+p-1)h).$
Now by Fermat’s little theorem, we have the identity
$x(x+y)\cdots(x+(p-1)y)\equiv x^{p}-xy^{p-1}\pmod{p}$
and so putting $x=1+\alpha_{j}h$, $y=h$ we obtain
$\displaystyle c(\mathcal{F})c(\mathcal{F}(1))\cdots c(\mathcal{F}(p-1))$
$\displaystyle\equiv\prod_{j=1}^{s}((1+\alpha_{j}h)^{p}-(1+\alpha_{j}h))h^{p-1}\pmod{p}$
$\displaystyle\equiv\prod_{j=1}^{s}(1-h^{p-1}+(\alpha_{j}^{p}-\alpha_{j})h^{p})\pmod{p}$
$\displaystyle\equiv 1-sh^{p-1}\pmod{(p,h^{p})}.$
_A priori_ , this is a congruence between polynomials with algebraic integer
coefficients. But if two rational integers are congruent mod $p$ as algebraic
integers then they are also congruent modulo $p$ as rational integers. This is
because their difference, divided by $p$, is both an algebraic integer and a
rational number, therefore an integer. ∎
We now restate and prove Theorem 1.2.
###### Theorem 5.3.
Suppose that $M$ has stable constant Jordan type $[1]^{s}$. Then $p$ divides
the Chern numbers $c_{m}(\mathcal{F}_{1}(M))$ for $1\leq m\leq p-2$.
###### Proof.
Since $M$ has stable Jordan type $[1]^{s}$, we have
$\mathcal{F}_{2}(M)=\ldots=\mathcal{F}_{p-1}(M)=0$. Hence, the trivial vector
bundle $\widetilde{M}$ has a filtration with filtered quotients (not in order)
$\mathcal{F}_{1}(M)$, $\mathcal{F}_{p}(M)$, $\mathcal{F}_{p}(M)(1)$, …,
$\mathcal{F}_{p}(M)(p-1)$. So we have
$\displaystyle 1$ $\displaystyle=c(\widetilde{M},h)$
$\displaystyle=c(\mathcal{F}_{1}(M),h)c(\mathcal{F}_{p}(M),h)c(\mathcal{F}_{p}(M)(1),h)\cdots
c(\mathcal{F}_{p}(M)(p-1),h)$ $\displaystyle\equiv
c(\mathcal{F}_{1}(M),h)\pmod{(p,h^{p-1})}$
by Lemma 5.2. It follows that the coefficients $c_{m}(\mathcal{F}_{1}(M))$ are
divisible by $p$ for $1\leq m\leq p-2$. ∎
###### Remark 5.4.
For $p=2$ this theorem says nothing. But for $p$ odd, it at least forces
$c_{1}(\mathcal{F}_{1}(M))$ to be divisible by $p$. As an explicit example,
the twists of the Horrocks–Mumford bundle $\mathcal{F}_{\mathsf{HM}}(i)$ have
$c_{1}=2i+5$ and $c_{2}=i^{2}+5i+10$ ([11]). For $p\geq 7$ these cannot both
be divisible by $p$, and so there is no module $M$ of stable constant Jordan
type $[1]^{2}$ and integer $i$ such that
$\mathcal{F}_{1}(M)\cong\mathcal{F}_{\mathsf{HM}}(i)$.
###### Remark 5.5.
The conclusion of the theorem is limited to the modules of stable constant
Jordan type $[1]^{s}$. For example, if $M_{n}$ is a “zig-zag” module of
dimension $2n+1$ for $\mathbb{Z}/p\times\mathbb{Z}/p$, then
$\mathcal{F}_{1}(M_{n})\simeq\mathcal{O}(-n)$ and
$\mathcal{F}_{1}(M_{n}^{*})\simeq\mathcal{O}(n)$ for any $n\geq 0$ (see [7,
§6]).
Acknowledgement. Both authors are grateful to MSRI for its hospitality during
the 2008 program “Representation theory of finite groups and related topics”
where much of this research was carried out.
## References
* [1] D. J. Benson, _Modules of constant Jordan type and the Horrocks–Mumford bundle_ , preprint, 2008.
* [2] by same author, _Modules of constant Jordan type with one non-projective block_ , Algebras and Representation Theory 13 (2010), 315–318.
* [3] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, _Algebraic vector bundles on $\mathbb{P}^{n}$ and problems of linear algebra_, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67.
* [4] J. F. Carlson and E. M. Friedlander, _Exact category of modules of constant Jordan type_ , Algebra, arithmetic and geometry: Manin Festschrift, Progr. in Math., vol. 269, Birkhäuser Verlag, Basel, 2009, pp. 259–281.
* [5] J. F. Carlson, E. M. Friedlander, and J. Pevtsova, _Modules of constant Jordan type_ , J. Reine & Angew. Math. 614 (2008), 191–234.
* [6] J. F. Carlson, E. M. Friedlander, and A. A. Suslin, _Modules for $\mathbb{Z}/p\times\mathbb{Z}/p$_, to appear.
* [7] E. M. Friedlander and J. Pevtsova, _Constructions for infinitesimal group schemes_ , to appear.
* [8] by same author, _Generalized support varieties for finite group schemes_ , Documenta Math. Extra Volume Suslin (2010), 197–222.
* [9] W. Fulton, _Intersection theory_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, Band 2, Springer-Verlag, Berlin/New York, 1984.
* [10] R. Hartshorne, _Algebraic geometry_ , Graduate Texts in Mathematics, vol. 52, Springer-Verlag, Berlin/New York, 1977.
* [11] G. Horrocks and D. Mumford, _A rank $2$ vector bundle on $\mathbb{P}^{4}$ with $15,000$ symmetries_, Topology 12 (1973), 63–81.
|
arxiv-papers
| 2010-07-22T08:35:16 |
2024-09-04T02:49:11.821569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David J. Benson and Julia Pevtsova",
"submitter": "Julia Pevtsova",
"url": "https://arxiv.org/abs/1007.3827"
}
|
1007.3846
|
# A First Glimpse into the far-IR properties of high-z UV-selected Galaxies:
Herschel/PACS observations of z$\sim$3 LBGs
G.E Magdis 11affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-
Yvette, France , D.Elbaz 11affiliation: CEA, Laboratoire AIM, Irfu/SAp,
F-91191 Gif-sur-Yvette, France , H.S. Hwang 11affiliation: CEA, Laboratoire
AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , E. Daddi 11affiliation: CEA,
Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , D. Rigopoulou
22affiliation: Department of Astrophysics, Oxford University, Keble Road,
Oxford, OX1 3RH , B. Altieri 33affiliation: European Space Astronomy Centre,
Villafrance del Castillo, Spain , P. Andreani 44affiliation: ESO, Karl-
Schwarzschild-Str. 2, D-85748 Garching, Germany , H. Aussel 11affiliation:
CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , S. Berta
55affiliation: Max-Planck-Institut für Extraterrestrische Physik (MPE),
Postfach 1312, 85741 Garching, Germany , A. Cava 66affiliation: Instituto de
Astrofísica de Canarias, 38205 La Laguna, Spain , A. Bongiovanni
66affiliation: Instituto de Astrofísica de Canarias, 38205 La Laguna, Spain ,
J. Cepa 66affiliation: Instituto de Astrofísica de Canarias, 38205 La Laguna,
Spain , A. Cimatti 77affiliation: Dipartimento di Astronomia, Università di
Padova, Vicolo dell’Osservatorio 3,Italy , M. Dickinson 88affiliation: NOAO,
950 N. Cherry Avenue, Tucson, AZ 85719, USA , H. Dominguez 99affiliation:
INAF-Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna,
Italy , N. Förster Schreiber 55affiliation: Max-Planck-Institut für
Extraterrestrische Physik (MPE), Postfach 1312, 85741 Garching, Germany , R.
Genzel 55affiliation: Max-Planck-Institut für Extraterrestrische Physik (MPE),
Postfach 1312, 85741 Garching, Germany , J-S Huang 1515affiliation: Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138,
USA , D. Lutz 55affiliation: Max-Planck-Institut für Extraterrestrische Physik
(MPE), Postfach 1312, 85741 Garching, Germany , R. Maiolino 1212affiliation:
INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte
Porzio Catone, Italy , B. Magnelli 55affiliation: Max-Planck-Institut für
Extraterrestrische Physik (MPE), Postfach 1312, 85741 Garching, Germany , G.E.
Morrison 1010affiliation: Institute for Astronomy, University of Hawaii,
Honolulu, HI 968226 1111affiliation: Canada-France-Hawaii Telescope, Kamuela,
HI 96743 , R. Nordon 55affiliation: Max-Planck-Institut für Extraterrestrische
Physik (MPE), Postfach 1312, 85741 Garching, Germany , A.M. Pérez García
66affiliation: Instituto de Astrofísica de Canarias, 38205 La Laguna, Spain ,
A. Poglitsch 55affiliation: Max-Planck-Institut für Extraterrestrische Physik
(MPE), Postfach 1312, 85741 Garching, Germany , P. Popesso 55affiliation: Max-
Planck-Institut für Extraterrestrische Physik (MPE), Postfach 1312, 85741
Garching, Germany , F. Pozzi 1212affiliation: INAF - Osservatorio Astronomico
di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy , L. Riguccini
11affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France
, G. Rodighiero 1313affiliation: Dipartimento di Astronomia, Università di
Padova, Vicolo dell’Osservatorio 3,35122 Padova, Italy , A. Saintonge
55affiliation: Max-Planck-Institut für Extraterrestrische Physik (MPE),
Postfach 1312, 85741 Garching, Germany , P. Santini 1212affiliation: INAF -
Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio
Catone, Italy , M. Sanchez-Portal 55affiliation: Max-Planck-Institut für
Extraterrestrische Physik (MPE), Postfach 1312, 85741 Garching, Germany , L.
Shao 55affiliation: Max-Planck-Institut für Extraterrestrische Physik (MPE),
Postfach 1312, 85741 Garching, Germany , E. Sturm 55affiliation: Max-Planck-
Institut für Extraterrestrische Physik (MPE), Postfach 1312, 85741 Garching,
Germany , L. Tacconi 55affiliation: Max-Planck-Institut für Extraterrestrische
Physik (MPE), Postfach 1312, 85741 Garching, Germany , I. Valtchanov
1414affiliation: Herschel Science Centre
###### Abstract
We present first insights into the far-IR properties for a sample of IRAC and
MIPS-24$\mu$m detected Lyman Break Galaxies (LBGs) at $z$ $\sim$ 3, as derived
from observations in the northern field of the Great Observatories Origins
Survey (GOODS-N) carried out with the PACS instrument on board the Herschel
Space Observatory. Although none of our galaxies are detected by Herschel, we
employ a stacking technique to construct, for the first time, the average
spectral energy distribution of infrared luminous LBGs from UV to radio
wavelengths. We derive a median IR luminosity of $L_{\rm IR}$ = 1.6 $\times$
1012 $L_{\odot}$, placing the population in the class of ultra luminous
infrared galaxies (ULIRGs). Complementing our study with existing multi-
wavelength data, we put constraints on the dust temperature of the population
and find that for their $L_{\rm IR}$, MIPS-LBGs are warmer than submm-luminous
galaxies while they fall in the locus of the $L_{\rm IR}$-$T\rm_{d}$ relation
of the local ULIRGs. This, along with estimates based on the average SED,
explains the marginal detection of LBGs in current sub-mm surveys and suggests
that these latter studies introduce a bias towards the detection of colder
ULIRGs in the high-$z$ universe, while missing high-$z$ ULIRGS with warmer
dust.
††slugcomment: to appear in ApJ
## 1 Introduction
Lyman break galaxies (LBGs), represent a significant fraction of z $\sim$ 3
star-forming galaxies that have been discovered up to now. Since LBGs are the
most common population at this epoch, investigating their properties is
crucial to enhance our view of the early universe. A large number of multi-
wavelength studies, spanning from X-rays to radio, have characterized the
nature of the population. One recent advance into our understanding of this
population was provided by the Infrared Array Camera (IRAC), the Multi-band
Imaging Photometer (MIPS) and the Infrared Spectrograph (IRS) on board the
Spitzer Space Telescope (Spitzer). In particular, IRAC measurements at rest-
frame near-infrared wavelengths (3.6-8$\mu$m) indicate that their stellar
masses are typically 109-1011M⊙ (e.g., Magdis et al. 2010a) while the
detection of a fraction ($\sim$20%) of these UV-selected galaxies at
MIPS-24$\mu$m unveiled the sub-population of the infrared luminous Lyman break
galaxies (Huang et al. 2005, Rigopoulou et al. 2006). Subsequent follow-up IRS
spectroscopy of MIPS detected LBGs, revealed prominent Polycyclic Aromatic
Hydrocarbon (PAH) features in their mid-IR spectrum, providing evidence for
the existence of significant amounts of dust and that they are mainly powered
by star-formation rather than an AGN (Huang et al. 2007, Rigopoulou et al.
2010).
The star-formation rate (SFR) of MIPS detected LBGs has been studied in detail
by Magdis et al. (2010b), where they found that UV, mid-IR and radio
measurements of the SFR for UV-selected galaxies are in good agreement,
indicating an average $<$SFR$>$=250$M_{\odot}yr^{-1}$. This SFR along with the
fact that these galaxies contain considerable amounts of dust as indicated by
their mid-IR emission, suggests that MIPS-LBGs should be detectable at sub-mm
wavelengths. Although the first attempts to detect far-IR counterparts were
not successful (e.g. Chapman et al. 2000), recently there has been a
significant progress in this front. Chapman et al. (2009) reported the
SCUBA-850$\mu$m sub-mm detection of Westphal-MM8 while Rigopoulou et al.
(2010) reported MAMBO 1.2mm detections of a further two LBGs, EGS-D49 and
EGS-M28 selected based on their strong MIPS 24$\mu$m emission. In addition to
those, two more lensed-LBGs have been detected in the sub-mm by Baker et al.
(2001) (MS1512-cB58) and by Siana et al. (2009) (Cosmic Eye). Despite these
promising detections though, the FIR properties of LBGs still remain elusive.
In this paper we use observations of the northern field of the Great
Observatories Origins Survey (GOODS-N) obtained by Photoconductor Array Camera
and Spectrometer (PACS) (Poglitsch et al. 2010) on board the Herschel Space
Observatory (Pilbratt et al. 2010). We complement our study with multi-
wavelength data and we aim to get a first insight into the FIR properties of a
sample of MIPS 24$\mu$m detected LBGs, construct the average spectral energy
distribution (SED) of the population, derive estimates of their dust
temperature ($T\rm_{d}$) and compare their properties to those of SMGs. The
Herschel data used in this study are part of the Herschel Science
Demonstration Phase of the PACS Evolutionary Probe (PEP, PI Lutz) program.
Throughout this paper we assume $\Omega_{\rm m}$=0.3, H0=71km sec-1 Mpc-1 and
$\Omega_{\Lambda}$=0.7
## 2 Sample Selection and Herschel Observations
GOODS-North was observed by PACS, at 100- and 160$\mu$m in the framework of
the Guaranteed Time Key Program, Pacs Evolutionary Probe (PEP). The area
covered by PACS was 10’$\times$15’ in a total exposure time of 30 hours and an
average integration time of $\sim$ 2.5sec per pixel. For source extraction two
independent methods were employed: the traditional blind source extraction
that was performed by using the Starfinder Point-spread function (PSF)-fitting
code (Diolaiti et al. 2000) and a guided extraction using 24$\mu$m priors. In
both methods, PSFs were extracted from the final science maps and were
subsequently used for a PSF-fitting photometric analysis. Flux uncertainties
were obtained by Monte-Carlo simulations, adding fake sources in the real
maps. The measured 1$\sigma$ noise is 1.00 mJy at 100$\mu$m and 1.90 mJy at
160$\mu$m. For more details we refer the reader to the appendix of Berta et
al. (2010).
Our sample of LBGs is identical to the one studied by Magdis et al. (2010b)
and we refer the reader to this paper for a detailed description. In brief we
consider 49 IRAC detected ([3.6]$\rm{}_{AB}<$25.0 or 0.36$\mu$Jy) LBGs in
GOODS-N, that have originally been optically selected (U,G,R, R$<$25.5) by
Steidel et al. (2003). All galaxies in our sample have spectroscopic redshift
(median z=2.95) derived by optical spectroscopy that has also been used to
determine the absence of AGN signatures (i.e., strong high ionization emission
lines) in their rest-frame UV spectrum (Steidel 2003, Shapley 2003). Among the
original sample, 9 LBGs are also detected at MIPS-24$\mu$m
(5$\sigma\sim$20$\mu$Jy) with a median 24$\mu$m flux density $f_{\rm
24}$=31.2$\mu$Jy. Henceforth, we will refer to these LBGs as the MIPS-LBGs
while to the remaining (40) as IRAC-LBGs. IRAC photometry for these objects is
provided in Magdis et al. (2010a), while an analysis of MIPS 24$\mu$m data is
described Magdis et al. (2010b). Matching our sample with the 100- and
160$\mu$m blind and prior catalogs returned no individual detections at the 3
sigma level (3mJy and $\sim$5mJy respectively). Since none of our LBGs is
individually detected we use the residual maps to do stacking. On stacking we
first considered the sample of MIPS-LBGs. We employed median stacking
analysis, cutting sub-images of the residual maps centered at the optical
position of each undetected LBG. The residual maps were created by Starfinder,
by subtracting all individually detected sources down to 3$\sigma$. To avoid
contaminating the stacked signal from residuals, we only added galaxies to the
stack if there were no bright sources within $\sim$4”(100$\mu$m) and
$\sim$6”(160$\mu$m). Then a stacked flux was measured in a manner similar to
the measurement of the detected PACS sources. To quantify the error of our
measurement we stacked at 9 random positions and repeated it 50.000 times. The
1$\sigma$ of the distribution of the derived fluxes was adopted as the
uncertainty of our measurement. Stacking at 100$\mu$m returned no detection
(f100=0.47$\pm$0.36 mJy) indicating a 3$\sigma$ upper of f${}_{100}<$1.1mJy,
while at 160$\mu$m we recovered a median flux density $f_{160}$=2.21 $\pm$
0.52mJy (S/N$\sim$4.2) for the MIPS-LBGs. The final 160$\mu$m stacked image is
shown in Figure 1 along with the distribution of the fluxes that correspond to
random stacked positions. This distribution indicates the probability to
recover a stacked flux density $f_{\rm 160}>$ 4.2mJy by chance, is
2.31$\times$10-5. Stacking the whole sample of LBGs, or just the IRAC LBGs
alone, returned no detection in either PACS band. For the IRAC LBGs we
determine a 3$\sigma$ upper limit of f${}_{100}<0.47$mJy and
f${}_{160}<0.79$mJy while the actual fluxes that correspond to the stacked
images are $f_{100}$=0.21$\pm$0.16 mJy and f160=0.32$\pm$ 0.26 mJy.
For the IRAC sample we also consider median stacked flux densities of
f1.1mm=0.41$\pm$0.11mJy and f1.4GHz=3.6$\pm$0.8$\mu$Jy as derived from
stacking at 1.1mm AzTEC and 1.4GHz VLA map, while for the MIPS sample we
consider a median stacked flux density of f1.4GHz=8.5$\pm$2.2$\mu$Jy and a
3$\sigma$ upper limit of f1.1mm=1.1mJy (Magdis et al. 2010b). We also add
optical (ground based UGR, ACS BViz) and near-IR (J,K) photometry provided by
the GOODS team.
## 3 Results and Discussion
### 3.1 Spectral Energy Distribution of LBGs
Using the multi-wavelength photometry (and the upper limits) described above
we attempt to construct the full, (rest-frame UV to radio) average SED of MIPS
selected LBGs. We fit the optical to near-IR part with model SEDs generated by
the Charlot and Bruzual 2007 (CB07) code and the mid-IR to radio with template
SEDs from the Chary & Elbaz 2001 (CE01) and Dale & Helou 2002 (DH02)
libraries. For the fitting procedure we allow for renormalization of the CE01
templates while for the DH02 models the luminosities are normalized as
described by Marcillac et al. (2006). Results based on the two methods are in
very close agreement indicating a median LIR=1.6($\pm$0.5)$\times$1012L⊙ and
that MIPS-LBGs belong to the class of Ultra-Luminous Infrared Galaxies
(ULIRGs). This luminosity translates to SFR$=275M_{\odot}yr^{-1}$ (Salpeter
IMF), in good agreement with the UV (250${}^{+35}_{-80}$M⊙yr-1) and radio
(280$\pm$85M⊙yr-1) SFR estimates presented in Magdis et al. (2010b). Adopting
a median stellar mass of MIPS LBGs M∗=7.9$\times 10^{10}\rm M_{\odot}$ (Magdis
et al. 2010a), we derive a specific star-formation rate of SSFR=3.5Gyr-1. This
value is lower than that derived for the IRAC-LBGs (4.3Gyr-1, Magdis et al.
2010a) but higher than that found for z$\sim$2 star-forming galaxies (2.5
Gyr-1, Daddi et al. 2007a). Finally, based on the best fit CE01 model, we
estimate a median $f_{\rm 850}$=1.36mJy, very close to the confusion limit of
current sub-mm surveys. We comment on this finding later on where we present a
comparison between LBGs and SMGs.
### 3.2 Indication of Warmer Dust in MIPS-LBGs
As we have already discussed, stacking IRAC-LBGs at 1.1mm has recovered a flux
density of $f_{1.1mm}$=0.41$\pm$0.1mJy while at 160$\mu$m there is no
detection. The situation is exactly the opposite for the MIPS-LBGs where
stacking at 1.1mm indicate no detection while at 160$\mu$m we recover a median
flux of $f_{160}$=2.21$\pm$0.52 mJy. This inversion could be interpreted as a
change in the shape of the SED of the two samples. Furthermore a detailed
study of IRAC-LBGs by Magdis et al. (2010b), indicated that the average IRAC
detected LBG is a Luminous Infrared Galaxy (LIRG) with $L_{\rm
IR}$=4.5($\pm$1.2)$\times$1011 $L_{\odot}$, less luminous than the current
sample of MIPS-LBGs. It has been observed that in the local universe ULIRGs
have warmer dust and their SEDs peak at shorter wavelengths than those of
LIRGs. Here we attempt to put constraints on the peak of the SED and the dust
temperature ($T_{\rm d}$) of LBGs and investigate these parameters as a
function of luminosity at z$\sim$3.
Although ideally such a task requires photometric points close to the peak of
the SED of a galaxy (which in this case translates into SPIRE data), here we
use PACS 160$\mu$m and AzTEC 1.1mm flux densities and upper limits, with the
advantage of sampling both sides around the peak of the SED. Using 3$\sigma$
upper limits for the $f_{160}$ of IRAC-LBGs could provide us with an upper
limit of the $T_{\rm d}$ of the sample while 3$\sigma$ upper limits for the
$f_{\rm 1.1mm}$ of the MIPS LBGs will yield an estimate of a lower limit of
the $T_{\rm d}$ of the sample.
We first employed Monte Carlo simulations to generate 10.000 combinations of
160$\mu$m and 1.1mm fluxes. For each of the two bands, the simulated fluxes
were generated following the flux distribution derived from the stacking
simulations and centered on the measured stacked flux. To derive the dust
temperature for each of the generated realizations, we used a single
temperature greybody fitting form,
$F_{\nu}\propto\nu^{3+\beta}/(exp(h\nu/kT_{d})-1)$, where $\beta$ is the dust
emissivity and Td is the effective or emission weighted dust temperature. We
fixed the $\beta$ value to 1.5 (Gordon et al. 2010) and considered the median
redshifts as representative for each sample (z=2.98 for IRAC-LBGs and z=2.92
for the MIPS-LBGs). The $T_{\rm d}$ for each realization was then obtained
from the best fit model that was derived based on the minimization of the
$\chi^{2}$ value. Since at 100$\mu$m (rest-frame 25$\mu$m) there is strong
contribution of Very Small Grains (VSGs) in the IR emission, $f_{100}$ upper
limits were not fitted but they were considered as a sanity check, i.e. that
the models do not violate the upper limits. Based on the distribution of the
obtained dust temperatures (Figure 3) we then derive an upper limit of $T_{\rm
d}$ = 38.2(39.3) K (IRAC-LBGs), and a lower limit of $T_{\rm d}$ = 40.1(38.9)
K (MIPS-LBGs) at a 2$\sigma$(3$\sigma$) confidence level. The best fit models
that correspond to the 2$\sigma$ temperature limits are depicted in Figure 4.
We note that adopting $\beta$ = 2.0 instead of 1.5, would result in a lower
dust temperature, by 2-3 K.
The above analysis indicates that the dust temperatures of the two samples are
different at a 2.6$\sigma$ confidence level, with MIPS-LBGs having warmer
dust. Since MIPS-LBGs and IRAC-LBGs are objects divided by a threshold in
their rest 6$\mu$m emission, which is due to warm, very small grains and/or
PAHs, if the presence of warm small grains correlates with the presence of
warm larger grains then our result would be object of selection bias. To
explore this possibility, we consider a sample of local ULIRGs with ISOCAM
6.75$\mu$m and IRAS 60- and 100$\mu$m data (Elbaz et al. 2002). Plotting the
$f_{\rm 100}/f_{\rm 60}$ which is a dust temperature indicator, against the
rest-frame 6.75$\mu$m luminosity (that corresponds to 24$\mu$m emission of
z$\sim$3), reveals that there is no obvious correlation between 6$\mu$m
luminosity and $T_{\rm d}$. Hence, assuming that this applies for galaxies at
z$\sim$3, it seems that our result is not subject to a selection bias. On the
other hand, as we have already shown, MIPS-LBGs are more infrared-luminous
than IRAC-LBGs with corresponding infrared luminosities
LIR=1.6($\pm$0.5)$\times$1012L⊙ for MIPS- and LIR=4.5($\pm$0.5)$\times$1011L⊙
for IRAC-LBGs. Therefore, it appears that we observe for the first time the
general $L_{\rm IR}$$-$$T_{\rm d}$ trend seen in the local universe, (with
more luminous galaxies having warmer dust), for UV-selected galaxies at $z$
$\sim$ 3.
### 3.3 Comparison with SMGs
Considering the large SFRs and substantial dust reddening that are inferred
for some LBGs, and the fact that their spectra exhibit characteristics of
local starbursts (Pettini et al. 2001), it is somewhat surprising that there
are only few examples of direct sub-millimeter detection for these galaxies.
MIPS-LBGs are the most rapidly star-forming, most luminous, and dustiest
galaxies among the high redshift UV-selected population, and therefore are the
best candidates for having far-IR emission that could be detected in current
sub-mm surveys. In what follows we discuss how the current data can shed light
on the mystery of this marginal detection of LBGs in sub-mm bands. Based on
the average SED of MIPS-LBGs that we presented here, we predict that the flux
density of the MIPS-LBGs emitted at 850$\mu$m is $f_{\rm 850}$=1.1-1.5mJy,
just below the current confusion limit. It could therefore be suggested that
MIPS-LBGs provide a link between SMGs and typical UV selected LBGs that are
faint in the IR. It has also been speculated that the fact that LBGs are not
detected in the sub-mm bands is due to warmer dust compared to that of SMGs
(Kaviani et al. 2003). Although to test this one needs both robust
measurements of the $T_{\rm d}$ (based on observations that probe the peak of
the SED of the galaxies) and to have a large sample of LBGs, the upper limits
that we derived in this study can provide a first insight.
Figure 5 compares dust temperature versus infrared luminosity for the MIPS and
IRAC LBGs as derived from this study. For comparison we also consider the
large compilation of $z$ $\sim$ 2 SMGs by Chapman et al. 2005, and the
local/intermediate-z samples of ULIRGs presented by Clements et al. (2010) and
Yang et al. (2007). In all these studies, the $T_{\rm d}$ was measured in
manner that is similar to ours, fitting modified black-body models with fixed
$\beta$=1.5 to the FIR photometric points of each sample, hence the comparison
between these studies and our sample is meaningful. We also plot, the
$3\sigma$ envelope of the $L_{\rm IR}$-$T_{\rm d}$ relation for local infrared
galaxies in SDSS (Hwang et al. 2010 in prep).
It is evident that for the $L_{\rm IR}$ of the MIPS-LBGs, the bulk of SMGs are
considerably colder, while MIPS-LBGs fall in the locus of the local ULIRGs and
are within the scatter observed in local galaxies. Based on modified black
body models, we also compute tracks of constant 850$\mu$m flux density for
galaxies at $z$=3, close to the confusion/detection limit of current sub-mm
surveys ($f_{\rm 850}$=1mJy and $f_{\rm 850}$=2mJy). MIPS-LBGs lie in between
the two tracks, indicating that a typical MIPS detected LBG emits at 1-2 mJy
level at the sub-mm bands, in excellent agreement with the median $f_{\rm
850}$ flux density derived from the average SED of the population. This
explains the small overlap between the LBGs and SMGs found in previous
studies. For a given star formation rate and optical depth for dust absorption
(and hence, for a given bolometric far-infrared luminosity), the warmer dust
temperature in LBGs leads to fainter sub-mm emission compared to that found in
typical SMGs. This finding also suggests that star-formation in LBGs is
spatially compact. On the other hand, recent studies based on minimal
excitation (CO J=1 $\rightarrow$ 0) emission that traces the bulk of the
metal-rich molecular gas provide evidence of a more extended distribution of
star-formation in SMGs (e.g. Ivison et al. 2010), that could explain their
colder dust component. Generalizing our results, we argue that ULIRGs with
warmer dust than that of SMGs exist in the high-$z$ universe and would not be
selected in current sub-mm surveys. This is further enforced by the study of
IRAC selected ULIRGs at $z\sim$ 2, (Magdis et al. 2010 in prep), where based
on Herschel PACS/SPIRE observations, we find that 60$\%$ of the sample would
be missed by current sub-mm surveys.
We have presented a first glimpse into the far-IR properties of $z\sim$ 3
LBGs, demonstrating the power of Herschel in the study even of high-z UV-
selected galaxies and its ability to address issues that we were unable to
resolve in the pre-Herschel era. We aim to further extent this study,
complementing PACS with SPIRE data that probe the peak of the SED of $z$
$\sim$ 3 galaxies and taking advantage of the Super-Deep and Ultra-Deep
surveys in GOODS-N and GOODS-S respectively as part of the Herschel-GOODS
program (PI D. Elbaz).
PACS has been developed by a consortium of institutes led by MPE (Germany) and
including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, OAMP (France); MPIA
(Germany); IFSI, OAP/AOT, OAA/CAISMI, LENS, SISSA (Italy); IAC (Spain). This
development has been supported by the funding agencies BMVIT (Austria), ESA-
PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI (Italy), and CICT/MCT
(Spain).
## References
* (1) Berta, S., et al. 2010, A$\&$A in press
* (2) Chapman et al. 2000, MNRAS, 319, 318
* (3) Chapman, S. C., Blain, A. W., Smail, Ian, Ivison, R. J. 2005, ApJ, 622, 772
* (4) Chapman, S. C. & Casey, C. M. 2009, MNRAS, 398, 1615
* (5) Chary & Elbaz 2001, ApJ,556,562
* (6) Clements, D. L., Dunne, L., Eales, S., 2010, MNRAS, 403, 274
* (7) Daddi, E., Dickinson, M., Morrison, G., Chary, R., Cimatti, A., Elbaz, D., Frayer, D., Renzini, A., et al. 2007, ApJ, 670, 156
* (8) Dale, Daniel A. & Helou, George 2002, ApJ, 576, 159
* (9) Diolaiti, E., Bendinelli, O., Bonaccini, D., Close, L., Currie, D., Parmeggiani, G., 2000, A$\&$AS, 147, 335
* (10) Elbaz, D., Cesarsky, C. J. Chanial, P., Aussel, H., Franceschini, A., Fadda, D., Chary, R. R., 2002, A$\&$A, 84, 848
* (11) Gordon, G. et al., 2010, A$\&$A in press
* (12) Huang, J.-S., et al. 2005, ApJ, 634, 137
* (13) Huang, J.-S., Rigopoulou, D., Papovich, C., Ashby, M. L. N., Willner, S. P., Ivison, R., Laird, E. S., Webb, T., et al. 2007, ApJ, 660, 69
* (14) R. J., Ivison, Ian Smail,P. P.,Papadopoulos,I.,Wold,J., Richard, A. M., Swinbank, J.-P.,Kneib, F.N.Owen, 2010, MNRAS, 404, 198
* (15) Kaviani, A., Haehnelt, M. G., Kauffmann, G., 2003, MNRAS, 340, 739
* (16) Magdis 2010a, G. E., Rigopoulou, D., Huang, J.-S., Fazio, G. G. 2010, MNRAS, 401, 1521
* (17) Magdis 2010b, G. E., Elbaz, D., Daddi, E., Morrison, G. E., Dickinson, M., Rigopoulou, D., Gobat, R., Hwang, H. S., 2010, ApJ, 714, 1740
* (18) Magdis, G. E., Rigopoulou, D., Huang, J.-S., Fazio, G. G., Willner, S. P., Ashby, M. L. N. 2008, MNRAS, 386, 11
* (19) Marcillac, D., Elbaz, D., Chary, R. R., Dickinson, M., Galliano, F., Morrison, G. 2006, 2006A$\&$A, 451, 57
* (20) Papovich, Casey, Dickinson, Mark, Ferguson, Henry C., 2001 ApJ, 559, 620
* (21) Pilbratt, G., et al. 2010, A$\&$A, in press (astro-ph/1005.5331)
* (22) Poglistch, A., et al. 2010, A$\&$A, in press (astro-ph/1005.1487)
* (23) Rigopoulou, D., Huang, J.-S., Papovich, C., Ashby, M. L. N., Barmby, P., Shu, C., Bundy, K., Egami, E.; Magdis, G., et al. 2006, ApJ, 648, 81
* (24) Rigopoulou et al. 2010 (submitted)
* (25) Shapley, Alice E., Steidel, Charles C., Adelberger, Kurt L., Dickinson, Mark, Giavalisco, Mauro, Pettini, Max, 2001, ApJ, 562, 95
* (26) Steidel, Charles C., Adelberger, Kurt L., Shapley, Alice E., Pettini, Max, Dickinson, Mark, Giavalisco, Mauro 2003ApJ, 592, 728S
* (27) Yang, M., Greve, T. R., Dowell, C. D.,Borys, 2007, ApJ, 660, 1198
* (28) Younger, J. D., Omont, A., Fiolet, N., Huang,J.-S., Fazio, G. G.; Lai, K., Polletta, M., Rigopoulou, D., Zylka,R, 2009, MNRAS, 394, 1685
Figure 1: Left) PACS-160$\mu$m stacked image (130”$\times$130”) of 9
MIPS-24$\mu$m detected LBGs in GOODS-N. Right) Stacking simulations at
160$\mu$m. Distribution of the measured fluxes derived from 50000 stackings at
9 random positions along with the best Gaussian fit (rms=0.52mJy). The red
line denotes the flux measurement of the stacking at the position MIPS-LBGs
indicating a $\sim$4.2$\sigma$ detection. Figure 2: Rest–frame average SED of
z$\sim$3, MIPS-LBGs. For the SED we use the median UGR+BViJK+IRAC+MIPS24
photometry of MIPS-LBGs, and the values (or upper limits) derived from
stacking 100$\mu$m, 160$\mu$m AzTEC and radio. Upper limits are indicated with
green squares. The rest-frame UV-NIR portion of the data is overlaid with the
best fit CB07 model (orange line), while the mid-IR to radio is shown with the
best–fit CE01 model (blue line). Black diamonds show the fluxes predicted from
the best fit CE01 model. Figure 3: Distribution of the derived $T_{\rm d}$ for
the IRAC- and MIPS-LBGs based on MC simulations. The two distributions begin
to overlap at a 2.6$\sigma$ confidence level. Blue lines illustrate a gaussian
fit to the left (MIPS-LBGs) and right (IRAC-LBGs) part of the distribution.
Figure 4: Derivation of lower and upper $T_{\rm d}$ limit estimates for MIPS-
and IRAC-LBGs respectively. Filled black squares indicate the median stacked
flux density of MIPS-LBGs at 160$\mu$m and the 3$\sigma$ upper limits of the
flux density at 1.1mm as estimated from stacking. The solid red line shows the
best fit modified black body model that corresponds to the 2$\sigma$ $T_{\rm
d}$ lower limit. For the model we assume fixed $\beta$=1.5. Similarly, filled
green circle indicate the median flux density of IRAC-LBGs at 1.1mm from
Magdis et al. 2010b and the 3$\sigma$ flux upper limit at 160$\mu$m as derived
from stacking in the present study. The solid blue line shows the best fit
modified black body model that corresponds to the 2$\sigma$ Td upper limit.
The open black squares and green circle indicate 3$\sigma$ flux upper limit at
100$\mu$m as derived from stacking for MIPS- and IRAC-LBGs respectively.
Vertical black dotted and dashed-dotted lines indicate the wavelength where
the far-IR emission peaks for MIPS-LBGs and IRAC-LBGS. Figure 5: The $L_{\rm
IR}-T_{\rm d}$ relation for MIPS and IRAC LBGs (red and black circles
respectively). We note that these values correspond to 3$\sigma$ upper and
lower $T_{\rm d}$ limits for IRAC and MIPS LBGs. Included are results for
0$<$z$<$0.98 ULIRGs (blue filled triangles, Clements et al. 2010, Yang et al.
2007). The green shaded region depicts the loci of high-z SMGs by Chapman et
al. (2005), while the orange shaded area shows the 2$\sigma$ envelope of the
$L_{\rm IR}-T_{\rm d}$ relation for local IR galaxies in SDSS (excluding AGNs)
adopted from Hwang et al. (2010 in prep). Black dashed lines represent tracks
of constant flux density at 850$\mu$m ($f_{\rm 850}$=1mJy and $f_{\rm 850}$=2
mJy) for galaxies at z=3. Objects at z=3 with higher $f_{\rm 850}$, lie on the
right of the lines. This plot demonstrates that for a given $L_{\rm IR}$, the
bulk of SMGs are considerably colder than MIPS-LBGs which fall in the locus of
the local ULIRGs. We also note that for their $T_{\rm d}$ and $L_{\rm IR}$,
MIPS-LBGs fall in between the 1- and 2 mJy constant flux density at 850$\mu$m,
indicating that they would be missed by current ground-based sub-mm surveys.
Finally, there is a hint that the LBGs at $z\sim$3 follow the general trend
observed in the local universe, with more luminous galaxies having warmer
dust.
|
arxiv-papers
| 2010-07-22T10:15:26 |
2024-09-04T02:49:11.829421
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G.E. Magdis, D. Elbaz, H.S. Hwang, E. Daddi, D. Rigopoulou, B.\n Altieri, P. Andreani, H. Aussel, S. Berta, A. Cava, A. Bongiovanni, J. Cepa,\n A. Cimatti, M. Dickinson, H. Dominguez, N. F\\\"orster Schreiber, R. Genzel,\n J.-S. Huang, D. Lutz, R. Maiolino, B. Magnelli, G.E. Morrison, R. Nordon,\n A.M. P\\'erez Garc\\'ia, A. Poglitsch, P. Popesso, F. Pozzi, L. Riguccini, G.\n Rodighiero, A. Saintonge, P. Santini, M. Sanchez-Portal, L. Shao, E. Sturm,\n L. Tacconi, I. Valtchanov",
"submitter": "Georgios Magdis E",
"url": "https://arxiv.org/abs/1007.3846"
}
|
1007.3858
|
# CHR(PRISM)-based Probabilistic Logic Learning
JON SNEYERS WANNES MEERT JOOST VENNEKENS
Dept. of Computer Science K.U.Leuven Belgium
{jon.sneyers,wannes.meert,joost.vennekens}@cs.kuleuven.be YOSHITAKA KAMEYA and
TAISUKE SATO
Tokyo Institute of Technology Japan
{kameya,sato}@mi.cs.titech.ac.jp
(21 April 2010; 20 March 2010)
###### Abstract
PRISM is an extension of Prolog with probabilistic predicates and built-in
support for expectation-maximization learning. Constraint Handling Rules (CHR)
is a high-level programming language based on multi-headed multiset rewrite
rules.
In this paper, we introduce a new probabilistic logic formalism, called
CHRiSM, based on a combination of CHR and PRISM. It can be used for high-level
rapid prototyping of complex statistical models by means of “chance rules”.
The underlying PRISM system can then be used for several probabilistic
inference tasks, including probability computation and parameter learning. We
define the CHRiSM language in terms of syntax and operational semantics, and
illustrate it with examples. We define the notion of ambiguous programs and
define a distribution semantics for unambiguous programs. Next, we describe an
implementation of CHRiSM, based on CHR(PRISM). We discuss the relation between
CHRiSM and other probabilistic logic programming languages, in particular
PCHR. Finally, we identify potential application domains.
## 1 Introduction
Constraint Handling Rules [Frühwirth (2009), Sneyers et al. (2010)] is a high-
level language extension based on multi-headed rules. Originally, CHR was
designed as a special-purpose language to implement constraint solvers, but in
recent years it has matured into a general purpose programming language. Being
a language _extension_ , CHR is implemented on top of an existing programming
language, which is called the _host language_. An implementation of CHR in
host language $X$ is called CHR($X$). For instance, several CHR(Prolog)
systems are available.
PRISM (PRogramming In Statistical Modeling) is a probabilistic extension of
Prolog [Sato (2008)]. It supports several probabilistic inference tasks,
including sampling, probability computation, and expectation-maximization (EM)
learning.
In this paper, we construct a new formalism, called CHRiSM — short for CHance
Rules induce Statistical Models. It is based on CHR(PRISM) and it combines the
advantages of CHR and those of PRISM. Like CHR, CHRiSM is a very concise and
expressive programming language. Like PRISM, CHRiSM has built-in support for
several probabilistic inference tasks. Furthermore, since CHRiSM is
implemented as a translation to CHR(PRISM) — which itself is translated to
PRISM and ultimately Prolog — CHRiSM rules can be freely mixed with CHR rules
and Prolog clauses.
This paper is based on an earlier workshop paper [Sneyers et al. (2009)].
Although it is mostly self-contained, some familiarity with CHR and PRISM is
recommended.
We use $\uplus$ for multiset union, $\subsetpluseq$ for multiset subset, and
$\bar{\exists}_{A}B$ to denote $\exists x_{1},\ldots,x_{n}:B$, with
$\\{x_{1},\ldots,x_{n}\\}=\mathit{vars}(B)\setminus\mathit{vars}(A)$, where
$\mathit{vars}(A)$ are the (free) variables in $A$; if $A$ is omitted it is
empty (so $\bar{\exists}B$ denotes the existential closure of $B$).
## 2 Syntax and Semantics of CHRiSM
In this section we define CHRiSM. The syntax is defined in Section 2.1 and the
(abstract) operational semantics is defined in Section 2.2. Finally, in
Section 2.3 the notion of _observations_ is introduced.
### 2.1 Syntax and Informal Semantics
A CHRiSM program $\mathcal{P}$ consists of a sequence of _chance rules_.
Chance rules rewrite a multiset $\mathbb{S}$ of data elements, which are
called (CHRiSM) _constraints_ (mostly for historical reasons). Syntactically,
a constraint c(X1,..,Xn) looks like a Prolog predicate: it has a functor c of
some arity $n$ and arguments X1,..,Xn which are Prolog terms. The multiset
$\mathbb{S}$ of constraints is called the _constraint store_ or just _store_.
The initial store is called the _query_ or _goal_ , the final store (obtained
by exhaustive rule application) is called the _answer_ or _result_.
#### Chance rules.
A chance rule is of the following form:
P ?? Hk \ Hr <=> G | B.
where P is a probability expression (as defined below), Hk is a conjunction of (kept head) constraints, Hr is a conjunction of (removed head) constraints, G is a guard condition (a Prolog goal to be satisfied), and B is the body of the rule. If Hk is empty, the rule is called a _simplification_ rule and the backslash is omitted; if Hr is empty, the rule is called a _propagation_ rule and it is written as “P ?? Hk ==> G | B”. If both Hk and Hr are non-empty, the rule is called a _simpagation_ rule. The guard G is optional; if it is removed, the “|” is also removed. The body B is recursively defined as a conjunction of CHRiSM constraints, Prolog goals, and probabilistic disjunctions (as defined below) of bodies.
Intuitively, the meaning of a chance rule is as follows: If the constraint
store $\mathbb{S}$ contains elements that match with the head of the rule
(i.e. if there is a matching substitution $\theta$ such that
$(\theta(\mathtt{Hk})\uplus\theta(\mathtt{Hr}))\subsetpluseq\mathbb{S}$), and
furthermore, the guard G is satisfied, then we can consider rule application.
The subset of $\mathbb{S}$ that corresponds to the head of the rule is called
a rule _instance_. Depending on the probability expression P, the rule
instance is either ignored or it actually leads to a rule application. Every
rule instance may only be considered once.
Rule application has the following effects: the constraints matching Hr are
removed from the constraint store, and then the body B is executed, that is,
Prolog goals are called and CHRiSM constraints are added into the store.
#### Probability expressions.
A probability expression P is one of the following:
* •
A number from 0 to 1, indicating the probability that the rule fires. A rule
of the form 1 ?? ... corresponds to a regular CHR rule; the “1 ??” may be
dropped. A rule of the form 0 ?? ... is never applied.
* •
An expression of the form eval(E), where E is an arithmetic expression (in
Prolog syntax). It should be ground when the rule is considered (otherwise a
runtime instantiation error occurs). The evaluated expression indicates the
probability that the rule fires.
* •
An experiment name. This is a Prolog term which should be ground when the rule
is considered. The probability distribution is unknown. Initially, unknown
probabilities are set to a uniform distribution (0.5 in the case of rule
probabilities). They can be changed manually using PRISM’s set_sw/2 builtin,
or automatically using PRISM’s EM-learning algorithm.
The arguments of the experiment name can include _conditions_ , which are of
the form “cond C”. Such arguments are evaluated at runtime and replaced by
either “yes” or “no”, depending on whether call(C) succeeded or failed.
These conditions are just syntactic sugar, so we may ignore them w.l.o.g. For example, the rule “foo(cond A>B) ?? c(A,B) <=> d” is syntactic sugar for “foo(X) ?? c(A,B) <=> (A>B -> X=yes ; X=no) | d”.
* •
Omitted (so the rule starts with “??”): this is a shorthand for a fresh zero-
arity experiment name.
#### Probabilistic disjunction.
The body B of a CHRiSM rule may contain probabilistic disjunctions. There are
two styles:
* •
LPAD-style probabilistic disjunctions [Vennekens et al. (2004)] of the form
“D1:P1 ; ... ; Dn:Pn”, where a disjunct D$i$ is chosen with probability P$i$.
The probabilities should sum to 1 (otherwise a compile-time error occurs).
* •
CHRiSM-style probabilistic disjunctions of the form “P ?? D1 ; ... ; Dn”,
where P is an experiment name determining the probability distribution.
The LPAD-style probabilistic disjunctions can be seen as a special case of
CHRiSM-style disjunctions for which the experiment name is implicit and the
distribution is given and fixed. Unlike CHR∨ disjunctions, which create a
choice point, both kinds of probabilistic disjunctions are _committed-choice_
: once a disjunct is chosen, the choice is not undone later. However, when
later on in a derivation the same experiment is sampled again, the result can
of course be different.
### 2.2 Operational Semantics
1\. Fail.
$\langle\\{b\\}\uplus\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.75pt\hbox{\raise
6.00006pt\hbox{\scriptsize$1$}}}\hskip-1.75pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \mbox{fail}$
where $b$ is a built-in (Prolog) constraint and
$\mathcal{D}_{\mathcal{H}}\models\lnot\bar{\exists}(\mathbb{B}\land b)$. 2\.
Solve.
$\langle\\{b\\}\uplus\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.75pt\hbox{\raise
6.00006pt\hbox{\scriptsize$1$}}}\hskip-1.75pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\langle\mathbb{G},\mathbb{S},b\land\mathbb{B},\mathbb{T}\rangle_{n}$
where $b$ is a built-in (Prolog) constraint and
$\mathcal{D}_{\mathcal{H}}\models\bar{\exists}(\mathbb{B}\land b)$. 3\.
Introduce.
$\langle\\{c\\}\uplus\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.75pt\hbox{\raise
6.00006pt\hbox{\scriptsize$1$}}}\hskip-1.75pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\langle\mathbb{G},\\{c\\#n\\}\cup\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n+1}$
where $c$ is a CHRiSM constraint. 4\. Probabilistic-Choice.
$\langle\\{d\\}\uplus\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-2.43619pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{i}$}}}\hskip-2.43619pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\langle\\{d_{i}\\}\uplus\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}$
where $d$ is a probabilistic disjunction of the form $d_{1}$:$p_{1}$ ;
$\ldots$ ; $d_{k}$:$p_{k}$ or of the form P ?? $d_{1}$ ; $\ldots$ ; $d_{k}$,
where the probability distribution given by P assigns the probability $p_{i}$
to the disjunct $d_{i}$. 5\. Maybe-Apply. $\langle\mathbb{G},H_{1}\uplus
H_{2}\uplus\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}5;3~{}5;3~{}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-6.89977pt\hbox{\raise
6.00006pt\hbox{\scriptsize$1-p$}}}\hskip-6.89977pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\langle\mathbb{G},H_{1}\uplus
H_{2}\uplus\mathbb{S},\mathbb{B},\mathbb{T}\cup\\{h\\}\rangle_{n}$
foo $\langle\mathbb{G},H_{1}\uplus
H_{2}\uplus\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\ \hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.76094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p$}}}\hskip-1.76094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \langle
B\uplus\mathbb{G},H_{1}\uplus\mathbb{S},\theta\land\mathbb{B},\mathbb{T}\cup\\{h\\}\rangle_{n}$
where the $r$-th rule of $\mathcal{P}$ is of the form P ?? $H^{\prime}_{1}\ \backslash\ H^{\prime}_{2}$ <=> G | B,
$\theta$ is a matching substitution such that
$\mathit{chr}(H_{1})=\theta(H^{\prime}_{1})$ and
$\mathit{chr}(H_{2})=\theta(H^{\prime}_{2})$,
$h=(r,\mathit{id}(H_{1}),\mathit{id}(H_{2}))\not\in\mathbb{T}$, and
$\mathcal{D}_{\mathcal{H}}\models\mathbb{B}\rightarrow\bar{\exists}_{\mathbb{B}}(\theta\wedge
G)$. If P is a number, then $p=\mathtt{P}$. Otherwise $p$ is the probability
assigned to the success branch of P.
Figure 1: Transition relation $\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$ $\mathcal{P}$ of the abstract operational semantics
$\maybebm{\omega_{t}^{\mathtt{??}}}$ of CHRiSM.
The abstract operational semantics of a CHRiSM program $\mathcal{P}$ is given
by a state-transition system that resembles the abstract operational semantics
$\maybebm{\omega_{t}}$ of CHR [Sneyers et al. (2010)]. The execution states
are defined analogously, except that we additionally define a unique failed
execution state which is denoted by “fail” (because we don’t want to
distinguish between different failed states). We use the symbol
$\maybebm{\omega_{t}^{\mathtt{??}}}$ to refer to the abstract operational
semantics of CHRiSM.
###### Definition 2.1 (identified constraint)
An _identified_ constraint $c\\#i$ is a CHRiSM constraint $c$ associated with
some unique integer $i$. This number serves to differentiate between copies of
the same constraint. We introduce the functions $\mathit{chr}(c\\#i)=c$ and
$\mathit{id}(c\\#i)=i$, and extend them to sequences and sets in the obvious
manner, e.g., $\mathit{id}(S)=\\{i|c\\#i\in S\\}$.
###### Definition 2.2 (execution state)
An _execution state_ $\sigma$ is a tuple
$\langle\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}$. The _goal_
$\mathbb{G}$ is a multiset of constraints to be rewritten to solved form. The
_store_ $\mathbb{S}$ is a set of _identified_ constraints that can be matched
with rules in the program $\mathcal{P}$. Note that $\mathit{chr}(\mathbb{S})$
is a multiset although $\mathbb{S}$ is a set. The _built-in store_
$\mathbb{B}$ is the conjunction of all Prolog goals that have been called so
far. The _history_ $\mathbb{T}$ is a set of tuples, each recording the
identifiers of the CHRiSM constraints that fired a rule and the rule number.
The history is used to prevent trivial non-termination: a rule instance is
allowed to be considered only once. Finally, the counter $n\in\mathbb{N}$
represents the next free identifier.
We use $\sigma,\sigma_{0},\sigma_{1},\ldots$ to denote execution states and
$\Sigma^{\mbox{chr}}$ to denote the set of all execution states. We use
$\mathcal{D}_{\mathcal{H}}$ to denote the theory defining the host language
(Prolog) built-ins and predicates used in the CHRiSM program. For a given CHR
program $\mathcal{P}$, the transitions are defined by the binary relation
$\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip 0.0pt\hbox{\raise
6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\subset\Sigma^{\mbox{chr}}\times\Sigma^{\mbox{chr}}$ shown in Figure
1. Every transition is annotated with a probability.
Execution proceeds by exhaustively applying the transition rules, starting
from an initial state (root) of the form $\langle
Q,\emptyset,\mbox{true},\emptyset\rangle_{0}$ and performing a random walk in
the directed acyclic graph defined by the transition relation $\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$ $\mathcal{P}$ , until a leaf node is reached, which is called a
final state. We consider only terminating programs (finite transition graphs).
Given a path from an initial state to the state $\sigma$, we define the
probability of $\sigma$ to be the product of the probabilities along the path.
We use $\sigma_{0}\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.76094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p$}}}\hskip-1.76094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\hskip-1.99997pt\hbox{\raise 3.00003pt\hbox{\scriptsize$*$}\hskip
3.00003pt}\sigma_{k}$ to denote a series of $k\geq 0$ transitions
$\sigma_{0}\ \hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{1}$}}}\hskip-2.74094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \sigma_{1}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{2}$}}}\hskip-2.74094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \sigma_{2}\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{3}$}}}\hskip-2.74094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \ldots\
\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-2.84302pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{k}$}}}\hskip-2.84302pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \sigma_{k}$
where $p=\prod_{i=1}^{k}p_{i}$ if $k>0$ and $p=1$ otherwise. If $\sigma_{0}$
is an initial state and $\sigma_{k}$ is a final state, then we call such a
series of transitions a _derivation_ of probability $p$. We define a function
$\mathit{prob}$ to give the probability of a derivation:
$\mathit{prob}(\sigma_{0}\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.76094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p$}}}\hskip-1.76094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\hskip-1.99997pt\hbox{\raise 3.00003pt\hbox{\scriptsize$*$}\hskip
3.00003pt}\sigma_{k})=p$.
Note that if all rule probabilities are 1 and the program contains no
probabilistic disjunctions — i.e. if the CHRiSM program is actually just a
regular CHR program — then the $\maybebm{\omega_{t}^{\mathtt{??}}}$ semantics
boils down to the $\maybebm{\omega_{t}}$ semantics of CHR.
### 2.3 Full and Partial Observations
A full observation Q <==> A denotes that there exist a series of probabilistic
choices such that a derivation starting with query Q results in the answer A.
A partial observation Q ===> A denotes that an answer for query Q contains at
least A: in other words, Q ===> A holds iff Q <==> B with A $\subsetpluseq$ B.
###### Definition 2.3 (observation)
A _full observation_ is of the form Q <==> A, where Q and A are conjunctions
of constraints. Given a program $\mathcal{P}$, a full observation refers to
derivations of the form
$\langle Q,\emptyset,\mbox{true},\emptyset\rangle_{0}\ \hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip-1.76094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p$}}}\hskip-1.76094pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\hskip-1.99997pt\hbox{\raise 3.00003pt\hbox{\scriptsize$*$}\hskip
3.00003pt}\ \langle\emptyset,A^{\prime},\mathbb{B},\mathbb{T}\rangle_{n}\hskip
10.00002pt\not\hskip-10.00002pt\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip 0.0pt\hbox{\raise
6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt$
such that $\mathtt{A}=\mathit{chr}(A^{\prime})$. A _partial observation_ is of
the form Q ===> A. It refers to derivations of the above form, such that
$\mathtt{A}\subsetpluseq\mathit{chr}(A^{\prime})$.
We also allow “negated” CHRiSM constraints in the right hand side:
Q ===> A,$\sim$N is a shorthand for Q <==> B with A $\subsetpluseq$ B and N
$\not\subsetpluseq$ B $\setminus$ A.
The following PRISM built-ins can be used to query a CHRiSM program:
* •
sample Q : probabilistically execute the query Q;
* •
prob Q <==> A : compute the probability that Q <==> A holds, i.e. the chance
that the choices are such that query Q results in answer A;
* •
prob Q ===> A : compute the probability that an answer for Q contains A;
* •
learn(L) : perform EM-learning based on a list L of observations
In observation lists, the syntax “$n$ times $X$” or “count($X$,$n$)” can be
used to denote that observation $X$ occurred $n$ times. This is simply a
shorthand for repeating the same observation ($X$) a number of times ($n$).
## 3 Example programs
As a first toy example, consider the following CHRiSM program for tossing a
coin:
toss <=> head:0.5 ; tail:0.5.
The query “sample toss” results in “head” or “tail”, with 50% chance each. The
query “sample toss,toss” has four possible outcomes, each with 25% chance:
“head,head”, “head,tail”, “tail,head”, and “tail,tail”.
Note that observations are not sensitive to the order in which the result is
given. As a result, the query “prob toss,toss <==> head,tail” returns a
probability of 50%, because the outcome “tail,head” also matches the
observation.
### 3.1 Rock-paper-scissors
Consider the following CHRiSM program simulating “rock-paper-scissors”
players:
player(P) <=> choice(P) ?? rock(P) ; scissors(P) ; paper(P).
rock(P1), scissors(P2) ==> winner(P1).
scissors(P1), paper(P2) ==> winner(P1).
paper(P1), rock(P2) ==> winner(P1).
We assume that each player has his own fixed probability distribution for
choosing between rock, scissors, and paper. This is denoted by using choice(P)
as the probability expression for the choice in the first rule: the
probability distribution depends on the value of P and thus every player has
his own distribution. However, these distributions are not known to us. By
default, the unknown probability distributions for, say, tom and jon are
therefore both set to the uniform distribution, which implies, among other
things, that each player should win one third of the time (cf. Figure 2). Here
is a possible interaction:
?- sample player(tom),player(jon)player(tom),player(jon) <==>
rock(jon),rock(tom).?- sample player(tom),player(jon)player(tom),player(jon)
<==> rock(jon),paper(tom),winner(tom).?- prob player(tom),player(jon) ===>
winner(tom)Probability of player(tom),player(jon)===>winner(tom) is: 0.333333
player(tom)
---
player(jon)
$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$ rock(tom)
---
player(jon)
$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$
scissors(tom)
---
player(jon)
$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$ paper(tom)
---
player(jon)
$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$$\scriptstyle{1\\!/3}$ rock(tom)
---
scissors(jon)
$\scriptstyle{1}$ rock(tom)
---
paper(jon)
$\scriptstyle{1}$ scissors(tom)
---
rock(jon)
$\scriptstyle{1}$ scissors(tom)
---
paper(jon)
$\scriptstyle{1}$ paper(tom)
---
rock(jon)
$\scriptstyle{1}$ paper(tom)
---
scissors(jon)
$\scriptstyle{1}$ rock(tom)
---
rock(jon)
...
---
winner(tom)
...
---
winner(jon)
...
---
winner(jon)
scissors(tom)
---
scissors(jon)
...
---
winner(tom)
...
---
winner(tom)
...
---
winner(jon)
paper(tom)
---
paper(jon)
Figure 2: A derivation tree for the rock-paper-scissors example.
Now suppose that we watch 100 games, and want to use our observations to
obtain a better model of the playing style of both players. If we can fully
observe these games, then this is easy: we can just use the frequency with
which each player played rock, paper or scissors as an estimate for the
probability of him making that particular move. The situation becomes more
difficult, however, if the games are only partly observable. For instance,
suppose that we do not know which moves the players made, but are only told
the final scores: tom won 50 games, jon won 20, and 30 games were a tie.
Deriving estimates for the probabilities of individual moves from this
information is less straightforward. For this reason, PRISM comes with a
built-in implementation of the EM-algorithm for performing parameter
estimation in the presence of missing information [Kameya and Sato (2000)]. We
can use this algorithm to find plausible corresponding distributions:
| ?- learn([ (50 times player(tom),player(jon) ===> winner(tom)), (20 times
player(tom),player(jon) ===> winner(jon)),(30 times player(tom),player(jon)
===> $\sim$winner(tom),$\sim$winner(jon))])The PRISM built-in show_sw shows
the learned probability distributions, which do indeed (approximately) lead to
the observation frequencies, e.g.:
| ?- show_swSwitch choice(jon): 1 (p: 0.60057) 2 (p: 0.06536) 3 (p:
0.33406)Switch choice(tom): 1 (p: 0.08420) 2 (p: 0.20973) 3 (p: 0.70605)| ?-
prob player(tom),player(jon) ===> winner(tom)Probability of
player(tom),player(jon)===>winner(tom) is: 0.499604
### 3.2 Random graphs
Suppose we want to generate a random directed graph, given its nodes. The
following rule generates every possible directed edge with probability 50%:
0.5 ?? node(A), node(B) ==> edge(A,B).
The above rule generates dense graphs; if we want to get a sparse graph, say
with an average (out-)degree of 3, we can use the following rule. The
auxiliary constraint nb_nodes$(n)$ contains the total number of nodes $n$; the
probability of the rule is such that each of the $n(n-1)$ possible edges is
generated with probability $3/(n-1)$, so on average it generates $3n$ edges:
eval(3/(N-1)) ?? nb_nodes(N), node(A), node(B) ==> edge(A,B).
### 3.3 Bayesian networks
Bayesian networks are one of the most widely used kinds of probabilistic
models. A classical example [Pearl (1988)] of a Bayesian network is that
describing the following alarm system. Suppose there is some probability that
there is a burglary, and also that there is some probability that an
earthquake happens. The probability that the alarm goes off depends on whether
those events happen. Also, the probability that John calls the police depends
on whether the alarm went off, and similarly for the probability that Mary
calls.
This Bayesian network can be described in CHRiSM in a straightforward way:
go ==> ?? burglary(yes) ; burglary(no).
go ==> ?? earthquake(yes) ; earthquake(no).
burglary(B), earthquake(E) ==> B,E ?? alarm(yes) ; alarm(no).
A ?? alarm(A) ==> johncalls.
A ?? alarm(A) ==> marycalls.
The probability distributions can be estimated given full observations (e.g.,
go <==> go, burglary(no), earthquake(yes), alarm(yes), marycalls.), or given
partial observations (e.g., go ===> johncalls, $\sim$marycalls.).
In this way, each Bayesian network can be represented in CHRiSM. We can derive
the same information from it as can be derived from the network itself.
## 4 Ambiguity and a Distribution Semantics for CHRiSM
In addition to the very nondeterministic abstract operational semantics
$\maybebm{\omega_{t}^{\mathtt{??}}}$, we can also define more deterministic
instantiations of $\maybebm{\omega_{t}^{\mathtt{??}}}$, just like
$\maybebm{\omega_{r}}$ and $\maybebm{\omega_{p}}$ are instantiations of
$\maybebm{\omega_{t}}$ (see also [Sneyers and Frühwirth (2008)]). In the
current implementation of CHRiSM we use the “refined semantics of CHRiSM”,
defined analogously to [Duck et al. (2004)]. Of course CHRiSM can also be
given a “priority semantics” [De Koninck et al. (2007)] in order to get a more
intuitive mechanism for execution control.
### 4.1 Instantiations of $\maybebm{\omega_{t}^{\mathtt{??}}}$
Any CHRiSM system uses a (computable) execution strategy in the sense of
[Sneyers and Frühwirth (2008)]. Note that in [Sneyers and Frühwirth (2008)],
an execution strategy completely fixes the derivation for a given input goal.
In the context of CHRiSM this is no longer the case because of the
probabilistic choices. However, we may assume that the derivation is fixed if
the same choices are made. In other words, the only choice is in the
probabilistic choices inside the transitions “Probabilistic Choice” and
“Maybe-Apply”; there is no nondeterminism in choosing which
$\maybebm{\omega_{t}^{\mathtt{??}}}$ transition to apply next.
###### Definition 4.1 (execution strategy)
An _execution strategy_ fixes the non-probabilistic choices during an
$\maybebm{\omega_{t}^{\mathtt{??}}}$ derivation. Formally,
$\xrightarrow{\hskip 6.97917pt}$ $\xi,\mathcal{P}$ is an execution strategy
for a program $\mathcal{P}$ if $\hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip 0.0pt\hbox{\raise
6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\subseteq\hbox{$\ext@arrow
0359\arrowfill@{\raisebox{1.07817pt}{\hbox{$\scriptscriptstyle\succ$}}}\relbar\rightarrow{}{\hskip
3.40279pt}$}\hbox{\hskip-28.27083pt\hskip 0.0pt\hbox{\raise
6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
28.27083pt\hskip-4.0pt\hskip-3.40279pt\hbox{\lower
3.57222pt\hbox{\tiny$\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt$ and for every
execution state $\sigma\in\Sigma^{\mbox{chr}}$, the set $S$ of all transitions
of the form $\sigma\hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip 0.0pt\hbox{\raise
6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\sigma^{\prime}$ corresponds to at most one of the five types of
transitions of $\maybebm{\omega_{t}^{\mathtt{??}}}$, that is, either
* •
$S=\emptyset$ and no $\maybebm{\omega_{t}^{\mathtt{??}}}$ transition is
applicable;
* •
or $S$ is a singleton corresponding to a Fail, Solve or Introduce transition;
* •
or $S$ is a set of transitions corresponding to the Probabilistic-Choice
transition for one specific disjunction;
* •
or $S$ is a set of transitions corresponding to the Maybe-Apply transition for
one specific rule instantiation.
It follows from this definition that for non-final states $\sigma$, the sum of
the probabilities of all transitions from $\sigma$ is one under any execution
strategy. We use $\sigma_{0}\hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip-1.76094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p$}}}\hskip-1.76094pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\hskip-1.99997pt\hbox{\raise 3.00003pt\hbox{\scriptsize$*$}\hskip
3.00003pt}\sigma_{k}$ to denote a series of $k\geq 0$ transitions
$\sigma_{0}\ \hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{1}$}}}\hskip-2.74094pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\sigma_{1}\ \hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{2}$}}}\hskip-2.74094pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\sigma_{2}\ \hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{3}$}}}\hskip-2.74094pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\ \ldots\
\hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip-2.84302pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{k}$}}}\hskip-2.84302pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\
\sigma_{k}$
where $p=\prod_{i=1}^{k}p_{i}$ if $k>0$ and $p=1$ otherwise, as before.
###### Definition 4.2 (strategy class)
A _strategy class_ $\Omega(\mathcal{P})$ is a set of execution strategies for
$\mathcal{P}$. The strategy class $\Omega_{t}^{\mathtt{??}}(\mathcal{P})$ is
the set of _all_ execution strategies for $\mathcal{P}$.
### 4.2 Distribution Semantics
Firstly, we define equivalence of execution states. We use a definition based
on [Raiser et al. (2009)] but adapted to our needs. Intuitively, we say two
states are equivalent if the constraint stores are equal and the built-in
stores are equivalent; we do not care about identifiers and propagation
histories.
###### Definition 4.3 (equivalent states)
Equivalence between execution states is the smallest equivalence relation
$\equiv$ s.t.:
1. 1.
$\langle\mathbb{G},\mathbb{S},{x=t\land\mathbb{B}},\mathbb{T}\rangle_{n}\equiv\langle\mathbb{G},{\mathbb{S}[x/t]},{x=t\land\mathbb{B}},\mathbb{T}^{\prime}\rangle_{n^{\prime}}$
2. 2.
$\langle\mathbb{G},\mathbb{S},{x=t\land\mathbb{B}},\mathbb{T}\rangle_{n}\equiv\langle{\mathbb{G}[x/t]},\mathbb{S},{x=t\land\mathbb{B}},\mathbb{T}^{\prime}\rangle_{n^{\prime}}$
3. 3.
$\langle\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\equiv\langle\mathbb{G},\mathbb{S}^{\prime},\mathbb{B},\mathbb{T}^{\prime}\rangle_{n^{\prime}}$
if $\mathit{chr}(\mathbb{S})=\mathit{chr}(\mathbb{S}^{\prime})$
4. 4.
$\langle\mathbb{G},\mathbb{S},\mathbb{B},\mathbb{T}\rangle_{n}\equiv\langle\mathbb{G},\mathbb{S},\mathbb{B}^{\prime},\mathbb{T}\rangle_{n}$
if
$\mathcal{D}_{\mathcal{H}}\models\bar{\exists}_{\mathbb{G},\mathbb{S}}\mathbb{B}\leftrightarrow\bar{\exists}_{\mathbb{G},\mathbb{S}}\mathbb{B}^{\prime}$
We now define the probability of getting some result (given an execution
strategy) as the sum of the probabilities of ending up in a final state
equivalent with it:
###### Definition 4.4 (observation probability)
Given a program $\mathcal{P}$ and an execution strategy
$\hbox{$\xrightarrow{\hskip 6.97917pt}$}\hbox{\hskip-5.9896pt\hskip
0.0pt\hbox{\raise 6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\in\Omega_{t}^{\mathtt{??}}(\mathcal{P})$, we write
$\sigma_{i}\hbox{$\ext@arrow 0359\Rightarrowfill@{}{\hskip
6.97917pt}$}\hbox{\hskip-22.55904pt\hskip-4.26538pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{\mathit{tot}}$}}}\hskip-4.26538pt\hskip
22.55904pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.41666pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip 1.00006pt\sigma_{f}$
if $\sigma_{f}$ is a final state and $p_{\mathit{tot}}=\sum_{d\in
D}\mathit{prob}(d)$ where $D=\\{\sigma_{i}\hbox{$\xrightarrow{\hskip
6.97917pt}$}\hbox{\hskip-5.9896pt\hskip-1.76094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p$}}}\hskip-1.76094pt\hskip
5.9896pt\hskip-4.0pt\hskip-6.97917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi,\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\hskip-1.99997pt\hbox{\raise 3.00003pt\hbox{\scriptsize$*$}\hskip
3.00003pt}\sigma^{\prime}_{f}\ |\ \sigma^{\prime}_{f}\equiv\sigma_{f}\\}$. We
say that $p_{\mathit{tot}}$ is the probability of observing the result
$\sigma_{f}$ for the query $\sigma_{i}$.
### 4.3 Ambiguity
Some programs are _ambiguous_ in the sense that they do not define a unique
probability distribution over the possible end states. Consider the following
example:
0.5 ?? a <=> b.
0.5 ?? a <=> c.
Suppose the query is “a”. If we use an execution strategy that starts with the
first rule, then with 50% chance this rule is applied and we get the final
result “b”, with 50% chance the second rule is considered resulting in “c”
with a probability of 25%, and when no rule is applied the result is “a” with
a probability of 25%. However, if we use an execution strategy that considers
the second rule first, then we get a different distribution: “c” has a
probability of 50%, and “b” a probability of 25%.
A program is unambiguous if the probability of an observation does not depend
on the execution strategy. The program in the above example is ambiguous in
general, but it is unambiguous w.r.t. the refined strategy class. Under the
refined semantics, the first rule is always considered first, thus the above
program defines only the first probability distribution on final states. In
general, we define ambiguity w.r.t. a strategy class — if the strategy class
is omitted, we assume it is the most general strategy class corresponding to
all execution strategies that instantiate
$\maybebm{\omega_{t}^{\mathtt{??}}}$.
###### Definition 4.5 (unambiguous program)
A CHRiSM program $\mathcal{P}$ is unambiguous (w.r.t. a strategy class
$\Omega$) if, for all states $\sigma_{i},\sigma_{f}\in\Sigma^{\mbox{chr}}$ and
all execution strategies $\hbox{$\xrightarrow{\hskip
8.37917pt}$}\hbox{\hskip-6.68959pt\hskip 0.0pt\hbox{\raise
6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
6.68959pt\hskip-4.0pt\hskip-8.37917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi_{1},\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt,\hbox{$\xrightarrow{\hskip 8.37917pt}$}\hbox{\hskip-6.68959pt\hskip
0.0pt\hbox{\raise 6.00006pt\hbox{\scriptsize$$}}}\hskip 0.0pt\hskip
6.68959pt\hskip-4.0pt\hskip-8.37917pt\hbox{\lower
3.57222pt\hbox{\tiny$\xi_{2},\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\in\Omega$, we have:
if $\sigma_{i}\hbox{$\ext@arrow 0359\Rightarrowfill@{}{\hskip
8.37917pt}$}\hbox{\hskip-23.25903pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{1}$}}}\hskip-2.74094pt\hskip
23.25903pt\hskip-4.0pt\hskip-8.37917pt\hbox{\lower
3.41666pt\hbox{\tiny$\xi_{1},\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\sigma_{f}$ and $\sigma_{i}\hbox{$\ext@arrow
0359\Rightarrowfill@{}{\hskip
8.37917pt}$}\hbox{\hskip-23.25903pt\hskip-2.74094pt\hbox{\raise
6.00006pt\hbox{\scriptsize$p_{2}$}}}\hskip-2.74094pt\hskip
23.25903pt\hskip-4.0pt\hskip-8.37917pt\hbox{\lower
3.41666pt\hbox{\tiny$\xi_{2},\mathcal{P}$}}\hskip 4.0pt\hskip
1.00006pt\sigma_{f}$ then $p_{1}=p_{2}$.
The distribution semantics (w.r.t. strategy class $\Omega$) of an unambiguous
(w.r.t. $\Omega$) CHRiSM program is defined for every query $Q$ as the
probability distribution over the equivalence classes of final states of
derivations (of $\Omega$).
Without specification of an execution strategy, ambiguous CHRiSM programs do
not have a well-defined meaning — they don’t define a unique probability
distribution over the final states, but _several_ distributions, depending on
which execution strategy is used. Ambiguity can be reduced by using a more
instantiated strategy class. The current CHRiSM system uses the refined
semantics. Many programs that are ambiguous in general are unambiguous w.r.t.
the refined strategy class, but not all of them. As a counterexample, consider
the program consisting of the rule “`0.5 ?? a, b(X) <=> c(X)`” with the query
“b(1), b(2), a”. There are two ways to execute this program in the refined
semantics: one in which the rule instantiation “a, b(1)” is considered first,
and one in which the rule instantiation “a, b(2)” is considered first.
According to the first execution strategy, the result is “c(1), b(2)” with a
probability of 50%, “c(2), b(1)” with a probability of 25%, and “a, b(1),
b(2)” with a probability of 25%. According to the second execution strategy
the probabilities of the first two outcomes are switched.
#### Ambiguity vs. confluence.
Ambiguity of CHRiSM programs is related to confluence [Abdennadher et al.
(1999)] of CHR programs. A CHR program is confluent if for every query, all
derivations (under the $\maybebm{\omega_{t}}$ semantics) lead to equivalent
final states. Confluent CHR programs tend to correspond to unambiguous CHRiSM
programs. For example, programs with only propagation rules are always
confluent and unambiguous. However, confluence and unambiguity do not
coincide. For example, a program consisting of the rule “a <=> b:0.5 ; c:0.5”
is not confluent (because for the query “a” it has two non-equivalent final
states) but it is unambiguous. Vice versa, some programs are confluent CHR
programs while they are ambiguous CHRiSM programs. For example, consider the
following program:
0.5 ?? a <=> b.
0.5 ?? a <=> c.
0.5 ?? c <=> b.
If we ignore the probabilities and consider this as a regular CHR program,
then we get a confluent program (all derivations for the query “a” end in the
result “b”). However, as a CHRiSM program, it is ambiguous. If the execution
strategy is such that the first rule is considered first for the query “a”,
then the probability of ending up with the result “b” is 67.5%. Using an
execution strategy that considers the second rule first, the probability of
getting “b” is only 50%. Therefore, the probability depends on the execution
strategy and the program is ambiguous.
## 5 Implementation of CHRiSM
The implementation of CHRiSM is based on a source-to-source transformation
from CHRiSM rules to CHR(PRISM) rules. PRISM is implemented on top of
B-Prolog, and several CHR systems are currently available for B-Prolog. In
[Sneyers et al. (2009)] we presented a prototype implementation of CHRiSM that
used a naive CHR(PRISM) system based on toychr111by Gregory J. Duck, 2004.
Download: http://www.cs.mu.oz.au/$\sim$gjd/toychr/, which is a rather naive
implementation of (ground) CHR in pure Prolog. The current implementation of
CHRiSM222 Download the CHRiSM system at
http://people.cs.kuleuven.be/jon.sneyers/chrism/ is based on the more advanced
Leuven CHR system [Schrijvers and Demoen (2004)].
### 5.1 PRISM
PRISM [Sato (2008)] is a probabilistic logic programming language. It is an
extension of Prolog with a probabilistic built-in _multi-valued random switch
(msw)_. A multi-valued switch atom msw(exp, Result) represents a probabilistic
experiment named exp (a ground Prolog term), which produces an outcome Result.
The set of possible outcomes for such an experiment is defined by means of a
predicate values(term,[v1,..., vn]) and term unifies with exp. By default, a
uniform distribution is assumed (all values are equally likely). Different
probabilities can be assigned by means of `set_sw(term, [p1, ..., pn])`.
A PRISM program consists out of two parts, rules $R$ and facts $F$. The facts
$F$ define a base probability distribution $P_{F}$ on msw-atoms, by means of
the values/2 and `set_sw`/2 predicates. The rules $R$ are a set of definite
clauses, which are allowed to contain the msw predicate in the body (but not
in the head). This set of clauses $R$ serves to extend the base distribution
$P$ to a distribution $P_{DB}(\cdot)$ over the set of Herbrand
interpretations: for each interpretation $M$ of the msw terms, the probability
$P_{F}(M)$ is assigned to the interpretation $I$ that is the least Herbrand
model of $R\cup M$ (_distribution semantics_).
### 5.2 Transformation to CHR(PRISM)
The transformation from CHRiSM to CHR(PRISM) is rather straightforward and can
be done efficiently (linear time). We illustrate it by example. Consider again
the rule “`player(P) <=> choice(P) ?? rock(P) ; scissors(P) ; paper(P)`” from
Section 3.1. It is translated to the following CHR(PRISM) code:
values(choice(_),[1,2,3]).
player(P) <=> msw(choice(P),X),
(X=1->rock(P); X=2->scissors(P); X=3->paper(P)).
Another example is the random graph rule from Section 3.2:
eval(3/(N-1)) ?? nb_nodes(N), node(A), node(B) ==> edge(A,B).
which gets translated to the following CHR(PRISM) code:
values(experiment1,[1,2]).
nb_nodes(N), node(A), node(B) ==>
P1 is 3/(N-1), P2 is 1-P1, set_sw(experiment1,[P1,P2]),
msw(experiment1,X), (X=1 -> edge(A,B) ; X=2 -> true).
Probabilistic simplification rules and simpagation rules are a bit more tricky
since it does not suffice to add a “nop”-disjunct like above. The reason is
that any removed heads are removed from the constraint store as soon as the
body is entered, and just reinserting the removed heads potentially causes
nontermination. Putting the msw-test in the guard of the rule also does not
work as expected. In sampling mode, this works fine, but when doing
probability computations or learning, an unwanted behavior emerges because of
the way PRISM implements explanation search. During explanation search, PRISM
essentially redefines msw/2 such that it creates a choice point and tries all
values. This causes the guard to always succeed and thus explanations that
involve _not_ firing a chance rule are erroneously missed. Hence some care has
to be taken to translate such rules to PRISM code that behaves correctly. The
solution we have adopted is to add a built-in to CHR to explicitly remove a
constraint from the head of a rule. All CHRiSM rules are translated to
propagation rules. The removed heads are explicitly removed in the body of the
rule, but only in the branch in which the rule instance is actually applied.
## 6 Related Work
The idea of a probabilistic version of CHR is not new. In [Frühwirth et al.
(2002)], a probabilistic variant of CHR, called PCHR, was introduced. In PCHR,
every rule gets a weight representing a relative probability. A rule is chosen
randomly from all applicable rules, according to a probability distribution
given by the normalized weights. For example, the following PCHR program
implements a coin toss:
toss <=>0.5: head.
toss <=>0.5: tail.
One of the conceptual advantages of PCHR, at least from a theoretical point of
view, is that its semantics instantiates the abstract operational semantics
$\maybebm{\omega_{t}}$ of CHR [Sneyers et al. (2010)]: every PCHR derivation
corresponds to some $\maybebm{\omega_{t}}$ derivation.
However, the semantics of PCHR may also lead to some confusion, since it is
not so clear what the meaning of the rule weight really is. For example,
consider again the above coin tossing example. For the query toss we get the
answer head with 50% chance and otherwise tail, so one may be tempted to
interpret weights as rule probabilities. However, if the second rule is
removed from the program, we do not get the answer head with 50% chance, but
with a probability of 100%. The reason is that the weights are normalized
w.r.t. the sum of the weights of all applicable rules. As a result of this
normalization, the actual probability of a rule can only be computed at
runtime and by considering the full program. In other words, the probabilistic
meaning of a single rule heavily depends on the rest of the PCHR program;
there is no localized meaning. Also, adding weights to _propagation_ rules is
not very useful in practice.
The abstract semantics $\maybebm{\omega_{t}}$ of CHR can be instantiated to
allow more execution control and more efficient implementations. However, the
PCHR semantics, even though it conforms to $\maybebm{\omega_{t}}$, cannot be
instantiated in a similar way. The reason is that the semantics of PCHR refers
to all applicable rules in order to randomly pick one. This conflicts
fundamentally with the purpose of instantiations like the refined semantics,
which consider only a small fragment of the set of applicable rules, e.g. only
rules for the current active constraint occurrence.
The $\maybebm{\omega_{t}^{\mathtt{??}}}$ semantics of CHRiSM differs from that
of PCHR. In particular, $\maybebm{\omega_{t}^{\mathtt{??}}}$ derivations do
not always correspond to $\maybebm{\omega_{t}}$ derivations (although they do,
in a sense, correspond to unfinished $\maybebm{\omega_{t}}$ derivations).
However, the semantics of CHRiSM can be instantiated since chance rules have a
localized meaning: the application probability does not depend on the set of
all applicable rules like in PCHR. As a result, it can be implemented
efficiently and more execution control can be obtained.
Another advantage of CHRiSM over PCHR are the features inherited from PRISM,
in particular probability computation and EM-learning. The existing PCHR
implementation only supports probabilistic execution, i.e. sampling.
#### Probabilistic Logic Programming.
There are numerous probabilistic extensions of logic programming. One
particular family of such extensions is formed by CP-logic or LPADs, ProbLog,
ICL, and PRISM itself [Sato (2008)]. All of these can be encoded in CHRiSM: in
[Sneyers et al. (2009)] we have shown that CP-logic (of which ProbLog, ICL,
etc. are sublogics) can be encoded in CHRiSM in a compact and modular way.
Next to these “logic programming flavored” languages, there are also a number
of formalisms that are inspired by Bayesian networks, such as BLP, RBN,
CLP(BN), and Blog. Based on the encoding of Bayesian networks that we gave in
Section 3.3, we can also translate BLPs to CHRiSM. RBNs, CLP(BN) and Blog
would be more difficult, because they allow more complex probability
distributions, for which CHRiSM currently does not offer support. (A more
detailed description of these formalisms can be found in [Getoor and Taskar
(2007)].)
## 7 Potential Applications
Both PRISM and CHR have been successfully applied in a wide range of research
fields. Since the features of PRISM and CHR are largely orthogonal, we can
expect CHRiSM to be extremely suitable for applications at the intersection of
the application areas of PRISM and CHR. One example of an application area at
the intersection is abduction, which has been studied in the context of PRISM
[Sato and Kameya (2002)] and also in the context of CHR (?), Section 7.3.2).
Computational linguistics and bio-informatics are two other domains in which
both PRISM and CHR have proven to be very valuable tools [Sato (2008),
Christiansen (2005), Christiansen and Lassen (2009)].
Furthermore, in many application domains of CHR, there is clearly a potential
for probabilistic extensions of the existing approaches, for instance to deal
with uncertain information. Examples are (section numbers refer to ?)):
scheduling (Section 7.1.1), spatio-temporal reasoning and robotics (Section
7.1.2), multi-agent systems (Section 7.1.3), the semantic web (Section 7.1.4),
type systems (Section 7.3.1), testing and verification (Section 7.3.5).
Another interesting application area is the automatic analysis and generation
of music. In the past, we have used PRISM to analyse and generate music in a
probabilistic setting [Sneyers et al. (2006)]. There are also several
deterministic approaches based on constraints and strict rules (e.g. ?)).
Preliminary results indicate that a combined approach, using CHRiSM, is very
promising. In this application, sampling of a probabilistic model corresponds
to music generation, while parameter learning from a training set corresponds
to tuning the model to a specific genre or composer, and probability
computation (or Viterbi computation) can be used for music classification.
## 8 Conclusion
In this exploratory paper, we have introduced a novel rule-based
probabilistic-logic formalism called CHRiSM, which is based on a combination
of CHR and PRISM. We have defined an operational semantics for arbitrary
CHRiSM programs and a distribution semantics for unambiguous CHRiSM programs.
We have illustrated the CHRiSM system by example and we have outlined some
potential application areas in which CHRiSM can be used. Finally, we have
sketched the implementation of the CHRiSM system and discussed related
languages, in particular PCHR.
In our opinion, CHR has important advantages over Prolog, including
complexity-wise completeness and the expressivity of multi-headed rules. We
expect CHRiSM to have the same advantages over plain PRISM.
There are several directions for future work. The notion of ambiguity and its
relation to confluence has to be explored; in particular, the existence of a
decidable ambiguity test (for terminating CHRiSM programs). Although the
current implementation is sufficiently efficient for sampling, it is too naive
for probability computation and learning, since those tasks require an
efficient mechanism to find explanations (sequences of probabilistic choices)
for observations. Improving the efficiency of explanation search is the topic
of ongoing work [Sneyers (2010)]. Another limitation of the current
implementation is that it only supports ground queries and observations.
Finally, it would be interesting to transfer automatic CHR program generation
techniques (e.g. ?)) to CHRiSM in order to obtain a system that supports not
only parameter learning but also structure learning (rule learning).
## References
* Abdennadher et al. (1999) Abdennadher, S., Frühwirth, T., and Meuss, H. 1999\. Confluence and semantics of constraint simplification rules. Constraints 4, 2, 133–165.
* Abdennadher et al. (2006) Abdennadher, S., Olama, A., Salem, N., and Thabet, A. 2006\. ARM: Automatic Rule Miner. In LOPSTR 2006, G. Puebla, Ed. LNCS, vol. 4407. Springer, 17–25.
* Boenn et al. (2008) Boenn, G., Brain, M., De Vos, M., and ffitch, J. 2008\. Automatic composition of melodic and harmonic music by answer set programming. In ICLP 2008, M. Garcia de la Banda and E. Pontelli, Eds. LNCS, vol. 5366. Springer, 160–174.
* Christiansen (2005) Christiansen, H. 2005\. CHR grammars. TPLP 5(4–5), 467–501.
* Christiansen and Lassen (2009) Christiansen, H. and Lassen, O. T. 2009\. Preprocessing for optimization of probabilistic-logic models for sequence analysis. In ICLP 2009, P. M. Hill and D. S. Warren, Eds. LNCS, vol. 5649. Springer, 70–83.
* De Koninck et al. (2007) De Koninck, L., Schrijvers, T., and Demoen, B. 2007\. User-definable rule priorities for CHR. In PPDP 2007, M. Leuschel and A. Podelski, Eds. ACM Press, 25–36.
* Duck et al. (2004) Duck, G. J., Stuckey, P. J., García de la Banda, M., and Holzbaur, C. 2004\. The refined operational semantics of Constraint Handling Rules. In ICLP 2004, B. Demoen and V. Lifschitz, Eds. LNCS, vol. 3132\. Springer, 90–104.
* Frühwirth (2009) Frühwirth, T. 2009\. Constraint Handling Rules. Cambridge University Press.
* Frühwirth et al. (2002) Frühwirth, T., Di Pierro, A., and Wiklicky, H. 2002\. Probabilistic Constraint Handling Rules. In WFLP 2002, M. Comini and M. Falaschi, Eds. ENTCS 76. Elsevier.
* Getoor and Taskar (2007) Getoor, L. and Taskar, B., Eds. 2007. Statistical Relational Learning. MIT Press.
* Kameya and Sato (2000) Kameya, Y. and Sato, T. 2000\. Efficient EM learning with tabulation for parameterized logic programs. In CL 2000, J. Lloyd, V. Dahl, et al., Eds. LNAI, vol. 1861. 269–294.
* Pearl (1988) Pearl, J. 1988\. Probabilistic Reasoning in Intelligent Systems : Networks of Plausible Inference. Morgan Kaufmann.
* Raiser et al. (2009) Raiser, F., Betz, H., and Frühwirth, T. 2009\. Equivalence of CHR states revisited. In CHR 2009, F. Raiser and J. Sneyers, Eds. 34–48.
* Sato (2008) Sato, T. 2008\. A glimpse of symbolic-statistical modeling by PRISM. Journal of Intelligent Information Systems 31, 161–176.
* Sato and Kameya (2002) Sato, T. and Kameya, Y. 2002\. Statistical abduction with tabulation. In Computational Logic: Logic Programming and Beyond, Essays in Honour of Robert A. Kowalski, Part II, A. Kakas and F. Sadri, Eds. LNCS, vol. 2408. Springer, 567–587.
* Schrijvers and Demoen (2004) Schrijvers, T. and Demoen, B. 2004\. The K.U.Leuven CHR system: Implementation and application. In CHR 2004, T. Frühwirth and M. Meister, Eds. 8–12.
* Sneyers (2010) Sneyers, J. 2010\. Result-directed CHR execution. In CHR 2010, P. Van Weert and L. De Koninck, Eds. To appear.
* Sneyers and Frühwirth (2008) Sneyers, J. and Frühwirth, T. 2008\. Generalized CHR machines. In CHR 2008, T. Schrijvers, T. Frühwirth, and F. Raiser, Eds. 143–157.
* Sneyers et al. (2009) Sneyers, J., Meert, W., and Vennekens, J. 2009\. CHRiSM: Chance rules induce statistical models. In CHR 2009, F. Raiser and J. Sneyers, Eds. 62–76.
* Sneyers et al. (2010) Sneyers, J., Van Weert, P., Schrijvers, T., and De Koninck, L. 2010\. As time goes by: Constraint Handling Rules — a survey of CHR research between 1998 and 2007. Theory and Practice of Logic Programming 10, 1 (January), 1–47.
* Sneyers et al. (2006) Sneyers, J., Vennekens, J., and De Schreye, D. 2006\. Probabilistic-logical modeling of music. In PADL 2006, P. Van Hentenryck, Ed. 60–72.
* Vennekens et al. (2004) Vennekens, J., Verbaeten, S., and Bruynooghe, M. 2004\. Logic programs with annotated disjunctions. In ICLP 2004, B. Demoen and V. Lifschitz, Eds. LNCS, vol. 3132\. Springer, 431–445.
|
arxiv-papers
| 2010-07-22T11:32:21 |
2024-09-04T02:49:11.835918
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jon Sneyers, Wannes Meert, Joost Vennekens, Yoshitaka Kameya and\n Taisuke Sato",
"submitter": "Jon Sneyers",
"url": "https://arxiv.org/abs/1007.3858"
}
|
1007.4276
|
# Systematic Correction for “Demonstration of the Casimir Force in the 0.6 to
6 $\mu$m Range”
S.K. Lamoreaux Yale University, Department of Physics, P.O. Box 208120, New
Haven, CT 06520, USA
###### Abstract
A new systematic correction for Casimir force measurements is proposed and
applied to the results of an experiment that was performed more than a decade
ago. This correction brings the experimental results into good agreement with
the Drude model of the metallic plates’ permittivity. The systematic is due to
time-dependent fluctuations in the distance between the plates caused by
mechanical vibrations or tilt, or position measurement uncertainty, and is
similar to the correction for plate roughness.
###### pacs:
12.20.Fv, 11.10.Wx, 73.40.Cg, 04.80. Cc
## I Introduction
This is a short note reporting a new type of correction to the Casimir force
Casimir , and the results of its application to my experiment that was
performed some 15 years ago Lamoreaux . One might question whether it is worth
reanalyzing an experiment that is so very dated, however, this work stands,
together with our work with Germanium kim , as the experiments with the
largest plate separations, and are particularly sensitive to a number of
fundamental, thermal, and systematic effects. Indeed, Lamoreaux led to a
resurgence on interest in the Casimir measurement field Casexp , and has been
discussed in works of varying sophistication, from those presenting
revolutionary new ideas Bostrom to those that indicate that the authors
cannot read a graph xxxx .
Although my experiment Lamoreaux was intended as a demonstration, the
deviation between its results and the theory presented by Böstrom and
Sernelius Bostrom remains as a puzzle. Despite years of theoretical work and
years of questioning, there has been no satisfactory explanation of the
discrepancy. In particular, it would appear that the Drude model must be the
correct one to describe the plates, because the so-called Plasma model implies
a superconducting boundary condition at zero frequency.
Unfortunately, the raw data that led to Lamoreaux is no longer available,
however there is enough information in that paper to make a good estimate of
the correction. As the correction depends on external factors that were not
measured, it is not clear that the raw data would help much in any case. The
work reported here was inspired by our recent and ongoing re-measurement of
the Casimir force between Au plates, which support the Drude model better than
the Plasma model. In the course of our recent work, it occurred to us that
time-dependent fluctuations between the plates can lead to a correction, much
like the surface roughness correction that has been studied by Prof.
Mostepanenko and collaborators xxxx .
The interesting features of Lamoreaux are as follows: The Drude model appears
to better describe the long-distance (greater that 1.5 $\mu$m) data; The
short-distance data appears to agree with the Plasma model prediction. The
roughness usually associated with optical surfaces is too small to account for
the deviation between the Drude theory and experiment, and seems to have the
wrong form as in this case, the discrepancy should falls, as a fraction of the
force, as $1/d^{2}$, where $d$ is the separation between the plates. Thus for
the Casimir force alone, the effect should be very short range. In Lamoreaux ,
a background potential existed, creating an electrostatic force that was
greater than the Casimir force over the measurement range. Thus, the effects
of both forces must be considered together.
## II Corrections due to Vibration and Distance Calibration Uncertainties
The force between two plates, for small variations $\delta(t)$ of the distance
$d$ is
$F(d+\delta(t))=F(d)+F^{\prime}(d)\delta(t)+{1\over
2}F^{\prime\prime}(d)\delta^{2}(t).$ (1)
If we assume that $\delta(t)$ represents a stationary random process with zero
mean, there are two cases to consider. First, if the correlation time of
$\delta(t)$ implies frequencies higher than the measurement bandwidth, the
term linear in $\delta(t)$ does not contribute to anything, while the second
order term results in a change in the apparent force,
$F_{a}(d)=F(d)+{1\over 2}F^{\prime\prime}(d)\langle\delta^{2}\rangle$ (2)
where $\langle\delta^{2}(t)\rangle=\delta_{rms}^{2}$ is the mean-square
fluctuation. It should be noted that $\delta_{rms}$ can have contributions
from multiple sources, which, if uncorrelated, can be added in quadrature. In
addition, a finite surface roughness can be included here; the form of Eq. (2)
does not distinguish between spatial or temporal roughness.
Second, if the fluctuations frequency is within the measurement bandwidth,
which is the case for uncertainties in the distance determination, there will
be an excess scatter associated with the force measurement,
$\sigma_{F_{a}}^{2}=\sigma_{F}^{2}+(F^{\prime}(d)\delta_{rms})^{2}$ (3)
while the apparent average force is given by Eq. (2), as before.
## III Application to the Experiment
In Lamoreaux , there is a large background electrostatic force that is used
for determining the absolute distance, obtained by fitting to
$\beta/(d-d_{0})$, using points at distances greater than 2 $\mu$m. For the
approximations here, let us assume that there are no significant corrections
at these long distances. Based on Fig. (3) of Lamoreaux , the background
electrostatic force is
$F_{e}(d)={\beta\over d};\ \ \ \ \beta=(215\pm 7)\ {\rm\mu dyne\ \mu m}.$ (4)
The total force is the electric plus Casimir force,
$F(d)=F_{e}(d)+F_{c}(d)$ (5)
and the first derivative at $d=0.62\ \mu$m is approximately $1000\
\mu$dyne$/\mu$m. Comparing the size of the error bars in Fig. 4 of Lamoreaux ,
where the 0.1 $\mu$m bins have $1/10$ the data of the 1 $\mu$m bins and should
be $\sqrt{10}$ time larger. It is observed that they are $1.32\sqrt{10}$
larger, so the contribution to the error due to fluctuations is 4 $\mu$dyne.
This implies that the rms fluctuations in position on a time scale of the
measurement of each point (50 seconds) is 40 nm. This is to be compared to 14
nm in our present experiment, where the intrinsic signal to noise is similar,
as is the applied calibration voltage. The excess noise in Lamoreaux is
likely due to the faster rate of drift in position. It is stated in Lamoreaux
that the fluctuations in absolute position measurement is less than 100 nm,
which is consistent with the result here. On the other hand, the quality of
the data in Fig. (3) of Lamoreaux suggest that perhaps 40 nm is optimistic;
unfortunately the original data set is not available to further investigate
this point. For fluctuations at this level, the apparent change in force is
less than 5% so these fluctuations do not contribute to the discrepancy.
Our recent work shows that, due to vibrations and tilt, there is an rms
position fluctuation of 20 nm in a .01 to 5 Hz bandwidth for our torsion
pendulum supported by a tungsten wire of a few cm length. The variations at
low frequency are dominated by tilt of the pendulum, and correspond to angular
fluctuations of order $1\times 10^{-7}$ radians. Given that the pendulum in
the present experiment has a length of 4 cm compared to an effective length of
nearly 80 cm for Lamoreaux , we might expect position rms fluctuations of
order 400 nm, as the change in position is roughly the pendulum length time
the tilt angle, which is not to be confused with the torsional motion angle.
However, the bandwidth of the swinging mode of the longer pendulum is lower
(it is relatively smaller by a approximately a factor $1/\sqrt{40}=1/6.3$, so
the effective noise should be a factor $1/\sqrt{6.3}=1/2.5$) implying an rms
noise of 400 nm$/2.5=160$ nm. We can therefore take $\delta_{rms}=100$ nm as a
lower limit. In operating the experiment Lamoreaux , the problems with tilt
noise were about an order of magnitude worse than our present experiment,
consistent with relative size of the rms position fluctuations as discussed
here.
Only angular fluctuations at frequencies below the swinging mode frequency
will contribute significantly to the relative separation fluctuation between
the plates because the magnetic damper tends to stabilize the relative
position of the plates. On the other hand, vibrations that cause a net
translational motion of the system couple in a different way, and frequencies
above the mode frequency can contribute. The angular noise dominates so we
neglect translational vibrations in this discussion.
Let us now consider the combined effect of $F_{e}(d)$ and $F_{c}(d)$ for
either the Plasma or Drude models of permittivity. The correction to the force
is given by Eq. (2), as
$F_{a}(d)-F(d)={1\over
2}(F_{e}^{\prime\prime}(d)+F^{\prime\prime}_{c}(d))\delta_{rms}^{2}.$ (6)
It’s easy to calculate $F_{e}^{\prime\prime}(d)$, while
$F^{\prime\prime}_{c}(d)$ can be numerically evaluated. The Plasma model and
Drude model forces were calculated using tabulated Au properties, with the
interesting result that the second derivative of the Drude model is about
twice as large, in the 0.5 to 1 $\mu$m range, as that for the Plasma model,
which isn’t too surprising as the Drude model force is falling off more
rapidly in this region. It should also be noted that
$F^{\prime\prime}_{e}\propto 1/d^{3}$ so there is a constant offset for large
$d$, when the correction is multiplied by $d^{3}$ as shown in the graph.
The results are shown in Fig. 1, where it is clear that the Drude model has
much better agreement. Furthermore, both the large and small distance data
agree with the theory, unlike the case of any other model. As an aside, a re-
measurement of the radius of curvature of the spherical plate used in
Lamoreaux shows $R=12.4\pm 0.1$ cm, which has lower error than the number
reported in the Erratum Lamoreaux .
The agreement can be made better by allowing $\delta{rms}$ vary with distance.
It is reasonable to assume that the runs which attained the lowest separation
were obtained when the system and environment was particularly quiet. For
example, if we let $\delta_{rms}=\sqrt{{d\over 3{\rm\mu m}}}\ \mu$m then
$\chi^{2}=0.79$ for the Drude model, while $\chi^{2}=6.9$ for the Plasma
model.
Figure 1: Results taking $\delta_{rms}=100$ nm. Green line: perfect conductor,
zero temperature; solid red: Plasma model with distance fluctuations; dashed
red: Plasma model without distance fluctuation correction; solid blue: Drude
model with distance fluctuations; dashed blue: Drude model without distance
fluctuation correction. Reduced $\chi^{2}=6.17$ (prob. $<10^{-5}$, 6 d.o.f.)
for the Plasma model with distance correction, while $\chi^{2}=1.75$ for the
Drude model with distance correction (prob. 10%, 6 d.o.f.).
## IV Discussion and Conclusion
By assuming that the relative separation between the plates of a Casimir
experiment is fluctuating on a short time scale, the results given in
Lamoreaux can be brought into agreement with theory. The required rms
fluctuations appear as large, of order 100 nm, but such a level is not
unreasonable. Given that the total pendulum length is nearly one meter, a tilt
of $10^{-7}$ radians is all that is required to generate the required
fluctuations. Such level of tilt is easily generated by air currents moving
past the apparatus, and by unavoidable oscillations of the floor. It should be
noted that the rms position fluctuations are due mainly to angle fluctuations
with frequency less than about 0.5 Hz, the natural frequency of the pendulum’s
swinging mode. Measurements with our new apparatus at Yale shows 20 nm rms
fluctuations, however, the pendulum is only a few cm in length. A full
analysis of the pendulum and how various tilts and vibrations affect the
relative positions of the plates is a tedious but elementary exercise. We do
note however that within the feedback bandwidth (which is greater than the
measurement bandwidth), the system compensates for a tilt by adjusting the
torsion pendulum angle, in order to keep the differential capacitor balanced.
Thus, for frequencies below about 0.1 Hz, a tilt position offset is
approximately doubled for the interplate separation. Indeed, the extreme
sensitivity of the apparatus to tilt and vibration was readily apparent; in
order to take sensible data, the experiment could only be operated between 11
pm and about 5 am, and the air conditioning ducts into the room had to be
blocked. The required rms position fluctuation of 100 nm is below the minimum
plate separation of 600 nm, and falls into what can be considered a reasonable
range. The angular fluctuations appeared as a very slow drift, causing a
changing in the distance offset between the plate, and was of order 1
$\mu$m/hour. On top of this slow drift, according to the calculations here,
were rapid fluctuations with periods up to 2 second (the frequency of the
pendulum swinging mode) with rms deviations of order $10^{-7}$ rad in the
.01-.5 Hz bandwidth.
It appears that $\delta_{rms}=100$ nm gives close to a best fit, and thus
should be considered as an adjustable parameter, the value of which is
verified by other means described in this note. Also to be noted that other
contributions to $\delta_{rms}$ can be included by adding all contributions in
quadrature. Of course, other systematic effects can contribute, such as a
contact potential that varies with distance. The contact potential was not
measured as a function of distance in Lamoreaux , however, our recent work
with Au suggests that the contact potential is nearly constant. This does not
preclude the possibility that there was such a variation in Lamoreaux .
If $\delta_{rms}(d)$ depends on distance, which is a very distinct
possibility, it is easy to see that the the data can be brought into better
agreement with the Drude model in particular. Certainly, $\delta_{rms}$
depends on time, and the data runs that attained the closest separation were
likely obtained when the system and the environment was the quietest. This
level of fine tuning is beyond the scope of the brief analysis presented here,
and beyond the scope of credibility.
Effects of vibration are important for all Casimir experiments. Even in the
absence of external perturbations, AFM cantilevers, for example, exhibit
Brownian motion and the effects of such need to be taken into account. The
implication is that very stiff springs should be used. In the case of the
torsion pendulum experiment, feedback is used to keep the torsion angle fixed;
this reduces position fluctuations due to Brownian motion, but can make the
system more sensitive to vibrations and tilts, as discussed above.
The apparent force due to the large $1/d$ electrostatic force varies as
$1/d^{3}$, with magnitude relative to the Casimir force (perfect conductors)
of $\beta\delta_{rms}^{2}/2=1.2\ \mu$dyne $\mu$m3, about 3% of the perfect
conducting force.
Again, the work described in Lamoreaux was presented as a demonstration; the
analysis here shows that there is a possible systematic effect that can lead
to a substantial increase in the apparent Casimir force. In this case, the
large electrostatic force, present for calibration, contributes substantially,
particularly at large separations. Because its contribution scales as
$1/d^{3}$, it appears as a scale factor for the Casimir force, for distances
around 0.5 $\mu$m. Our recent work at Yale led to the consideration of these
effects, and also appears to support the Drude model for the permittivity. We
hope to complete these studies in the very near future.
Finally, it must be emphasized that Lamoreaux was intended as a
demonstration; the results presented here should not be considered as a
verification of the Boström/Sernelius theory Bostrom , or as evidence against
the Plasma model, but the discovery of a systematic effect that brings the
experimental results into agreement with the theory described in Bostrom . It
is unclear whether additional systematic effects exist, however, it had always
been my impression that my experimental result was likely contaminated by
additional, possible large systematics patch . I have never considered the
results of this experiment as suitable for constraining possible new long
range forces; had I felt such was meaningful I would have produced those
limits in the context of Lamoreaux . Here I have presented what I consider a
very likely systematic effect. We now know to pay careful attention to
position fluctuations in our ongoing work.
## V Acknowledgements
The author thanks Alex Sushkov for helpful discussions.
This was funded by Yale University, and DARPA/MTO s Casimir Effect Enhancement
project under SPAWAR contract no. N66001-09-1-2071.
## References
* (1) H. G. B. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
* (2) S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); Phys. Rev. Lett. 81, 5475 (1998) (Erratum).
* (3) W.-J. Kim et al., Phys. Rev. Lett. 103, 060401 (2009)
* (4) U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998); H. B. Chan et al., Science 291, 1941 (2001); G. Bressi et al., Phys. Rev. Lett. 88, 041804 (2002); R. S. Decca et al., ibid. 91, 050402 (2003).
* (5) M. Boström and Bo. E. Sernelius, Phys. Rev. Lett. 84, 4757 (2000).
* (6) M. Bordag, G.L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko Advances in the Casimir Effect (Oxford, 2009) (Roughness is treated in Chapter 17).
* (7) W.-J. Kim et al., Phys. Rev. A 81, 022505 (2010).
|
arxiv-papers
| 2010-07-24T16:48:38 |
2024-09-04T02:49:11.849782
|
{
"license": "Public Domain",
"authors": "S.K. Lamoreaux",
"submitter": "Steve K. Lamoreaux",
"url": "https://arxiv.org/abs/1007.4276"
}
|
1007.4365
|
$B$-submodules of ${\mathfrak{g}}/{\mathfrak{b}}$ and
Smooth Schubert Varieties in $G/B$
James B. Carrell
Abstract Let $G$ be a semisimple linear algebraic group over ${\mathbb{C}}$
without $G_{2}$ factors, $B$ a Borel subgroup of $G$ and $T\subset B$ a
maximal torus. The flag variety $G/B$ is a projective $G$-homogeneous variety
whose tangent space at the identity coset is isomorphic, as a $B$-module, to
${\mathfrak{g}}/{\mathfrak{b}}$, where ${\mathfrak{g}}=\operatorname{Lie}(G)$
and ${\mathfrak{b}}=\operatorname{Lie}(B)$. Recall that if $w$ is an element
of the Weyl group $W$ of the pair $(G,T)$, the Schubert variety $X(w)$ in
$G/B$ is by definition the closure of the Bruhat cell $BwB$. In this note we
prove that $X(w)$ is nonsingular if and only if the following two conditions
hold: 1) its Poincaré polynomial is palindromic and 2) the tangent space
$TE(X(w))$ to the set $T$-stable curves in $X(w)$ through the identity is a
$B$-submodule of ${\mathfrak{g}}/{\mathfrak{b}}$. This gives two criteria in
terms of the combinatorics of $W$ which are necessary and sufficient for
$X(w)$ to be smooth: $\sum_{x\leq w}t^{\ell(x)}$ is palindromic, and every
root of $(G,T)$ in the convex hull of the set of negative roots whose
reflection is less than $w$ (in the Bruhat order on $W$) has the property that
its $T$-weight space (in ${\mathfrak{g}}/{\mathfrak{b}}$) is contained in
$TE(X(w))$. owever, these conditions don’t characterize the smooth Schubert
varieties when $G$ has type $G_{2}$.
## 1\. Introduction
Schubert varieties in the flag variety of a linear algebraic group $G$ were
originally defined by Chevalley in a famous unpublished paper [9], where it
was remarked offhandedly that all Schubert varieties were probably smooth.
This was an oversight, of course, because it was already known that there
exist singular Schubert varieties in Grassmannians. The question of actually
determining the smooth Schubert varieties in an arbitrary flag variety has
subsequently been treated in many places. For example, when $G$ is of type
$A$, Lakshmibai and Seshadri [14] computed the tangent spaces of all the
Schubert varieties, and subsequently Deodhar [10] used this to show that in
type $A$, the rationally smooth Schubert varieties are all smooth. Dale
Peterson showed this result holds whenever $G$ is simply laced (see [8] for a
proof). In [13], Lakshmibai and Sandhya showed that, in type $A$, the smooth
Schubert varieties are exactly the ones whose defining permutation avoids a
certain pattern, and, in [5], Billey and Postnikov extended pattern avoidance
to all classical $G$. Other results concerning globally smooth Schubert
varieties in $G/B$ in various settings include [1], [4], [8], [12], [15],
[16]. The main result of this note extends Peterson’s criterion for smoothness
to the non-simply laced setting by adding a condition which is vacuously
satisfied in the simply laced setting.
## 2\. Preliminary Remarks
Before stating the our result, we will review some elementary definitions and
well known facts about Schubert varieties. Let $G$ denote a semisimple linear
algebraic group over ${\mathbb{C}}$ with a fixed Borel subgroup $B$ and
maximal torus $T\subset B$, and let ${\mathfrak{g}},{\mathfrak{b}}$ and
${\mathfrak{t}}$ denote their respective Lie algebras. Recall the Cartan
decomposition
${\mathfrak{g}}={\mathfrak{t}}\oplus\sum_{{\alpha}\in\Phi}{\mathfrak{g}}_{\alpha}$
of ${\mathfrak{g}}$ into $T$-weight spaces, where $\Phi$ is the root system of
the pair $(G,T)$. The set of positive roots consists of the roots
corresponding to the $T$-module ${\mathfrak{b}}$. This set is denoted by
$\Phi^{+}$. One has $\Phi=\Phi^{+}\cup\Phi^{-}$, where $\Phi^{-}=-\Phi^{+}$.
Thus, the set of $T$-weights on ${\mathfrak{g}}/{\mathfrak{b}}$ is $\Phi^{-}$.
The flag variety $G/B$ of $G$ is a $G$-homogeneous projective variety, and it
is well known its tangent space $T_{e}(G/B)$ at the identity coset $e=B/B$ is
isomorphic with ${\mathfrak{g}}/{\mathfrak{b}}$. Note, that the identity coset
in $G/B$ is fixed by $B$, so $T_{e}(G/B)$ is in fact a $B$-module, and, by
considering the $B$-equivariant projection $\pi:G\to G/B$, one obtains that
$T_{e}(G/B)$ and ${\mathfrak{g}}/{\mathfrak{b}}$ are isomorphic as
$B$-modules. We will henceforth make the identification
$T_{e}(G/B)={\mathfrak{g}}/{\mathfrak{b}}=\sum_{{\alpha}<0}{\mathfrak{g}}_{\alpha}.$
Let $W=N_{G}(T)/T$ be the Weyl group of $(G,T)$, and recall that $W$ is a
finite reflection group generated by the reflections $r_{\alpha}$ (of
${\mathfrak{h}}$) through roots ${\alpha}$. By the Bruhat decomposition
$G=BWB,$ $G/B$ is the union of the $B$-orbits of the cosets $wB=n_{w}B$ as $w$
ranges over $W$, where $n_{w}\in N_{G}(T)$ is a representative of $w$. One of
the basic results of [9] says that the $B$-orbit $Bw\subset G/B$ is isomorphic
to affine space ${\mathbb{C}}^{\ell(w)}$, where $\ell$ is the length function
on $W$. The Zariski closure $X(w)$ of the $B$-orbit $Bw$ is called the
Schubert variety associated to $w$. Thus $\dim X(w)=\ell(w)$. Another basic
result of [9] is that the Bruhat order $\leq$ on $W$, defined combinatorially
in terms of reflections, is compatible with the natural geometric order on the
$B$-orbits: $x\leq w$ if and only if $X(x)\subseteq X(w)$. Moreover, the
$T$-fixed point set $X(w)^{T}$ is precisely the set $\\{x\in W\mid x\leq
w\\}$. It follows from these remarks that the Poincaré polynomial of $X(w)$,
with respect to ordinary rational homolgy, has the well known expression
(1) $P(X(w),t)=\sum_{x\leq w}t^{2\ell(x)}.$
## 3\. Statements of Results
The purpose of this paper is to give a simple constructive criterion which
describes which Schubert varieties with palindromic Poincaré polynomial in
$G/B$ are smooth that holds unless $G$ has a $G_{2}$ factor and, in fact, is
false in $G_{2}/B$. The condition that the Poincaré polynomial of a Schubert
variety $X(w)$ is palindromic is equivalent to the more difficult to formulate
condition that $X(w)$ is rationally smooth(cf. [6]). A variety is said to be
rationally smooth at a point if it satisfies local Poincaré duality at the
point and globally rationally smooth if it is rationally smooth at every point
[11]. We will state the smoothness critierion in three successive ways. The
first involves only the linear span $\Theta(w)\subset T_{e}(X(w))$ of the
reduced tangent cone to $X(w)$ at the identity element.
###### Theorem 1.
Suppose $G$ has no $G_{2}$ factors and $X(w)$ is a Schubert variety in $G/B$
whose Poincaré polynomial is palindromic (i.e. $X(w)$ is rationally smooth).
Then $X(w)$ is smooth if and only if $\dim\Theta(w)=\ell(w)$.
In type $A$, it follows readily from a result of Lakshmibai and Seshadri [14]
that $\Theta(w)=T_{e}(X(w))$. By combining a result of the author [7] and Polo
[15], this equality also holds in type $C$. Therefore, Theorem 1 follows
easily in types $A$ and $C$ from the Borel Fixed Point Theorem. Indeed, since
$X(w)$ is $B$-stable and the identity coset $e$ is fixed by $B$, $X(w)$ is
smooth if and only if it is smooth at $e$.
The second formulation of the smoothness criterion requires that we bring in
the tangent space $TE(X(w))$ to the set $E(X(w))$ of $T$-curves to $X(w)$ at
$e$ and discuss its relationship with $\Theta(w)$. Unless otherwise stated,
proofs of the assertions here are in [6]. Let
$TE(X(w))=\sum_{C\in E(X(w))}T_{e}(C).$
Each $T$-curve in $G/B$ is smooth, and hence if $C\in E(X(w))$,
$T_{e}(C)={\mathfrak{g}}_{\alpha}$ for some ${\alpha}<0$. One also has that
$\dim TE(X(w))=|E(X(w))|$.
###### Lemma 1.
Let $\Phi(w)$ denote the set of ${\alpha}<0$ such that $T$ has weight
${\alpha}$ on $T_{e}(C)$ for some $C\in E(X(w))$. Then
$\Phi(w)=\\{{\alpha}<0\mid r_{\alpha}\leq w\\}.$
Thus
(2)
$TE(X(w))=\sum_{{\alpha}\in\Phi(w)}{\mathfrak{g}}_{\alpha}\subseteq\Theta(w).$
By Deodhar’s inequality, $|E(X(w))|\geq\ell(w)$, so $\dim
TE(X(w))\geq\ell(w)$. It follows that $\dim\Theta(w)\geq\ell(w)$ with equality
if and only if $TE(X(w))=\Theta(w)$. Finally, if $X(w)$ is rationally smooth
at $e$, then $\dim TE(X(w))=\ell(w)$. However, knowing $\dim TE(X(w))=\ell(w)$
does not guarantee that $X(w)$ is rationally smooth at $e$.
Note that if $G$ is simply laced, then Theorem 1 is vacuously true since a
result of Dale Peterson’s says that every rationally smooth Schubert variety
in $G/B$ is smooth (see [8] for a discussion and proof). The assumption that
$\dim\Theta(w)=\ell(w)$ is automatically guaranteed when $X(w)$ is rationally
smooth, since $TE(X(w))=\Theta(w)$ for all $w$ in the simply laced case. In
other words, Theorem 1 gives the additional condition under which Peterson’s
result holds for the non $G_{2}$ setting.
The second formulation of Theorem 1 doesn’t involve $\Theta(w)$. Since $X(w)$
is $B$-stable and the identity coset $e$ is fixed by $B$, $T_{e}(X(w))$ and
$\Theta(w)$ are $B$-stable submodules of ${\mathfrak{g}}/{\mathfrak{b}}$. On
the other hand, $TE(X(w))$ in general isn’t $B$-stable.
###### Theorem 2.
Suppose $G$ has no factors of type $G_{2}$, and assume the Poincaré polynomial
of the Schubert variety $X(w)$ in $G/B$ is palindromic, i.e. $X(w)$ is
rationally smooth. Then $X(w)$ is smooth if and only if $TE(X(w))$ is a
$B$-submodule of ${\mathfrak{g}}/{\mathfrak{b}}$.
The proof of this version uses the result that $\Theta(w)$ is the $B$-module
span of $TE(X(w)$ (see [7, Theorem X]). Hence if $X(w)$ is rationally smooth
at $e$ and $TE(X(w))$ is a $B$-submodule of ${\mathfrak{g}}/{\mathfrak{b}}$,
then clearly $\dim\Theta(w)=\ell(w)$.
The third formulation uses a description of the weights occuring in
$\Theta(w)$. This description involves the following convexity condition. Note
that here, the root system $\Phi$ is assumed to lie in the dual space
${\mathfrak{t}}^{*}$.
###### Theorem 3.
Let $w\in W$ be arbitrary, and suppose ${\mathcal{H}}(w)$ is the convex hull
of $\Phi(w)$ in ${\mathfrak{t}}^{*}$ over ${\mathbb{R}}$. Then
(3)
$\Theta(w)=\sum_{{\alpha}\in{\mathcal{H}}(w)\cap\Phi^{-}}{\mathfrak{g}}_{\alpha}.$
Thus $TE(X(w))=\Theta(w)$ if and only if
$\Phi(w)={\mathcal{H}}(w)\cap\Phi^{-}$. Moreover, if $TE(X(w))$ is $B$-stable,
then $TE(X(w))=\Theta(w)$.
###### Proof.
All one needs to do is quote Theorem 3 and Corollary 1 of [7].∎
Our final formulation of the main result involves only $\Phi(w)$.
###### Theorem 4.
Suppose $G$ has no factors of type $G_{2}$, and assume the Poincaré polynomial
of the Schubert variety $X(w)$ in $G/B$ is palindromic, i.e. $X(w)$ is
rationally smooth. Then $X(w)$ is smooth if and only if
$\Phi(w)={\mathcal{H}}(w)\cap\Phi^{-}$.
Let us conclude this introduction with a few remarks about the inclusions
$TE(X(w))\subseteq\Theta(w)\subseteq T_{e}(X(w)).$
In the $ADE$ setting, $TE(X(w))=\Theta(w)$. This is due to two facts: first,
every $T$-line in the reduced tangent cone to $X(w)$ at $e$ arises as
$T_{e}(C)$ for a unique $C\in E(X(w))$, and, second, every root is long (hence
$\Phi(w)={\mathcal{H}}(w)\cap\Phi^{-}$).
## 4\. Proofs of Main Theorems
It is clear from the above discussion that we only need to prove one version
of the main result. Perhaps surprisingly, it turns out that the easiest
version to deal with is the $B$-module version of Theorem 2. We already noted
that the main results hold in types $ACDE$, so it suffices to check the $B$
and $F_{4}$ cases. We will see below that type $B$ is easy to conclude from a
result of Billey. Thus the only sticking point is type $F_{4}$. In order to
get around this difficulty, we use the notion of a stellar root system as
introduced in [5].
Let us begin by recalling what a stellar root system is. A reduced root system
$\Phi$ distinct from $A_{1}$ and $A_{2}$ is called stellar if its Dynkin
diagram is star shaped. That is, there exists a vertex which is on every edge.
Thus the stellar root systems are $B_{2},C_{2},G_{2},A_{3},B_{3},C_{3},D_{4}$.
Let ${\mathbb{R}}\Phi$ denote the real subspace of ${\mathfrak{t}}^{*}$
generated by $\Phi$. A subroot system of $\Phi$ is by definition a subset
$\Psi$ of $\Phi$ of the form $\Psi=\Phi\cap V$, where $V$ is a subspace of
${\mathbb{R}}\Phi$. A subroot system is a root system. Given $w\in W$, the
inversion set of $w$ is the set $I_{\Phi}(w)=\Phi^{+}\cap w(\Phi^{-})$. The
inversion set $I_{\Phi}(w)$ uniquely determines $w$, and for any root
subsystem $\Psi=\Phi\cap V$, $I_{\Phi}(w)\cap V$ is the inversion set of a
unique element $v\in W_{\Psi}$, the Weyl group of $\Psi$. The flattening map
${fl}:W\to W_{\Psi}$
is the assignment $w\to v$. That is, $fl(w)=v$.
Let $\Psi$ be a subsystem of $\Phi$, and let $G_{\Psi}$ (resp. $B_{\Psi}$) be
the subgroup of $G$ generated by $T$ and the root subgroups $U_{\alpha}$,
where ${\alpha}\in\Psi$ (resp. the $U_{\alpha}$ with ${\alpha}\in\Psi^{+}$).
The following result of Billey and Postnikov classifies the rationally smooth
(resp. smooth) Schubert varieties for arbitrary $G$ in terms of stellar
subsystems.
###### Theorem 5.
A Schubert variety $X(w)$ in $G/B$ is rationally smooth (resp. smooth) if and
only if for every stellar subsystem $\Psi$ of $\Phi$, the Schubert variety in
$G_{\Psi}/B_{\Psi}$ corresponding to $fl(w)$ is also rationally smooth (resp.
smooth), where $(G_{\Psi},B_{\Psi})$ is the unique pair of subgroups of
$(G,B)$ determined by $\Psi$.
Let now prove Theorem 2. Suppose $X(w)$ is a rationally smooth Schubert
variety in $G/B$, and let $\Psi$ be a stellar subsystem of $\Phi$. If $v\in
W_{\Psi}$, let $Y(v)$ denote the corresponding Schubert variety in
$G_{\Psi}/B_{\Psi}$. Put $v=fl(w)$. Then $Y(v)$ is rationally smooth. By the
discussion in [2, Section 4],
$Y(v)=X(w)\cap(G_{\Psi}/B_{\Psi}).$
We claim this implies that $TE(Y(v))$ is $B_{\Psi}$-stable. To see this,
suppose ${\alpha}\in\Psi^{-}$ is a weight of $TE(Y(v))$, and suppose that
${\beta}\in\Psi^{+}$ is such that ${\alpha}+{\beta}$ is also a weight of the
tangent space $T_{e}(Y(v))$. In particular, ${\alpha}+{\beta}\in\Psi^{-}$, so
it follows that ${\alpha}+{\beta}$ is a weight in $T_{e}(X(w))$. By
assumption, ${\alpha}+{\beta}$ a weight of $TE(X(w))$, since $TE(X(w))$ is
$B$-stable. Thus there exists a $T$-invariant curve $C$ in $X(w)$ such that
$C^{T}=\\{e,r_{{\alpha}+{\beta}}\\}$ with weight ${\alpha}+{\beta}$ at $e$.
Since ${\alpha}+{\beta}\in\Psi^{-}$, $C\subset G_{\Psi}/B_{\Psi}$ as well, so
$C\subset Y(v)$. Consequently ${\alpha}+{\beta}$ is a weight of $TE(Y(v))$,
hence $TE(Y(v))$ is $B_{\Psi}$-stable. Since $Y(v)$ is rationally smooth and
$\Psi$ is stellar, it suffices to verify that $Y(v)$ is smooth if $\Psi$ is of
type $B_{n}$ for $n=2,3$ or $C_{3}$. This can be checked directly, but it’s
more efficient to use the following lemma.
###### Lemma 2.
Theorem 2 holds when $G$ is of type $B$ or $C$.
###### Proof.
We have already verified this for type $C$. Thus let $X(w)$ be a rationally
smooth Schubert variety in $G/B$ such that $TE(X(w))$ is a $B$-submodule of
$T_{e}(X(w))$, where $G$ is type $B$. By the remarks in Section 3, it suffices
to show $T_{e}(X(w))=TE(X(w)).$ since $\dim TE(X(w))=\ell(w)$. If $X(w)$ is
singular, then one can apply Billey’s pattern avoidance criterion [1, Theorem
3]. Namely, $w$ contains the pattern $\bar{2}\bar{1}$ in signed permutation
notation. By the argument on p.113 of [3], it follows that there exists an
${\alpha}\in\Phi(w)$ and a ${\beta}>0$ such that
${\mathfrak{g}}_{{\alpha}+{\beta}}\subset T_{e}(X(w))$ for which the
inequality $r_{{\alpha}+{\beta}}\leq w$ fails. This says that
${\mathfrak{g}}_{{\alpha}}\subset TE(X(w))$, while
${\mathfrak{g}}_{{\alpha}+{\beta}}\not\subset TE(X(w))$, contradicting the
assumption that $TE(X(w))$ is $B$-stable. Thus $X(w)$ must be smooth at $e$,
consequently smooth. ∎
Theorem 2 now follows from Theorem 5 and the lemma. ∎
Let us next prove Theorem 3. To establish (3), we note the following result
[7, Theorem 2]: For any $x\leq w$, let $\Theta(w,x)$ denote the linear span of
the reduced tangent cone to $X(w)$ at $x$. Let ${\mathcal{H}}(w,x)$ be the
convex hull of $\Phi(w,x)=\\{{\alpha}\in\Phi\mid
x^{-1}({\alpha})<0,~{}r_{\alpha}x\leq w\\}$. Then
$\Theta(w,x)\subset\sum_{{\alpha}\in{\mathcal{H}}(w,x)\cap\Phi}{\mathfrak{g}}_{\alpha},$
and any $\gamma\in{\mathcal{H}}(w,x)\cap\Phi$ which isn’t a $T$-weight of
$\Theta(w,x)$ has the form $\gamma={\beta}+\epsilon\mu,$ where
${\beta}\in\Phi(w,x)$, $\mu>0$, $\epsilon\in\\{1,2\\}$ and $x^{-1}(\mu)<0$.
Thus, if $x=e$, $\gamma$ cannot exist. ∎
The fact that, in general, $\Theta(w)$ is the $B$-module span of $TE(X(w))$ is
proved explicitly in Theorem 3 of [7].
## 5\. Two Examples
The first example shows that Theorem 2 fails without the $G_{2}$ hypothesis.
That is, there exists a rationally smooth but singular Schubert variety $X(w)$
in $G_{2}/B$ for which $TE(X(w))$ is a $B$-submodule of
${\mathfrak{g}}_{2}/{\mathfrak{b}}$. Recall that all Schubert varieties in
$G/B$ are rationally smooth when $G$ has rank two.
Example 1. Let ${\alpha}$ and ${\beta}$ denote respectively the negatives of
long and short simple roots for $G_{2}$ corresponding to $B$, and let
$r=r_{\alpha}$ and $s=r_{\beta}$ be the corresponding reflections. Let
$w=srsrs$ and consider $X(w)$. Now $\ell(w)=5$ and it is not hard to see that
$\Phi(w)=\\{{\alpha},{\beta},{\alpha}+{\beta},{\alpha}+2{\beta},{\alpha}+3{\beta}\\}.$
Thus $TE(X(w))$ is indeed a $B$-submodule of $T_{e}(X(w))$. However, it is
well known that $X(w)$ is singular: for example, see [12]. ∎
Example 2. In this example, we consider a singular Schubert variety in the
flag variety $SO(5)/B$ of type $B_{2}$. Let ${\alpha}$ and ${\beta}$ denote
respectively the negatives of the long and short simple roots as in the
previous example, and let $w=srs$. We claim
$\Phi(w)=\\{{\alpha},{\beta},{\alpha}+2{\beta}\\}$, so $\Theta(w)$ is the
$B$-module with weights
$\\{{\alpha},{\beta},{\alpha}+{\beta},{\alpha}+2{\beta}\\}.$ Thus
$TE(X(w))\neq\Theta(w)$, hence, by Theorem 2, $X(w)$ is singular. Note that in
the signed permutation notation for the elements of $W(B_{2})$ (cf. [1]),
$w=\bar{2}\bar{1}$. It is well known and easy to see that $w$ is the unique
element of $W(B_{2})$ such that $X(w)$ is singular. ∎
It would be interesting to determine which $B$-submodules of
${\mathfrak{g}}/{\mathfrak{b}}$ are tangent spaces at the identity to a smooth
Schubert variety in $G/B$. This might lead to an efficient counting procedure
for enumerating the smooth Schubert varieties.
References
## References
* [1] S. Billey: Pattern avoidance and rational smoothness of Schubert varieties, Adv. Math. 139 (1998), no. 1, 141–156.
* [2] S. Billey and T. Braden:Lower bounds for Kazhdan-Lusztig polynomials from patterns, Transform. Groups 8 (2003), no. 4, 321–332.
* [3] S. Billey and V. Lakshmibai: Singular loci of Schubert varieties, Progress in Mathematics, 182 Birkh user Boston, Inc., Boston, MA, 2000.
* [4] S. Billey and S. A. Mitchell: Smooth and palindromic Schubert varieties in affine Grassmannians J. Algebraic Combin. 31 (2010), no. 2, 169–216.
* [5] S. Billey and A. Postnikov: Smoothness of Schubert varieties via patterns in root subsystems. Adv. in Appl. Math. 34 (2005), no. 3, 447–466.
* [6] J. B. Carrell: The Bruhat Graph of a Coxeter Group, a Conjecture of Deodhar, and Rational Smoothness of Schubert Varieties, Proc. Symp. in Pure Math. 56, No. 2, (1994), Part 1, 53-61.
* [7] J. B. Carrell: The span of the tangent cone of a Schubert variety, Algebraic groups and Lie groups, 51–59, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997.
* [8] J. B. Carrell and J. Kuttler: Singular points of $T$-varieties in $G/P$ and the Peterson map, Invent. Math. 151 (2003), 353–379.
* [9] C. Chevalley: Sur les decompositions cellulaires des espaces $G/B$, Proc. Symp. in Pure Math. 56 (1994), Part I, 1-25.
* [10] V. V. Deodhar: Local Poincar duality and nonsingularity of Schubert varieties, Comm. Algebra 13 (1985), no. 6, 1379–1388.
* [11] D. Kazhdan and G. Lusztig: Schubert varieties and Poincard́uality. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), 185–203, Proc. Sympos. Pure Math., 36, Amer. Math. Soc., Providence, R.I., 1980.
* [12] S. Kumar: Nil Hecke ring and singularity of Schubert varieties, Inventiones Math., 123 (1996), 471–506.
* [13] V. Lakshmibai and B. Sandhya: Criterion for smoothness of Schubert varieties in ${\rm SL}(n)/B$, Proc. Indian Acad. Sci. Math. Sci. 100 (1990) 45–52.
* [14] V. Lakshmibai and C. S. Seshadri: Singular locus of a Schubert variety, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 363–366.
* [15] P. Polo: On Zariski tangent spaces of Schubert varieties, and a proof of a conjecture of Deodhar, Indag. Math. (N.S.) 5 (1994), no. 4, 483–493.
* [16] K. M. Ryan: On Schubert varieties in the flag manifold of ${\rm Sl}(n,C)$, Math. Ann. 276 (1987) 205–224.
James B. Carrell
Department of Mathematics
University of British Columbia
Vancouver, Canada V6T 1Z2
carrell$@$math.ubc.ca
|
arxiv-papers
| 2010-07-26T01:16:31 |
2024-09-04T02:49:11.858346
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "James B. Carrell",
"submitter": "James B. Carrell",
"url": "https://arxiv.org/abs/1007.4365"
}
|
1007.4454
|
Particle Physics Group, School of Physics and Astronomy, University of
Manchester, UK
# Top Quark Spin Correlations at the Tevatron
Tim Head on behalf of the CDF and D0 collaborationsins:manc thead@fnal.gov
###### Abstract
Recent measurements of the correlation between the spin of the top and the
spin of the anti-top quark produced in proton anti-proton scattering at a
centre of mass energy of $\sqrt{s}=1.96\,\mathrm{TeV}$ by the CDF and D0
collaborations are discussed. Using up to $4.3\,\mathrm{fb^{-1}}$ of data
taken with the CDF and D0 detectors the spin correlation parameter $C$, the
degree to which the spins are correlated, is measured in dileptonic and
semileptonic final states. The measurements are found to be in agreement with
Standard Model predictions.
## 1 Introduction
Top quark physics at hadron colliders plays an important role in testing the
Standard Model of particle physics and its possible extensions.
In the Standard Model the top quark has a very short lifetime,
$\tau_{\nicefrac{{1}}{{2}}}\approx 5\times 10^{\textrm{-}25}\,\mathrm{s}$,
therefore the definite spin state in which the top anti-top pair is produced
is not spoilt by hadronisation effects. As a result, the direction of the spin
of the top quark is reflected in the angular distributions of its decay
products. In contrast to this, the spin of light quarks will flip before they
decay, making the spin state they are produced in unobservable. Furthermore,
the theoretical calculations necessary in order to predict the angular
distributions can be performed for top pairs, resulting in precise theoretical
predictions which can be tested by experiment. New physics in either the
production or decay mechanism would modify these angular distributions, making
spin correlations sensitive to new physics.
Until recently only one measurement of spin correlations has been performed.
Using $125\,\mathrm{pb^{\textrm{-}1}}$ of data taken during Run I of the
Tevatron collider at Fermilab the D0 collaboration measured a correlation
coefficient in agreement with the Standard Model [1]. However, since the
sample contained only six events, the sensitivity was too low to rule out the
hypothesis of no spin correlations. Recently the CDF and D0 collaborations
performed measurements using up to $4.3\,\mathrm{fb^{\textrm{-}1}}$ of data
taken with the CDF and D0 [2] detectors, the results of which are discussed
below.
## 2 Observables
In strong interactions the top and anti-top quark are produced unpolarised at
hadron colliders, however the $t\bar{t}$ system is in a definite spin state.
At the Tevatron about $85\,\mathrm{\%}$, at next to leading order, of top
quark pairs are produced via quark anti-quark annihilation. At threshold these
$t\bar{t}$ systems will be in a ${}^{3}S_{1}$ state, whereas the
$15\,\mathrm{\%}$ of top quark pairs produced via gluon fusion will be in a
${}^{1}S_{0}$ state. In the first case the top and anti-top quark will tend to
have their spins parallel, in the second case they tend to be anti-parallel.
One therefore expects to observe a correlation between the direction of the
spins.
The strength of the correlation due to the production mechanism can be
expressed as the asymmetry $A$,
$A=\frac{N_{\uparrow\uparrow}+N_{\downarrow\downarrow}-N_{\downarrow\uparrow}-N_{\uparrow\downarrow}}{N_{\uparrow\uparrow}+N_{\downarrow\downarrow}+N_{\downarrow\uparrow}+N_{\uparrow\downarrow}}$
(1)
between the number of events with spins parallel, $N_{\uparrow\uparrow}$ and
$N_{\downarrow\downarrow}$, and the number of events with spins anti-parallel,
$N_{\uparrow\downarrow}$ and $N_{\downarrow\uparrow}$.
In order to measure the direction of the spin vector a quantisation axis needs
to be defined. At the Tevatron three sets of quantisation axes, referred to as
“spin basis”, are commonly used. They are shown in Figure 1.
The simplest is the so called “beamline basis” in which the direction of one
of the incoming hadrons is used as quantisation axis. This basis is easy to
construct and is optimal for $t\bar{t}$ systems produced at threshold. The
production asymmetry has been calculated at next to leading order (NLO) in QCD
as $A=0.777$ [3].
The second basis is the “helicity basis” in which the momentum of the
(anti)top quark in the top-anti-top quark zero momentum frame is used to
quantise the (anti)top quark spin. At the Tevatron the strength of the
correlation is smaller than in the “beamline basis”, in NLO QCD
$A=\textrm{-}0.352$. The opposite sign arises due to the fact that the spins
tend to be anti-parallel in this basis.
Finally the third basis is the “off-diagonal basis”. The direction of the
quantisation axes are defined by the angle $\omega$ with respect to the
(anti)top quark momentum. The angle $\omega$ is given by
$\tan\omega=\sqrt{1-\beta^{2}}\tan\theta$, where $\beta$ is the speed of the
top quark and $\theta$ is its scattering angle. This basis interpolates
between the “beamline basis” close to threshold (low $\beta$) and the
“helicity basis” above threshold (large $\beta$). The production asymmetry is
$A=0.782$. While this is slightly larger than in the “beamline basis” it is
more complex to reconstruct.
Figure 1: The three choices of quantisation axis used at the Tevatron. The
“beamline basis” (left) is optimal for top pairs produced at threshold, the
“helicity basis” (centre) is used for above threshold top pairs and the
“helicity basis” (right) interpolates between the two.
The angular distribution of decay product $i$ in the top quark rest frame is
given by:
$\frac{1}{\sigma}\frac{\textrm{d}\sigma}{\textrm{d}\cos\theta_{i}}=\frac{1}{2}\left(1-\alpha_{i}\cdot\cos\theta_{i}\right)$
(2)
where $\theta_{i}$ is the angle between the direction of flight of decay
product $i$ and the direction of the spin vector; $\alpha_{i}$ is the so-
called spin analysing power. From Equation 2 it is clear that the angular
distribution of a decay product with $\alpha_{i}=0$ contains no information
about the direction of the top quark spin and the angular distribution of a
decay product with $\alpha_{i}=\pm 1$ will contain most information. The spin
analysing power of the various top quark decay products are listed in Table 1.
The particles with the highest spin analysing power are the lepton and the
down type quark from the W boson decay.
Table 1: Spin analysing power of the top quark decay products. The up type quark, down type quark, neutrino and lepton are the decay products of the W boson. For the antiparticles the sign is reversed. | lepton, down type quark | neutrino | up type quark | b quark
---|---|---|---|---
analysing power $\alpha$ | +1 | +0.31 | +0.31 | +0.41
In order to observe a correlation between the direction of the spin of the top
and anti-top quark one must consider the angle $\theta$ of a decay product of
the top quark and the angle of a decay product of the anti-top quark
simultaneously. The double differential distribution for a top quark decay
product $i$ and anti-top quark decay product $j$ is given by:
$\frac{1}{\sigma}\frac{\mathrm{d^{2}}\sigma}{\mathrm{d}\cos\theta_{i}\mathrm{d}\cos\theta_{j}}=\frac{1}{4}\left(1-A\alpha_{i}\alpha_{j}\cos\theta_{i}\cos\theta_{j}\right)$
(3)
where $\sigma$ is the total cross section, $A$ is the production asymmetry,
and $\alpha_{i,\,j}$ is the spin analysing power of the $i,\,j$-th decay
product. In all analyses presented here the spin correlation parameter
$C=A\alpha_{i}\alpha_{j}$ is measured. A measurement of the distribution given
in Equation 3 should be performed as follows:
1. 1.
Reconstruct the top and anti-top quark momenta in the laboratory frame,
2. 2.
Perform a boost from the laboratory frame to the rest frame of the $t\bar{t}$
system. Define the vectors $\hat{b}_{i}$ and $\hat{b}_{j}$ along which to
quantise the top and anti-top quark spins respectively.
3. 3.
Boost the top (anti-top) quark decay product to the top (anti-top) quark rest
frame and calculate $cos\theta_{i,\,j}=\hat{b}_{i,\,j}\cdot\hat{q}_{i,\,j}$.
The difference between the case of no spin correlations, $A=0$, and SM spin
correlations as measured in the “beamline basis”, $A=0.777$, using leptons as
spin analysers is shown in Figure 2.
Figure 2: The distribution $\cos\theta_{1}\cos\theta_{2}$ for a sample of top
anti-top quark events using generated partons. With spin correlations (red
dashed) and without (black solid). Here both top quarks decayed to leptons
which subsequently were used as spin analysers [4].
## 3 Measurements
While in theory the down type quark is as powerful a spin analyser as the
lepton, it is more difficult to identify in practise. This leads to two
different approaches. In the first one, one selects a pure sample of top pairs
in which both the top and anti-top quark decay to leptons. In the second a
sample with higher statistics is selected by requiring only one top quark to
decay to a lepton. In the following the advantages, challenges and results are
discussed for both approaches.
### 3.1 Dilepton final states
The advantages of the dilepton final state are that it is simple to identify
the final state particles of interest and the high purity of the sample. The
disadvantage is that one suffers from a low branching ratio and needs to deal
with two neutrinos when reconstructing the kinematics of the event. Both CDF
and D0 select events with two high $p_{t}$ leptons of opposite charge and at
least two jets. The detailed event selections are described in References [5,
4] for CDF and D0, respectively. In final states with same flavour leptons
($e^{+}e^{\textrm{-}}$ and $\mu^{+}\mu^{\textrm{-}}$) the main background
arises from Drell-Yan,
$Z\textrm{/}\gamma*\rightarrow\ell^{\textrm{-}}\ell^{\textrm{+}}$, production.
In the $e\mu$ final state the main background is instrumental, this occurs
mainly due to W+jets events in which a jet is misidentified as a lepton. The
second largest background arises due to semileptonic decays of
$Z\textrm{/}\gamma*\rightarrow\tau^{\textrm{-}}\tau^{\textrm{+}}$. Further
sources of background in all three final states are the diboson processes WW,
WZ and ZZ. Both signal and background are modelled using Monte Carlo
simulation, except for the instrumental background which is estimated from
data.
In order to reconstruct the momentum of the top and anti-top quark, one needs
to deal with the two neutrinos in the final state. To fully characterise the
kinematics of the final state one needs 18 quantities, assuming the masses of
the final state particles are known. While the leptons and jets are observable
in the detector, the two neutrinos escape detection. It is possible to infer
the sum of the momenta of the neutrinos in the $x$ and $y$ plane from the
missing transverse energy, ${\displaystyle{\not}E_{T}^{x}}$ and
${\displaystyle{\not}E_{T}^{y}}$. Using this information and making an
assumption about the mass of the W boson and the top quark it is possible to
write down a set of quartic equations which fully describe the final state.
Solving them yields up to four solutions per event. Additionally one needs to
try both lepton-jet pairings which increases the number of possible solutions
to eight.
In the CDF measurement a likelihood function is constructed from several
observables and maximised with respect to the unknown neutrino momenta
($\vec{p}_{\nu}$, $\vec{p}_{\bar{\nu}}$) and the energies of the bottom quark
jets ($E_{b}^{\textrm{guess}}$, $E_{\bar{b}}^{\textrm{guess}}$):
$\begin{array}[]{l}L\left(\vec{p}_{\nu},\,\vec{p}_{\bar{\nu}},\,E_{b}^{\textrm{guess}},\,E_{\bar{b}}^{\textrm{guess}}\right)=P\left(p_{z}^{t\bar{t}}\right)P\left(p_{T}^{t\bar{t}}\right)P\left(M_{t\bar{t}}\right)\times\\\
\frac{1}{\sigma_{b}}\exp\left(-\frac{1}{2}\left(\frac{E_{b}^{\textrm{meas}}-E_{b}^{\textrm{guess}}}{\sigma_{b}}\right)^{2}\right)\times\frac{1}{\sigma_{\bar{b}}}\exp\left(-\frac{1}{2}\left(\frac{E_{\bar{b}}^{\textrm{meas}}-E_{\bar{b}}^{\textrm{guess}}}{\sigma_{\bar{b}}}\right)^{2}\right)\times\\\
\frac{1}{\sigma_{x}^{\textrm{MET}}}\exp\left(-\frac{1}{2}\left(\frac{{\displaystyle\not}E_{x}^{\textrm{meas}}-{\displaystyle\not}E_{x}^{\textrm{guess}}}{\sigma_{x}^{\textrm{MET}}}\right)^{2}\right)\times\frac{1}{\sigma_{y}^{\textrm{MET}}}\exp\left(-\frac{1}{2}\left(\frac{{\displaystyle\not}E_{y}^{\textrm{meas}}-{\displaystyle\not}E_{y}^{\textrm{guess}}}{\sigma_{y}^{\textrm{MET}}}\right)^{2}\right)\end{array}$
where $P\left(p_{z}^{t\bar{t}}\right)$, $P\left(p_{T}^{t\bar{t}}\right)$ and
$P\left(M_{t\bar{t}}\right)$ are probability density functions obtained from
Pythia $t\bar{t}$ Monte Carlo events, $E_{b,\,\bar{b}}^{\textrm{meas}}$ the
measured energies of the bottom/anti-bottom quark jets,
${\displaystyle\not}E_{x,\,y}^{\textrm{meas}}$ the measured components of
${\displaystyle\not}E_{T}$, and $\sigma_{i}$ the respective resolutions. The
maximisation is performed for both lepton-jet pairings and the combination
with the larger $L$ is kept.
As the Pythia Monte Carlo simulation does not contain spin correlations,
templates for values of $C=-1,\,0.8,\dots,\,1$ are obtained by reweighting the
signal Monte Carlo at the generator level using a weight $w\sim
1-C\cdot\cos\theta_{1}\cos\theta_{2}$. For each value of $C$ a two dimensional
template in the decay angles of the lepton and anti-lepton,
$\cos\theta_{\ell^{+}}$ , $\cos\theta_{\ell^{\textrm{-}}}$ and a template in
the decay angles of the bottom quark jets, $\cos\theta_{b}$ ,
$\cos\theta_{\bar{b}}$ is created. The two templates are fit with an
analytical function $f^{\ell,\,b}\left(x,\,y;\,C\right)$. The measurement is
then performed on the $N$ candidate events by maximising the likelihood
function:
$L\left(C\right)=\prod_{i=0}^{N}f^{l}\left(x,\,y;\,C\right)f^{b}\left(x,\,y;\,C\right).$
In order to extract limits from the measurement, a confidence belt according
to the Feldman-Cousins prescription [6] is created. This naturally includes
both statistical and systematic uncertainties and allows one to decide before
looking at the data whether to quote a one or two sided limit. Using
$2.8\,\mathrm{fb^{\textrm{-}1}}$ of data the best fit value is
$C=0.32_{-0.78}^{+0.55}\textrm{(stat + syst)}$ and the corresponding
confidence belts are shown in Figure 3. The measurement was performed in the
“helicity basis”. The result is consistent with the expected value of
$C=0.782$. The largest contributions to the systematic uncertainty come from
evaluating the PDF uncertainties and the finite number of Monte Carlo events
used to form the templates.
Figure 3: The 68% (stat only), 68% and 95% Confidence Level intervals
constructed according to the Feldman-Cousins prescription including
statistical and all systematic uncertainties for the CDF measurement. The best
fit value is $C=0.32_{-0.78}^{+0.55}$ [5].
At D0 the neutrino weighting technique is used to solve for the event
kinematics. By making an assumption about the rapidity, $\eta$, of the
neutrino and anti-neutrino, it is possible to solve the event kinematics,
while not using ${\displaystyle{\not}E_{T}^{x}}$ and
${\displaystyle{\not}E_{T}^{y}}$ in the process but instead to assign a
weight, $w$, to each solution given by:
$w=\exp\left(-\frac{\left({\displaystyle{\not}E_{T}^{x}-\nu_{x}-\bar{\nu}_{x}}\right)^{2}}{\sigma^{2}}\right)\times\exp\left(-\frac{\left({\displaystyle{\not}E_{T}^{y}-\nu_{y}-\bar{\nu}_{y}}\right)^{2}}{\sigma^{2}}\right)$
where $\nu_{x,\,y}$ and $\bar{\nu}_{x,\,y}$ are the x and y components of the
neutrino and anti-neutrino momentum for a given solution and $\sigma$ is the
${\not}E_{T}^{x}$ resolution. Many solutions are obtained by sampling the
neutrino and anti-neutrino rapidity based on Monte Carlo simulation. No
dependence of the neutrino rapidity on the presence of spin correlations is
observed. The weighted mean of all solutions for an event is used as estimator
for the true value of $\cos\theta_{\ell^{+}}\cos\theta_{\ell^{-}}$.
As for the CDF measurement, the Pythia Monte Carlo simulation is used to model
the signal sample. A one dimensional template in the variable
$\cos\theta_{\ell^{+}}\cos\theta_{\ell^{-}}$ is created for $C=0$ and
$C=0.777$ by reweighting the distribution at the generator level. In order to
extract a value of $C$ a linear combination of the two templates is fit to the
data.
Pseudo-experiments are created for each value of $C$ and fit with signal and
background templates. Each source of systematic uncertainty is considered as a
nuisance parameter during the fit. Feldman-Cousins confidence belts are
constructed from the pseudo experiments. Using up to
$4.2\,\mathrm{fb^{\textrm{-}1}}$ of data the best fit value is
$C=-0.17_{-0.53}^{+0.64}\textrm{(stat + syst)}$. In this measurement the
“beamline basis” was used and the measured value is consistent with the
Standard Model expectation of $C=0.777$ at the two sigma confidence level.
The two main sources of systematic uncertainty are the variation of the
assumed top mass during the event reconstruction from $175\,\mathrm{GeV}$ to
$170\,\mathrm{GeV}$ and the test of the reweighting method. For the latter,
the two Pythia signal templates were replaced by Alpgen, which contains spin
correlations, and MC@NLO where spin correlations were turned off.
Figure 4: Left: the 68%, 95% and 99% Feldman-Cousins confidence belts are
shown. The best fit value can be read of at the intersection of the dashed
black line and the thin blue line. Right: The sum of all dilepton channels is
shown. The open black histogram shows the expected distribution for the case
of no spin correlations, $C=0$ and the filled red histogram the expected
distribution for Standard Model spin correlations, $C=0.777$ [4].
### 3.2 Semileptonic final states
Selecting semileptonic events results in a higher yield, but the challenge is
to identify the down type quark. This is done probabilistically by choosing
the jet closest to the bottom type jet in the W boson rest frame [7], which
will result in picking the correct jet about 60% of the time.
Events are selected by requiring at least one high $p_{T}$, central lepton,
large missing transverse energy and four or more jets, one of which must be
identified as a b-jet. The backgrounds are estimated both from simulation and
data. For details of the selection see Reference [8]. Using
$4.3\,\mathrm{fb^{\textrm{-}1}}$ of data a total of 1001 events are selected
of which 786 are expected to be top pair events.
When produced in pairs the top and anti-top quark either have the same
helicity or opposite helicity. The fraction of top pairs with opposite
helicity is given by:
$f_{O}=\frac{\sigma\left(\bar{t}_{R}t_{L}\right)+\sigma\left(\bar{t}_{L}t_{R}\right)}{\sigma\left(\bar{t}_{R}t_{R}+\bar{t}_{L}t_{L}+\bar{t}_{R}t_{L}+\bar{t}_{L}t_{R}\right)},$
where $\sigma\left(\bar{t}_{L,\,R}t_{L,\,R}\right)$ denotes the cross section
for each possible helicity configuration. Using Equation 1 one can show that a
measurement of $f_{O}$ is equivalent to a measurement of $A$ in the helicity
basis.
One template for top pairs with same helicity and one template for top pairs
of opposite helicity are created using a modified version of the Herwig event
generator. The opposite helicity fraction is extracted with a binned maximum
likelihood fit of the two templates to the data, with contributions from
backgrounds taken into account. The best fit value is $f_{O}=0.80\pm
0.26\textrm{(stat + syst)}$ or equivalently $A=2f_{O}-1=0.60\pm
0.52\textrm{(stat + syst)}$. This is consistent with the Standard Model
expectation of $A=0.4$. The two main systematic uncertainties are Monte Carlo
statistics and jet energy scale.
Figure 5: The best fit of same helicity, opposite helicity and background
templates for the CDF semileptonic decay channel. On the (left) the
distribution of the product of the decay angle of the lepton and the bottom
quark. On the (right) the distribution of the product of the decay angle of
the lepton and the down type quark. The best fit value from a simultaneous fit
to both distributions is $f_{O}=0.8\pm 0.26\textrm{(stat + syst)}$ or
$C=0.6\pm 0.52\textrm{(stat + syst)}$ [8].
## 4 Conclusions
The spin correlation parameter $C$ has been measured in dilepton and
semileptonic decays of top and anti-top quark pairs using up to
$4.3\,\mathrm{fb^{-1}}$ of data collected with the CDF and D0 detectors.
Measurements were performed in the “beamline”, “helicity” and “off-diagonal”
bases. The measurements are found to be in agreement with the Standard Model
predictions. All three measurements are still statistically limited.
Considering that the Tevatron collider has delivered nearly twice as much
integrated luminosity since the analyses have been performed, updates of all
measurements can be expected soon.
## References
* [1] Abbott B. et al., Phys. Rev. Lett.852000256.
* [2] Abazov V. M. et al., Nucl. Instrum. Meth.A5652006463.
* [3] Bernreuther W., Brandenburg A., Si Z. G. Uwer P., Nucl. Phys.B690200481.
* [4] The D0 collaboration, Spin Correlations in $t\bar{t}$ Production in Dilepton Final States, D0 note 5950-CONF.
* [5] The CDF Collaboration, A Measurement of $t\bar{t}$ Spin Correlations Coefficient in 2.8 fb${}^{\textrm{-}1}$ Dilepton Candidates, CDF note 9824.
* [6] Feldman G. J. Cousins R. D., Phys. Rev.D5719983873.
* [7] Mahlon G. Parke S. J., Phys. Rev.D5319964886.
* [8] The CDF Collaboration, Measurement of $t\bar{t}$ Helicity Fractions and Spin Correlation Using Reconstructed Lepton+Jets Events, CDF note 10048.
|
arxiv-papers
| 2010-07-26T13:37:38 |
2024-09-04T02:49:11.865031
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tim Head",
"submitter": "Tim Head",
"url": "https://arxiv.org/abs/1007.4454"
}
|
1007.4634
|
# Decoherence-Based Quantum Zeno Effect in a Cavity-QED System
D. Z. Xu Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing, 100190, China Q. Ai Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing, 100190, China C. P. Sun suncp@itp.ac.cn
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing,
100190, China
###### Abstract
We present a decoherence-based interpretation for the quantum Zeno effect
(QZE) where measurements are dynamically treated as dispersive couplings of
the measured system to the apparatus, rather than the von Neumann’s
projections. It is found that the explicit dependence of the survival
probability on the decoherence time quantitatively distinguishes this dynamic
QZE from the usual one based on projection measurements. By revisiting the
cavity-QED experiment of the QZE [J. Bernu, et al., Phys. Rev. Lett, 101,
180402 (2008)], we suggest an alternative scheme to verify our theoretical
consideration that frequent measurements slow down the increase of photon
number inside a microcavity due to the nondemolition couplings with the atoms
in large detuning.
###### pacs:
03.65.Xp, 03.65.Yz, 42.50.Ct
Introduction – It usually follows from the von Neumann’s postulate of wave
packet collapse (WPC) that the frequent measurements about whether the system
stays in its initial unstable state would inhibit the transitions to other
states Misra77 . This inhibition phenomenon is now called the quantum Zeno
paradox or the quantum Zeno effect (QZE). Some experiments, which claimed the
verifications of the QZE for various physical systems Itano90 ; Fischer01 ;
Streed06 , seemed to provide clear evidence supporting the necessariness of
the WPC in the logical system of quantum mechanics. However, many physicists
wondered whether the QZE phenomena were really rooted in the WPC-based
measurement (or called the projection measurement) Frerichs91 ; Schulman98 ;
Perse ; Cook88 ; Ballentine91 ; Petrosky90 ; Pascazio94 ; Sun95 .
In the early days of the discovery of the QZE, Asher Peres demonstrated that
the QZE-like phenomenon could also be explained in terms of the strong
interaction between the observed system and an external agent Perse . When
Itano et al. carried out a QZE experiment based on the theoretical proposal of
Cook Cook88 and claimed the role of the projection measurement Itano90 , some
authors argued that no WPC really happened since the existing experimental
data could also be recovered by unitary dynamic calculations without invoking
the WPC Ballentine91 ; Petrosky90 . Furthermore, a recent experiment in
cavity-QED system for freezing the growth of the photon number in a cavity was
explained in terms of the WPC-based QZE Bernu08 . It awakened us to seriously
revisit the problem whether this QZE phenomenon depends on the von Neumann’s
postulate Neumann55 , which lies in the core of Copenhagen’s quantum mechanics
interpretation (QMI). We expect the similar experiment and its extension could
provide an accessible way to well clarify the physical distinguishability of
different QMIs in accounting for the QZE.
In this Letter, We generally describe the QZE by a unitary evolution regarding
the quantum measurement as a dispersive coupling for decoherence Zurek03 ;
Sun93 . With respect to the system’s eigenstates being measured, the
decoherence-based quantum measurement is generally formulated by a diagonal
normal operator valued in the apparatus’ observable (we call it the
measurement operator) Zeh03 ; Sun95 ; Sun94 . Then we show the frequent “bang-
bang” insertions of such measurement operators in the original time evolution
decohere the system. These frequent measurements cancel the off-diagonal
elements of the system’s density matrix through the destructive interference.
Therefore, the transitions among the eigenstates of the system are inhibited.
This universal proof deals with quantum measurement as a dynamic dephasing
process, rather than an instantaneous collapse. Thus the measurement time is
introduced as a crucial parameter to signature our decoherence-based model in
contrary to the conventional WPC-based one. By re-considering the cavity-QED
experiment Bernu08 where the periodically driven cavity field is measured by
the nondemolition dispersive couplings to the injected off-resonant atoms, we
calculate the two-dimensional “phase diagrams” of an alternative experimental
scheme with respect to the measurement time and the “bang-bang” time interval.
Characterizing the dynamic nature of the QZE, the dependence of the survival
probability on the measurement time explicitly reflect the experimentally
testable difference between two QMIs related to the WPC and dispersive
couplings respectively.
_Decoherence-induced quantum Zeno effect –_ Now we develop a general approach
for QZE based on dynamic description of quantum measurement Pascazio94 ; Sun94
; Sun95 . The dispersive couplings of the measured system $S$ to the apparatus
$A$ lead to a time evolution of the total system $S$ plus $A$ from the initial
state
$\left|\varphi(0)\right\rangle=\sum_{j}c_{j}\left|s_{j}\right\rangle\otimes\left|a\right\rangle$
to an entangled state
$\left|\varphi(t)\right\rangle=M(t)\left|\varphi(0)\right\rangle\equiv\sum_{j}c_{j}\left|s_{j}\right\rangle\otimes\left|a_{j}\right\rangle$.
Here $\left|s_{j}\right\rangle$ $\left(j=1,2,\ldots\right)$ serves as an
orthonormal basis of the Hilbert space $\mathcal{H}_{S}$ of $S$, while
$\left|a\right\rangle$ is the initial state of $A$. The unitary measurement
operator $M(t)$ is a diagonal normal matrix with elements
$M_{jj}=\exp(-i\hat{h}_{j}t)$ for the branch Hamiltonian $\hat{h}_{j}$ being a
Hermitian operator on the Hilbert space $\mathcal{H}_{A}$ of $A$. The final
state
$\left|a_{j}\right\rangle\equiv\left|a_{j}(t)\right\rangle=\exp(-i\hat{h}_{j}t)\left|a\right\rangle$
of $A$ corresponds to the system’s state $\left|s_{j}\right\rangle$.
Obviously, $M(t)$ is capable of defining a nondemolition measurement ndm . An
ideal measurement could well distinguish the apparatus state
$\left|a_{j}\right\rangle$ from $\left|a_{j^{\prime}}\right\rangle$, i.e.,
$\left\langle a_{j^{\prime}}|a_{j}\right\rangle=\delta_{jj^{\prime}}$. In this
ideal case, the reduced density matrix of the system is depicted by
$\rho_{s}(t)=\text{Tr}_{A}(\left|\varphi(t)\right\rangle\left\langle\varphi(t)\right|)$
with vanishing off-diagonal elements.
$U(t)$ is defined as the unitary evolution operator of $S$ in the absence of
the above “measurement”. Then we generally describe the QZE phenomenon by a
unitary evolution matrix
$U_{c}(t)=U_{c}(\tau,\tau_{m})=[M(\tau_{m})U(\tau)]^{N}$ (see Fig. 1) with a
fixed duration $t=N\tau$. Here $\tau$ indicates a small time interval for
which the system evolves freely, and a measurement with shorter time
$\tau_{m}$ is performed at the end of each $U(\tau)$. Actually, the free
evolution co-exists with the measurements through the whole QZE process, but
it could be ignored when measurement is turned on since the apparatus induces
a fast decoherence. An ideal measurement requires a very short $\tau_{m}$, but
a finite $\tau_{m}$ will reflect the dynamic feature of the realistic
measurements. Usually, $U(\tau)$ does not commutate with $M(\tau_{m})$ so that
it can induce the transitions among states $\left|s_{j}\right\rangle$. We re-
write $U_{c}(t)$ as a $N$-multi-product
$U_{c}(\tau,\tau_{m})=\left[\prod_{n=1}^{N}U_{n}(\tau)\right]M^{N}\text{,}$
(1)
where the factors $U_{n}(\tau)=M^{n}U(\tau)M^{-n}$ for$\ M\equiv M(\tau_{m})$
and $n=1,2,...,N$. For a very short $\tau$ or a very large $N$, it could be
approximated as $U_{n}(\tau)\simeq 1-i\tau M^{n}HM^{-n}\equiv 1-i\tau H_{n}$.
If $M$ is not degenerate, we have
$U_{c}(\tau,\tau_{m})\simeq\left(1-itH_{d}-i\frac{t}{N}S\right)M^{N}\text{,}$
(2)
where $A_{d}$ and $A_{\text{off}}$ denote the diagonal and off-diagonal parts
of matrix $A$, respectively. The summation $S=\sum_{n}(H_{n})_{\text{off}}$ is
convergent as $N\rightarrow\infty$ or $\tau\rightarrow 0$ for fixed $t=N\tau$,
since $S=\sum_{j\neq
j^{\prime}}\Lambda_{jj^{\prime}}H_{jj^{\prime}}\left|s_{j}\right\rangle\langle
s_{j^{\prime}}|$, where
$\Lambda_{jj^{\prime}}=\frac{\sin(\frac{1}{2}\tau_{m}N\Delta_{jj^{\prime}})}{\sin(\frac{1}{2}\tau_{m}\Delta_{jj^{\prime}})}e^{-i\tau_{m}(N+1)\Delta_{jj^{\prime}}/2}$
(3)
for $\Delta_{jj^{\prime}}=\hat{h}_{j}-\hat{h}_{j^{\prime}}$.
$\Lambda_{jj^{\prime}}$ is a finite number when $\Delta_{jj^{\prime}}\neq 0$,
then in the large-$N$ limit, the QZE is achieved as
$\lim_{N\rightarrow\infty}U_{c}(\tau,\tau_{m})\rightarrow
e^{-iH_{d}t}[1-i\mathcal{O}(\frac{t}{N})]M^{N}\text{.}$ (4)
Therefore, the time evolution with very frequent $M$-kicks will keep the
system in its initial state because $U_{c}(\tau,\tau_{m})$ approaches a
diagonalized unitary matrix $\exp(-iH_{d}t)$.
This argument generally proves the QZE in a dynamic version. Thus the frequent
measurements (for $N\rightarrow\infty$) based on the decoherence model indeed
result in the QZE even though no WPC is used. We remark that the similar
arguments for the QZE have been given by making use of the von Neumann’s
quantum ergodic theorem Facchi04 .
_Cavity-QED setup for testing decoherence-based quantum Zeno effect –_ The
experiment based on high-$Q$ superconducting cavity has explicitly
demonstrated the increase of the photon number inside the cavity is suppressed
by the continuous measurements Bernu08 . In this experiment, a series of
microwave pulses resonant with the cavity are injected into the cavity, which
corresponds to the $U$-process; between every two adjacent pulses an ensemble
of off-resonant atoms are sent into the cavity to probe the average photon
number, playing the part of the $M$-process. A single QND probe is actually a
dynamic process and changes the cavity field by a phase factor instead of its
photon number. Even we do not read out the photon number after each probe, the
QND coupling of the cavity field to the off-resonant atom can result in the
phase random in the accumulation of these phase factors thus leads to freezing
the photon number in its initial state. We propose an alternative cavity-QED
scheme to verify this illustration.
Figure 1: Controlled evolution process containing $N$ unitary evolution $U$
-processes and $N$ dynamic measurement $M$-processes. The $y$-axis represents
the strength of the interaction.
Let the cavity be initially prepared in the vacuum state
$\left|0\right\rangle$ with an ensemble of off-resonant atoms located in it.
Then classical driving laser pulses are sequentially injected into the cavity.
Each pulse is applied for a duration $\tau$. This unitary evolution of the
cavity field is described by the Hamiltonian
$H_{U}(t)=\omega a^{\dagger}a+fe^{-i\omega_{F}t}a^{\dagger}+h.c.\text{,}$ (5)
where $\omega$ is the frequency of the cavity, $f$ and $\omega_{F}$ the
strength and the frequency of the driving field respectively, $a$ and
$a^{\dagger}$ the annihilation and creation operators of the cavity field. The
driving pulse is peaked at the frequency resonant with the cavity, i.e.,
$\omega_{F}\approx\omega$. Compared to the strength of the driving field, the
interaction between the atom and the cavity field is rather weak, and thus can
be omitted when the pulse is switched on. In the interval when we turn off the
driving field, the atom-field interaction becomes important. Since the energy
level spacing $\omega_{a}$ of the atom and the frequency $\omega$ of the
cavity are largely detuned, adiabatic elimination results in an effective
measurement Hamiltonian
$H_{M}=\frac{g^{2}}{\Delta}a^{\dagger}a\left(\left|+\right\rangle\left\langle+\right|-\left|-\right\rangle\left\langle-\right|\right)\text{.}$
(6)
Here $\left|\pm\right\rangle$ are the two atomic energy levels, $g$ the vacuum
Rabi frequency defining the atom-cavity coupling and
$\Delta=\omega-\omega_{a}$ the atom-cavity detuning. The unitary evolution
dominated by $H_{M}$ is regarded as a QND measurement, for the atom records
the information of the photon number of the cavity field by its phase of the
$\left|\pm\right\rangle$ superposition. The whole experimental procedure
consists of a series of dynamic processes described by $H_{U}$ and $H_{M}$
alternatively the same as demonstrated in Fig. 1, but the strengths of $U$ and
$M$ processes are reversed. The probe of the photon number is only carried out
after the last driving pulse.
_Free evolution and decoherence-based measurement –_ The time evolution of the
cavity field governed by $H_{U}\left(\tau\right)$ is described by phase-
modulated displacement operator
$U(\tau)=e^{i\omega
a^{\dagger}a\tau}e^{i\phi\left(\tau\right)}D\left[\alpha\left(\tau\right)\right]\text{,}$
(7)
where
$D\left[\alpha\left(\tau\right)\right]=\exp[\alpha\left(\tau\right)a^{\dagger}-\alpha^{\ast}\left(\tau\right)a]$
with the displacement parameter
$\alpha\left(\tau\right)=\left[\exp\left(-i\delta\tau\right)-1\right]f/\delta$,
and the phase factor is
$\phi\left(\tau\right)=\left(\sin\delta\tau-\delta\tau\right)f^{2}/\delta^{2}$,
$\delta=\omega_{F}-\omega$. Here the Wei-Norman algebra method Wei63 is used
in deriving $U(\tau)$.
In a cavity in the vacuum state $\left|0\right\rangle$, the atom is initially
prepared in the superposition state
$\left|\phi(0)\right\rangle=\left(\left|+\right\rangle+\left|-\right\rangle\right)/\sqrt{2}$.
After the first driving pulse applied for time $\tau$, the total system
evolves into
$\left|\psi(\tau)\right\rangle=\left|\phi(0)\right\rangle\otimes\left|\alpha\left(\tau\right)\exp\left(-i\omega\tau\right)\right\rangle$.
We can see that the average photon number
$\bar{n}=\left|\alpha\left(\tau\right)\right|^{2}\approx f^{2}\tau^{2}$
quadratically depends on $\tau$, for $\tau$ is a sufficiently short interval.
Then the pulse is turned off and the atom-cavity field interaction $H_{M}$
dominates the unitary evolution by
$M\left(\tau_{m}\right)=\exp\left(-i\tau_{m}H_{M}\right)$ for the measurement
interval $\tau_{m}$. After the first measurement, the state
$\left|\psi(\tau)\right\rangle$ evolves into an atom-cavity field entangled
state,
$\left|\psi(\tau+\tau_{m})\right\rangle=\frac{1}{\sqrt{2}}e^{i\phi\left(\tau\right)}\sum_{j=\pm}\left|j\right\rangle\otimes\left|\alpha_{j}\right\rangle\text{,}$
(8)
with $\alpha_{\pm}\equiv\alpha\left(\tau\right)\exp\left(-i\omega\tau\mp
ig^{2}\tau_{m}/\Delta\right)$. The average photon number does not change due
to the QND nature of the measurement, but the cavity field acquires different
phases corresponding to the two atomic states.
_Continuous measurement process for QZE –_ During the free evolution, we
insert the decoherence-based measurements for $N$ times at instants $n\tau$
$\left(n=1,2,\ldots,N\right)$. Mathematically, we apply
$\left[M\left(\tau_{m}\right)U\left(\tau\right)\right]^{N}$ to the initial
state, and then the quantum state evolves into
$\left|\psi_{N}\right\rangle\\!\\!\\!\equiv\\!\\!\\!\left|\psi\left[N\left(\tau+\tau_{m}\right)\right]\right\rangle$=$\sum_{j=\pm}\left[M_{j}\left(\tau_{m}\right)U\left(\tau\right)\right]^{N}\left|j\right\rangle\\!\otimes\\!\left|0\right\rangle$.
Here $M\left(\tau_{m}\right)$ acts on the cavity field as two operators
$M_{\pm}\left(\tau_{m}\right)\\!\\!=\\!\\!\exp\left(\mp
i\xi_{m}a^{\dagger}a\right)$ corresponding to the two atomic states
respectively, where $\xi_{m}\\!\\!=\\!\\!g^{2}\tau_{m}/\Delta$. From the
calculations of the explicit expression for
$\left[M_{\pm}\left(\tau_{m}\right)U\left(\tau\right)\right]^{N}$, we finally
obtain the evolution wavefunction
$\left|\psi_{N}\right\rangle=\sum_{j=\pm}\frac{e^{i\phi_{j}}}{\sqrt{2}}\left|j\right\rangle\otimes\left|\alpha_{jN}e^{-i\omega
t}\right\rangle\text{,}$ (9)
where $\phi_{\pm}=N\phi\left(\tau\right)+\theta_{\pm}\left(N\right)$, and
$\displaystyle\theta_{\pm}\left(N\right)$ $\displaystyle=$
$\displaystyle\pm\frac{\left|\alpha\left(\tau\right)\right|^{2}}{2}\frac{N\sin\xi_{m}-\sin\left(N\xi_{m}\right)}{1-\cos\xi_{m}}\text{,}$
$\displaystyle\alpha_{\pm N}$ $\displaystyle=$
$\displaystyle\alpha\left(\tau\right)e^{\mp
i\left(N+1\right)\xi_{m}}\frac{\sin\left(N\xi_{m}/2\right)}{\sin\left(\xi_{m}/2\right)}\text{.}$
Accordingly the average photon number is calculated as
$\bar{n}=\left|\alpha\left(\tau\right)\right|^{2}\frac{\sin^{2}\left(N\xi_{m}/2\right)}{\sin^{2}\left(\xi_{m}/2\right)}\text{.}$
(10)
We can see in the continuous measurement limit, i.e., $\tau\rightarrow 0$,
$\left|\alpha\left(\tau\right)\right|^{2}\approx f^{2}\tau^{2}$ .
Figure 2: (color online) Average photon number $\bar{n}$ as a function of the
pulse number $N$. We choose $g^{2}/\Delta=10$kHz, $\delta=0.5$Hz, $f=400$Hz,
and $\tau=50\mu$s. Without the QND probe, $\bar{n}$ grows quadratically with
$N$ (red solid line). The QZE emerges as $\bar{n}$ is frozen at zero with
$\tau_{m}=5$ms (bule dashed line). If the measurement time is chosen
specifically at $\tau_{m}=(2\pi\Delta/g^{2}+3.5)\mu$s, $\bar{n}$ increases
obviously (green dashdotted line) which is not explained in terms of the WPC
interpretation.
Except for certain measurement time interval $\tau_{m}^{\ast}$ chosen as
$\xi_{m}=g^{2}\tau_{m}^{\ast}/\Delta=2k\pi$, with $k$ integral, $\bar{n}$
approaches zero with $\tau$ decreasing.
As illustrated in Fig.2, $\bar{n}$ shows the similar inhibition phenomenon
(blue dashed line) to Ref. Bernu08 , with $\tau$ chosen as $50\mu$s. The
reason for the photon number ceasing increase is that the dynamic measurements
interrupt the coherent accumulation of photons by adding a phase factor to the
cavity field corresponding to $\xi_{m}$. The total phase factor after
$N$-times measurement destroys the quantum interference of the cavity field,
thus leads to the QZE. This deocoherence-based process in the existing
experiment Bernu08 reveals that the QZE can be completely interpreted from
the dynamic aspect. To compare with the situation with only free evolution and
no measurements, we set the atom-cavity coupling $g=0$, and $\bar{n}$ is also
depicted in Fig. 2 (red solid line), which indeed grows quadratically with
$t=N\tau$.
The above argument is coincident with the existing experimental data, but this
theoretical description implies the difference between the dynamical
measurement and the projection one. We notice that, when the measurement time
interval is set at critical values $\tau_{m}^{\ast}=2k\pi\Delta/g^{2}$
$(k=1,2,3,...)$, $\bar{n}$ is no longer bounded and increases linearly with
$N$. In Fig. 2, $\bar{n}$ increases clearly shown as the green dashdotted
line, with $\tau_{m}$ chosen around $\tau_{m}^{\ast}$ as
$\left(2\pi\Delta/g^{2}+3.5\right)\mu$s. Fixing the total free evolution time
$t$, we illustrate the variation of the average photon number in the cavity
field corresponding to the time interval $\tau$ and $\tau_{m}$ in Fig. 3. For
a given $\tau_{m}$ far from the critical value $\tau_{m}^{\ast}$, $\bar{n}$
approaches to zero as $\tau$ decreases, which recovers the conventional QZE
phenomenon based on the projection measurement. However, $\bar{n}$ mounts up
evidently when $\tau_{m}$ approaches to $\tau_{m}^{\ast}$. This
$\tau_{m}$-dependent decoherernce-based QZE could not be predicted by the WPC
interpretation, but can be testified by the realizable cavity-QED experiment.
If we observe the rise up of the average photon number at certain
$\tau_{m}^{\ast}$ in continuous measurement limit, then we can conclude that
the dynamic measurement model is more compatible with the physical reality in
comparison with the projection measurement in respect of the QZE.
Figure 3: The average photon number as a function of the free evolution time
interval $\tau$ and the measurement time interval $\tau_{m}$, where
$g^{2}/\Delta=10$kHz, $\delta=0.5$Hz, $f=400$Hz, and the total free evolution
time $t$ is fixed at $1$ms. The result is normalized by the maximum.
_Conclusion –_ In this Letter, we provided a general algebraic proof that QZE
could be induced by frequent decoherence-based measurements, which are unitary
processes without reference to the WPC postulate (projection measurement).
This approach essentially shows the general QZE phenomenon can be explained
independent of the quantum mechanics-interpretation for the measurement.
Projection measurement provides us a neat description of the QZE, beyond
which, the decoherenced-based model contains more physical detail. In tht
quantum open system, the same model can be extended to predict the QZE or
anti-QZE Ai10 . Associated with a recent cavity QED experiment Bernu08 , we
predict an observable effect of the decoherence-based measurements to
distinguish it from the one based on projection measurement: the survival
probability after finite $N$ measurements will explicitly depend on the
measurement time even in the continuous limit. At certain critical measurement
times, the survival probability will deviate from its initial value predicted
in the WPC-based explanation of the QZE.
We thank P. Zhang for valuable discussions. This work was supported by NSFC
through grants 10974209 and 10935010 and by the National 973 program (Grant
No. 2006CB921205).
## References
* (1) B. Misra and E. C. G. Sudarshan, J. Math. Phys. (N.Y.) 18, 756 (1977).
* (2) W. M. Itano, D. J. Heinzen, J. J. Bollinger and D. J. Wineland, Phys. Rev. A 41, 2295 (1990).
* (3) M. C. Fischer, B. Gutierrez-Medina, and M. G. Raizen, Phys. Rev. Lett. 87, 040402 (2001).
* (4) E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 97, 260402 (2006).
* (5) V. Frerichs and A. Schenzle, Phys. Rev. A 44, 1962 (1991).
* (6) L. S. Schulman, Phys. Rev. A 57, 1509 (1998).
* (7) A. Peres, Am. J. Phys. 48, 931 (1980).
* (8) R. J. Cook, Phys. Scr. T 21, 49 (1988).
* (9) L. E. Ballentine, Phys. Rev. A 43, 5165 (1991).
* (10) T. Petrosky, S. Tasaki, and I. Prigogine, Phys. Lett. A 151, 109 (1990); T. Petrosky, S. Tasaki, and I. Prigogine, Physica A 170, 306 (1991).
* (11) S. Pascazio and M. Namiki, Phys. Rev. A 50, 4582 (1994).
* (12) C. P. Sun, X. X. Yi and X. J. Liu, Fort. Phys. 43, 585 (1995).
* (13) J. Bernu, S. Deléglise, C. Sayrin, S. Kuhr, I. Dotsenko, M. Brune, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 101 , 180402 (2008).
* (14) J. von Neumann, Mathematical Foundations of Quantum Mechanics, translated by E. T. Beyer (Princeton University Press, Princeton, 1955).
* (15) W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003); W. H. Zurek, Phys. Today. 44, 36 (1991).
* (16) C. P. Sun, Phys. Rev. A 48, 898 (1993).
* (17) C. P. Sun, in Quantum Classical Correspondence: The 4th Drexel Symposium on Quantum Nonintegrability, 1994, edited by D. H. Feng and B. L. Hu (International Press, Cambridge, MA), p. 99.
* (18) E. Joos, H. D. Zeh, C. Kiefer, D. J. W. Giulini, J. Kupsch, I. O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer, Berlin, 1996)
* (19) V. B. Braginsky, F. Y. Khalili, K. S. Thorne, Quantum Measurement (Cambridge University Press, Cambridge, UK, 1995)
* (20) P. Facchi, D. A. Lidar, and S. Pascazio, Phys. Rev. A 69, 032314 (2004).
* (21) J. Wei and E. Norman, J. Math. Phys. 4A, 575 (1963).
* (22) Q. Ai, D. Z. Xu, S. Yi, A. G. Kofman, C. P. Sun and F. Nori, in preparation.
|
arxiv-papers
| 2010-07-27T07:58:59 |
2024-09-04T02:49:11.875380
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Z. Xu, Q. Ai, C. P. Sun",
"submitter": "Dazhi Xu",
"url": "https://arxiv.org/abs/1007.4634"
}
|
1007.4658
|
# Extending a valuation centered in a local domain to the formal completion.
F. J. Herrera Govantes111These authors are partially supported by
MTM2007-66929, MTM2010-19298 and FEDER.
Departamento de Álgebra
Facultad de Matemáticas
Avda. Reina Mercedes, s/n
Universidad de Sevilla
41012 Sevilla, Spain
email: jherrera@algebra.us.es M. A. Olalla Acosta∗
Departamento de Álgebra
Facultad de Matemáticas
Avda. Reina Mercedes, s/n
Universidad de Sevilla
41012 Sevilla, Spain
email: miguelolalla@algebra.us.es M. Spivakovsky
Institut de Mathématiques de Toulouse
UMR 5219 du CNRS,
Université Paul Sabatier
118, route de Narbonne
31062 Toulouse cedex 9, France.
email: mark.spivakovsky@math.univ-toulouse.fr B. Teissier222This author is
grateful for the hospitality of the RIMS in Kyoto, where a part of this
project was completed.
Equipe “ Géométrie et Dynamique”,
Institut Mathématique de Jussieu,
UMR 7586 du CNRS
175 Rue du Chevaleret
F-75013 Paris, France.
email: teissier@math.jussieu.fr
## 1 Introduction
All the rings in this paper will be commutative with 1.
Let $(R,m,k)$ be a local noetherian domain with field of fractions $K$ and
$R_{\nu}$ a valuation ring, dominating $R$ (not necessarily birationally). Let
$\nu|_{K}:K^{*}\twoheadrightarrow\Gamma$ be the restriction of $\nu$ to $K$;
by definition, $\nu|_{K}$ is centered at $R$. Let $\hat{R}$ denote the
$m$-adic completion of $R$. In the applications of valuation theory to
commutative algebra and the study of singularities, one is often induced to
replace $R$ by its $m$-adic completion $\hat{R}$ and $\nu$ by a suitable
extension $\hat{\nu}_{-}$ to $\frac{\hat{R}}{P}$ for a suitably chosen prime
ideal $P$, such that $P\cap R=(0)$ (one specific application we have in mind
has to do with the approaches to proving the Local Uniformization Theorem in
arbitrary characteristic such as [13] and [14]). The first reason is that the
ring $\hat{R}$ is not in general an integral domain, so that we can only hope
to extend $\nu$ to a pseudo-valuation on $\hat{R}$, which means precisely a
valuation $\hat{\nu}_{-}$ on a quotient $\frac{\hat{R}}{P}$ as above. The
prime ideal $P$ is called the support of the pseudo-valuation. It is well
known and not hard to prove that such extensions $\hat{\nu}_{-}$ exist for
some minimal prime ideals $P$ of $\hat{R}$. Although, as we shall see, the
datum of a valuation $\nu$ determines a unique minimal prime of $\hat{R}$ when
$R$ is excellent, in general there are many possible primes $P$ as above and
for a fixed $P$ many possible extensions $\hat{\nu}_{-}$. This is the second
reason to study extensions $\hat{\nu}_{-}$.
The purpose of this paper is to give, assuming that $R$ is excellent, a
systematic description of all such extensions $\hat{\nu}_{-}$ and to identify
certain classes of extensions which are of particular interest for
applications. In fact, the only assumption about $R$ we ever use in this paper
is a weaker and more natural condition than excellence, called the G
condition, but we chose to talk about excellent rings since this terminology
seems to be more familiar to most people. For the reader’s convenience, the
definitions of excellent and G-rings are recalled in the Appendix. Under this
assumption, we study extensions to (an integral quotient of) the completion
$\hat{R}$ of a valuation $\nu$ and give descriptions of the valuations with
which such extensions are composed. In particular we give criteria for the
uniqueness of the extension if certain simple data on these composed
valuations are fixed.
We conjecture (see statement 5.19 in [14] and Conjecture 1.11 below for a
stronger and more precise statement) that
given an excellent local ring $R$ and a valuation $\nu$ of $R$ which is
positive on its maximal ideal $m$, there exists a prime ideal $H$ of the
$m$-adic completion $\hat{R}$ such that $H\bigcap R=(0)$ and an extension of
$\nu$ to $\frac{\hat{R}}{H}$ which has the same value group as $\nu$.
When studying extensions of $\nu$ to the completion of $R$, one is led to the
study of its extensions to the henselization $\tilde{R}$ of $R$ as a natural
first step. This, in turn, leads to the study of extensions of $\nu$ to
finitely generated local strictly étale extensions $R^{e}$ of $R$. We
therefore start out by letting $\sigma:R\rightarrow R^{\dagger}$ denote one of
the three operations of completion, (strict) henselization, or a finitely
generated local strictly étale extension:
$\displaystyle R^{\dagger}$ $\displaystyle=$ $\displaystyle\hat{R}\quad\mbox{
or}$ (1) $\displaystyle R^{\dagger}$ $\displaystyle=$
$\displaystyle\tilde{R}\quad\mbox{ or}$ (2) $\displaystyle R^{\dagger}$
$\displaystyle=$ $\displaystyle R^{e}.$ (3)
The ring $R^{\dagger}$ is local; let $m^{\dagger}$ denote its maximal ideal.
The homomorphisms
$R\rightarrow\tilde{R}\quad\text{ and }\quad R\rightarrow R^{e}$
are regular for any ring $R$; by definition, if $R$ is an excellent ring then
the completion homomorphism is regular (in fact, regularity of the completion
homomorphism is precisely the defining property of G-rings; see the Appendix
for the definition of regular homomorphism).
Let $r$ denote the (real) rank of $\nu$. Let
$(0)=\Delta_{r}\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_{0}=\Gamma$
be the isolated subgroups of $\Gamma$ and $P_{0}=(0)\subsetneqq
P_{1}\subseteq\dots\subseteq P_{r}=m$ the prime valuation ideals of $R$, which
need not, in general, be distinct. In this paper, we will assume that $R$ is
excellent. Under this assumption, we will canonically associate to $\nu$ a
chain $H_{1}\subset H_{3}\subset\dots\subset H_{2r+1}=mR^{\dagger}$ of ideals
of $R^{\dagger}$, numbered by odd integers from 1 to $2r+1$, such that
$H_{2\ell+1}\cap R=P_{\ell}$ for $0\leq\ell\leq r$. We will show that all the
ideals $H_{2\ell+1}$ are prime. We will define $H_{2\ell}$ to be the unique
minimal prime ideal of $P_{\ell}R^{\dagger}$, contained in $H_{2\ell+1}$ (that
such a minimal prime is unique follows from the regularity of the homomorphism
$\sigma$).
We will thus obtain, in the cases (1)–(3), a chain of $2r+1$ prime ideals
$H_{0}\subset H_{1}\subset\dots\subset H_{2r}=H_{2r+1}=mR^{\dagger},$
satisfying $H_{2\ell}\cap R=H_{2\ell+1}\cap R=P_{\ell}$ and such that
$H_{2\ell}$ is a minimal prime of $P_{\ell}R^{\dagger}$ for $0\leq\ell\leq r$.
Moreover, if $R^{\dagger}=\tilde{R}$ or $R^{\dagger}=R^{e}$, then
$H_{2\ell}=H_{2\ell+1}$. We call $H_{i}$ the $i$-th implicit prime ideal of
$R^{\dagger}$, associated to $R$ and $\nu$. The ideals $H_{i}$ behave well
under local blowing ups along $\nu$ (that is, birational local homomorphisms
$R\to R^{\prime}$ such that $\nu$ is centered in $R^{\prime}$), and more
generally under $\nu$-extensions of $R$ defined below in subsection 1.1. This
means that given any local blowing up along $\nu$ or $\nu$-extension
$R\rightarrow R^{\prime}$, the $i$-th implicit prime ideal $H^{\prime}_{i}$ of
${R^{\prime}}^{\dagger}$ has the property that $H^{\prime}_{i}\cap
R^{\dagger}=H_{i}$. This intersection has a meaning in view of Lemma 1.2
below.
For a prime ideal $P$ in a ring $R$, $\kappa(P)$ will denote the residue field
$\frac{R_{P}}{PR_{P}}$.
Let
$(0)\subsetneqq\mathbf{m}_{1}\subsetneqq\dots\subsetneqq\mathbf{m}_{r-1}\subsetneqq\mathbf{m}_{r}=\mathbf{m}_{\nu}$
be the prime ideals of the valuation ring $R_{\nu}$. By definitions, our
valuation $\nu$ is a composition of $r$ rank one valuations
$\nu=\nu_{1}\circ\nu_{2}\dots\circ\nu_{r}$, where $\nu_{\ell}$ is a valuation
of the field $\kappa(\mathbf{m}_{\ell-1})$, centered at
$\frac{(R_{\nu})_{\mathbf{m}_{\ell}}}{\mathbf{m}_{\ell-1}}$ (see [18], Chapter
VI, §10, p. 43 for the definition of composition of valuations; more
information and a simple example of composition is given below in subsection
1.1, where we interpret each $\mathbf{m}_{\ell}$ as the limit of a tree of
ideals).
If $R^{\dagger}=\tilde{R}$, we will prove that there is a unique extension
$\tilde{\nu}_{-}$ of $\nu$ to $\frac{\tilde{R}}{H_{0}}$. If
$R^{\dagger}=\hat{R}$, the situation is more complicated. First, we need to
discuss the behaviour of our constructions under $\nu$-extensions.
### 1.1 Local blowings up and trees.
We consider extensions $R\rightarrow R^{\prime}$ of local rings, that is,
injective morphisms such that $R^{\prime}$ is an $R$-algebra essentially of
finite type and $m^{\prime}\cap R=m$. In this paper we consider only
extensions with respect to $\nu$; that is, both $R$ and $R^{\prime}$ are
contained in a fixed valuation ring $R_{\nu}$. Such extensions form a direct
system $\\{R^{\prime}\\}$. We will consider many direct systems of rings and
of ideals indexed by $\\{R^{\prime}\\}$; direct limits will always be taken
with respect to the direct system $\\{R^{\prime}\\}$. Unless otherwise
specified, we will assume that
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}R^{\prime}=R_{\nu}.$ (4)
Note that by the fundamental properties of valuation rings ([18], §VI),
assuming the equality (4) is equivalent to assuming that
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}K^{\prime}=K_{\nu}$, where
$K^{\prime}$ stands for the field of fractions of $R^{\prime}$ and $K_{\nu}$
for that of $R_{\nu}$, and that
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}R^{\prime}$ is a valuation
ring.
###### Definition 1.1
A tree of $R^{\prime}$-algebras is a direct system $\\{S^{\prime}\\}$ of
rings, indexed by the directed set $\\{R^{\prime}\\}$, where $S^{\prime}$ is
an $R^{\prime}$-algebra. Note that the maps are not necessarily injective. A
morphism $\\{S^{\prime}\\}\to\\{T^{\prime}\\}$ of trees is the datum of a map
of $R^{\prime}$-algebras $S^{\prime}\to T^{\prime}$ for each $R^{\prime}$
commuting with the tree morphisms for each map $R^{\prime}\to
R^{\prime\prime}$.
###### Lemma 1.2
Let $R\rightarrow R^{\prime}$ be an extension of local rings. We have:
1) The ideal $N:=m^{\dagger}\otimes_{R}1+1\otimes_{R}m^{\prime}$ is maximal in
the $R$-algebra $R^{\dagger}\otimes_{R}R^{\prime}$.
2) The natural map of completions (resp. henselizations)
$R^{\dagger}\to{R^{\prime}}^{\dagger}$ is injective.
Proof.- 1) follows from that fact that $R^{\dagger}/m^{\dagger}=R/m$. The
proof of 2) relies on a construction which we shall use often: the map
$R^{\dagger}\to{R^{\prime}}^{\dagger}$ can be factored as
$R^{\dagger}\to\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{N}\to{R^{\prime}}^{\dagger},$
(5)
where the first map sends $x$ to $x\otimes 1$ and the second is determined by
$x\otimes x^{\prime}\mapsto\hat{b}(x).c(x^{\prime})$ where $\hat{b}$ is the
natural map $R^{\dagger}\to{R^{\prime}}^{\dagger}$ and $c$ is the canonical
map $R^{\prime}\to{R^{\prime}}^{\dagger}$. The first map is injective because
$R^{\dagger}$ is a flat $R$-algebra and it is obtained by tensoring the
injection $R\to R^{\prime}$ by the $R$-algebra $R^{\dagger}$; furthermore,
elements of $R^{\dagger}$ whose image in $R^{\dagger}\otimes_{R}R^{\prime}$
lie outside of $N$ are precisely units of $R^{\dagger}$, hence they are not
zero divisors in $R^{\dagger}\otimes_{R}R^{\prime}$ and $R^{\dagger}$ injects
in every localization of $R^{\dagger}\otimes_{R}R^{\prime}$.
Since $m^{\prime}\cap R=m$, we see that the inverse image by the natural map
of $R^{\prime}$-algebras
$\iota\colon R^{\prime}\to(R^{\dagger}\otimes_{R}R^{\prime})_{N},$
defined by $x^{\prime}\mapsto 1\otimes_{R}x^{\prime}$, of the maximal ideal
$M=(m^{\dagger}\otimes_{R}1+1\otimes_{R}m^{\prime})(R^{\dagger}\otimes_{R}R^{\prime})_{N}$
of $(R^{\dagger}\otimes_{R}R^{\prime})_{N}$ is the ideal $m^{\prime}$ and that
$\iota$ induces a natural isomorphism
$\frac{R^{\prime}}{{m^{\prime}}^{i}}\overset{\sim}{\rightarrow}\frac{(R^{\dagger}\otimes_{R}R^{\prime})_{N}}{M^{i}}$
for each $i$. From this it follows by the universal properties of completion
and henselization that the second map in the sequence (5) is the completion
(resp. the henselization inside the completion) of
$R^{\dagger}\otimes_{R}R^{\prime}$ with respect to the ideal $M$. It is
therefore also injective.
###### Definition 1.3
Let $\\{S^{\prime}\\}$ be a tree of $R^{\prime}$-algebras. For each
$S^{\prime}$, let $I^{\prime}$ be an ideal of $S^{\prime}$. We say that
$\\{I^{\prime}\\}$ is a tree of ideals if for any arrow
$b_{S^{\prime}S^{\prime\prime}}\colon S^{\prime}\rightarrow S^{\prime\prime}$
in our direct system, we have
$b^{-1}_{S^{\prime}S^{\prime\prime}}I^{\prime\prime}=I^{\prime}$. We have the
obvious notion of inclusion of trees of ideals. In particular, we may speak
about chains of trees of ideals.
###### Examples 1.4
The maximal ideals of the local rings of our system $\\{R^{\prime}\\}$ form a
tree of ideals.
For any non-negative element $\beta\in\Gamma$, the valuation ideals
$\mathcal{P}^{\prime}_{\beta}\subset R^{\prime}$ of value $\beta$ form a tree
of ideals of $\\{R^{\prime}\\}$. Similarly, the $i$-th prime valuation ideals
$P^{\prime}_{i}\subset R^{\prime}$ form a tree. If $rk\ \nu=r$, the prime
valuation ideals $P^{\prime}_{i}$ give rise to a chain
$P^{\prime}_{0}=(0)\subsetneqq P^{\prime}_{1}\subseteq\dots\subseteq
P^{\prime}_{r}=m^{\prime}$ (6)
of trees of prime ideals of $\\{R^{\prime}\\}$.
We discuss this last example in a little more detail and generality in order
to emphasize our point of view, crucial throughout this paper: the data of a
composite valuation is equivalent to the data of its components. Namely,
suppose we are given a chain of trees of ideals as in (6), where we relax our
assumptions of the $P^{\prime}_{i}$ as follows. We no longer assume that the
chain (6) is maximal, nor that $P^{\prime}_{i}\subsetneqq P^{\prime}_{i+1}$,
even for $R^{\prime}$ sufficiently large; in particular, for the purposes of
this example we momentarily drop the assumption that $rk\ \nu=r$. We will
still assume, however, that $P^{\prime}_{0}=(0)$ and that
$P^{\prime}_{r}=m^{\prime}$.
Taking the limit in (6), we obtain a chain
$(0)=\mathbf{m}_{0}\subsetneqq\mathbf{m}_{1}\subseteqq\dots\subseteqq\mathbf{m}_{r}=\mathbf{m}_{\nu}$
(7)
of prime ideals of the valuation ring $R_{\nu}$.
Similarly, for each $1\leq\ell\leq r$ one has the equality
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\frac{R^{\prime}}{P^{\prime}_{\ell}}}=\frac{R_{\nu}}{\bf
m_{\ell}}.$
Then specifying the valuation $\nu$ is equivalent to specifying valuations
$\nu_{0},\nu_{1}$, …, $\nu_{r}$, where $\nu_{0}$ is the trivial valuation of
$K$ and, for $1\leq\ell\leq r$, $\nu_{\ell}$ is a valuation of the residue
field $k_{\nu_{\ell-1}}=\kappa(\mathbf{m}_{\ell-1})$, centered at the local
ring
$\lim\limits_{\longrightarrow}\frac{R^{\prime}_{P^{\prime}_{\ell}}}{P^{\prime}_{\ell-1}R^{\prime}_{P^{\prime}_{\ell}}}=\frac{(R_{\nu})_{\mathbf{m}_{\ell}}}{\mathbf{m}_{\ell-1}}$
and taking its values in the totally ordered group
$\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$.
The relationship between $\nu$ and the $\nu_{\ell}$ is that $\nu$ is the
composition
$\nu=\nu_{1}\circ\nu_{2}\circ\dots\circ\nu_{r}.$ (8)
For example, the datum of the valuation $\nu$, or of its valuation ring
$R_{\nu}$, is equivalent to the datum of the valuation ring
$\frac{R_{\nu}}{\mathbf{m}_{r-1}}\subset\frac{(R_{\nu})_{\mathbf{m}_{r-1}}}{\mathbf{m}_{r-1}(R_{\nu})_{\mathbf{m}_{r-1}}}=\kappa(\mathbf{m}_{r-1})$
of the valuation $\nu_{r}$ of the field $\kappa(\mathbf{m}_{r-1})$ and the
valuation ring $(R_{\nu})_{\mathbf{m}_{r-1}}$. If we assume, in addition, that
for $R$ sufficiently large the chain (6) (equivalently, (7)) is a maximal
chain of distinct prime ideals then $rk\ \nu=r$ and $rk\ \nu_{\ell}=1$ for
each $\ell$.
###### Remark 1.5
Another way to describe the same property of valuations is that, given a prime
ideal $H$ of the local integral domain $R$ one builds all valuations centered
in $R$ having $H$ as one of the $P_{\ell}$ by choosing a valuation $\nu_{1}$
of $R$ centered at $H$, so that $\mathbf{m}_{\nu_{1}}\cap R=H$ and choosing a
valuation subring $\overline{R}_{\overline{\nu}}$ of the field
$\frac{R_{\nu_{1}}}{\mathbf{m}_{\nu_{1}}}$ centered at $R/H$. Then
$\nu=\nu_{1}\circ\overline{\nu}$.
Note that choosing a valuation of $R/H$ determines a valuation of its field of
fractions $\kappa(H)$, which is in general much smaller than
$\frac{R_{\nu_{1}}}{\mathbf{m}_{\nu_{1}}}$. Given a valuation of $R$ with
center $H$, in order to determine a valuation of $R$ with center $m$ inducing
on $R/H$ a given valuation $\mu$ we must choose an extension $\overline{\nu}$
of $\mu$ to $\frac{R_{\nu_{1}}}{\mathbf{m}_{\nu_{1}}}$, and there are in
general many possibilities.
This will be used in the sequel. In particular, it will be applied to the case
where a valuation $\nu$ of $R$ extends uniquely to a valuation $\hat{\nu}_{-}$
of $\frac{\hat{R}}{H}$ for some prime $H$ of $\hat{R}$. Assuming that
$\hat{R}$ is an integral domain, this determines a unique valuation of
$\hat{R}$ only if the height ${\operatorname{ht}}\ H$ of $H$ in $\hat{R}$ is
at most one. In all other cases the dimension of $\hat{R}_{H}$ is at least $2$
and we have infinitely many valuations with which to compose $\hat{\nu}_{-}$.
This is the source of the height conditions we shall see in §6.
###### Example 1.6
Let $k_{0}$ be a field and $K=k_{0}((u,v))$ the field of fractions of the
complete local ring $R=k_{0}[[u,v]]$. Let $\Gamma=\mathbf{Z}^{2}$ with
lexicographical ordering. The isolated subgroups of $\Gamma$ are
$(0)\subsetneqq(0)\oplus\mathbf{Z}\subsetneqq\mathbf{Z}^{2}$. Consider the
valuation $\nu:K^{*}\rightarrow\mathbf{Z}^{2}$, centered at $R$, given by
$\displaystyle\nu(v)$ $\displaystyle=$ $\displaystyle(0,1)$ (9)
$\displaystyle\nu(u)$ $\displaystyle=$ $\displaystyle(1,0)$ (10)
$\displaystyle\nu(c)$ $\displaystyle=$ $\displaystyle 0\quad\text{ for any
}c\in k_{0}^{*}.$ (11)
This information determines $\nu$ completely; namely, for any power series
$f=\sum\limits_{\alpha,\beta}c_{\alpha\beta}u^{\alpha}v^{\beta}\in
k_{0}[[u,v]],$
we have
$\nu(f)=\min\\{(\alpha,\beta)\ |\ c_{\alpha\beta}\neq 0\\}.$
We have $rk\ \nu=rat.rk\ \nu=2$. Let $\Delta=(0)\oplus\mathbf{Z}$. Let
$\Gamma_{+}$ denote the semigroup of all the non-negative elements of
$\Gamma$. Let $k_{0}[[\Gamma_{+}]]$ denote the $R$-algebra of power series
$\sum c_{\alpha,\beta}u^{\alpha}v^{\beta}$ where $c_{\alpha,\beta}\in k_{0}$
and the exponents $(\alpha,\beta)$ form a well ordered subset of $\Gamma_{+}$.
By classical results (see [7], [8]), it is a valuation ring with maximal ideal
generated by all the monomials $u^{\alpha}v^{\beta}$, where
$(\alpha,\beta)>(0,0)$ (in other words, either $\alpha>0,\beta\in\mathbf{Z}$
or $\alpha=0,\beta>0$). Then
$R_{\nu}=k_{0}[[\Gamma_{+}]]\bigcap k_{0}((u,v))$
is a valuation ring of $K$, and contains $k[[u,v]]$; it is the valuation ring
of the valuation $\nu$. The prime ideal $\mathbf{m}_{1}$ is the ideal of
$R_{\nu}$ generated by all the $uv^{\beta}$, $\beta\in\mathbf{Z}$. The
valuation $\nu_{1}$ is the discrete rank 1 valuation of $K$ with valuation
ring
$(R_{\nu})_{\mathbf{m}_{1}}=k_{0}[[u,v]]_{(u)}$
and $\nu_{2}$ is the discrete rank 1 valuation of $k_{0}((v))$ with valuation
ring $\frac{R_{\nu}}{\mathbf{m}_{1}}\cong k_{0}[[v]]$.
###### Example 1.7
To give a more interesting example, let $k_{0}$ be a field of characteristic
zero and
$K=k_{0}(x,y,z)$
a purely transcendental extension of $k_{0}$ of degree 3. Let $w$ be an
independent variable and put
$k=\bigcup\limits_{j=1}^{\infty}k_{0}\left(w^{\frac{1}{j}}\right)$. Let
$\Gamma=\mathbf{Z}\oplus\mathbf{Q}$ with the lexicographical ordering and
$\Delta=(0)\oplus\mathbf{Q}$ the non-trivial isolated subgroup of $\Gamma$.
Let $u,v$ be new variables and let
$\mu_{1}:k((u,v))\rightarrow\mathbf{Z}^{2}_{lex}$ be the valuation of the
previous example. Let $\mu_{2}$ denote the $x$-adic valuation of $k$ and put
$\mu=\mu_{1}\circ\mu_{2}$. Consider the map $\iota:k_{0}[x,y,z]\rightarrow
k[[u,v]]$ which sends $x$ to $w$, $y$ to $v$ and $z$ to
$u-\sum\limits_{j=1}^{\infty}w^{\frac{1}{j}}v^{j}$. Let
$\nu_{1}=\left.\mu_{1}\right|_{K}$ and $\nu=\mu|_{K}$.
The valuation $\nu:K^{*}\rightarrow\Gamma$ is centered at the local ring
$R=k_{0}[x,y,z]_{(x,y,z)}$; we have
$\displaystyle\nu(x)$ $\displaystyle=$ $\displaystyle(0,1)$ (12)
$\displaystyle\nu(y)$ $\displaystyle=$ $\displaystyle(1,0),$ (13)
$\displaystyle\nu(z)$ $\displaystyle=$ $\displaystyle(1,1).$ (14)
Write as a composition of two rank 1 valuations: $\nu=\nu_{1}\circ\nu_{2}$. We
have natural inclusions $R_{\nu_{1}}\subset R_{\mu_{1}}$ and
$k_{\nu_{1}}\subset k_{\mu_{1}}=k$. We claim that $k_{\nu_{1}}$ is not
finitely generated over $k_{0}$. Indeed, if this were not the case then there
would exist a prime number $p$ such that $w^{\frac{1}{p}}\mbox{$\in$
/}k_{\nu_{1}}$. Let $k^{\prime}=k_{0}\left(x^{\frac{1}{(p-1)!}}\right)$. Let
$L=k^{\prime}(y,z)$. Consier the tower of field extensions $K\subset L\subset
k[[u,v]]$ and let $\nu^{\prime}$ denote the restriction of $\mu$ to $L$. Let
$\Gamma^{\prime}$ be the value group of $\nu^{\prime}$ and $k_{\nu^{\prime}}$
the residue field of its valuation ring. Now, $L$ contains the element
$z_{p}:=z-\sum\limits_{j=1}^{p-1}x^{\frac{1}{j}}y^{j}$ as well as
$\frac{z_{p}}{y^{p}}$. We have
$\nu^{\prime}(z_{p})=\left(p,\frac{1}{p}\right),$ (15)
$\nu^{\prime}\left(\frac{z_{p}}{y^{p}}\right)=0$ and the natural image of
$\frac{z_{p}}{y^{p}}$ in $k_{\mu_{1}}=k$ is $w^{\frac{1}{p}}$. Now, $p\not|\
[L:K]$, $\left.[\Gamma^{\prime}:\Gamma]\ \right|\ [L:K]$ and
$\left.[k_{\nu^{\prime}}:k_{\nu}]\ \right|\ [L:K]$. This implies that
$z_{p}\in L$ and $w^{\frac{1}{p}}\in k_{\nu_{1}}$, which gives the desired
contradicion.
It is not hard to show that for each $j$, there exists a local blowing up
$R\rightarrow R^{\prime}$ of $R$ such that, in the notation of (6), we have
$\kappa(P^{\prime}_{1})=k_{0}\left(w^{\frac{1}{j!}}\right)$ and that
$\kappa(\mathbf{m}_{1})=\lim\limits_{j\to\infty}\kappa(P^{\prime}_{1})=k$. The
first one is the blowing up of the ideal $(y,z)R$, localized at the point
$y=0,z/y=x$. Then one blows up the ideal $(z/y-x,y)$, and so on.
Another way to see the valuation $\nu=\nu_{1}\circ\nu_{2}$ is to note that
$\nu_{1}$ is the restriction to $K$ of the $v$-adic valuation under the
inclusion of fields deduced from the inclusion of rings
$k_{0}[[x,y,z]]_{(y,z)}\hookrightarrow
k\left[\left[v^{{\mathbf{Z}}_{+}}\right]\right]$
which sends $x$ to $w$, $y$ to $v$ and $z$ to
$\sum\limits_{j=1}^{\infty}w^{\frac{1}{j}}v^{j}$. Recall that the ring on the
right is made of power series with non negative rational exponents whose set
of exponents is well ordered. We have $k_{\nu_{1}}=k$.
###### Remark 1.8
The point of the last example is to show that, given a composed valuation as
in (8), $\nu_{\ell}$ is a valuation of the field $k_{\nu_{\ell-1}}$, which may
properly contain $\kappa(P^{\prime}_{\ell-1})$ for every
$R^{\prime}\in\mathcal{T}$. This fact will be a source of complication later
on and we prefer to draw attention to it from the beginning.
Coming back to the implicit prime ideals, we will see that the implicit prime
ideals $H^{\prime}_{i}$ form a tree of ideals of $R^{\dagger}$.
We will show that if $\nu$ extends to a valuation of $\hat{\nu}_{-}$ centered
at $\frac{\hat{R}}{P}$ with $P\cap R=(0)$ then the prime $P$ must contain the
minimal prime $H_{0}$ of $\hat{R}$. We will then show that specifying an
extension $\hat{\nu}_{-}$ of $\nu$ as above is equivalent to specifying a
chain of prime valuation ideals
$\tilde{H}^{\prime}_{0}\subset\tilde{H}^{\prime}_{1}\subset\dots\subset\tilde{H}^{\prime}_{2r}=m^{\prime}\hat{R}^{\prime}$
(16)
of $\hat{R}^{\prime}$ such that
$H^{\prime}_{\ell}\subset\tilde{H}^{\prime}_{\ell}$ for all
$\ell\in\\{0,\dots,2r\\}$, and valuations
$\hat{\nu}_{1},\hat{\nu}_{2},\dots,\hat{\nu}_{2r}$, where $\hat{\nu}_{i}$ is a
valuation of the field $k_{\hat{\nu}_{i-1}}$ (the residue field of the
valuation ring $R_{\hat{\nu}_{i-1}}$), arbitrary when $i$ is odd and
satisfying certain conditions, coming from the valuation $\nu_{\frac{i}{2}}$,
when $i$ is even.
The prime ideals $H_{i}$ are defined as follows.
Recall that given a valued ring $(R,\nu)$, that is a subring $R\subseteq
R_{\nu}$ of the valuation ring $R_{\nu}$ of a valuation with value group
$\Gamma$, one defines for each $\beta\in\Gamma$ the valuation ideals of $R$
associated to $\beta$:
$\begin{array}[]{lr}{\cal P}_{\beta}(R)=&\\{x\in R/\nu(x)\geq\beta\\}\cr{\cal
P}^{+}_{\beta}(R)=&\\{x\in R/\nu(x)>\beta\\}\end{array}$
and the associated graded ring
$\hbox{\rm gr}_{\nu}R=\bigoplus_{\beta\in\Gamma}\frac{{\cal
P}_{\beta}(R)}{{\cal
P}^{+}_{\beta}(R)}=\bigoplus_{\beta\in\Gamma_{+}}\frac{{\cal
P}_{\beta}(R)}{{\cal P}^{+}_{\beta}(R)}.$
The second equality comes from the fact that if
$\beta\in\Gamma_{-}\setminus\\{0\\}$, we have ${\cal P}^{+}_{\beta}(R)={\cal
P}_{\beta}(R)=R$. If $R\to R^{\prime}$ is an extension of local rings such
that $R\subset R^{\prime}\subset R_{\nu}$ and $m_{\nu}\cap
R^{\prime}=m^{\prime}$, we will write ${\cal P}^{\prime}_{\beta}$ for ${\cal
P}_{\beta}(R^{\prime})$.
Fix a valuation ring $R_{\nu}$ dominating $R$, and a tree ${\cal
T}=\\{R^{\prime}\\}$ of nœtherian local $R$-subalgebras of $R_{\nu}$, having
the following properties: for each ring $R^{\prime}\in\cal{T}$, all the
birational $\nu$-extensions of $R^{\prime}$ belong to $\cal{T}$. Moreover, we
assume that the field of fractions of $R_{\nu}$ equals
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}K^{\prime}$, where
$K^{\prime}$ is the field of fractions of $R^{\prime}$. The tree $\cal{T}$
will stay constant throughout this paper. In the special case when $R$ happens
to have the same field of fractions as $R_{\nu}$, we may take $\cal{T}$ to be
the tree of all the birational $\nu$-extensions of $R$.
###### Notation 1.9
For a ring $R^{\prime}\in\cal T$, we shall denote by ${\cal T}(R^{\prime})$
the subtree of $\cal T$ consisting of all the $\nu$-extensions
$R^{\prime\prime}$ of $R^{\prime}$.
We now define
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left(\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\cal
P}^{\prime}_{\beta}{R^{\prime}}^{\dagger}\right)\bigcap R^{\dagger}\right),\
0\leq\ell\leq r-1$ (17)
(in the beginning of §3 we provide some motivation for this definition and
give several elementary examples of $H^{\prime}_{i}$ and
$\tilde{H}^{\prime}_{i}$).
The questions answered in this paper originally arose from our work on the
Local Uniformization Theorem, where passage to completion is required in both
the approaches of [13] and [14]. In [14], one really needs to pass to
completion for valuations of arbitrary rank. One of the main intended
applications of the theory of implicit prime ideals is the following
conjecture. Let
$\Gamma\hookrightarrow\hat{\Gamma}$ (18)
be an extension of ordered groups of the same rank. Let
$(0)=\Delta_{r}\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_{0}=\Gamma$
(19)
be the isolated subgroups of $\Gamma$ and
$(0)=\hat{\Delta}_{r}\subsetneqq\hat{\Delta}_{r-1}\subsetneqq\dots\subsetneqq\hat{\Delta}_{0}=\hat{\Gamma}$
the isolated subgroups of $\hat{\Gamma}$, so that the inclusion (18) induces
inclusions
$\displaystyle\Delta_{\ell}$ $\displaystyle\hookrightarrow$
$\displaystyle\hat{\Delta}_{\ell}\quad\text{ and}$ (20)
$\displaystyle\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$
$\displaystyle\hookrightarrow$
$\displaystyle\frac{\hat{\Delta}_{\ell}}{\hat{\Delta}_{\ell+1}}.$ (21)
Let $G\hookrightarrow\hat{G}$ be an extension of graded algebras without zero
divisors, such that $G$ is graded by $\Gamma_{+}$ and $\hat{G}$ by
$\hat{\Gamma}_{+}$. The graded algebra $G$ is endowed with a natural valuation
with value group $\Gamma$ and similarly for $\hat{G}$ and $\hat{\Gamma}$.
These natural valuations will both be denoted by $ord$.
###### Definition 1.10
We say that the extension $G\hookrightarrow\hat{G}$ is scalewise birational if
for any $x\in\hat{G}$ and $\ell\in\\{1,\dots,r\\}$ such that $ord\
x\in\hat{\Delta}_{\ell}$ there exists $y\in G$ such that $ord\
y\in\Delta_{\ell}$ and $xy\in G$.
Of course, scalewise birational implies birational and also that
$\hat{\Gamma}=\Gamma$.
While the main result of this paper is the primality of the implicit ideals
associated to a valuation, and the subsequent description of the extensions of
the valuation to the completion, the main conjecture stated here is the
following:
###### Conjecture 1.11
Assume that $\dim\ R^{\prime}=\dim\ R$ for all $R^{\prime}\in\mathcal{T}$.
Then there exists a tree of prime ideals $H^{\prime}$ of $\hat{R}^{\prime}$
with $H^{\prime}\cap R^{\prime}=(0)$ and a valuation $\hat{\nu}_{-}$, centered
at $\lim\limits_{\to}\frac{\hat{R}^{\prime}}{H^{\prime}}$ and having the
following property:
For any $R^{\prime}\in\cal{T}$ the graded algebra
$\mbox{gr}_{\hat{\nu}_{-}}\frac{\hat{R}^{\prime}}{H^{\prime}}$ is a scalewise
birational extension of $\mbox{gr}_{\nu}R^{\prime}$.
The example given in remark 5.20, 4) of [14] shows that the morphism of
associated graded rings is not an isomorphism in general.
The approach to the Local Uniformization Theorem taken in [13] is to reduce
the problem to the case of rank 1 valuations. The theory of implicit prime
ideals is much simpler for valuations of rank 1 and takes only a few pages in
Section 2.
The paper is organized as follows. In §3 we define the odd-numbered implicit
ideals $H_{2\ell+1}$ and prove that $H_{2\ell+1}\cap R=P_{\ell}$. We observe
that by their very definition, the ideals $H_{2\ell+1}$ behave well under
$\nu$-extensions; they form a tree. Proving that $H_{2\ell+1}$ is indeed prime
is postponed until later sections; it will be proved gradually in §4–§8. In
the beginning of §3 we will explain in more detail the respective roles played
by the odd-numbered and the even-numbered implicit ideals, give several
examples (among other things, to motivate the need for taking the limit with
respect to $R^{\prime}$ in (17)) and say one or two words about the techniques
used to prove our results.
In §4 we prove the primality of the implicit prime ideals assuming a certain
technical condition, called stability, about the tree $\cal T$ and the
operation ${\ }^{\dagger}$. It follows from the noetherianity of $R^{\dagger}$
that there exists a specific $R^{\prime}$ for which the limit in (17) is
attained. One of the main points of §4 is to prove properties of stable rings
which guarantee that this limit is attained whenever $R^{\prime}$ is stable.
We then use the excellence of $R$ to define the even-numbered implicit prime
ideals: for $i=2\ell$ the ideal $H_{2\ell}$ is defined to be the unique
minimal prime of $P_{\ell}R^{\dagger}$, contained in $H_{2\ell+1}$ (in the
case $R^{\dagger}=\hat{R}$ it is the excellence of $R$ which implies the
uniqueness of such a minimal prime). We have
$H_{2\ell}\cap R=P_{\ell}$
for $\ell\in\\{0,\dots,r\\}$. The results of §4 apply equally well to
completions, henselizations and other local étale extensions; to complete the
proof of the primality of the implicit ideals in various contexts such as
henselization or completion, it remains to show the existence of stable
$\nu$-extensions in the corresponding context.
In §5 we describe the set of extensions $\nu^{\dagger}_{-}$ of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}}{P^{\prime}{R^{\prime}}^{\dagger}}$,
where $P^{\prime}$ is a tree of prime ideals of ${R^{\prime}}^{\dagger}$ such
that $P^{\prime}\cap R^{\prime}=(0)$. We show (Theorem 5.6) that specifying
such a valuation $\nu^{\dagger}_{-}$ is equivalent to specifying the following
data:
(1) a chain (16) of trees of prime ideals $\tilde{H}^{\prime}_{i}$ of
${R^{\prime}}^{\dagger}$ (where $\tilde{H}^{\prime}_{0}=P^{\prime}$), such
that $H^{\prime}_{i}\subset\tilde{H}^{\prime}_{i}$ for each $i$ and each
$R^{\prime}\in\mathcal{T}$, satisfying one additional condition (we will refer
to the chain (16) as the chain of trees of ideals, determined by the extension
$\nu^{\dagger}_{-}$)
(2) a valuation $\nu^{\dagger}_{i}$ of the residue field
$k_{\nu^{\dagger}_{i-1}}$ of $\nu^{\dagger}_{i-1}$, whose restriction to the
field
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})$
is centered at the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$.
If $i=2\ell$ is even, the valuation $\nu^{\dagger}_{i}$ must be of rank 1 and
its restriction to $\kappa(\mathbf{m}_{\ell-1})$ must coincide with
$\nu_{\ell}$.
Notice the recursive nature of this description of $\nu^{\dagger}_{-}$: in
order to describe $\nu^{\dagger}_{i}$ we must know $\nu^{\dagger}_{i-1}$ in
order to talk about its residue field $k_{\nu^{\dagger}_{i-1}}$.
In §6 we address the question of uniqueness of $\nu^{\dagger}_{-}$. We
describe several classes of extensions $\nu^{\dagger}_{-}$ which are
particularly useful for the applications: minimal and evenly minimal
extensions, and also those $\nu^{\dagger}_{-}$ for which, denoting by
${\operatorname{ht}}\ I$ the height of an ideal, we have
${\operatorname{ht}}\ \tilde{H}^{\prime}_{2\ell+1}-{\operatorname{ht}}\
\tilde{H}^{\prime}_{2_{\ell}}\leq 1\quad\text{ for }0\leq\ell\leq r;$ (22)
in fact, the special case of (22) which is of most interest for the
applications is
$\tilde{H}^{\prime}_{2\ell}=\tilde{H}^{\prime}_{2\ell+1}\quad\text{ for
}1\leq\ell\leq r.$ (23)
We prove some necessary and some sufficient conditions under which an
extension $\nu^{\dagger}_{-}$ whose corresponding ideals
$\tilde{H}^{\prime}_{i}$ satisfy (23) is uniquely determined by the ideals
$\tilde{H}^{\prime}_{i}$. We also give sufficient conditions for the graded
algebra $gr_{\nu}R^{\prime}$ to be scalewise birational to
$gr_{\hat{\nu}_{-}}\hat{R}^{\prime}$ for each $R^{\prime}\in\cal{T}$. These
sufficient conditions are used in §9 to prove some partial results towards
Conjecture 1.11.
In §7 we show the existence of $\nu$-extensions in $\cal T$, stable for
henselization, thus reducing the proof of the primality of $H_{2\ell+1}$ to
the results of §4. We study the extension of $\nu$ to $\tilde{R}$ modulo its
first prime ideal and prove that such an extension is unique.
In §8 we use the results of §7 to prove the existence of $\nu$-extensions in
$\cal T$, stable for completion. Combined with the results of §4 this proves
that the ideals $H_{2\ell+1}$ are prime.
In §9 we describe a possible approach and prove some partial results towards
constructing a chain of trees (16) of prime ideals of $\hat{R}^{\prime}$
satisfying (23) and a corresponding valuation $\hat{\nu}_{-}$ which satisfies
the conclusion of Conjecture 1.11. We also prove a necessary and a sufficient
condition for the uniqueness of $\hat{\nu}_{-}$, assuming Conjecture 1.11.
We would like to acknowledge the paper [5] by Bill Heinzer and Judith Sally
which inspired one of the authors to continue thinking about this subject, as
well as the work of S.D. Cutkosky, S. El Hitti and L. Ghezzi: [3] (which
contains results closely related to those of §2) and [2].
## 2 Extending a valuation of rank one centered in a local domain to its
formal completion.
Let $(R,M,k)$ be a local noetherian domain, $K$ its field of fractions, and
$\nu:K\rightarrow{\Gamma}_{+}\cup\\{\infty\\}$ a rank one valuation, centered
at $R$ (that is, non-negative on $R$ and positive on $M$).
Let $\hat{R}$ denote the formal completion of $R$. It is convenient to extend
$\nu$ to a valuation centered at $\frac{\hat{R}}{H}$, where $H$ is a prime
ideal of $\hat{R}$ such that $H\cap R=(0)$. In this section, we will assume
that $\nu$ is of rank one, so that the value group ${\Gamma}$ is archimedian.
We will explicitly describe a prime ideal $H$ of $\hat{R}$, canonically
associated to $\nu$, such that $H\cap R=(0)$ and such that $\nu$ has a unique
extension $\hat{\nu}_{-}$ to $\frac{\hat{R}}{H}$.
Let ${\Phi}=\nu(R\setminus(0))$, let $\mathcal{P}_{\beta}$ denote the
$\nu$-ideal of $R$ of value $\beta$ and $\mathcal{P}_{\beta}^{+}$ the greatest
$\nu$-ideal, properly contained in $\mathcal{P}_{\beta}$. We now define the
main object of study of this section. Let
$H:=\bigcap\limits_{\beta\in{\Phi}}(\mathcal{P}_{\beta}\hat{R}).$ (24)
###### Remark 2.1
Since $R$ is noetherian, we have $\nu(M)>0$ and since the ordered group
$\Gamma$ is archimedian, for every $\beta\in{\Phi}$ there exists
$n\in\mathbf{N}$ such that $M^{n}\subset\mathcal{P}_{\beta}$. In other words,
the $M$-adic topology on $R$ is finer than (or equal to) the $\nu$-adic
topology. Therefore an element $x\in\hat{R}$ lies in
$\mathcal{P}_{\beta}\hat{R}\iff$ there exists a Cauchy sequence
$\\{x_{n}\\}\subset R$ in the $M$-adic topology, converging to $x$, such that
$\nu(x_{n})\geq\beta$ for all $n\iff$ for every Cauchy sequence
$\\{x_{n}\\}\subset R$, converging to $x$, $\nu(x_{n})\geq\beta$ for all $n\gg
0$. By the same token, $x\in H\iff$ there exists a Cauchy sequence
$\\{x_{n}\\}\subset R$, converging to $x$, such that
$\lim\limits_{n\to\infty}\nu(x_{n})=\infty\iff$ for every Cauchy sequence
$\\{x_{n}\\}\subset R$, converging to $x$,
$\lim\limits_{n\to\infty}\nu(x_{n})=\infty$.
###### Example 2.2
Let $R=k[u,v]_{(u,v)}$. Then $\hat{R}=k[[u,v]]$. Consider an element
$w=u-\sum\limits_{i=1}^{\infty}c_{i}v^{i}\in\hat{R}$, where $c_{i}\in k^{*}$
for all $i\in\mathbf{N}$, such that $w$ is transcendental over $k(u,v)$.
Consider the injective map $\iota:k[u,v]_{(u,v)}\rightarrow k[[t]]$ which
sends $v$ to $t$ and $u$ to $\sum\limits_{i=1}^{\infty}c_{i}t^{i}$. Let $\nu$
be the valuation induced from the $t$-adic valuation of $k[[t]]$ via $\iota$.
The value group of $\nu$ is $\mathbf{Z}$ and ${\Phi}=\mathbf{N}_{0}$. For each
$\beta\in\mathbf{N}$,
$\mathcal{P}_{\beta}=\left(v^{\beta},u-\sum\limits_{i=1}^{\beta-1}c_{i}v^{i}\right)$.
Thus $H=(w)$.
We come back to the general theory. Since the formal completion homomorphism
$R\rightarrow\hat{R}$ is faithfully flat,
$\mathcal{P}_{\beta}\hat{R}\cap R=\mathcal{P}_{\beta}\quad\text{for all
}\beta\in{\Phi}.$ (25)
Taking the intersection over all $\beta\in{\Phi}$, we obtain
$H\cap
R=\left(\bigcap\limits_{\beta\in{\Phi}}\left(\mathcal{P}_{\beta}\hat{R}\right)\right)\cap
R=\bigcap\limits_{\beta\in{\Phi}}\mathcal{P}_{\beta}=(0),$ (26)
In other words, we have a natural inclusion
$R\hookrightarrow\frac{\hat{R}}{H}$.
###### Theorem 2.3
1. 1.
$H$ is a prime ideal of $\hat{R}$.
2. 2.
$\nu$ extends uniquely to a valuation $\hat{\nu}_{-}$, centered at
$\frac{\hat{R}}{H}$.
Proof.- Let $\bar{x}\in\frac{\hat{R}}{H}\setminus\\{0\\}$. Pick a
representative $x$ of $\bar{x}$ in $\hat{R}$, so that $\bar{x}=x\
{\operatorname{mod}}\ H$. Since $x\mbox{$\in$ /}H$, we have $x\mbox{$\in$
/}\mathcal{P}_{\alpha}\hat{R}$ for some $\alpha\in{\Phi}$.
###### Lemma 2.4
(See [18], Appendix 5, lemma 3) Let $\nu$ be a valuation of rank one centered
in a local noetherian domain $(R,M,k)$. Let
${\Phi}=\nu(R\setminus(0))\subset{\Gamma}.$
Then ${\Phi}$ contains no infinite bounded sequences.
Proof.- An infinite ascending sequence $\alpha_{1}<\alpha_{2}<\dots$ in
${\Phi}$, bounded above by an element $\beta\in{\Phi}$, would give rise to an
infinite descending chain of ideals in $\frac{R}{\mathcal{P}_{\beta}}$. Thus
it is sufficient to prove that $\frac{R}{\mathcal{P}_{\beta}}$ has finite
length.
Let $\delta:=\nu(M)\equiv\min({\Phi}\setminus\\{0\\})$. Since ${\Phi}$ is
archimedian, there exists $n\in\mathbf{N}$ such that $\beta\leq n\delta$. Then
$M^{n}\subset\mathcal{P}_{\beta}$, so that there is a surjective map
$\frac{R}{M^{n}}\twoheadrightarrow\frac{R}{\mathcal{P}_{\beta}}$. Thus
$\frac{R}{\mathcal{P}_{\beta}}$ has finite length, as desired.
By Lemma 2.4, the set $\\{\beta\in{\Phi}\ |\ \beta<\alpha\\}$ is finite. Hence
there exists a unique $\beta\in{\Phi}$ such that
$x\in\mathcal{P}_{\beta}\hat{R}\setminus\mathcal{P}_{\beta}^{+}\hat{R}.$ (27)
Note that $\beta$ depends only on $\bar{x}$, but not on the choice of the
representative $x$. Define the function
$\hat{\nu}_{-}:\frac{\hat{R}}{H}\setminus\\{0\\}\rightarrow{\Phi}$ by
$\hat{\nu}_{-}(\bar{x})=\beta.$ (28)
By (25), if $x\in R\setminus\\{0\\}$ then
$\hat{\nu}_{-}(x)=\nu(x).$ (29)
It is obvious that
$\hat{\nu}_{-}(x+y)\geq\min\\{\hat{\nu}_{-}(x),\hat{\nu}_{-}(y)\\}$ (30)
$\hat{\nu}_{-}(xy)\geq\hat{\nu}_{-}(x)+\hat{\nu}_{-}(y)$ (31)
for all $x,y\in\frac{\hat{R}}{H}$. The point of the next lemma is to show that
$\frac{\hat{R}}{H}$ is a domain and that $\hat{\nu}_{-}$ is, in fact, a
valuation (i.e. that the inequality (31) is, in fact, an equality).
###### Lemma 2.5
For any non-zero $\bar{x},\bar{y}\in\frac{\hat{R}}{H}$, we have
$\bar{x}\bar{y}\neq 0$ and
$\hat{\nu}_{-}(\bar{x}\bar{y})=\hat{\nu}_{-}(\bar{x})+\hat{\nu}_{-}(\bar{y})$.
Proof.- Let $\alpha=\hat{\nu}_{-}(\bar{x})$, $\beta=\hat{\nu}_{-}(\bar{y})$.
Let $x$ and $y$ be representatives in $\hat{R}$ of $\bar{x}$ and $\bar{y}$,
respectively. We have $M\mathcal{P}_{\alpha}\subset\mathcal{P}_{\alpha}^{+}$,
so that
$\frac{\mathcal{P}_{\alpha}}{\mathcal{P}_{\alpha}^{+}}\cong\frac{\mathcal{P}_{\alpha}}{\mathcal{P}_{\alpha}^{+}+M\mathcal{P}_{\alpha}}\cong\frac{\mathcal{P}_{\alpha}}{\mathcal{P}_{\alpha}^{+}}\otimes_{R}k\cong\frac{\mathcal{P}_{\alpha}}{\mathcal{P}_{\alpha}^{+}}\otimes_{R}\frac{\hat{R}}{M\hat{R}}\cong\frac{\mathcal{P}_{\alpha}\hat{R}}{(\mathcal{P}_{\alpha}^{+}+M\mathcal{P}_{\alpha})\hat{R}}\cong\frac{\mathcal{P}_{\alpha}\hat{R}}{\mathcal{P}_{\alpha}^{+}\hat{R}},$
(32)
and similarly for $\beta$. By (32) there exist $z\in\mathcal{P}_{\alpha}$,
$w\in\mathcal{P}_{\beta}$, such that $z\equiv x\ {\operatorname{mod}}\
\mathcal{P}_{\alpha}^{+}\hat{R}$ and $w\equiv y\ {\operatorname{mod}}\
\mathcal{P}_{\beta}^{+}\hat{R}$. Then
$xy\equiv zw\ {\operatorname{mod}}\ \mathcal{P}_{\alpha+\beta}^{+}\hat{R}.$
(33)
Since $\nu$ is a valuation, $\nu(zw)=\alpha+\beta$, so that
$zw\in\mathcal{P}_{\alpha+\beta}\setminus\mathcal{P}_{\alpha+\beta}^{+}$. By
(25) and (33), this proves that
$xy\in\mathcal{P}_{\alpha+\beta}\hat{R}\setminus\mathcal{P}_{\alpha+\beta}^{+}\hat{R}$.
Thus $xy\mbox{$\in$ /}H$ (hence $\bar{x}\bar{y}\neq 0$ in $\frac{\hat{R}}{H}$)
and $\hat{\nu}_{-}(\bar{x}\bar{y})=\alpha+\beta$, as desired.
By Lemma 2.5, $H$ is a prime ideal of $\hat{R}$. By (30) and Lemma 2.5,
$\hat{\nu}_{-}$ is a valuation, centered at $\frac{\hat{R}}{H}$. To complete
the proof of Theorem 2.3, it remains to prove the uniqueness of
$\hat{\nu}_{-}$. Let $x$, $\bar{x}$, the element $\alpha\in{\Phi}$ and
$z\in\mathcal{P}_{\alpha}\setminus\mathcal{P}_{\alpha}^{+}$ (34)
be as in the proof of Lemma 2.5. Then there exist
$\begin{array}[]{rl}u_{1},\ldots,u_{n}&\in\mathcal{P}_{\alpha}^{+}\text{
and}\\\ v_{1},\ldots,v_{n}&\in\hat{R}\end{array}$ (35)
such that $x=z+\sum\limits_{i=1}^{n}u_{i}v_{i}$. Letting $\bar{v}_{i}:=v_{i}\
{\operatorname{mod}}\ H$, we obtain
$\bar{x}=\bar{z}+\sum\limits_{i=1}^{n}\bar{u}_{i}\bar{v}_{i}$ in
$\frac{\hat{R}}{H}$. Therefore, by (34)–(35), for any extension of $\nu$ to a
valuation $\hat{\nu}^{\prime}_{-}$, centered at $\frac{\hat{R}}{H}$, we have
$\hat{\nu}^{\prime}_{-}(\bar{x})=\alpha=\hat{\nu}_{-}(\bar{x}),$ (36)
as desired. This completes the proof of Theorem 2.3.
###### Definition 2.6
The ideal $H$ is called the implicit prime ideal of $\hat{R}$, associated to
$\nu$. When dealing with more than one ring at a time, we will sometimes write
$H(R,\nu)$ for $H$.
More generally, let $\nu$ be a valuation centered at $R$, not necessarily of
rank one. In any case, we may write $\nu$ as a composition
$\nu=\mu_{2}\circ\mu_{1}$, where $\mu_{2}$ is centered at a non-maximal prime
ideal $P$ of $R$ and $\mu_{1}\left|{}_{\frac{R}{P}}\right.$ is of rank one.
The valuation $\mu_{1}\left|{}_{\frac{R}{P}}\right.$ is centered at
$\frac{R}{P}$. We define the implicit prime ideal of $R$ with respect to
$\nu$, denoted $H(R,\nu)$, to be the inverse image in $\hat{R}$ of the
implicit prime ideal of $\frac{\hat{R}}{P}$ with respect to
$\mu_{1}\left|{}_{\frac{R}{P}}\right.$. For the rest of this section, we will
continue to assume that $\nu$ is of rank one.
###### Remark 2.7
By (32), we have the following natural isomorphisms of graded algebras:
$\begin{array}[]{rl}\mbox{gr}_{\nu}R&\cong\mbox{gr}_{\hat{\nu}_{-}}\frac{\hat{R}}{H}\\\
G_{\nu}&\cong G_{\hat{\nu}_{-}}.\end{array}$
We will now study the behaviour of $H$ under local blowings up of $R$ with
respect to $\nu$ and, more generally, under local homomorphisms. Let
$\pi:(R,M)\rightarrow(R^{\prime},M^{\prime})$ be a local homomorphism of local
noetherian domains. Assume that $\nu$ extends to a rank one valuation
$\nu^{\prime}:R^{\prime}\setminus\\{0\\}\rightarrow{\Gamma}^{\prime}$, where
${\Gamma}^{\prime}\supset{\Gamma}$. The homomorphism $\pi$ induces a local
homomorphism $\hat{\pi}:\hat{R}\rightarrow\hat{R}^{\prime}$ of formal
completions. Let ${\Phi}^{\prime}=\nu^{\prime}(R^{\prime}\setminus\\{0\\})$.
For $\beta\in{\Phi}^{\prime}$, let $\mathcal{P}^{\prime}_{\beta}$ denote the
$\nu^{\prime}$-ideal of $R_{\nu^{\prime}}$ of value $\beta$, as above. Let
$H^{\prime}=H(R^{\prime},\nu^{\prime})$.
###### Lemma 2.8
Let $\beta\in{\Phi}$. Then
$\left(\mathcal{P}^{\prime}_{\beta}\hat{R}^{\prime}\right)\cap\hat{R}=\mathcal{P}_{\beta}\hat{R}.$
(37)
Proof.- Since by assumption $\nu^{\prime}$ extends $\nu$ we have
$\mathcal{P}^{\prime}_{\beta}\cap R=\mathcal{P}_{\beta}$ and the inclusion
$\left(\mathcal{P}^{\prime}_{\beta}\hat{R}^{\prime}\right)\cap\hat{R}\supseteq\mathcal{P}_{\beta}\hat{R}.$
(38)
We will now prove the opposite inclusion. Take an element
$x\in\left(\mathcal{P}^{\prime}_{\beta}\hat{R}^{\prime}\right)\cap\hat{R}$.
Let $\\{x_{n}\\}\subset R$ be a Cauchy sequence in the $M$-adic topology,
converging to $x$. Then $\\{\pi(x_{n})\\}$ converge to $\hat{\pi}(x)$ in the
$M^{\prime}$-adic topology of $\hat{R}^{\prime}$. Applying remark 2.1 to
$R^{\prime}$, we obtain
$\nu(x_{n})\equiv\nu^{\prime}(\pi(x_{n}))\geq\beta\quad\text{for }n\gg 0.$
(39)
By (39) and Remark 2.7, applied to $R$, we have
$x\in\mathcal{P}_{\beta}\hat{R}$. This proves the opposite inclusion in (38),
as desired.
###### Corollary 2.9
We have
$H^{\prime}\cap\hat{R}=H.$
Proof.- Since $\nu^{\prime}$ is of rank one, ${\Phi}$ is cofinal in
${\Phi}^{\prime}$. Now the Corollary follows by taking the intersection over
all $\beta\in{\Phi}$ in (37).
Let $J$ be a non-zero ideal of $R$ and let $R\rightarrow R^{\prime}$ be the
local blowing up along $J$ with respect to $\nu$. Take an element $f\in J$,
such that $\nu(f)=\nu(J)$. By the strict transform of $J$ in
$\hat{R}^{\prime}$ we will mean the ideal
$J^{\text{str}}:=\bigcup\limits_{i=1}^{\infty}\left(\left(J\hat{R}^{\prime}\right):f^{i}\right)\equiv\left(J\hat{R}^{\prime}_{f}\right)\cap\hat{R}^{\prime}.$
If $g$ is another element of $J$ such that $\nu(g)=\nu(J)$ then
$\nu\left(\frac{f}{g}\right)=0$, so that $\frac{f}{g}$ is a unit in
$R^{\prime}$. Thus the definition of strict transform is independent of the
choice of $f$.
###### Corollary 2.10
$H^{\text{str}}\subset H^{\prime}$.
Proof.- Since $H\hat{R}^{\prime}\subset H^{\prime}$, we have
$H^{\text{str}}=\left(H\hat{R}^{\prime}_{f}\right)\cap\hat{R}^{\prime}\subset\left(H^{\prime}\hat{R}^{\prime}_{f}\right)\cap\hat{R}^{\prime}=H^{\prime}$,
where the last equality holds because $H^{\prime}$ is a prime ideal of
$\hat{R}^{\prime}$, not containing $f$.
Using Zariski’s Main Theorem, it can be proved that $H^{\text{str}}$ is prime.
Since this fact is not used in the sequel, we omit the proof.
###### Corollary 2.11
Let the notation and assumptions be as in corollary 2.10. Then
${\operatorname{ht}}\ H^{\prime}\geq{\operatorname{ht}}\ H.$ (40)
In particular,
$\dim\frac{\hat{R}^{\prime}}{H^{\prime}}\leq\dim\frac{\hat{R}}{H}.$ (41)
Proof.- Let
$\bar{R}:=\left(\hat{R}\otimes_{R}R^{\prime}\right)_{M^{\prime}\hat{R}^{\prime}\cap(\hat{R}\otimes_{R}R^{\prime})}$.
Let $\phi$ denote the natural local homomorphism
$\bar{R}\rightarrow\hat{R}^{\prime}.$
Let $\bar{H}:=H^{\prime}\cap\bar{R}$. Now, take $f\in J$ such that
$\nu(f)=\nu(J)$. Then $f\mbox{$\in$ /}H^{\prime}$ and, in particular,
$f\mbox{$\in$ /}\bar{H}$. Since $R^{\prime}_{f}\cong R_{f}$, we have
$\hat{R}_{f}=\bar{R}_{f}$. In view of Corollary 2.9, we obtain
$H\hat{R}_{f}\cong\bar{H}\bar{R}_{f}$, so
${\operatorname{ht}}\ H={\operatorname{ht}}\ \bar{H}.$ (42)
Now, $\bar{R}$ is a local noetherian ring, whose formal completion is
$\hat{R}^{\prime}$. Hence $\phi$ is faithfully flat and therefore satisfies
the going down theorem. Thus we have ${\operatorname{ht}}\
H^{\prime}\geq{\operatorname{ht}}\ \bar{H}$. Combined with (42), this proves
(40). As for the last statement of the Corollary, it follows from the well
known fact that dimension does not increase under blowing up ([12], Lemma
2.2): we have $\dim\ R^{\prime}\leq\dim\ R$, hence
$\dim\ \hat{R}^{\prime}=\dim R^{\prime}\leq\dim\ R=\dim\ \hat{R},$
and (41) follows from (40) and from the fact that complete local rings are
catenarian.
It may well happen that the containment of corollary 2.10 and the inequality
in (40) are strict. The possibility of strict containement in corollary 2.10
is related to the existence of subanalytic functions, which are not analytic.
We illustrate this statement by an example in which $H^{\text{str}}\subsetneqq
H^{\prime}$ and ${\operatorname{ht}}\ H<{\operatorname{ht}}\ H^{\prime}$.
###### Example 2.12
Let $k$ be a field and let
$\begin{array}[]{rl}R&=k[x,y,z]_{(x,y,z)},\\\
R^{\prime}&=k[x^{\prime},y^{\prime},z^{\prime}]_{(x^{\prime},y^{\prime},z^{\prime})},\end{array}$
where $x^{\prime}=x$, $y^{\prime}=\frac{y}{x}$ and $z^{\prime}=z$. We have
$K=k(x,y,z)$, $\hat{R}=k[[x,y,z]]$,
$\hat{R}^{\prime}=k[[x^{\prime},y^{\prime},z^{\prime}]]$. Let $t_{1},t_{2}$ be
auxiliary variables and let $\sum\limits_{i=1}^{\infty}c_{i}t_{1}^{i}$ (with
$c_{i}\in k$) be an element of $k[[t_{1}]]$, transcendental over $k(t_{1})$.
Let $\theta$ denote the valuation, centered at $k[[t_{1},t_{2}]]$, defined by
$\theta(t_{1})=1$, $\theta(t_{2})=\sqrt{2}$ (the value group of $\theta$ is
the additive subgroup of $\mathbf{R}$, generated by 1 and $\sqrt{2}$). Let
$\iota:R^{\prime}\hookrightarrow k[[t_{1},t_{2}]]$ denote the injective map
defined by $\iota(x^{\prime})=t_{2}$, $\iota(y^{\prime})=t_{1}$,
$\iota(z^{\prime})=\sum\limits_{i=1}^{\infty}c_{i}t_{1}^{i}$. Let $\nu$ denote
the restriction of $\theta$ to $K$, where we view $K$ as a subfield of
$k((t_{1},t_{2}))$ via $\iota$. Let ${\Phi}=\nu(R\setminus\\{0\\})$;
${\Phi}^{\prime}=\nu(R^{\prime}\setminus\\{0\\})$. For
$\beta\in{\Phi}^{\prime}$, $P^{\prime}_{\beta}$ is generated by all the
monomials of the form ${x^{\prime}}^{\alpha}{y^{\prime}}^{\gamma}$ such that
$\sqrt{2}\alpha+\gamma\geq\beta$, together with
$z^{\prime}-\sum\limits_{j=1}^{i}c_{j}{y^{\prime}}^{j}$, where $i$ is the
greatest non-negative integer such that $i<\beta$.
Let $w^{\prime}:=z^{\prime}-\sum\limits_{i=1}^{\infty}c_{i}{y^{\prime}}^{i}$.
Then $H^{\prime}=(w^{\prime})$, but $H=H^{\prime}\cap\hat{R}=(0)$, so that
$H^{\text{str}}=(0)\subsetneqq H^{\prime}$ and ${\operatorname{ht}}\
H=0<1={\operatorname{ht}}\ H^{\prime}$.
Recall the following basic result of the theory of G-rings:
###### Proposition 2.13
Assume that $R$ is a reduced G-ring. Then $\hat{R}_{H}$ is a regular local
ring.
Proof.- Let $K=R_{\mathcal{P}_{\infty}}=\kappa(\mathcal{P}_{\infty})$ (here we
are using that $R$ is reduced and that $\mathcal{P}_{\infty}$ is a minimal
prime of $R$). By definition of G-ring, the map $R\rightarrow\hat{R}$ is a
regular homomorphism. Then by (26) $\hat{R}_{H}$ is geometrically regular over
$K$, hence regular.
###### Remark 2.14
Having extended in a unique manner the valuation $\nu$ to a valuation
$\hat{\nu}_{-}$ of $\frac{\hat{R}}{H}$, we see that if $R$ is a G-ring, by
Proposition 2.13 there is a unique minimal prime $\hat{\mathcal{P}}_{\infty}$
of $\hat{R}$ contained in $H$, corresponding to the ideal $(0)$ in
$\hat{R}_{H}$. Since $H\cap R=(0)$, we have the equality
$\hat{\mathcal{P}}_{\infty}\cap R=(0)$. Choosing a valuation $\mu$ of the
fraction field of $\frac{\hat{R}_{H}}{\hat{\mathcal{P}}_{\infty}\hat{R}_{H}}$
centered at $\frac{\hat{R}_{H}}{\hat{\mathcal{P}}_{\infty}\hat{R}_{H}}$ whose
value group $\Psi$ is a free abelian group produces a composed valuation
$\hat{\nu}_{-}\circ\mu$ on $\frac{\hat{R}}{\hat{\mathcal{P}}_{\infty}}$ with
value group $\Psi\bigoplus\Gamma$ ordered lexicographically, as follows:
Given $x\in\frac{\hat{R}}{\hat{\mathcal{P}}_{\infty}}$, let $\psi=\mu(x)$ and
blow up in $R$ the ideal $\mathcal{P}_{\psi}$ along our original valuation,
obtaining a local ring $R^{\prime}$. According to what we have seen so far in
this section, in its completion $\hat{R}^{\prime}$ we can write $x=ye$ with
$\mu(e)=\psi$ and $y\in\hat{R}^{\prime}\setminus H^{\prime}$. The valuation
$\nu$ on $R^{\prime}$ extends uniquely to a valuation of
$\frac{\hat{R}^{\prime}}{H^{\prime}}$, which we may still denote by
$\hat{\nu}_{-}$ because it induces $\hat{\nu}_{-}$ on $\frac{\hat{R}}{H}$. Let
us consider the image $\overline{y}$ of $y$ in
$\frac{\hat{R}^{\prime}}{H^{\prime}}$. Setting
$(\hat{\nu}_{-}\circ\mu)(x)=\psi\bigoplus\hat{\nu}_{-}(\overline{y})\in\Psi\bigoplus\Gamma$
determines a valuation of $\frac{\hat{R}}{\hat{\mathcal{P}}_{\infty}}$ as
required.
If we drop the assumption that $\Psi$ is a free abelian group, the above
construction still works, but the value group $\hat{\Gamma}$ of
$\hat{\nu}_{-}\circ\mu$ need not be isomorphic to the direct sum
$\Psi\bigoplus\Gamma$. Rather, we have an exact sequence
$0\rightarrow\Gamma\rightarrow\bar{\Gamma}\rightarrow\Psi\rightarrow 0$, which
need not, in general, be split; see [15], Proposition 4.3.
In the sequel we shall reduce to the case where $\hat{R}$ is an integral
domain, so that $\hat{\mathcal{P}}_{\infty}=(0)$ and we will have constructed
a valuation of $\hat{R}$.
## 3 Definition and first properties of implicit ideals.
Let the notation be as above. Before plunging into technical details, we would
like to give a brief and informal overview of our constructions and the
motivation for them. Above we recalled the well known fact that if $rk\ \nu=r$
then for every $\nu$-extension $R\rightarrow R^{\prime}$ the valuation $\nu$
canonically determines a flag (6) of $r$ subschemes of $\mbox{Spec}\
R^{\prime}$. This paper shows the existence of subschemes of $\mbox{Spec}\
\hat{R}$, determined by $\nu$, which are equally canonical and which become
explicit only after completion. To see what they are, first of all note that
the ideal $P^{\prime}_{l}\hat{R}^{\prime}$, for $R^{\prime}\in\cal{T}$ and
$0\leq\ell\leq r-1$, need not in general be prime (although it is prime
whenever $R^{\prime}$ is henselian). Another way of saying the same thing is
that the ring $\frac{R^{\prime}}{P^{\prime}_{\ell}}$ need not be analytically
irreducible in general. However, we will see in §8 (resp. §7) that the
valuation $\nu$ picks out in a canonical way one of the minimal primes of
$P^{\prime}_{l}\hat{R}^{\prime}$ (resp. $P^{\prime}_{l}\tilde{R}^{\prime}$).
We call this minimal prime $H^{\prime}_{2\ell}$ for reasons which will become
apparent later. By the flatness of completion (resp. henselization), we have
$H^{\prime}_{2\ell}\cap R^{\prime}=P^{\prime}_{\ell}$. We will show that the
ideals $H^{\prime}_{2\ell}$ form a tree.
Let
$(0)=\Delta_{r}\subsetneqq\Delta_{r-1}\subsetneqq\dots\subsetneqq\Delta_{0}=\Gamma$
(43)
be the isolated subgroups of $\Gamma$. There are other ideals of $\hat{R}$,
apart from the $H_{2\ell}$, canonically associated to $\nu$, whose
intersection with $R$ equals $P_{\ell}$, for example, the ideal
$\bigcap\limits_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}\hat{R}$. The same
is true of the even larger ideal
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left(\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\cal
P}^{\prime}_{\beta}\hat{R}^{\prime}\right)\bigcap\hat{R}\right),$ (44)
(that $H_{2\ell+1}\cap R=P_{\ell}$ is easy to see and will be shown later in
this section, in Proposition 3.5). While the examples below show that the
ideal $\bigcap\limits_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}\hat{R}$ need
not, in general, be prime, the ideal $H_{2\ell+1}$ always is (this is the main
theorem of this paper; it will be proved in §8). The ideal $H_{2\ell+1}$
contains $H_{2\ell}$ but is not, in general equal to it. To summarize, we will
show that the valuation $\nu$ picks out in a canonical way a generic point
$H_{2\ell}$ of the formal fiber over $P_{\ell}$ and also another point
$H_{2\ell+1}$ in the formal fiber, which is a specialization of $H_{2\ell}$.
The main technique used to prove these results is to to analyze the set of
zero divisors of
$\frac{{R^{\prime}}^{\dagger}}{P^{\prime}_{\ell}{R^{\prime}}^{\dagger}}$
(where $R^{\dagger}$ stands for either $\hat{R}$, $\tilde{R}$, or a finite
type étale extension $R^{e}$ of $R$), as follows. We show that the
reducibility of
$\frac{{R^{\prime}}^{\dagger}}{P^{\prime}_{\ell}{R^{\prime}}^{\dagger}}$ is
related to the existence of non-trivial algebraic extensions of
$\kappa(P_{\ell})$ inside $\kappa(P_{\ell})\otimes_{R}R^{\dagger}$. More
precisely, in the next section we define $R$ to be stable if
$\frac{R^{\dagger}}{P_{\ell+1}R^{\dagger}}$ is a domain and there does not
exist a non-trivial algebraic extension of $\kappa(P_{\ell+1})$ which embeds
both into $\kappa(P_{\ell+1})\otimes_{R}R^{\dagger}$ and into
$\kappa(P^{\prime}_{\ell+1})$ for some $R^{\prime}\in\mathcal{T}$. We show
that if $R$ is stable then
$\frac{{R^{\prime}}^{\dagger}}{P^{\prime}_{\ell+1}{R^{\prime}}^{\dagger}}$ is
a domain for all $R^{\prime}\in\cal T$. For
$\overline{\beta}\in\frac{\Gamma}{\Delta_{\ell+1}}$, let
$\mathcal{P}_{\overline{\beta}}=\left\\{x\in R\ \left|\
\nu(x)\mod\Delta_{\ell+1}\geq\overline{\beta}\right.\right\\}$ (45)
If $\Phi$ denotes the semigroup $\nu(R\setminus\\{0\\})\subset\Gamma$, which
is well ordered since $R$ is noetherian (see [ZS], Appendix 4, Proposition 2),
and
$\beta(\ell)=\min\\{\gamma\in\Phi\ |\ \beta-\gamma\in\Delta_{\ell+1}\\}$
then $\mathcal{P}_{\overline{\beta}}=\mathcal{P}_{\beta(l)}$.
We have the inclusions
$P_{\ell}\subset\mathcal{P}_{\overline{\beta}}\subset P_{\ell+1},$
and $\mathcal{P}_{\overline{\beta}}$ is the inverse image in $R$ by the
canonical map $R\to\frac{R}{P_{\ell}}$ of a valuation ideal
$\overline{\mathcal{P}}_{\overline{\beta}}\subset\frac{R}{P_{\ell}}$ for the
rank one valuation
$\frac{R}{P_{\ell}}\setminus\\{0\\}\to\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$
induced by $\nu_{\ell+1}$.
We will deduce from the above that if $R$ is stable then for each
$\overline{\beta}\in\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$ and each
$\nu$-extension $R\rightarrow R^{\prime}$ we have
$\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}\cap
R^{\dagger}=\mathcal{P}_{\overline{\beta}}R^{\dagger}$, which gives us a very
good control of the limit in the definition of $H_{2\ell+1}$ and of the
$\nu$-extensions $R^{\prime}$ for which the limit is attained.
We then show, separately in the cases when $R^{\dagger}=\tilde{R}$ (§7) and
$R^{\dagger}=\hat{R}$ (§8), that there always exists a stable $\nu$-extension
$R^{\prime}\in\cal{T}$.
We are now ready to go into details, after giving several examples of implicit
ideals and the phenomena discussed above.
Let $0\leq\ell\leq r$. We define our main object of study, the $(2\ell+1)$-st
implicit prime ideal $H_{2\ell+1}\subset R^{\dagger}$, by
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left(\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\cal
P}^{\prime}_{\beta}{R^{\prime}}^{\dagger}\right)\bigcap R^{\dagger}\right),$
(46)
where $R^{\prime}$ ranges over $\mathcal{T}$. As usual, we think of (46) as a
tree equation: if we replace $R$ by any other $R^{\prime\prime}\in{\cal T}$ in
(46), it defines the corresponding ideal
$H^{\prime\prime}_{2\ell+1}\subset\hat{R}^{\prime\prime{\dagger}}$. Note that
for $\ell=r$ (46) reduces to
$H_{2r+1}=mR^{\dagger}.$
We start by giving several examples of the ideals $H^{\prime}_{i}$ (and also
of $\tilde{H}^{\prime}_{i}$, which will appear a little later in the paper).
###### Example 3.1
Let $R=k[x,y,z]_{(x,y,z)}$. Let $\nu$ be the valuation with value group
$\Gamma=\mathbf{Z}^{2}_{lex}$, defined as follows. Take a transcendental power
series $\sum\limits_{j=1}^{\infty}c_{j}u^{j}$ in a variable $u$ over $k$.
Consider the homomorphism $R\hookrightarrow k[[u,v]]$ which sends $x$ to $v$,
$y$ to $u$ and $z$ to $\sum\limits_{j=1}^{\infty}c_{j}u^{j}$. Consider the
valuation $\nu$, centered at $k[[u,v]]$, defined by $\nu(v)=(0,1)$ and
$\nu(u)=(1,0)$; its restriction to $R$ will also be denoted by $\nu$, by abuse
of notation. Let $R_{\nu}$ denote the valuation ring of $\nu$ in $k(x,y,z)$
and let $\mathcal{T}$ be the tree consisting of all the local rings
$R^{\prime}$ essentially of finite type over $R$, birationally dominated by
$R_{\nu}$. Let ${}^{\dagger}=\hat{\ }$ denote the operation of formal
completion. Given $\beta=(a,b)\in\mathbf{Z}^{2}_{lex}$, we have
${\mathcal{P}}_{\beta}=x^{b}\left(y^{a},z-c_{1}y-\dots-c_{a-1}y^{a-1}\right)$.
The first isolated subgroup $\Delta_{1}=(0)\oplus\mathbf{Z}$. Then
$\bigcap\limits_{\beta\in(0)\oplus\mathbf{Z}}\left({\cal
P}_{\beta}\hat{R}\right)=(y,z)$ and
$\bigcap\limits_{\beta\in\Gamma=\Delta_{0}}\left({\cal
P}_{\beta}\hat{R}\right)=\left(z-\sum\limits_{j=1}^{\infty}c_{j}y^{j}\right)$.
It is not hard to show that for any $R^{\prime}\in\mathcal{T}$ we have
$H^{\prime}_{1}=\left(z-\sum\limits_{j=1}^{\infty}c_{j}y^{j}\right)\hat{R}^{\prime}$
and that $H_{3}=(y,z)\hat{R}$. It will follow from the general theory
developed in §6 that $\nu$ admits a unique extension $\hat{\nu}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\hat{R}^{\prime}$. This
extension has value group $\hat{\Gamma}=\mathbf{Z}^{3}_{lex}$ and is defined
by $\hat{\nu}(x)=(0,0,1)$, $\hat{\nu}(y)=(0,1,0)$ and
$\hat{\nu}\left(z-\sum\limits_{j=1}^{\infty}c_{j}y^{j}\right)=(1,0,0)$. For
each $R^{\prime}\in\mathcal{T}$ the ideal $H^{\prime}_{1}$ is the prime
valuation ideal corresponding to the isolated subgroup
$(0)\oplus\mathbf{Z}^{2}_{lex}$ of $\hat{\Gamma}$ (that is, the ideal whose
elements have values outside of $(0)\oplus\mathbf{Z}^{2}_{lex}$) while
$H^{\prime}_{3}$ is the prime valuation ideal corresponding to the isolated
subgroup $(0)\oplus(0)\oplus\mathbf{Z}$.
###### Example 3.2
Let $R=k[x,y,z]_{(x,y,z)}$, $\Gamma=\mathbf{Z}^{2}_{lex}$, the power series
$\sum\limits_{j=1}^{\infty}c_{j}u^{j}$ and the operation ${}^{\dagger}=\hat{\
}$ be as in the previous example. This time, let $\nu$ be defined as follows.
Consider the homomorphism $R\hookrightarrow k[[u,v]]$ which sends $x$ to $u$,
$y$ to $\sum\limits_{j=1}^{\infty}c_{j}u^{j}$ and $z$ to $v$. Consider the
valuation $\nu$, centered at $k[[u,v]]$, defined by $\nu(v)=(1,0)$ and
$\nu(u)=(0,1)$; its restriction to $R$ will be also denoted by $\nu$. Let
$R_{\nu}$ denote the valuation ring of $\nu$ in $k(x,y,z)$ and let
$\mathcal{T}$ be the tree consisting of all the local rings $R^{\prime}$
essentially of finite type over $R$, birationally dominated by $R_{\nu}$.
Given $\beta=(a,b)\in\mathbf{Z}^{2}_{lex}$, we have
$\mathcal{P}_{\beta}=z^{a}\left(x^{b},y-c_{1}x-\dots-c_{b-1}x^{b-1}\right)$.
The first isolated subgroup $\Delta_{1}=(0)\oplus\mathbf{Z}$. Then
$\bigcap\limits_{\beta\in(0)\oplus\mathbf{Z}}\left({\mathcal{P}}_{\beta}\hat{R}\right)=\left(y-\sum\limits_{j=1}^{\infty}c_{j}x^{j},z\right)$
and $\bigcap\limits_{\beta\in\Gamma=\Delta_{0}}\left({\cal
P}_{\beta}\hat{R}\right)=(0)$. It is not hard to show that for any
$R^{\prime}\in\cal{T}$ we have $H^{\prime}_{1}=(0)$ and that
$H_{3}=\left(y-\sum\limits_{j=1}^{\infty}c_{j}x^{j},z\right)\hat{R}^{\prime}$.
In this case, the extension $\hat{\nu}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\hat{R}^{\prime}$ is not
unique. Indeed, one possible extension $\hat{\nu}^{(1)}$ has value group
$\hat{\Gamma}=\mathbf{Z}^{3}_{lex}$ and is defined by
$\hat{\nu}^{(1)}(x)=(0,0,1)$,
$\hat{\nu}^{(1)}\left(y-\sum\limits_{j=1}^{\infty}c_{j}x^{j}\right)=(0,1,0)$
and $\hat{\nu}^{(1)}(z)=(1,0,0)$. In this case, for any $R^{\prime}\in\cal{T}$
the ideal $H^{\prime}_{3}$ is the prime valuation ideal corresponding to the
isolated subgroup $(0)\oplus(0)\oplus\mathbf{Z}$ of $\hat{\Gamma}$.
Another extension $\hat{\nu}^{(2)}$ of $\nu$ is defined by
$\hat{\nu}^{(2)}(x)=(0,0,1)$,
$\hat{\nu}^{(2)}\left(y-\sum\limits_{j=1}^{\infty}c_{j}x^{j}\right)=(1,0,0)$
and $\hat{\nu}^{(2)}(z)=(0,1,0)$. In this case, the tree of ideals
corresponding to the isolated subgroup $(0)\oplus(0)\oplus\mathbf{Z}$ is
$H^{\prime}_{3}$ (exactly the same as for $\hat{\nu}^{(1)}$) while that
corresponding to $(0)\oplus\mathbf{Z}^{2}_{lex}$ is
$\tilde{H}^{\prime}_{1}=\left(y-\sum\limits_{j=1}^{\infty}c_{j}x^{j}\right)$.
The tree $\tilde{H}^{\prime}_{1}$ of prime $\hat{\nu}^{(2)}$-ideals determines
the extension $\hat{\nu}^{(2)}$ completely.
The following two examples illustrate the need for taking the limit over the
tree $\mathcal{T}$.
###### Example 3.3
Let us consider the local domain
$S=\frac{k[x,y]_{(x,y)}}{(y^{2}-x^{2}-x^{3})}$. There are two distinct
valuations centered in $(x,y)$. Let $a_{i}\in k,\ i\geq 2$ be such that
$\left(y+x+\sum_{i\geq 2}a_{i}x^{i}\right)\left(y-x-\sum_{i\geq
2}a_{i}x^{i}\right)=y^{2}-x^{2}-x^{3}.$
We shall denote by $\nu_{+}$ the rank one discrete valuation defined by
$\nu_{+}(x)=\nu_{+}(y)=1,$ $\nu_{+}(y+x)=2,$
$\nu_{+}\left(y+x+\sum_{i=2}^{b-1}a_{i}x^{i}\right)=b.$
Now let $R=\frac{k[x,y,z]_{(x,y,z)}}{(y^{2}-x^{2}-x^{3})}$. Let
$\Gamma=\mathbf{Z}^{2}$ with the lexicographical ordering. Let $\nu$ be the
composite valuation of the $(z)$-adic one with $\nu_{+}$, centered in
$\frac{R}{(z)}$. The point of this example is to show that
$H^{*}_{2\ell+1}=\bigcap_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}{\hat{R}}$
does not work as the definition of the $(2\ell+1)$-st implicit prime ideal
because the resulting ideal $H^{*}_{2\ell+1}$ is not prime. Indeed, as
$\mathcal{P}_{(a,0)}=(z^{a})$, we have
$H_{1}^{*}=\bigcap_{(a,b)\in\mathbf{Z}^{2}}\mathcal{P}_{(a,b)}{\hat{R}}=(0).$
Let $f=y+x+\sum\limits_{i\geq 2}a_{i}x^{i},g=y-x-\sum\limits_{i\geq
2}a_{i}x^{i}\in\hat{R}$. Clearly $f,g\mbox{$\in$ /}H^{*}_{1}=(0)$, but $f\cdot
g=0$, so the ideal $H^{*}_{1}$ is not prime.
One might be tempted (as we were) to correct this problem by localizing at
$H^{*}_{2\ell+3}$. Indeed, if we take the new definition of $H^{*}_{2\ell+1}$
to be, recursively in the descending order of $\ell$,
$H^{*}_{2\ell+1}=\left(\bigcap_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}{\hat{R}}_{H^{*}_{2\ell+3}}\right)\cap{\hat{R}},$
(47)
then in the present example the resulting ideals $H^{*}_{3}=(z,f)$ and
$H^{*}_{1}=(f)$ are prime. However, the next example shows that the definition
(47) also does not, in general, give rise to prime ideals.
###### Example 3.4
Let $R=\frac{k[x,y,z]_{(x,y,z)}}{(z^{2}-y^{2}(1+x))}$. Let
$\Gamma=\mathbf{Z}^{2}$ with the lexicographical ordering. Let $t$ be an
independent variable and let $\nu$ be the valuation, centered in $R$, induced
by the $t$-adic valuation of $k\left[\left[t^{\Gamma}\right]\right]$ under the
injective homomorphism $\iota:R\hookrightarrow
k\left[\left[t^{\Gamma}\right]\right]$, defined by $\iota(x)=t^{(0,1)}$,
$\iota(y)=t^{(1,0)}$ and $\iota(z)=t^{(1,0)}\sqrt{1+t^{(0,1)}}$. The prime
$\nu$-ideals of $R$ are $(0)\subsetneqq P_{1}\subsetneqq m$, with
$P_{1}=(y,z)$. We have
$\bigcap\limits_{\beta\in\Delta_{1}}\mathcal{P}_{\beta}\hat{R}=(y,z)\hat{R}=P_{1}\hat{R}$
and
$\bigcap\limits_{\beta\in\Gamma}\mathcal{P}_{\beta}\hat{R}_{(y,z)}=\bigcap\limits_{\beta\in\Gamma}\mathcal{P}_{\beta}\hat{R}=(0)$.
Note that the ideal $(0)$ is not prime in $\hat{R}$. Now, let
$R^{\prime}=R\left[\frac{z}{y}\right]_{m^{\prime}}$, where
$m^{\prime}=\left(x,y,\frac{z}{y}-1\right)$ is the center of $\nu$ in
$R\left[\frac{z}{y}\right]$. We have
$z-y\sqrt{1+x}\in\hat{R}\setminus\mathcal{P}_{(2,0)}\hat{R}$. On the other
hand, $z-y\sqrt{1+x}=y\left(\frac{z}{y}-\sqrt{1+x}\right)=0$ in
$\hat{R}^{\prime}$; in particular,
$z-y\sqrt{1+x}\in\bigcap\limits_{\beta\in\Gamma}\mathcal{P}^{\prime}_{\beta}\hat{R}^{\prime}$.
Thus this example also shows that the ideals $\mathcal{P}_{\beta}\hat{R}$,
$\bigcap\limits_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}\hat{R}$ and
$\bigcap\limits_{\beta\in\Delta_{\ell}}\mathcal{P}_{\beta}\hat{R}_{H_{2\ell+3}}$
do not behave well under blowing up.
Note that both Examples 3.3 and 3.4 occur not only for the completion
$\hat{R}$ but also for the henselization $\tilde{R}$.
We come back to the general theory of implicit ideals.
###### Proposition 3.5
We have $H_{2\ell+1}\cap R=P_{\ell}$.
Proof.- Recall that $P_{\ell}=\left\\{x\in R\ \left|\ \ \nu(x)\mbox{$\in$
/}\Delta_{\ell}\right.\right\\}$. If $x\in P_{\ell}$ then, since
$\Delta_{\ell}$ is an isolated subgroup, we have $x\in{\cal P}_{\beta}$ for
all $\beta\in\Delta_{\ell}$. The same inclusion holds for the same reason in
all extensions $R^{\prime}\subset R_{\nu}$ of $R$, and this implies the
inclusion $P_{\ell}\subseteq H_{2\ell+1}\cap R$. Now let $x$ be in
$H_{2\ell+1}\cap R$ and assume $x\mbox{$\in$ /}P_{\ell}$. Then there is a
$\beta\in\Delta_{\ell}$ such that $x\mbox{$\in$ /}{\cal P}_{\beta}$. By
faithful flatness of $R^{\dagger}$ over $R$ we have ${\cal
P}_{\beta}R^{\dagger}\cap R={\cal P}_{\beta}$. This implies that $x\mbox{$\in$
/}{\cal P}_{\beta}R^{\dagger}$, and the same argument holds in all the
extensions $R^{\prime}\in\mathcal{T}$, so $x$ cannot be in $H_{2\ell+1}\cap
R$. This contradiction shows the desired equality.
###### Proposition 3.6
The ideals $H^{\prime}_{2\ell+1}$ behave well under $\nu$-extensions
$R\rightarrow R^{\prime}$ in $\mathcal{T}$. In other words, let $R\rightarrow
R^{\prime}$ be a $\nu$-extension in $\mathcal{T}$ and let
$H^{\prime}_{2\ell+1}$ denote the $(2\ell+1)$-st implicit prime ideal of
$\hat{R}^{\prime}$. Then $H_{2\ell+1}=H^{\prime}_{2\ell+1}\cap R^{\dagger}$.
Proof.- Immediate from the definitions.
To study the ideals $H_{2\ell+1}$, we need to understand more explicitly the
nature of the limit appearing in (46). To study the relationship between the
ideals $P_{\beta}R^{{\dagger}}$ and
$P^{\prime}_{\beta}{R^{\prime}}^{\dagger}\bigcap R^{\dagger}$, it is useful to
factor the natural map $R^{\dagger}\rightarrow{R^{\prime}}^{\dagger}$ as
$R^{\dagger}\rightarrow(R^{\dagger}\otimes_{R}R^{\prime})_{M^{\prime}}\overset{\phi}{\rightarrow}{R^{\prime}}^{\dagger}$
as we did in the proof of Lemma 1.2. In general, the ring
$R^{\dagger}\otimes_{R}R^{\prime}$ is not local (see the above examples), but
it has one distinguished maximal ideal $M^{\prime}$, namely, the ideal
generated by $mR^{\dagger}\otimes 1$ and $1\otimes m^{\prime}$, where
$m^{\prime}$ denotes the maximal ideal of $R^{\prime}$. The map $\phi$ factors
through the local ring
$\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}$ and the resulting
map
$\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}\rightarrow{R^{\prime}}^{\dagger}$
is either the formal completion or the henselization; in either case, it is
faithfully flat. Thus
$P^{\prime}_{\beta}{R^{\prime}}^{\dagger}\cap\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}=P^{\prime}_{\beta}\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}$.
This shows that we may replace ${R^{\prime}}^{\dagger}$ by
$\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}$ in (46) without
affecting the result, that is,
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left(\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\cal
P}^{\prime}_{\beta}\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}\right)\bigcap
R^{\dagger}\right).$ (48)
From now on, we will use (48) as our working definition of the implicit prime
ideals. One advantage of the expression (48) is that it makes sense in a
situation more general than the completion and the henselization. Namely, to
study the case of the henselization $\tilde{R}$, we will need to consider
local étale extensions $R^{e}$ of $R$, which are contained in $\tilde{R}$
(particularly, those which are essentially of finite type). The definition
(48) of the implicit prime ideals makes sense also in that case.
## 4 Stable rings and primality of their implicit ideals.
Let the notation be as in the preceding sections. As usual, $R^{\dagger}$ will
denote one of $\hat{R}$, $\tilde{R}$ or $R^{e}$ (a local étale $\nu$-extension
essentially of finite type). Take an $R^{\prime}\in\mathcal{T}$ and
$\overline{\beta}\in\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$. We have the
obvious inclusion of ideals
$\mathcal{P}_{\overline{\beta}}R^{\dagger}\subset\mathcal{P}_{\overline{\beta}}{R^{\prime}}^{\dagger}\cap
R^{\dagger}$ (49)
(where $\mathcal{P}_{\overline{\beta}}$ is defined in (45)). A useful subtree
of $\mathcal{T}$ is formed by the $\ell$-stable rings, which we now define. An
important property of stable rings, proved below, is that the inclusion (49)
is an equality whenever $R^{\prime}$ is stable.
###### Definition 4.1
A ring $R^{\prime}\in\mathcal{T}(R)$ is said to be $\ell$-stable if the
following two conditions hold:
(1) the ring
$\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$
(50)
is an integral domain and
(2) there do not exist an $R^{\prime\prime}\in\mathcal{T}(R^{\prime})$ and a
non-trivial algebraic extension $L$ of $\kappa(P^{\prime}_{\ell})$ which
embeds both into
$\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$
and $\kappa(P^{\prime\prime}_{\ell})$.
We say that $R$ is stable if it is $\ell$-stable for each
$\ell\in\\{0,\dots,r\\}$.
###### Remark 4.2
(1) Rings of the form (50) will be a basic object of study in this paper.
Another way of looking at the same ring, which we will often use, comes from
interchanging the order of tensor product and localization. Namely, let
$T^{\prime}$ denote the image of the multiplicative system
$\left(R^{\prime}\otimes_{R}R^{\dagger}\right)\setminus M^{\prime}$ under the
natural map
$R^{\prime}\otimes_{R}R^{\dagger}\rightarrow\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}R^{\dagger}$.
Then the ring (50) equals the localization
$(T^{\prime})^{-1}\left(\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}R^{\dagger}\right)$.
(2) In the special case $R^{\prime}=R$ in Definition 4.1, we have
$\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}=\kappa\left(P_{\ell}\right)\otimes_{R}R^{\dagger}.$
If, moreover, $\frac{R}{P_{\ell}}$ is analytically irreducible then the
hypothesis that $\kappa\left(P_{\ell}\right)\otimes_{R}R^{\dagger}$ is a
domain holds automatically; in fact, this hypothesis is equivalent to analytic
irreducibility of $\frac{R}{P_{\ell}}$ if $R^{\dagger}=\hat{R}$ or
$R^{\dagger}=\tilde{R}$.
(3) Consider the special case when $R^{\prime}$ is Henselian and ${\
}^{\dagger}=\hat{\ }$. Excellent Henselian rings are algebraically closed
inside their formal completions, so both (1) and (2) of Definition 4.1 hold
automatically for this $R^{\prime}$. Thus excellent Henselian local rings are
always stable.
In this section we study $\ell$-stable rings. We prove that if $R$ is
$\ell$-stable then so is any $R^{\prime}\in\mathcal{T}(R)$ (justifying the
name “stable”). The main result of this section, Theorem 4.9, says that if $R$
is stable then the implicit ideal $H^{\prime}_{2\ell+1}$ is prime for each
$\ell\in\\{0,\dots,r\\}$ and each $R^{\prime}\in\mathcal{T}(R)$.
###### Remark 4.3
In the next two sections we will show that there exist stable rings
$R^{\prime}\in\cal T$ for both $R^{\dagger}=\hat{R}$ and $R^{\dagger}=R^{e}$.
However, the proof of this is different depending on whether we are dealing
with completion or with an étale extension, and will be carried out separately
in two separate sections: one devoted to henselization, the other to
completion.
###### Proposition 4.4
Fix an integer $\ell$, $0\leq\ell\leq r$. Assume that $R^{\prime}$ is
$\ell$-stable and let $R^{\prime\prime}\in\mathcal{T}(R^{\prime})$. Then
$R^{\prime\prime}$ is $\ell$-stable.
Proof.- We have to show that (1) and (2) of Definition 4.1 for $R^{\prime}$
imply (1) and (2) of Definition 4.1 for $R^{\prime\prime}$. The ring
$\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime\prime}}$
(51)
is a localization of
$\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}\left(\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}\right)$.
Hence (1) and (2) of Definition 4.1, applied to $R^{\prime}$, imply that
$\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime\prime}}$
is an integral domain, so (1) of Definition 4.1 holds for $R^{\prime\prime}$.
Replacing $R^{\prime}$ by $R^{\prime\prime}$ clearly does not affect the
hypotheses about the non-existence of the extension $L$, so (2) of Definition
4.1 also holds for $R^{\prime\prime}$.
Next, we prove a technical result on which much of the rest of the paper is
based. For $\overline{\beta}\in\frac{\Gamma}{\Delta_{\ell+1}}$, let
$\mathcal{P}_{\overline{\beta}+}=\left\\{x\in R\ \left|\
\nu(x)\mod\Delta_{\ell+1}>\overline{\beta}\right.\right\\}.$ (52)
As usual, $\mathcal{P}^{\prime}_{\overline{\beta}+}$ will stand for the
analogous notion, but with $R$ replaced by $R^{\prime}$, etc.
###### Proposition 4.5
Assume that $R$ itself is $(\ell+1)$-stable and let
$R^{\prime}\in\mathcal{T}(R)$.
1. 1.
For any $\overline{\beta}\in\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$
$\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}\cap
R^{\dagger}=\mathcal{P}_{\overline{\beta}}R^{\dagger}.$ (53)
2. 2.
For any $\overline{\beta}\in\frac{\Gamma}{\Delta_{\ell+1}}$ the natural map
$\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}\rightarrow\frac{\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}}{\mathcal{P}^{\prime}_{\overline{\beta}+}{R^{\prime}}^{\dagger}}$
(54)
is injective.
Proof.- As explained at the end of the previous section, since
${R^{\prime}}^{\dagger}$ is faithfully flat over the ring
$\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}$, we may replace
${R^{\prime}}^{\dagger}$ by
$\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}$ in both 1 and 2
of the Proposition.
Proof of 1 of the Proposition: It is sufficient to prove that
$\mathcal{P}^{\prime}_{\overline{\beta}}\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}\bigcap
R^{\dagger}=\mathcal{P}_{\overline{\beta}}R^{\dagger}.$ (55)
One inclusion in (55) is trivial; we must show that
$\mathcal{P}^{\prime}_{\overline{\beta}}\left(R^{\dagger}\otimes_{R}R^{\prime}\right)_{M^{\prime}}\bigcap
R^{\dagger}\subset\mathcal{P}_{\overline{\beta}}R^{\dagger}.$ (56)
###### Lemma 4.6
Let $T^{\prime}$ denote the image of the multiplicative set
$\left(R^{\prime}\otimes_{R}R^{\dagger}\right)\setminus M^{\prime}$ under the
natural map of $R$-algebras
$R^{\prime}\otimes_{R}R^{\dagger}\rightarrow\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}$.
Then the map of $R$-algebras
$\bar{\pi}:\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline{\beta}}R_{P_{\ell+1}}}\otimes_{R}R^{\dagger}\rightarrow(T^{\prime})^{-1}\left(\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}\right)$
(57)
induced by $\pi:R\rightarrow R^{\prime}$ is injective.
Proof.-(of Lemma 4.6) We start with the field extension
$\kappa(P_{\ell+1})\hookrightarrow\kappa(P^{\prime}_{\ell+1})$
induced by $\pi$. Since $R^{\dagger}$ is flat over $R$, the induced map
$\pi_{1}:\kappa(P_{\ell+1})\otimes_{R}R^{\dagger}\rightarrow\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}$
is also injective. By (1) of Definition 4.1,
$\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}$ is a domain. In
particular,
$\kappa\left(P^{\prime}_{\ell+1}\right)\otimes_{R}R^{\dagger}=\left(\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}\right)_{red}.$
(58)
The local ring
$\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}$
is artinian because it can be seen as the quotient of
$\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{P^{\prime}_{\ell}R^{\prime}_{P^{\prime}_{\ell+1}}}$
by a valuation ideal corresponding to a rank one valuation. Since the ring is
noetherian the valuation of the maximal ideal is positive, and since the group
is archimedian, a power of the maximal ideal is contained in the valuation
ideal.
Therefore, its only associated prime is its nilradical, the ideal
$\frac{P^{\prime}_{\ell+1}R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}$;
in particular, the $(0)$ ideal in this ring has no embedded components. Since
$R^{\dagger}$ is flat over $R$,
$\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}$
is flat over
$\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}$
by base change. Hence the $(0)$ ideal of
$\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}$
has no embedded components. In particular, since the multiplicative system
$T^{\prime}$ is disjoint from the nilradical of
$\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}$,
the set $T^{\prime}$ contains no zero divisors, so localization by
$T^{\prime}$ is injective.
By the definition of $\mathcal{P}_{\overline{\beta}}$, the map
$\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline{\beta}}R_{P_{\ell+1}}}\rightarrow\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}$
is injective, hence so is
$\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline{\beta}}R_{P_{\ell+1}}}\otimes_{R}R^{\dagger}\rightarrow\frac{R^{\prime}_{P^{\prime}_{\ell+1}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell+1}}}\otimes_{R}R^{\dagger}$
by the flatness of $R^{\dagger}$ over $R$. Combining this with the injectivity
of the localization by $T^{\prime}$, we obtain that $\bar{\pi}$ is injective,
as desired. Lemma 4.6 is proved.
Again by the definition of $\mathcal{P}_{\overline{\beta}}$, the localization
map
$\frac{R}{\mathcal{P}_{\overline{\beta}}}\rightarrow\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline{\beta}}R_{P_{\ell+1}}}$
is injective, hence so is the map
$\frac{R}{\mathcal{P}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\rightarrow\frac{R_{P_{\ell+1}}}{\mathcal{P}_{\overline{\beta}}R_{P_{\ell+1}}}\otimes_{R}R^{\dagger}$
(59)
by the flatness of $R^{\dagger}$ over $R$. Combining this with Lemma 4.6, we
see that the composition
$\frac{R}{\mathcal{P}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\rightarrow(T^{\prime})^{-1}\left(\frac{R^{\prime}_{P^{\prime}_{\ell}}}{\mathcal{P}^{\prime}_{\overline{\beta}}R^{\prime}_{P^{\prime}_{\ell}}}\otimes_{R}R^{\dagger}\right)$
(60)
of (59) with $\bar{\pi}$ is also injective. Now, (60) factors through
$\left(\frac{R^{\prime}}{\mathcal{P}^{\prime}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$
(here we are guilty of a slight abuse of notation: we denote the natural image
of $M^{\prime}$ in
$\frac{R^{\prime}}{\mathcal{P}^{\prime}_{\overline{\beta}}}\otimes_{R}R^{\dagger}$
also by $M^{\prime}$). Hence the map
$\frac{R}{\mathcal{P}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\rightarrow\left(\frac{R^{\prime}}{\mathcal{P}^{\prime}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$
(61)
is injective. Since
$\frac{R}{\mathcal{P}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\cong\frac{R^{\dagger}}{\mathcal{P}_{\overline{\beta}}R^{\dagger}}$
and
$\left(\frac{R^{\prime}}{\mathcal{P}^{\prime}_{\overline{\beta}}}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}\cong\frac{\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}}{\mathcal{P}^{\prime}_{\overline{\beta}}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}}$,
the injectivity of (61) is the same as (56). This completes the proof of 1 of
the Proposition.
Proof of 2 of the Proposition: We start with the injective homomorphism
$\frac{\mathcal{P}_{\overline{\beta}}}{\mathcal{P}_{\overline{\beta}+}}\otimes_{R}\kappa(P_{\ell+1})\rightarrow\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R}\kappa(P^{\prime}_{\ell+1})$
(62)
of $\kappa(P_{\ell+1})$-vector spaces. Since $R^{\dagger}$ is flat over $R$,
tensoring (62) produces an injective homomorphism
$\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}\otimes_{R}\kappa(P_{\ell+1})\rightarrow\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R}\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}$
(63)
of $R^{\dagger}$-modules. Now, the $\kappa(P_{\ell+1})$-vector space
$\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R}\kappa(P^{\prime}_{\ell+1})$
is, in particular, a torsion-free $\kappa(P_{\ell+1})$-module. Since
$\kappa(P_{\ell+1})\otimes_{R}R^{\dagger}$ is a domain by definition of
$(\ell+1)$-stable and by the flatness of
$R^{\dagger}\otimes_{R}\kappa(P_{\ell+1})$ over $\kappa(P_{\ell+1})$, the
$R^{\dagger}\otimes_{R}\kappa(P_{\ell+1})$-module
$\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R}\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}$
is also torsion-free; in particular, its localization map by any
multiplicative system is injective. Let $S^{\prime}$ denote the image of the
multiplicative set $\left(R^{\prime}\otimes_{R}R^{\dagger}\right)\setminus
M^{\prime}$ under the natural map of $R$-algebras
$R^{\prime}\otimes_{R}R^{\dagger}\rightarrow\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}$.
By the above, the composition
$\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}\otimes_{R}\kappa(P_{\ell+1})\rightarrow(S^{\prime})^{-1}\left(\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R}\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}\right)$
(64)
of (63) with the localization by $S^{\prime}$ is injective.
By the definition of $\mathcal{P}_{\overline{\beta}}$, the localization map
$\frac{\mathcal{P}_{\overline{\beta}}}{\mathcal{P}_{\overline{\beta}+}}\rightarrow\frac{\mathcal{P}_{\overline{\beta}}}{\mathcal{P}_{\overline{\beta}+}}\otimes_{R}\kappa(P_{\ell+1})$
is injective, hence so is the map
$\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}=\frac{\mathcal{P}_{\overline{\beta}}}{\mathcal{P}_{\overline{\beta}+}}\otimes_{R}R^{\dagger}\rightarrow\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}\otimes_{R}\kappa(P_{\ell+1})$
(65)
by the flatness of $R^{\dagger}$ over $R$. Combining this with the injectivity
of (64), we see that the composition
$\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}\rightarrow(S^{\prime})^{-1}\left(\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R}\kappa(P^{\prime}_{\ell+1})\otimes_{R}R^{\dagger}\right)$
(66)
of (65) with (64) is also injective. Now, (66) factors through
$\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R^{\prime}}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$.
Hence the map
$\frac{\mathcal{P}_{\overline{\beta}}R^{\dagger}}{\mathcal{P}_{\overline{\beta}+}R^{\dagger}}\rightarrow\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R^{\prime}}\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$
(67)
is injective. Since
$\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R^{\prime}}{R^{\prime}}^{\dagger}\cong\frac{\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}}{\mathcal{P}^{\prime}_{\overline{\beta}+}{R^{\prime}}^{\dagger}}$
and by faithful flatness of ${R^{\prime}}^{\dagger}$ over
$\left(R^{\prime}\otimes_{R}R^{\dagger}\right)_{M^{\prime}}$, the injectivity
of (67) implies the injectivity of the map (54) required in 2 of the
Proposition.
This completes the proof of the Proposition.
###### Corollary 4.7
Take an integer $\ell\in\\{0,\dots,r-1\\}$ and assume that $R$ is
$(\ell+1)$-stable. Then
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}{\cal
P}_{\beta}R^{\dagger}.$ (68)
Proof.- By Lemma 4 of Appendix 4 of [18], the ideals
$\mathcal{P}_{\overline{\beta}}$ are cofinal among the ideals
$\mathcal{P}_{\beta}$ for $\beta\in\Delta_{\ell}$.
###### Corollary 4.8
Assume that $R$ is stable. Take an element $\beta\in\Gamma$. Then
$\mathcal{P}^{\prime}_{\beta}{R^{\prime}}^{\dagger}\cap
R^{\dagger}=\mathcal{P}_{\beta}$.
Proof.- It is sufficient to prove that for each $\ell\in\\{0,\dots,r-1\\}$ and
$\bar{\beta}\in\frac{\Gamma}{\Delta_{\ell+1}}$, we have
$\mathcal{P}^{\prime}_{\bar{\beta}}{R^{\prime}}^{\dagger}\cap
R^{\dagger}=\mathcal{P}_{\bar{\beta}};$ (69)
the Corollary is just the special case of (69) when $\ell=r-1$. We prove (69)
by contradiction. Assume the contrary and take the smallest $\ell$ for which
(69) fails to be true. Let $\Phi^{\prime}=\nu(R^{\prime}\setminus\\{0\\})$. We
will denote by $\frac{\Phi}{\Delta_{\ell+1}}$ the image of $\Phi$ under the
composition of natural maps
$\Phi\hookrightarrow\Gamma\rightarrow\frac{\Gamma}{\Delta_{\ell+1}}$ and
similarly for $\frac{\Phi^{\prime}}{\Delta_{\ell+1}}$. Clearly, if (69) fails
for a certain $\bar{\beta}$, it also fails for some
$\bar{\beta}\in\frac{\Phi^{\prime}}{\Delta_{\ell+1}}$; take the smallest
$\bar{\beta}\in\frac{\Phi^{\prime}}{\Delta_{\ell+1}}$ with this property. If
we had
$\bar{\beta}=\min\left\\{\left.\tilde{\beta}\in\frac{\Phi^{\prime}}{\Delta_{\ell+1}}\
\right|\ \tilde{\beta}-\bar{\beta}\in\Delta_{\ell}\right\\}$, then (69) would
also fail with $\bar{\beta}$ replaced by
$\bar{\beta}\mod\Delta_{\ell}\in\frac{\Gamma}{\Delta_{\ell}}$, contradicting
the minimality of $\ell$. Thus
$\bar{\beta}>\min\left\\{\left.\tilde{\beta}\in\frac{\Phi^{\prime}}{\Delta_{\ell+1}}\
\right|\ \tilde{\beta}-\bar{\beta}\in\Delta_{\ell}\right\\}.$ (70)
Let $\bar{\beta}-$ denote the immediate predecessor of $\bar{\beta}$ in
$\frac{\Phi^{\prime}}{\Delta_{\ell+1}}$. By (70), we have
$\bar{\beta}-\bar{\beta}-\in\Delta_{\ell}$. By the choice of $\bar{\beta}$, we
have
$\mathcal{P}_{\bar{\beta}-}=\mathcal{P}^{\prime}_{\bar{\beta}-}{R^{\prime}}^{\dagger}\cap
R^{\dagger}$ but
$\mathcal{P}_{\bar{\beta}}\subsetneqq\mathcal{P}^{\prime}_{\bar{\beta}}{R^{\prime}}^{\dagger}\cap
R^{\dagger}$. This contradicts Proposition 4.5, applied to $\bar{\beta}-$. The
Corollary is proved.
Now we are ready to state and prove the main Theorem of this section.
###### Theorem 4.9
(1) Fix an integer $\ell\in\\{1,\dots,r+1\\}$. Assume that there exists
$R^{\prime}\in\mathcal{T}(R)$ which is $(\ell+1)$-stable. Then the ideal
$H_{2\ell+1}$ is prime.
(2) Let $i=2\ell+2$. There exists an extension $\nu^{\dagger}_{i0}$ of
$\nu_{\ell+1}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$,
with value group
$\Delta_{i-1,0}=\frac{\Delta_{\ell}}{\Delta_{\ell+1}},$ (71)
whose valuation ideals are described as follows. For an element
$\overline{\beta}\in\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$, the
$\nu^{\dagger}_{i0}$-ideal of $\frac{R^{\dagger}}{H_{i-1}}$ of value
$\overline{\beta}$, denoted by $\mathcal{P}^{\dagger}_{\overline{\beta}\ell}$,
is given by the formula
$\mathcal{P}^{\dagger}_{\overline{\beta},\ell+1}=\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}}{H^{\prime}_{i-1}}\right)\cap\frac{R^{\dagger}}{H_{i-1}}.$
(72)
###### Remark 4.10
Once the even-numbered implicit prime ideals $H^{\prime}_{2\ell}$ are defined
below, we will show that $\nu^{\dagger}_{i0}$ is the unique extension of
$\nu_{\ell+1}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$,
centered in the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{R^{\prime{\dagger}}_{H^{\prime}_{2\ell+2}}}{H^{\prime}_{2\ell+1}R^{\prime{\dagger}}_{H^{\prime}_{2\ell+2}}}$.
Proof.-(of Theorem 4.9) Let $R^{\prime}$ be a stable ring in $\mathcal{T}(R)$.
Once Theorem 4.9 is proved for $R^{\prime}$, the same results for $R$ will
follow easily by intersecting all the ideals of ${R^{\prime}}^{\dagger}$ in
sight with $R^{\dagger}$. Therefore from now on we will replace $R$ by
$R^{\prime}$, that is, we will assume that $R$ itself is stable.
Let $\Phi_{\ell}$ denote the image of the semigroup $\nu(R\setminus\\{0\\})$
in $\frac{\Gamma}{\Delta_{\ell+1}}$. As we saw above, $\Phi_{\ell}$ is well
ordered. For an element $\overline{\beta}\in\Phi_{\ell}$, let
$\overline{\beta}+$ denote the immediate successor of $\overline{\beta}$ in
$\Phi_{\ell}$.
Take any element $x\in R^{\dagger}\setminus H_{i-1}$. By Corollary 4.7, there
exists (a unique)
$\overline{\beta}\in\Phi_{\ell}\cap\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$ such
that
$x\in{\cal P}_{\overline{\beta}}R^{\dagger}\setminus{\cal
P}_{\overline{\beta}+}R^{\dagger}$ (73)
(where, of course, we allow $\overline{\beta}=0$). Let $\bar{x}$ denote the
image of $x$ in $\frac{R^{\dagger}}{H_{i-1}}$. We define
$\nu^{\dagger}_{i0}(\bar{x})=\overline{\beta}.$
Next, take another element $y\in R^{\dagger}\setminus H_{2\ell+1}$ and let
$\gamma\in\Phi_{\ell}\cap\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$ be such that
$y\in{\cal P}_{\overline{\gamma}}R^{\dagger}\setminus{\cal
P}_{\overline{\gamma}+}R^{\dagger}.$ (74)
Let $(a_{1},...,a_{n})$ be a set of generators of ${\cal
P}_{\overline{\beta}}$ and $(b_{1},...,b_{s})$ a set of generators of ${\cal
P}_{\overline{\gamma}}$, with $\nu_{\ell+1}(a_{1})=\overline{\beta}$ and
$\nu_{\ell+1}(b_{1})=\overline{\gamma}$. Let $R^{\prime}$ be a local blowing
up along $\nu$ such that $R^{\prime}$ contains all the fractions
$\frac{a_{i}}{a_{1}}$ and $\frac{b_{j}}{b_{1}}$. By Proposition 4.4 and
Definition 4.1 (1), the ideal $P^{\prime}_{\ell+1}{R^{\prime}}^{\dagger}$ is
prime. By construction, we have $a_{1}\ |\ x$ and $b_{1}\ |\ y$ in
${R^{\prime}}^{\dagger}$. Write $x=za_{1}$ and $y=wb_{1}$ in
${R^{\prime}}^{\dagger}$. The equality (53), combined with (73) and (74),
implies that $z,w\mbox{$\in$ /}P^{\prime}_{\ell+1}{R^{\prime}}^{\dagger}$,
hence
$zw\mbox{$\in$ /}P^{\prime}_{\ell+1}{R^{\prime}}^{\dagger}$ (75)
by the primality of $P^{\prime}_{\ell+1}{R^{\prime}}^{\dagger}$. We obtain
$xy=a_{1}b_{1}zw.$ (76)
Since $\nu$ is a valuation on $R^{\prime}$, we have $\left({\cal
P}^{\prime}_{\overline{\beta}+\overline{\gamma}+}:(a_{1}b_{1})R^{\prime}\right)\subset
P^{\prime}_{\ell+1}$. By faithful flatness of ${R^{\prime}}^{\dagger}$ over
$R^{\prime}$ we obtain
$\left({\cal
P}^{\prime}_{\overline{\beta}+\overline{\gamma}+}{R^{\prime}}^{\dagger}:(a_{1}b_{1}){R^{\prime}}^{\dagger}\right)\subset
P^{\prime}_{\ell+1}{R^{\prime}}^{\dagger}.$ (77)
Combining this with (75) and (76), we obtain
$xy\mbox{$\in$ /}{\cal P}_{\overline{\beta}+\overline{\gamma}+}R^{\dagger},$
(78)
in particular, $xy\mbox{$\in$ /}H_{2\ell+1}$. We started with two arbitrary
elements $x,y\in R^{\dagger}\setminus H_{2\ell+1}$ and showed that
$xy\mbox{$\in$ /}H_{2\ell+1}$. This proves (1) of the Theorem.
Furthermore, (78) shows that
$\nu^{\dagger}_{i0}(\bar{x}\bar{y})=\overline{\beta}+\overline{\gamma}$, so
$\nu^{\dagger}_{i0}$ induces a valuation of $\kappa(H_{i-1})$ and hence also
of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$.
Equality (71) holds by definition and (72) by the assumed stability of $R$.
Next, we define the even-numbered implicit prime ideals $H^{\prime}_{2\ell}$.
The only information we need to use to define the prime ideals
$H^{\prime}_{2\ell}\subset H^{\prime}_{2\ell+1}$ and to prove that
$H^{\prime}_{2\ell-1}\subset H^{\prime}_{2\ell}$ are the facts that
$H_{2\ell+1}$ is a prime lying over $P_{\ell}$ and that the ring homomorphism
$R^{\prime}\rightarrow{R^{\prime}}^{\dagger}$ is regular.
###### Proposition 4.11
There exists a unique minimal prime ideal $H_{2\ell}$ of
$P_{\ell}R^{\dagger}$, contained in $H_{2\ell+1}$.
Proof.- Since $H_{2\ell+1}\cap R=P_{\ell}$, $H_{2\ell+1}$ belongs to the fiber
of the map $Spec\ R^{\dagger}\rightarrow Spec\ R$ over $P_{\ell}$. Since $R$
was assumed to be excellent, $S:={R^{\dagger}}\otimes_{R}\kappa(P_{\ell})$ is
a regular ring (note that the excellence assumption is needed only in the case
$R^{\dagger}=\hat{R}$; the ring homomorphism $R\rightarrow R^{\dagger}$ is
automatically regular if $R^{\dagger}=\tilde{R}$ or $R^{\dagger}=R^{e}$).
Hence its localization
$\bar{S}:=S_{H_{2\ell+1}S}\cong\frac{R^{\dagger}_{H_{2\ell+1}}}{P_{\ell}R^{\dagger}_{H_{2\ell+1}}}$
is a regular local ring. In particular, $\bar{S}$ is an integral domain, so
$(0)$ is its unique minimal prime ideal. The set of minimal prime ideals of
$\bar{S}$ is in one-to-one correspondence with the set of minimal primes of
$P_{\ell}$, contained in $H_{2\ell+1}$, which shows that such a minimal prime
$H_{2\ell}$ is unique, as desired.
We have $P_{\ell}\subset H_{2\ell}\cap R\subset H_{2\ell+1}\cap R=P_{\ell}$,
so $H_{2\ell}\cap R\subset P_{\ell}$.
###### Proposition 4.12
We have $H_{2\ell-1}\subset H_{2\ell}$.
Proof.- Take an element $\beta\in\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$ and a
stable ring $R^{\prime}\in\cal T$. Then $\mathcal{P}^{\prime}_{\beta}\subset
P^{\prime}_{\ell}$, so
$H^{\prime}_{2\ell-1}\subset\mathcal{P}^{\prime}_{\beta}{R^{\prime}}^{\dagger}\subset
P^{\prime}_{\ell}{R^{\prime}}^{\dagger}\subset H^{\prime}_{2\ell}.$ (79)
Intersecting (79) back with $R^{\dagger}$ we get the result.
In §7 we will see that if $R^{\dagger}=\tilde{R}$ or $R^{\dagger}=R^{e}$ then
$H_{2\ell}=H_{2\ell+1}$ for all $\ell$.
Let the notation be the same as in Theorem 4.9.
###### Proposition 4.13
The valuation $\nu_{i0}^{\dagger}$ is the unique extension of $\nu_{\ell}$ to
a valuation of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$,
centered in the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}$.
Proof.- As usual, without loss of generality we may assume that $R$ is stable.
Take an element $x\in R^{\dagger}\setminus H_{2\ell-1}$. Let
$\beta=\nu^{\dagger}_{i0}(\bar{x})$ and let $R^{\prime}$ be the blowing up of
the ideal $\mathcal{P}_{\beta}=(a_{1},\dots,a_{n})$, as in the proof of
Theorem 4.9. Write
$x=za_{1}$ (80)
in $R^{\prime}$. We have $z\in{R^{\prime}}^{\dagger}\setminus
P^{\prime}_{\ell}{R^{\prime}}^{\dagger}$, hence
$\bar{z}\in\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}\setminus\frac{P^{\prime}_{\ell}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}=\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}\setminus\frac{H^{\prime}_{2\ell}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}.$
(81)
If $\nu^{*}$ is any other extension of $\nu_{\ell}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$,
centered in
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}$,
then $\nu^{*}(\bar{a}_{1})=\beta$, $\nu^{*}(z)=0$ by (81), so
$\nu^{*}(\bar{x})=\beta=\nu_{i0}^{\dagger}(\bar{x})$. This completes the proof
of the uniqueness of $\nu_{i0}^{\dagger}$.
###### Remark 4.14
If $R^{\prime}$ is stable, we have a natural isomorphism of graded algebras
$\mbox{gr}_{\nu^{\dagger}_{i0}}\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}\cong\mbox{gr}_{\nu_{\ell}}\frac{R^{\prime}_{P^{\prime}_{\ell}}}{P^{\prime}_{\ell-1}R^{\prime}_{P^{\prime}_{\ell}}}\otimes_{R^{\prime}}\kappa(H^{\prime}_{2\ell}).$
In particular, the residue field of $\nu^{\dagger}_{i0}$ is
$k_{\nu^{\dagger}_{i0}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{2\ell})$.
## 5 A classification of extensions of $\nu$ to $\hat{R}$.
The purpose of this section is to give a systematic description of all the
possible extensions $\nu^{\dagger}_{-}$ of $\nu$ to a quotient of
$R^{\dagger}$ by a minimal prime as compositions of $2r$ valuations,
$\nu^{\dagger}_{-}=\nu^{\dagger}_{1}\circ\dots\circ\nu^{\dagger}_{2r},$ (82)
satisfying certain conditions. One is naturally led to consider the more
general problem of extending $\nu$ not only to rings of the form
$\frac{R^{\dagger}}{P}$ but also to the ring
$\lim\limits_{\to}\frac{{R^{\prime}}^{\dagger}}{P^{\prime}}$, where
$P^{\prime}$ is a tree of prime ideals of ${R^{\prime}}^{\dagger}$, such that
$P^{\prime}\cap R^{\prime}=(0)$. We deal in a uniform way with all the three
cases $R^{\dagger}=\hat{R}$, $R^{\dagger}=\tilde{R}$ and $R^{\dagger}=R^{e}$,
in order to be able to apply the results proved here to all three later in the
paper. However, the reader should think of the case $R^{\dagger}=\hat{R}$ as
the main case of interest and the cases $R^{\dagger}=\tilde{R}$ and
$R^{\dagger}=R^{e}$ as auxiliary and slightly degenerate, since, as we shall
see, in these cases the equality $H_{2\ell}=H_{2\ell+1}$ is satisfied for all
$\ell$ and the extension $\nu^{\dagger}_{-}$ will later be shown to be unique.
We will associate to each extension $\nu^{\dagger}_{-}$ of $\nu$ to
$R^{\dagger}$ a chain
$\tilde{H}^{\prime}_{0}\subset\tilde{H}^{\prime}_{1}\subset\dots\subset\tilde{H}^{\prime}_{2r}=m^{\prime}{R^{\prime}}^{\dagger}$
(83)
of prime $\nu^{\dagger}_{-}$-ideals, corresponding to the decomposition (82)
and prove some basic properties of this chain of ideals.
Now for the details. We wish to classify all the pairs
$\left(\left\\{\tilde{H}^{\prime}_{0}\right\\},\nu^{\dagger}_{+}\right)$,
where $\left\\{\tilde{H}^{\prime}_{0}\right\\}$ is a tree of prime ideals of
${R^{\prime}}^{\dagger}$, such that $\tilde{H}^{\prime}_{0}\cap
R^{\prime}=(0)$, and $\nu^{\dagger}_{+}$ is an extension of $\nu$ to the ring
$\lim\limits_{\to}\frac{{R^{\prime}}^{\dagger}}{\tilde{H}^{\prime}_{0}}$.
Pick and fix one such pair
$\left(\left\\{\tilde{H}^{\prime}_{0}\right\\},\nu^{\dagger}_{+}\right)$. We
associate to it the following collection of data, which, as we will see, will
in turn determine the pair
$\left(\left\\{\tilde{H}^{\prime}_{0}\right\\},\nu^{\dagger}_{+}\right)$.
First, we associate to
$\left(\left\\{\tilde{H}^{\prime}_{0}\right\\},\nu^{\dagger}_{-}\right)$ a
chain (83) of $2r$ trees of prime $\nu^{\dagger}_{-}$-ideals. Let
$\Gamma^{\dagger}$ denote the value group of $\nu^{\dagger}_{-}$. Defining
(83) is equivalent to defining a chain
$\Gamma^{\dagger}=\Delta^{\dagger}_{0}\supset\Delta^{\dagger}_{1}\supset\dots\supset\Delta^{\dagger}_{2r}=\Delta^{\dagger}_{2r+1}=(0)$
(84)
of $2r$ isolated subroups of $\Gamma^{\dagger}$ (the chain (84) will not, in
general, be maximal, and $\Delta^{\dagger}_{2\ell+1}$ need not be distinct
from $\Delta^{\dagger}_{2\ell}$).
We define the $\Delta^{\dagger}_{i}$ as follows. For $0\leq\ell\leq r$, let
$\Delta^{\dagger}_{2\ell}$ and $\Delta^{\dagger}_{2\ell+1}$ denote,
respectively, the greatest and the smallest isolated subgroups of
$\Gamma^{\dagger}$ such that
$\Delta^{\dagger}_{2\ell}\cap\Gamma=\Delta^{\dagger}_{2\ell+1}\cap\Gamma=\Delta_{\ell}.$
(85)
###### Lemma 5.1
We have
$rk\ \frac{\Delta^{\dagger}_{2\ell-1}}{\Delta^{\dagger}_{2\ell}}=1$ (86)
for $1\leq\ell\leq r$.
Proof.- Since by construction
$\Delta^{\dagger}_{2\ell}\neq\Delta^{\dagger}_{2\ell-1}$, equality (86) is
equivalent to saying that there is no isolated subgroup $\Delta^{\dagger}$ of
$\Gamma^{\dagger}$ which is properly contained in $\Delta^{\dagger}_{2\ell-1}$
and properly contains $\Delta^{\dagger}_{2\ell}$. Suppose such an isolated
subgroup $\Delta^{\dagger}$ existed. Then
$\Delta_{\ell}=\Delta^{\dagger}_{2\ell}\cap\Gamma\subsetneqq\Delta^{\dagger}\cap\Gamma\subsetneqq\Delta^{\dagger}_{2\ell-1}\cap\Gamma=\Delta_{\ell-1},$
(87)
where the first inclusion is strict by the maximality of
$\Delta^{\dagger}_{2\ell}$ and the second by the minimality of
$\Delta^{\dagger}_{2\ell-1}$. Thus $\Delta^{\dagger}\cap\Gamma$ is an isolated
subgroup of $\Gamma$, properly containing $\Delta_{\ell}$ and properly
contained in $\Delta_{l-1}$, which is impossible since $rk\
\frac{\Delta_{\ell-1}}{\Delta_{\ell}}=1$. This is a contradiction, hence $rk\
\frac{\Delta^{\dagger}_{2\ell-1}}{\Delta^{\dagger}_{2\ell}}=1$, as desired.
###### Definition 5.2
Let $0\leq i\leq 2r$. The $i$-th prime ideal determined by $\nu^{\dagger}_{-}$
is the prime $\nu^{\dagger}_{-}$-ideal $\tilde{H}^{\prime}_{i}$ of
${R^{\prime}}^{\dagger}$, corresponding to the isolated subgroup
$\Delta^{\dagger}_{i}$ (that is, the ideal $\tilde{H}^{\prime}_{i}$ consisting
of all the elements of ${R^{\prime}}^{\dagger}$ whose values lie outside of
$\Delta^{\dagger}_{i}$). The chain of trees (83) of prime ideals of
${R^{\prime}}^{\dagger}$ formed by the $\tilde{H}^{\prime}_{i}$ is referred to
as the chain of trees determined by $\nu^{\dagger}_{-}$.
The equality (85) says that
$\tilde{H}^{\prime}_{2\ell}\cap R^{\prime}=\tilde{H}^{\prime}_{2\ell+1}\cap
R^{\prime}=P^{\prime}_{\ell}$ (88)
By definitions, for $1\leq i\leq 2r$, $\nu^{\dagger}_{i}$ is a valuation of
the field $k_{\nu^{\dagger}_{i-1}}$. In the sequel, we will find it useful to
talk about the restriction of $\nu^{\dagger}_{i}$ to a smaller field, namely,
the field of fractions of the ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$;
we will denote this restriction by $\nu^{\dagger}_{i0}$. The field of
fractions of
$\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$
is $\kappa(\tilde{H}^{\prime}_{i-1})$, hence that of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$
is
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})$,
which is a subfield of $k_{\nu^{\dagger}_{i-1}}$. The value group of
$\nu^{\dagger}_{i0}$ will be denoted by $\Delta_{i-1,0}$; we have
$\Delta_{i-1,0}\subset\frac{\Delta^{\dagger}_{i-1}}{\Delta^{\dagger}_{i}}$. If
$i=2\ell$ is even then
$\frac{R^{\prime}_{P^{\prime}_{l}}}{P^{\prime}_{l-1}R^{\prime}_{P^{\prime}_{l}}}<\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$,
so
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{R^{\prime}_{P^{\prime}_{l}}}{P^{\prime}_{l-1}R^{\prime}_{P^{\prime}_{l}}}<\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$.
In this case $rk\ \nu^{\dagger}_{i}=1$ and $\nu^{\dagger}_{i}$ an extension of
the rank 1 valuation $\nu_{\ell}$ from $\kappa(P_{\ell-1})$ to
$k_{\nu^{\dagger}_{i-1}}$; we have
$\frac{\Delta_{\ell-1}}{\Delta_{\ell}}\subset\Delta_{i-1,0}\subset\frac{\Delta^{\dagger}_{i-1}}{\Delta^{\dagger}_{i}}.$
(89)
###### Proposition 5.3
Let $i=2\ell$. As usual, for an element
$\overline{\beta}\in\left(\frac{\Delta_{\ell}}{\Delta_{\ell+1}}\right)_{+}$,
let $\mathcal{P}_{\overline{\beta}}$ (resp.
$\mathcal{P}^{\prime}_{\overline{\beta}}$) denote the preimage in $R$ (resp.
in $R^{\prime}$) of the $\nu_{\ell+1}$-ideal of $\frac{R}{P_{\ell}}$ (resp.
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$) of value greater than or equal to
$\overline{\beta}$. Then
$\bigcap\limits_{\overline{\beta}\in\left(\frac{\Delta_{\ell}}{\Delta_{\ell+1}}\right)_{+}}\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\left(\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}+\tilde{H}^{\prime}_{i+1}\right){R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i+2}}\cap{R}^{\dagger}\subset\tilde{H}_{i+1}.$
(90)
The inclusion (90) should be understood as a condition on the tree of ideals.
In other words, it is equally valid if we replace $R^{\prime}$ by any other
ring $R^{\prime\prime}\in\cal T$.
Proof.-(of Proposition 5.3) Since
$rk\frac{\Delta^{\dagger}_{i+1}}{\Delta^{\dagger}_{i+2}}=1$ by Lemma 5.1,
$\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$ is cofinal in
$\frac{\Delta^{\dagger}_{i+1}}{\Delta^{\dagger}_{i+2}}$. Then for any
$x\in\bigcap\limits_{\overline{\beta}\in\left(\frac{\Delta_{\ell}}{\Delta_{\ell+1}}\right)_{+}}\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\left(\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}+\tilde{H}^{\prime}_{i+1}\right){R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i+2}}\cap{R}^{\dagger}$
we have $\nu^{\dagger}_{-}(x)\mbox{$\in$ /}\Delta^{\dagger}_{i}$, hence
$x\in\tilde{H}_{i+1}$, as desired.
From now to the end of §6, we will assume that $\cal T$ contains a stable ring
$R^{\prime}$, so that we can apply the results of the previous section, in
particular, the primality of the ideals $H^{\prime}_{i}$.
###### Proposition 5.4
We have
$H^{\prime}_{i}\subset\tilde{H}^{\prime}_{i}\qquad\text{ for all
}i\in\\{0,\dots,2r\\}.$ (91)
Proof.- For $\beta\in\Gamma^{\dagger}$ and $R^{\prime}\in\cal T$, let
${\mathcal{P}^{\prime}_{\beta}}^{\dagger}$ denote the
$\nu^{\dagger}_{-}$-ideal of ${R^{\prime}}^{\dagger}$ of value $\beta$. Fix an
integer $\ell\in\\{0,\dots,r\\}$. For each $R^{\prime}\in\cal T$, each
$\beta\in\Delta_{\ell}$ and $x\in\mathcal{P}^{\prime}_{\beta}$ we have
$\nu^{\dagger}_{-}(x)=\nu(x)\geq\beta$, hence
$\mathcal{P}^{\prime}_{\beta}{R^{\prime}}^{\dagger}\subset{\mathcal{P}^{\prime}_{\beta}}^{\dagger}.$
(92)
Taking the inductive limit over all $R^{\prime}\in\cal T$ and the intersection
over all $\beta\in\Delta_{\ell}$ in (92), and using the cofinality of
$\Delta_{\ell}$ in $\Delta^{\dagger}_{2\ell+1}$ and the fact that
$\bigcap\limits_{\beta\in\Delta^{\dagger}_{2\ell}}\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\mathcal{P}^{\prime}_{\beta}}^{\dagger}\right)=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\tilde{H}^{\prime}_{2\ell+1}$,
we obtain the inclusion (91) for $i=2\ell+1$. To prove (91) for $i=2\ell$,
note that $\tilde{H}^{\prime}_{2\ell}\cap
R^{\prime}=\tilde{H}^{\prime}_{2\ell+1}\cap R^{\prime}=P_{\ell}$. By the same
argument as in Proposition 4.11, excellence of $R^{\prime}$ implies that there
is a unique minimal prime $H^{*}_{2\ell}$ of
$P^{\prime}_{\ell}{R^{\prime}}^{\dagger}$, contained in
$\tilde{H}^{\prime}_{2\ell+1}$ and a unique minimal prime $H^{**}_{2\ell}$ of
$P^{\prime}_{\ell}{R^{\prime}}^{\dagger}$, contained in
$\tilde{H}^{\prime}_{2\ell}$. Now, Proposition 4.11 and the facts that
$H^{\prime}_{2\ell+1}\subset\tilde{H}^{\prime}_{2\ell+1}$ and
$\tilde{H}^{\prime}_{2\ell}\subset\tilde{H}^{\prime}_{2\ell+1}$ imply that
$H^{\prime}_{2\ell}=H^{*}_{2\ell}=H^{**}_{2\ell}$, hence
$H^{\prime}_{2\ell}=H^{**}_{2\ell}\subset\tilde{H}^{\prime}_{2\ell}$, as
desired.
###### Definition 5.5
A chain of trees (83) of prime ideals of ${R^{\prime}}^{\dagger}$ is said to
be admissible if $H^{\prime}_{i}\subset\tilde{H}^{\prime}_{i}$ and (88) and
(90) hold.
Equalities (88), Proposition 5.3 and Proposition 5.4 say that a chain of trees
(83) of prime ideals of ${R^{\prime}}^{\dagger}$, determined by
$\nu^{\dagger}_{-}$, is admissible.
Summarizing all of the above results, and keeping in mind the fact that
specifying a composition of $2r$ valuation is equivalent to specifying all of
its $2r$ components, we arrive at one of the main theorems of this paper:
###### Theorem 5.6
Specifying the valuation $\nu^{\dagger}_{-}$ is equivalent to specifying the
following data. The data will be described recursively in $i$, that is, the
description of $\nu^{\dagger}_{i}$ assumes that $\nu^{\dagger}_{i-1}$ is
already defined:
(1) An admissible chain of trees (83) of prime ideals of
${R^{\prime}}^{\dagger}$.
(2) For each $i$, $1\leq i\leq 2r$, a valuation $\nu^{\dagger}_{i}$ of
$k_{\nu^{\dagger}_{i-1}}$ (where $\nu^{\dagger}_{0}$ is taken to be the
trivial valuation by convention), whose restriction to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})$
is centered at the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$.
The data $\left\\{\nu^{\dagger}_{i}\right\\}_{1\leq i\leq 2r}$ is subject to
the following additional condition: if $i=2\ell$ is even then $rk\
\nu^{\dagger}_{i}=1$ and $\nu^{\dagger}_{i}$ is an extension of $\nu_{\ell}$
to $k_{\nu^{\dagger}_{i-1}}$ (which is naturally an extension of
$k_{\nu_{\ell-1}}$).
In particular, note that such extensions $\nu^{\dagger}_{-}$ always exist, and
usually there are plenty of them. The question of uniqueness of
$\nu^{\dagger}_{-}$ and the related question of uniqueness of
$\nu^{\dagger}_{i}$, especially in the case when $i$ is even, will be
addressed in the next section.
## 6 Uniqueness properties of $\nu^{\dagger}_{-}$.
In this section we address the question of uniqueness of the extension
$\nu^{\dagger}_{-}$. One result in this direction, which will be very useful
here, was already proved in §4: Proposition 4.13. We give some necessary and
some sufficient conditions both for the uniqueness of $\nu^{\dagger}_{-}$ once
the chain (83) of prime ideals determined by $\nu^{\dagger}_{-}$ has been
fixed, and also for the unconditional uniqueness of $\nu^{\dagger}_{-}$. In §7
we will use one of these uniqueness criteria to prove uniqueness of
$\nu^{\dagger}_{-}$ in the cases $R^{\dagger}=\tilde{R}$ and
$R^{\dagger}=R^{e}$. At the end of this section we generalize and give a new
point of view of an old result of W. Heinzer and J. Sally (Proposition 6.16),
which provides a sufficient condition for the uniqueness of
$\nu^{\dagger}_{-}$; see also [14], Remarks 5.22.
For a ring $R^{\prime}\in\cal T$ let $K^{\prime}$ denote the field of
fractions of $R^{\prime}$. For some results in this section we will need to
impose an additional condition on the tree $\cal T$: we will assume that there
exists $R_{0}\in\cal T$ such that for all $R^{\prime}\in{\cal T}(R_{0})$ the
field $K^{\prime}$ is algebraic over $K_{0}$. This assumption is needed in
order to be able to control the height of all the ideals in sight. Without
loss of generality, we may take $R_{0}=R$.
###### Proposition 6.1
Assume that for all $R^{\prime}\in\cal T$ the field $K^{\prime}$ is algebraic
over $K$. Consider a ring homomorphism $R^{\prime}\rightarrow
R^{\prime\prime}$ in $\cal T$. Take an $\ell\in\\{0,\dots,r\\}$. We have
${\operatorname{ht}}\ H^{\prime\prime}_{2\ell}\leq{\operatorname{ht}}\
H^{\prime}_{2\ell}.$ (93)
If equality holds in (93) then
${\operatorname{ht}}\ H^{\prime\prime}_{2\ell+1}\geq{\operatorname{ht}}\
H^{\prime}_{2\ell+1}.$ (94)
Proof.- We start by recalling a well known Lemma (for a proof see [18],
Appendix 1, Propositions 2 and 3, p. 326):
###### Lemma 6.2
Let $R\hookrightarrow R^{\prime}$ be an extension of integral domains,
essentially of finite type. Let $K$ and $K^{\prime}$ be the respective fields
of fractions of $R$ and $R^{\prime}$. Consider prime ideals $P\subset R$ and
$P^{\prime}\subset R^{\prime}$ such that $P=P^{\prime}\cap R$. Then
${\operatorname{ht}}\ P^{\prime}+tr.deg.(\kappa(P^{\prime})/\kappa(P))\leq\
{\operatorname{ht}}\ P+tr.deg.(K^{\prime}/K).$ (95)
Moreover, equality holds in (95) whenever $R$ is universally catenarian.
Apply the Lemma to the rings $R^{\prime}$ and $R^{\prime\prime}$ and the prime
ideals $P^{\prime}_{\ell}\subset R^{\prime}$ and
$P^{\prime\prime}_{\ell}\subset R^{\prime\prime}$. In the case at hand we have
$tr.deg.(K^{\prime\prime}/K^{\prime})=0$ by assumption. Hence
${\operatorname{ht}}\ P^{\prime\prime}_{\ell}\leq\ {\operatorname{ht}}\
P^{\prime}_{\ell}.$ (96)
Since $H^{\prime}_{2\ell}$ is a minimal prime of
$P^{\prime}_{\ell}{R^{\prime}}^{\dagger}$ and ${R^{\prime}}^{\dagger}$ is
faithfully flat over $R^{\prime}$, we have $ht\ P^{\prime}_{\ell}=ht\
H^{\prime}_{2\ell}$. Similarly, ${\operatorname{ht}}\
P^{\prime\prime}_{\ell}={\operatorname{ht}}\ H^{\prime\prime}_{2\ell}$, and
(93) follows. Furthermore, equality in (93) is equivalent to equality in (96).
To prove (94), let
$\bar{R}=(R^{\prime\prime}\otimes_{R^{\prime}}{R^{\prime}}^{\dagger})_{M^{\prime\prime}}$,
where $M^{\prime\prime}=(m^{\prime\prime}\otimes 1+1\otimes
m^{\prime}{R^{\prime}}^{\dagger})$ and let $\bar{m}$ denote the maximal ideal
of $\bar{R}$. We have the natural maps
${R^{\prime}}^{\dagger}\overset{\iota}{\rightarrow}\bar{R}\overset{\sigma}{\rightarrow}{R^{\prime\prime}}^{\dagger}$.
The homomorphism $\sigma$ is nothing but the formal completion of the local
ring $\bar{R}$; in particular, it is faithfully flat. Let
$\bar{H}=H^{\prime\prime}_{2\ell+1}\cap\bar{R},$ (97)
$\bar{H}_{0}=H^{\prime\prime}_{0}\cap\bar{R}$. Since $H^{\prime\prime}_{0}$ is
a minimal prime of ${R^{\prime\prime}}^{\dagger}$ and $\sigma$ is faithfully
flat, $\bar{H}_{0}$ is a minimal prime of $\bar{R}$.
Assume that equality holds in (93) (and hence also in (96)). Since equality
holds in (96), by Lemma 6.2 (applied to the ring extension
$R^{\prime}\rightarrow R^{\prime\prime}$) the field $\kappa(P^{\prime\prime})$
is algebraic over $\kappa(P^{\prime})$.
Apply Lemma 6.2 to the ring extension
$\frac{{R^{\prime}}^{\dagger}}{H^{\prime}_{0}}\hookrightarrow\frac{\bar{R}}{\bar{H}_{0}}$
and the prime ideals $\frac{H^{\prime}_{2\ell+1}}{H^{\prime}_{0}}$ and
$\frac{\bar{H}}{\bar{H}_{0}}$. Since $K^{\prime\prime}$ is algebraic over
$K^{\prime}$, $\kappa(\bar{H}_{0})$ is algebraic over
$\kappa(H^{\prime}_{0})$. Since $\kappa(P^{\prime\prime})$ is algebraic over
$\kappa(P^{\prime})$, $\kappa(\bar{H})$ is algebraic over
$\kappa(H^{\prime}_{2\ell+1})$. Finally, $\hat{R}^{\prime}$ is universally
catenarian because it is a complete local ring. Now in the case ${\
}^{\dagger}=\hat{\ }$ Lemma 6.2 says that ${\operatorname{ht}}\
\frac{H^{\prime}_{2\ell+1}}{H^{\prime}_{0}}={\operatorname{ht}}\
\frac{\bar{H}}{\bar{H}_{0}}$. Since both $\hat{R}^{\prime}$ and $\bar{R}$ are
catenarian, this implies that
${\operatorname{ht}}\ H^{\prime}_{2\ell+1}={\operatorname{ht}}\ \bar{H}.$ (98)
In the case where ${\ }^{\dagger}$ stands for henselization or a finite étale
extension, (98) is an immediate consequence of (97). Thus (98) is true in all
the cases. Since $\sigma$ is faithfully flat and in view of (97),
${\operatorname{ht}}\bar{H}\leq{\operatorname{ht}}\
H^{\prime\prime}_{2\ell+1}$. Combined with (98), this completes the proof.
###### Corollary 6.3
For each $i$, $0\leq i\leq 2r$, the quantity ${\operatorname{ht}}\
H^{\prime}_{i}$ stabilizes for $R^{\prime}$ sufficiently far out in $\cal T$.
The next Proposition is an immediate consequence of Theorem 5.6.
###### Proposition 6.4
Suppose given an admissible chain of trees (83) of prime ideals of
${R^{\prime}}^{\dagger}$. For each $\ell\in\\{0,\dots,r-1\\}$, consider the
set of all $R^{\prime}\in\cal T$ such that
${\operatorname{ht}}\ \tilde{H}^{\prime}_{2\ell+1}-{\operatorname{ht}}\
\tilde{H}^{\prime}_{2_{\ell}}\leq 1\qquad\text{ for all even }i$ (99)
and, in case of equality, the 1-dimensional local ring
$\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{2\ell+1}}}{\tilde{H}^{\prime}_{2\ell}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{2\ell+1}}}$
is unibranch (that is, analytically irreducible). Assume that for each $\ell$
the set of such $R^{\prime}$ is cofinal in $\cal T$. Let
$\nu^{\dagger}_{2\ell+1,0}$ denote the unique valuation centered at
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{2\ell+1}}}{\tilde{H}^{\prime}_{2\ell}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{2\ell+1}}}$.
Assume that for each even $i=2\ell$, $\nu_{\ell}$ admits a unique extension
$\nu^{\dagger}_{i0}$ to a valuation of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})$,
centered in
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$.
Then specifying the valuation $\nu^{\dagger}_{-}$ is equivalent to specifying
for each odd $i$, $2<i<2r$, an extension $\nu_{i}^{\dagger}$ of the valuation
$\nu^{\dagger}_{i0}$ of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})$
to its field extension $k_{\nu^{\dagger}_{i-1}}$ (in particular, such
extensions $\nu^{\dagger}_{-}$ always exist). If for each odd $i$, $2<i<2r$,
the field extension
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})\rightarrow
k_{\nu^{\dagger}_{i-1}}$ is algebraic and the extension $\nu_{i}^{\dagger}$ of
$\nu^{\dagger}_{i0}$ to $k_{\nu^{\dagger}_{i-1}}$ is unique then there is a
unique extension $\nu^{\dagger}_{-}$ of $\nu$ such that the
$\tilde{H}^{\prime}_{i}$ are the prime ideals, determined by
$\nu^{\dagger}_{-}$.
Conversely, assume that $K^{\prime}$ is algebraic over $K$ and that there
exists a unique extension $\nu^{\dagger}_{-}$ of $\nu$ such that the
$\tilde{H}^{\prime}_{i}$ are the prime $\nu^{\dagger}_{-}$-ideals, determined
by $\nu^{\dagger}_{-}$. Then for each $\ell\in\\{0,\dots,r-1\\}$ and for all
$R^{\prime}$ sufficiently far out in $\cal T$ the inequality (99) holds. For
each even $i=2\ell$, $\nu_{\ell}$ admits a unique extension
$\nu^{\dagger}_{i0}$ to a valuation of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa\left(\tilde{H}^{\prime}_{i-1}\right)$,
centered in
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$;
we have $rk\ \nu^{\dagger}_{i0}=1$. For each odd $i$, the ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$
is a valuation ring of a (not necessarily discrete) rank 1 valuation. For each
odd $i$, $1\leq i<2r$, the field extension
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})\rightarrow
k_{\nu^{\dagger}_{i-1}}$ is algebraic and the extension $\nu_{i}^{\dagger}$ of
$\nu^{\dagger}_{i0}$ to $k_{\nu^{\dagger}_{i-1}}$ is unique.
###### Remark 6.5
We do not know of a simple criterion to decide when, given an algebraic field
extension $K\hookrightarrow L$ and a valuation $\nu$ of $K$, is the extension
of $\nu$ to $L$ unique. See [6], [16] for more information about this question
and an algorithm for arriving at the answer using MacLane’s key polynomials.
Next we describe three classes of extensions of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{R^{\prime}}^{\dagger}$,
which are of particular interest for applications, and which we call minimal,
evenly minimal and tight extensions.
###### Definition 6.6
Let $\nu^{\dagger}_{-}$ be an extension of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{R^{\prime}}^{\dagger}$
and let the notation be as above. We say that $\nu^{\dagger}_{-}$ is evenly
minimal if whenever $i=2\ell$ is even, the following two conditions hold:
(1)
$\Delta_{i-1,0}=\frac{\Delta_{\ell-1}}{\Delta_{\ell}}.$ (100)
(2) For an element $\overline{\beta}\in\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$,
the $\nu^{\dagger}_{i0}$-ideal of
$\frac{R^{\dagger}_{\tilde{H}_{i}}}{\tilde{H}_{i-1}R^{\dagger}_{\tilde{H}_{i}}}$
of value $\overline{\beta}$, denoted by
$\mathcal{P}^{\dagger}_{\overline{\beta},\ell}$, is given by the formula
$\mathcal{P}^{\dagger}_{\overline{\beta},\ell}=\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}\right)\cap\frac{R^{\dagger}_{\tilde{H}_{i}}}{\tilde{H}_{i-1}R^{\dagger}_{\tilde{H}_{i}}}.$
(101)
We say that $\nu^{\dagger}_{-}$ is minimal if
$\tilde{H}^{\prime}_{i}=H^{\prime}_{i}$ for each $R^{\prime}$ and each
$i\in\\{0,\dots,2r+1\\}$. We say that $\nu^{\dagger}_{-}$ is tight if it is
evenly minimal and
$\tilde{H}^{\prime}_{i}=\tilde{H}^{\prime}_{i+1}\qquad\text{ for all even }i.$
(102)
###### Remark 6.7
(1) The valuation $\nu^{\dagger}_{i0}$ is uniquely determined by conditions
(100) and (101). Recall also that if $i=2\ell$ is even and we have:
$\displaystyle\tilde{H}^{\prime}_{i}$ $\displaystyle=$ $\displaystyle
H^{\prime}_{i}\qquad\text{ and}$ (103) $\displaystyle\tilde{H}^{\prime}_{i-1}$
$\displaystyle=$ $\displaystyle H^{\prime}_{i-1}$ (104)
then $\nu^{\dagger}_{i0}$ is uniquely determined by $\nu_{\ell}$ by
Proposition 4.13. In particular, if $\nu^{\dagger}_{-}$ is minimal (that is,
if (103)–(104) hold for all $i$) then $\nu^{\dagger}_{-}$ is evenly minimal.
(2) The definition of evenly minimal extensions can be rephrased as follows in
terms of the associated graded algebras of $\nu_{\ell}$ and
$\nu^{\dagger}_{i0}$. First, consider a homomorphism in $\mathcal{T}$ and the
following diagram, composed of natural homomorphisms:
$\textstyle{{\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R^{\prime}}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{\prime}}$$\textstyle{{\frac{\mathcal{P}^{\prime}_{\overline{\beta}}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{\frac{{\mathcal{P}^{\prime}_{\overline{\beta}}}^{\dagger}}{{\mathcal{P}^{\prime}}_{\overline{\beta}+}^{\dagger}}}$
$\textstyle{{\frac{\mathcal{P}_{\overline{\beta}}^{\dagger}}{\mathcal{P}_{\overline{\beta}+}^{\dagger}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\prime}}$
(105)
It follows from Nakayama’s Lemma that equality (101) is equivalent to saying
that
$\frac{\mathcal{P}_{\overline{\beta}}^{\dagger}}{\mathcal{P}_{\overline{\beta}+}^{\dagger}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}{\psi^{\prime}}^{-1}\left((\phi^{\prime}\circ\lambda^{\prime})\left(\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R^{\prime}}\kappa\left(\tilde{H}^{\prime}_{i}\right)\right)\right).$
(106)
Taking the direct sum in (106) over all
$\overline{\beta}\in\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$ and passing to the
limit on the both sides, we see that the extension $\nu^{\dagger}_{-}$ is
evenly minimal if and only if we have the following equality of graded
algebras:
$\mbox{gr}_{\nu_{\ell}}\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{R^{\prime}}{P^{\prime}_{\ell}}\right)\otimes_{R^{\prime}}\kappa\left(\tilde{H}^{\prime}_{i}\right)=\mbox{gr}_{\nu^{\dagger}_{i0}}\left(\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}\right).$
###### Examples 6.8
The extension $\hat{\nu}$ of Example 3.1 (page 3.1) is minimal, but not tight.
The valuation $\nu$ admits a unique tight extension
$\hat{\nu}_{2}\circ\hat{\nu}_{3}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{1}}$;
the valuation $\hat{\nu}$ is the composition of the discrete rank 1 valuation
$\hat{\nu}_{1}$, centered in
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\hat{R}^{\prime}_{H^{\prime}_{1}}$
with $\hat{\nu}_{2}\circ\hat{\nu}_{3}$.
The extension $\hat{\nu}^{(1)}$ of Example 3.2 (page 3.2) is minimal. The
extension $\hat{\nu}^{(2)}$ is evenly minimal but not minimal. Neither
$\hat{\nu}^{(1)}$ nor $\hat{\nu}^{(2)}$ is tight. The valuation $\nu$ admits a
unique tight extension $\hat{\nu}_{2}\circ\hat{\nu}_{3}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{R^{\prime}}{\tilde{H}^{\prime}_{1}}$,
where
$\tilde{H}^{\prime}_{1}=\left(y-\sum\limits_{j=1}^{\infty}c_{j}x^{j}\right)$;
the valuation $\hat{\nu}^{(2)}$ is the composition of the discrete rank 1
valuation $\hat{\nu}_{1}$, centered in
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\hat{R}^{\prime}_{\tilde{H}^{\prime}_{1}}$
with $\hat{\nu}_{2}\circ\hat{\nu}_{3}$.
###### Remark 6.9
As of this moment, we do not know of any examples of extensions
$\hat{\nu}_{-}$ which are not evenly minimal. Thus, formally, the question of
whether every extension $\hat{\nu}_{-}$ is evenly minimal is open, though we
strongly suspect that counterexamples do exist.
###### Proposition 6.10
Let $i=2\ell$ be even and let $\nu^{\dagger}_{i0}$ be the extension of
$\nu_{\ell}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i-1})$,
centered at the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$,
defined by (101). Then
$k_{\nu^{\dagger}_{i0}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i}).$
(107)
Proof.- Take two elements
$x,y\in\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$,
such that $\nu_{i0}^{\dagger}(x)=\nu_{i0}^{\dagger}(y)$. We must show that the
image of $\frac{x}{y}$ in $k_{\nu^{\dagger}_{i0}}$ belongs to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{i})$.
Without loss of generality, we may assume that
$x,y\in\frac{R^{\dagger}_{\tilde{H}_{i}}}{\tilde{H}_{i-1}{R}^{\dagger}_{\tilde{H}_{i}}}$.
Let $\beta=\nu_{i0}^{\dagger}(x)=\nu_{i0}^{\dagger}(y)$. Choose
$R^{\prime}\in\cal T$ sufficiently far out in the direct system so that
$x,y\in\frac{\mathcal{P}^{\prime}_{\beta}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$.
Let $R^{\prime}\rightarrow R^{\prime\prime}$ be the blowing up of the ideal
$\mathcal{P}^{\prime}_{\beta}R^{\prime}$. Then in
$\frac{\mathcal{P}^{\prime\prime}_{\beta}{R^{\prime\prime}}^{\dagger}_{\tilde{H}^{\prime\prime}_{i}}}{\tilde{H}^{\prime\prime}_{i-1}{R^{\prime\prime}}^{\dagger}_{\tilde{H}^{\prime\prime}_{i}}}$
we can write
$\displaystyle x$ $\displaystyle=$ $\displaystyle az\qquad\text{ and }$ (108)
$\displaystyle y$ $\displaystyle=$ $\displaystyle aw,$ (109)
where $\nu_{i0}^{\dagger}(a)=\beta$ and
$\nu_{i0}^{\dagger}(z)=\nu_{i0}^{\dagger}(w)=0$. Let $\bar{z}$ be the image of
$z$ in $\kappa(\tilde{H}^{\prime\prime}_{i})$ and similarly for $\bar{w}$.
Then the image of $\frac{x}{y}$ in $k_{\nu^{\dagger}_{i0}}$ equals
$\frac{\bar{z}}{\bar{w}}\in\kappa(\tilde{H}^{\prime\prime}_{i})$, and the
result is proved.
###### Remark 6.11
Theorem 5.6 and the existence of the extension $\nu^{\dagger}_{2\ell,0}$ of
$\nu_{\ell}$ in the case when $\tilde{H}^{\prime}_{2\ell}=H^{\prime}_{2\ell}$
and $\tilde{H}^{\prime}_{2\ell-1}=H^{\prime}_{2\ell-1}$ guaranteed by Theorem
4.9 (2) allow us to give a fairly explicit description of the totality of
minimal extensions as compositions of $2r$ valuations and, in particular, to
show that they always exist. Indeed, minimal extensions $\nu^{\dagger}_{-}$
can be constructed at will, recursively in $i$, as follows. Assume that the
valuations $\nu^{\dagger}_{1},\dots,\nu^{\dagger}_{i-1}$ are already
constructed. If $i$ is odd, let $\nu^{\dagger}_{i}$ be an arbitrary valuation
of the residue field $k_{\nu^{\dagger}_{i-1}}$ of the valuation ring
$R_{\nu^{\dagger}_{i-1}}$. If $i=2\ell$ is even, let $\nu^{\dagger}_{i0}$ be
the extension of $\nu_{\ell}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$,
centered at the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{i}}}{H^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{H^{\prime}_{i}}}$,
whose existence and uniqueness are guaranteed by Theorem 4.9 (2) and
Proposition 4.13, respectively. Let $\nu^{\dagger}_{i}$ be an arbitrary
extension of $\nu^{\dagger}_{i0}$ to the field $k_{\nu^{\dagger}_{i-1}}$. It
is clear that all the minimal extensions $\nu^{\dagger}_{-}$ of $\nu$ are
obtained in this way.
In the next section we will use this remark to show that if
$R^{\dagger}=\tilde{R}$ or $R^{\dagger}=R^{e}$ then $\nu$ admits a unique
extension to $\frac{R^{\dagger}}{H_{0}}$, which is necessarily minimal.
We end this section by giving some sufficient conditions for the uniqueness of
$\nu^{\dagger}_{-}$.
###### Proposition 6.12
Suppose given an admissible chain of trees (83) of prime ideals of
${R^{\prime}}^{\dagger}$. For each $\ell\in\\{0,\dots,r-1\\}$, consider the
set of all $R^{\prime}\in\cal T$ such that
$ht\ \tilde{H}^{\prime}_{2\ell+1}-ht\ \tilde{H}^{\prime}_{2\ell}\leq
1\qquad\text{ for all even }i$ (110)
and, in case of equality, the 1-dimensional local ring
$\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{2\ell+1}}}{\tilde{H}^{\prime}_{2\ell}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{2\ell+1}}}$
is unibranch (that is, analytically irreducible). Assume that for each $\ell$
the set of such $R^{\prime}$ is cofinal in $\cal T$.
Let $\nu^{\dagger}_{-}$ be an extension of $\nu$ such that the
$\tilde{H}^{\prime}_{i}$ are prime $\nu^{\dagger}_{-}$-ideals. Assume that
$\nu^{\dagger}_{-}$ is evenly minimal. Then there is at most one such
extension $\nu^{\dagger}_{-}$ and exactly one such $\nu^{\dagger}_{-}$ if
$\tilde{H}^{\prime}_{i}=H^{\prime}_{i}\quad\text{ for all }i.$ (111)
(in the latter case $\nu^{\dagger}_{-}$ is minimal by definition).
Proof.- By Theorem 4.9 (2) and Proposition 4.13, if (111) holds then
$\nu^{\dagger}_{-}$ is minimal and for each even $i$ the extension
$\nu^{\dagger}_{i0}$ exists and is unique. Therefore we may assume that in all
the cases $\nu^{\dagger}_{-}$ is evenly minimal and that $\nu^{\dagger}_{i0}$
exists whenever (111) holds.
The valuation $\nu^{\dagger}_{-}$, if it exists, is a composition of $2r$
valuations:
$\nu^{\dagger}_{-}=\nu^{\dagger}_{1}\circ\nu^{\dagger}_{2}\circ\dots\circ\nu^{\dagger}_{2r}$,
subject to the conditions of Theorem 5.6. We prove the uniqueness of
$\nu^{\dagger}_{-}$ by induction on $r$. Assume the result is true for $r-1$.
This means that there is at most one evenly minimal extension
$\nu^{\dagger}_{3}\circ\nu^{\dagger}_{4}\circ\dots\circ\nu^{\dagger}_{2r}$ of
$\nu_{2}\circ\nu_{3}\circ\dots\circ\nu_{r}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{2})$,
and exactly one in the case when (111) holds. To complete the proof of
uniqueness of $\nu^{\dagger}_{-}$, it is sufficient to show that both
$\nu_{1}^{\dagger}$ and $\nu_{2}^{\dagger}$ are unique and that the residue
field of $\nu_{2}^{\dagger}$ equals
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{2})$.
We start with the uniqueness of $\nu_{1}^{\dagger}$. If (102) holds then
$\nu_{1}^{\dagger}$ is the trivial valuation. Suppose, on the other hand, that
equality holds in (110). Then the restriction of $\nu_{1}^{\dagger}$ to each
$R^{\prime}\in\cal T$ such that the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}_{\tilde{H}^{\prime}_{1}}}{\tilde{H}^{\prime}_{0}\hat{R}^{\prime}_{\tilde{H}^{\prime}_{1}}}$
is one-dimensional and unibranch is the unique divisorial valuation centered
in that ring (in particular, its residue field is
$\kappa(\tilde{H}^{\prime}_{1})$). By the assumed cofinality of such
$R^{\prime}$, the valuation $\nu^{\dagger}_{1}$ is unique and its residue
field equals
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{1})$.
Thus, regardless of whether or not the inequality in (110) is strict,
$\nu^{\dagger}_{1}$ is unique and we have the equality of residue fields
$k_{\nu_{1}^{\dagger}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{1})$
(112)
This equality implies that $\nu_{2}^{\dagger}=\nu^{\dagger}_{20}$. Now, the
valuation $\nu^{\dagger}_{2}=\nu^{\dagger}_{20}$ is uniquely determined by the
conditions (100) and (101), and its residue field is
$k_{\nu^{\dagger}_{2}}=k_{\nu^{\dagger}_{20}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(\tilde{H}^{\prime}_{2}).$
(113)
by Proposition 6.10. Furthermore, by Theorem 4.9 exactly one such
$\nu_{2}^{\dagger}$ exists whenever (111) holds. This proves that there is at
most one possibility for $\nu^{\dagger}_{-}$: the composition of
$\nu_{1}^{\dagger}\circ\nu^{\dagger}_{2}$ with
$\nu^{\dagger}_{3}\circ\nu^{\dagger}_{4}\circ\dots\circ\nu^{\dagger}_{2r}$,
and exactly one if (111) holds.
###### Proposition 6.13
The extension $\nu^{\dagger}_{-}$ is tight if and only if for each
$R^{\prime}$ in our direct system the natural graded algebra extension
$\mbox{gr}_{\nu}R^{\prime}\rightarrow\mbox{gr}_{\nu^{\dagger}_{-}}{R^{\prime}}^{\dagger}$
is scalewise birational.
###### Remark 6.14
Proposition 6.13 allows us to rephrase Conjecture 1.11 as follows: the
valuation $\nu$ admits at least one tight extension $\nu^{\dagger}_{-}$.
Proof.-(of Proposition 6.13) “If” Assume that for each $R^{\prime}$ in our
direct system the natural graded algebra extension
$\mbox{gr}_{\nu}R^{\prime}\rightarrow\mbox{gr}_{\nu^{\dagger}_{-}}{R^{\prime}}^{\dagger}$
is scalewise birational. Then
$\Gamma^{\dagger}=\Gamma.$ (114)
Together with (85) this implies that for each $l\in\\{1,\dots,r+1\\}$ we have
$\Delta^{\dagger}_{2\ell-2}=\Delta^{\dagger}_{2\ell-1}=\Delta_{\ell-1}$ under
the identification (114). Then
$\frac{\Delta^{\dagger}_{2\ell-1}}{\Delta^{\dagger}_{2\ell}}=(0)$, so for all
odd $i$ the valuation $\nu^{\dagger}_{i}$ is trivial. This proves the equality
(102) in the definition of tight. It remains to show that $\nu^{\dagger}_{-}$
is evenly minimal. We will prove the even minimality in the form of equality
(106) for each $\bar{\beta}\in\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$. The
right hand side of (106) is trivially contained in the left hand side; we must
prove the opposite inclusion. To do that, take a non-zero element
$x\in\frac{\mathcal{P}_{\overline{\beta}}^{\dagger}}{\mathcal{P}_{\overline{\beta}+}^{\dagger}}$.
By scalewise birationaliy, there exist non-zero elements
$\bar{y},\bar{z}\in\mbox{gr}_{\nu}R$, with $ord\ \bar{y},ord\
\bar{z}\in\Delta_{\ell}$, such that $x\bar{y}=\bar{z}$. Let $y$ be a
representative of $\bar{y}$ in $R$, and similarly for $z$. Let $R\rightarrow
R^{\prime}$ be the local blowing up with respect to $\nu$ along the ideal
$(y,z)$. Then, in the notation of (106), we have
$x={\psi^{\prime}}^{-1}\left(\left(\phi^{\prime}\circ\lambda^{\prime}\right)\left(\frac{\bar{z}}{\bar{y}}\otimes_{R^{\prime}}1\right)\right)\in{\psi^{\prime}}^{-1}\left((\phi^{\prime}\circ\lambda^{\prime})\left(\frac{\mathcal{P}^{\prime}_{\overline{\beta}}}{\mathcal{P}^{\prime}_{\overline{\beta}+}}\otimes_{R^{\prime}}\kappa\left(\tilde{H}^{\prime}_{i}\right)\right)\right).$
(115)
This proves (106). “If” is proved.
“Only if”. Assume that $\nu^{\dagger}_{-}$ is tight (that is, it is evenly
minimal and (102) holds) and take $R^{\prime}\in\cal T$. Then the valuation
$\nu_{2\ell+1}$ is trivial for all $\ell$, so
$\nu^{\dagger}_{-}=\nu^{\dagger}_{2}\circ\nu^{\dagger}_{4}\circ\dots\circ\nu^{\dagger}_{2r}$.
We must show that the graded algebra extension
$\mbox{gr}_{\nu}R^{\prime}\rightarrow\mbox{gr}_{\nu^{\dagger}_{-}}{R^{\prime}}^{\dagger}$
is scalewise birational. Again, we use induction on $r$. Take an element
$x\in{R^{\prime}}^{\dagger}$. If $\nu^{\dagger}_{-}(x)\in\Delta_{1}$ then
$\mbox{in}_{\nu^{\dagger}_{-}}x\in\mbox{gr}_{\nu^{\dagger}_{4}\circ\dots\circ\nu^{\dagger}_{2r}}\frac{{R^{\prime}}^{\dagger}}{\tilde{H}^{\prime}_{2}}$,
hence by the induction assumption there exists $y\in R^{\prime}$ with
$\nu^{\dagger}_{-}(x)\in\Delta_{1}$ and
$\mbox{in}_{\nu^{\dagger}_{-}}(xy)\in\mbox{gr}_{\nu}\frac{R^{\prime}}{P_{1}}$.
In this case, there is nothing more to prove. Thus we may assume that
$\nu^{\dagger}_{-}(x)\mbox{$\in$ /}\Delta_{1}$. It remains to show that there
exists $y\in R^{\prime}$ such that
$\mbox{in}_{\nu^{\dagger}_{-}}(xy)\in\mbox{gr}_{\nu}R^{\prime}$. Since the
natural map sending each element of the ring to its image in the graded
algebra behaves well with respect to multiplication and division, local
blowings up induce birational transformations of graded algebras, and it is
enough to find a local blowing up $R^{\prime\prime}\in{\cal T}(R^{\prime})$
and $y\in R^{\prime\prime}$ such that
$\mbox{in}_{\nu^{\dagger}_{-}}(xy)\in\mbox{gr}_{\nu}R^{\prime\prime}$.
Now, Proposition 6.10 shows that there exists a local blowing up
$R^{\prime}\rightarrow R^{\prime\prime}$ such that $x=az$ (108), with $z\in
R^{\prime\prime}$ and $\nu^{\dagger}_{2}(a)=\nu^{\dagger}_{2,0}(a)=0$. The
last equality means that $\nu^{\dagger}_{-}(a)\in\Delta_{1}$, and the result
follows from the induction assumption, applied to $a$.
The argument above also shows the following. Let
${\Phi^{\prime}}^{\dagger}=\nu^{\dagger}_{-}\left({R^{\prime}}^{\dagger}\setminus\\{0\\}\right)$,
take an element $\beta\in{\Phi^{\prime}}^{\dagger}$ and let ${\cal
P^{\prime}}^{\dagger}_{\beta}$ denote the $\nu^{\dagger}_{-}$-ideal of
${R^{\prime}}^{\dagger}$ of value $\beta$.
###### Corollary 6.15
Take an element $x\in{\cal P^{\prime}}^{\dagger}_{\beta}$. There exists a
local blowing up $R^{\prime}\rightarrow R^{\prime\prime}$ such that
$\beta\in\nu(R^{\prime\prime})\setminus\\{0\\}$ and $x\in{\cal
P}^{\prime\prime}_{\beta}{R^{\prime\prime}}^{\dagger}$.
The next Proposition gives a sufficient condition for the uniqueness of
$\nu^{\dagger}_{-}$ (this result is due to Heinzer and Sally [5]).
###### Proposition 6.16
Assume that $K^{\prime}$ is algebraic over $K$ for all $R^{\prime}\in\cal T$
and that the following conditions hold:
(1) $ht\ H^{\prime}_{1}\leq 1$
(2) $ht\ H^{\prime}_{1}+rat.rk\ \nu=\dim\ R^{\prime}$, where $R^{\prime}$ is
taken to be sufficiently far out in the direct system.
Let $\nu^{\dagger}_{-}$ be an extension of $\nu$ to a ring of the form
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}}{\tilde{H}^{\prime}_{0}}$.
Then either
$\displaystyle\tilde{H}^{\prime}_{0}$ $\displaystyle=$ $\displaystyle
H^{\prime}_{0}\qquad\text{ or }$ (116) $\displaystyle\tilde{H}^{\prime}_{0}$
$\displaystyle=$ $\displaystyle H^{\prime}_{1}.$ (117)
The valuation $\nu$ admits a unique extension to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}}{H^{\prime}_{0}}$
and a unique extension to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}}{H^{\prime}_{1}}$.
The first extension is minimal and the second is tight.
Proof.- For $1\leq\ell\leq r$, let $r_{\ell}$ denote the rational rank of
$\nu_{\ell}$. Let $\nu^{\dagger}_{-}$ be an extension of $\nu$ to a ring of
the form
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}}{\tilde{H}^{\prime}_{0}}$,
where $\tilde{H}^{\prime}_{0}$ is a tree of prime ideals of
${R^{\prime}}^{\dagger}$ such that $\tilde{H}^{\prime}_{0}\cap
R^{\prime}=(0)$.
By Corollary 6.3 $ht\ H^{\prime}_{i}$ stabilizes for $1\leq i\leq 2r$ and
$R^{\prime}$ sufficiently far out in the direct system. From now on, we will
assume that $R^{\prime}$ is chosen sufficiently far so that the stable value
of $ht\ H^{\prime}_{i}$ is attained. Now, let $i=2\ell$. The valuation
$\nu^{\dagger}_{i0}$ is centered in the local noetherian ring
$\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}$,
hence by Abhyankar’s inequality
$rat.rk\
\nu^{\dagger}_{i0}\leq\dim\frac{{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}{\tilde{H}^{\prime}_{i-1}{R^{\prime}}^{\dagger}_{\tilde{H}^{\prime}_{i}}}\leq
ht\ \tilde{H}^{\prime}_{i}-ht\ \tilde{H}^{\prime}_{i-1}.$ (118)
Since this inequality is true for all even $i$, summing over all $i$ we
obtain:
$\begin{array}[]{rcl}\dim R^{\prime}&=&\dim\
{R^{\prime}}^{\dagger}=\sum\limits_{i=1}^{2r}(ht\ \tilde{H}^{\prime}_{i}-ht\
\tilde{H}^{\prime}_{i-1})\geq ht\
\tilde{H}^{\prime}_{1}+\sum\limits_{\ell=1}^{r}(ht\
\tilde{H}^{\prime}_{2\ell}-ht\ \tilde{H}^{\prime}_{2\ell-1})\geq\\\ &\geq&ht\
H^{\prime}_{1}+\sum\limits_{\ell=1}^{r}rat.rk\ \nu^{\dagger}_{2\ell,0}\geq ht\
H^{\prime}_{1}+\sum\limits_{\ell=1}^{r}r_{\ell}=ht\ H^{\prime}_{1}+rat.rk\
\nu=\dim R^{\prime}.\end{array}$ (119)
Hence all the inequalities in (118) and (119) are equalities. In particular,
we have
$ht\ \tilde{H}^{\prime}_{1}=ht\ H^{\prime}_{1};$
combined with Proposition 5.4 this shows that
$\tilde{H}^{\prime}_{1}=H^{\prime}_{1}.$ (120)
Together with the hypothesis (1) of the Proposition, this already proves that
at least one of (116)–(117) holds. Furthermore, equalities in (118) and (119)
prove that
$ht\ \tilde{H}^{\prime}_{i}=ht\ \tilde{H}^{\prime}_{i-1}$
for all odd $i>1$, so that
$\tilde{H}^{\prime}_{i}=\tilde{H}^{\prime}_{i-1}\qquad\text{whenever $i>1$ is
odd}$ (121)
and that
$r_{i}=ht\ \tilde{H}^{\prime}_{i}-ht\ \tilde{H}^{\prime}_{i-1}$ (122)
whenever $i$ is even.
Now, consider the special case when $\tilde{H}^{\prime}_{i}=H^{\prime}_{i}$
for $i\geq 1$ and $\tilde{H}^{\prime}_{0}$ is as in (116)–(117). According to
Proposition 4.13 for each even $i=2\ell$ there exists a unique extension
$\nu^{\dagger}_{i0}$ of $\nu_{l}$ to a valuation of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i-1})$,
centered in the local ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{2\ell}}}{H^{\prime}_{2\ell-1}}$.
Moreover, we have
$k_{\nu^{\dagger}_{2\ell,0}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{2\ell})$
(123)
by Remark 4.14. By Theorem 5.6, there exists an extension $\nu^{\dagger}_{-}$
of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}}{\tilde{H}^{\prime}_{0}}$
such that the $\\{\tilde{H}^{\prime}_{i}\\}$ as above is the chain of trees of
prime ideals, determined by $\nu^{\dagger}_{-}$. In particular, (121) and
(122) hold with $\tilde{H}^{\prime}_{i}$ replaced by $H^{\prime}_{i}$.
Now (122) and Proposition 5.4 imply that for any extension $\nu^{\dagger}_{-}$
we have $\tilde{H}^{\prime}_{i}=H^{\prime}_{i}$ for $i>0$, so that the special
case above is, in fact, the only case possible. Furthermore, by (121) we have
$H^{\prime}_{2\ell+1}=H^{\prime}_{2\ell}$ for all $\ell\in\\{1,\dots,r\\}$.
This implies that for all such $\ell$ the valuation
$\nu^{\dagger}_{2\ell+1,0}$ is the trivial valuation of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{2\ell})$;
in particular,
$k_{\nu^{\dagger}_{2\ell+1,0}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{2\ell})$
(124)
for all $\ell\in\\{1,\dots,r-1\\}$.
If $\tilde{H}^{\prime}_{0}=H^{\prime}_{1}=\tilde{H}^{\prime}_{1}$ then the
only possibility for $\nu^{\dagger}_{10}=\nu^{\dagger}_{1}$ is the trivial
valuation of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{1})$;
we have
$k_{\nu^{\dagger}_{1}}=k_{\nu^{\dagger}_{10}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{1}).$
(125)
If $\tilde{H}^{\prime}_{0}=H^{\prime}_{0}$ then by the hypothesis (1) of the
Proposition and the excellence of $R$ the ring
$\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{1}}}{H^{\prime}_{0}{R^{\prime}}^{\dagger}_{H^{\prime}_{1}}}$
is a regular one-dimensional local ring (in particular, unibranch), hence the
valuation $\nu^{\dagger}_{1}=\nu^{\dagger}_{10}$ centered at
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{{R^{\prime}}^{\dagger}_{H^{\prime}_{1}}}{H^{\prime}_{0}{R^{\prime}}^{\dagger}_{H^{\prime}_{1}}}$
is unique and (125) holds also in this case.
By induction on $i$, it follows from (123), (124), the uniqueness of
$\nu^{\dagger}_{2\ell,0}$ and the triviality of $\nu^{\dagger}_{2\ell+1,0}$
for $\ell\geq 1$ that $\nu_{i}^{\dagger}$ is uniquely determined for all $i$
and
$k_{\nu^{\dagger}_{i}}=\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\kappa(H^{\prime}_{i})$.
This proves that in both cases (116) and (117) the valuation
$\nu^{\dagger}_{-}=\nu^{\dagger}_{1}\circ\dots\circ\nu^{\dagger}_{2r}$ is
unique. The last statement of the Proposition is immediate from definitions.
A related necessary condition for the uniqueness of $\nu^{\dagger}_{-}$ will
be proved in §9.
## 7 Extending a valuation centered in an excellent local domain to its
henselization.
Let $\tilde{R}$ denote the henselization of $R$, as above. The completion
homomorphism $R\rightarrow\hat{R}$ factors through the henselization:
$R\rightarrow\tilde{R}\rightarrow\hat{R}$. In this section, we will show that
$H_{1}$ is a minimal prime of $\tilde{R}$, that $\nu$ extends uniquely to a
valuation $\tilde{\nu}_{-}$ of rank $r$ centered at $\frac{\tilde{R}}{H_{1}}$,
and that $H_{1}$ is the unique prime ideal $P$ of $\tilde{R}$ such that $\nu$
extends to a valuation of $\frac{\tilde{R}}{P}$. Furthermore, we will prove
that $H_{2\ell+1}$ is a minimal prime of $P_{\ell}\tilde{R}$ for all $\ell$
and that these are precisely the prime $\tilde{\nu}$-ideals of $\tilde{R}$.
Studying the implicit prime ideals of $\tilde{R}$ and the extension of $\nu$
to $\tilde{R}$ is a logical intermediate step before attacking the formal
completion, for the following reason. As we will show in the next section, if
$R$ is already henselian in (46) then
$\mathcal{P}^{\prime}_{\beta}\hat{R}^{\prime}_{H^{\prime}_{2\ell+1}}\cap\hat{R}=\mathcal{P}_{\beta}\hat{R}$
for all $\beta$ and $R^{\prime}$ and thus we have
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left({\cal
P}_{\beta}\hat{R}\right)$.
We state the main result of this section. In the case when $R^{e}$ is an étale
extension of $R$, contained in $\tilde{R}$, we use (48) with
$R^{\dagger}=R^{e}$ as our definition of the implicit prime ideals.
###### Theorem 7.1
Let $R^{e}$ be a local étale extension of $R$, contained in $\tilde{R}$. Then:
(1) The ideal $H_{2\ell+1}$ is prime for $0\leq l\leq r$; it is a minimal
prime of $P_{\ell}R^{e}$. In particular, $H_{1}$ is a minimal prime of
$R^{e}$. We have $H_{2\ell}=H_{2\ell+1}$ for $0\leq l\leq r$.
(2) The ideal $H_{1}$ is the unique prime $P$ of $R^{e}$ such that there
exists an extension $\nu^{e}_{-}$ of $\nu$ to $\frac{R^{e}}{P}$; the extension
$\nu^{e}_{-}$ is unique. The graded algebra
$\mbox{gr}_{\nu^{e}_{-}}\frac{R^{e}}{H_{1}}$ is scalewise birational to
$\mbox{gr}_{\nu}R$; in particular, $rk\ \nu^{e}_{-}=r$.
(3) The ideals $H_{2\ell+1}$ are precisely the prime $\nu^{e}_{-}$-ideals of
$R^{e}$.
Proof.- By assumption, the ring $R^{e}$ is a direct limit of local, strict
étale extensions of $R$ which are essentially of finite type. All the
assertions (1)–(3) behave well under taking direct limits, so it is sufficient
to prove the Theorem in the case when $R^{e}$ is essentially of finite type
over $R$. From now on, we will restrict attention to this case.
The next step is to describe explicitly those local blowings up $R\rightarrow
R^{\prime}$ for which $R^{\prime}$ is $\ell$-stable. Their interest to us is
that, according to Proposition 4.5, if $R^{\prime}$ is $\ell$-stable then for
all $R^{\prime\prime}\in{\cal T}(R^{\prime})$ and all
$\beta\in\frac{\Delta_{\ell}}{\Delta_{\ell+1}}$, we have the equality
${\cal P}^{\prime\prime}_{\beta}(R^{\prime\prime}\otimes_{R}R^{e})\cap
R^{e}={\cal P}_{\beta}R^{e};$ (126)
in particular, the limit in (48) is attained, that is, we have the equality
$H_{2\ell+1}=\bigcap\limits_{\beta\in\Delta_{\ell}}\left(\left({\cal
P}^{\prime}_{\beta}\left(R^{e}\otimes_{R}R^{\prime}\right)_{M^{\prime}}\right)\bigcap
R^{e}\right).$ (127)
###### Lemma 7.2
Let $\frac{R}{P_{\ell}}\to T$ be a finitely generated extension of
$\frac{R}{P_{\ell}}$, contained in $\frac{R_{\nu}}{\bf m_{\ell}}$. Let
${\bf q}=\frac{\bf m_{\nu}}{\bf m_{\ell}}\cap T.$
There exists a $\nu$-extension $R\to R^{\prime}$ of $R$ such that
$\frac{R^{\prime}}{P^{\prime}_{\ell}}=T_{\bf q}$.
Proof.- Write
$T=\frac{R}{P_{\ell}}\left[\overline{a}_{1},\ldots,\overline{a}_{k}\right]$,
with $\overline{a}_{i}\in\frac{R_{\nu}}{\bf m_{\ell}}$, that is,
$\nu_{\ell+1}\left(\overline{a}_{i}\right)\geq 0,\ 1\leq i\leq k$. We can lift
the $\overline{a}_{i}$ to elements $a_{i}$ in $R_{\nu}$ such that
$\nu\left(a_{i}\right)\geq 0$. Let us consider the ring
$R^{\prime\prime}=R\left[a_{1},\ldots,a_{k}\right]\subset R_{\nu}$ and its
localization $R^{\prime}=R^{\prime\prime}_{{\bf m}_{\nu}\cap
R^{\prime\prime}}$. The ideal $P^{\prime}_{\ell}$ is the kernel of the natural
map $R^{\prime}\rightarrow\frac{R_{\nu}}{\bf m_{\ell}}$. Thus both
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$ and $T_{\bf q}$ are equal to the
$\frac{R}{P_{l}}$-subalgebra of $\frac{R_{\nu}}{\bf m_{\ell}}$, obtained by
adjoining $\overline{a}_{1},\ldots,\overline{a}_{k}$ to $\frac{R}{P_{l}}$
inside $\frac{R_{\nu}}{\bf m_{\ell}}$ and then localizing at the preimage of
the ideal $\frac{\bf m_{\nu}}{\bf m_{\ell}}$. This proves the Lemma.
Let us now go back to our étale extension $R\to R^{e}$.
###### Lemma 7.3
Fix an integer $l\in\\{0,\dots,r\\}$. There exists a local blowing up
$R\rightarrow R^{\prime}$ along $\nu$ having the following property: let
$P^{\prime}_{\ell}$ denote the $\ell$-th prime $\nu$-ideal of $R^{\prime}$.
Then the ring $\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is analytically
irreducible; in particular,
$\frac{R^{\prime}}{P^{\prime}_{\ell}}\otimes_{R}R^{e}$ is an integral domain.
###### Remark 7.4
We are not claiming that there exists $R^{\prime}\in\cal T$ such that
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is analytically irreducible for all
$\ell$ (and we do not know how to prove such a claim), only that for each
$\ell$ there exists an $R^{\prime}$, which may depend on $\ell$, such that
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is analytically irreducible. On the
other hand, below we will prove that there exists an $\ell$-stable
$R^{\prime}\in\cal T$. According to Definition 4.1 (2) and Proposition 4.4,
such a stable $R^{\prime}$ has the property that
$\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}\left(R^{\prime\prime}\otimes_{R}R^{e}\right)_{M^{\prime\prime}}$
is a domain for all $R^{\prime\prime}\in{\cal T}(R^{\prime})$. For a given
$R^{\prime\prime}$, this property is weaker than the analytic irreducibility
of $R^{\prime\prime}/P^{\prime\prime}_{\ell}$. The latter is equivalent to
saying that
$\kappa(P^{\prime\prime}_{\ell})\otimes_{R}(R^{\prime\prime}\otimes_{R}R^{\sharp})_{M^{\prime\prime}}$
is a domain for every local étale extension $R^{\sharp}$ of
$R^{\prime\prime}$.
Proof.-(of Lemma 7.3) Since $R$ is an excellent local ring, every homomorphic
image of $R$ is Nagata [10] (Theorems 72 (31.H), 76 (33.D) and 78 (33.H)). Let
$\pi:\frac{R}{P_{\ell}}\rightarrow S$ be the normalization of
$\frac{R}{P_{\ell}}$. Then $S$ is a finitely generated
$\frac{R}{P_{\ell}}$-algebra contained in $\frac{R_{\nu}}{\bf m_{\ell}}$, to
which we can apply Lemma 7.2. We obtain a $\nu$-extension $R\to R^{\prime}$
such that the ring
$\frac{R^{\prime}}{P^{\prime}_{\ell}}\cong\frac{R^{\prime}}{P_{\ell}R^{\prime}}$
is a localization of $S$ at a prime ideal, hence it is an excellent normal
local ring. In particular, it is analytically irreducible ([11], Theorem
(43.20), p. 187 and Corollary (44.3), p. 189), as desired.
Next, we fix $\ell\in\\{0,\dots,r\\}$ and study the ring
$(T^{\prime})^{-1}(\kappa(P^{\prime}_{\ell})\otimes_{R}R^{e})$, in particular,
the structure of the set of its zero divisors, as $R^{\prime}$ runs over
${\cal T}(R)$ (here $T^{\prime}$ is as in Remark 4.2). Since $R^{e}$ is
separable algebraic, essentially of finite type over $R$, the ring
$(T^{\prime})^{-1}(\kappa(P^{\prime}_{\ell})\otimes_{R}R^{e})$ is finite over
$\kappa(P^{\prime}_{\ell})$; this ring is reduced, but it may contain zero
divisors. In fact, it is a direct product of fields which are finite separable
extensions of $\kappa(P^{\prime}_{\ell})$ because $R^{e}$ is separable and
essentially of finite type over $R$.
Consider a chain $R\rightarrow R^{\prime}\rightarrow R^{\prime\prime}$ of
$\nu$-extensions in $\cal T$. Let
$\displaystyle\kappa(P_{\ell})\otimes_{R}R^{e}$ $\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{n}K_{j}$ (128)
$\displaystyle(T^{\prime})^{-1}\left(\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}R^{e}\right)$
$\displaystyle=$ $\displaystyle\prod\limits_{j=1}^{n^{\prime}}K^{\prime}_{j}$
(129)
$\displaystyle(T^{\prime\prime})^{-1}\left(\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}R^{e}\right)$
$\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{n^{\prime\prime}}K^{\prime\prime}_{j}$ (130)
be the corresponding decompositions as products of finite field extensions of
$\kappa(P_{\ell})$ (resp. $\kappa(P^{\prime}_{\ell})$, resp.
$\kappa(P^{\prime\prime}_{\ell})$). We want to compare
$(T^{\prime})^{-1}\left(\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}R^{e}\right)$
with
$(T^{\prime\prime})^{-1}\left(\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}R^{e}\right)$.
###### Remark 7.5
The ring $\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}R^{e}$ is itself a
direct product of finite extensions of $\kappa\left(P^{\prime}_{\ell}\right)$;
say $\kappa\left(P^{\prime}_{\ell}\right)=\prod\limits_{j\in
S^{\prime}}K^{\prime}_{j}$ for a certain set $S^{\prime}$. In this situation,
localization is the same thing as the natural projection to the product of the
$K^{\prime}_{j}$ over a certain subset $\\{1,\dots,n^{\prime}\\}$ of
$S^{\prime}$. Thus the passage from
$(T^{\prime})^{-1}\left(\kappa\left(P^{\prime}_{\ell}\right)\otimes_{R}R^{e}\right)$
to
$(T^{\prime\prime})^{-1}\left(\kappa\left(P^{\prime\prime}_{\ell}\right)\otimes_{R}R^{e}\right)$
can be viewed as follows: first, tensor each $K^{\prime}_{j}$ with
$\kappa\left(P^{\prime\prime}_{\ell}\right)$ over
$\kappa\left(P^{\prime}_{\ell}\right)$; then, in the resulting direct product
of fields, remove a certain number of factors.
Let $\bar{K}^{\prime}_{1},\dots,\bar{K}^{\prime}_{\bar{n}^{\prime}}$ be the
distinct isomorphism classes of finite extensions of
$\kappa\left(P^{\prime}_{\ell}\right)$ appearing among
$K^{\prime}_{1},\dots,K^{\prime}_{n^{\prime}}$, arranged in such a way that
$\left[\bar{K}^{\prime}_{j}:\kappa\left(P^{\prime}_{\ell}\right)\right]$ is
non-increasing with $j$, and similarly for
$\bar{K}^{\prime\prime}_{1},\dots,\bar{K}^{\prime\prime}_{\bar{n}^{\prime\prime}}$.
###### Lemma 7.6
We have the inequality
$\left(\left[\bar{K}^{\prime\prime}_{1}:\kappa\left(P^{\prime\prime}_{\ell}\right)\right],\dots,\left[\bar{K}^{\prime\prime}_{\bar{n}^{\prime\prime}}:\kappa\left(P^{\prime\prime}_{\ell}\right)\right],n^{\prime\prime}\right)\leq\left(\left[\bar{K}^{\prime}_{1}:\kappa\left(P^{\prime}_{\ell}\right)\right],\dots,\left[\bar{K}^{\prime}_{\bar{n}^{\prime}}:\kappa\left(P^{\prime}_{\ell}\right)\right],n^{\prime}\right)$
(131)
in the lexicographical ordering. Furthermore, either $R^{\prime}$ is
$\ell$-stable or there exists $R^{\prime\prime}\in\cal T$ such that strict
inequality holds in (131).
Proof.- Fix a $q\in\\{1,\dots,\bar{n}^{\prime}\\}$ and consider the tensor
product
$\bar{K}^{\prime}_{q}\otimes_{R}\kappa\left(P^{\prime\prime}_{\ell}\right)$.
Since $\bar{K}^{\prime}_{q}$ is separable over
$\kappa\left(P^{\prime}_{\ell}\right)$, the ring
$\bar{K}^{\prime}_{q}\otimes_{R}\kappa\left(P^{\prime\prime}_{\ell}\right)=\prod\limits_{j\in
S^{\prime\prime}_{q}}K^{\prime\prime}_{j}$ is a product of fields. Moreover,
two cases are possible:
(a) there exists a non-trivial extension $L$ of
$\kappa\left(P^{\prime}_{\ell}\right)$ which embeds both into
$\kappa\left(P^{\prime\prime}_{\ell}\right)$ and $\bar{K}^{\prime}_{q}$. In
this case
$\left[K^{\prime\prime}_{j}:\kappa\left(P^{\prime\prime}_{\ell}\right)\right]<\left[\bar{K}^{\prime}_{q}:\kappa\left(P^{\prime}_{\ell}\right)\right]\quad\text{
for all }j\in S^{\prime\prime}_{q}.$ (132)
(b) there is no field extension $L$ as in (a). In this case
$\bar{K}^{\prime}_{q}\otimes_{R}\kappa\left(P^{\prime\prime}_{\ell}\right)$ is
a field, so
$\\#S^{\prime\prime}_{q}=1$ (133)
and
$\left[K^{\prime\prime}_{j}:\kappa\left(P^{\prime\prime}_{\ell}\right)\right]=\left[\bar{K}^{\prime}_{q}:\kappa\left(P^{\prime}_{\ell}\right)\right]\quad\text{
for }j\in S^{\prime\prime}_{q}.$ (134)
Now, if there exists $q\in\\{1,\dots,\bar{n}^{\prime}\\}$ for which (a) holds,
take the smallest such $q$. Then (132)–(134) imply that strict inequality
holds in (131). On the other hand, if (b) holds for all
$q\in\\{1,\dots,\bar{n}^{\prime}\\}$ then (133) and (134) imply that
$\left(\left[\bar{K}^{\prime\prime}_{1}:\kappa\left(P^{\prime\prime}_{\ell}\right)\right],\dots,\left[\bar{K}^{\prime\prime}_{\bar{n}^{\prime\prime}}:\kappa\left(P^{\prime\prime}_{\ell}\right)\right]\right)=\left(\left[\bar{K}^{\prime}_{1}:\kappa\left(P^{\prime}_{\ell}\right)\right],\dots,\left[\bar{K}^{\prime}_{\bar{n}^{\prime}}:\kappa\left(P^{\prime}_{\ell}\right)\right]\right)$
(135)
and $n^{\prime\prime}\leq n^{\prime}$, so again (131) holds.
Finally, assume that $R^{\prime}$ is not $\ell$-stable. If there exists
$R^{\prime\prime}\in\cal T$ and $q\in\\{1,\dots,\bar{n}^{\prime}\\}$ for which
(a) holds, then by the above we have strict inequality in (131) and there is
nothing more to prove. Assume there are no such $R^{\prime\prime}$ and $q$.
Then $(T^{\prime})^{-1}(\kappa(P^{\prime}_{\ell})\otimes_{R}R^{e})$ is not a
domain, so $n^{\prime}>1$.
Take $R^{\prime\prime}\in{\cal T}(R^{\prime})$ such that
$\left(\frac{R^{\prime\prime}}{P^{\prime\prime}_{l}}\otimes_{R}R^{e}\right)_{M^{\prime\prime}}$
is an integral domain; such an $R^{\prime\prime}$ exists by Lemma 7.3. Then
$n^{\prime\prime}=1<n^{\prime}$, as desired.
###### Corollary 7.7
There exists a stable $R^{\prime}\in\cal T$. The limit in (48) is attained for
this $R^{\prime}$.
Proof.- In view of Proposition 4.4, it is sufficient to prove that there
exists $R^{\prime}\in\cal T$ which which is $\ell$-stable for all
$\ell\in\\{0,1,\dots,r\\}$. First, we fix $\ell\in\\{0,1,\dots,r\\}$. Lemma
7.6 implies that there exists $R^{\prime}\in\mathcal{T}(R)$ which is
$\ell$-stable.
By Proposition 4.4, repeating the procedure above for each $\ell$ we can
successively enlarge $R^{\prime}$ in such a way that it becomes stable.
The last statement follows from Proposition 4.5.
We are now in the position to prove Theorem 7.1.
By Theorem 4.9 (1), $H_{2\ell-1}$ is prime. By Proposition 3.5, $H_{2\ell+1}$
maps to $P_{\ell}$ under the map $\pi^{e}:\mbox{Spec}\
R^{e}\rightarrow\mbox{Spec}\ R$. Since this map is étale, its fibers are zero-
dimensional, which shows that $H_{2\ell+1}$ is a minimal prime of $P_{\ell}$.
This proves (1) of Theorem 7.1.
By Proposition 5.4, for $0\leq i\leq 2r$, $\tilde{H}_{i}$ is a prime ideal of
$R^{e}$, containing $H_{i}$. Since the fibers of $\pi^{e}$ are zero-
dimensional, we must have $\tilde{H}_{i}=H_{i}$, so
$\tilde{H}_{2\ell}=\tilde{H}_{2\ell+1}=H_{2\ell}=H_{2\ell+1}$ for
$0\leq\ell\leq r$. In particular, $\tilde{H}_{0}=H_{1}$. This shows that the
unique prime $\tilde{H}_{0}$ of $R^{e}$ such that there exists an extension
$\nu^{e}_{-}$ of $\nu$ to $\frac{R^{e}}{\tilde{H}_{0}}$ is
$\tilde{H}_{0}=H_{1}$. Now (2) of the Theorem is given by Proposition 6.12.
(3) of Theorem 7.1 is now immediate. This completes the proof of Theorem 7.1.
We note the following corollary of the proof of (2) of Theorem 7.1 and
Corollary 6.15. Let $\Phi^{e}=\nu^{e}_{-}(R^{e}\setminus\\{0\\})$, take an
element $\beta\in\Phi^{e}$ and let ${\cal P}^{e}_{\beta}$ denote the
$\nu^{e}_{-}$-ideal of $R^{e}$ of value $\beta$.
###### Corollary 7.8
Take an element $x\in{\cal P}^{e}_{\beta}$. There exists a local blowing up
$R\rightarrow R^{\prime}$ such that $\beta\in\nu(R^{\prime})\setminus\\{0\\}$
and $x\in{\cal P}^{\prime}_{\beta}{R^{\prime}}^{e}$.
## 8 The Main Theorem: the primality of implicit ideals.
In this section we study the ideals $H_{j}$ for $\hat{R}$ instead of
$\tilde{R}$. The main result of this section is
###### Theorem 8.1
The ideal $H_{2\ell-1}$ is prime.
Proof.- For the purposes of this proof, let $H_{2\ell-1}$ denote the implicit
ideals of $\hat{R}$ and $\tilde{H}_{2\ell-1}$ the implicit prime ideals of the
henselization $\tilde{R}$ of $R$.
Let $S$ be a local domain. By [11] (Theorem (43.20), p. 187) there exists
bijective maps between the set of minimal prime ideals of the henselization
$\tilde{S}$ and the maximal ideals of the normalization $S^{n}$. If, in
addition, $S$ is excellent, the two above sets also admit a natural bijection
to the set of minimal primes of $\hat{S}$ [11] (Corollary (44.3), p. 189). If
$S$ is a henselian local domain, its only minimal prime is the (0) ideal,
hence by the above the same is true of $\hat{S}$. Thus $\hat{S}$ is also a
domain.
This shows that any excellent henselian local domain is analytically
irreducible, hence $\tilde{H}_{2\ell-1}\hat{R}$ is prime for all
$\ell\in\\{1,\dots,r+1\\}$. Let $\tilde{\nu}_{-}$ denote the unique extension
of $\nu$ to $\frac{\tilde{R}}{\tilde{H}_{1}}$, constructed in the previous
section. Let $H^{*}_{2\ell-1}\subset\frac{\tilde{R}}{\tilde{H}_{1}}$ denote
the implicit ideals associated to the henselian ring
$\frac{\tilde{R}}{\tilde{H}_{1}}$ and the valuation $\tilde{\nu}_{-}$.
Claim. We have $H^{*}_{2\ell-1}=\frac{H_{2\ell-1}}{\tilde{H}_{1}}$.
Proof of the claim: For $\beta\in\Gamma$, let $\tilde{P}_{\beta}$ denote the
$\tilde{\nu}_{-}$-ideal of $\frac{\tilde{R}}{\tilde{H}_{1}}$ of value $\beta$.
For all $\beta$, we have
$\frac{P_{\beta}}{\tilde{H}_{1}}\subset\tilde{P}_{\beta}$, and the same
inclusion holds for all the local blowings up of $R$, hence
$\frac{H_{2\ell-1}}{\tilde{H}_{1}}\subset H^{*}_{2\ell-1}$. To prove the
opposite inclusion, we may replace $\tilde{R}$ by a finitely generated strict
étale extension $R^{e}$ of $R$. Now let
$\Phi^{e}=\nu^{e}_{-}\left(R^{e}\setminus\\{0\\}\right)$ and take an element
$\beta\in\Phi^{e}\cap\Delta_{\ell-1}$. By Corollary 7.8, there exists a local
blowing up $R\rightarrow R^{\prime}$ such that $x\in
P^{\prime}_{\beta}{R^{\prime}}^{e}$. Letting $\beta$ vary over
$\Phi^{e}\cap\Delta_{\ell-1}$, we obtain that if $x\in H^{*}_{2\ell-1}$ then
$x\in\frac{H_{2\ell-1}}{\tilde{H}_{1}}$, as desired. This completes the proof
of the claim.
The Claim shows that replacing $R$ by $\frac{\tilde{R}}{\tilde{H}_{1}}$ in
Theorem 8.1 does not change the problem. In other words, we may assume that
$R$ is a henselian domain and, in particular, that $\hat{R}$ is also a domain.
Similarly, the ring
$\frac{R}{P_{\ell}}\otimes_{R}\hat{R}\cong\frac{\hat{R}}{P_{\ell}}$ is a
domain, hence so is its localization $\kappa(P_{\ell})\otimes_{R}\hat{R}$.
Since $R$ is a henselian excellent ring, it is algebraically closed in
$\hat{R}$ ([11], Corollary (44.3), p. 189 and Corollary .20 of the Appendix);
of course, the same holds for $\frac{R}{P_{\ell}}$ for all $\ell$. Then
$\kappa(P_{\ell})$ is algebraically closed in
$\kappa(P_{\ell})\otimes_{R}\hat{R}$. This shows that the ring $R$ is stable.
Now the Theorem follows from Theorem 4.9. This completes the proof of Theorem
8.1.
## 9 Towards a proof of Conjecture 1.11, assuming local uniformization in
lower dimension
Let the notation be as in the previous sections. In this section, we assume
that the Local Uniformization Theorem holds and propose an approach to proving
Conjecture 1.11. We prove a Corollary of Conjecture 1.11 which gives a
sufficient condition for $\hat{\nu}_{-}$ to be unique, which also turns out to
be necessary under the additional assumption that $\hat{\nu}_{-}$ is minimal.
We will assume that all the $R^{\prime}\in\mathcal{T}$ are birational to each
other, so that all the fraction fields $K^{\prime}=K$ and the homomorphisms
$R^{\prime}\rightarrow R^{\prime\prime}$ are local blowings up with respect to
$\nu$. Finally, we assume that $R$ contains a field $k_{0}$ and a local
$k_{0}$-subalgebra $S$ essentailly of finite type, over which $R$ is strictly
étale. In particular, all the rings in sight are equicharacteristic.
First, we state the Local Uniformization Theorem in the precise form in which
we are going to use it.
###### Definition 9.1
We say that the embedded Local Uniformization theorem holds in $\mathcal{T}$
if the following conditions are satisfied.
Take an integer $\ell\in\\{1,\dots,r-1\\}$. Let
$\mu_{\ell+1}:=\nu_{\ell+1}\circ\nu_{\ell+2}\circ\dots\circ\nu_{r}$. Consider
a tree $\\{H^{\prime}\\}$ of prime ideals of
$\frac{\hat{R}^{\prime}}{P^{\prime}_{\ell}}$ such that
$H^{\prime}\cap\frac{R^{\prime}}{P^{\prime}_{\ell}}=(0)$ and a tight extension
$\hat{\mu}_{2\ell+2}$ of $\mu_{\ell+1}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}}$.
(1) There exists a local blowing up $\pi:R\rightarrow R^{\prime}$ in
$\mathcal{T}$, which induces an isomorphism at the center of $\nu_{\ell}$,
such that $\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is a regular local ring.
(2) Assume that $\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is a regular local
ring. Then there exists in $\mathcal{T}$ a sequence $\pi:R\rightarrow
R^{\prime}$ of local blowings up along non-singular centers not containing the
center of $\nu_{l}$ such that $\frac{\hat{R}^{\prime}}{H^{\prime}}$ is a
regular local ring.
It is well known ([1], [9], [17]) that the embedded Local Uniformization
theorem holds if $R$ is an excellent local domain such that either $char\ k=0$
or $\dim\ R\leq 3$ (to be precise, (1) of Definition 9.1 is well known and (2)
is an easy consequence of known results). While the Local Uniformization
theorem in full generality is still an open problem, it is widely believed to
hold for arbitrary quasi-excellent local domains. Proving this is an active
field of current research in algebraic geometry. Proving local uniformization
for rings of arbitrary characteristic is one of the intended applications of
Conjecture 1.11. Note that in Definition 9.1 we require only local
uniformization of rings of dimension strictly less than $\dim\ R$; the idea is
to use induction on $\dim\ R$ to prove local uniformization of rings of
dimension $\dim\ R$.
We begin by stating a strengthening of Conjecture 1.11 (using Remark 6.14):
###### Conjecture 9.2
The valuation $\nu$ admits at least one tight extension $\hat{\nu}_{-}$. This
tight extension $\hat{\nu}_{-}$ can be chosen to have the following additional
property: for rings $R^{\prime}$ sufficiently far in the tree $\mathcal{T}$ we
have the equality of semigroups
$\hat{\nu}_{-}\left(\frac{\hat{R}^{\prime}}{\tilde{H}^{\prime}_{0}}\setminus\\{0\\}\right)=\nu(R^{\prime}\setminus\\{0\\})$
and for $\beta\in\nu(R^{\prime}\setminus\\{0\\})$ the $\hat{\nu}_{-}$-ideal of
value $\beta$ is
$\frac{\mathcal{P}_{\beta}\hat{R}^{\prime}}{\tilde{H}^{\prime}_{0}}$. In
particular, we have the equality of graded algebras
$\mbox{gr}_{\nu}R^{\prime}=\mbox{gr}_{\hat{\nu}_{-}}\frac{\hat{R}^{\prime}}{\tilde{H}^{\prime}_{0}}$.
Below, we give an explicit construction of a valuation $\hat{\nu}_{-}$ whose
existence is asserted in the Conjecture by describing the trees of ideals
$\tilde{H}^{\prime}_{i}$, $0\leq i\leq 2r$ and, for each $i$, a valuations
$\hat{\nu}_{i}$ of the residue field $k_{\nu_{i-1}}$, such that
$\hat{\nu}_{-}=\hat{\nu}_{1}\circ\dots\hat{\nu}_{2r}$. More precisely, for
$\ell\in\\{0,\dots,r-1\\}$, we will construct, recursively in the descending
order of $\ell$, a tree $J^{\prime}_{2\ell+1}$ of prime ideals of
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell}}$, $R^{\prime}\in\mathcal{T}$,
such that $J^{\prime}_{2\ell+1}\cap\frac{R^{\prime}}{P_{\ell}}=(0)$, and an
extension $\hat{\mu}_{2\ell+2}$ of $\mu_{\ell+1}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}\in\mathcal{T}}}\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}\hat{R}^{\prime}}$;
the valuation $\hat{\mu}_{2}$ will be our candidate for the desired tight
extension $\hat{\nu}_{-}$ of $\mu_{1}=\nu$. Unfortunately, two steps in this
construction still remain conjectural, namely, proving that
$\hat{\mu}_{2\ell+2}$ is, indeed, a valuation, and that it is tight (this is
essentially the content of Conjectures 9.7 and 9.8 below). Once these
conjectures are proved, our recursive construction will be complete and
Conjecture 9.2 will follow by setting $\hat{\nu}_{-}=\hat{\mu}_{2}$.
Let us now describe the construction in detail. According to Corollary 6.3, we
may assume that $ht\ H^{\prime}_{i}$ is constant for each $i$ after replacing
$R$ by some other ring sufficiently far in $\mathcal{T}$. From now on, we will
make this assumption without always stating it explicitly.
By (1) of Definition 9.1, applied successively to the trees of ideals
$P^{\prime}_{\ell}\subset R^{\prime},\quad\ell\in\\{1,\dots,r-1\\},$
there exists $R^{\prime\prime}\in\cal T$ such that
$\frac{R^{\prime\prime}}{P^{\prime\prime}_{\ell}}$ is regular for all
$\ell\in\\{1,\dots,r-1\\}$. Without loss of generality, we may also assume
that $R^{\prime\prime}$ is stable.
For $\ell\in\\{1,\dots,r-1\\}$ and $R^{\prime\prime}\in\mathcal{T}$, let
$\mathcal{T}_{\ell}(R^{\prime\prime})$ denote the subtree of $\mathcal{T}$,
consisting of all the local blowings up of $R^{\prime\prime}$ along ideals not
contained in $P^{\prime\prime}_{\ell}$ (such local blowings up induce an
isomorphism at the point $P^{\prime\prime}_{\ell}\in\mbox{Spec}\
R^{\prime\prime}$). Below, we will sometimes work with trees of rings and
ideals indexed by $\mathcal{T}_{\ell}(R^{\prime\prime})$ for suitable $\ell$
and $R^{\prime\prime}$ (instead of trees indexed by all of $\cal T$); the
precise tree with which we are working will be specified in each case.
For $\ell=r-1$, we define $J^{\prime}_{2r-1}:=H^{\prime}_{2r-1}$ and
$\hat{\mu}_{2r}:=\hat{\nu}_{2r,0}$; according to Proposition 4.13,
$\hat{\nu}_{2r,0}=\hat{\nu}_{2r}$ is the unique extension of $\nu_{r}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{2r-1}\hat{R}^{\prime}}$.
Next, assume that $\ell\in\\{1,\dots,r-1\\}$, that the tree
$J^{\prime}_{2\ell+1}$ of prime ideals of
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell}\hat{R}^{\prime}}$ and a tight
extension $\hat{\mu}_{2\ell+2}$ of $\mu_{\ell+1}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}$
are already constructed for $R^{\prime}\in\mathcal{T}$ and that
$J^{\prime}_{2\ell+1}\cap\frac{R^{\prime}}{P_{\ell}}=(0)$. It remains to
construct the ideals
$J^{\prime}_{2\ell-1}\subset\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}\hat{R}^{\prime}}$
and a tight extension $\hat{\mu}_{2\ell}$ of $\mu_{\ell}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell-1}}$
for $R^{\prime}\in\mathcal{T}$.
We will assume, inductively, that for all $R^{\prime}\in\mathcal{T}$ the
quantity $ht\ J^{\prime}_{2\ell+1}$ is constant and the following conditions
hold:
1. 1.
We have the equality of semigroups
$\hat{\mu}_{2\ell+2}\left(\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}\setminus\\{0\\}\right)\cong\mu_{\ell+1}\left(\frac{R^{\prime}}{P^{\prime}_{\ell}}\setminus\\{0\\}\right)$.
2. 2.
For all
$\beta\in\mu_{\ell+1}\left(\frac{R^{\prime}}{P^{\prime}_{\ell}}\setminus\\{0\\}\right)$
the $\hat{\mu}_{2\ell+2}$-ideal of
$\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}$ of value $\beta$ is the
extension to $\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}$ of the
$\mu_{\ell+1}$-ideal of $\frac{R^{\prime}}{P^{\prime}_{\ell}}$ of value
$\beta$.
3. 3.
In particular, we have a canonical isomorphism
$gr_{\hat{\mu}_{2\ell+2}}\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}\cong
gr_{\mu_{\ell+1}}\frac{R^{\prime}}{P^{\prime}_{\ell}}$ of graded algebras.
By (2) of Definition 9.1 applied to the prime ideals
$J^{\prime}_{2\ell+1}\subset\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell}\hat{R}^{\prime}}$,
there exists $R^{\prime}\in\cal T$ such that both
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$ and
$\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}$ are regular. The fact that
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is regular implies that so is
$\frac{\hat{R}^{\prime}}{P^{\prime}_{\ell}\hat{R}^{\prime}}$. In particular,
$\frac{\hat{R}^{\prime}}{P^{\prime}_{\ell}\hat{R}^{\prime}}$ is a domain, so
$H^{\prime}_{2\ell}=P^{\prime}_{\ell}\hat{R}^{\prime}$. Take a regular system
of parameters
$\bar{u}^{\prime}=(\bar{u}^{\prime}_{1},\dots,\bar{u}^{\prime}_{n_{\ell}})$
of $\frac{R^{\prime}}{P^{\prime}_{\ell}}$. Let $k^{\prime}$ denote the common
residue field of $R^{\prime}$, $\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ and
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$. Fix an isomorphism
$\frac{R^{\prime}}{P^{\prime}_{\ell}}\cong k^{\prime}[[\bar{u}^{\prime}]]$.
Renumbering the variables, if necessary, we may assume that there exists
$s_{\ell}\in\\{1,\dots,n_{\ell}\\}$ such that
$\bar{u}^{\prime}_{1},\dots,\bar{u}^{\prime}_{s_{\ell}}$ are
$k^{\prime}$-linearly independent modulo
$({m^{\prime}}^{2}+J^{\prime}_{2\ell+1})\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell}}$.
Since $\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}}$ is regular, the ideal
$J^{\prime}_{2\ell+1}$ is generated by a set of the form
$\bar{v}^{\prime}=(\bar{v}^{\prime}_{s_{\ell}+1},\dots,\bar{v}^{\prime}_{n_{\ell}})$,
where
$\bar{v}^{\prime}_{j}=\bar{u}^{\prime}_{j}-\bar{\phi}_{j}(\bar{u}^{\prime}_{1},\dots,\bar{u}^{\prime}_{s_{\ell}}),\
\bar{\phi}_{j}(\bar{u}^{\prime}_{1},\dots,\bar{u}^{\prime}_{s_{\ell}})\in
k^{\prime}[[\bar{u}^{\prime}_{1},\dots,\bar{u}^{\prime}_{s_{\ell}}]].$
Let
$\bar{w}^{\prime}=(\bar{w}^{\prime}_{1},\dots,\bar{w}^{\prime}_{s_{\ell}})=(\bar{u}^{\prime}_{1},\dots,\bar{u}^{\prime}_{s_{\ell}})$.
Let $z^{\prime}$ be a minimal set of generators of
$\frac{P^{\prime}_{\ell}}{P^{\prime}_{\ell-1}}$. Let $k^{\prime}_{0}$ be a
quasi-coefficient field of $R^{\prime}$ (that is, a subfield of $R^{\prime}$
over which $k^{\prime}$ is formally étale; such a quasi-coefficient field
exists by [10], moreover, since $R^{\prime}$ is algebraic over a finite type
algebra over a field by hypotheses, $k^{\prime}$ is finite over
$k^{\prime}_{0}$). By the hypotheses on $R$ and since
$\frac{R^{\prime}}{P^{\prime}_{\ell}}$ is a regular local ring and
$\bar{u}^{\prime}$ is a minimal set of generators of its maximal ideal
$\frac{m^{\prime}}{P^{\prime}_{\ell}}$, there exists an ideal $I\subset
k^{\prime}_{0}[z^{\prime}]$ such that $\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$
is an étale extension of
$\frac{k^{\prime}_{0}[z^{\prime},\bar{u}^{\prime}]_{(z^{\prime},\bar{u}^{\prime})}}{I}$.
By assumptions, we have $ht\ P^{\prime}_{\ell-1}<ht\ P^{\prime}_{\ell}$, so
$0<ht\ P^{\prime}_{\ell}-ht\ P^{\prime}_{\ell-1}=ht(z^{\prime})-ht\ I$, in
other words,
$ht\ I<ht(z^{\prime}).$ (136)
Next, we prove two general lemmas about ring extensions.
###### Notation 9.3
Let $k_{0}$ be a field and $(S,m,k)$ a local noetherian $k_{0}$-algebra. For a
field extension
$k_{0}\hookrightarrow L$ (137)
such that $k\otimes_{k_{0}}L$ is a domain, let $S(L)$ denote the localization
of the ring $S\otimes_{k_{0}}L$ at the prime ideal $m(S\otimes_{k_{0}}L)$.
###### Lemma 9.4
Let $k_{0}$, $(S,m,k)$ and $L$ be as above. The ring $S(L)$ is noetherian.
Proof.- If the field extension (137) is finitely generated, the Lemma is
obvious. In the general case, write $L=\lim\limits_{\overrightarrow{i}}L_{i}$
as a direct limit of its finitely generated subextensions. For each $L_{i}$,
let $k_{i}$ denote the residue field of $S(L_{i})$; $k_{i}$ is nothing but the
field of fractions of $k\otimes_{k_{0}}L_{i}$. Write
$\hat{S}=\frac{k[[x]]}{H}$, where $x$ is a set of generators of $m$ and $H$ a
certain ideal of $k[[x]]$. Then
$\widehat{S(L_{i})}\cong\frac{k_{i}[[x]]}{Hk_{i}[[x]]}$. Given two finitely
generated extensions $L_{i}\subset L_{j}$ of $k_{0}$, contained in $L$, we
have a commutative diagram
$\begin{matrix}S(L_{j})&\overset{\pi_{j}}{\rightarrow}&\widehat{S(L_{j})}&\cong&\frac{k_{j}[[x]]}{Hk_{j}[[x]]}&\\\
\psi_{ij}\uparrow&\ &\uparrow&\ &\uparrow\phi_{ij}\\\
S(L_{i})&\overset{\pi_{i}}{\rightarrow}&\widehat{S(L_{i})}&\cong&\frac{k_{i}[[x]]}{Hk_{i}[[x]]}\end{matrix}$
where $\phi_{ij}$ is the map induced by the natural inclusion
$k_{i}\hookrightarrow k_{j}$ and the identity map of $x$ to itself. Let
$k_{\infty}=\lim\limits_{\overrightarrow{i}}k_{i}$. Then, for each $i$, we
have the obvious faithfully flat map
$\rho_{i}:\widehat{S(L_{i})}\rightarrow\frac{k_{\infty}[[x]]}{Hk_{\infty}[[x]]}$,
defined by the natural inclusion $k_{i}\hookrightarrow k_{\infty}$ and the
identity map of $x$ to itself; the maps $\rho_{i}$ commute with the
$\phi_{ij}$. Thus, we have constructed a faithfully flat map
$\rho_{i}\circ\pi_{i}$ from each element of the direct system $S(L_{i})$ to
the fixed noetherian ring $\frac{k_{\infty}[[x]]}{Hk_{\infty}[[x]]}$;
moreover, the maps $\rho_{i}\circ\pi_{i}$ are compatible with the
homomorphisms $\psi_{ij}$ of the direct system. This implies that the ring
$S(L)=\lim\limits_{\overrightarrow{i}}S(L_{i})$ is noetherian.
###### Lemma 9.5
Let $(S,m,k)$ be a local noetherian ring. Let $t$ be an arbitrary collection
of independent variables. Consider the rings $S[t]$ and $S(t):=S[t]_{mS[t]}$.
Let $I$ be an ideal of $S$. Then
$IS(t)\cap S[t]=IS[t].$ (138)
Proof.- First, assume the collection $t$ consists of a single variable.
Consider elements $f,g\in S[t]$ such that
$f\mbox{$\in$ /}mS[t]$ (139)
and
$fg\in IS[t].$ (140)
Proving the equation (138) amounts to proving that
$g\in IS[t].$ (141)
We prove (141) by contradiction. Assume that $g\mbox{$\in$ /}IS[t]$. Then
there exists $n\in\mathbb{N}$ such that $g\mbox{$\in$ /}(I+m^{n})S[t]$. Take
the smallest such $n$, so that
$g\in\left(I+m^{n-1}\right)S[t]\setminus(I+m^{n})S[t].$ (142)
Write $f=\sum\limits_{j=0}^{q}a_{j}t^{j}$ and
$g=\sum\limits_{j=0}^{l}b_{j}t^{j}$. Let
$\displaystyle l_{0}:$ $\displaystyle=$
$\displaystyle\max\\{j\in\\{0,\dots,l\\}\ |\ b_{j}\mbox{$\in$
/}I+m^{n}\\}\quad\text{ and}$ (143) $\displaystyle q_{0}:$ $\displaystyle=$
$\displaystyle\max\\{j\in\\{0,\dots,q\\}\ |\ a_{j}\mbox{$\in$ /}m\\}.$ (144)
Let $c_{l_{0}+q_{0}}$ denote the $(l_{0}+q_{0})$-th coefficient of $fg$. We
have
$c_{l_{0}+q_{0}}=\sum\limits_{i+j=l_{0}+q_{0}}a_{i}b_{j}=a_{q_{0}}b_{l_{0}}+\sum\limits_{\begin{array}[]{c}i+j=l_{0}+q_{0}\\\
i>q_{0}\end{array}}a_{i}b_{j}+\sum\limits_{\begin{array}[]{c}i+j=l_{0}+q_{0}\\\
j>l_{0}\end{array}}a_{i}b_{j}.$
By definition of $l_{0}$ and $q_{0}$ and (142) we have:
$\displaystyle a_{q_{0}}b_{l_{0}}$ $\in$ / $\displaystyle
I+m^{n}\quad\text{and}$ (145)
$\displaystyle\sum\limits_{\begin{array}[]{c}i+j=l_{0}+q_{0}\\\
i>q_{0}\end{array}}a_{i}b_{j}+\sum\limits_{\begin{array}[]{c}i+j=l_{0}+q_{0}\\\
j>l_{0}\end{array}}a_{i}b_{j}$ $\displaystyle\in$ $\displaystyle I+m^{n}.$
(150)
Hence $c_{l_{0}+q_{0}}\mbox{$\in$ /}I+m^{n}$, which contradicts (140). This
completes the proof of Lemma 9.5 in the case when $t$ is a single variable.
The case of a general $t$ now follows by transfinite induction on the
collection $t$.
###### Lemma 9.6
There exist sets of representatives
$u^{\prime}=(u^{\prime}_{1},\dots,u^{\prime}_{n_{\ell}})$
of $\bar{u}^{\prime}$ and $\phi_{j}$ of $\bar{\phi}_{j}$, $s_{\ell}<j\leq
n_{\ell}$, in $\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}\hat{R}^{\prime}}$,
having the following properties. Let
$\displaystyle w^{\prime}=(w^{\prime}_{1},\dots,w^{\prime}_{s_{\ell}})$
$\displaystyle=$ $\displaystyle(u^{\prime}_{1},\dots,u^{\prime}_{s_{\ell}}),$
(151) $\displaystyle
v^{\prime}=(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{n_{\ell}})$
$\displaystyle=$
$\displaystyle(u^{\prime}_{s_{\ell}+1}-\phi_{s_{\ell}+1},\dots,u^{\prime}_{n_{\ell}}-\phi_{n_{\ell}}).$
(152)
Let
$J^{\prime}_{2\ell-1}=\frac{H^{\prime}_{2\ell-1}}{H^{\prime}_{2\ell-2}}+(v^{\prime})\subset\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$.
Then
$w^{\prime}\subset\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ (153)
and
$J^{\prime}_{2\ell-1}\cap\frac{R^{\prime}}{P^{\prime}_{\ell-1}}=(0).$ (154)
Proof.-(of Lemma 9.6) There is no problem choosing $w^{\prime}$ to satisfy
(153).
As for (154), we first prove the Lemma under the assumption that $k$ is
countable. We choose the representatives $u^{\prime}$ arbitrarily and let
$\bar{\phi}_{j}(u^{\prime})\in k[[u^{\prime}]]$ denote the formal power series
obtained by substituting $u^{\prime}$ for $\bar{u}^{\prime}$ in
$\bar{\phi}_{j}$. Any representative $\phi_{j}$ of $\bar{\phi}_{j}$,
$s_{\ell}<j\leq n_{\ell}$ has the form
$\phi_{j}=\bar{\phi}_{j}(u^{\prime})+h_{j}$ with
$h_{j}\in(z^{\prime})\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$. We define
the $h_{j}$ required in the Lemma recursively in $j$. Take
$j\in\\{s_{\ell}+1,\dots,n_{\ell}\\}$. Assume that
$h_{s_{\ell+1}},\dots,h_{j-1}$ are already defined and that
$(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})\cap\frac{R^{\prime}}{P^{\prime}_{\ell-1}}=(0),$
(155)
where we view $\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ as a subring of
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}}$. Since the ring
$\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ is countable, there are countably
many ideals in
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$,
not contained in
$\frac{(z^{\prime})\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$,
which are minimal primes of ideals of the form
$(f)\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$,
where $f$ is a non-zero element of $\frac{m^{\prime}}{P^{\prime}_{\ell-1}}$.
Let us denote these ideals by $\\{I_{q}\\}_{q\in\mathbb{N}}$; we have
$ht\ I_{q}=1\quad\text{ for all }q\in\mathbb{N}.$ (156)
We note that
$\frac{(z^{\prime})\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}\not\subset
I_{q}\quad\text{ for all }q\in\mathbb{N}.$ (157)
Indeed, by (136) and (155) we have $ht\
\frac{(z^{\prime})\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}\geq
1$. In view of (156), containment in (157) would imply equality, which
contradicts the definition of $I_{q}$.
Since $H^{\prime}_{2\ell-1}\subsetneqq H^{\prime}_{2\ell}$ and
$J^{\prime}_{2\ell+1}\subsetneqq\frac{m^{\prime}\hat{R}^{\prime}}{H^{\prime}_{2\ell}}$,
we have
$\dim\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}\geq(ht\
H^{\prime}_{2\ell}-ht\ H^{\prime}_{2\ell-1})+ht\
\frac{m^{\prime}\hat{R}^{\prime}}{H^{\prime}_{2\ell}}-(j-s_{\ell}-1)\geq$
$(ht\ H^{\prime}_{2\ell}-ht\ H^{\prime}_{2\ell-1})+ht\
\frac{m^{\prime}\hat{R}^{\prime}}{H^{\prime}_{2\ell}}-ht\
J^{\prime}_{2\ell+1}+1\geq 3.$ (158)
Let $\tilde{u}_{j}$ denote the image of
$u^{\prime}_{j}-\bar{\phi}_{j}(u^{\prime})$ in
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$.
Next, we construct an element
$\tilde{h}_{j}\in\frac{(z^{\prime})\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$
such that
$\tilde{u}_{j}-\tilde{h}_{j}\mbox{$\in$ /}\bigcup\limits_{q=1}^{\infty}I_{q}.$
(159)
The element $\tilde{h}_{j}$ will be given as the sum of an infinite series
$\sum\limits_{t=0}^{\infty}h_{jt}^{t}$ in
$(z^{\prime})\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$,
convergent in the $m^{\prime}$-adic topology, which we will now construct
recursively in $t$. Put $h_{j0}=0$. Assume that $t>0$, that
$h_{j0},\dots,h_{j,t-1}$ are already defined and that for
$q\in\\{1,\dots,t-1\\}$ we have
$u^{\prime}_{j}-\bar{\phi}_{j}(u^{\prime})-\sum\limits_{l=0}^{q}h_{jl}\mbox{$\in$
/}\bigcup\limits_{l=1}^{q}I_{l}$ and
$h_{jq}\in(z^{\prime})\bigcap\left(\bigcap\limits_{l=1}^{q-1}I_{l}\right)$. If
$u^{\prime}_{j}-\bar{\phi}_{j}(u^{\prime})-\sum\limits_{l=0}^{t-1}h_{jl}\mbox{$\in$
/}I_{t}$, put $h_{jt}=0$. If
$u^{\prime}_{j}-\bar{\phi}_{j}(u^{\prime})-\sum\limits_{l=0}^{t-1}h_{jl}\in
I_{t}$, let $h_{jt}$ be any element of
$(z^{\prime})\bigcap\left(\bigcap\limits_{l=1}^{t-1}I_{l}\right)\setminus
I_{t}$ (such an element exists because $I_{t}$ is prime, in view of (157)).
This completes the definition of $\tilde{h}_{j}$. Let $h_{j}$ be an arbitrary
representative of $\tilde{h}_{j}$ in
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$.
We claim that
$\left(H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j})\right)\cap\frac{R^{\prime}}{P^{\prime}_{\ell-1}}=(0).$
(160)
Indeed, suppose the above intersection contained a non-zero element $f$. Then
any minimal prime $\tilde{I}$ of the ideal
$(v^{\prime}_{j})\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$
is also a minimal prime of
$(f)\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$.
Since $v_{j}\mbox{$\in$
/}\frac{(z^{\prime})\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$,
we have
$\tilde{I}\not\subset\frac{(z^{\prime})\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}+(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{j-1})}$.
Hence $\tilde{I}=I_{q}$ for some $q\in\mathbb{N}$. Then $v_{j}\in I_{q}$,
which contradicts (159).
Carrying out the above construction for all
$j\in\\{s_{\ell}+1,\dots,n_{\ell}\\}$ produces the elements $\phi_{j}$
required in the Lemma. This completes the proof of Lemma 9.6 in the case when
$k$ is countable.
Next, assume that $k$ is uncountable. Let $u^{\prime}$ be chosen as above.
By assumption, $\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ contains a
$k_{0}$-subalgebra $S$ essentially of finite type, over which
$\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ is strictly étale. Take a countable
subfield $L_{1}\subset k_{0}$ such that the algebra $S$ is defined already
over $L_{1}$ (this means that $S$ has the form
$S=(S^{\prime}_{1}\otimes_{L_{1}}k_{0})_{m^{\prime}_{1}(S^{\prime}_{1}\otimes_{L_{1}}k_{0})},$
(161)
where $(S^{\prime}_{1},m^{\prime}_{1},k^{\prime}_{1})$ is a local
$L_{1}$-algebra essentially of finite type). Next, let $L_{1}\subset
L_{2}\subset...$ be an increasing chain of finitely generated field extensions
of $L_{1}$, contained in $k_{0}$, having the following property. Let
$(S^{\prime}_{q},m^{\prime}_{q},k^{\prime}_{q})$ denote the localization of
$S^{\prime}_{1}\otimes_{k^{\prime}_{1}}L_{q}$ at the maximal ideal
$m^{\prime}_{1}(S^{\prime}_{1}\otimes_{k^{\prime}_{1}}L_{q})$. We require that
$k^{\prime}_{\infty}:=\bigcup\limits_{q=1}^{\infty}k^{\prime}_{q}$
contain all the coefficients of all the formal power series
$\bar{\phi}_{s_{\ell}+1},\dots,\bar{\phi}_{n_{\ell}}$ and such that the ideal
$\frac{H^{\prime}_{2\ell-1}}{H^{\prime}_{2\ell-2}}$ is generated by elements
of $\frac{k^{\prime}_{\infty}[[z^{\prime}]]}{I_{\infty}}[[u^{\prime}]]$, where
$I_{\infty}$ is the kernel of the natural homomorphism
$k^{\prime}_{\infty}[[z^{\prime}]]\rightarrow\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$.
Let
$H^{\prime}_{2\ell-1,\infty}=\frac{H^{\prime}_{2\ell-1}}{H^{\prime}_{2\ell-2}}\cap\frac{k^{\prime}_{\infty}[[z^{\prime}]]}{I_{\infty}}[[u^{\prime}]]$.
We have constructed an increasing chain $S^{\prime}_{1}\subset
S^{\prime}_{2}\subset...$ of local $L_{1}$-algebras essentially of finite type
such that $k^{\prime}_{q}$ is the residue field of $S^{\prime}_{q}$. Then
$S^{\prime}_{\infty}:=\bigcup\limits_{q=1}^{\infty}S^{\prime}_{q}$ is a local
noetherian ring whose completion is
$\frac{k^{\prime}_{\infty}[[z^{\prime},u^{\prime}]]}{\left(P^{\prime}_{\ell-1}\cap
S^{\prime}_{\infty}\right)k^{\prime}_{\infty}[[z^{\prime},u^{\prime}]]}$. Let
$m^{\prime}_{\infty}$ denote the maximal ideal of $S^{\prime}_{\infty}$. The
above argument in the countable case shows that there exist representatives
$\phi_{s_{\ell}+1},\dots,\phi_{n_{\ell}}$ of
$\bar{\phi}_{s_{\ell}+1},\dots,\bar{\phi}_{n_{\ell}}$ in
$\frac{\hat{S}^{\prime}_{\infty}}{H^{\prime}_{2\ell-2}\cap\hat{S}^{\prime}_{\infty}}$
such that, defining
$v^{\prime}=(v^{\prime}_{s_{\ell}+1},\dots,v^{\prime}_{n_{\ell}})$ as in
(152), we have
$\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\cap
S^{\prime}_{\infty}=(0).$ (162)
Let $L_{\infty}=\bigcup\limits_{q=1}^{\infty}L_{q}$ and let $t$ denote a
transcendence base of $k_{0}$ over $L_{\infty}$. Let the notation be as in
Lemma 9.4 with $k_{0}$ replaced by $L_{\infty}$. For example,
$S^{\prime}_{\infty}(L_{\infty}(t))$ will denote the localization of the ring
$S^{\prime}_{\infty}\otimes_{L_{\infty}}L_{\infty}(t)$ at the prime ideal
ideal
$m^{\prime}_{\infty}(S^{\prime}_{\infty}\otimes_{L_{\infty}}L_{\infty}(t))$.
By (162),
$\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}[t]\cap
S^{\prime}_{\infty}[t]=(0).$ (163)
Now Lemma 9.5 and the fact that $S^{\prime}_{\infty}[t]$ is a domain imply
that
$\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}(L_{\infty}(t))\cap
S^{\prime}_{\infty}(L_{\infty}(t))=(0).$ (164)
Next, let $\tilde{L}$ be a finite extension of $L_{\infty}(t)$, contained in
$k_{0}$; then $S^{\prime}_{\infty}(\tilde{L})$ is finite over
$S^{\prime}_{\infty}(L_{\infty}(t))$. Since
$\hat{S}^{\prime}_{\infty}(\tilde{L})$ is faithfully flat over
$\hat{S}^{\prime}_{\infty}(L_{\infty}(t))$ and in view of (164), we have
$\left(\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}(\tilde{L})\cap
S^{\prime}_{\infty}(\tilde{L})\right)\cap
S^{\prime}_{\infty}(L_{\infty}(t))=(0).$
Hence $ht\
\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}(\tilde{L})\cap
S^{\prime}_{\infty}(\tilde{L})=0$. Since $S^{\prime}_{\infty}(\tilde{L})$ is a
domain, this implies that
$\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}(\tilde{L})\cap
S^{\prime}_{\infty}(\tilde{L})=(0).$ (165)
Since $k_{0}$ is algebraic over $L_{\infty}(t)$, it is the limit of the direct
system of all the finite extensions of $L_{\infty}(t)$ contained in it. We
pass to the limit in (165). By (161), we have $S=S^{\prime}_{\infty}(k_{0})$;
we also note that $\hat{S}=\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$.
Since the natural maps
$\hat{S}^{\prime}_{\infty}(\tilde{L})\rightarrow\hat{S}^{\prime}_{\infty}(k_{0})$
are all faithfully flat, we obtain
$\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}(k_{0})\cap
S=(0).$ (166)
Since $\hat{S}=\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$ is also the
formal completion of $\hat{S}^{\prime}_{\infty}(k_{0})$, it is faithfully flat
over $\hat{S}^{\prime}_{\infty}(k_{0})$. Hence
$J^{\prime}_{2\ell-1}\cap\hat{S}^{\prime}_{\infty}(k_{0})=\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}\cap\hat{S}^{\prime}_{\infty}(k_{0})=\left((v^{\prime})+H^{\prime}_{2\ell-1,\infty}\right)\hat{S}^{\prime}_{\infty}(k_{0}).$
(167)
Combining this with (166), we obtain
$J^{\prime}_{2\ell-1}\cap S=(0).$ (168)
Thus the ideal
$J^{\prime}_{2\ell-1}\cap\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ contracts to
$(0)$ in $S$. Since $\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$ is étale over
$S$, this implies the desired equality (154). This completes the proof of
Lemma 9.6.
Since $\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell}\hat{R}^{\prime}}$ is a
complete regular local ring and $(w^{\prime},v^{\prime})$ is a set of
representatives of a minimal set of generators of its maximal ideal
$\frac{m^{\prime}\hat{R}^{\prime}}{H^{\prime}_{2\ell}}$, there exists a
complete local domain $R^{\prime}_{\ell}$ (not necessarily regular) such that
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-1}}\cong
R^{\prime}_{\ell}[[w^{\prime},v^{\prime}]]$. Consider the ring homomorphism
$R^{\prime}_{\ell}[[w^{\prime},v^{\prime}]]\rightarrow
R^{\prime}_{\ell}[[w^{\prime}]],$ (169)
obtained by taking the quotient modulo $(v^{\prime})$. By (169), the quotient
of $\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}$ by $J^{\prime}_{2\ell-1}$
is the integral domain $R^{\prime}_{\ell}[[w^{\prime}]]$, hence
$J^{\prime}_{2\ell-1}$ is prime.
Consider a local blowing up $R^{\prime}\rightarrow R^{\prime\prime}$ in
$\mathcal{T}$. Because of the stability assumption on $R$, the ring
$\frac{\hat{R}^{\prime\prime}}{H^{\prime\prime}_{2\ell-2}}\otimes_{R}\kappa(P^{\prime\prime}_{l-1})$
is finite over
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}\otimes_{R}\kappa(P^{\prime}_{l-1})$;
hence the ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime\prime}\in\mathcal{T}}}\left(\frac{\hat{R}^{\prime\prime}}{H^{\prime\prime}_{2\ell-2}}\otimes_{R}\kappa(P^{\prime\prime}_{l-1})\right)$
is integral over
$\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}\otimes_{R}\kappa(P^{\prime}_{l-1})$.
In particular, there exists a prime ideal in
$\lim\limits_{\overset{\longrightarrow}{R^{\prime\prime}\in\mathcal{T}}}\left(\frac{\hat{R}^{\prime\prime}}{H^{\prime\prime}_{2\ell-2}}\otimes_{R}\kappa(P^{\prime\prime}_{l-1})\right),$
lying over
$J^{\prime}_{2\ell-1}\frac{\hat{R}^{\prime}}{H^{\prime}_{2\ell-2}}\otimes_{R}\kappa(P^{\prime}_{l-1})$.
Pick and fix one such prime ideal. Intersecting this ideal with
$\frac{\hat{R}^{\prime\prime}}{H^{\prime\prime}_{2\ell-2}}$ for each
$R^{\prime\prime}\in\mathcal{T}$, we obtain a tree
$J^{\prime\prime}_{2\ell-1}$ of prime ideals of
$\frac{\hat{R}^{\prime\prime}}{H^{\prime\prime}_{2\ell-2}}$,
$R^{\prime\prime}\in\mathcal{T}$.
Our next task is to define the restriction of the valuation
$\hat{\mu}_{2\ell}$ to the ring
$\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell-1}}$. By the induction assumption,
$\hat{\mu}_{2\ell+2}$ is already defined on
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}\in\mathcal{T}}}\frac{\hat{R}^{\prime}}{J^{\prime}_{2\ell+1}\hat{R}^{\prime}}$.
For all stable $R^{\prime\prime}\in\mathcal{T}$ we have the isomorphism
$gr_{\hat{\mu}_{2\ell+2}}\frac{\hat{R}^{\prime\prime}}{J^{\prime\prime}_{2\ell+1}}\cong
gr_{\mu_{\ell+1}}\frac{R^{\prime\prime}}{P^{\prime\prime}_{\ell}}$ of graded
algebras (in particular,
$gr_{\hat{\mu}_{2\ell+2}}\frac{\hat{R}^{\prime\prime}}{J^{\prime\prime}_{2\ell+1}}$
is scalewise birational to
$gr_{\mu_{\ell+1}}\frac{R^{\prime\prime}}{P^{\prime\prime}_{\ell}}$ for any
$R^{\prime\prime}\in\mathcal{T}$ and $\hat{\mu}_{2\ell+2}$ has the same value
group $\Delta_{\ell}$ as $\mu_{\ell+1}$).
Define the prime ideals
$\tilde{H}^{\prime\prime}_{2\ell-2}=\tilde{H}^{\prime\prime}_{2\ell-1}$ to be
equal to the preimage of $J^{\prime\prime}_{2\ell-1}$ in
$\hat{R}^{\prime\prime}$ and
$\tilde{H}^{\prime\prime}_{2\ell}=\tilde{H}^{\prime\prime}_{2\ell+1}$ the
preimage of $J^{\prime\prime}_{2\ell+1}$ in $\hat{R}^{\prime\prime}$. By
definition of tight extensions, the valuation $\hat{\nu}_{2\ell+1}$ must be
trivial. It remains to describe the valuation $\hat{\mu}_{2\ell}$ on
$\frac{\hat{R}^{\prime\prime}}{J^{\prime\prime}_{2\ell-1}}$,
$R^{\prime\prime}\in\mathcal{T}$. We will first define $\hat{\nu}_{2\ell}$ and
then put $\hat{\mu}_{2\ell}=\hat{\nu}_{2\ell}\circ\hat{\mu}_{2\ell+2}$.
By definition of tight extensions, the value group of $\hat{\mu}_{2\ell}$ must
be equal to $\Delta_{\ell-1}$ and that of $\hat{\nu}_{2\ell}$ to
$\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$. For a positive element
$\bar{\beta}\in\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$, define the candidate
for $\hat{\nu}_{2\ell}$-ideal of
$\frac{\hat{R}^{\prime\prime}_{\tilde{H}^{\prime\prime}_{2\ell}}}{\tilde{H}^{\prime\prime}_{2\ell-1}\hat{R}^{\prime\prime}_{\tilde{H}^{\prime\prime}_{2\ell}}}$
of value $\bar{\beta}$, denoted by
$\hat{\mathcal{P}}^{\prime\prime}_{\beta\ell}$, by the formula
$\hat{\mathcal{P}}^{\prime\prime}_{\bar{\beta}\ell}=\frac{\mathcal{P}^{\prime\prime}_{\bar{\beta}}\hat{R}^{\prime\prime}_{\tilde{H}^{\prime\prime}_{2\ell}}}{\tilde{H}^{\prime\prime}_{2\ell-1}\hat{R}^{\prime\prime}_{\tilde{H}^{\prime\prime}_{2\ell}}}.$
(170)
###### Conjecture 9.7
The elements $\phi_{j}$ of Lemma 9.6 can be chosen in such a way that the
following condition holds. For each positive element
$\beta\in\frac{\Delta_{\ell-1}}{\Delta_{\ell}}$ and each tree morphism
$R^{\prime}\rightarrow R^{\prime\prime}$ in $\mathcal{T}$, we have
$\hat{\mathcal{P}}^{\prime\prime}_{\beta\ell}\cap\hat{R}^{\prime}_{\tilde{H}^{\prime}_{2\ell}}=\hat{\mathcal{P}}^{\prime}_{\beta\ell}.$
###### Conjecture 9.8
The elements $\phi_{j}$ of Lemma 9.6 can be chosen in such a way that
$\bigcap\limits_{\bar{\beta}\in\left(\frac{\Delta_{\ell-1}}{\Delta_{\ell}}\right)_{+}}\left(\mathcal{P}^{\prime}_{\bar{\beta}}+\tilde{H}^{\prime}_{2\ell-1}\right)\hat{R}^{\prime}_{\tilde{H}^{\prime}_{2\ell}}\subset\tilde{H}^{\prime}_{2\ell-1}.$
(171)
For the rest of this section assume that Conjectures 9.7 and 9.8 are true.
For all
$\bar{\beta}\in\left(\frac{\Delta_{\ell-1}}{\Delta_{\ell}}\right)_{+}$, we
have the natural isomorphism
$\lambda_{\bar{\beta}}:\frac{\mathcal{P}^{\prime}_{\bar{\beta}}}{\mathcal{P}^{\prime}_{\bar{\beta}+}}\otimes_{\kappa(P^{\prime}_{\ell-1})}\kappa(\tilde{H}^{\prime}_{2\ell})\longrightarrow\frac{\hat{\mathcal{P}}^{\prime}_{\bar{\beta}}}{\hat{\mathcal{P}}^{\prime}_{\bar{\beta}+}}$
of $\kappa(\tilde{H}^{\prime}_{2\ell})$-vector spaces. The following fact is
an easy consequences of Conjecture 9.7:
###### Corollary 9.9
(conditional on Conjecture 9.7) If the elements $\phi_{j}$ of Lemma 9.6 can be
chosen as in Conjecture 9.7 then the graded algebra
$\mbox{gr}_{\nu_{\ell}}\frac{R^{\prime}_{P^{\prime}_{\ell}}}{P^{\prime}_{\ell-1}R^{\prime}_{P^{\prime}_{\ell}}}\otimes_{\kappa(P^{\prime}_{\ell-1})}\kappa(\tilde{H}^{\prime}_{2\ell})\cong\bigoplus\limits_{\bar{\beta}\in\left(\frac{\Delta_{\ell-1}}{\Delta_{\ell}}\right)_{+}}\frac{\hat{\mathcal{P}}^{\prime}_{\bar{\beta}}}{\hat{\mathcal{P}}^{\prime}_{\bar{\beta}+}}$
is an integral domain.
For a non-zero element
$x\in\frac{\hat{R}^{\prime}_{\tilde{H}^{\prime}_{2\ell}}}{\tilde{H}^{\prime}_{2\ell-1}}$,
let
$\operatorname{Val}_{\ell}(x)=\left\\{\left.\beta\in\nu_{\ell}\left(\frac{R^{\prime}}{P^{\prime}_{\ell-1}}\setminus\\{0\\}\right)\
\right|\ x\in\hat{\mathcal{P}}^{\prime}_{\beta\ell}\right\\}$. We define
$\hat{\nu}_{2\ell}$ by the formula
$\hat{\nu}_{2\ell}(x)=\max\ \operatorname{Val}_{\ell}(x).$ (172)
Since $\nu_{\ell}$ is a rank 1 valuation, centered in a local noetherian
domain $\frac{R^{\prime}}{P^{\prime}_{\ell-1}}$, the semigroup
$\nu_{\ell}\left(\frac{R^{\prime}}{P^{\prime}_{\ell-1}}\setminus\\{0\\}\right)$
has order type $\mathbb{N}$, so by (171) the set $Val_{\ell}(x)$ contains a
maximal element. This proves that the valuation $\hat{\nu}_{2\ell}$ is well
defined by the formula (172), and that we have a natural isomorphism of graded
algebras
$\mbox{gr}_{\nu_{\ell}}\frac{R^{\prime}_{P^{\prime}_{\ell}}}{P^{\prime}_{\ell-1}R^{\prime}_{P^{\prime}_{\ell}}}\otimes_{\kappa(P^{\prime}_{\ell-1})}\kappa(\tilde{H}^{\prime}_{2\ell})\cong\mbox{gr}_{\hat{\nu}_{2\ell}}\frac{\hat{R}_{\tilde{H}^{\prime}_{2\ell}}}{\tilde{H}^{\prime}_{2\ell-1}}.$
Since the above construction is valid for all $R\in\mathcal{T}$,
$\hat{\nu}_{2\ell}$ extends naturally to a valuation centered in the ring
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime\prime}}{\tilde{H}^{\prime\prime}_{2\ell-1}\hat{R}^{\prime\prime}}$
(by abuse of notation, this extension will also be denoted by
$\hat{\nu}_{2\ell}$).
The extension $\hat{\mu}_{2\ell}$ of $\mu_{\ell}$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime\prime}\in\mathcal{T}(R^{\prime})}}\frac{\hat{R}^{\prime\prime}}{\tilde{H}^{\prime\prime}_{2\ell-1}\hat{R}^{\prime}}$
is defined by $\hat{\mu}_{2\ell}=\hat{\nu}_{2\ell}\circ\hat{\mu}_{2\ell+2}$.
This completes the proof of Conjecture 9.2 (assuming Conjectures 9.7 and 9.8)
by descending induction on $\ell$. $\Box$
The next Corollary of Conjecture 9.2 gives necessary conditions for
$\hat{\nu}_{-}$ to be uniquely determined by $\nu$; it also shows that the
same conditions are sufficient for $\hat{\nu}_{-}$ to be the unique minimal
extension of $\nu$, that is, to satisfy
$\tilde{H}^{\prime}_{i}=H^{\prime}_{i},\quad 0\leq i\leq 2r.$ (173)
Suppose given a tree $\left\\{\tilde{H}^{\prime}_{0}\right\\}$ of minimal
prime ideals of $\hat{R}^{\prime}$ (in particular,
$R^{\prime}\cap\tilde{H}^{\prime}_{0}=(0)$). If the valuation $\nu$ admits an
extension to a valuation $\hat{\nu}_{-}$ of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{\tilde{H}^{\prime}_{0}}$,
then $\tilde{H}^{\prime}_{0}$ is the 0-th prime ideal of $\hat{R}^{\prime}$,
determined by $\hat{\nu}_{-}$. Since $\tilde{H}^{\prime}_{0}$ is assumed to be
a minimal prime, we have $\tilde{H}^{\prime}_{0}=H^{\prime}_{0}$ by
Proposition 5.4.
###### Remark 9.10
Let the notation be as in Conjecture 9.2. Denote the tree of prime ideals
$\\{\tilde{H}^{\prime}_{0}\\}$ by $\\{H^{\prime}\\}$ for short. Consider a
homomorphism
$R^{\prime}\rightarrow R^{\prime\prime}$ (174)
in $\mathcal{T}$. Assume that the local rings $R^{\prime}$ and
$\frac{\hat{R}^{\prime}}{H^{\prime}}$ are regular, and let
$V=(V_{1},\dots,V_{s})$ be a minimal set of generators of $H^{\prime}$. Then
$V$ can be extended to a regular system of parameters for $\hat{R}^{\prime}$.
We have an isomorphism
$\hat{R}^{\prime}\cong\frac{\hat{R}^{\prime}}{H^{\prime}}[[V]]$. The morphism
(174) induces an isomorphism
$\hat{R}^{\prime}_{H^{\prime}}\cong\hat{R}^{\prime\prime}_{H^{\prime\prime}}$,
so that $V$ induces a regular system of parameters of
$\hat{R}^{\prime\prime}_{H^{\prime\prime}}$. In particular, the
$H^{\prime\prime}$-adic valuation of
$\hat{R}^{\prime\prime}_{H^{\prime\prime}}$ coincides with the
$H^{\prime}$-adic valuation of $\hat{R}^{\prime}_{H^{\prime}}$. On the other
hand, we do not know, assuming that $R^{\prime\prime}$ and
$\frac{\hat{R}^{\prime\prime}}{H^{\prime\prime}}$ are regular and $ht\
H^{\prime\prime}=ht\ H^{\prime}$, whether $V$ induces a minimal set of
generators of $H^{\prime\prime}$; we suspect that the answer is “no”.
###### Corollary 9.11
(conditional on Conjecture 9.2) If the valuation $\nu$ admits a unique
extension to a valuation $\hat{\nu}_{-}$ of
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{0}}$,
then the following conditions hold:
(1) $ht\ H^{\prime}_{1}\leq 1$
(2) $H^{\prime}_{i}=H^{\prime}_{i-1}$ for all odd $i>1$. Moreover, this unique
extension $\hat{\nu}_{-}$ is minimal.
Conversely, assume that (1)–(2) hold. Then there exists a unique minimal
extension $\hat{\nu}_{-}$ of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{0}}$.
Proof.- The fact that conditions (1), (2) and equations (173) determine
$\hat{\nu}_{-}$ uniquely is nothing but Proposition 6.12. Conversely, assume
that there exists a unique extension $\hat{\nu}_{-}$ of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{0}}$.
By Remark 6.11, there exist minimal extensions of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{0}}$,
hence $\hat{\nu}_{-}$ must be minimal.
Next, by Conjecture 9.2, there exists a tree of prime ideals
$\tilde{H}^{\prime}$ with $H^{\prime}\cap R^{\prime}=(0)$ and a tight
extension $\hat{\mu}_{-}$ of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}}$.
The ideals $H^{\prime}$ are both the the 0-th and the 1-st ideals determined
by $\hat{\mu}_{-}$; in particular, we have
$H^{\prime}_{0}\subset H^{\prime}_{1}\subset H^{\prime}$ (175)
by Proposition 5.4. Now, take any valuation $\theta$, centered in the regular
local ring $\frac{R^{\prime}_{H^{\prime}}}{H^{\prime}_{0}}$, such that the
residue field $k_{\theta}=\kappa(H^{\prime})$. Then the composition
$\hat{\mu}_{-}\circ\theta$ is an extension of $\nu$ to
$\lim\limits_{\overset{\longrightarrow}{R^{\prime}}}\frac{\hat{R}^{\prime}}{H^{\prime}_{0}}$,
hence
$\hat{\mu}_{-}\circ\theta=\hat{\nu}_{-}$ (176)
by uniqueness. For $i\geq 1$, the $i$-th prime ideal, determined by
$\hat{\mu}_{-}\circ\theta=\hat{\nu}_{-}$ coincides with that determined by
$\hat{\mu}_{-}$. Since $\nu$ is minimal and $\hat{\mu}_{-}$ is tight, we
obtain condition (2) of the Corollary. Finally, if we had $ht\ H^{\prime}>1$,
there would be infinitely many choices for $\theta$, contradicting 176 and the
uniqueness of $\hat{\nu}_{-}$. Thus $ht\ H^{\prime}\leq 1$. Combined with 175,
this proves (1) of the Corollary. This completes the proof of Corollary
(9.11), assuming Conjecture 9.2.
Regular morphisms and G-rings.
In this Appendix we recall the definitions of regular homomorphism, G-rings
and excellent and quasi-excellent rings. We also summarize some of their basic
properties used in the rest of the paper.
###### Definition .12
([10], Chapter 13, (33.A), p. 249) Let $\sigma:A\rightarrow B$ be a
homomorphism of noetherian rings. We say that $\sigma$ is regular if it is
flat, and for every prime ideal $P\subset A$, the ring $B\otimes_{A}\kappa(P)$
is geometrically regular over $\kappa(P)$ (this means that for any finite
field extension $\kappa(P)\rightarrow k^{\prime}$, the ring
$B\otimes_{A}k^{\prime}$ is regular).
###### Remark .13
If $\kappa(P)$ is perfect, the ring $B\otimes_{A}\kappa(P)$ is geometrically
regular over $\kappa(P)$ if and only if it is regular.
###### Remark .14
It is known that a morphism of finite type is regular in the above sense if
and only if it is smooth (that is, formally smooth in the sense of
Grothendieck with respect to the discrete topology), though we do not use this
fact in the present paper.
Regular morphisms come up in a natural way when one wishes to pass to the
formal completion of a local ring:
###### Definition .15
([10], (33.A) and (34.A)) Let $R$ be a noetherian ring. For a maximal ideal
$m$ of $R$, let $\hat{R}_{m}$ denote the $m$-adic completion of $R$. We say
that $R$ is a G-ring if for every maximal ideal $m$ of $R$, the natural map
$R\rightarrow\hat{R}_{m}$ is a regular homomorphism.
The property of being a G-ring is preserved by localization and passing to
rings essentially of finite type over $R$.
###### Definition .16
([10], Definition 2.5, (34.A), p. 259) Let $R$ be a noetherian ring. We say
that $R$ is quasi-excellent if the following two conditions hold:
(1) $R$ is J-2, that is, for any scheme $X$, which is reduced and of finite
type over $\mbox{Spec}\ R$, $Reg(X)$ is open in the Zariski topology.
(2) For every maximal ideal $m\subset R$, $R_{m}$ is a G-ring.
It is known [10] that a local G-ring is automatically J-2, hence automatically
quasi-excellent. Thus for local rings “G-ring” and “quasi-excellent” are one
and the same thing. A ring is said to be excellent if it is quasi-excellent
and universally catenary, but we do not need the catenary condition in this
paper.
Both excellence and quasi-excellence are preserved by localization and passing
to rings of finite type over $R$ ([10], Chapter 13, (33.G), Theorem 77, p.
254). In particular, any ring essentially of finite type over a field,
$\mathbf{Z}$, $\mathbf{Z}_{(p)}$, $\mathbf{Z}_{p}$, the Witt vectors or any
other excellent Dedekind domain is excellent. See [11] (Appendix A.1, p. 203)
for some examples of non-excellent rings.
Rings which arise from natural constructions in algebra and geometry are
excellent. Complete and complex-analytic local rings are excellent (see [10],
Theorem 30.D) for a proof that any complete local ring is excellent).
Finally, we remark that the category of quasi-excellent rings is a natural one
for doing algebraic geometry, since it is the largest reasonable class of
rings for which resolution of singularities can hold. Namely, let $R$ be a
noetherian ring. Grothendieck ([4], IV.7.9) proves that if all of the
irreducible closed subschemes of $\mbox{Spec}\ R$ and all of their finite
purely inseparable covers admit resolution of singularities, then $R$ must be
quasi-excellent. Grothendieck’s result means that the largest class of
noetherian rings, closed under homomorphic images and finite purely
inseparable extensions, for which resolution of singularities could possibly
exist, is quasi-excellent rings.
We now summarize the specific uses we make of quasi-excellence in the present
paper. We begin by recalling three results from [10] and [11]. As a point of
terminology, we note that Nagata’s “pseudo-geometric” rings are now commonly
known as Nagata rings. Quasi-excellent rings are Nagata ([10], (33.H), Theorem
78).
###### Theorem .17
([10], (34.C), Theorem 79) Let $R$ be an excellent normal local ring. Then $R$
is analytically normal (this means that its formal completion $\hat{R}$ is
normal).
###### Theorem .18
([11], (43.20), p. 187) Let $R$ be a local integral domain, $\tilde{R}$ its
Henselization and $R^{\prime}$ its normalization. There is a natural one-to-
one correspondence between the minimal primes of $\tilde{R}$ and the maximal
ideals of $R^{\prime}$.
###### Proposition .19
([11], Corollary (44.3), p. 189) Let $R$ be a quasi-excellent analytically
normal local ring. Then its Henselization $\tilde{R}$ is analytically
irreducible and is algebraically closed in its formal completion.
From the above results we deduce
###### Corollary .20
Let $(R,\mathbf{m})$ be a Henselian excellent local domain. Then $R$ is
analytically irreducible and is algebraically closed in $\hat{R}$.
Proof.- If, in addition, we assume $R$ to be normal, the result follows from
Theorems .17 and .19. In the general case, let $R^{\prime}$ denote the
normalization of $R$. Then $R^{\prime}$ is a Henselian normal quasi-excellent
local ring, so it satisfies the conclusions of the Corollary. Consider the
commutative diagram
$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$$\textstyle{{\hat{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\psi}}$$\textstyle{R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{\hat{R}^{\prime}}$
(177)
where $\hat{R}^{\prime}$ stands for the formal completion of $R^{\prime}$.
Since $R$ is Nagata, $R^{\prime}$ is a finite $R$-module. Thus $\phi^{\prime}$
coincides with the $\mathbf{m}$-adic completion of $R^{\prime}$, viewed as an
$R$-module. Hence $\hat{R}^{\prime}\cong R^{\prime}\otimes_{R}\hat{R}$. Since
$\psi$ is injective and $\hat{R}$ is flat over $r$, the map $\hat{\phi}$ is
also injective. Since $R^{\prime}$ is analytically irreducible,
$\hat{R}^{\prime}$ is a domain, and therefore so is its subring $\hat{R}$.
This proves that $R$ is analytically irreducible.
To prove that $R$ is algebraically closed in $\hat{R}$, take an element
$x\in\hat{R}$, algebraic over $R$. Since all the maps in 177 are injective,
let us view all the rings involved as subrings of $\hat{R}^{\prime}$. Since
$R^{\prime}$ is algebraically closed in $\hat{R}^{\prime}$, we have $x\in
R^{\prime}$, in particular, we may write $x=\frac{a}{b}$ with $a,b\in R$. Now,
since $(a)\hat{R}\subset(b)\hat{R}$ and $\hat{R}$ is faithfully flat over $R$,
we have $(a)\subset(b)$ in $R$, so $x=\frac{a}{b}\in R$. This proves that $R$
is algebraically closed in $\hat{R}$. The Corollary is proved.
Next we summarize, in a more specific manner, the way in which these results
are applied in the present paper. The main applications are as follows.
(1) Let $R$ be an excellent local domain, $P$ a prime ideal of $R$ and
$H_{i}\subset H_{i+1}$ two prime ideals of $\hat{R}$ such that
$H_{i}\cap R=H_{i+1}\cap R=P.$ (178)
Then $\frac{R}{P}$ is also excellent. Definitions .12, .15 and .16 imply that
the ring $\hat{R}\otimes_{R}\kappa(P)$ is geometrically regular over $P$, in
particular, regular. Moreover, (178) implies that the ideal
$\frac{H_{i+1}}{P\hat{R}}$ is a prime ideal of $\frac{\hat{R}}{P\hat{R}}$,
disjoint from the natural image of $R\setminus P$ in
$\frac{\hat{R}}{P\hat{R}}$. Thus the local ring
$\frac{\hat{R}_{H_{i+1}}}{P\hat{R}_{H_{i+1}}}$ is a localization of
$\hat{R}\otimes_{R}\kappa(P)$ at the prime ideal
$H_{i+1}(\hat{R}\otimes_{R}\kappa(P))$ and so is a local ring, geometrically
regular over $\kappa(P)$, in particular, a regular local ring and, in
particular, a domain.
(2) Assume, in addition, that $H_{i}$ is a minimal prime of $P\hat{R}$. Since
$\frac{\hat{R}_{H_{i+1}}}{P\hat{R}_{H_{i+1}}}$ is a domain, $H_{i}$ is the
only minimal prime of $P\hat{R}$, contained in $H_{i+1}$. We have
$P\hat{R}_{H_{i+1}}=H_{i}\hat{R}_{H_{i+1}}$.
## References
* [1] S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic $p\neq 0$, Ann. of Math., 63 (1956) 491–526.
* [2] S. D. Cutkosky and S. El Hitti, Formal prime ideals of infinite value and their algebraic resolution, preprint, arXiv:0905.4518
* [3] S. Cutkosky and L. Ghezzi, Completions of valuation rings, Contemp. Math. 386 (2005), 13–34.
* [4] A. Grothendieck, J. Dieudonné, Eléments de Géométrie algébrique, Chap. IV, Pub. Math. IHES, No. 24, 1965.
* [5] W. Heinzer and J. Sally, Extensions of valuations to the completion of a local domain. J. Pure Appl. Algebra, Vol. 71, no 2–3, pp. 175–186, (1991).
* [6] J. Herrera, M.A. Olalla, M. Spivakovsky, Valuations in algebraic field extensions, J. Algebra 312 (2007), no. 2, 1033–1074.
* [7] I. Kaplansky, Maximal fields with valuations I. Duke Math. J., 9:303–321 (1942).
* [8] I. Kaplansky, Maximal fields with valuations II. Duke Math. J., 12:243–248 (1945).
* [9] J. Lipman Desingularization of two-dimensional schemes, Ann. Math. 107 (1978) 151–207.
* [10] H. Matsumura, Commutative Algebra. Benjamin/Cummings Publishing Co., Reading, Mass., 1970.
* [11] M. Nagata, Local Rings., Interscience Publishers, 1960.
* [12] M. Spivakovsky, Valuations in function fields of surfaces. Amer. J. Math 112, 1, 107–156 (1990).
* [13] M. Spivakovsky, Resolution of singularities I: Local Uniformization of an equicharacteristic quasi-excellent local domain whose residue field $k$ satisfies $\left[k:k^{p}\right]<\infty$, in preparation.
* [14] B. Teissier, Valuations, deformations, and toric geometry, Proceedings of the Saskatoon Conference and Workshop on valuation theory (second volume), F-V. Kuhlmann, S. Kuhlmann, M. Marshall, editors, Fields Institute Communications, 33, 2003, 361-459.
* [15] M. Vaquié, Valuations, in “Resolution of Singularities, a research textbook in tribute to Oscar Zariski”, Birkhäuser, Progress in Math. No. 181, 2000.
* [16] M. Vaquié, Famille admissible de valuations et défaut d’une extension, J. Algebra 311 (2007), no. 2, 859–876.
* [17] O. Zariski, Local uniformization theorem on algebraic varieties, Ann. of Math., 41 (1940), 852–896.
* [18] O. Zariski, P. Samuel Commutative Algebra, Vol. II, Springer-Verlag (1960).
|
arxiv-papers
| 2010-07-27T09:53:04 |
2024-09-04T02:49:11.885197
|
{
"license": "Public Domain",
"authors": "F. J. Herrera Govantes, M. A. Olalla Acosta, M. Spivakovsky and B.\n Teissier",
"submitter": "Miguel Angel Olalla Acosta",
"url": "https://arxiv.org/abs/1007.4658"
}
|
1007.4877
|
# Observation of universal behaviour of ultracold quantum critical gases
Hongwei Xiong State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy
of Sciences, Wuhan 430071, P. R. China Xinzhou Tan State Key Laboratory of
Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of
Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, P. R.
China Graduate School of the Chinese Academy of Sciences, P. R. China Bing
Wang State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of
Sciences, Wuhan 430071, P. R. China Graduate School of the Chinese Academy of
Sciences, P. R. China Lijuan Cao State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences, Wuhan 430071, P. R. China Graduate School of the
Chinese Academy of Sciences, P. R. China Baolong Lű State Key Laboratory of
Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of
Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, P. R.
China
###### Abstract
Quantum critical matter has already been studied in many systems, including
cold atomic gases. We report the observation of a universal behaviour of
ultracold quantum critical Bose gases in a one-dimensional optical lattice. In
the quantum critical region above the Berezinskii-Kosterlitz-Thouless
transition, the relative phase fluctuations between neighboring subcondensates
and spatial phase fluctuations of quasi-2D subcondensates coexist. We study
the density probability distribution function when both these two phase
fluctuations are considered. A universal exponential density probability
distribution is demonstrated experimentally, which agrees well with a simple
theoretical model by considering these two phase fluctuations.
The nature of quantum criticality Hertz ; Sachdev ; Cole driven by quantum
fluctuations is still a great puzzle, despite of the remarkable advances in
heavy-fermion metals and rare-earth-based intermetallic compounds, etc Gegen .
New understanding of quantum criticality is widely believed to be a key to
resolving open questions in metal-insulator transitions Imada , high
temperature superconductivity HS and novel material design, etc. Cold atoms
in optical lattices provide a unique chance to not only simulate other
strongly correlated systems RMP-Bloch , but also study some models
unaccessible in solid state systems, particularly for the Bose-Hubbard model
Fisher ; Jaksch ; Bloch . Despite of its complexity, a strongly correlated
system in quantum critical regime is expected to exhibit a universal behaviour
described by a certain physical quantity.
Here we report the observation of universal behaviour for ultracold quantum
critical Bose gases in a one-dimensional optical lattice. Density probability
distributions of the released gases are measured for different depths of the
lattice potential. It was found that the density probability follows a simple
exponential law when the Bose gases reach the quantum critical region above
the Berezinskii-Kosterlitz-Thouless (BKT) transition BKTtheory1 ; BKTtheory2 ;
Pol ; BKTexp . This universal behaviour can be well understood in terms of our
theoretical model considering both the relative phase fluctuations of quasi-2D
subcondensates and spatial phase fluctuations of individual subcondensates
above the BKT transition. The method of density probability distribution
should provide a unique tool for identifying certain quantum phases of optical
lattice systems.
Ultracold Bose atoms in 1D, 2D and 3D optical lattices are widely studied by
the Bose-Hubbard model Fisher . At zero temperature, there is a continuous
quantum phase transition from superfluid (SF) to Mott insulator (MI). Because
of the strongly correlated quantum behaviour, the quantum critical point (QCP)
at absolute zero temperature distorts strongly the structure of the phase
diagram at finite temperatures, leading to the emergence of an unconventional
‘V-shaped’ quantum critical region Sachdev ; Ho ; Cap ; Kato ; Chin ; Haz
(Fig. 1). In analogy with a black hole, the crossover to quantum critical gas
involves crossing a ‘material event horizon’, which implies strongly that the
quantum critical gas has a simple and universal behaviour Cole .
Figure 1: Finite temperature phase diagram of ultracold Bose gases in a one-
dimensional optical lattice. $U$ and $J$ represent the strengths of the on-
site repulsion and of the nearest neighbor hopping in the Bose-Hubbard model,
respectively. By increasing the strength of an optical lattice, $U/J$ can be
increased by several orders of magnitude. The existence of the quantum
critical point (QCP) at zero temperature leads to a ‘V-shaped’ quantum
critical (QC) region. Apart from the ordinary SF-QC-MI transition, the BKT
transition for quasi-2D subcondensates can also occur at sufficiently low
temperatures. In the QC region above the BKT transition (Type-I QC region),
there exists a universal behaviour in the density probability distribution.
All the phase boundary lines, except for the dashed line between QC and MI
regions, are calculated with a total atom number of $1.1\times 10^{5}$ and
trap parameters used in the experiment. $T_{har}$ is the critical temperature
for the purely harmonic trap in absence of the optical lattice. The gray line
displays a phase transition path from SF to Type-I QC region. Data points of
(a4)-(a6) along the gray line are obtained from the corresponding absorption
images in Fig. 2
In the commonly used 3D optical lattice systems, the atoms in a lattice site
are strongly confined in all directions, with a mean occupation number per
lattice site of about $1\sim 3$ Bloch . Thus the spatial phase fluctuations
for the atoms in a single lattice site can be omitted, and the relevant
theoretical calculations based on the Bose-Hubbard model Fisher ; Jaksch and
Wannier function Kohn can give quantitative descriptions of almost all
experimental phenomenaRMP-Bloch . In contrast, for ultracold Bose atoms in a
1D optical lattice, unique characteristics may arise due to the following two
reasons:
(1) Quasi-2D Bose gas: In an experiment of 1D optical lattice system as ours,
there can be hundreds of atoms in a lattice site. The Bose gas in a lattice
site becomes quasi-two-dimensional (quasi-2D) Pedri ; Burger ; Had , if the
local trapping frequency $\widetilde{\omega}_{z}$ of a lattice site in the
lattice direction satisfies the condition
$\hbar\widetilde{\omega}_{z}>>k_{B}T$ Pethick .
(2) BKT transition: For such a quasi-2D Bose gas, the critical temperature in
the occupied lattice site is given by
$T_{2D}\approx\hbar\omega_{\perp}\left(N_{l}\zeta\left(2\right)\right)^{1/2}/k_{B}$,
with $N_{l}$ being the atomic number in a lattice site. Well below $T_{2D}$,
the whole system becomes a chain of subcondensates. If there is no correlation
between the subcondensates in different lattice sites, a quasi-2D gas
undergoes a BKT transition at $T_{BKT}=T_{2D}/4$ BKTtheory1 ; BKTtheory2 .
Beyond this critical value of $T_{BKT}$, it is favorable to create vortices in
quasi-2D subcondensates, and the unbinding of bound vortices will lead to
strong spatial phase fluctuations within each subcondensate.
Since the BKT transition is crucial to revealing the property of cold atoms in
a 1D optical lattice, its critical temperature is also plotted in the phase
diagram (red line in Fig. 1). The BKT transition line divides the QC region
into two parts (Type I and Type II). Our experiments focus on the transition
from SF region to QC region above the BKT transition.
Figure 2: Density probability distribution for different lattice depths.
(a1)-(a6) in the left column are absorption images showing the density
distribution of the released atomic clouds. The field of view is $0.9\times
0.9$ mm2, and the pixel size is $\Delta^{2}=9.0\times 9.0$ $\mathrm{\mu
m}^{2}$. $T_{c}$ denotes the critical temperature of the atomic gas in the
combined trap. For moderate $s$, the interference fringes are prominent.
Further increasing $s$, however, we see complete disappearance of the
interference fringes (a6). This is due to the crossover from SF to QC region
as $s$ is increased. (b1)-(b6) in the right column are the corresponding
density probability distributions. Open circles are the data points calculated
from the density distributions of the images in the left column. The blue
solid curves give the exponential density probability distribution $P_{0}^{e}$
defined in text. When $s$ becomes large enough, the data points agree well
with the exponential curve (see b5 and b6), which shows clearly a universal
behaviour in the QC region above the BKT transition.
The experiments started with pure 87Rb condensates confined in a magnetic trap
with axial and transverse trapping frequencies of
$\left\\{\omega_{\bot},\omega_{z}\right\\}=2\pi\left\\{83.7,7.6\right\\}$ Hz.
The 1D optical lattice was formed by a retroreflected laser beam of
$\lambda=800\,$nm along the axis ($z$ direction) of the condensate. This laser
beam was ramped up to a given intensity over a time of $50$ ms, yielding a
lattice potential $V_{opt}=sE_{R}\sin^{2}\left(2\pi z/\lambda\right)$, with
$E_{R}$ being the recoil energy of an atom absorbing one lattice photon. After
a holding time of $10$ ms, we suddenly switched off the combined potential and
allowed the cold atomic cloud to expand freely for a time of $30$ ms.
The expanded atomic gases were probed using the conventional absorption
imaging technique. Since the probe beam is applied along $x$ direction, what
an absorption image records is the two-dimensional ($y-z$) column density
profile of the atomic cloud. We denote by $N_{1}^{ph}\left(y,z\right)$ and
$N_{2}^{ph}\left(y,z\right)$ the number of photons detected in the pixel at
position $\left(y,z\right)$ with and without the atomic cloud, respectively.
Then the density distribution is written as
$n_{2D}\left(y,z\right)=\ln\left[N_{2}^{ph}\left(y,z\right)/N_{1}^{ph}\left(y,z\right)\right]\Delta^{2}/\sigma_{e}$,
with $\sigma_{e}$ being the absorption cross section of a single atom and
$\Delta$ the pixel size.
The left column of Fig. 2 ((a1)-(a6)) displays the density distributions for
different lattice depths $s$. The temperatures of the system were inferred
from the condensate fractions as described in the Appendix. From the top three
images ((a1)-(a3)), we see an increasing of the side peaks in the interference
patterns as $s$ is increased. Further increasing $s$, we find the interference
fringes become blurred and disappear eventually (see (a4)-(a6)), similar to
the experimental observation in Ref. Orzel . The gradual disappearance of the
interference fringes can be partially explained by the increasing of the
random relative phase Orzel during the SF-QC transition. In Figs. 2(a5)-(a6),
except for the density fluctuations along $z$ direction associated with the
random relative phase between different subcondensates, we also see
significant density fluctuations along $y$ direction. The simultaneous
existence of density fluctuations in these two directions is further
demonstrated in Fig. 3 (c) and (d) for a typical case of $s=30.7$. These
experimental results suggest that in the QC region above the BKT transition,
both the random relative phases and spatial phase fluctuations play important
roles in the expanded density distribution.
Figure 3: Density fluctuations of the released atomic clouds. (a)-(d) are one-
dimensional cuts through the corresponding images. For $s=30.7$, the density
distribution is highly fluctuated along both the horizontal ($z$) and vertical
($y$) directions.
To qualitatively analyze the density fluctuations, we consider the following
density probability distribution
$P\left(n\right)=\frac{S\left(n-\delta n/2\right)-S\left(n+\delta
n/2\right)}{\delta n\cdot S_{\mathrm{total}}}\,,$ (1)
where $\delta n$ is the width of a density interval, $S_{\mathrm{total}}$ is
the total area of the region occupied by the atomic gas, while
$S\left(n\right)$ is the area of the region where the density larger than $n$.
The averaged density is $n_{s}=1/S_{\mathrm{total}}$. For convenience, we
define a dimensionless density as $n_{0}=n/n_{s}$. The dimensionless density
probability distribution is then
$P_{0}\left(n_{0}\right)=n_{s}P\left(n\right)$.
The right column of Fig. 2 displays $P_{0}\left(n_{0}\right)$ calculated from
the corresponding density distribution. In each image, the pixel region chosen
for the calculation of $P_{0}\left(n_{0}\right)$ is that occupied by the cold
atoms, as enclosed by the red dashed ellipse. In all our calculations of
$P_{0}\left(n_{0}\right)$, $\delta n_{0}$ is just the horizontal spacing of
discrete data points. Due to the optical noise, $P_{0}\left(n_{0}\right)$ can
be nonzero even for negative $n_{0}$. This can be understood from the atomic
density formula based on the absorption imaging signal. Assuming $n_{0b}$ is
the minimum negative density for nonzero $P_{0}\left(n_{0}\right)$, optical
noise concentrates in the region of $n_{0b}<n_{0}<\left|n_{0b}\right|$ in each
$P_{0}\left(n_{0}\right)$ plot. Thus, $n_{0}>\left|n_{0b}\right|$ gives the
effective region where the optical noise is negligible and
$P_{0}\left(n_{0}\right)$ reflects the true density probability distribution.
The blue solid lines in Fig. 2(b1)-(b6) represent the following exponential
density probability distribution
$P_{0}^{e}=e^{-\left|n_{0}\right|}.$ (2)
It is obvious that, with increased $s$ (and hence $U/J$),
$P_{0}\left(n_{0}\right)$ has a tendency to $P_{0}^{e}$. In Fig. 2(b6), we see
that $P_{0}\left(n_{0}\right)$ agrees well with $P_{0}^{e}$ in the whole
effective region. Despite of the extremely complex many-body state in the QC
region above the BKT transition, our results show a simple universal
behaviour.
We have numerically simulated our experiments to explain the exponential
density probability distribution. When a released gas has experienced a free
expansion over a time of $t$, the 2D density distribution is given by
$n_{2D}\left(y,z,t\right)=\int
dx\left|\sum_{k}\sqrt{N_{k}}e^{i\left(\phi_{k\perp}\left(x,y,t\right)+\phi_{k}^{s}\left(z,t\right)\right)}\varphi_{k\perp}\left(x,y,t\right)\varphi_{kz}\left(z,t\right)\right|^{2}+n_{2D}^{opt}\left(x,y\right).$
(3)
Here $\varphi_{k\perp}\left(x,y,t\right)\varphi_{kz}\left(z,t\right)$ is the
wave packet of the expanded subcondensate initially in the $k$th lattice site,
with $|\varphi_{k\perp}\left(x,y,t\right)|^{2}$ being the Thomas-Fermi density
distribution in transverse ($x$-$y$) directions, and
$|\varphi_{kz}\left(z,t\right)|^{2}$ the Gaussian density distribution along
$z$ direction. Two classes of phase fluctuations enter the expression.
$\phi_{k\perp}\left(x,y,t\right)$ represents the spatial phase fluctuations in
transverse directions, while $\phi_{k}^{s}\left(z,t\right)$ comprises the
relative phase fluctuations between different subcondensates. $N_{k}$ denotes
the atomic number of the $k$th subcondensate. The term $n_{2D}^{opt}$ is added
to account for the optical noise. In numerical calculations, it was simulated
with the realistic optical noise distribution in the region having no atoms.
Figure 4: Theoretical simulation. Left column: simulated density distributions
based on Eq. (3). (a1), (a2) and (a3) are three typical cases in which an
atomic gas initially stays in SF, Type-I QC and Type-II QC regions,
respectively. In (a1), $s=11.2$, $\delta\phi_{k\perp}=0.2\pi$. In (a2),
$s=30.7$, both $\phi_{k}^{s}$ and $\phi_{k\perp}$ are completely random. In
(a3), $s=30.7$, $\phi_{k\perp}=0$, but $\phi_{k}^{s}$ is completely random.
Each of the color map has a size of $0.9\times 0.9$ mm2. Right column: Black
lines are the density probability distribution calculated from the
corresponding density distributions in the left column. Red lines are obtained
under the same conditions except that the optical noise is not included. The
exponential density probability distribution (blue lines) is also plotted for
comparison. The data in (b2) clearly shows an exponential density probability
distribution in the QC region above the BKT transition.
Figs. 4(a1) and (b1) give the simulated density distribution and density
probability distribution for an atomic gas initially in the combined trap with
$s=11.2$. Spatial phase fluctuations ($\phi_{k\perp}$) have been assumed to be
zero. Figs. 4(a2) and (b2) display similar calculations for $s=30.7$, but with
completely random phases $\phi_{k\perp}$ and $\phi_{k}^{s}$. The theoretical
results in Figs. 4(b2) match the exponential density probability distribution,
in agreement with our experimental results. The coexistence of two random
phases of $\phi_{k\perp}$ and $\phi_{k}^{s}$ makes the system similar to the
situation of the interference of randomly scattered waves Sheng , where
exponential distribution is also found. More general analytical derivations
for Eq. (3) and the exponential density probability distribution are given in
the Appendix. Figs. 4(a3) and (b3) give the simulation of $s=30.7$ for a cold
atomic gas in the QC region below the BKT transition. Accordingly,
$\phi_{k\perp}$ is assumed to be zero, while there exists a completely random
relative phase in $\phi_{k}^{s}$. No exponential density probability
distribution is found in this simulation. In addition, we do not notice
regular interference fringes either. Note that in Ref. Had , the QC region far
below the BKT transition was studied experimentally. However, because there
are only about $30$ subcondensates, high-contrast interference fringes were
still observed. Since our calculation is based on the experimental parameters
in this work, the total number of subcondensates is much higher (up to $250$).
To further explain the experimental phenomena, we stress that there are two
completely different situations in determining the BKT transition temperature
for cold atoms in 1D optical lattices.
(1) BKT transition in QC and MI regions: In these two regions, the quasi-2D
Bose gases in different lattice sites can be regarded as independent
subcondensates as far as the BKT transition is concerned. By minimizing the
free energy $F=E-TS$ ($E$ and $S$ are the energy and entropy of the Bose gas
in a lattice site when vortices are considered), one can get the BKT
transition temperature $T_{BKT}=T_{2D}/4$ Pethick . In the MI region above the
BKT transition, Eq. (3) can be used to simulate the density distribution by
including a completely random relative phase in $\phi_{k}^{s}$ Had , if the
averaged particle number per lattice site is much larger than $1$. Thus, in
the MI region above the BKT transition, one still expects the exponential
density probability distribution for a large number of more than one hundred
independent subcondensates.
(2) BKT transition in SF region: In the SF region, although the Bose gases in
the lattice sites become quasi-2D, they are highly correlated. For a series of
$M$ highly correlated quasi-2D Bose gases, the free energy becomes $F=ME-TS$.
If there are vortices in the quasi-2D subcondensates, vortices in different
lattice sites are highly correlated in spatial locations. Thus, $S$ can be
approximated as the entropy of a single subcondensate. Then the BKT transition
temperature becomes $T_{BKT}^{M}=MT_{BKT}$. Since $M>100$, we see that
$T_{BKT}^{M}>>T_{2D}$. This means that the temperatures of quasi-2D
condensates are always much lower than the BKT transition temperature. For the
situation of $s=11.2$ in our experiment, based on the estimation of the
particle number fluctuations $\delta N_{k}$ Jav and the relation of
$\delta\phi_{k}^{s}\sim 2\pi/\delta N_{k}$ Orzel , we deduced a phase
fluctuation of $\delta\phi_{k}^{s}\approx 0.2\pi<<2\pi$, which shows strong
correlation between different lattice sites. As shown in Fig. 3(b), there are
no noticeable density fluctuations along $y$ direction, which confirms that
the initially confined gas is well below the BKT transition even though $T$ is
already beyond $T_{BKT}$.
We should also emphasize that, in the QC region above the BKT transition, the
system cannot be regarded as a classical thermal cloud although the
temperature is higher than the critical temperature $T_{c}$. A classical
thermal cloud features density fluctuations as $\delta^{2}n\approx n$. In
contrast, the exponential density probability distribution corresponds to much
larger density fluctuations of $\delta^{2}n\approx n^{2}$ XiongWu . This
exponential density probability distribution physically originates from the
superposition of quantum sates with a completely random phase, while the
universal behaviour lies in that the distribution is always exponential for
completely random superposition of quantum state XiongWu . The present work
supports the long-standing belief that the quantum states in the QC region are
strongly correlated, and there should be a simple universal behaviour, once
appropriate physical quantity is found Sachdev ; Cole ; Gegen .
###### Acknowledgements.
H. W. thank the cooperation with Biao Wu about the dynamical universal
behaviour of quantum chaos, which stimulates the present work. This work was
supported by National Key Basic Research and Development Program of China
under Grant No. 2006CB921406 and NSFC under Grant No. 10634060.
## References
* (1) J. Hertz, Phys. Rev. B 14, 1165 (1976).
* (2) S. Sachdev, Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999).
* (3) P. Coleman, A. J. Schofield, Nature 433, 226 (2005).
* (4) P. Gegenwart, Q. Si, F. Steglich, Nature Phys. 4, 186 (2008).
* (5) M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).
* (6) S. Sachdev, Rev. Mod. Phys. 75, 913 (2003).
* (7) I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
* (8) M. P. A. Fisher, P. B. Weichman, G. Grinstein, D. S. Fisher, Phys. Rev. B 40, 546 (1989).
* (9) D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).
* (10) M. Greiner, O. Mandel, T. Esslinger, T. W. Ha̋nsch, I. Bloch, Nature 415, 39 (2002).
* (11) V. L. Berezinskii, Sov. Phys. JETP 34, 610 (1972).
* (12) J. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1973).
* (13) A. Polkovnikov, E. Altman, E. Demler, Proc. Natl Acad. Sci. USA 103, 6125 (2006).
* (14) Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, J. Dalibard, Nature 441, 1118 (2006).
* (15) R. B. Diener, Q. Zhou, H. Zhai, T. L. Ho, Phys. Rev. Lett. 98, 180404 (2007); Q. Zhou and T. L. Ho, Phys. Rev. Lett. 105, 245702 (2010).
* (16) B. Capogrosso-Sansone, N. V. Prokof ’ev, B. V. Svistunov, Phys. Rev. B 75, 134302 (2007).
* (17) Y. Kato, Q. Zhou, N. Kawashima, N. Trivedi, Nature Phys. 4, 617 (2008).
* (18) N. Gemelke, X. B. Zhang, C. L. Hung, C. Chin, Nature 460, 995 (2009).
* (19) K. R. A. Hazzard, E. J. Mueller, arXiv: 1006.0969 (2010).
* (20) W. Kohn, Phys. Rev. 115, 809 (1959).
* (21) P. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001).
* (22) S. Burger et al., Europhys. Lett. 57, 1 (2002).
* (23) Z. Hadzibabic et al., Phys. Rev. Lett. 93, 180403 (2004).
* (24) C. J. Pethick, H. Smith, Bose-Einstein condensation in dilute gases. (Cambridge University Press, Cambridge, 2008).
* (25) C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich, Science 291, 2386 (2001).
* (26) P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena. Chap. 10 (Academic Press. Inc., California, 1995).
* (27) J. Javanainen, M. Y. Ivanov, Phys. Rev. A 60, 2351 (1999).
* (28) H. Xiong, B. Wu, arXiv: 1007.2771 (2010).
Appendix
Bose-Hubbard model. The combined potential of a one-dimensional (1D) optical
lattice and a harmonic trap is given by
$V=\frac{1}{2}m\omega_{\perp}^{2}\left(x^{2}+y^{2}\right)+\frac{1}{2}m\omega_{z}^{2}z^{2}+sE_{R}\sin^{2}\left(\frac{2\pi
z}{\lambda}\right).$ (4a) Here $E_{R}$ is the recoil energy of an atom
absorbing one lattice photon with wavelength $\lambda$. For an atom in a
lattice site, it experiences an effective harmonic potential along $z$
direction with angular frequency $\widetilde{\omega}_{z}\simeq
2\sqrt{s}E_{R}/\hbar$.
Bose-condensed gases in 1D, 2D and 3D optical lattices are widely studied by
the following Bose-Hubbard model FisherS
$\widehat{H}=-\sum\limits_{\left\langle
ij\right\rangle}J\widehat{a}_{i}^{{\dagger}}\widehat{a}_{j}+\frac{U}{2}\sum\limits_{i}\widehat{n}_{i}\left(\widehat{n}_{i}-1\right)+\sum\limits_{i}\left(\varepsilon_{i}-\mu\right)\widehat{n}_{i}.$
(4b)
Here $\widehat{n}_{i}=\widehat{a}_{i}^{{\dagger}}\widehat{a}_{i}$ is the
number operator at the $i$th site, with $\widehat{a}_{i}^{{\dagger}}$
($\widehat{a}_{i})$ being the boson creation (annihilation) operator. $U$ and
$J$ represent the strengths of the on-site repulsion and of the nearest
neighbor hopping, respectively. $\varepsilon_{i}$ describes an energy offset
due to the harmonic trap, and $\mu$ is the chemical potential. The phase
diagram given by Fig. 1 in the text is calculated with the above combined
potential and Bose-Hubbard model.
Universal exponential density probability distribution. The many-body quantum
state in the QC region can be written as in a general way
$\left|\Psi_{qcr}\right\rangle=\sum_{\left\\{\Sigma
N_{k}=N\right\\}}C\left(N_{1},\cdot\cdot\cdot,N_{2k_{M}+1}\right)\prod_{k}\frac{1}{\sqrt{N_{k}!}}\left(\widehat{a}_{k}^{{\dagger}}\right)^{N_{k}}\left|0\right\rangle.$
(4c)
Here $N_{k}$ denotes the atomic number in the $k$th lattice site. The total
atomic number is then $N=\Sigma_{k=-k_{M}}^{k=k_{M}}N_{k}$, with $2k_{M}+1$
being the total number of occupied lattice sites. For strongly correlated
quantum gases in the QC region, the function
$C\left(N_{1},\cdot\cdot\cdot,N_{2k_{M}+1}\right)$ is extremely complex, and
no analytical expression exists. After switching off the combined potential,
the density distribution is then
$n\left(x,y,z,t\right)=\left\langle\Psi_{qcr}\left(t\right)\right|\widehat{\Psi}^{{\dagger}}\widehat{\Psi}\left|\Psi_{qcr}\left(t\right)\right\rangle.$
(4d)
Assuming that the initial wave function in the $k$th lattice site is
$\varphi_{k}\left(x,y,z,t=0\right)=\varphi_{k\perp}\left(x,y,t=0\right)\varphi_{kz}\left(z,t=0\right)$,
for long-time evolution so that the expanded subcondensates have sufficient
overlapping with each other, we have
$n\left(x,y,z,t\right)\approx\left|\sum_{k}\sqrt{N_{k}}e^{i\phi_{k}^{s}\left(z,t\right)}\varphi_{k\perp}\left(x,y,t\right)\right|^{2}\left|\varphi_{0z}\left(z,t\right)\right|^{2}.$
(4e)
This formula holds in the QC region, because there are non-negligible particle
number fluctuations in each lattice site. Here
$\phi_{k}^{s}\left(z,t\right)=\phi_{k}^{r}+\phi_{k}^{0}$, with $\phi_{k}^{r}$
being a completely random relative phase in the QC region.
$\phi_{k}^{0}=m\left(z-k\lambda/2\right)^{2}/2\hbar t$ can be obtained
directly from the free expansion along $z$ direction. Without the random phase
$\phi_{k}^{r}$, the phase $\phi_{k}^{0}$ will lead to clear interference
fringes along $z$ direction with a period of $4\pi\hbar t/m\lambda$ PedriS .
It is understood that there would be no interference fringes along $z$
direction if $\phi_{k}^{s}$ is a completely random relative phase, as shown in
Figs. 2(a5)-(a6) in the text.
In the QC region above the BKT transition, there are two sorts of random
phases: (i) the completely random phase $\phi_{k}^{s}\left(z,t\right)$ in the
QC region; (ii) the spatially relevant random phase
$\phi_{k\perp}\left(x,y,t\right)$ above the BKT transition. In this region,
the density distribution can be further written as
$\displaystyle n\left(x,y,z,t\right)$ $\displaystyle\approx$
$\displaystyle\left|\sum_{k}\sqrt{N_{k}}e^{i\phi_{k\perp}\left(x,y,t\right)}e^{i\phi_{k}^{s}\left(z,t\right)}\right|^{2}$
(4f)
$\displaystyle\left|\varphi_{0\perp}\left(x,y,t\right)\varphi_{0z}\left(z,t\right)\right|^{2}.$
Here $\phi_{k\perp}\left(x,y,t\right)$ physically originates from the spatial
phase fluctuations of quasi-2D Bose gas above the BKT transition.
The density distribution recorded by a CCD is a 2D density distribution
$n_{2D}\left(y,z,t\right)=\int dxn\left(x,y,z,t\right)$. By assuming that
$\left|\varphi_{0\perp}\left(x,y,t\right)\varphi_{0z}\left(z,t\right)\right|^{2}$
is locally constant (i.e. the local density approximation), after integrating
out the $x$ variable, we have
$\displaystyle n_{2D}\left(y,z,t\right)$ $\displaystyle\approx$
$\displaystyle\left|\frac{1}{\sqrt{2k_{M}+1}}\sum_{k}\alpha_{k}\left(y,z,t\right)e^{i\beta_{k}\left(y,z,t\right)}\right|^{2}$
(4g) $\displaystyle n_{2D}^{0}\left(y,z,t\right),$
with $n_{2D}^{0}\left(y,z,t\right)=\int
dx\left|\sqrt{N}\varphi_{0\perp}\left(x,y,t\right)\varphi_{0z}\left(z,t\right)\right|^{2}$.
$\alpha_{k}\left(y,z,t\right)$ and $\beta_{k}\left(y,z,t\right)$ are
statistically independent random real function, because they originates from
two different random phases $\phi_{k}^{s}\left(z,t\right)$ and
$\phi_{k\perp}\left(x,y,t\right)$. $\beta_{k}\left(y,z,t\right)$ distributes
uniformly between $-\pi$ and $\pi$. The normalization condition requests
$\left\langle\alpha_{k}^{2}\right\rangle=1$.
Based on the central limit theorem Feller , we get the following exponential
density probability distribution
$P\left(n_{2D}\left(x,y,t\right)\right)=\frac{e^{-n_{2D}\left(x,y,t\right)/n_{2D}^{0}\left(x,y,t\right)}}{n_{2D}^{0}\left(x,y,t\right)}.$
(4h)
For uniform $n_{2D}^{0}$, it is straightforward to get Eq. (2) in the text.
For nonuniform $n_{2D}^{0}$, with the local density approximation, Eq. (2) in
the text is a very good approximation obtained from Eq. (4h).
In Figs. 2(b5)-(b6) in the text, we see that the density probability
distribution agrees well with the exponential density probability
distribution. However, because the system temperature is slightly smaller than
the critical temperature for Fig. 2(b5) ($s=22.3$), we expect Fig. 2(b6)
agrees better with the exponential density probability distribution. Here we
give the semilog plot of the density probability distribution for $s=22.3$ and
$s=30.7$, respectively. We do find that the situation of $s=30.7$ agrees
better with the exponential density probability distribution.
Figure 5: Semilog plot of the density probability distribution for $s=22.3$
and $s=30.7$. The blue line gives the exponential density probability
distribution.
System temperature. In our experiment, the total atomic number can be
accurately measured. The atomic numbers of Figs. 2(a4)-(a6) in the text are
$1.10\times 10^{5}$, $1.19\times 10^{5}$ and $1.13\times 10^{5}$,
respectively. To drive the system into the deeper region of superfluidity, we
have to further lower the temperature of an atomic sample before loading it to
the lattice. This is implemented by kicking more atoms out of the magnetic
trap during the evaporative cooling stage. Specifically, for Figs. 2(a2) and
(a3) in the text, the atomic numbers are reduced by about a factor of two
compared with Figs. 2(a4)-(a6) in the text.
However, the measurement of the system temperature is hampered by the lack of
accurate thermometry in optical lattices. Actually, the ratio $T/T_{c}$ is a
more important quantity than $T$ in characterizing the transition from
superfluid to quantum critical regions. Therefore, the inaccuracy of system
temperature does not affect the physics of the present work, as long as
$T/T_{c}$ can be somehow determined with a satisfying accuracy.
As predicted in Ref. Duan and demonstrated in a recent experiment Phillips ,
the appearance of superfluid in an inhomogeneous lattice is associated with a
bimodal structure in the density profile. The condensate fraction $N_{0}/N$
can be obtained by fits to the bimodal distribution. As we have known,
$N_{0}/N$ shows a temperature dependence having a characteristic shape as
$N_{0}/N=1-(T/T_{c})^{\alpha}.$ (4i)
Apparently, $T/T_{c}$ can be directly determined if the parameter $\alpha$ is
known. Blakie _et al._ Tc-Blakie had performed numerical calculations of the
critical temperature as well as the condensate fraction, for a combined trap
similar to that in our work. Their results show that $\alpha$ is close to $3$
at a moderate lattice depth, but it may drop slightly with increasing lattice
depth. Nevertheless, at a high depth level, the condensate fraction is usually
very small, and, as a result, the value of $T/T_{c}$ derived from Eq. (4i)
becomes insensitive to the potential errors of $\alpha$. Therefore, we just
set $\alpha=3$ in the calculation of $T/T_{c}$, which seems to be a reasonable
assumption.
Figure 6: A bimodal fit to the interference pattern in Fig. 2(a3) in the text.
The total atom number is measured as $N\simeq 5.9\times 10^{4}$, and the
atomic cloud was released from the combined trap with a lattice strength of
$s=11.2$. Black line, the measured atomic density which is obtained by
integrating over each column of pixels. Red line, a fitting curve including
six Gaussian peaks, where the three narrow peaks at the upper part are
attributed to superfluid. The condensate fraction is $N_{0}/N\approx 0.58$,
corresponding to $T/T_{c}\approx 0.75$.
For each absorption image, we first obtain the atomic density along the $z$
dimension by integrating pixels in each column. The narrow peaks riding over
broad ones are regarded as the condensate parts. We then fit both the broad
and the narrow peaks using Gaussian profiles, so as to get the condensate
fraction. Apparently, this method works only when $T<T_{c}$. In Fig. 2(a6) in
the text, the interference peaks vanish completely, indicating that $T>T_{c}$.
In this case, we are unable to extract the quantity $T/T_{c}$ any more with
this method. The universal exponential density probability distribution found
in this work also implies that the accurate measurement of system temperature
in the QC region above BKT transition is difficult.
## References
* (1) M. P. A. Fisher, P. B. Weichman, G. Grinstein, D. S. Fisher, Phys. Rev. B 40, 546 (1989).
* (2) P. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001).
* (3) G. D. Lin, W. Zhang, L. M. Duan, Phys. Rev. A 77, 043626 (2008).
* (4) I. B. Spielman, W. D., Phillips, J. V. Porto, Phys. Rev. Lett. 100, 120402 (2008).
* (5) P. B. Blakie, W. X. Wang, Phys. Rev. A 76, 053620 (2007).
* (6) W. Feller, An Introduction to Probability Theory and its Applications Vol. 1 (John Wiley & Sons, New York, 1957).
|
arxiv-papers
| 2010-07-28T06:44:34 |
2024-09-04T02:49:11.909380
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hongwei Xiong, Xinzhou Tan, Bing Wang, Lijuan Cao, Baolong L\\\"u",
"submitter": "Hongwei Xiong",
"url": "https://arxiv.org/abs/1007.4877"
}
|
1007.4921
|
11institutetext: Graduate School of Science and Technology, Hirosaki
University, Hirosaki, 036-8561, Japan 22institutetext: Department of Science,
School of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502,
Japan 33institutetext: Department of Physics, Moscow State University, Moscow,
119992, Russia 44institutetext: Department of Physics, Saitama University,
Saitama, 338-8570, Japan 55institutetext: Kyowa Interface Science Co.,Ltd.,
Saitama, 351-0033, Japan 66institutetext: Inovative Research Organization,
Saitama University, Saitama, 338-8570, Japan 77institutetext: Research
Institute for Science and Engineering, Waseda University, Tokyo, 169-0092,
Japan
77email: konish@si.hirosaki-u.ac.jp
# On the Sensitivity of L/E Analysis of Super-Kamiokande Atmospheric Neutrino
Data to Neutrino Oscillation Part 2
Four Possible L/E Analyses for the Maximum Oscillation by the Numerical
Computer Experiment
E. Konishi 11 Y. Minorikawa 22 V.I. Galkin 33 M. Ishiwata 44 I. Nakamura 44 N.
Takahashi 11 M. Kato 55 A. Misaki 6677
###### Abstract
In the previous paper (Part 1), we have verified that the SK assumption on the
direction does not hold in the analysis of neutrino events occurred inside the
SK detector. Based on the correlation between $L_{\nu}$ and $L_{\mu}$ (Figures
12 and 13 in Part 1) and the correlation between $E_{\nu}$ and $E_{\mu}$
(Figure 14 in Part 1), we have made four possible $L/E$ analyses, namely
$L_{\nu}/E_{\nu}$, $L_{\nu}/E_{\mu}$, $L_{\mu}/E_{\nu}$ and $L_{\mu}/E_{\mu}$.
Among four kinds of $L/E$ analyses, we have shown that only $L_{\nu}/E_{\nu}$
analysis can give the signature of maximum oscillations clearly, not only the
first maximum oscillation but also the second and third maximum oscillation,
while the $L_{\mu}/E_{\mu}$ analysis which are really done by Super-Kamiokande
Collaboration cannot give the maximum oscillation at all. It is thus concluded
from those results that the experiments with the use of the cosmic-ray beam
for neutrino oscillation, such as Super-Kamiokande type experiment, cannot
find the maximum oscillation from $L/E$ analysis, because the incident
neutrino cannot be observed due to its neutrality. Therefore, we would suggest
Super-Kamiokande Collaboration to re-analyze the zenith angle distribution of
the neutrino events which occur inside the detector carefully, because
$L_{\nu}$ and $L_{\mu}$ are alternative expressions of the cosine of the
zenith angle for the incident neutrino and that for the emitted muon,
respectively.
###### pacs:
13.15.+g, 14.60.-z
## 1 Introduction
In Figures 12 and 13 of the preceding paperKonishi2 , we have shown that the
SK assumption on the direction that the directions of the incident neutrinos
are the same as those of the emitted muons does not hold. Also, in Figure 14
of the same paper, we have shown that the energies of the incident neutrinos
cannot be determined from the those of the emitted muons, uniquely. However,
the discrepancies between two variables in Figures 12 and 13 are distinctively
large compared with those in Figure 14. Therefore, non-holding of the SK
assumption on the direction plays an essential role in the $L/E$ analysis for
finding the maximum oscillation (oscillation pattern in neutrino oscillation).
The survival probability of a given flavor is given in Eq.(1), in the case of
Super-Kamiokande Collaboration. The variables for the $L/E$ analysis are
$L_{\nu}$ and $E_{\nu}$, where $L_{\nu}$ denotes the flight length for the
incident neutrino between the generation point of the incident neutrino and
the interaction point of the neutrino concerned in the detector, and $E_{\nu}$
is the energy of the incident neutrino.
Figure 1: $L_{\nu}/E_{\nu}$ distribution without oscillation for 1489.2 live
days (one SK live day).
Figure 2: $L_{\nu}/E_{\nu}$ distribution without oscillation for 37230 live
days (25 SK live days).
Figure 3: Survival probability of $P(\nu_{\mu}\rightarrow\nu_{\mu})$ as a
function of $L_{\nu}/E_{\nu}$ under the neutrino oscillation parameters
obtained by Super-Kamiokande Collaboration .
## 2 $L/E$ Distributions in Our Numerical Computer Experiment
Our computer numerical experiments are carried out in the unit of 1489.2 days.
Hereafter, we call 1489.2 live days as one SK live day. The live days of
1489.2 is the total live days for the analysis of the neutrino events
generated inside the detector by Super-Kamiokande Collaboration Ashie2 . We
repeat one SK live day experiment as much as 25 times, namely, the total live
days for our computer numerical experiments is 37230 live days (25 SK live
days). In Figure 2, we show $L_{\nu}/E_{\nu}$ distribution without oscillation
for one experiment (1489.2 live days) among twenty five computer numerical
experiments. In those numerical experiments, there are statistical
uncertainties only which are due to both the stochastic character in the
physical processes concerned and the geometry of the detectors. Therefore we
add the standard deviation as for the statistical uncertainty around their
average in the forthcoming graphs, if neccessary. In Figure 2, we show the
statistical uncertainty, the standard deviations around their average values
through twenty five experiments. Similarly for other possible combinations of
$L$ and $E$ ( $L_{\nu}/E_{\mu}$, $L_{\mu}/E_{\nu}$ and $L_{\mu}/E_{\mu}$) for
37230 live days (25 SK live days) we did so.
### 2.1 $L_{\nu}/E_{\nu}$ distribution
#### 2.1.1 For null oscillation
In Figures 2 and 2, both distributions show the sinusoidal-like character for
$L_{\nu}/E_{\nu}$ distribution, namely, the appearance of the top and the
bottom, even for null oscillation. The uneven histograms in Figure 2,
comparing with those in Figure 2, show that the statistics of Figure 2 is not
enough compared with that of Figure 2. Roughly speaking, smaller
$L_{\nu}/E_{\nu}$ correspond to the contribution from downward neutrinos,
larger $L_{\nu}/E_{\nu}$ correspond to that from upward neutrinos and
$L_{\nu}/E_{\nu}$ near the minimum correspond to the horizontal neutrinos,
although the real situation is more complicated, because the backscattering
effect in QEL as well as the azimuthal angle effect in QEL could not be
neglected. From Figure 2, we understand that the bottom around 70 km/GeV
denotes the contribution from the horizontal direction and has no relation
with neutrino oscillation in any sence.
Figure 4: $L_{\nu}/E_{\nu}$ distribution with oscillation for 1489.2 live days
(one SK live day), sample No.1.
Figure 5: $L_{\nu}/E_{\nu}$ distribution with oscillation for 1489.2 live days
(one SK live day), sample No.2.
Figure 6: $L_{\nu}/E_{\nu}$ distribution with oscillation for 14892 live days
(10 SK live days).
Figure 7: $L_{\nu}/E_{\nu}$ distribution with standard deviations with
oscillation for 37230 live days (25 SK live days).
Figure 8: The correlation diagram between $L_{\nu}$ and $E_{\nu}$ with
oscillation for 1489.2 live days (one SK live day).
Figure 9: The correlation diagram between $L_{\nu}$ and $E_{\nu}$ with
oscillation for 14892 live days (10 SK live days).
Figure 10: $L_{\nu}/E_{\nu}$ distribution with and without oscillation for
14892 live days (10 SK live days).
Figure 11: The ratio of $(L_{\nu}/E_{\nu})_{osc}/(L_{\nu}/E_{\nu})_{null}$ for
1489.2 live days (one SK live day).
Figure 12: The ratio of $(L_{\nu}/E_{\nu})_{osc}/(L_{\nu}/E_{\nu})_{null}$ for
14892 live days (10 SK live days).
#### 2.1.2 For oscillation (SK oscillation parameters)
The survival probability of a given flavor, such as $\nu_{\mu}$, is given by
$P(\nu_{\mu}\rightarrow\nu_{\mu})=1-sin^{2}2\theta\cdot sin^{2}(1.27\Delta
m^{2}L_{\nu}/E_{\nu}).\,\,\,(1)$
Then, for maximum oscillations under SK neutrino oscillation parameters, we
have
$1.27\Delta m^{2}L_{\nu}/E_{\nu}=(2n+1)\times\frac{\pi}{2},\,\,\,\,\,\,(2)$
where $\Delta m^{2}=2.4\times 10^{-3}\rm{eV^{2}}$. From Eq.(2), we have the
following values of $L_{\nu}/E_{\nu}$ for maximum oscillations.
$\displaystyle L_{\nu}/E_{\nu}$ $\displaystyle=$ $\displaystyle 515{\rm
km/GeV}\,\,\,\,\,for\,\,n=0\,\,\,(3-1)$ $\displaystyle=$ $\displaystyle
1540{\rm km/GeV}\,\,\,for\,\,n=1\,\,\,(3-2)$ $\displaystyle=$ $\displaystyle
2575{\rm km/GeV}\,\,\,for\,\,n=2\,\,\,(3-3)$ $\displaystyle{\rm and}\,{\rm
so}\,{\rm on.}$
In Figure 3, we give the survival probability
$P(\nu_{\mu}\rightarrow\nu_{\mu})$ as a function of $L_{\nu}/E_{\nu}$ under
the neutrino oscillation parameters obtained by Super-Kamiokande
Collaboration. In cosmic ray experiments, the energy spectrum of the incident
neutrino, is convoluted into the survival probability.
In Figure 6, we give one example of the $L_{\nu}/E_{\nu}$ distribution for one
SK live day (1489.2 live days)Ashie2 among twenty five sets of the computer
numerical experiments in the unit of one SK live day. In Figure 6, we give
another example for one SK live day. Arrows A, B and C represent the first,
the second and the third maximum oscillation which are given in Eq. (3-1),
(3-2) and (3-3), respectively. By the definition of our computer numerical
experiments, there are no experimental error bars in $L_{\nu}/E_{\nu}$
distributions in Figures 6 and 6.
In Figure 6, we show the $L_{\nu}/E_{\nu}$ distribution for 14892 live days
(10 SK live days). Compared Figure 620 with Figures 6 and 6, it is clear that
$L_{\nu}/E_{\nu}$ distribution in Figure 6 becomes smoother due to larger
statistics. In Figure 7, we can add the statistical uncertainty (standard
deviation in this case) around their average, because every one SK live day
experiment among twenty five sets of the experiments fluctuates one by one due
to their stochastic character in their physical processes and geometrical
conditions of the detectors concerned. In order to make the image of the
maximum oscillations in $L_{\nu}/E_{\nu}$ distributions clearer, we show the
correlations between $L_{\nu}$ and $E_{\nu}$ in Figures 10 and 10, which
correspond to Figures 6 and 6, respectively. In Figure 10 for one SK live day,
we can observe vacant regions for the events concerned assigned by A, B and C.
In Figure 10 for ten SK live days, the existence of the vacant regions for the
events concerned becomes clearer due to larger statistics.
In Figure 10, we give $L_{\nu}/E_{\nu}$ distribtution with 14892 live days (10
SK live days) in the linear scale which is another expression of the same
content as in Figure 10. Also, it is the survival probability convoluted with
the incident neutrino energy spectrum. If we compare Figure 10 with Figure 3,
then, we clearly see the series of maximum oscillations characterized with n=0
(A), 1(B), 2(C) and so on which are given by Eq.(2). It is clear from Figure
10 that the maximum oscillations with n=0,1 and 2 have the almost same
frequencies 111 Super-Kamiokande Collaboration never mentioned existence of
the second and the third maximum oscillations (n=1 and 2) under the incident
neutrino energy spectrum utilized by Super-Kamiokande Collaboration Honda
(see footnote 1). The situation shown in Figures 10 to 10 shows definitely
that our computer numerical experiment are carried out as exactly as possible
from the view point of the stochastic treatment to the matter.
We have repeated the computer numerical experiment for one SK live day as much
as twenty five times independently, in both cases with oscillation and without
oscillation. Consequently, there are 625 ($=25\times 25$) sets of ratios of
$(L_{\nu}/E_{\nu})_{osc}/(L_{\nu}/E_{\nu})_{null}$ for one SK live day which
correspond to Eq.(1). In Figure 12, we show one example among 625
combinations. In Figure 12, we show the same ratio for 14892 live days (10 SK
live days). In conclusion, from Figures 6 to 12, we can reproduce the minimum
extrema for neutrino oscillation in our $L_{\nu}/E_{\nu}$ analysis. This fact
shows doubtlessly that our computer numerical experiments are done in the
correct manner.
Figure 13: $L_{\mu}/E_{\mu}$ distribution without oscillation for 1489.2 live
days (one SK live day).
Figure 14: $L_{\mu}/E_{\mu}$ distribution without oscillation for 37230 live
days (25 SK live days).
Figure 15: $L_{\mu}/E_{\mu}$ distribution with oscillation for 1489.2 live
days (one SK live day).
Figure 16: $L_{\mu}/E_{\mu}$ distribution with oscillation for 37230 live days
(25 SK live days).
### 2.2 $L_{\mu}/E_{\mu}$ distribution
As physical quantities which can really be observed are $L_{\mu}$ and
$E_{\mu}$ instead of $L_{\nu}$ and $E_{\nu}$, therefore we examine
$L_{\mu}/E_{\mu}$ distribution focusing the existence of the maximum
oscillation.
#### 2.2.1 For null oscillation
In Figure 14, we give one sample for one SK live day (1489.2 live days) from
the totally 37230 live days (25 SK live days) events, each of which has 1489.2
live days. Figure 14 shows the average distribution accompanied by the
statistical uncertainty bar (not experimental error bar). It is clear from
these figures that the existence of the dip or bottom, namely the sinusoidal
character, means the contribution merely from horizontal contribution, having
no relation with any neutrino oscillation character.
Figure 17: The correlation diagram between $L_{\mu}$ and $E_{\mu}$ with
oscillation for 1489.2 live days (one SK live day).
Figure 18: The correlation diagram between $L_{\mu}$ and $E_{\mu}$ with
oscillation for 14892 live days (10 SK live days).
Figure 19: $L_{\mu}/E_{\mu}$ distribution with and without oscillation for
14892 live days (10 SK live days).
Figure 20: The ratio of $(L_{\mu}/E_{\mu})_{osc}/(L_{\mu}/E_{\mu})_{null}$ for
1489.2 live days (one SK live day).
Figure 21: The ratio of $(L_{\mu}/E_{\mu})_{osc}/(L_{\mu}/E_{\mu})_{null}$ for
14892 live days (10 SK live days).
Figure 22: $L_{\mu}/E_{\nu}$ distribution without oscillation for 1489.2 live
days (one SK live day).
Figure 23: $L_{\mu}/E_{\nu}$ distribution without oscillation for 37230 live
days (25 SK live days).
#### 2.2.2 For oscillation (SK oscillation parameters)
In Figures 16 and 16, we give the $L_{\mu}/E_{\mu}$ distributions with
oscillation for 1489.2 live days (one SK live day) and 37230 live days (25 SK
live days), respectively. In Figure 16, we may observe the uneven histogram,
something like curious bottoms coming from neutrino oscillation. However, in
Figure 16 where the statistics is 25 times as much as that of Figure 16, the
histogram becomes smoother and such bottoms disappear, which turns out finally
for the bottoms to be pseudo. It is impossible to extract the neutrino
oscillation parameters from the comparison of Figure 16 with Figure 14.
In Figures 19 and 19, correspondingly, we give the correlation between
$L_{\mu}$ and $E_{\mu}$ for 1489.2 live days (one SK live day) and 14892 live
days (10 SK live days), respectively.
In Figure 19, we give the $L_{\mu}/E_{\mu}$ distribution for 14892 live days
(10 SK live days) in the linear scale which is another expression of the same
content as in Figure 19. As in Figure 19, we cannot find any maximum
oscillation-like phenomena in Figure 19, which is contrast to Figure 10.
It is clear from the figures that we can not observe the maximum oscillation
in $L_{\mu}/E_{\mu}$, on the contrary to Figures 6 to 10 which give the
maximum oscillations. Namely, we may conclude that we can not observe the
sinusoidal flavor transition probability of neutrino oscillation against the
claim by Super-Kamiokande CollaborationAshie1 when we adopt physically
observable quantities, such as $L_{\mu}$ and $E_{\mu}$.
Figure 24: $L_{\mu}/E_{\nu}$ distribution with oscillation for 1489.2 live
days (one SK live day).
Figure 25: $L_{\mu}/E_{\nu}$ distribution with oscillation for 37230 live days
(25 SK live days).
Figure 26: The $L_{\mu}/E_{\nu,SK}$ distribution in comparison with
$L_{\mu}/E_{\nu}$ distribution with oscillation for 14892 day (10 SK live
days).
Figure 27: The $L_{\nu}/E_{\mu}$ distribution without oscillation for 37230
days (25 SK live days).
Figure 28: The $L_{\nu}/E_{\mu}$ distribution with oscillation for 37230 days
(25 SK live days).
In order to confirm the disappearance of the psuedo maximum oscillations, in
Figures 21 and 21, we give the survival probability of a given flavor for
$L_{\mu}/E_{\mu}$ distribution, namely,
$(L_{\mu}/E_{\mu})_{osc}/(L_{\mu}/E_{\mu})_{null}$, for 1489.2 live days (one
SK live day) and 14892 live days (10 SK live days), respectively. In Figure
21, we show one example of $(L_{\mu}/E_{\mu})_{osc}/(L_{\mu}/E_{\mu})_{null}$
among 625 sets of ratios. Comparing Figure 21 with Figure 21, the pseudo dips
in Figure 21 disappear in Figure 21. Thus the histogram becomes a rather
decreasing function of $L_{\mu}/E_{\mu}$ in Figure 21. If we further make
statistics higher, the survival probability for $L_{\mu}/E_{\mu}$ distribution
should be a monotonously decreasing function of $L_{\mu}/E_{\mu}$, whithout
showing any characteristics of the maximum oscillation, which is in contrast
to Figures 12 and 12.
In conclusion, we should say that we can not find any maximum oscillation for
the neutrino oscillation in the $L_{\mu}/E_{\mu}$ distribution.
### 2.3 $L_{\mu}/E_{\nu}$ distribution
Now, we examine the $L_{\mu}/E_{\nu}$ distribution which Super-Kamiokande
Collaboration treat in the thier paper, expecting the evidence for the
oscillatory signatuture in atmospheric neutrino oscillations.
#### 2.3.1 For null oscillation
In Figures 23 and 23, we give the $L_{\mu}/E_{\nu}$ distribution without
oscillation for 1489.2 live days (one SK live day) and 37230 live days (25 SK
live days), respectively. Comparing Figure 23 with Figure 23, the larger
statistics makes the distribution smoother. Also, there is a sinusoidal-like
bottom which has no relation with neutrino oscillation.
#### 2.3.2 For oscillation (SK oscillation parameters)
In Figures 25 and 25, we give the $L_{\mu}/E_{\nu}$ distribution with
oscillation for 1489.2 live days (one SK live day) and 37230 live days (25 SK
live days), respectively. In Figure 25, we may find something like a bottom
which corresponds to the first maximum oscillation near $\sim$200 (km/GeV).
However, such the dip disappears, by making the statistics larger as shown in
Figure 25.
#### 2.3.3 $L_{\mu}/E_{\nu,SK}$ distribution for the oscillation
Instead of $E_{\nu}$ which is correctly sampled from the corresponding
probability functions, let us utilize $E_{\nu,SK}$ which is obtained from the
”approximate” formula (Eq.(6) in the preceding paperKonishi2 ).
We express $E_{\nu}$ described in Eq.(6) of the preceding paperKonishi2
utilized by Super-Kamiokande Collaboration as $E_{\nu,SK}$ to discriminate our
$E_{\nu}$ obtained in the stochastic manner correctly.
In Figure 26, we give $L_{\mu}/E_{\nu,SK}$ distribution for 14892 live days
(10 SK live days), comparing with $L_{\mu}/E_{\nu}$ distribution. It is
understood from the comparison that there is no significant difference between
$L_{\mu}/E_{\nu,SK}$ distribution and $L_{\mu}/E_{\nu}$ one. This fact tells
us that the ”aproximate” formula for $E_{\nu}$ by Super-Kamiokande
Collaboration does not produce so significant error practically. Although this
kind of foumula is not suitable for the treatment of stochastic quantities,
the result is understandable from Figure 14 in the preceding paperKonishi2 .
Also, we can conclude that we do not find any hole corresponding to the
maximum oscillation in $L_{\mu}/E_{\nu}$ or $L_{\mu}/E_{\nu,SK}$
distributions. The reason why the Figure 25 can not show such dip structure as
shown in Figures 6 and 6, comes from the situation that the role of $L_{\nu}$
is much more crucial than that of $E_{\nu}$ in the $L/E$ analysis. Namely,
$L_{\nu}$ cannot be replaced by $L_{\mu}$ at all. Also, see the discussion in
the following subsection 4.4.
Figure 29: The correlation diagram between $L_{\nu}$ and $E_{\mu}$ with
oscillation for 14892 days (10 SK live days).
### 2.4 $L_{\nu}/E_{\mu}$ distribution
#### 2.4.1 For null oscillation
In Figure 28, we give $L_{\nu}/E_{\mu}$ distribution without oscillation for
37230 live days (25 SK live days) of Super-Kamiokande Experiment to consider
statistical fluctuation effect as precisely as possible. It is clear from the
figure that there is not any dip corresponding to the maximum oscillation
which is expected to appear in presence of neutrino oscillation, as it must
be.
#### 2.4.2 For oscillation (SK oscillation parameters)
In Figure 28, we give the corresponding distribution with the oscillation. In
Figure 29, we give the correlation diagram between $L_{\nu}$ and $E_{\mu}$ for
14823 live days (10 SK live days). On the contrary to Figure 25, there are
surely some kinds of holes in Figure 28, and furthermore we can discriminate
the strip pattern in Figure 29, similarly as in Figure 10.
Therefore, we surmise from Figures 28 and 29 that we may observe some ”maximum
oscillation like” quantities which are related to the maximum oscillations in
the $L_{\nu}/E_{\nu}$ distribution through the correlation between $E_{\mu}$
and $E_{\nu}$ shown in Figure 14 in the preceding paperKonishi2 . However, it
seems to be difficult to extract a pair of concrete values of $L_{\nu}$ and
$E_{\nu}$ through the analysis of the $L_{\nu}/E_{\mu}$ distribution. In
Figure 31, we make a comparison between $L_{\nu}/E_{\nu}$ distribution and
$L_{\nu}/E_{\mu}$ distribution where the correlation between $E_{\nu}$ and
$E_{\mu}$ is shown in Figure 14 in the preceding paperKonishi2 . It is clear
from the figure that the $L_{\nu}/E_{\nu}$ distribution demonstrates the
maximum oscillation as already shown in Figures 6 to 10 and the
$L_{\nu}/E_{\mu}$ distribution also demonstrates the maximum oscillation-like
as already shown in Figure 28 and 29. In Figure 31, we give the relation
between $L_{\mu}/E_{\nu}$ distribution and $L_{\mu}/E_{\mu}$ distribution
where the same correlation between $E_{\nu}$ and $E_{\mu}$ holds in the case
of Figure 31. It is also clear from the figures that both the distributions
demonstrate neither the maximum oscillation nor the maximum oscillation-like,
which is also clear from Figures 16 to 19 and Figures 25 to 25. Thus, it can
be concluded from Figures 13 and 14 in the preceding paperKonishi2 and Figure
31 and Figure 31 in the present paper that $L_{\nu}$ plays an essential role
compared with others $L_{\mu}$, $E_{\nu}$ or $E_{\mu}$. In other words, it
should be noticed that $L_{\nu}$ cannot be approximated by $L_{\mu}$, while
$E_{\nu}$ can be obtained approximately from $E_{\mu}$ through some procedure.
Also, such a serious discrepancy between $L_{\nu}$-$L_{\mu}$ relation and
$E_{\nu}$-$E_{\mu}$ relation is shown in the comparison of Figure 31 with
Figure 31.
Figure 30: Comparison between $L_{\nu}/E_{\nu}$ distribution and
$L_{\nu}/E_{\mu}$ distribution with oscillation for 37230 days (25 SK live
days).
Figure 31: Comparison between $L_{\mu}/E_{\nu}$ distribution and
$L_{\mu}/E_{\mu}$ distribution with oscillation for 37230 days (25 SK live
days).
Figure 32: The comparison of $L/E$ distribution for single-ring muon events
due to QEL among Fully Contained Events with the corresponding one by the
Super-Kamiokande Experiment.
## 3 Comparison of $L/E$ Distribution in the Super-Kamiokande Experiment with
our Results
In our classification, the $L/E$ distribution by Super-
Kamiokande Collaboration Ashie2 Ashie1 should be compared directly with our
$L_{\mu}/E_{\nu}$ distributiton. Taking account of their assertion of
existence of the maximum oscillation we compare their results with our results
on $L_{\nu}/E_{\nu}$ in Figure 32 222We read out Fully Contained Events among
total events from Super-Kamiokande Collaboration Ashie2 Ashie1 .. It is clear
from the figure that there are two big differences between them.
One is that we observe the first maximum oscillation ($L_{\nu}/E_{\nu}=515$
km/GeV under the SK oscillation parameters) sharply, while SK observe it in
the wider range of $L_{\nu}/E_{\nu}=100\sim 800$ km/GeV.
Such the lack of the neutrino events over the wide range may be due to their
measurement of $L_{\mu}$, but not $L_{\nu}$, because the given definite
$L_{\nu}$ corresponds to $L_{\mu}$ over a wide range and vice versa (See also
the correlation between $L_{\nu}$ and $L_{\mu}$ in Figure 35 and 38)
Figure 33: Correlation diagram between $L_{\nu}$ and $E_{\nu}$ for
$1.0<L_{\nu}/E_{\nu}<1.26$ (km/GeV) which corresponds to the maximum frequency
of the neutrino events for $L_{\nu}/E_{\nu}$ distribution in our computer
numerical experiment for 14892 live days (10 SK live days).
Figure 34: Correlation diagram between $L_{\nu}$ and $L_{\mu}$ for
$1.0<L_{\nu}/E_{\nu}<1.26$ (km/GeV) under the neutrino oscillation parmeters
obtained by Super-Kamiokande Collaboration for 14892 live days (10 SK live
days).
Figure 35: Correlation diagram between $L_{\mu}$ and $E_{\mu}$ for
$1.0<L_{\nu}/E_{\nu}<1.26$ (km/GeV) which corresponds to the maximum frequency
of the neutrino events for $L_{\nu}/E_{\nu}$ distribution in our computer
numerical experiment for 14892 live days (10 SK live days).
Figure 36: Correlation diagram between $L_{\nu}$ and $E_{\nu}$ for
$20<L_{\nu}/E_{\nu}<25$ (km/GeV) which corresponds to the maximum frequency of
the neutrino events for $L_{\mu}/E_{\nu}$ distribution in SK experiment for
14892 live days (10 SK live days).
Figure 37: Correlation diagram between $L_{\nu}$ and $L_{\mu}$ for
$20<L_{\nu}/E_{\nu}<25$ (km/GeV) under the neutrino oscillation parmeters
obtained by Super-Kamiokande Collaboration for 14892 live days (10 SK live
days).
Figure 38: Correlation diagram between $L_{\mu}$ and $E_{\mu}$ for
$20<L_{\nu}/E_{\nu}<25$ (km/GeV) which corresponds to the maximum frequency of
the neutrino events for $L_{\nu}/E_{\nu}$ distribution in SK experiment for
14892 live days (10 SK live days).
The other is that there is big difference between them as for the position
which give the maximum frequency for the events concerned. Here, we do not
mention to the existence of the maximum oscillation which is derived from the
measurement of $L_{\mu}$ utilized in Super-Kamiokande Collaboration, because
one cannot observe the maximum oscillation, if we utilize $L_{\mu}$ (see
Figures 14 to 19). Conseqently, we examine the second point as for the maximum
frequency for the events concerned. Our computer numerical experiment gives
the maximum frequency for interval $1.0<L_{\nu}/E_{\nu}<1.26$ (km/GeV) as
shown in Figure 32.
In Figure 35, we give the correlation between $L_{\nu}$ and $E_{\nu}$ for
interval $1.0<L_{\nu}/E_{\nu}<1.26$ (km/GeV). It is clear from the figure that
the larger part of the incident neutrino events is occupied by the vertically
downward ones and the smaller part is occupied by the horizontally downward
neutrino events. This is quite reasonable, because more intensive downward
flux contribute to the maximum frequency for the events concerned, compared
with weaker upward flux under the Super-Kamokande neutrino oscillation
parameters.
Figure 39: Correlation diagram between $L_{\mu}$ and $E_{\nu}$ for
$15.8<L_{\mu}/E_{\nu}<31.6$ (km/GeV) which correspond to the maximum frequency
of the neutrino events for $L_{\mu}/E_{\nu}$ distribution in SK experiment for
14892 live days (10 SK live days).
Figure 40: Correlation diagram between $L_{\nu}$ and $L_{\mu}$ for
$15.8<L_{\mu}/E_{\nu}<31.6$ (km/GeV) under the neutrino oscillation parmeters
obtained by Super-Kamiokande Collaboration for 14892 live days (10 SK live
days).
Figure 41: Correlation diagram between $L_{\nu}$ and $E_{\mu}$ for
$15.8<L_{\mu}/E_{\nu}<31.6$ (km/GeV) which correspond to the maximum frequency
of the neutrino events for $L_{\mu}/E_{\nu}$ distribution in SK experiment for
14892 live days (10 SK live days).
In Figure 35, we give the correlation diagram between $L_{\nu}$ and $L_{\mu}$
for the same intervals as in Figure 35. It is clear from Figure 35 that the
majority of the events is concentrated into the squared regions with
$L_{\nu}<10$ km and $L_{\mu}<10$ km. This denotes that the downward incident
neutrinos produce muons toward the forward direction with either smaller or
larger angles and only the smaller part of the downward incident neutrino
events produce the upward muons due to backscattering (1000 to 10,000 km in
$L_{\mu}$) as well as the azimuthal angle effect in QEL. In Figure 35, we give
the correlation diagram between $L_{\mu}$ and $E_{\mu}$ for the same intervals
as in Figure 35. It is clear from this figure that the produced muons with
higher energies are ejected toward the forward and vertical-like directions,
while the produced muon with lower energies may be ejected toward the backward
or holizontal-like direction. Namely, it is concluded from Figures 35, 35 and
35 that the vertically downward neutrino events can contribute to the maximum
frequency, because they are free from neutrino oscillation. It is further
noted that the direction of the produced muon does not coincide with the
original direction of the neutrino.
Now, we examine $L$-$E$ relation at the position for our computer numerical
expermiment where Super-Kamiokande Collaboration give the maximum frequency
for the events ($20<L_{\nu}/E_{\nu}<25$ (km/GeV)). In Figure 38, we show the
correlation between $L_{\nu}$ and $E_{\nu}$ for interval
$20<L_{\nu}/E_{\nu}<25$ (km/GeV).
It is clear from Figure 38 that $L_{\nu}$ distribute over 27 $\sim$ 120 km,
corresponding to $cos\theta_{\nu}=$ \- 0.1 $\sim$ 0, which denotes the
horizontal-like downward neutrino events. The frequency of the horizontal-like
downward neutrino events in Figure 38 are pretty smaller than that of the
vertical-like downward neutrino events in Figure 35 due to smaller solid
angles. In Figure 38, we give the correlation diagram between $L_{\nu}$ and
$L_{\mu}$ for $20<L_{\nu}/E_{\nu}<25$ (km/GeV). It is impressive from the
figure that $L_{\mu}$ distribute over four orders of magnitude (2 km to
$1.2\times 10^{4}$ km), while $L_{\nu}$ cover within one order of magnitude
(20 $\sim$ 120 km). This fact denotes that the effect of the azimuthal angles
in QEL is pretty strong even in the horizontal-like downward neutrino events
in which the produced muons are apparently judged to come from the upward
direction (see Figure 3-c and Figures 8 to 10 in the preceding paperKonishi2
).
In Figure 38, we give the correlation diagram between $L_{\mu}$ and $E_{\mu}$
for the same intervals as in Figure 38. If we compare Figure 38 with Figure
38, then we can find the following interesting situation. As it is clearly
understandable from Figure 38, horizontal-like downward neutrinos produce the
muons in the three different regions, namely, vertical-like downward muons,
horizontal-like downward muons and upward muons. From horizontal-like downward
neutrinos with rather low energies, the vertically downward muons are ejected
with rather large scattering angles. On the other hand, the horizontal-like
downward muons are ejected with rather small angles whose energies are close
to the incident neutrinos energies. Furthermore, the upward muons are produced
either due to backscattering or due to the azimuthal effect in QEL for
horizontal-like incident neutrinos (see Figures 8 and 9 in the preceding
paperKonishi2 ). Thus, from the comparison of Figures 35, 35 and 35 with
Figures 38, 38 and 38, it is reasonable for the the maximun frequency of the
$L_{\nu}/E_{\nu}$ events to occur for $1.0<L_{\nu}/E_{\nu}<1.26$ (km/GeV), and
not to occur for $20<L_{\nu}/E_{\nu}<25$ (km/GeV) where Super-Kamiokande
Collaboration ”assert”.
Finally, we examine the correlation between $L_{\mu}$ and $E_{\nu}$ for
$15.8<L_{\mu}/E_{\nu}<31.6$ (km/GeV) where Super-Kamiokande Collaboration give
the maximum frequency of $L/E$ neutrino events as shown in Figure 32. Although
we compare their frequency with that of our $L_{\nu}/E_{\nu}$ in Figure 32, we
can compare their frequency with that of our $L_{\mu}/E_{\nu}$ in Figure 31,
which shows big difference between them. In Figure 41, we give the correlation
diagram between $L_{\mu}$ and $E_{\nu}$ for $15.8<L_{\mu}/E_{\nu}<31.6$
(km/GeV). In Figures 41 and 41, we give the corresponding correlation diagrams
between $L_{\nu}$ and $L_{\mu}$, and $L_{\nu}$ and $E_{\mu}$, respectively.
It is clear from Figure 41 that Super-Kamiokande Collaboration measure the
vertical-like downward muons. It is also clear from Figures 41 and 41 that
these vertical-like downward muon are produced by the incident neutrinos whose
$L_{\nu}$ are distributed over four orders of magnitude. These incident
neutrinos are classified into two parts. One is the downward incident
neutrinos ($1.0<L_{\nu}<100$ km) and the other ($L_{\nu}>100$ km) is the
upward incident neutrinos. The majority of the incident neutrino is occupied
by the vertical-like downward. However, the frequency of the upward neutrinos
is in the same order of the magnitude as the horizontal-like downward. The
upward incident neutrinos may produce downward muons due to both
backscattering and the azithumal angle effect in QEL. At any rate, for the
measured muons in the case of the maximun frequency of the events, $L_{\nu}$
of the corresponding incident neutrinos distribute over four orders of
magnitude. Shortly speaking, for the maximum frequency of the neutrino events
$15.8<L_{\mu}/E_{\nu}<31.6$ (km/GeV), the magnitude of the $L_{\mu}$ of the
produced muons lie within one order of magnitude (see Figure 41), although the
$L_{\nu}$ of the incident neutrinos which produce these muons distribute over
four orders of magnitude. In other words, it is concluded that Super-
Kamiokande Collaboration do not measure the definite direction of the incident
neutrinos as far as they measure $L_{\mu}$. It is furthermore noticed from the
comparison of Figure 41 with Figure 29 that Figure 41 is obtained from Figure
29 by cutting off the stripe of $15.8<L_{\mu}/E_{\nu}<31.6$ (km/GeV).
Therefore ,we can recognize the vacant region of the neutrino events faintly
in the part between 100 and 1000 (km/GeV) in Figure 41 which is clearly shown
in Figure 29. The vacant region of the events shows indication of neutrino
oscillation.
The summary on Figures from 35 to 41 are as follows; Figures from 35 to 35
represent the mutual relations among $L_{\nu}$, $L_{\mu}$, $E_{\nu}$ and
$E_{\mu}$ near our maximum frequency of $L_{\nu}/E_{\nu}$ distribution. Here,
all the incident neutrinos are occupied by the downward vertical-like
neutrinos, while the majority of the emitted muons is occupied by the downward
muon and the minority is occupied by the upward muon. Figures from 38 to 38
represent the similar mutual relations for our $L_{\nu}/E_{\nu}$ distribution
which correspond to the near the maximum frequency of $L_{\mu}/E_{\nu}$
distribution obtained by Super-Kamiokande Collaboration. Here, almost the
incident neutrinos are occupied by the downward holizontal-like neutrinos,
while about the half of emitted muons is recognized as the downward muon and
the other half is done as the upward muon. Figures from 41 to 41 represent the
mutual similar relations, assuming the numerical values of the maximum
frequency of $L_{\mu}/E_{\nu}$ distribution obtained by Super-Kamiokande
Collaboration. Here, the majority of the emitted muons is occupied by the
horizontal like muons, while their parent neutrinos come from both the
downward neutrinos and the upward ones. The common characteristics through
Figures from 35 to 41 is that for given definite $L_{\nu}$($L_{\mu}$) we find
$L_{\mu}$($L_{\nu}$) which distribute over the four order of magnitudes.
## 4 Conclusion
The assumption made by Super-Kamiokande Collaboration that the direction of
the reconstructed lepton approximately represents the direction of the
original neutrino does not hold even approximately Konishi1 . This is
logically equivalent to the statement that $L_{\nu}$ cannot be replaced by
$L_{\mu}$ even if approximately. This is really clarified in Figures 12 and 13
in the preceding paperKonishi2 .
Although the derivation of $E_{\nu}$ from $E_{\mu}$ (Eq.(6) of the preceding
paperKonishi2 ) is theoretically, irrelevant to the stochastic plobrem,
because of the neglect of the stochastic character in physical processes
concerned, such the approximation does not induce so practically serious error
compared with the assumption of $L_{\nu}\approx L_{\mu}$. As clarified in
Figures 6 to 12, the maximum oscillation in $L/E$ analysis can be observed
only in the $L_{\nu}/E_{\nu}$ distribution and it is quite natural by the
definition of the probability for a given favor whose argument is
$L_{\nu}/E_{\nu}$ (Eq.(1)). As clarified in Figures 16 to 19 and Figures 25 to
26 the maximum oscillation for the presence of neutrino oscillation cannot be
observed from both $L_{\mu}/E_{\mu}$ and $L_{\mu}/E_{\nu}$. The relation
between $L_{\nu}$ and $L_{\mu}$ is too complicate to extract similar
expression to Eq.(6) of the preceding paperKonishi2 for the argument on
$L_{\mu}/E_{\mu}$ and $L_{\mu}/E_{\nu}$ . Similarly in the case of argument of
$L_{\nu}/E_{\nu}$, we can indicate something like the maximum oscillation in
$L_{\nu}/E_{\mu}$ distribution which are shown in Figures 2841 and 28. The
situation is derived from the fact that what plays a decisive role in $L/E$
analysis is $L_{\nu}$, but not $E_{\nu}$, which are clearly shown by comparing
Figures 12 and 13 with Figure 14 in the preceding paperKonishi2 .
As for $L/E$ distribution obtained by Super-Kamiokande Collaboration, we
definitely indicate that the maximum oscillation cannot be observed through
the measurement of $L_{\mu}$. Consequently, we cannot observe the maximum
oscillation in $L/E$ analysis which is carried out in Super-Kamiokande
Collaboration. Furthermore, one cannot find the maximum frequency of $L/E$
events at the position where Super-Kamiokande Collaboration observe, even if
one can observe $L_{\nu}$.
In conclusion, the maximum oscillation in $L/E$ analysis can be observed only
in $L_{\nu}/E_{\nu}$, but not in any other combinations of $L$ with $E$.
However, $L_{\nu}$ is physically unobservable quantities and it cannot be
approximated by $L_{\mu}$, because the assumption between $L_{\nu}$ and
$L_{\mu}$, does not hold even if statistically. Consequently, it should be
concluded that Super-Kamiokande cannot observe the maximum oscillation in
their $L_{\mu}/E_{\nu,SK}$ analysis.
Finally, our conclusion that $L_{\nu}$ cannot be approximate by $L_{\mu}$ is
logically equivalent to the statement that $cos\theta_{\nu}$ cannot be
approximated by $cos\theta_{\mu}$, where $cos\theta_{\nu}$ denotes cosine of
the zenith angle of the incident neutrino and $cos\theta_{\mu}$ denotes that
of the produced muon, respectively Konishi1 . In Super-Kamiokande
Collaboration, they approximate $cos\theta_{\nu}$ as $cos\theta_{\mu}$ ( See
the reproduction of their statements in the 2 page in the present paper ). The
analysis of the zenith angle distribution of the atmospheric neutrino events
by Super-Kamiokande Collaboration will be re-examined in our subsequent
papers.
## References
* (1) Konishi,E et al., arXiv hep-ex/1007.3812v1
* (2) Ashie,Y. et al., Phys. Rev. D 71 (2005) 112005.
* (3) Honda, M., et al., Phys. Rev. D 52 (1996) 4985.
Honda, M., et al., Phys. Rev. D 70 (2004)043008-1.
* (4) Ashie,Y et al., Phys.Rev.Lett.93 (2004)101801-1.
* (5) Konishi,E et al., arXiv hep-ex/0808.0664v2
|
arxiv-papers
| 2010-07-28T10:30:25 |
2024-09-04T02:49:11.916443
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Konishi (1), Y. Minorikawa (2), V.I. Galkin (3), M. Ishiwata (4),\n I. Nakamura (4), N. Takahashi (1), M. Kato (5) and A. Misaki (6) ((1)\n Graduate School of Science and Technology, Hirosaki University, Hirosaki,\n Japan, (2) Department of Science, School of Science and Engineering, Kinki\n University, Higashi-Osaka, Japan, (3) Department of Physics, Moscow State\n University, Moscow, Russia, (4) Department of Physics, Saitama University,\n Saitama, Japan, (5) Kyowa Interface Science Co., Ltd., Saitama, Japan, (6)\n Research Institute for Science and Engineering, Waseda University, Tokyo,\n Japan)",
"submitter": "Eiichi Konishi",
"url": "https://arxiv.org/abs/1007.4921"
}
|
1007.5008
|
# Constraints on Possible Monopole-Dipole Interactions of WISPs from the
Transverse Relaxation Time of Polarized 3He Gas
Changbo Fua,b Thomas R. Gentilea and William M. Snowb aNational Institute of
Standards and Technology, Gaithersburg, MD 20899
bCenter for Exploration of Energy and Matter, Indiana Univ., Bloomington, IN
47408
###### Abstract
Various theories beyond the Standard Model predict new particles with masses
in the sub-eV range with very weak couplings to ordinary matter. A $P$-odd,
$T$-odd, spin-dependent interaction between polarized and unpolarized matter
is one such possibility. Such a monopole-dipole interaction can be induced by
the exchange of spin-$0$ particles. The presence of a possible monopole-dipole
interaction between fermion spins and unpolarized matter would cause an
decreased transverse spin relaxation time $T_{2}$ for a confined gas of
polarized nuclei. By reanalyzing previously existing data on the spin
relaxation times of polarized 3He in gas cells with pressure in the millibar
range and applying the well-established theory of spin relaxation for magnetic
field gradients to gradients in a possible monopole-dipole field, we present
new laboratory constraints on the strength and range of such an interaction.
These constraints represent to our knowledge the best limits on such
interactions for the neutron with ranges between $0.01$ cm and 1 cm.
## 1 Introduction
The possible existence of new particles beyond the Standard Model with masses
in the sub-eV range and very weak couplings to ordinary matter is starting to
attract increased attention[1]. These possible particles are now starting to
be referred to in the literature as WISPs (Weakly-Interacting sub-eV
Particles)[1, 2]. A $P$-odd and $T$-odd interaction between polarized and
unpolarized matter proportional to ${\vec{s}}\cdot{\vec{r}}$ (spin and
distance) is one such possibility. An interaction involving a scalar coupling
$g_{s}$ at one vertex and a pseudoscalar coupling $g_{p}$ at the other vertex
(sometimes referred to as a monopole-dipole interaction in the literature) can
be induced by the exchange of spin-$0$ particles between two vertexes. The
axion is one possible example of such a particle, and many laboratory
experiments and astrophysical observations have searched for it[3].
The monopole-dipole interaction potential between nucleon spins and
unpolarized matter is given by[4]
$V=\hbar^{2}g_{s}g_{p}\frac{\hat{\mathbf{\sigma}}\cdot\mathbf{\hat{r}}}{8\pi
m_{n}}\left(\frac{1}{r\lambda}+\frac{1}{r^{2}}\right)e^{-r/\lambda},$ (1)
where $m_{n}$ is the mass of the nucleon at the polarized vertex, $\hbar$ is
the reduced Planck constant, $\hbar\hat{\sigma}/2$ is the fermion spin,
$\lambda=\hbar/m_{a}c$ is the range of the interaction, $m_{a}$ is the mass of
axion, and $\hat{\mathbf{r}}={\mathbf{r}}/r$ is the unit vector between the
dipole and the mass.
Measurements of the longitudinal relaxation time $T_{1}$ of polarized 3He were
recently used to constrain the interaction strength and range of such a
possible monopole-dipole interaction coupling to neutrons[5, 6]. The method
takes the advantage of the fact that the motion of a polarized species through
a possible monopole-dipole field gradient will cause the polarization to
decay. In Ref. [Serebrov2009423] the author constrained the monopole-dipole
interaction by using the depolarization rate of ultracold neutrons (UCN) in a
material trap. In Ref. [Yuri-T1-3He] the author employed the longitudinal
relaxation time $T_{1}$ of polarized 3He gas at pressures of a few bar to set
a constraint on the monopole-dipole interaction strength.
In this work we constrain the strength of a possible monopole-dipole
interaction of neutrons by using previous measurements of the transverse spin
relaxation time $T_{2}$ of polarized 3He gas at pressures of a few mbar. We
find that this is the best constraint to our knowledge in the distance range
from $0.01$ cm to $1$ cm for a monopole-dipole interaction involving neutrons.
This constraint could be greatly improved in dedicated experiments.
## 2 Spin Relaxation in a Low Pressure Cell
Consider a dipole in a uniform magnetic field $\textbf{B}_{0}$ along the z
axis and a very large mass block of thickness $d$ at position $z$. If a
monopole-dipole interaction exists, the potential has the form[5]
$V(\mathbf{r})=\frac{g_{s}g_{p}N\hbar^{2}\lambda}{4m_{n}}e^{-z/\lambda}(1-e^{-d/\lambda}),$
(2)
where $N$ is the nucleon density of the matter. Consider a spherical cell with
radius $R$ containing low density polarized gas. The polarized gas in the cell
sees both the monopole-dipole field $V$ and the normal magnetic field
$\textbf{B}_{0}$. If $R\gg\lambda$, the average variation of the monopole-
dipole interaction potential over the whole cell can be written as
$\langle\Delta V\rangle\approx\frac{4\pi R^{2}\lambda}{\frac{4}{3}\pi
R^{3}}\frac{g_{s}g_{p}N\hbar^{2}\lambda}{4m_{n}}(1-e^{-d/\lambda}).$ (3)
Because both the monopole-dipole interaction and the magnetic interaction
$\vec{\mu}\cdot\vec{B}$ are of the form $\vec{s}\cdot\vec{\hat{r}}f(r)$, one
would expect an extra contribution to the spin relaxation due to the monopole-
dipole interaction.[7] In analogy with a magnetic field gradient, the
transverse relaxation rate induced by the monopole-dipole field gradient can
be written as[8]
$\frac{1}{T_{2}^{{}^{\prime}}}=(\delta\omega)^{2}\,\tau_{c},$ (4)
where $\tau_{c}$ is the correlation time for the relaxation mechanism under
consideration and $\hbar\,\delta\omega$ is the interaction energy in the
monopole-dipole interaction field. The decrease in the transverse relaxation
rate $\delta\omega$ by the factor $\delta\omega\,\tau_{c}$ due to the motion
of the dipole is known as motional narrowing. Using the correlation time for
the random motion of polarized atoms in an ideal gas
$\tau_{c}\approx\frac{R^{2}}{2D}$, where $D$ is diffusion constant, the
transverse relaxation rate from the monopole-dipole interaction becomes
$\frac{1}{T_{2}^{{}^{\prime}}}\approx\left[\frac{3\lambda^{2}\,g_{s}g_{p}N\hbar^{2}}{4R\,m_{n}}(1-e^{-d/\lambda})\right]^{2}\frac{R^{2}}{2D}.$
(5)
The relaxation rates from independent processes add, yielding a total rate
$1/T_{2}$ giving by
$\frac{1}{T_{2}}=\frac{1}{T_{2}^{(w)}}+\frac{1}{T_{2}^{(\partial
B)}}+\frac{1}{T_{2}^{(dd)}}+\frac{1}{T_{2}^{{}^{\prime}}},$ (6)
where we list the dominant sources: wall collision ($1/T_{2}^{(w)}$),
inhomogeneity of the external magnetic field ($1/T_{2}^{(\partial B)}$),
relaxation related to the dipole-dipole interaction ($1/T_{2}^{(dd)}$), and
inhomogeneity of the monopole-dipole field ($1/T_{2}^{{}^{\prime}}$). By
assuming that the difference between the measured $T_{2}$ and that calculated
from theory is due to the monopole-dipole interaction, one can set a limit on
$g_{s}g_{p}$ using
$g_{s}g_{p}\leq\frac{4\,m_{n}}{3N\hbar\lambda^{2}}\sqrt{\frac{2D}{T_{2}}}\frac{1}{(1-e^{-d/\lambda})},\
\ (\lambda\ll R).$ (7)
## 3 Monopole-dipole interaction coupling constant constraints on the neutron
from previously existing measurements of $T_{2}$ spin relaxation measurements
of 3He
We now use these relations to set limits on the monopole-dipole interaction on
the neutron spins in polarized 3He. In Ref. [long-T2-Gemmel-low-P], $T_{2}$
measurements are reported for a spherical aluminosilicate glass cell of radius
$R=3$ cm filled with 3He to a pressure of $4.5$ mbar. The transverse
relaxation time was measured to be $T_{2}=(60.2\pm 0.1)$ h. The diffusion
constant is $D=0.04$ m2/s. The nucleon density of the aluminosilicate glass is
$N=1.31\times 10^{30}$ m-3. Assuming a cell wall thickness of $1.5$ mm,
typical for glass cells used in magnetometry work of this type, from the
relations above the limit on a monopole-dipole interaction involving the
neutron is
$\displaystyle g_{s}^{(n)}g_{p}^{(n)}\leq\frac{9.8\times
10^{-23}}{\lambda^{2}(1-e^{-0.15/\lambda})},\,\ \lambda[{\rm cm}]\ll 3\ {\rm
cm}.$ (8)
In Ref. [McGregor-T2], a 13.1 mbar, 4.96-cm-diameter spherical cell was used
to study transverse spin relaxation times for polarized 3He in measured
magnetic field gradients. The diffusion constant of 3He in the cell is
$D=0.0137$ m2/s. They measured $T_{2}$ in a nonmagnetic building with low
magnetic field gradients to be 14.86 h. Assuming the cell wall nucleon density
was $1.3\times 10^{30}$ m-3 and the wall thickness is 2 mm, the constraint on
the monopole-dipole interaction strength is
$\displaystyle g_{s}^{(n)}g_{p}^{(n)}\leq\frac{1.2\times
10^{-22}}{\lambda^{2}(1-e^{-0.2/\lambda})},\,\ \lambda[{\rm cm}]\ll 5\ {\rm
cm},$ (9)
which is very close to the constraint above.
file=LowPgraph-1.eps,width=4.1in
Figure 1: Constraints on the monopole-dipole coupling strength
$g_{s}^{(n)}g_{p}^{(n)}$: 1. Solid line, this work, by reanalyzing the data in
Ref. [long-T2-Gemmel-low-P]; 2. Dot line, this work, by reanalyzing the data
in Ref. [McGregor-T2]; 3. Short dash line, from Ref. [Serebrov2009423]; 4.
Dash-dot line, from Ref. [Yuri-T1-3He]; 5. Dash-dot-dot line, from Ref.
[Youdin-HgCs]; 6\. Short-dot line, from Ref. [Bab-UCN].
Finally, we note that there are unpublished reports of even longer $T_{2}$
(140 h) for polarized 3He gas in glass cells[10]. Unfortunately the pressure
and the wall material information are not shown in the reports, hence we do
not use these results to set a limit.
From measurements and theoretical calculations[11], it is known that the
polarization of the 3He nucleus is dominated by the neutron polarization, with
only a small contribution from orbital motion and other effects. Therefore
these limits are to a good approximation interpretable simply as constraints
on a neutron monopole-dipole interaction. With the inclusion of constraints
from $T_{2}$ measurements in other nuclei, such as 129Xe, it would be possible
to place separate constraints on the monopole-dipole interactions of neutrons
and protons. This work is in progress.
## 4 Summary
In this work we present constraints on the strength and range of a possible
monopole-dipole interaction involving the neutron. These new laboratory
limits, set by reanalysis of previous measurements of the transverse spin
relaxation time $T_{2}$ in polarized 3He cells, are the best in existence for
the neutron to our knowledge in the range $0.01$ cm to 1 cm. Constraints on
the monopole-dipole interaction using this method can be significantly
improved in dedicated experiments.
This work was supported in part by the National Science Foundation under award
PHY-0116146.
## References
* [1] G. L. Kane, Nucl. Phys. Procc. Suppl. 51, 178(1996).
* [2] J. Jaeckel, A. Ringwald, arXiv:1002.0329 (2010).
* [3] G. G. Raffelt, Phys. Rep. 198, 1(1990).
* [4] J. E. Moody and F. Wilczek, Phys. Rev. D. 30, 130 (1984).
* [5] A. P. Serebrov, Phys. Lett. B 680, 423(2009).
* [6] Y. N. Pokotilovski, Phys. Lett. B 686, 114(2010).
* [7] D. D. McGregor, Phys. Rev. A 41, 2631(1990).
* [8] D. Pines and C. P. Slichter, Phys. Rev. 100, 1014(1955).
* [9] C. Gemmel,et al., Eur. Phys. J. D 57, 303(2010).
* [10] D. Colegrove, and B. Marton, private communication.
* [11] J. L. Friar, Phys. Rev. C 42, 2310 (1990).
* [12] A. N. Youdin et al., Phys. Rev. Lett. 77, 2170(1996).
* [13] S. Baeßle et al., Phys. Rev. D 75, 075006(2007).
|
arxiv-papers
| 2010-07-28T15:31:17 |
2024-09-04T02:49:11.923746
|
{
"license": "Public Domain",
"authors": "Changbo Fu, Thomas R. Gentile, and William M. Snow",
"submitter": "Changbo Fu",
"url": "https://arxiv.org/abs/1007.5008"
}
|
1007.5037
|
# Current-induced torques in the presence of spin-orbit coupling
Paul M. Haney and M. D. Stiles Center for Nanoscale Science and Technology,
National Institute of Standards and Technology, Gaithersburg, Maryland
20899-6202, USA
###### Abstract
In systems with strong spin-orbit coupling, the relationship between spin-
transfer torque and the divergence of the spin current is generalized to a
relation between spin transfer torques, total angular momentum current, and
mechanical torques. In ferromagnetic semiconductors, where the spin-orbit
coupling is large, these considerations modify the behavior of the spin
transfer torques. One example is a persistent spin transfer torque in a spin
valve: the spin transfer torque does not decay away from the interface, but
approaches a constant value. A second example is a mechanical torque at single
ferromagnetic-nonmagnetic interface.
###### pacs:
85.35.-p, 72.25.-b,
Introduction— Since the prediction bergerdw ; slonc ; berger of spin transfer
torques in non-collinear ferromagnetic metal circuits, they have been the
subject of extensive research ralph ; miltat . The possibility of using spin
transfer torque to improve the commercial viability of magnetic random access
memory (MRAM) katine , and the rich non-equilibrium physics involved establish
the topic as one of practical and fundamental interest. These torques arise
from the exchange interaction between non-equilibrium, current-carrying
electrons and the spin-polarized electrons that make up the magnetization. In
systems where the spin-orbit coupling is weak, the torque on the magnetization
can be computed from the change in the spins flowing through the region
containing the magnetization. This relation is a consequence of conservation
of total spin. Here, we consider systems in which the spin-orbit coupling
cannot be neglected (and hence total spin is no longer conserved).
In systems where spin angular momentum is not conserved, the relationship
between the spin transfer torque and the flow of spins needs to be
generalized. Conservation of total angular momentum implies that mechanical
torques on the lattice of the material accompany changes in the magnetization
richardson ; einstein . This effect has been used for decades to measure the
$g$-factor of metals. More recent theoretical mohanty ; kovalev and
experimental guiti work considers the current-induced mechanical torques
present at the interface of a ferromagnet and non-magnet, similar in spirit to
the spin transfer torques on the magnetization present in spin valves.
In this article we develop a theory for current-induced torques (both spin
transfer torques and mechanical torques) in systems with strong spin-orbit
coupling, and apply it to a model of dilute magnetic semiconductors. We find
that by accounting for the orbital angular momentum of the electrons, we can
relate the change in total angular momentum flow to spin transfer torques and
mechanical torques. We study two system geometries where these torques play
important roles. The first is a spin-valve geometry, which is used to study
the features of spin transfer torques in the presence of spin-orbit coupling.
The second is a single interface between a ferromagnet and non-magnet, which
elucidates the physics underlying current-induced mechanical torques.
Formalism — We consider a Hamiltonian consisting of a spin-independent kinetic
and potential energy $H_{0}=\frac{-\hbar^{2}\nabla^{2}}{2m}+V({\bf r})$, an
exchange splitting $\Delta$, and an atomic-like spin-orbit interaction
parameterized by $\alpha$:
$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{0}+\frac{\Delta}{\hbar}\frac{\left({\bf M}\cdot\hat{\bf
s}\right)}{M_{s}}+\frac{\alpha}{\hbar^{2}}\left(\hat{\bf L}\cdot\hat{\bf
s}\right),$ (1)
where $\hat{\bf L}$ and $\hat{\bf s}$ are the electron angular momentum and
spin operators, respectively footnote1 . The exchange splitting arises from a
magnetization ${\bf M}$, with magnitude $M_{s}$. We treat the magnetization
within mean field theory.
We consider the torque on the magnetization due to electric current flow. The
spin transfer torque $\bm{\tau}_{\rm STT}$ at position ${\bf r}$ from
electronic states with spin density ${\bf s}({\bf r})$ is proportional to the
component of spin transverse to the magnetization nunez : $\bm{\tau}_{\rm
STT}({\bf r})=\frac{d{\bf M}({\bf r})}{dt}=\frac{-\Delta}{\hbar^{2}}\left({\bf
M({\bf r})}\times{\bf s({\bf r})}\right)$. In the absence of spin-orbit
coupling, this torque can be related to the divergence of a spin current,
which offers conceptual and computational simplicity stiles . In the following
we analyze how spin-orbit coupling changes this simple result. One consequence
is an expression for the mechanical torque $\bm{\tau}_{\rm lat}$.
We develop an expression for $\bm{\tau}_{\rm STT}$ by evaluating the time-
dependence of the electron spin and angular momentum densities. To do so, we
adopt a Heisenberg picture of time evolution, and evaluate
$\frac{d\hat{O}({\bf r})}{dt}=\frac{i}{\hbar}\left[H,\hat{\psi}^{\dagger}({\bf
r})\hat{O}\hat{\psi}({\bf r})\right],$ where $\hat{\psi}({\bf r})$ is the
position operator, for the operators $\hat{O}={\bf\hat{s}}$, ${\bf\hat{L}}$.
This procedure leads to ralph :
$\displaystyle\frac{d{\bf\hat{s}}}{dt}=\nabla\cdot\hat{\bf Q}_{\bf s}({\bf
r})-\hat{\bm{\tau}}_{\rm STT}+\frac{\alpha}{\hbar^{2}}\left(\hat{\bf
L}\times\hat{\bf s}\right)$ (2)
where $\hat{\bf Q}_{\bf s}({\bf r})=\hat{\psi}^{\dagger}({\bf r}){\hat{\bf
v}}\otimes{\bf{\hat{s}}}\hat{\psi}({\bf r})$, and the velocity operator is
given by
${\bf{\hat{v}}}=\frac{i\hbar}{2m}\left(\overleftarrow{\nabla}-\overrightarrow{\nabla}\right)$;
here the arrow superscript specifies the direction in which the gradient acts.
In addition:
$\displaystyle\frac{d{\bf\hat{L}}}{dt}=\nabla\cdot\hat{\bf Q}_{\bf L}({\bf
r})-\hat{\bm{\tau}}_{\rm lat}+\frac{\alpha}{\hbar^{2}}\left(\hat{\bf
s}\times\hat{\bf L}\right)$ (3)
where $\hat{\bf Q}_{\bf L}({\bf r})=\hat{\psi}^{\dagger}({\bf
r})\frac{1}{2}\left({\hat{\bf v}}{\bf{\hat{L}}}+{\hat{\bf
L}}{\bf{\hat{v}}}\right)\hat{\psi}({\bf r})$ (the product of non-commuting
operators ${\bf{\hat{L}}}$ and ${\bf{\hat{v}}}$ is symmetrized). We’ve defined
$\hat{\bm{\tau}}_{\rm lat}({\bf r})=\frac{i}{\hbar}\hat{\psi}^{\dagger}({\bf
r})\left[H_{0},{\bf{\hat{L}}}\right]\hat{\psi}({\bf r})$, which is nonzero for
a potentials $V({\bf r})$ which break rotational symmetry footnote2 .
We define a total angular momentum $\hat{\bf J}=\hat{\bf L}+\hat{\bf s}$, a
total angular momentum current $\hat{\bf Q}_{\bf J}=\hat{\bf Q}_{\bf
L}+\hat{\bf Q}_{\bf s}$, and combine Eqs. (2) and (3) to obtain:
$\displaystyle\frac{d\hat{\bf J}}{dt}-\bm{\nabla}\cdot\hat{\bf Q}_{\bf
J}=-\hat{\bm{\tau}}_{\rm STT}-\hat{\bm{\tau}}_{\rm lat}.$ (4)
Finally, we take the expectation value of Eqs. (2-4), replacing operators by
densities. Eq. (4) is our main formal result. When spin-orbit coupling is
important, the total angular momentum in the conduction electrons couples both
to the magnetization and the lattice. The coupling of electron spin to the
lattice requires both spin-orbit coupling and crystal field potential. The
term $\bm{\tau}_{\rm lat}$ changes the physical picture of spin transfer
torque substantially, as is illustrated by considering Eq. (4) for a single
bulk eigenstate: $\frac{d{\bf J}}{dt}$ and $\nabla\cdot{\bf Q_{\bf J}}$
vanish, however $\bm{\tau}_{\rm STT}$ and $\bm{\tau}_{\rm lat}$ may both be
non-zero, implying a coupling from the angular momentum of the lattice to the
magnetization. This coupling flows from the lattice to the orbital subsystem
through the crystal field, which then couples to the spin through spin-orbit
coupling, and finally to the magnetization through the exchange interaction.
Figure 1: Left and right panels shows GaMnAs band structure without and with
spin-orbit, respectively (for $\gamma_{2}=\gamma_{3}=2.4$). (arrows indicate
spin direction of eigenstates). The inset shows the direction of bulk
magnetization, and spin, velocity, and k vectors for a single state (in black,
red, blue, and green). The torque from the misalignment between magnetization
and spin equals the torque from the misalignment between velocity and k
vectors.
Application to DMS — We apply this general formalism to a model of a dilute
magnetic semiconductor (DMS). DMSs are semiconductor host materials which
become ferromagnetic when doped with magnetic atoms. ${\rm Ga_{1-x}Mn_{x}As}$
is the archetype for these materials, and can be described as a system of
local moments of ${\rm Mn}$ $d$-electrons, whose interaction is mediated by
holes in the semiconductor valence band jungwirth . The valence states are
described by the Kohn-Luttinger Hamiltonian $H_{0}^{\rm KL}$, which represents
a small-${\bf k}$ expansion for a periodic $H_{0}$, acting in the $\ell=1$
subspace (describing valence states). It is given by:
$\displaystyle H_{0}^{\rm KL}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\left(\left(\gamma_{1}+4\gamma_{2}\right)k^{2}-\frac{6\gamma_{2}}{\hbar^{2}}\left({\bf
L}\cdot{\bf k}\right)^{2}\right.$ (5)
$\displaystyle\left.-\frac{6}{\hbar^{2}}\left(\gamma_{3}-\gamma_{2}\right)\sum_{i\neq
j}k_{i}k_{j}L_{i}L_{j}\right),$
where ${\bf L}$ are the spin-1 matrices for the $p$-state orbitals,
$\gamma_{1},~{}\gamma_{2},~{}\gamma_{3}$ are Luttinger parameters, and ${\bf
k}$ is the Bloch wave-vector. Figure 1 shows how the presence of spin-orbit
coupling affects the band structure.
For periodic systems the velocity operator can be written as:
${\bf{\hat{v}}}=\frac{1}{\hbar}\frac{\partial H}{\partial{\bf k}}$, and spin
and angular momentum current densities are again defined as symmetrized
products of ${\hat{\bf v}}$ and ${\hat{\bf L}}$, and ${\hat{\bf v}}$ and
${\hat{\bf s}}$. The dynamics of the magnetization occur on a much longer time
scale than that of the electronic states, so we compute the dynamics from a
sum over scattering states, for which $\frac{d{\bf s}}{dt}=\frac{d{\bf
L}}{dt}=0$. For the Luttinger Hamiltonian, the $z$-component of
$\bm{\hat{\tau}}_{\rm lat}$ is:
$\displaystyle{\hat{\tau}}_{\rm lat}^{z}$ $\displaystyle=$
$\displaystyle\left({\hat{\bf v}}\times\hbar{\bf
k}\right)_{z}+\frac{6(\gamma_{2}-\gamma_{3})}{\hbar
m}\left\\{\left(k_{x}L_{y}+k_{y}L_{x}\right),\right.$ (6)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left(k_{x}L_{x}-k_{y}L_{y}\right)\right\\}~{}~{}~{}~{}$
where the brackets on the second term indicate an anticommutator. Other
components are given by cyclic permutation of indices. The first term of Eq.
(6) can be written as ${\hat{\bf v}}\times\hbar{\bf
k}=\frac{d}{dt}\left({\hat{\bf r}}\times\hbar{\bf k}\right)$. This term can be
interpreted as a torque on the crystal angular momentum ${\hat{\bf
r}}\times\hbar{\bf k}$, and results from the misalignment between wave vector
and velocity. It is generically nonzero for any material with a non-spherical
Fermi surface. In the spherical approximation ($\gamma_{2}=\gamma_{3}$), Eq.
(4) implies that the net flux of total angular momentum into a volume is equal
to the change of magnetization plus the crystal angular momentum inside the
volume.
STT in spin-valves — We first consider a system to study the $\bm{\tau}_{\rm
STT}$ term of Eq. (4). Figure 2(a) shows the geometry; current flows in the
$\hat{z}$-direction, perpendicular to the magnetization of both layers. We
focus on the component of torque which is in the plane spanned by the two
magnetization directions. This in-plane torque is determined by the out-of-
plane (or $\hat{z}$-component) spin density nunez . For the results presented
here, we use the parameter values:
$(\gamma_{1},\gamma_{2},\gamma_{3})=(6.85,2.1,2.9)$, $\Delta=0.27~{}{\rm eV}$,
$\alpha=0.11~{}{\rm eV}$, $E_{\rm F}=0.16~{}{\rm eV}$ ($E_{\rm F}$ is measured
from the top of the valence band). The tunnel barrier is described by Eqs. (1)
and (5), with $\Delta=0$, and with an energy offset so that the top of the
valence band is 0.1 eV below $E_{\rm F}$. We calculate the eigenstates
numerically and apply boundary conditions as described in Ref. malik .
Figure 2: (a)Spin valve geometry: FM layers’ magnetization points in the
$\hat{x}$ and $\hat{y}$ (out-of-page) directions. (b) The spin transfer torque
versus position away from the left normal metal-FM interface, which decays to
zero in the absence of spin-orbit coupling, and does not in its presence. (c)
Plot of the total spin transfer torque, the net flux of spin current, and net
flux of total angular momentum current versus FM thickness. The linear
dependence for large thickness is due to a persistent spin transfer torque.
Figure 2(b) shows the spin transfer torque density as a function of distance
away from the interface. We find that for $\alpha=0$ (no spin-orbit coupling),
the torque decays to zero away from the interface, as expected stiles . For
$\alpha\neq 0$, the torque oscillates around a nonzero value, and extends into
the bulk. Figure 2(c) shows that the total spin transfer torque as a function
of ferromagnetic (FM) layer thickness $L_{\rm FM}$ is proportional to
thickness for large $L_{\rm FM}$. This is in contrast to the metallic spin
valve, where the torque is an interface effect and becomes constant for large
$L_{\rm FM}$.
This persistent spin transfer torque arises because the spins of individual
eigenstates are not aligned with the magnetization (see Fig. 1) in the
presence of spin-orbit coupling. The misalignment gives rise to a torque
between the lattice and the magnetization. In equilibrium, these torques
cancel when summed over all occupied states. However, the presence of a
current changes the occupation of the bulk states and can give rise to a
torque manchon ; garate in systems without inversion symmetry. Inversion
symmetry is only very weakly broken in bulk GaMnAs, and is not included in the
Kohn Luttinger Hamiltonian, Eq. (5). Here, interfaces between materials breaks
inversion symmetry.
The combination of an interface and a current flow changes the occupation of
the bulk states near the Fermi energy (depending on the transmission
probabilities of individual states across the interface) and induces coherence
between these states. The change in the occupation probabilities gives rise to
a persistent transverse spin accumulation, which only decays through other
scattering mechanisms not included here (e.g. defect scattering). This spin
accumulation gives rise to the persistent spin transfer torque. The coherence
between the states modifies the spin accumulation and the torque near the
interface but these corrections decay away from the interface due to
dephasing.
Figure 3 shows, as a function of the spin-orbit coupling constant $\alpha$,
the values of total spin transfer torque, the angular momentum current flux,
-$\bm{\tau}_{\rm lat}$, and the persistent contribution to spin transfer
torque (for $L_{\rm FM}=30~{}{\rm nm}$). We determine the persistent
contribution from the slope of the integrated total versus FM width $L_{\rm
FM}$ at large $L_{\rm FM}$ (see Fig. 2c). This procedure neglects the
contributions from coherence near the interface. In this example, the spin
transfer torque increases with the addition of spin-orbit coupling, largely
because of the addition of the persistent term. This qualitative behavior
depends on system parameters: for $E_{\rm F}=0.34~{}{\rm eV}$, for example,
the spin-orbit coupling decreases the total torque.
Figure 3: The total spin transfer torque, the net flux of total angular
momentum, $-\bm{\tau}_{\rm lat}$, and the persistent component of the spin
transfer torque on a FM layer with $L_{\rm FM}=30~{}{\rm nm}$ as $\alpha$ is
increased from 0 to $\alpha_{0}=0.11~{}{\rm eV}$. Figure 4: (a) shows the
system geometry. We take electron particle flow from left to right, and
consider the mechanical torque $\bm{\tau}_{\rm lat}$ in the z direction. (b)
shows slices of the Fermi surface for the different layers, with $\langle{\bf
J}({\bf k})\rangle$ superimposed, and also shows the ${\bf J}$-character of
the states specified by the black circle. Also shown is the transmission
probability for each of the states in the GaAs. (c) shows the total mechanical
torque density in the GaAs as a function of distance from the interface (dark
curve), and the contribution from the single incoming state specified in (b)
(dashed curve).
Nanomechanical torques in wires — We next consider a system which exemplifies
that physics of the $\bm{\tau}_{\rm lat}$ term of Eq. (4): a single interface
between GaMnAs and GaAs, with the direction of the magnetization parallel to
the current flow (see Fig. 4a). This is similar to the geometry considered in
previous theoretical and experimental work mohanty ; kovalev ; guiti . The
vanishing magnetization in GaAs implies $\bm{\tau}_{\rm STT}=0$, so that
$\bm{\tau}_{\rm lat}=\nabla\cdot{\bf Q}_{\bf J}$, and its total value can be
deduced from ${\bf Q}_{\bf J}^{\rm in}-{\bf Q}_{\bf J}^{\rm out}$. We use the
same parameters as before, except $E_{\rm F}=0.06~{}{\rm eV}$, and the top of
the valence band of both layers coincide.
Figure 4c shows $\bm{\tau}_{\rm lat}$ in the GaAs layer as a function of
distance away from the interface (assuming electron particle flow from left to
right). The total torque (dark curve) shows oscillatory decay, while the
torque from a particular channel (light curve) shows simple oscillation. The
behavior of the single channel is illustrated in Fig. 4b. We assume specular
scattering, so that the incident state chosen (black circle) transmits into
the four states of GaAs with equal $k_{x},~{}k_{y}$ (also shown with black
circles). The character of these states, along with the transmission
probability, is shown in Fig. 4b. The incoming state couples most strongly to
the state with similar ${\bf J}$ character, but also partially transmits into
other states with different ${\bf J}$ character and wave vector $k_{z}$. These
different scattering channels interfere with each other, leading to an
oscillatory ${\bf J}(z)$, with an oscillation period inversely proportional to
the splitting of $k_{z}$ wave-vectors of the different sheets of the Fermi
surface. This splitting is from the lattice crystal field and spin-orbit
coupling, the agents responsible for $\bm{\tau}_{\rm lat}$. Different channels
have different oscillation periods, so that their total decays away from the
interface, as happens for spin transfer torques in ferromagnets stiles . For
the parameters used here, we find $Q_{J^{z}}^{\rm in}=1.20\hbar\frac{I}{e}$,
due to the polarization of the states from the magnetization, while
$Q_{J^{z}}^{\rm out}=0.46\hbar\frac{I}{e}$. Mechanisms not considered here,
such as spin-flip scattering, ensure that $Q_{J^{z}}^{\rm out}$ decays to zero
away from the interface.
The mechanical torque is $Q_{J^{z}}^{\rm in}-Q_{J^{z}}^{\rm
out}=0.74\hbar\frac{I}{e}$. For appropriate experimental conditions, this
torque is greater than the thermal fluctuations and is a measurable effect. We
refer the reader to Ref. mohanty ; kovalev ; guiti for details of treatment
of the torsion dynamics and experimental details. The formalism developed here
generalizes previous work to allow for microscopic evaluation of the
electronic structure contribution to the current-induced mechanical torque.
For systems with nonzero magnetization, the microscopic form of
$\bm{\tau}_{\rm lat}$ is necessary to determine the partitioning of total
angular momentum flux between torques on the magnetization and torques on the
lattice. Our theory neglects other mechanisms of spin relaxation, such as
disorder-induced spin-flip scattering, so that full calculations will require
microscopic calculations like these to be embedded in diffusive transport
calculations.
Conclusion— We have shown how atomic-like spin-orbit coupling affects current-
induced torques: both the spin transfer torque on the magnetization and the
mechanical torque on the lattice. In GaMnAs spin valves, we find a
contribution to the spin transfer torque that persists throughout the bulk.
This result may explain experiments which find critical currents which are up
to an order of magnitude smaller than the value expected from a simple
accounting of the net spin current flux chiba ; elsen . For a single interface
between GaMnAs and GaAs, we microscopically compute the mechanical torque due
to scattering from the interface. These results highlight important,
qualitatively different physics at play when spin-orbit coupling is strong.
The authors acknowledge helpful conversations with A. H. MacDonald.
## References
* (1) L. Berger, J. Appl. Phys. 3, 2156 (1978); ibid. 3, 2137 (1979).
* (2) J. Slonczewski, J. Magn. Magn. Mat. 62, 123, (1996).
* (3) L. Berger, Phys. Rev. B 54, 9353 (1996).
* (4) D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2007).
* (5) M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225 (2006).
* (6) J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2007).
* (7) O. W. Richardson, Phys. Rev. 26, 248 (1908).
* (8) A. Einstein and A. de Hass, Verhandlungen der Deutschen Physikalischen Gesellschaft, 17, 152 (1915).
* (9) P. Mohanty et al., Phys. Rev. B 70, 195301 (2004).
* (10) A. A. Kovalev et al., Phys. Rev. B 75, 014430 (2007).
* (11) G. Zolfagharkhani et al., Nature Nanotech. 3, 720 (2008).
* (12) In addition to atomic-like angular momentum, there is a contribution to the total orbital angular momentum from itinerant motion through the lattice. The distinction between “local” and “itinerant” orbital angular momentum is discussed in Ref. vanderbilt . In this work, we consider only the atomic-like contribution.
* (13) T. Thonhauser et al., Phys Rev. Lett. 95, 137205 (2005).
* (14) A. S. Núñez and A. H. MacDonald, Solid State. Comm. 139, 31 (2006).
* (15) M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 (2002).
* (16) For $H_{0}=-\hbar^{2}\nabla^{2}/2m+V({\bf r})$, our definition of $\hat{\tau}_{\rm lat}$ is equivalent to $\left[V({\bf r}),\hat{\bf L}\right]$. We use $\hat{\tau}_{\rm lat}=\left[H_{0},\hat{\bf L}\right]$ in anticipation of other forms of $H_{0}$, in particular the $\bf k\cdot\bf p$ form of the Luttinger Hamiltonian.
* (17) T. Jungwirth et al., Rev. Mod. Phys. 78, 809 (2006).
* (18) A. M. Malik et al., Phys. Rev. B 59, 2861 (1999).
* (19) A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).
* (20) Ion Garate and A. H. MacDonald, Phys. Rev. B 80, 134403 (2009).
* (21) D. Chiba et al., Phys. Rev. Lett. 93, 216602 (2004).
* (22) M. Elsen et al., Phys. Rev B 73, 035303 (2006).
|
arxiv-papers
| 2010-07-28T17:29:33 |
2024-09-04T02:49:11.928447
|
{
"license": "Public Domain",
"authors": "Paul M. Haney and M. D. Stiles",
"submitter": "Paul Haney Mr.",
"url": "https://arxiv.org/abs/1007.5037"
}
|
1007.5147
|
# Jacob’s ladders and the nonlocal interaction of the function $Z^{2}(t)$ with
the function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for the
collections of disconnected sets
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
jan.mozer@fmph.uniba.sk
###### Abstract.
It is shown in this paper that there is a fine correlation of the fourth order
between the functions $Z^{2}[\varphi_{1}(t)]$ and $\tilde{Z}^{2}(t)$,
respectively. This correlation is with respect to two collections of
disconnected sets. Corresponding new asymptotic formulae cannot be obtained
within known theories of Balasubramanian, Heath-Brown and Ivic.
###### Key words and phrases:
Riemann zeta-function
## 1\. The result
In this paper we obtain some new properties of the signal
$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$
generated by the Riemann zeta-function, where
(1.1)
$\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right).$
Let (see [3])
(1.2) $\begin{split}&G_{3}(x)=G_{3}(x;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu}\leq
T+U}\\{t:\ g_{2\nu}(-x)\leq t\leq g_{2\nu}(x)\\},\ 0<x\leq\frac{\pi}{2},\\\
&G_{4}(y)=G_{4}(y;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu+1}\leq T+U}\\{t:\
g_{2\nu+1}(-y)\leq t\leq g_{2\nu+1}(y)\\},\ 0<y\leq\frac{\pi}{2},\end{split}$
(1.3) $U=T^{5/12}\ln^{3}T,$
and the collection of sequences $\\{g_{\nu}(\tau)\\}$, $\tau\in[-\pi,\pi]$,
$\nu=1,2,\dots$ is defined by the equation (see [2], [3], (6))
$\vartheta_{1}[g_{\nu}(\tau)]=\frac{\pi}{2}\nu+\frac{\tau}{2};\
g_{\nu}(0)=g_{\nu},$
where (comp. (1.1)
$\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}.$
Let
(1.4) $G_{3}(x)=\varphi_{1}[\mathring{G}_{3}(x)],\
G_{4}(y)=\varphi_{1}[\mathring{G}_{4}(y)],$
where $y=\varphi_{1}(T),\ T\geq T_{0}[\varphi_{1}]$ is the Jacob’s ladder. The
following theorem holds true.
###### Theorem.
(1.5)
$\begin{split}&\int_{\mathring{G}_{3}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\\\ &\frac{x}{\pi}U\ln\frac{T}{2\pi}+\frac{2x}{\pi}\left(c+\frac{\sin
x}{x}\right)U+\mathcal{O}(xT^{5/12}\ln^{2}T),\\\
&\int_{\mathring{G}_{4}(y)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm d}t=\\\
&\frac{y}{\pi}U\ln\frac{T}{2\pi}+\frac{2y}{\pi}\left(c-\frac{\sin
y}{y}\right)U+\mathcal{O}(yT^{5/12}\ln^{2}T),\end{split}$
where
(1.6) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\ t\to\infty,$
and $c$ is the Euler’s constant and $\pi(t)$ is the prime-counting function.
Let (comp. (1.4)) $T=\varphi_{1}(\mathring{T}),\
T+U=\varphi_{1}(\widering{T+U})$. Similarly to [14], (1.8) we have
(1.7) $\rho\\{[T,T+U];[\mathring{T},\widering{T+U}]\\}\sim(1-c)\pi(T),$
where $\rho$ stands for the distance of the corresponding segments.
###### Remark 1.
Some nonlocal interaction of the functions $Z^{2}[\varphi_{1}(t)],\
\tilde{Z}^{2}(t)$ is expressed by eq. (1.5). This interaction is connected
with two collections of disconnected sets unboundedly receding each from other
(see (1.6), (1.7); $\rho\to\infty$ as $T\to\infty$) - like mutually receding
galaxies (the Hubble law).
###### Remark 2.
The asymptotic formulae (1.5) (comp. (1.3)) cannot be received by methods of
Balasubramanian, Heath-Brown and Ivic (comp. [1]).
This paper is a continuation of the series of works [4]-[18].
## 2\. On big asymmetry in the behaviour of the function
$Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t)$ relatively to the sets
$\mathring{G}_{3}(x)$ and $\mathring{G}_{4}(x)$
We obtain from (1.5)
###### Corollary.
(2.1)
$\begin{split}&\int_{\mathring{G}_{3}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{4}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\\\ &\frac{4}{\pi}U\sin x+\mathcal{O}(xT^{5/12}\ln^{2}T),\
x\in(0,\pi/2],\end{split}$
especially, in the case $x=\pi/2$, we have
(2.2) $\int_{\mathring{G}_{3}(\pi/2)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t-\int_{\mathring{G}_{4}(\pi/2)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t\sim\frac{4}{\pi}U,$
where
$[\mathring{T},\widering{T+U}]\subset\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$.
###### Remark 3.
The formulae (2.1), (2.2) represent the big difference of the areas (measures)
of the figures which correspond to the functions
$Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t),\ t\in\mathring{G}_{3}(x);\qquad
Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t),\ t\in\mathring{G}_{4}(x).$
The reason for this big asymmetry given by (2.2) is probably the fact that the
zeroes of
$\zeta\left(\frac{1}{2}+it\right),\
t\in\mathring{G}_{3}\left(\frac{\pi}{2}\right)\bigcup\mathring{G}_{4}\left(\frac{\pi}{2}\right)$
lie preferably in the set $\mathring{G}_{4}(\pi/2)$.
## 3\. Proof of the Theorem
### 3.1.
The following lemma holds true (see [6], (2.5); [7], (3.3); [14], (4.1))
###### Lemma.
For every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ the following is true
(3.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\
U\in\left(\left.0,\frac{T}{\ln T}\right]\right.,$
where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$, $c$ is the Euler’s constant, $\pi(t)$
is the prime-counting function and
$\begin{split}&\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t),\\\
&\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t},\end{split}$
(see [7], (1.1), (3.1), (3.2)).
###### Remark 4.
The formula (3.1) remains true also in the case when the integral on the
right-hand side of (3.1) is only relatively convergent integral of the second
kind (in the Riemann sense).
In the case (comp. (1.4) $T=\varphi_{1}(\mathring{T}),\
T+U=\varphi_{1}(\widering{T+U})$) we obtain from (3.1)
(3.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$
### 3.2.
First of all, we have from (3.2), for example
(3.3)
$\int_{\mathring{g}_{2\nu(-x)}}^{\mathring{g}_{2\nu(x)}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{g_{2\nu(-x)}}^{g_{2\nu(x)}}f(t){\rm d}t$
(see (1.4)). Next, in the case $f(t)=Z^{2}(t)$, we have the following
$\tilde{Z}^{2}$-transformation
(3.4)
$\begin{split}&\int_{\mathring{G}_{3}(x)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{3}(x)}Z^{2}(t){\rm d}t,\\\
&\int_{\mathring{G}_{4}(y)}Z^{2}[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{G_{4}(y)}Z^{2}(t){\rm d}t.\end{split}$
Let us remind that we have proved in the paper [3] the following mean-value
formulae
(3.5) $\begin{split}&\int_{G_{3}(x)}Z^{2}(t){\rm
d}t=\frac{x}{\pi}U\ln\frac{T}{2\pi}+\frac{2x}{\pi}\left(c+\frac{\sin
x}{x}\right)U+\mathcal{O}(xT^{5/12}\ln^{2}T),\\\ &\int_{G_{4}(y)}Z^{2}(t){\rm
d}t=\frac{y}{\pi}U\ln\frac{T}{2\pi}+\frac{2y}{\pi}\left(c-\frac{\sin
y}{y}\right)U+\mathcal{O}(yT^{5/12}\ln^{2}T).\end{split}$
Now, the formula (1.5) follows from (3.4), (3.5).
###### Remark 5.
The formulae (3.5) are the consequences of their discrete form
$\begin{split}&\sum_{T\leq g_{2\nu}\leq T+U}Z^{2}[g_{2\nu}(\tau)]=\\\
&\frac{1}{2\pi}U\ln^{2}\frac{T}{2\pi}+\frac{1}{\pi}(c+\cos\tau)U\ln\frac{T}{2\pi}+\mathcal{O}(T^{5/12}\ln^{3}T),\\\
&\sum_{T\leq g_{2\nu+1}\leq T+U}Z^{2}[g_{2\nu+1}(\tau)]=\\\
&\frac{1}{2\pi}U\ln^{2}\frac{T}{2\pi}+\frac{1}{\pi}(c-\cos\tau)U\ln\frac{T}{2\pi}+\mathcal{O}(T^{5/12}\ln^{3}T)\end{split}$
(see [3], (10)).
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
## References
* [1] A. Ivic, ‘The Riemann zeta-function‘, A Willey-Interscience Pub., New York, 1985.
* [2] J. Moser, ‘An improvement of the theorem of Hardy-Littlewood on density of zeroes of the function $\zeta\left(\frac{1}{2}+it\right)$‘, Acta Arithmetica, 43 (1983), 21-47 (in russian).
* [3] J. Moser, ‘New mean-value theorems for the function $|\zeta\left(\frac{1}{2}+it\right)|^{2}$‘, Acta Math. Univ. Comen., 46-47 (1985), pp. 21-40, (in russian).
* [4] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, (2008), arXiv:0901.3973.
* [5] J. Moser, ‘Jacob’s ladders and the tangent law for short parts of the Hardy-Littlewood integral’, (2009), arXiv:0906.0659.
* [6] J. Moser, ‘Jacob’s ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral’, (2009), arXiv:0907.0301.
* [7] J. Moser, ‘Jacob’s ladders and the quantization of the Hardy-Littlewood integral’, (2009), arXiv:0909.3928.
* [8] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the sixth order $|\zeta(1/2+i\varphi(t)/2)|^{4}|\zeta(1/2+it)|^{2}$’, (2009), arXiv:0911.1246.
* [9] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the fifth order $Z[\varphi(t)/2+\rho_{1}]Z[\varphi(t)/2+\rho_{2}]Z[\varphi(t)/2+\rho_{3}]\hat{Z}^{2}(t)$ for the collection of disconnected sets‘, (2009), arXiv:0912.0130.
* [10] J. Moser, ‘Jacob’s ladders, the iterations of Jacob’s ladder $\varphi_{1}^{k}(t)$ and asymptotic formulae for the integrals of the products $Z^{2}[\varphi^{n}_{1}(t)]Z^{2}[\varphi^{n-1}(t)]\cdots Z^{2}[\varphi^{0}_{1}(t)]$ for arbitrary fixed $n\in\mathbb{N}$‘ (2010), arXiv:1001.1632.
* [11] J. Moser, ‘Jacob’s ladders and the asymptotic formula for the integral of the eight order expression $|\zeta(1/2+i\varphi_{2}(t))|^{4}|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.2114.
* [12] J. Moser, ‘Jacob’s ladders and the asymptotically approximate solutions of a nonlinear diophantine equation‘, (2010), arXiv: 1001.3019.
* [13] J. Moser, ‘Jacob’s ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.4007.
* [14] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $|\zeta(1/2+it)|$ with $\arg\zeta(1/2+it)$ on the distance $\sim(1-c)\pi(t)$‘, (2010), arXiv: 1004.0169.
* [15] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$ \- transformation of polynomials in $\ln\varphi_{1}(t)$‘, (2010), arXiv: 1005.2052.
* [16] J. Moser, ‘Jacob’s ladders and the oscillations of the function $|\zeta\left(\frac{1}{2}+it\right)|^{2}$ around the main part of its mean-value; law of the almost exact equality of the corresponding areas‘, (2010), arXiv: 1006.4316
* [17] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $Z(t)$ with the function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for a collection of disconneted sets‘, (2010), arXiv: 1006.5158
* [18] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$-transformation of the orthogonal system of trigonometric functions‘, (2010), arXiv: 1007.0108.
|
arxiv-papers
| 2010-07-29T08:42:50 |
2024-09-04T02:49:11.935648
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Michal Demetrian",
"url": "https://arxiv.org/abs/1007.5147"
}
|
1007.5264
|
# Pythagorean triangles within Pythagorean triangles
Konstantine Zelator
Department of Mathematics
301 Thackeray Hall
139 University Place
The University of Pittsburgh
Pittsburgh, PA 15260
USA
and
P.O. Box 4280
Pittsburgh, PA 15203
kzet159@pitt.edu
e-mails: konstantine zelator@yahoo.com
## 1 Introduction
Suppose that $CBA$ is a Pythagorean triangle with sidelengths
$\left|\overline{AB}\right|=c,\ \left|\overline{CA}\right|=b$, and
$\left|\overline{CB}\right|=a$; that is, a right triangle with the right angle
at $C$; and with $a,b,c$ being positive integers such that
$a^{2}+b^{2}=c^{2}$. Then (without loss of generality – $a$ and $b$ may be
switched),
$\left\\{\begin{array}[]{l}a=d(m^{2}-n^{2}),\ b=d(2mn),\ c=d(m^{2}+n^{2})\\\
\\\ {\rm where}\ d,m,n\ {\rm are\ positive\ integers\ such\ that}\\\ \\\ m>n,\
(m,n)=1,\ {\rm and}\ m+1\equiv 1({\rm mod}\ 2)\end{array}\right\\}$ (1)
Note: Throughout this paper, $(X,Y)$ will stand for the greatest common
divisor of two integers $X$ and $Y$.
Thus, the condition $(m,n)=1$ says that $m$ and $n$ are relatively prime,
their greatest common divisor is $1$. Also, the condition $m+n\equiv 1({\rm
mod}2)$ says that $m$ and $n$ have different parities; one of them is even,
the other odd. The formulas in (1), are the well known parametric formulas
describing the entire family of Pythagorean triangles or triples.
A derivation of the formulas can be found in references [1] and [2]. For a
wealth of historic information on Pythagorean triangles see [2] or [3].
Now, consider a point $P$ on the hypotenuse $\overline{AB}$, and let $D$ and
$E$ be the intersection points of the two lines through $P$ and parallel to
$\overline{CA}$ and $\overline{CB}$; with the sides $\overline{CB}$ and
$\overline{CA}$ respectively. Two right triangles are formed; the triangles
$BDP$ and $APE$. Let $x$ and $y$ denote the lengths of line segments
$\overline{DP}$ and $\overline{PE}$ respectively. Also, let
$h_{1}=\left|\overline{BP}\right|$ and $h_{2}=\left|\overline{AP}\right|$.
Then,
$\left\\{\begin{array}[]{l}\left|\overline{DP}\right|=\left|\overline{CE}\right|=x\
{\rm and}\ \left|\overline{PE}\right|=\left|\overline{DC}\right|=y.\\\ \\\
{\rm Thus},\
\left|\overline{BD}\right|=\left|\overline{BC}\right|-\left|\overline{DC}\right|=a-y;\\\
\\\ {\rm and}\
\left|\overline{AE}\right|=\left|\overline{AC}\right|-\left|\overline{CE}\right|=b-x\end{array}\right\\}$
(2)
Both right triangles $BDP$ and $APE$ are similar to the right triangle of
$CBA$. We have the similarity ratios,
$\left\\{\begin{array}[]{ccccc}\dfrac{x}{b}&=&\dfrac{a-y}{a}&=&\dfrac{h_{1}}{c}\\\
\\\ \dfrac{y}{a}&=&\dfrac{b-x}{b}&=&\dfrac{h_{2}}{c}\end{array}\right\\}$ (3i)
(3ii)
Since $a,b,c$ are (positive) integers, it follows, by inspection, from (3i)
that if one of $x,y$, or $h_{1}$ is a rational number, then all three of them
must be rational numbers. Hence, either all three $x,y,h_{1}$ are rationals,
or otherwise, all three of them must be irrational. Likewise, it follows from
(3ii) that either all three $x,y,h_{2}$ are rational or all three are
irrational. Combining these two observations, we infer that
Either all four $x,y,h_{1},h_{2}$ are rational numbers or, otherwise, all four
of them are irrationals.
In Section 2, we state three lemmas from number theory. One of them (Euclid’s
Lemma) is well known. We offer proofs for the other two.
In Section 3, we prove Theorems 1 and 2; Theorem 2 is a corollary of Theorem
1.
In Section 4,we consider and analyze three special cases. These are the cases
when the point $P$ is the midpoint $M$ of the hypotenuse $\overline{AB}$; when
$P$ is the point $I$ where the angle bisector of the $90^{\circ}$ angle at $C$
intersects the hypotenuse $\overline{AB}$, and when the point $P$ is the foot
$F$ of the perpendicular from $C$ to the hypotenuse $\overline{AB}$.
Back to Section 3. In Theorem 1 we prove that the two right triangles $BDP$
and $PEA$ in Figure 1 are either both Pythagorean or neither of them is a
Pythagorean triangle (assuming, of course, that $BCA$ is a Pythagorean
triangle). It then follows, and this is part of Theorem 2, that when the
triangle $BCA$ is a primitive Pythagorean triangle, neither of the triangles,
$BDP$ and $PEA$ are Pythagorean for any position of the point $P$ along the
hypotenuse $\overline{AB}$.
In Section 5 (Theorem 6), we postulate that given a Pythagorean triangle with
side lengths $a=d(m^{2}-n^{2}),\ b=d(2mn)$, and $c=d(m^{2}+n^{2})$, where
$d,m,n$ are positive integers such that $d\geq 2$, $(m,n)=1$, $m>n$, and
$m+n\equiv 1({\rm mod}\ 2)$. Then there are exactly $d-1$ positions of the
point $P$, such that triangles $BDP$ and $PEA$ are both Pythagorean.
In Section 6, we will examine the general question of when, in addition to the
two triangles $BDP$ and $APE$ being Pythagorean, the four congruent right
triangles (within the rectangle $CDPE$) $CDP,\ CEP,\ DCE$, and $EPD$ are also
Pythagorean. We derive a family of non-primitive Pythagorean triangles $CBA$
with that property.
Note: In addition to the notation $(k,\ell)$ denoting the greatest common
divisor of two integers, $k$ and $\ell$, the notation $t|v$, will stand for
“The integer $t$ is a divisor of the integer $v$”.
## 2 Three lemmas from number theory
###### Lemma 1.
(Euclid’s Lemma): Suppose that $a,b,c$, are natural numbers such that $c|ab$
(i.e., $c$ is a divisor of the product $ab$). If $(c,a)=1$, then $c|b$.
For a proof of this well-known result, the reader may refer to [1] or [2].
###### Lemma 2.
Let $m,n$ be positive integers such that $m>n,\ (m,n)=1$, and $m+n\equiv
1({\rm mod}2)$. Then
1. (i)
$(m^{2}+n^{2},2mn)=1$
2. (ii)
$(m^{2}+n^{2},m^{2}-n^{2})=1$
3. (iii)
$(m^{2}-n^{2},2mn)=1$
###### Proof.
1. (i)
We show that $m^{2}+n^{2}$ and $2mn$ have no prime divisors in common. If, to
the contrary, $p$ were a prime divisor of both $m^{2}+n^{2}$ and $2mn$, then
$p$ would be odd, since $m^{2}+n^{2}\equiv 1({\rm mod}2)$, by virtue of the
hypothesis $m+n\equiv 1({\rm mod}2)$. Thus, $p|2mn$ implies, since $(p,2)=1$,
that $p|mn$ (by Lemma 1). But $p$ is a prime, so $p|mn$ implies that $p$ must
divide at least one of $m,n$. If $p|m$, then from $p|m^{2}+n^{2}$, it follows
that $p|n^{2}$, and so $p|n$. Thus, $p|m$ and $p|n$ contradicting the
hypothesis that $(m,n)-1$
2. (ii)
A similar argument left to the reader ($p$ must divide the sum of
$m^{2}+n^{2}$ and $m^{2}-n^{2}$, and their difference. Hence, $p|2n^{2}$ and
$p|2m^{2}$, which since $p$ is odd, eventually implies $p|n$ and $p|m$, a
contradiction).
3. (iii)
A similar argument as in (i).
∎
## 3 A theorem and a corollary
###### Theorem 1.
Suppose that $ABC$ is a Pythagorean triangle with the right angle at $C$; and
with the three sidelengths satisfying the formulas in (1), namely
$a=d(m^{2}-n^{2}),\ b=d(2mn),\ c=d(m^{2}+n^{2})$, where $d,m,n$ are positive
integers such that $m>n,\ (m,n)=1$, and $m+n\equiv 1({\rm mod}\ 2)$.
Let $P$ be a point on the hypotenuse $\overline{AB}$, distinct from $A$ and
$B$. Furthermore, suppose that $D$ is the foot of the perpendicular from $P$
to the side $\overline{CB}$; and $E$ the foot of the perpendicular from $P$ to
the side $\overline{CA}$, as in Figure 1. Then, either both right triangles
$BDP$ and $APE$ are Pythagorean or neither of them is.
Moreover, if they are both Pythagorean, then the
sidelengths$\left|\overline{BD}\right|=a-y,\ \left|\overline{DP}\right|=x$,
and $\left|\overline{BP}\right|=h_{1}$ of the triangle $BDP$ satisfy the
formulas,
$a-y=\delta(m^{2}-n^{2}),\ x=\delta(2mn),\ h_{1}=\delta(m^{2}+n^{2}).$
While the sidelengths $\left|\overline{PE}\right|=y,\
\left|\overline{EA}\right|=b-x,\ \left|\overline{PA}\right|=h_{2}$ of the
triangle $PEA$ satisfy the formulas
$y=(d-\delta)(m^{2}-n^{2}),\ b-x=(d-\delta)(2mn),\
h_{2}=(d-\delta)(m^{2}+n^{2})$
where $\delta$ is a positive integer such that $1\leq\delta\leq d-1$
###### Proof.
Suppose that the triangle $BDP$ is Pythagorean. We will prove that the tringle
$APE$ must also be Pythagorean; then so must the triangle $BDP$ be.
Since the triangle $BDP$ is Pythagorean, its three sidelengths, $x,\ a-y$, and
$h_{1}$ (see Figure 1) must be natural numbers. From (3i)
$\begin{array}[]{rlll}\Rightarrow&x&=&\dfrac{b\cdot h_{1}}{c}\underset{{\rm
by}\ (1)}{=}\dfrac{d(2mn)}{d(m^{2}+n^{2})}\cdot h_{1};\\\ \\\
&x&=&\dfrac{2mnh_{1}}{m^{2}+n^{2}}\end{array}$ (4)
From the conditions $(m,n)=1$ and $m+n\equiv 1({\rm mod}\ 2)$, it follows by
Lemma 2(i) that
$(m^{2}+n^{2},2mn)=1$ (4i)
Since $x$ is a natural number, equation (4) says that the integer
$m^{2}+n^{2}$ must be a divisor of the product $2mnh_{1}$, which clearly
implies, by (4i) and Lemma 1, that $h_{1}$ must be divisible by $m^{2}+n^{2}$.
$h_{1}=\delta\cdot(m^{2}+n^{2})$ (4ii)
for some positive integer $\delta$; and since $h_{1}$ is the length hypotenuse
$\overline{BP}$ (triangle $BDP$), and the point $P$ lies strictly between $A$
and $B$, it is clear that
$h_{1}=\left|\overline{BP}\right|<c=\left|\overline{BA}\right|=d(m^{2}+n^{2}),$
which together with (4ii) clearly show that
$\begin{array}[]{l}1\leq\delta<d;\ {\rm or\ equivalently},\\\ \\\
1\leq\delta\leq d-1\end{array}$ (4iii)
Note that by (4iii), we must have $d\geq 2$. Going back to (4) and using (4ii)
we get
$x=(2mn)\delta$ (4iv)
and so, by (4iv), (3i), (4ii), and (1), we further obtain
$\begin{array}[]{rcl}a-y&=&\delta(m^{2}-n^{2});\ y=a-\delta(m^{2}-n^{2});\\\
\\\ y&=&d(m^{2}-n^{2})-\delta(m^{2}-n^{2})=(d-\delta)(m^{2}-n^{2})\end{array}$
(4v)
By using (1), (3i), and (4v) we also get
$b-x=(d-\delta)(2mn)$
and
$h_{2}=(d-\delta)(m^{2}+n^{2}).$
The proof is complete. ∎
###### Theorem 2.
Let $CBA$ be a Pythagorean triangle, with the 90 degree angle at $C$. Also,
let $\left|\overline{CB}\right|=a,\ \left|\overline{CA}\right|=b$ and
$\left|\overline{BA}\right|=c$, be the three sidelengths so that
$a=d(m^{2}-n^{2}),\ b=d(2mn),c=d(m^{2}+n^{2})$ where $m,n,d$ are positive
integers such that $m>n,\ (m,n)=1$, and $m+n\equiv 1({\rm mod}\ 2)$.
Let $P$ be a point on the hypotenuse $\left|\overline{BA}\right|$ and strictly
between the end-points $B$ and $A$.
Let $D$ and $E$ be the feet of the perpendiculars from the point $P$ to the
sides $\overline{CB}$ and $\overline{CA}$ respectively.
1. (i)
If $d=1$. i.e., if the Pythagorean triangle $CBA$ is primitive, then neither
of the right triangles $PDB$ and $PEA$ is Pythagorean.
2. (ii)
If $d=2$, and the point $P$ is coincident with the midpoint $M$ of the
hypotenuse $\overline{BA}$, then both triangles $PDB$ and $PEA$ are
Pythagorean. Otherwise, if $P\neq M$, neither of these two triangles is
Pythagorean.
3. (iii)
If $d=3$, and the point $P$ is such that
$\dfrac{\left|\overline{PB}\right|}{\left|\overline{PA}\right|}=\dfrac{1}{3}$
or $\dfrac{2}{3}$, then both triangles $PDB$ and $PEA$ are Pythagorean.
Otherwise, if
$\dfrac{\left|\overline{PB}\right|}{\left|\overline{PA}\right|}\neq\dfrac{1}{3},\
\dfrac{2}{3}$, then neither of these triangles are Pythagorean.
###### Proof.
1. (i)
If $d=1$, then neither of the two right triangles, $BDP$ and $PEA$ can be
Pythagorean since according to Theorem 1, the natural number $\delta$ must
satisfy $1\leq\delta\leq d-1$, which is impossible when $d=1$.
2. (ii)
Suppose that $d=2$.
If the point $P$ coincides with the midpoint $M$ of the hypotenuse
$\overline{BA}$, then each of the triangles $BDP$ and $PEA$ is half the size
of the triangle $CBA$. So, by inspection,
$\begin{array}[]{rcl}\left|\overline{BD}\right|=\left|\overline{PE}\right|&=&\dfrac{a}{2}=\dfrac{2(m^{2}-n^{2})}{2}=m^{2}-n^{2}\\\
\\\
\left|\overline{DP}\right|=\left|\overline{EA}\right|&=&\dfrac{b}{2}=\dfrac{2(2mn)}{2}=2mn\\\
\\\
\left|\overline{BP}\right|=\left|\overline{PA}\right|&=&\dfrac{c}{2}=\dfrac{2(m^{2}+n^{2})}{2}=m^{2}+n^{2},\end{array}$
which proves that both triangles $BDP$ and $PEA$ are (in fact primitive)
Pythagorean triangles. Conversely, if both triangles are Pythagorean, then by
Theorem 1, it follows that $1\leq\delta\leq d-1=2-1=1,$ $1\leq\delta\leq 1,\
\delta=1$ which establishes that each of the triangles is half the size of
triangle of $CBA$; which implies that $P$ is the midpoint of $\overline{BA}$.
3. (iii)
Assume that $d=3$.
Suppose
$\dfrac{\left|\overline{PB}\right|}{\left|\overline{PA}\right|}=\dfrac{1}{3}$
or $\dfrac{2}{3}$. If
$\dfrac{\left|\overline{PB}\right|}{\left|\overline{PA}\right|}=\dfrac{1}{3}$,
then the triangle $BDP$ is $\dfrac{1}{3}$ the size of triangle $CBA$ and the
triangle $PEA$ is $\dfrac{2}{3}$ the size of $CBA$. We have,
$\begin{array}[]{rcll}\left|\overline{BD}\right|&=&\dfrac{a}{3}=\dfrac{3(m^{2}-n^{2})}{3}=m^{2}-n^{2},&\left|\overline{DP}\right|=\dfrac{b}{3}=\dfrac{3(2mn)}{3}=2mn,\\\
\\\
\left|\overline{PB}\right|&=&\dfrac{c}{3}=\dfrac{3(m^{2}+n^{2})}{3}=m^{2}+n^{2}&\end{array}$
and $\left|\overline{PE}\right|=\dfrac{2a}{3}=2(m^{2}-n^{2}),\
\left|\overline{EA}\right|=\dfrac{2b}{3}=2(2mn),$
$\left|\overline{PA}\right|=\dfrac{2c}{3}=2(m^{2}+n^{2})$. It is clear that
both triangles $BDE$ and $PEA$ are Pythagorean.
The argument for the case
$\dfrac{\left|\overline{PB}\right|}{\left|\overline{PA}\right|}=\dfrac{2}{3}$
is similar (we omit the details).
Now, the converse. Assume that both triangles, $BDP$ and $PEA$, are
Pythagorean. Then by Theorem 1 we must have,
$1\leq\delta\leq d-1=3-1=2;\ \ \delta=1\ {\rm or}\ 2.$
Using the formulas for the sidelengths (of triangles $BDP$ and $PEA$), found
in Theorem 1 we easily see that
$\dfrac{\left|\overline{PA}\right|}{\left|\overline{PB}\right|}=\dfrac{1}{3}$,
if $\delta=1$. While
$\dfrac{\left|\overline{PA}\right|}{\left|\overline{PB}\right|}=\dfrac{2}{3}$,
if $\delta=2$. The proof is complete.
∎
## 4 Three special cases
1. A.
Case 1: When the point $P$ is the midpoint $M$ of the hypotenuse
$\overline{BA}$
By inspection, it is clear that all six right triangles $BDP,\ PEA,\ CDP$,
$EPD,\ DCE$, and $PEC$ are all congruent and each of them is half the size of
triangle $BCA$. Clearly then, by (1), these six triangles will be Pythagorean
if and only if the integer $d$ in (1) is even.
###### Theorem 3.
Let $BCA$ be a Pythagorean triangle with the 90 degree angle at $C$ and
$\left|\overline{CB}\right|=a=d(m^{2}-n^{2}),\
\left|\overline{CA}\right|=b=d(2mn),$
$\left|\overline{BA}\right|=c=d(m^{2}+n^{2})$, where $d,m,n$ are postive
integers such that $m>n,\ (m,n)=1$ and $m+n\equiv 1({\rm mod}\ 2)$.
Let $M$ be the midpoint of the hypotenuse $\overline{BA}$ and $D,E$ the feet
of the perpendiculars from $M$ to the sides $\overline{CB}$ and
$\overline{CA}$ respectively (so $D$ and $E$ are the midpoints of
$\overline{CB}$ and $\overline{CA}$). Then, the six right angles, $BDM,\ MEA,\
CDM,\ EMD,\ DCE$, and $MEC$ are congruent and have sidelengths as follows:
length of horizontal side | $=\dfrac{d}{2}(2mn)=dmn$
---|---
length of veritical side | $=\dfrac{d(m^{2}-n^{2})}{2}$
length of hypotenuse | $=\dfrac{d(m^{2}+n^{2})}{2}$
If $d$ is an even natural number, then the above six triangles are
Pythagorean; otherwise, if $d$ is odd, they are non-Pythagorean.
2. B.
Case 2: When the point $P$ is the foot $I$ of the angle bisector of the 90∘
angle at $C$
Using the notation of Theorem 1, we have
$\left|\overline{BD}\right|=a-y,\left|\overline{DI}\right|=x,\\\ \\\
\left|\overline{BP}\right|=h_{1},\ \left|\overline{EA}\right|=b-x,\\\ \\\
\left|\overline{EI}\right|=y,\ {\rm and}\ \left|\overline{IA}\right|=h_{2}.$
Clearly, we have $x=y$ in this case. Note that the four congruent isosceles
right triangles $DCI,\ IEC,\ DCE,\ DIE$ cannot be Pythagorean (no Pythagorean
triangle is isosceles).
By Theorem 1, the two right triangles $BDI$ and $IEA$ are either both
Pythagorean or neither of them are. If they are both Pythagorean, then by
Theorem 1 we have, in particular, $x=\delta(2mn)$ and
$y=(d-\delta)(m^{2}-n^{2})$ with $m,n,d,\delta$ being positive integers such
that $m>n,\ (m,n)=1,\ m+n\equiv 1({\rm mod}\ 2)$ and $1\leq\delta\leq d-1$
(and so $d\geq 2$).
Since $x=y$, we must have
$\delta(2mn)=(d-\delta)(m^{2}-n^{2})$ (5)
By Lemma 2(iii), we know that $(m^{2}-n^{2},2mn)=1$. So, by Lemma 1 and (5) it
follows that $2mn|d-\delta$ and $m^{2}-n^{2}|\delta$ which, in turn, leads to
(when we go back to (5))
$\left\\{\begin{array}[]{l}\delta=K\cdot(m^{2}-n^{2})\\\ \\\
d-\delta=K\cdot(2mn),\\\ \\\ {\rm for\ some\ positive\ integer}\ K.\\\ \\\
{\rm Hence}\ d=k\cdot(m^{2}-n^{2}+2mn)\end{array}\right\\}$ (5i)
Note that clearly, from (5i), $1\leq\delta\leq d-1$. In fact, the smallest
possible value of $d$ is $7$; obtained for $K=1$ and $m=2,n=1$. Moreover,
$1\leq\delta\leq d-4$ since the smallest possible value of $K\cdot(2mn)$ is
$4$.
Using (5i) and Theorem 1, one can compute in terms of $m,n$, and $K$. The
other four sidelengths of the triangles $BDI$ and $IEA$. Also, by (5i) we get
$x=\delta(2mn)=K(2mn)(m^{2}-n^{2})=y$. We now state the following theorem.
###### Theorem 4.
Let $CBA$ be a Pythagorean triangle with the $90^{\circ}$ angle at $C$ and
sidelengths given by
$\left|\overline{CB}\right|=a=d(m^{2}-n^{2}),\
\left|\overline{CA}\right|=b=d(2mn),\
\left|\overline{BA}\right|=c=d(m^{2}+n^{2}),$
where $d,m,n$ are positive integers such that, $m>n,\ m+n\equiv 1({\rm mod}\
2)$, and $(m,n)=1$. Let $I$ be the foot of the perpendicular of the angle
bisector (of the $90^{\circ}$ angle at $C$) to the hypotenuse $\overline{BA}$.
Also, let $D$ and $E$ be the feet of the perpendiculars from the point $I$ to
the sides $\overline{CB}$ and $\overline{CA}$ respectively. Then, the two
right triangles $BDI$ and $IEA$ are both Pythagorean precisely when (i.e., if
and only if), $d=K\cdot(m^{2}-n^{2}+2mn)$ for some integer $K$. If
$d=K\cdot(m^{2}-n^{2}+2mn)$, then the sidelengths of triangle $BDI$ are given
by $\left|\overline{DI}\right|=x=K\cdot(2mn)(m^{2}-n^{2}),\ \
h_{1}=\left|\overline{BI}\right|=K(m^{2}-n^{2})(m^{2}+n^{2})=K(m^{4}-n^{4})$,
and $\left|\overline{BD}\right|=a-y=K\cdot(m^{2}-n^{2})^{2}$ and the
sidelengths of triangle $IEA$ are given by
$\begin{array}[]{l}\left|\overline{IE}\right|=y=K(2mn)(m^{2}-n^{2}),\\\ \\\
\left|\overline{EA}\right|=b-x=K(2mn)(2mn)=K\cdot(2mn)^{2},\end{array}$
and $h_{2}=\left|\overline{IA}\right|=K\cdot(2mn)(m^{2}+n^{2})$.
If the integer $d$ is not divisible by $m^{2}-n^{2}+2mn$, then neither of the
triangles, $BDI$ and $IEA$, is Pythagorean.
3. C.
Case 3: When the point $P$ is the foot $F$ of the perpendicular from the
vertex $C$ to the hypotenuse $\overline{BA}$
In this part, instead of using Theorem 1, we will first compute the
sidelengths of the triangles $BDF,\ FEA$, and the four congruent triangles
$FDC$, $DFE,\ DCE$, and $CFE$ in terms of (the sidelengths) $a,b,c$. After
that we will implement the formulas in (1) in order to express the above
sidelengths in terms of the integers $d,m,n$.
After that we will implement Lemma 2 to be able to draw the conclusions which
will lead to Theorem 5. Note that since $F$ is the foot of the perpendicular
from $C$ to the hypotenuse $\overline{BA}$, the aforementioned six right
triangles are all similr to the triangle $CBA$. Let $\omega$ and $\varphi$ be
the degree measures of the angles $\angle CBA$ and $\angle CAB$ respectively
(see Figure 3).
We have (and we set)
$\left\\{\begin{array}[]{l}\left|\overline{CB}\right|=a,\
\left|\overline{CA}\right|=b,\ \left|\overline{BA}\right|=c\\\ \\\
\left|\overline{DF}\right|=\left|\overline{CE}\right|=x,\
\left|\overline{EA}\right|=b-x\\\ \\\
\left|\overline{DC}\right|=\left|\overline{FE}\right|=y,\
\left|\overline{BD}\right|=a-y\\\ \\\ \left|\overline{BF}\right|=h_{1},\
\left|\overline{FA}\right|=h_{2},\
\left|\overline{CF}\right|=\left|\overline{DE}\right|=h\end{array}\right\\}$
(6)
Furthermore,
$\sin\omega=\dfrac{y}{h}=\cos\varphi=\dfrac{b}{c}\ {\rm and}\
\cos\omega=\dfrac{h}{b}=\dfrac{a}{c};$
and thus $h=\dfrac{ab}{c}$, which implies
$y=h\cdot\cos\varphi=h\cdot\dfrac{b}{c}=\dfrac{ab}{c}\cdot\dfrac{b}{c}=\dfrac{ab^{2}}{c^{2}}$.
So, $a-y=a-\dfrac{ab^{2}}{c^{2}}=\dfrac{a(c^{2}-b^{2})}{c^{2}}=({\rm since}\
c^{2}=a^{2}+b^{2})\dfrac{a\cdot a^{2}}{c^{2}};\ a-y=\dfrac{a^{3}}{c^{2}}$
Next we calculate the lengths $x$ and $b-x$. We have
$\tan\omega=\cot\varphi=\dfrac{y}{x}$; $\cot\omega=\tan\varphi=\dfrac{x}{y}$,
and $\tan\varphi=\dfrac{a}{b}$ which gives
$\dfrac{x}{y}=\dfrac{a}{b},x=\dfrac{a}{b}\cdot y$. Since
$y=\dfrac{ab^{2}}{c^{2}}$ (see above), we obtain
$x=\dfrac{a}{b}\cdot\dfrac{ab^{2}}{c^{2}}=\dfrac{ba^{2}}{c^{2}}$. From this we
get $b-x=b-\dfrac{ba^{2}}{c^{2}}=\dfrac{b(c^{2}-a^{2})}{c^{2}}=\dfrac{b\cdot
b^{2}}{c^{2}}=\dfrac{b^{3}}{c^{2}}$, since $c^{2}-a^{2}=b^{2}$.
Also, $\sin\omega=\dfrac{x}{h_{1}};\ h_{1}=\dfrac{1}{\sin\omega}\cdot
x=\dfrac{c}{b}\cdot\dfrac{ba^{2}}{c^{2}}=\dfrac{a^{2}}{c}$. Similarly, we have
$\sin\varphi=\dfrac{y}{h_{2}};\ h_{2}=\dfrac{1}{\sin\varphi}\cdot
y=\dfrac{c}{a}\cdot\dfrac{ab^{2}}{c^{2}}=\dfrac{b^{2}}{c}$. We summarize these
lengths as follows:
Sidelengths of triangle $BDF$
$\left(\left|\overline{BD}\right|=a-y=\dfrac{a^{3}}{c^{2}},\
\left|\overline{DF}\right|=x=\dfrac{ba^{2}}{c^{2}},\
\left|\overline{BF}\right|=h_{1}=\dfrac{a^{2}}{c}\right)$ (6i)
Sidelengths of triangle $FEA$
$\left(\left|\overline{FE}\right|=y=\dfrac{ab^{2}}{c^{2}},\
\left|\overline{EA}\right|=b-x=\dfrac{b^{3}}{c^{2}},\
\left|\overline{FA}\right|=h_{2}=\dfrac{b^{2}}{c}\right)$ (6ii)
Sidelengths of the four congruent triangles $FDC,\ DFE,\ DCE,\ CFE$
$\begin{array}[]{rcl}\left(\left|\overline{DC}\right|\right.&=&\left|\overline{FE}\right|=y=\dfrac{ab^{2}}{c^{2}},\
\left|\overline{DF}\right|=\left|\overline{CE}\right|\\\ \\\
&=&\left.x=\dfrac{ba^{2}}{c^{2}},\
\left|\overline{CF}\right|=\left|\overline{DE}\right|=\dfrac{ab}{c}=h\right)\end{array}$
(6iii)
Next, we combine the length formulas in (6i), (6ii), and (6iii) with the
formulas in (1), since $CBA$ is a Pythagorean triangle, to obtain the
following.
$\left\\{\begin{array}[]{rcl}a-y&=&\dfrac{d\cdot(m^{2}-n^{2})^{3}}{(m^{2}+n^{2})^{2}},\
x=\dfrac{d\cdot(m^{2}-n^{2})^{2}\cdot(2mn)}{(m^{2}+n^{2})^{2}}\\\ \\\
y&=&\dfrac{d\cdot(m^{2}-n^{2})\cdot(2mn)^{2}}{(m^{2}+n^{2})^{2}},\
b-x=\dfrac{d\cdot(2mn)^{3}}{(m^{2}+n^{2})^{2}}\\\ \\\
h_{1}&=&\dfrac{d\cdot(m^{2}-n^{2})^{2}}{m^{2}+n^{2}},\
h_{2}=\dfrac{d\cdot(2mn)^{2}}{m^{2}+n^{2}}\\\ \\\
h&=&\dfrac{d\cdot(2mn)\cdot(m^{2}-n^{2})}{m^{2}+n^{2}}\end{array}\right\\}$
(7)
The following lemma from number theory is well-known and comes in handy.
###### Lemma 3.
Let $i_{1},\ i_{2},\ i_{3},\ e_{1},\ e_{2},\ e_{3}$ be positive integers such
that $(i_{1},i_{2})=1=(i_{1},i_{3})$. Then,
1. (a)
$\left(i^{e_{1}}_{1},\ i^{e_{2}}_{2}\right)=1$
2. (b)
$\left(i^{e_{1}}_{1},\ i^{e_{2}}_{2}\cdot i^{e_{3}}_{3}\right)=1$
It follows from Lemmas 2 and 3 that
$\left\\{\begin{array}[]{l}\left(\left(m^{2}+n^{2}\right)^{2},\
\left(m^{2}-n^{2}\right)^{3}\right)=1,\\\ \\\
\left(\left(m^{2}+n^{2}\right)^{2},\ \left(2mn\right)^{2}\right)=1\\\ \\\
\left(m^{2}+n^{2},\ \left(m^{2}-n^{2}\right)^{2}\right)=1,\\\ \\\
\left(m^{2}+n^{2},\ \left(2mn\right)^{2}\right)=1\\\ \\\ \left(m^{2}+n^{2},\
\left(2mn\right)\cdot\left(m^{2}-n^{2}\right)\right)=1\\\ \\\
\left(\left(m^{2}+n^{2}\right)^{2},\
\left(m^{2}-n^{2}\right)^{2}\cdot\left(2mn\right)\right)=1\\\ \\\
\left(\left(m^{2}+n^{2}\right)^{2},\
\left(m^{2}-n^{2}\right)\cdot\left(2mn\right)^{2}\right)=1\end{array}\right\\}$
(8)
A careful look at formulas (7) and the coprimeness conditions in (8), in
conjunction with Lemma 1, reveals that either all six triangles, $BDF,\ FEA,\
FDC,\ DFE,\ DCE$, and $CFE$ are Pythagorean; or none of them are.
They are all Pythagorean precisely (i.e., if and only if) the integer $d$ is
divisible by $(m^{2}+n^{2})^{2}$, i.e., when
$\left\\{\begin{array}[]{l}d=K\cdot(m^{2}+n^{2})^{2}\\\ \\\ {\rm for\ some\
positive\ integer}\ K\end{array}\right\\}$ (9)
This is precisely when all seven numbers $y,\ a-y,\ x,\ b-x,\ h_{1},\ h_{2}$,
and $h$ are integers. When (9) holds true, we can compute, via (7) and (61),
(6ii), and (6iii) all the sidelengths in terms of the integers $m,n$, and $K$.
We have the following theorem.
###### Theorem 5.
Let $CBA$ be a Pythagorean triangle, with the $90$-degree angle at $C$. With
$\left|\overline{CB}\right|=a=d(m^{2}-n^{2}),\
\left|\overline{CA}\right|=b=d(2mn),\
\left|\overline{BA}\right|=d(m^{2}+n^{2})=c$, where $d,m,n$ are positive
integers such that $m>n,\ (m,n)=1$, and $m+n\equiv 1({\rm mod}2)$.
Also, let $F$ be the foot of the perpendicular from the vertex $C$ to the
hypotenuse $\overline{BA}$. Then, the six similar triangles $BDF,\ FEA$, (and
the four congruent ones) $FDC,\ DFE,\ DCE,\ CFE$ are either all Pythagorean or
none of them are. They are all Pythagorean precisely when (i.e., if and only
if) $d=K\cdot(m^{2}+n^{2})^{2}$, for some positive integer $K$. When $d$
satisfies the said condition, the sidelengths of the above six triangles are
given by the following formulas.
For triangle $BDF$
$\left|\overline{BD}\right|=a-y=K\cdot\left(m^{2}-n^{2}\right),\
\left|\overline{DF}\right|=x=K\cdot\left(m^{2}-n^{2}\right)^{2}\cdot(2mn),\
{\rm and}\
h_{1}=K\cdot\left(m^{2}+n^{2}\right)\cdot\left(m^{2}-n^{2}\right)^{2}$
For triangle $FEA$:
$\left|\overline{FE}\right|=y=K\cdot\left(m^{2}-n^{2}\right)\cdot(2mn)^{2},\
\left|\overline{EA}\right|=b-x=K\cdot(2mn)^{3}$, and
$h_{2}=K\cdot\left(m^{2}+n^{2}\right)\cdot(2mn)^{2}$.
For the four congruent triangles $FDC,\ DFE,\ DCE,CFE$:
$\left|\overline{DC}\right|=\left|\overline{FE}\right|=y=K\cdot\left(m^{2}-n^{2}\right)\cdot\left(2mn\right)^{2}$,
$\left|\overline{DF}\right|=\left|\overline{CE}\right|=x=K\cdot\left(2mn\right)\cdot\left(m^{2}-n^{2}\right)^{2},$
and
$h=\left|\overline{CF}\right|=\left|\overline{DE}\right|=K\cdot(2mn)\cdot\left(m^{2}-n^{2}\right)\cdot\left(m^{2}+n^{2}\right)=K\cdot(2mn)\left(m^{4}-n^{4}\right)$.
Numerical Examples
If we take $K=1$ and $mn\leq 4$, then $K=1$ and $m=2,\ n=1$; or $K=1$ and
$m=4,n=1$.
1. (a)
$K=1,\ m=2,\ n=1$. We obtain the following:
$\begin{array}[]{l}d=1\cdot\left(2^{2}+1^{2}\right)^{2}=5^{2}=25,\ h=60,\
h_{1}=45,\ h_{2}=80,\\\ \\\ y=48,\ a-y=75-48=27,\ x=36,\ b-x=100-36=64,\\\ \\\
a=75,\ b=100,\ c=125\end{array}$
2. (b)
$K=1,\ m=4,\ n=1$. We have the following:
$\begin{array}[]{l}d=289,\ a-y=15,\ x=1800,\ h_{1}=3825,\\\ \\\ y=960,\
b-x=512,\ h_{2}=1088,\ h=1404\\\ \\\ a=4335,\ b=2312,\ c=4913\end{array}$
## 5 Exactly $(d-1)$ positions of $P$
Given a Pythagorean triangle $CBA$, as in Figure 1, and with the point $P$ on
the hypotenuse $\overline{BA}$, and $D$ and $E$ being the perpendicular
projections of $P$ on the sides $\overline{CB}$ and $\overline{CA}$
respectively. We know from Theorem 1 that either both triangles $BDP$ and
$PEA$ are Pythagorean, or neither of them are. The integer $\delta$, as
described in Theorem 1 must satisfy $1\leq\delta\leq d-1$; which means that
$d\geq 2$ is a necessary condition. There are $(d-1)$ choices for $\delta$. If
we subdivide the hypotenuse $\overline{BA}$ into $d$ equal length segments,
each segment having length $m^{2}+n^{2}$, it is easily seen that for each such
position of the point $P$ both triangles $BDP$ and $PEA$ are Pythagorean.
There are exactly $(d-1)$ such positions for the point $P$ along the
hypotenuse $\overline{BA}$. These are the points $P_{1},\ldots,P_{d-1}$; so
that each of the consecutive line segments
$\overline{BP}_{1},\overline{P_{1}P_{2}},\ldots,\overline{P_{d-1}A}$ (exactly
$d$ line segments) has length $m^{2}+n^{2}$.
We postulate the following theorem.
###### Theorem 6.
Let $CBA$ be a Pythagorean triangle with the $90^{\circ}$ angle at the vertex
$C$. With sidelengths given by $\left|\overline{CB}\right|=a=d(m^{2}-n^{2})$,
$\left|\overline{CA}\right|=b=d(2mn),\
\left|\overline{BA}\right|=d(m^{2}+n^{2})$, where $d,m,n$ are positive
integers such that $d\geq 2,\ m>n,\ (m,n)=1$, and $m+n\equiv({\rm mod}\ 2)$.
Also, let $P_{1},\ldots,P_{d-1}$ be the $(d-1)$ points on the hypotenuse
$\overline{BA}$ such that the $d$ consecutive line segments
$\overline{BP}_{1},\ \overline{P_{1}P_{2}},\ldots,\overline{P_{d-1}A}$ have
equal lengths; each having length $m^{2}+n^{2}$. Then there are exactly
$(d-1)$ points $P$ on the hypotenuse $\overline{BA}$ such that both trangles
$BDP$ and $PEA$ are Pythagorean where $D$ and $E$ are the feet of the
perpendiculars from $P$ to the sides $\overline{CB}$ and $\overline{CA}$
respectively. These $(d-1)$ points are precisely the points
$P_{1},\ldots,P_{d-1}$ described above. Furthermore, each pair of Pythagorean
triangles $BD_{i}P_{i}$ and $P_{i}E_{i}A$ have sidelengths given by
$\left|\overline{BD}_{i}\right|=i\cdot\left(m^{2}-n^{2}\right),\
\left|\overline{D_{i}P_{i}}\right|=i(2mn),\
\left|\overline{BP_{i}}\right|=i\left(m^{2}+n^{2}\right),\
\left|\overline{P_{i}E_{i}}\right|=(d-i)\left(m^{2}-n^{2}\right),\
\left|\overline{E_{i}A}\right|=(d-i)(2mn),\
\left|\overline{P_{i}A}\right|=(d-i)\left(m^{2}+n^{2}\right)$, for
$i=1,\ldots,d-1$; and where $D_{i}$ and $E_{i}$ are the perpendicular
projections of the point $P_{i}$ onto the sides $\overline{CB}$ and
$\overline{CA}$ respectively.
## 6 Other cases
In this section, we explore the following question. If in addition to the two
triangles in Figure 1, $BDP$ and $PEA$ being Pythagorean, we require that the
four congruent triangles $DCE,\ PEC,\ CDP,\ EPD$, also be Pythagorean. What
are the necessary and sufficient conditions for this to occur?
For these four congruent triangles to be Pythagorean, the
integers$x=\left|\overline{DP}\right|=\left|\overline{CE}\right|$ and
$y=\left|\overline{DC}\right|=\left|\overline{PE}\right|$ must satisfy the
condition,
$x^{2}+y^{2}=\ {\rm perfect\ square}.$
Combining this with Theorem 6 leads to the following theorem.
###### Theorem 7.
Let $CBA$ be a Pythagorean triangle with the $90$ degree angle at the vertex
$C$; and with sidelengths,
$a=\left|\overline{CB}\right|=d\left(m^{2}-n^{2}\right)$,
$b=\left|\overline{CA}\right|=d(2mn),\
c=\left|\overline{BA}\right|=d\left(m^{2}+n^{2}\right)$ where $d,m,n$ are
positive integers such that $d\geq 2,\ m>n,\ (m,n)=1$, and $m+n\equiv 1({\rm
mod}\ 2)$. Let $P$ be a point on the hypotenuse $\overline{BA}$, and $D$ and
$E$ be the feet of the perpendiculars from the point $P$ onto the sides
$\overline{CB}$ and $\overline{CA}$ respectively. Also, let
$x=\left|\overline{DP}\right|=\left|\overline{CD}\right|,\
y=\left|\overline{DC}\right|=\left|\overline{PE}\right|$ so that
$a-y=\left|\overline{BD}\right|\ {\rm and}\ b-x=\left|\overline{EA}\right|.$
Then, the two right triangles $BDP$ and $PEA$, as well as the four congruent
triangles, $DCE,\ PEC,\ CDP,\ EPD$, are all (six triangles) are Pythagorean if
and only if there exist positive integers $D,M,N$ such that
$M>N,\ (M,N)=1,\ M+N\equiv 1({\rm mod}\ 2)$
and with either
$\left\\{\begin{array}[]{l}y=\delta\left(m^{2}-n^{2}\right)=D\cdot\left(M^{2}-N^{2}\right)\\\
\\\ x=(d-\delta)\cdot(2mn)=D\cdot(2MN)\\\ \\\ \delta\ {\rm a\ positive\
integer\ such\ that\ }1\leq\delta\leq d-1\end{array}\right\\}$ (10i)
or
$\left\\{\begin{array}[]{l}y=\delta\left(m^{2}-n^{2}\right)=D\cdot(2MN)\\\ \\\
x=(d-\delta)(2mn)=D\cdot\left(M^{2}-N^{2}\right)\\\ \\\ \delta\ {\rm a\
positive\ integer\ such\ that\ }1\leq\delta\leq d-1\end{array}\right\\}$
(10ii)
The following example shows that there exist nonprimitive Pythagorean
triangles such that there is no point $P$ on the hypotenuse $\overline{BA}$
such that all six triangles $BDP,\ PEA,\ DCE,\ PEC,\ CDP,\ EPD$, are
Pythagorean.
Example: Take $d=5,\ m=2,\ n=1$. Then the sidelengths of triangle $CBA$ are
$a=5\cdot\left(2^{2}-1^{2}\right)=15,\ b=5\cdot(2\cdot 2\cdot 1)=20$, and
$c=5\cdot\left(2^{2}+1^{1}\right)=25$. The possible values of the integer
$\delta$ are $\delta=1,2,\ldots,d-1=1,2,3,4$. Using the formulas
$y=\delta\left(m^{2}-n^{2}\right)$ and $x=(d-\delta)(2mn)$ we have the
following.
1. 1.
$\delta=1:\ y=3,\ x=(5-1)\cdot 4=16$
and $y^{2}+x^{2}=9+256=265$ not an integer square.
2. 2.
$\delta=2,\ y=2\cdot 3=6,\ x=(5-2)\cdot 4=12$
and $y^{2}+x^{2}=36+144=180$, not a perfect square.
3. 3.
$\delta=3,\ y=3\cdot 3=9,\ x=(5-3)\cdot 4=8$
and $y^{2}+x^{2}=81+64=145$, not an integer square.
4. 4.
$\delta=4,\ y=4\cdot 3=12,\ x=(5-4)\cdot 4=4$
and $y^{2}+x^{2}=144+16=160$, not a perfect square.
There are many ways in which one can use the conditions (10i) or (10ii) of
Theorem 7 in order to produce families of Pythagorean triangles such that each
member (of those families) has the property that there is a point $P$ on its
hypotenuse such that all six triangles (as described in Theorem 7) are
Pythagorean. We produce such a family.
Family 1: Consider (10i):
$\left.\begin{array}[]{rcl}y&=&\delta\left(m^{2}-n^{2}\right)=D\left(M^{2}-N^{2}\right)\\\
\\\ x&=&(d-\delta)(2mn)=D(2MN)\end{array}\right\\}$ (10i)
Let $K$ be a positive integer.
Take $D=K\cdot mn\left(m^{2}-n^{2}\right)$.
From the second equation in (10i) we obtain
$d-\delta=K\cdot MN\left(m^{2}-n^{2}\right)$
and from the first equation in (10i) we get
$\delta=Kmn\left(M^{2}-N^{2}\right).$
Hence,
$d=\delta+KMN\left(m^{2}-n^{2}\right)=K\cdot\left[mn\left(M^{2}-N^{2}\right)+MN\left(m^{2}-n^{2}\right)\right]$.
Obviously $1\leq\delta\leq d-1$ and $d\geq 2$, as required. We have the
following.
Family 1
Let $m,n,M,N$ be positive integrs such that $m>n,\ (m,n)=1$, $m+n\ \equiv({\rm
mod}\ 2),\ M>N,\ (M,N)=1\ M+N\equiv 1({\rm mod}\ 2)$. Also, let $K$ be a
positive integer and $\delta=Kmn\left(M^{2}-N^{2}\right),\
d=K\cdot\left[mn\left(M^{2}-N^{2}\right)+MN\left(m^{2}-n^{2}\right)\right]$.
Consider the Pythagorean triangle $CBA$ with sidelengths
$\left|\overline{CB}\right|=a=d\left(m^{2}-n^{2}\right),$$\left|\overline{CA}\right|=b=d(2mn),\
\left|\overline{BA}\right|=c=d\left(m^{2}+n^{2}\right)$. Let $P$ be the point
on the hypotenuse $\left|\overline{BA}\right|$ such that
$\left|\overline{BP}\right|=h_{1}=\delta\left(m^{2}+n^{2}\right)$; and let $D$
and $E$ be the perpendicular projections of $P$ onto the sides $\overline{CB}$
and $\overline{CA}$ respectively. Then all six right triangles $BDP,\ PEA,\
DCE,\ PEC,\ CDP$ and $EPD$ are Pythagorean.
## References
* [1] Rosen, Kenneth H., Elementary Number Theory and Its Applications, Fifth edition (2005). ISBN: 0-321-23707-2, Pearson, Addison Wesley. For Pythagorean triangles see pp. 510-515. For Lemma 1, see page 109.
* [2] Sierpinski, W., Elementary Theory of Numbers, original edition, Warsaw, Poland (1964). ISBN: 0-568-52758-3, Elsevier Publishing (1988) For Pythagorean triangles see pp. 38-42. For Lemma 1, see page 14. For Lemma 3, see page 15.
* [3] Dickson, L. E., History of the Theory of Numbers, Vol. II, AMS Chelsea Publishing, Providence, RI (1992). (unaltered textual reprint of the original book, first published by Carnegie Institute of Washinton in 1919, 1920, and 1923.) ISBN: 0-8218-1935-6 For Pythagorean triangles see pp. 165-190.
|
arxiv-papers
| 2010-07-29T16:58:32 |
2024-09-04T02:49:11.942025
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/1007.5264"
}
|
1007.5338
|
FERMILAB-FN-0298
# Very Big Accelerators as Energy Producers
R.R. Wilson Fermi National Accelerator Laboratory
Batavia, IL 60510 USA
(August 9, 1976)
One consequence of the application of superconductivity to accelerator
construction is that the power consumption of accelerators will become much
smaller. This raises the old possibility of using high energy protons to make
neutrons which are then absorbed by fertile uranium or thorium to make a
fissionable material like plutonium that can be burned in a nuclear reactor
111Shades of E. O. Lawrence’s MTA project! See Atomic Shield, page 425, by R.
Hewlett and F. Duncan. I have also learned that there is a project in Canada
to use an accelerator this way. See W. Metz, Science, July 23, 1976, Page
307.. The Energy Doubler/Saver being constructed at Fermilab is to be a
superconducting accelerator that will produce 1000 GeV protons. The expected
intensity of about $10^{12}$ protons per second corresponds to a beam power of
about 0.2 MW. The total power requirements of the Doubler will be about 20 MW
of which the injector complex will use approximately 13 MW, and the
refrigeration of the superconducting magnets will use about 7 MW. Thus the
beam power as projected is only a few orders of magnitude less than the accele
ator power. But each 1000 GeV proton will produce about 60,000 neutrons in
each nuclear cascade shower that is releaseq in a block of uranium, and then
most of these neutrons will be absorbed to produce 60,000 plutonium a toms.
Each of these when burned will Subsequently release about 0.2 GeV of fission
energy to make a total energy of 12,000 GeV (20 ergs) for each 1000 GeV
proton. Inasmuch as megawatts are involved, it appears to be worthwhile to
consider the cost of making the protons to see if there could be an overall
energy production.
A high energy proton accelerator seems at first glance to be an unlikely
device for producing energy in this manner but high energy does have a few
special advantages. One is that once the protons have been injected and
accepted by the accelerator at the low energy of injection, then the
subsequent losses during the acceleration to high energy are essentially zero.
Another is that space charge limitations of intensity become less important at
high energy. A third advantage is that the betatron oscillations of the beam
are damped as the energy increases. This means that the emittance of the beam
gets better as the energy grows and hence the beam can be transferred to rings
of smaller aperture (lower refrigeration costs) as the energy and hence ring
size increases. This assumes that the accelerator consists of a series of
magnet rings of ever increasing radius. Alternatively, more protons can be
injected into a given aperture and accelerated. The main point is that the
number of neutrons produced is roughly proportional to the beam power and this
can be made large by increasing both the intensity and the energy of the
protons. The intensity of an accelerator usually runs into a hard limit
imposed by space charge and resonance phenomenon, but the energy can be
increased without limit.
A disadvantage of very high proton energy is that the relative number of
neutrons produced in the nuclear shower gets somewhat less as the energy of
the initial proton increases because the fraction of the electromagnetic
component increases with energy. This adverse effect could be vitiated in part
by accelerating deuterons (or particles of even higher mass) so that the
energy per nucleon is lower. In any case, once a proton has gone through the
expensive and inefficient business of being produced and accelerated to about
10 GeV energy, then all the energy possible should be pumped into it during
the efficient part of the acceleration process that brings it to high energy.
Now let us consider the process a little more quantitatively. We assume that
the number of neutrons produced in a nuclear shower in U238 when $N$ protons
of energy $E$ are incident is proportional to the energy in the shower, i.e.,
to $aNE$, where the constant $a$ will have a value of roughly 60 neutrons per
GeV, about half of which come from fission of U238 Then the potential power
$P$ produced overall when these neutrons are nearly all absorbed to form
plutonium is given by
$P=0.2aNE-P_{0}-bNE,$
where 0.2 is the energy per fission in GeV, $P_{0}$ is the power required to
run the accelerator when no protons are accelerated, and $bNE$ is the power
used by the RF system to accelerate the protons. The constant $b$ is a measure
of the inefficiency of producing RF power from the electrical mains; it has a
value of roughly 2 although this might be improved to a value of about 1.5.
For a given accelerator, if $0.2a$ is greater than $b$, as it is for the
Energy Doubler, then there will be a proton intensity $N_{0}$ above which more
power will be available from the plutonium burned than will be used by the
accelerator. This value is given by
$N_{0}\approx P_{0}/10E.$
For the Energy Doubler at Fermilab plus all its injector stages, $P_{0}$ will
be roughly 20 MW, and $N_{0}$ then comes out to be about $2\times 10^{13}$
protons per second. This is about twenty times the expected intensity - but it
is far from being unattainable. An intensity of $10^{13}$ protons per second
will make about 15 Megawatts of fission energy available; this does not count
the energy put into the accelerator. For an overall production of 15 MW, an
intensity of $3\times 10^{13}$ protons would be required; or more generally
$P\approx 10(N-N_{0})E.$
Now the above calculations are pessimistic in that they assume that the beam
energy is thrown away. Furthermore there will be considerable fission energy
produced by fast neutrons being absorbed in the U238 during the course of the
cascade shower. Let us now assume that both of these forms of energy are
available for use. Mr. Andreas VanGinneken has made a rough computer
calculation following the shower development and assuming that each U238
fission leads to 3 Pu239 nuclei, but that the Pu239 fission is not used as a
source of neutrons. His results are tabulated in Table I and illustrate the
relative increase of the electromagnetic component with energy. Remaking the
calculations for the rougher break-even intensity made above, we find $N_{0}$
to be close to $10^{13}$ protons per second, an intensity which might be
attainable even in the Doubler by increasing the rate of pulsing, or by
resorting to a stacking process, or by adding a second ring of superconducting
magnets. He has also made the same calculation at 100 and 300 GeV.
It appears then that an accelerator that is similar to the Energy Doubler
could be made to be energy productive. Starting from the beginning, perhaps
the injector system could be made to use less ertergy and produce higher
intensities. One could also optimize the size of the rings so as to reduce the
refrigeration requirement and to raise the intensity capability. An important
improvement (and one assumed in all the above) would be to make at least two
magnet-rings back-to-back so that the electric energy could oscillate from one
to the other rather than just being put back into the power mains. This would
double the intensity and would make efficient use of the injection system, for
one ring could be loaded while the other was accelerating the protons
previously loaded into it. My guess is that the optimum energy of the
accelerator will be smaller, but this will depend on details of the injection
system and on the shape of the real estate that is available. Values of
$N_{0}$ for other energies are given in Table I: they vary roughly inversely
with the energy but 100 GeV gives fractionally about 30% more energy per
proton than does 1000 GeV.
Capital costs, of course, are equally significant. The bare-bones accelerator
might cost, as a very rough guess, about $200 million for a plant that would
produce the fuel to power a 100 MW fission plant. A larger installation might
of course cost relatively less per MW.
When the kind of intense proton beam considered above is to be absorbed on a
target, a serious problem arises. Even with our present intensities at 400
BeV, i.e., about $10^{12}$ protons/sec, targets tend to disappear. One can
imagine that the target might be a slurry of uranium oxide and heavy water.
Each pulse of protons will last about twenty microseconds, then the water will
turn to stearn and start to explode. This could move a piston in the classical
manner until the stearn has cooled (or has passed to a next stage) and has
been replaced again by the slurry. Such a dramatic device need not be used;
rather the stearn could be taken off in the more usual manner of any power
reactor and then used for the generation of energy.
If undepleted uranium were to be used (or were slightly enriched with Pu) then
a magnification of the neutrons would occur, but still without having a
supercritical reactor. This might bring the break-even intensity of protons
needed for power production down by another factor of ten, which would then
put it within range of the present Energy Doubler accelerator at Fermilab.
There are probably better ways of producing plutonium, but it does appear that
it would be feasible to construct an intense proton accelerator that would
produce more energy than it consumes. A further more careful study of the
accelerator as well as of the plutonium production in the cascade showers
would determine the optimum proton energy for such a device. One happy result
of all that intensity would be that a truly magnificent neutrino source could
be produced!
I wish to thank Mr. A. Van Ginneken for making the nuclear shower calculation.
Table I
---
Calculation of A. VanGinneken for protons on
a large U238 beam dump
| 100 Gev | 300 Gev | 1000 Gev
Ioniz. of Hadron Shower* | .34 | .29 | .23
E. M. Showers* | .32 | .40 | .48
U238 Fission* | 2.2 | 1.9 | 1.6
Pu239 Fission* | 13.8 | 11.8 | 10.2
Total* | 16.6 | 14.3 | 12.5
$N^{0}$ | $8.6\times 10^{13}$ | $3.4\times 10^{13}$ | $1.2\times 10^{13}$
(Pu nuclei)/proton | 7000 | 18,000 | 62,000
Total Energy/1013 protons | 2.7 | 6.9 | 20 MW
* Fraction of incident energy recovered: calculations assume that each U238 fission makes three Pu239 nuclei. The fission of Pu239 has not been considered as a source of neutrons.
|
arxiv-papers
| 2010-07-29T21:35:16 |
2024-09-04T02:49:11.949168
|
{
"license": "Public Domain",
"authors": "R. R. Wilson",
"submitter": "David Ritchie",
"url": "https://arxiv.org/abs/1007.5338"
}
|
1007.5376
|
# Optimal control of a big financial company with debt liability under
bankrupt probability constraints
Zongxia Liang
Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China. Email: zliang@math.tsinghua.edu.cn
Bin Sun
Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China. Email: nksunbin@yahoo.com.cn
###### Abstract.
This paper considers an optimal control of a big financial company with debt
liability under bankrupt probability constraints. The company, which faces
constant liability payments and has choices to choose various
production/business policies from an available set of control policies with
different expected profits and risks, controls the business policy and
dividend payout process to maximize the expected present value of the
dividends until the time of bankruptcy. However, if the dividend payout
barrier is too low to be acceptable, it may result in the company’s bankruptcy
soon. In order to protect the shareholders’ profits, the managements of the
company impose a reasonable and normal constraint on their dividend strategy,
that is, the bankrupt probability associated with the optimal dividend payout
barrier should be smaller than a given risk level within a fixed time horizon.
This paper aims at working out the optimal control policy as well as optimal
return function for the company under bankrupt probability constraint by
stochastic analysis, PDE methods and variational inequality approach.
Moreover, we establish a risk-based capital standard to ensure the capital
requirement of can cover the total given risk by numerical analysis and give
reasonable economic interpretation for the results. MSC(2000): Primary 91B30,
93E20, 65K10 ; Secondary 60H05, 60H10. Keywords: Regular-singular stochastic
optimal control; Stochastic differential equations with reflection; Debt
liability; Bankrupt probability constraints; Optimal dividend barrier;
Dividend payout process.
## 1\. Introduction
In this paper, we study a model of a big financial company, which has the
possibility to choose various production/business policies with different
expected profits and associated risks. But at the same time, the company has
liability in which it has to pay out at a constant rate no matter what the
business plan is. The company controls the business policy and dividend payout
process to maximize the expected present value of the dividends until the time
of bankruptcy. Recently, there has been an upsurge of interest in dealing with
such optimal dividend control problems. We refer readers to Avanzi [4](2009)
and references therein, Højgaard and Taksar[19, 20](1999, 2001), Asmussen et
al.[2, 3](1997, 2000), He and Liang[17, 18](2008,2009). For results in the
model with debt liability, see Choulli, Taksar and Zhou [7, 8, 6, 31,
32](2008, 2004, 2003, 2001, 2000). Guo, Liu and Zhou [13](2004) is a
theoretical work on constrained nonlinear singular-regular stochastic control
problem. The optimal dividend payout for diffusions with solvency constraints
is solved in Paulsen [29](2003). According to Miller Modigliani, the
managements of the company choose the maximum of shareholders’ return as their
goals. We see from these literatures that the optimality is achieved by using
an optimal dividend payout barrier $b$, i.e. whenever the liquid reserve of
the company goes above $b$, the excess is paid out to the shareholders as
dividends. However, the optimal dividend barrier $b$ may be too low to be
acceptable because this low dividend payout barrier may result in the
company’s bankruptcy soon. Thus, the company may be prohibited to pay
dividends at such a low level in order to avoid bankruptcy. So the managements
of the company impose some constraints on its dividends payout strategy. One
reasonable and normal constraint is that the optimal dividend barrier $b$
should be such that the bankrupt probability is not larger than some
predetermined risk level $\varepsilon$ within a fixed time horizon $T$ and the
cost for the safety is minimal. Based on the new idea, He, Hou and
Liang[16](2008) studied the linear regular-singular optimal control problem of
insurance company with proportional reinsurance policy under bankrupt
probability constraints as we state above. They succeeded to find the optimal
control policy under bankrupt probability constraints by proving the bankrupt
probability is decreasing in the dividend barrier and the existence of the
dividend barrier satisfying any given bankrupt probability constraints.
Furthermore, Liang, Huang and Yao[25, 26, 24](2010) gave a exact result of
such a control problem by proving the bankrupt probability is continuous and
strictly increasing w.r.t. the dividend barrier $b$. These new results are
mainly about the insurance company with proportional reinsurance policy.
Motivated by these works, under any given bankrupt probability constraint, we
are interested in a common company, such as a big financial company, insurance
company, …, facing constant liability payments, which controls the business
policy and dividend payout process to maximize the expected present value of
the dividends until the time of bankruptcy. Based on the relationship between
those parameters that govern the company’s reserve process, we try to derive
the optimal control policy as well as optimal return function as well as a
risk-based capital standard to ensure the capital requirement of can cover the
total risk in several distinct cases of the qualitative behavior of the
company under some bankrupt probability constraints. Moreover, we also give a
robust analysis of the optimal return function and the optimal dividend
strategy w.r.t. the model parameters and the constrained risk level of
bankrupt probability. The paper is organized as follows. In section 2, we
established a rigorous stochastic control model of a big financial company
facing constant liability payments with some bankrupt probability constraints.
In section 3, we present main result of this paper and its reasonable economic
interpretation. In section 4, we give risk analysis of the model we deal with
to state the setting treated in this paper is well defined and why we study
such regular-singular stochastic optimal control of the company. In section 5,
we give some numerical examples to portray how the debt rate $\delta$, the
constrained risk level of bankrupt probability, the initial capital $x$ , the
volatility rate $\sigma^{2}$ and the profit rate $\mu$ impact on the optimal
return function and the optimal dividend strategy. We will list the properties
of the optimal return function and bankrupt probability in section 6. The
proof of main result will be given in section 7. The procedure of solving the
HJB equations and proofs of lemmas which are used to prove the main result
will be presented in the appendix.
## 2\. Mathematical Model
To give a mathematical formulation of our optimal control problem treated in
this paper, We start with a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbf{P})$ and a one-
dimensional standard Brownian motion $\\{\mathcal{W}_{t}\\}_{t\geq 0}$ on it,
adapted to the filtration $\mathcal{F}_{t}$. $\mathcal{F}_{t}$ represents the
information available at time $t$ and any decision made up to time $t$ is
based on this information. For the intuition of our diffusion model we start
from the classical Cramér-Lundberg model of a reserve(risk) process. In this
model claims arrive according to a Poisson process $N_{t}$ with intensity
$\lambda$ on $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbf{P})$. The size of each claim is $X_{i}$. Random variables $X_{i}$
are i.i.d. and are independent of the Poisson process $N_{t}$ with finite
first and second moments given by $\mu$ and $\sigma^{2}$ respectively. If
there is no reinsurance and dividend payout, the reserve (risk) process of
insurance company is described by
$r_{t}=r_{0}+pt-\sum^{N_{t}}_{i=1}X_{i},$
where $p$ is the premium rate. If $\eta>0$ denotes the safety loading, the $p$
can be calculated via the expected value principle as
$p=(1+\eta)\lambda\mu.$
In a case where the insurance company shares risk with the reinsurance, the
sizes of the claims held by the insurer become $X^{a}_{i}$, where $a$ is a
(fixed) retention level. For proportional reinsurance, $a$ denotes the
fraction of the claim covered by the insurance company . Consider the case of
cheap reinsurance for which the reinsuring company uses the same safety
loading as the insurance company, the reserve process of the insurance company
is given by
$r^{(a,\eta)}_{t}=u+p^{(a,\eta)}t-\sum^{N_{t}}_{i=1}X^{(a)}_{i},$
where
$\displaystyle p^{(a,\eta)}=(1+\eta)\lambda\mathbb{E}\\{X^{(a)}_{i}\\}.$
Then by center limit theorem it is well known that for large enough $\lambda$
$\displaystyle\\{r_{t}^{(a,\eta)}\\}_{t\geq 0}\stackrel{{\scriptstyle
d}}{{\approx}}BM(\mu at,\sigma^{2}a^{2}t).$
in $\mathcal{D}[0,\infty)$ (the space of right continuous functions with left
limits endowed with the skorohod topology), where $\mu=\eta\lambda E(X_{i})$,
$\sigma=\sqrt{\lambda E(X_{i}^{2})}$ and $BM(\mu,\sigma^{2})$ stands for
Brownian motion with the drift coefficient $\mu$ and diffusion coefficient
$\sigma$ on $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$.
So the passage to the limit works well in the presence of a big portfolios,
the reserve (risk) process of the insurance company can be approximated by the
following diffusion process
$\displaystyle dR_{t}=\mu a(t)dt+\sigma a(t)d\mathcal{W}_{t},$ (2.1)
where $a(t)$ denotes retention level. We refer readers for this fact and for
the specifies of the diffusion approximations to Emanuel, Harrison and Taylor
[9](1975), Grandell[10](1977), Grandell[11](1978), Grandell[12](1990),
Harrison[15](1985), Iglehart[21](1969), Schmidli[30](1994). In this paper, we
consider a common company which faces constant liability payments. In view of
the diffusion approximations for the classical Cramér-Lundberg model described
above, we assume that the reserve process of the company facing constant
liability payments is given by the following diffusion process
$\displaystyle dR_{t}=(a(t)\mu-\delta)dt+a(t)\sigma d{\mathcal{W}}_{t},\quad
R_{0}=x$ (2.2)
where $x$ is the initial reserve of the company, $\mu$ is the expected profit
per unit time (profit rate), $\sigma$ is the volatility rate of the reserve
process in the absence of any risk control, $\delta$ represents the amount of
money the company has to pay per unit time (the debt rate) irrespective of
what business activities rate it chooses. In this model, the business
activities rate that the company chooses at time $t$ are modeled by $a(t)$,
which takes values in the interval $[\alpha,\beta]$, where
$0<\alpha<\beta<+\infty$. The restriction $\alpha>0$ reflects the fact that
there are statutory reasons that its business activities rate cannot be
reduced to zero, unless the company faces bankruptcy. In our model, a policy
$\pi$ is a pair of non-negative càdlàg $\mathcal{F}_{t}$-adapted processes
$\\{a_{\pi}(t),L_{t}^{\pi}\\}$, where $a_{\pi}(t)\in[\alpha,\beta]$
corresponds to the risk exposure at time $t$ and $L_{t}^{\pi}$ corresponds to
the cumulative amount of dividend pay-outs distributed up to time $t$. A
policy $\pi=\\{a_{\pi}(t),L_{t}^{\pi}\\}$ is called admissible if $\alpha\leq
a_{\pi}(t)\leq\beta$ and $L_{t}^{\pi}$ is a nonnegative, non-decreasing,
right-continuous function. We denote $\Pi$ the set of all admissible policies.
When a admissible policy $\pi$ is applied, the resulting reserve process is
denoted by $\\{R_{t}^{\pi}\\}$. In view of (2.2) we rewrite $R_{t}^{\pi}$ as
follows
$\displaystyle dR^{\pi}_{t}=(a_{\pi}(t)\mu-\delta)dt+a_{\pi}(t)\sigma
d{\mathcal{W}}_{t}-dL^{\pi}_{t},\quad R^{\pi}_{0}=x,$ (2.3)
where $x(>0)$ is the initial (capital) reserve. In addition, we assume the
company needs to keep its reserve above 0 to avoid bankruptcy. For the given
control policy $\pi$, we define the time of bankruptcy as
$\tau^{\pi}_{x}=\inf\big{\\{}t\geq 0:R^{\pi}_{t}\leq 0\big{\\}}$.
$\tau^{\pi}_{x}$ is an $\mathcal{F}_{t}$ -stopping time. The objective of the
company is to maximize the expected present value of the dividends payout
until the time of bankrupt by choosing control policy $\pi$ from the
admissible set $\Pi$. Choulli, Taksar and Zhou[7](2003) proved that there
exists an optimal dividend barrier $b_{0}$ and the optimal policy
$\pi^{*}_{b_{0}}=\\{a_{\pi^{*}_{b_{0}}}(\cdot),L^{\pi^{*}_{b_{0}}}_{\cdot}\\}$,
which maximize the expected present value of the dividends payout before
bankruptcy, i.e., $b_{0}$ is such that
$\displaystyle
J(x,\pi)=\mathbf{E}\big{\\{}\int_{0}^{\tau^{\pi}_{x}}e^{-ct}dL_{t}^{\pi}\big{\\}},$
(2.4) $\displaystyle
V(x,b_{0})=\sup_{\pi\in\Pi}\\{J(x,\pi)\\}=J(x,\pi^{*}_{b_{0}}),$ (2.5)
where $c$ denotes the discount rate. If the optimal dividend barrier is too
low, the bankrupt probability within a fixed time will be bigger than a given
level. This is not acceptable by the management of the company. Taking the
balance of profit and risk into consideration, we impose small bankrupt
probability constraint on the company’s control policy. We describe our
optimal control problem as follows. Let
$\Pi_{b}=\big{\\{}\pi\in\Pi:\int_{0}^{\infty}{I}_{\\{s:R^{\pi}(s)<b\\}}dL_{s}^{\pi}=0\big{\\}}$
for $b\geq 0$ . Then it is easy to see that $\Pi=\Pi_{0}$ and
$b_{1}>b_{2}\Rightarrow\Pi_{b_{1}}\subset\Pi_{b_{2}}$. For a given admissible
policy $\pi$, we define the optimal return function $V(x)$ by
$\displaystyle J(x,\pi)$ $\displaystyle=$ $\displaystyle{\bf
E}\big{\\{}\int_{0}^{\tau^{\pi}_{x}}e^{-ct}dL_{t}^{\pi}\big{\\}},$
$\displaystyle V(x,b)$ $\displaystyle=$
$\displaystyle\sup_{\pi\in\Pi_{b}}\\{J(x,\pi)\\},$ (2.6) $\displaystyle V(x)$
$\displaystyle=$ $\displaystyle\sup_{b\in\mathfrak{B}}\\{V(x,b)\\}$ (2.7)
and the optimal policy $\pi^{*}$ by
$\displaystyle J(x,\pi^{*})=V(x),$ (2.8)
where
$\displaystyle\mathfrak{B}:=\big{\\{}b\ :\ \mathbb{P}[\tau_{b}^{\pi_{b}}\leq
T]\leq\varepsilon\ ,\ J(x,\pi_{b})=V(x,b)\mbox{ and}\
\pi_{b}\in\Pi_{b}\big{\\}},$
$c>0$ is a discount rate, $\tau_{b}^{\pi_{b}}$ is the time of bankruptcy
$\tau_{x}^{\pi_{b}}$ when the initial reserve $x=b$ and the control policy is
$\pi_{b}$. $\varepsilon$ is a given constrained risk level of bankrupt
probability. $1-\varepsilon$ is the standard of security and less than
solvency for a given risk level $\varepsilon>0$. $\mathfrak{B}$ is called the
risk constrained set. The main purpose of this paper is to derive the optimal
return function $V(x)$, the optimal policy $\pi^{*}$ as well as the optimal
dividend payout barrier $b^{*}$ and try to give a reasonable economic
interpretation and discuss effect of the debt rate $\delta$, the constrained
risk level $\varepsilon$ of bankrupt probability, the initial capital $x$, the
volatility rate $\sigma^{2}$ and the profit rate $\mu$ on the optimal return
function and the optimal dividend strategy $\pi^{*}$.
## 3\. Main Result
In this section we first present main result of this paper, then, together
with numerical results in section 5 below, give its reasonable economic
interpretation. The proof of the main result will be given in section 7.
###### Theorem 3.1.
Let $\varepsilon\in(0,1)$ be the constrained risk level of bankrupt
probability and time horizon $T$ be given. (i) If
$\mathbf{P}[\tau_{b_{0}}^{\pi^{*}_{b_{0}}}\leq T]\leq\varepsilon$, then the
optimal return function $V(x)$ is $f(b_{0},x)$ defined by (6.5) below, and
$V(x)=f(b_{0},x)=J(x,\pi_{b_{o}}^{\ast})$. The optimal policy
$\pi_{b_{o}}^{\ast}$ is
$\\{a^{\ast}_{b_{o}}(R^{\pi_{b_{o}}^{\ast}}_{t}),L^{\pi_{b_{o}}^{\ast}}_{t}\\}$,
where $\\{R^{\pi_{b_{o}}^{\ast}}_{t},L^{\pi_{b_{o}}^{\ast}}_{t}\\}$ is
uniquely determined by the following stochastic differential equation
$\displaystyle\left\\{\begin{array}[]{l l
l}dR_{t}^{\pi_{b_{o}}^{\ast}}=(a^{*}_{b_{o}}(R^{\pi_{b_{o}}^{\ast}}_{t})\mu-\delta)dt+\sigma
a^{*}_{b_{o}}(R^{\pi_{b_{o}}^{\ast}}_{t})d{\mathcal{W}}_{t}-dL_{t}^{\pi_{b_{o}}^{\ast}},\\\
R_{0}^{\pi_{b_{o}}^{\ast}}=x,\\\ 0\leq R^{\pi_{b_{o}}^{\ast}}_{t}\leq
b_{0},\\\
\int^{\infty}_{0}I_{\\{t:R^{\pi_{b_{o}}^{\ast}}_{t}<b_{0}\\}}(t)dL_{t}^{\pi_{b_{o}}^{\ast}}=0.\end{array}\right.$
(3.5)
The solvency of the company is bigger than $1-\varepsilon$. (ii) If
$\mathbf{P}[\tau_{b_{0}}^{\pi^{*}_{b_{0}}}\leq T]>\varepsilon$, there is a
unique optimal dividend $b^{\ast}(\geq b_{0})$ satisfying
$\mathbf{P}[\tau_{b^{\ast}}^{\pi_{b^{*}}^{\ast}}\leq T]=\varepsilon$. The
optimal return function $V(x)$ is $g(x,b^{*})$ defined by (6.20), that is,
$\displaystyle V(x)=g(x,b^{*})=\sup_{b\in\mathfrak{B}}\\{V(x,b)\\}$ (3.6)
and
$\displaystyle
b^{*}\in\mathfrak{B}:=\big{\\{}b:\mathbb{P}[\tau_{b}^{\pi^{*}_{b}}\leq
T]\leq\varepsilon,\ J(x,\pi^{*}_{b})=V(x,b)\mbox{ and}\ \pi^{*}_{b}\in\Pi_{b}\
\big{\\}}.$ (3.7)
The optimal policy
$\pi_{b^{*}}^{\ast}=\\{a^{\ast}_{b^{*}}(R^{\pi_{b^{*}}^{\ast}}_{t}),L^{\pi_{b^{*}}^{\ast}}_{t}\\}$,
where $\\{R^{\pi_{b^{*}}^{\ast}}_{t},L^{\pi_{b^{*}}^{\ast}}_{t}\\}$ is
uniquely determined by the following stochastic differential equation
$\displaystyle\left\\{\begin{array}[]{l l l}dR_{t}^{\pi_{b^{*}}^{\ast}}=(\mu
a^{*}_{b^{*}}(R^{\pi_{b^{*}}^{\ast}}_{t})-\delta)dt+\sigma
a^{*}_{b^{*}}(R^{\pi_{b^{*}}^{\ast}}_{t})d{W}_{t}-dL_{t}^{\pi_{b^{*}}^{\ast}},\\\
R_{0}^{\pi_{b^{*}}^{\ast}}=x,\\\ 0\leq R^{\pi_{b^{*}}^{\ast}}_{t}\leq
b^{*},\\\
\int^{\infty}_{0}I_{\\{t:R^{\pi_{b^{*}}^{\ast}}_{t}<b^{*}\\}}(t)dL_{t}^{\pi_{b^{*}}^{\ast}}=0.\end{array}\right.$
(3.12)
The solvency of the company is $1-\varepsilon$. (iii) Moreover,
$\displaystyle\frac{g(x,b^{*})}{g(x,b_{0})}\leq 1,$ (3.13)
where $a^{*}_{b}(x)$ is defined by (6.25) below.
Economic interpretation of Theorem 3.1 is as follows. (1) For a given
constrained risk level $\varepsilon$ of bankrupt probability and time horizon
$T$, if the company’s bankruptcy probability is less than this given risk
constraint level $\varepsilon$, the optimal control problem of (2) and (2.7)
is the traditional problem (2.4) and (2.5). The bankrupt probability
constraints here do not work. (2) If the company’s bankruptcy probability is
larger than this given constrained risk level $\varepsilon$, the traditional
optimal control policy fails to meet the requirement of bankrupt probability
constraint. So the company has to find an optimal policy $\pi_{b^{*}}^{\ast}$
to improve its solvency ability and ensure the company operates safely. The
optimal reserve process $R^{\pi_{b^{*}}^{\ast}}_{t}$ is a diffusion process
reflected at a dividend barrier $b^{*}$, and the process
$L^{\pi_{b^{*}}^{\ast}}_{t}$ is the dividend payout process that ensures the
reflection. $a^{*}_{b^{*}}$ is the optimal feedback control function. The
optimal policy means that the company will pay out the amount of reserve in
excess of $b^{*}$ as dividend once its reserve is bigger than $b^{*}$. Under
this control policy, we can guarantee that the company’s bankrupt probability
can stay below $\varepsilon$. (3)The inequality (3.13) shows that the optimal
control policy $\pi_{b^{*}}^{\ast}$ will decrease the company’s profit-earning
capability. We can treat this decrease of the profit as the cost of keeping
the company’s risk at a low level and the cost, $g(x,b_{0})-g(x,b^{*})$, is
minimal in view of ( 3.7), Lemma 6.3 and Lemma 6.20 below. Thus
$\pi_{b^{*}}^{\ast}$ is the best equilibrium control policy between making
profit and improving solvency. (4) From the figure 1 in Example 5.1 below we
see that the optimal return function $g(x)$ is decreasing w.r.t. the debt rate
$\delta$. Figure 1 shows that the higher debt rate will lessen the company’s
profit, so the company should keep its debt rate at a appropriate level.(5)We
can see from figure 2 in Example 5.2 below that the optimal dividend barrier
$b^{*}$ is decreasing w.r.t. the constrained risk level $\varepsilon$ of
bankrupt probability. And the optimal dividend barrier $b^{*}$ is uniquely
decided by $\mathbf{P}[\tau_{b^{\ast}}^{\pi_{b^{*}}^{\ast}}\leq
T]=\varepsilon$, i.e., $1-\phi^{b^{*}}(b^{*},b^{*})=\varepsilon$ (see Lemma
6.5 below). The the optimal dividend barrier $b^{*}$ is also increasing
function of the volatility $\sigma^{2}$ (see the figure 3 in Example 5.3 below
). (6) We call $R_{0}:=x_{b^{*}}(\varepsilon)$ the risk-based capital standard
to ensure the capital requirement of can cover the total given risk
$\varepsilon$, where $x_{b^{*}}(\varepsilon)$ is determined by
$1-\phi^{b^{*}}(x,b^{*})=\varepsilon$ (see Lemma 6.5). We see from the figure
4 in Example 5.4 below that risk-based capital $x_{b^{*}}(\varepsilon)$
decreases with risk level $\varepsilon$. Since the optimal feedback control
function $a^{*}_{b^{*}}(x)$ is increasing w.r.t. $x$, in view of comparison
theorem for SDE, the constrained risk level $\varepsilon$ lessens the optimal
business activities rate $a^{*}(t)$, but improves dividend payout process
$L^{*}_{t}(\varepsilon)$. It also lessens the optimal return function(see
Example 5.7 below). (7) We can see from the figures 5 and 6 below that the
optimal return function $g(x)$ is increasing in both the profit rate $\mu$ and
the volatility rate $\sigma$.
## 4\. Risk Analysis
In this section, we proceed a risk analysis on the model we are studying. We
first work out the lower boundary of bankrupt probability when we applied
$b_{0}$ as the dividend barrier. It proves that the risk constrained set
$\mathfrak{B}$ is not $\Re_{+}=[0,+\infty)$. So the topic of this paper is
fundamental to studying the optimal control problem under bankrupt probability
constraints. It also states that the company has to find optimal policy to
improve its solvency.
###### Theorem 4.1.
Let $\\{R^{\pi_{b_{0}}^{\ast}}_{t},L^{\pi_{b_{0}}^{\ast}}_{t}\\}$ be the
unique solution of the following SDE( see Lions and Sznitman [27](1984))
$\displaystyle\left\\{\begin{array}[]{l l l}dR_{t}^{\pi_{b_{o}}^{\ast}}=(\mu
a^{*}_{b_{o}}(R^{\pi_{b_{o}}^{\ast}}_{t})-\delta)dt+\sigma
a^{*}_{b_{o}}(R^{\pi_{b_{o}}^{\ast}}_{t})d{W}_{t}-dL_{t}^{\pi_{b_{o}}^{\ast}},\\\
R_{0}^{\pi_{b_{o}}^{\ast}}=b_{0},\\\ 0\leq R^{\pi_{b_{o}}^{\ast}}_{t}\leq
b_{0},\\\
\int^{\infty}_{0}I_{\\{t:R^{\pi_{b_{o}}^{\ast}}_{t}<b_{0}\\}}(t)dL_{t}^{\pi_{b_{o}}^{\ast}}=0.\end{array}\right.$
(4.5)
Then
$\displaystyle{\bf P}(\tau_{b_{0}}^{\pi^{*}_{b_{0}}}\leq T)$
$\displaystyle\geq$
$\displaystyle\frac{4[1-\Phi(\frac{b_{0}}{\alpha\sigma\sqrt{T}})]^{2}}{\exp\\{\frac{T}{\sigma^{2}}{\max\\{\mu-\frac{\delta}{\beta},|\mu-\frac{\delta}{\alpha}|\\}}^{2}\\}}$
(4.6) $\displaystyle\equiv$
$\displaystyle\varepsilon_{0}(b_{0},\mu,\delta,\sigma,T,\alpha,\beta)>0,$
where $\tau^{\pi^{*}_{b_{o}}}_{b_{o}}=\inf\big{\\{}t\geq
0:R^{\pi^{*}_{b_{o}}}_{t}\leq 0\big{\\}}$ and $\Phi(\cdot)$ is the standard
normal distribution function.
###### Proof.
Since $a^{*}_{b_{o}}(x)$ (defined by (6.25)) is a bounded Lipschitz continuous
function w.r.t. $x$, the following SDE
$\displaystyle\left\\{\begin{array}[]{l l l}dX_{t}=(\mu
a^{*}_{b_{0}}(X_{t})-\delta)dt+\sigma a^{*}_{b_{0}}(X_{t})d{W}_{t},\\\
X_{0}=b_{0}\\\ \end{array}\right.$ (4.9)
has a unique solution $X_{t}$. By comparison theorem for one-dimensional Itô
process, we have
$\displaystyle{\bf{P}}[R^{\pi_{b_{o}}^{\ast}}_{t}\leq X_{t}]=1$ (4.10)
Define a measure ${\bf Q}$ on $\mathcal{F}_{T}$ by
$\displaystyle d{\bf{Q}}(\omega)=M(T)d{\bf P}(\omega)$ (4.11)
where
$\displaystyle M(t)$ $\displaystyle\equiv$
$\displaystyle\exp\big{\\{}-\int_{0}^{t}\frac{(\mu
a^{*}_{b_{0}}(X_{t})-\delta)}{\sigma a^{*}_{b_{0}}(X_{t})}dW_{s}$
$\displaystyle-$ $\displaystyle\frac{1}{2}\int_{0}^{t}\frac{(\mu
a^{*}_{b_{0}}(X_{t})-\delta)^{2}}{[\sigma
a^{*}_{b_{0}}(X_{t})]^{2}}ds\big{\\}}.$
Since $\\{M(t)\\}$ is a martingale w.r.t. $\mathcal{F}_{t}$, we have ${\bf
E}\big{[}M(T)\big{]}=1$. Moreover, noticing that $a^{*}_{b_{o}}(X_{t})$
belongs to $[\alpha,\beta]$, we obtain
$\displaystyle{\bf{E}}^{\bf{P}}[M(T)^{2}]\leq\exp\\{\frac{T}{\sigma^{2}}{\max\\{\mu-\frac{\delta}{\beta},|\mu-\frac{\delta}{\alpha}|\\}}^{2}\\}$
(4.12)
Using Girsanov theorem, ${\bf Q}$ is a probability measure on
$\mathcal{F}_{T}$ and the process $\\{X_{t}\\}$ satisfies the following SDE
$\displaystyle dX_{t}=\sigma a^{*}_{b_{0}}(X_{t})d\tilde{W}_{t},\ X_{0}=b_{0}$
(4.13)
where $\tilde{W}_{t}=W_{t}+\int_{0}^{t}\frac{(\mu
a^{*}_{b_{0}}(X_{s})-\delta)}{\sigma a^{*}_{b_{0}}(X_{s})}d{s}$. It is easy to
see that $\tilde{W}_{t}$ is a Brownian motion on
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},{\bf Q})$. Define a time
changes $\rho(t)$ by
$\displaystyle\dot{\rho}(t)=\frac{1}{{a^{*}_{b_{0}}}^{2}(X_{t})\sigma^{2}},$
(4.14)
then $\rho(t)$ is a strictly increasing function. If we denote $X_{\rho(t)}$
by $\hat{X}_{t}$, then we have
$\displaystyle\hat{X}_{t}=b_{0}+\hat{W}_{t}.$
Noticing that $0<\alpha\leq{a^{*}_{b_{0}}}(R_{t})\leq\beta<+\infty$, we get
$\displaystyle\frac{1}{\beta^{2}\sigma^{2}}\leq\dot{\rho}(t)\leq\frac{1}{\alpha^{2}\sigma^{2}}.$
(4.15)
Due to the fact $\rho(t)=\int_{0}^{t}\dot{\rho(s)}ds$, we can deduce that
$\rho(t)\leq\frac{t}{\alpha^{2}\sigma^{2}}$ and
$\rho^{-1}(t)\geq\alpha^{2}\sigma^{2}t$, where $\rho^{-1}$ denotes the inverse
of $\rho$. Then we have
$\displaystyle{\bf{Q}}[\inf\\{t:X_{t}\leq 0\\}\leq T]$ $\displaystyle=$
$\displaystyle{\bf{Q}}[\inf\\{t:\hat{X}_{\rho^{-1}(t)}\leq 0\\}\leq T]$ (4.16)
$\displaystyle=$ $\displaystyle{\bf{Q}}[\inf\\{\rho(t):b_{0}+\hat{W}_{t}\leq
0\\}\leq T]$ $\displaystyle=$ $\displaystyle{\bf{Q}}[\inf\\{t:\hat{W}_{t}\leq-
b_{0}\\}\leq\rho^{-1}(T)]$ $\displaystyle\geq$
$\displaystyle{\bf{Q}}[\inf\\{t:\hat{W}_{t}\leq-
b_{0}\\}\leq\alpha^{2}\sigma^{2}T]$ $\displaystyle=$ $\displaystyle
2[1-\Phi(\frac{b_{0}}{\alpha\sigma\sqrt{T}})]>0.$
Using Hölder inequalities as well as (4.11),
$\displaystyle{\bf{Q}}[\inf\\{t:X_{t}\leq 0\\}\leq T]$ $\displaystyle=$
$\displaystyle\int_{\Omega}{\bf{1}}_{[\inf\\{t:X_{t}\leq 0\\}\leq
T]}d\bf{Q}(\omega)$ $\displaystyle=$
$\displaystyle\int_{\Omega}{\bf{1}}_{[\inf\\{t:X_{t}\leq 0\\}\leq
T]}M_{T}d{\bf{P}}(\omega)$ $\displaystyle=$
$\displaystyle{\bf{E}}^{{\bf{P}}}[M_{T}{\bf{1}}_{[\inf\\{t:X_{t}\leq 0\\}\leq
T]}]$ $\displaystyle\leq$
$\displaystyle{\bf{E}}^{{\bf{P}}}[M^{2}_{T}]^{\frac{1}{2}}{\bf{P}}[\inf\\{t:X_{t}\leq
0\\}\leq T]^{\frac{1}{2}}.$
Substituting (4.12) and (4.16) into (4), we get
$\displaystyle{\bf{P}}[\inf\\{t:X_{t}\leq 0\\}\leq T]$ $\displaystyle\geq$
$\displaystyle\frac{{\bf{Q}}[\inf\\{t:X_{t}\leq 0\\}\leq
T]^{2}}{{\bf{E}}^{\bf{P}}[M_{T}^{2}]}$ $\displaystyle\geq$
$\displaystyle\frac{4[1-\Phi(\frac{b_{0}}{\alpha\sigma\sqrt{T}})]^{2}}{\exp\\{\frac{T}{\sigma^{2}}{\max\\{\mu-\frac{\delta}{\beta},|\mu-\frac{\delta}{\alpha}|\\}}^{2}\\}}.$
By virtue of (4.10), we have
$\displaystyle{\bf P}[\tau_{b_{0}}^{\pi^{*}_{b_{0}}}\leq T]$ $\displaystyle=$
$\displaystyle{\bf{P}}[\inf\\{t:R^{\pi^{*}_{b_{o}}}_{t}\leq 0\\}\leq T]$
(4.18) $\displaystyle\geq$ $\displaystyle{\bf{P}}[\inf\\{t:X_{t}\leq 0\\}\leq
T]$ $\displaystyle\geq$
$\displaystyle\frac{4[1-\Phi(\frac{b_{0}}{\alpha\sigma\sqrt{T}})]^{2}}{\exp\\{\frac{T}{\sigma^{2}}{\max\\{\mu-\frac{\delta}{\beta},|\mu-\frac{\delta}{\alpha}|\\}}^{2}\\}}$
$\displaystyle\equiv$
$\displaystyle\varepsilon_{0}(b_{0},\mu,\delta,\sigma,T,\alpha,\beta)$
$\displaystyle>$ $\displaystyle 0.$
∎
The economic interpretation of theorem 4.1 is the following.Assume
$\frac{2\delta}{\mu}<\alpha$, then we have $\mu-\frac{\delta}{\alpha}>0$. So
the lower boundary of the bankrupt probability
$\varepsilon_{0}(b_{0},\mu,\delta,\sigma,T,\alpha,\beta)$ becomes to
$\frac{4[1-\Phi(\frac{b_{0}}{\alpha\sigma\sqrt{T}})]^{2}}{\exp\\{\frac{T}{\sigma^{2}}(\mu-\frac{\delta}{\beta})^{2}\\}}$.
Based on the assumption, we have the following explanations.(1) The lower
boundary of bankrupt probability for the company
$\varepsilon_{0}(b_{0},\mu,\delta,\sigma,\\\ T,\alpha,\beta)$ is an increasing
function of $(\sigma,\delta,\alpha)$, which means that higher volatility rate
$\sigma$ and debt rate $\delta$ will make the company face larger risk. In
addition, risk will increase as the lower boundary $\alpha$ of control
function $a(x)$ increases. (2) The lower boundary of bankrupt probability for
the company $\varepsilon_{0}(b_{0},\mu,\delta,\sigma,\\\ T,\alpha,\beta)$ is
decreasing in $(b_{0},\mu,\beta)$, which means that paying dividends at a
lower barrier will cause larger bankrupt probability. On the other hand, the
higher the profit rate is, the lower the risk is. Improving the upper boundary
$\beta$ of the control function $a(x)$ can also reduce the company’s risk. (3)
The company has a positive bankrupt probability within the time interval
$[0,T]$ if we set $b_{0}$ as the dividends barrier. In order to keep the
company’s risk at a low level, we need adjust our control policy and find the
optimal dividends barrier $b^{*}$ under lower constrained risk level of
bankrupt probability. The second result is the following, which states that
the risk constrained set $\mathfrak{B}$ defined in section 2 is non-empty for
any $\varepsilon>0$, together with the first result, also guarantees our
problem (2), (2.7), (2.8) is well defined.
###### Theorem 4.2.
Let $(R^{\pi_{b}^{\ast}}_{t},L_{t}^{\pi_{b}^{\ast}})$ be defined by
$\displaystyle\left\\{\begin{array}[]{l l l}dR_{t}^{\pi_{b}^{\ast}}=(\mu
a^{*}_{b}(R^{\pi_{b}^{\ast}}_{t})-\delta)dt+\sigma
a^{*}_{b}(R^{\pi_{b}^{\ast}}_{t})d{W}_{t}-dL_{t}^{\pi_{b}^{\ast}},\\\
R_{0}^{\pi_{b}^{\ast}}=b,\\\ 0\leq R^{\pi_{b}^{\ast}}_{t}\leq b,\\\
\int^{\infty}_{0}I_{\\{t:R^{\pi_{b}^{\ast}}_{t}<b\\}}(t)dL_{t}^{\pi_{b}^{\ast}}=0,\end{array}\right.$
(4.23)
and $\tau_{b}^{\pi_{b}^{*}}=\inf\\{t\geq 0:R^{\pi_{b}^{\ast}}_{t}<0\\}$. Then
$\displaystyle\lim_{b\rightarrow\infty}{\bf P}[\tau_{b}^{b}\leq T]=0.$ (4.24)
###### Proof.
For any $b\geq 1$, we have $b\geq\sqrt{b}$. By comparison theorem on SDE (see
Ikeda and Watanabe [23](1981)), we have
$\displaystyle\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq
T]\leq\mathbf{P}[\tau_{\sqrt{b}}^{\pi^{*}_{b}}\leq T].$ (4.25)
Let $R^{(1)}_{t}$ satisfy the following SDE,
$\displaystyle\left\\{\begin{array}[]{l l
l}dR_{t}^{(1)}=(a^{*}(R_{t}^{(1)})\mu-\delta)dt+a^{*}(R_{t}^{(1)})\sigma
d\mathcal{W}_{t},\\\ R_{0}^{(1)}=\sqrt{b}.\end{array}\right.$ (4.28)
Then, we have
$\displaystyle\mathbf{P}[\tau_{\sqrt{b}}^{\pi^{*}_{b}}\leq T]$
$\displaystyle\leq$ $\displaystyle\mathbf{P}[R_{t}^{(1)}=0\ \mbox{or}\
R_{t}^{(1)}=b\ \mbox{for some $0\leq t\leq T$ }]$ (4.29) $\displaystyle\leq$
$\displaystyle\mathbf{P}[\sup_{0\leq t\leq T}R_{t}^{(1)}\geq
b]+\mathbf{P}[\inf_{0\leq t\leq T}R_{t}^{(1)}\leq 0].$
Next, we estimate $\mathbf{P}[\sup_{0\leq t\leq T}R_{t}^{(1)}\geq b]$ and
$\mathbf{P}[\inf_{0\leq t\leq T}R_{t}^{(1)}\leq 0]$, respectively. Hölder’s
inequality and $a^{*}(x)\leq\beta$ yield that
$\displaystyle\sup_{0\leq t\leq T}(R_{t}^{(1)})^{2}$ $\displaystyle\leq$
$\displaystyle 3(\sqrt{b})^{2}+3\sup_{0\leq t\leq
T}(\int_{0}^{t}(a^{*}(R_{s}^{(1)})\mu-\delta)ds)^{2}+$ (4.30) $\displaystyle
3\sup_{0\leq t\leq T}(\int_{0}^{t}a^{*}(R_{t}^{(1)})\sigma
d\mathcal{W}_{s})^{2}$ $\displaystyle\leq$ $\displaystyle
3b+3(\beta\mu-\delta)^{2}T^{2}+$ $\displaystyle 3\sup_{0\leq t\leq
T}(\int_{0}^{t}a^{*}(R_{t}^{(1)})\sigma d\mathcal{W}_{s})^{2}.$
By Markov inequality, B-D-G inequalities and (4.30), we obtain
$\displaystyle\mathbf{P}[\sup_{0\leq t\leq T}R_{t}^{(1)}\geq b]$
$\displaystyle\leq$ $\displaystyle\frac{\mathbf{E}[\sup_{0\leq t\leq
T}(R_{t}^{(1)})^{2}]}{b^{2}}$ (4.31) $\displaystyle\leq$
$\displaystyle\frac{3b+3(\beta\mu-\delta)^{2}T^{2}+12{\beta}^{2}{\sigma}^{2}T}{b^{2}}$
$\displaystyle\rightarrow$ $\displaystyle 0,\ \ \mbox{as}\
b\rightarrow\infty.$
Now we turn to estimating $\mathbf{P}[\inf_{0\leq t\leq T}R_{t}^{(1)}\leq 0]$.
Let $R_{t}^{(2)}$ satisfy the following SDE
$\displaystyle\left\\{\begin{array}[]{l l
l}dR_{t}^{(2)}=(a^{*}(R_{t}^{(1)})\mu-\delta)dt+a^{*}(R_{t}^{(1)})\sigma
d\mathcal{W}_{t},\\\ R_{0}^{(2)}=0.\end{array}\right.$ (4.34)
Thus we have
$\displaystyle R_{t}^{(1)}=\sqrt{b}+R_{t}^{(2)}.$ (4.35)
Therefore, by using the same argument as in (4.31) , we get
$\displaystyle\mathbf{P}[\inf_{0\leq t\leq T}R_{t}^{(1)}\leq 0]$
$\displaystyle=$ $\displaystyle\mathbf{P}[\inf_{0\leq t\leq
T}R_{t}^{(2)}\leq-\sqrt{b}]$ (4.36) $\displaystyle=$
$\displaystyle\mathbf{P}[\sup_{0\leq t\leq T}(-R_{t}^{(2)})\geq\sqrt{b}]$
$\displaystyle\leq$ $\displaystyle\frac{\mathbf{E}[\sup_{0\leq t\leq
T}(-R_{t}^{(2)})^{2}]}{(\sqrt{b})^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{2(\beta\mu-\delta)^{2}T^{2}+8{\beta}^{2}{\sigma}^{2}T}{b}$
$\displaystyle\rightarrow$ $\displaystyle 0,\ \ \mbox{as}\
b\rightarrow\infty.$
Hence, (4.25), (4.29), (4.31) and (4.36) yield that
$\displaystyle\lim\limits_{b\rightarrow\infty}\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq
T]=0.$
∎
## 5\. Numerical examples
In this section, we present some numerical examples to give the readers an
intuitive impression on the relations between the results and model
parameters. Setting the parameters at suitable level, we portray how the debt
rate $\delta$, the constrained risk level of bankrupt probability, the initial
capital $x$ , the volatility rate $\sigma^{2}$ and the profit rate $\mu$
impact on the optimal return function and the optimal dividend strategy based
on the PDE (6.29) below. we also show the figures of the optimal return
function $g(x)$ and the associated optimal feedback control function
$a^{*}(x)$.
###### Example 5.1.
Let $\mu=2$, $\sigma^{2}=50$, $c=0.05$, $\alpha=0.5$, $\beta=8$, $T=300$ and
$b=100$. Figure 1 shows that the optimal return function $g_{\delta}(x)$
decreases with the debt rate $\delta$.
Figure 1. The optimal return function $g_{\delta}(x)$ as a function of
$\delta$. (Parameters: $\mu=2,\sigma=50,c=0.05,\alpha=0.5,\beta=8,T=300,b=100$
)
###### Example 5.2.
Let $\mu=2$, $\sigma^{2}=50$, $\delta=0.2$, $c=0.05$, $\alpha=0.5$, $\beta=8$,
$T=300$. Let $b(\varepsilon)$ be the solution of $1-\phi(T,b)=\varepsilon$,
where $\phi(T,b)$ is defined in Lemma 6.5. Thus given a constrained risk level
$\varepsilon$ of bankrupt probability, $b(\varepsilon)$ is the associated
dividends barrier. Figure 2 shows that the dividends barrier $b(\varepsilon)$
decreases with the constrained risk level $\varepsilon$.
Figure 2. Dividends barrier $b(\varepsilon)$ as a function of $\varepsilon$.
(Parameters: $\mu=2,\sigma^{2}=50,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300$
)
###### Example 5.3.
Let $\mu=2$, $\delta=0.2$, $c=0.05$, $\alpha=0.5$, $\beta=8$, $T=300$. Let
$b_{\sigma}(\varepsilon)$ be the solution of $1-\phi(T,b)=\varepsilon$, where
$\phi(T,b)$ is defined in Lemma 6.5. We see from Figure 3 that at the same
constrained risk level, the bigger the volatility rate $\sigma$ is, the higher
the dividends barrier $b_{\sigma}(\varepsilon)$ is.
Figure 3. Dividends barrier $b_{\sigma}(\varepsilon)$ as a function of
$\sigma^{2}$. (Parameters: $\mu=2,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300$
)
###### Example 5.4.
Let $\mu=2$, $\sigma^{2}=50$ $\delta=0.2$, $c=0.05$, $\alpha=0.5$, $\beta=8$,
$T=300$. Let $b_{\sigma}(\varepsilon)$ be the solution of
$1-\phi(T,b)=\varepsilon$, where $\phi(T,b)$ is defined in Lemma 6.5.
$R_{0}=x$ is the initial reserve and $\varepsilon$ is the constrained risk
level of bankrupt probability. We see from figure 4 that the lower the initial
reserve $x$ is, the higher the constrained risk level $\varepsilon$ is.
Figure 4. Initial reserve $x(\varepsilon)$ as a function of the risk
restrained level $\varepsilon$. (Parameters:
$\mu=2,\sigma^{2}=50,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300$ )
###### Example 5.5.
Let $\sigma^{2}=50$, $\delta=0.2$, $c=0.05$, $\alpha=0.5$, $\beta=8$, $T=300$
and $b=100$. Figure 5 shows that the optimal return function $g_{\mu}(x)$
increases with the profit rate $\mu$.
Figure 5. The optimal return function $g_{\mu}(x)$ as a function of $\mu$.
(Parameters: $\sigma^{2}=50,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300,b=100$
)
###### Example 5.6.
Let $\mu=2$, $\delta=0.2$, $c=0.05$, $\alpha=0.5$, $\beta=8$, $T=300$ and
$b=100$. Figure 6 shows that the optimal return function $g_{\sigma}(x)$
increases with the volatility rate $\sigma^{2}$.
Figure 6. The optimal return function $g_{\sigma}(x)$ as a function of
$\sigma^{2}$. (Parameters:
$\mu=2,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300,b=100$ )
###### Example 5.7.
Let $\mu=2$, $\sigma^{2}=50$, $\delta=0.2$, $c=0.05$, $\alpha=0.5$, $\beta=8$,
$T=300$ and $b=100$. Set $x_{\alpha}=4.72$, $x_{\beta}=94.79$, the images of
the optimal return function $g(x)$ as well as the optimal feedback control
function $a^{*}(x)$ are as follows (see Figure 7 and Figure 8).
Figure 7. The optimal return function $g(x)$. (Parameters:
$\mu=2,\sigma^{2}=50,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300,b=100$ )
Figure 8. The optimal feedback control function $a(x)$. (Parameters:
$\mu=2,\sigma^{2}=50,\delta=0.2,c=0.05,\alpha=0.5,\beta=8,T=300,b=100$ )
## 6\. Properties of $V(x,b)$ and Bankrupt Probability
In this section, we will discuss some important properties of the optimal
return function $V(x,b)$ and bankrupt probability, which are used to prove the
main result of this paper. The rigorous proofs of these properties will be
given in the appendix. In view of Lemma 8.1 in the appendix, different value
of $\frac{2\delta}{\mu}$ can lead to three different cases. When
$\frac{2\delta}{\mu}<\alpha$, this case is the most complicated. We select
this case as the basis of our discussion throughout the paper, and the results
of the other two cases are almost same.
###### Lemma 6.1.
If $f(x)\in C^{2}$ and satisfies the following HJB equation and boundary
conditions,
$\displaystyle\left\\{\begin{array}[]{l l
l}\max\limits_{a\in[\alpha,\beta]}[\frac{1}{2}\sigma^{2}a^{2}f^{{}^{\prime\prime}}(x)+(\mu
a-\delta)f^{{}^{\prime}}(x)-cf(x)]=0,\ \mbox{for}\ 0\leq x\leq b_{0},\\\
f^{{}^{\prime}}(x)=1,\ \mbox{ for}\ x\geq b_{0},\\\
f^{{}^{\prime\prime}}(x)=0,\ \mbox{ for}\ x\geq b_{0},\\\
f(0)=0,\end{array}\right.$ (6.5)
then we have
$\displaystyle b_{0}=\inf\\{x\geq 0:f^{\prime\prime}(x)=0\\}$
and
$\displaystyle\left\\{\begin{array}[]{l l l}\max\mathcal{L}f(x)\leq 0\
\mbox{and }\ f^{{}^{\prime}}(x)\geq 1\ \mbox{for}\ x\geq 0,\\\
f(0)=0,\end{array}\right.$
where $\mathcal{L}=\frac{1}{2}\sigma^{2}a^{2}\frac{d^{2}}{dx^{2}}+(\mu
a-\delta)\frac{d}{dx}-c$.
###### Lemma 6.2.
Let $b>b_{0}$ be a predetermined variable. If $g\in C^{1}(R_{+})$, $g\in
C^{2}(R_{+}\setminus\\{b\\})$ and satisfies the following HJB equation and
boundary conditions,
$\displaystyle\left\\{\begin{array}[]{l l
l}\max\limits_{a\in[\alpha,\beta]}[\frac{1}{2}\sigma^{2}a^{2}g^{{}^{\prime\prime}}(x)+(\mu
a-\delta)g^{{}^{\prime}}(x)-cg(x)]=0,\ \mbox{for}\ 0\leq x\leq b,\\\
g^{{}^{\prime}}(x)=1,\ \ \mbox{ for}\ \ x\geq b,\\\
g^{{}^{\prime\prime}}(x)=0,\ \ \mbox{ for}\ \ x>b,\\\
g(0)=0,\end{array}\right.$ (6.11)
then we have
$\displaystyle\left\\{\begin{array}[]{l l l}\max\mathcal{L}g(x)\leq 0,\
\mbox{for}\ \ x\geq 0,\\\ g^{{}^{\prime}}(x)\geq 1,\ \mbox{for}\ \ x\geq b,\\\
g(0)=0,\end{array}\right.$ (6.15)
where $b_{0}$ and $\mathcal{L}$ are the same as in Lemma 6.1,
$g^{\prime\prime}(b):=g^{\prime\prime}(b-)$. The expression of $g(x)$ can be
written as
$\displaystyle g(x,b)=\left\\{\begin{array}[]{l l
l}k_{1}(e^{r_{+}(\alpha)}x-e^{r_{-}(\alpha)x}),\ 0\leq x<x_{\alpha},\\\
k_{2}[\frac{\alpha\mu-2\delta}{2c}+\int_{x_{\alpha}}^{x}exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{y}\frac{dv}{a(v)})dy],\
x_{\alpha}\leq x<x_{\beta},\\\
k_{3}e^{r_{+}(\beta)(x-b_{0})}+k_{4}e^{r_{-}(\beta)(x-b_{0})},\ x_{\beta}\leq
x<b,\\\ x-b+g_{3}(b),\ x\geq b,\end{array}\right.$ (6.20)
where $r_{\pm}(x),x_{\alpha},x_{\beta},k_{1},k_{2},k_{3}$ and $k_{4}$ are
given by (8.9), (8.10), (8.17), (8.21), (8.37) and (8.36), respectively.
###### Lemma 6.3.
Let $g(x,b)$ be as the same as in Lemma 6.2. Then $\frac{\partial}{\partial
b}g(x,b)\leq 0$ holds for $b\geq b_{0}$.
###### Lemma 6.4.
The bankrupt probability $\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq T]$ is a
strictly decreasing function w.r.t. the dividends barrier $b$ on
$[x_{\beta},D)$, $D:=\inf\\{b:\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq T]=0\\}$,
and $x_{\beta}$ is defined by (8.17).
From the proof of Lemma 6.2, for each $x\leq b$, if we define
$\displaystyle a^{*}(x):=arg\
\max\limits_{a\in[\alpha,\beta]}[\frac{1}{2}\sigma^{2}a^{2}g^{{}^{\prime\prime}}(x)+(\mu
a-\delta)g^{{}^{\prime}}(x)-cg(x)],$ (6.21)
then it follows that $a^{*}(x)$ can be represented as
$\displaystyle a^{*}(x)=\left\\{\begin{array}[]{l l l}\alpha,\ \ 0\leq
x<x_{\alpha},\\\ a(x),\ \ x_{\alpha}\leq x<x_{\beta},\\\ \beta,\ \ x\geq
x_{\beta},\end{array}\right.$ (6.25)
where $a(x)$ and $x_{\alpha},x_{\beta}$ are specified by (8.16), (8.10),
(8.17), respectively. We now have the following lemma.
###### Lemma 6.5.
Let $a^{*}(x)$ be defined by (6.25), and define
$\psi^{b}(T,x):={\bf{P}}[\tau^{\pi^{*}_{b}}_{x}\leq T]$, i.e., $\psi^{b}(T,x)$
is the bankrupt probability when the initial reserve of
$\\{R^{\pi^{*}_{b}}_{t}\\}_{t\geq 0}$ is $x$ and dividends barrier is $b$. Let
$\phi^{b}(t,y)\in C^{1}(0,\infty)\cap C^{2}(0,b)$ and satisfy the following
partial differential equation and the boundary conditions,
$\displaystyle\left\\{\begin{array}[]{l l
l}\phi_{t}^{b}(t,x)=\frac{1}{2}[a^{*}(x)]^{2}\sigma^{2}\phi_{xx}^{b}(t,x)+(a^{*}(x)\mu-\delta)\phi_{x}^{b}(t,x),\\\
\phi^{b}(0,x)=1,\ \mbox{for}\ \ 0<x\leq b,\\\
\phi^{b}(t,0)=0,\phi_{x}^{b}(t,b)=0,\ \mbox{for}\ t>0.\end{array}\right.$
(6.29)
Then $\phi^{b}(T,x)=1-\psi^{b}(T,x)$, i.e., $\phi^{b}(T,x)$ is probability
that the company will survive on $[0,T]$.
###### Lemma 6.6.
Let $\phi^{b}(t,x)$ solve the equation(6.29). Then $\phi^{b}(T,b)$ is
continuous with respect to the dividends barrier $b$ on $[b_{0},+\infty)$.
## 7\. Proof of Main Result
In this section, we prove the main result of this paper, which is described in
Theorem 3.1. In order to do this, we first need the following.
###### Theorem 7.1.
Let $a^{*}_{b}(x)$ be defined by (6.25), and $f(x)$, $g(x,b)$ be as the same
as in Lemma 6.1 and Lemma 6.2, respectively. Then
(i) If $b\leq b_{0}$, we have $V(x,b)=V(x,b_{0})=V(x)=f(x)$, the optimal
policy associated with $V(x)$ is
$\pi_{b_{o}}^{\ast}=\\{a^{*}_{b_{0}}(R^{\pi_{b_{o}}^{\ast}}_{\cdot}),L^{\pi_{b_{o}}^{\ast}}_{\cdot}\\}$,
where the process
$\\{R^{\pi_{b_{o}}^{\ast}}_{t},L^{\pi_{b_{o}}^{\ast}}_{t}\\}$ is uniquely
determined by the following SDE,
$\displaystyle\left\\{\begin{array}[]{l l l}dR_{t}^{\pi_{b_{o}}^{\ast}}=(\mu
a^{*}_{b_{0}}(R^{\pi_{b_{o}}^{\ast}}_{t})-\delta)dt+\sigma
a^{*}_{b_{0}}(R^{\pi_{b_{o}}^{\ast}}_{t})d{W}_{t}-dL_{t}^{\pi_{b_{o}}^{\ast}},\\\
R_{0}^{\pi_{b_{o}}^{\ast}}=x,\\\ 0\leq R^{\pi_{b_{o}}^{\ast}}_{t}\leq
b_{0},\\\
\int^{\infty}_{0}I_{\\{t:R^{\pi_{b_{o}}^{\ast}}_{t}<b_{0}\\}}(t)dL_{t}^{\pi_{b_{o}}^{\ast}}=0.\end{array}\right.$
(7.5)
(ii) If $b>b_{0}$, we have $V(x,b)=g(x,b)$ and the optimal policy
$\pi_{b}^{\ast}$ is
$\\{a^{\ast}_{b}(R^{\pi_{b}^{\ast}}_{t}),L^{\pi_{b}^{\ast}}_{t}\\}$, where
$\\{R^{\pi_{b}^{\ast}}_{t},L^{\pi_{b}^{\ast}}_{t}\\}$ is uniquely determined
by the following SDE
$\displaystyle\left\\{\begin{array}[]{l l l}dR_{t}^{\pi_{b}^{\ast}}=(\mu
a^{*}_{b}(R^{\pi_{b}^{\ast}}_{t})-\delta)dt+\sigma
a^{*}_{b}(R^{\pi_{b}^{\ast}}_{t})d{W}_{t}-dL_{t}^{\pi_{b}^{\ast}},\\\
R_{0}^{\pi_{b}^{\ast}}=x,\\\ 0\leq R^{\pi_{b}^{\ast}}_{t}\leq b,\\\
\int^{\infty}_{0}I_{\\{t:R^{\pi_{b}^{\ast}}_{t}<b\\}}(t)dL_{t}^{\pi_{b}^{\ast}}=0.\end{array}\right.$
(7.10)
###### Proof.
(i) If $b\leq b_{0}$, since $\pi^{*}_{b_{0}}\in\Pi_{b_{0}}\subset\Pi_{b}$, we
have $V(x,b_{0})\leq V(x,b)\leq V(x)$. It suffices to show $V(x)\leq
f(x)=V(x,b_{0})$. Since its proof is similar to [7], we omit it here. (ii) If
$b\geq b_{0}$, denote $g(x,b)$ by $g(x)$ for simplicity, for any admissible
policy $\pi=\\{a_{\pi},L^{\pi}\\}$, we assume that $(R^{\pi}_{t},L^{\pi}_{t})$
is the process (2.3) associated with $\pi$. Let $\Lambda=\\{s:L_{s-}^{\pi}\neq
L_{s}^{\pi}\\}$, $\hat{L}=\sum_{s\in\Lambda,s\leq
t}(L_{s}^{\pi}-L_{s-}^{\pi})$ be the discontinuous part of $L_{s}^{\pi}$ and
$\tilde{L}_{t}^{\pi}=L_{t}^{\pi}-\hat{L}_{t}^{\pi}$ be the continuous part of
$L_{s}^{\pi}$. Applying generalized Itô formula to
$e^{-c(t\wedge\tau^{\pi}_{x})}g(R_{t\wedge\tau^{\pi}_{x}}^{\pi})$, we have
$\displaystyle
e^{-c(t\wedge\tau^{\pi}_{x})}g(R_{t\wedge\tau^{\pi}_{x}}^{\pi})$
$\displaystyle=$ $\displaystyle
g(x)+\int_{0}^{t\wedge\tau^{\pi}_{x}}e^{-cs}\mathcal{L}g(R_{s}^{\pi})ds$
(7.11) $\displaystyle+\int_{0}^{t\wedge\tau^{\pi}_{x}}a_{\pi}\sigma
e^{-cs}g^{{}^{\prime}}(R_{s}^{\pi})d\mathcal{W}_{s}\mathcal{-}\int_{0}^{t\wedge\tau^{\pi}_{x}}e^{-cs}g^{{}^{\prime}}(R_{s}^{\pi})dL_{s}^{\pi}$
$\displaystyle+\sum\limits_{s\in\Lambda,s\leq
t\wedge\tau^{\pi}_{x}}e^{-cs}[g(R_{s}^{\pi})-g(R_{s-}^{\pi})$
$\displaystyle-g^{{}^{\prime}}(R_{s-}^{\pi})(R_{s}^{\pi}-R_{s-}^{\pi})]$
$\displaystyle=$ $\displaystyle
g(x)+\int_{0}^{t\wedge\tau^{\pi}_{x}}e^{-cs}\mathcal{L}g(R_{s}^{\pi})ds$
$\displaystyle+\int_{0}^{t\wedge\tau^{\pi}_{x}}a_{\pi}\sigma
e^{-cs}g^{{}^{\prime}}(R_{s}^{\pi})d\mathcal{W}_{s}\mathcal{-}\int_{0}^{t\wedge\tau^{\pi}_{x}}e^{-cs}g^{{}^{\prime}}(R_{s}^{\pi})d\tilde{L}_{s}^{\pi}$
$\displaystyle+\sum\limits_{s\in\Lambda,s\leq
t\wedge\tau^{\pi}_{x}}e^{-cs}[g(R_{s}^{\pi})-g(R_{s-}^{\pi}))],$
where
$\displaystyle\mathcal{L}=\frac{1}{2}a^{2}\sigma^{2}\frac{d^{2}}{dx^{2}}+(\mu
a-\delta)\frac{d}{dx}-c.$
In view of the HJB equation (6.11), $\mathcal{L}g(R_{s}^{\pi})$ is always non-
positive, so is the second term on the right hand side of(7.11). By taking
mathematical expectations at both sides of (7.11), we get
$\displaystyle{\bf
E}\big{[}e^{-c(t\wedge\tau^{\pi}_{x})}g(R_{t\wedge\tau^{\pi}_{x}}^{\pi})\big{]}$
$\displaystyle\leq$ $\displaystyle g(x)-{\bf
E}\big{[}\int_{0}^{t\wedge\tau^{\pi}_{x}}e^{-cs}g^{{}^{\prime}}(R_{s}^{\pi})d\tilde{L}_{s}^{\pi}\big{]}$
$\displaystyle+{\bf E}\big{[}\sum\limits_{s\in\Lambda,s\leq
t\wedge\tau^{\pi}_{x}}e^{-cs}[g(R_{s}^{\pi})-g(R_{s-}^{\pi})]\big{]}.$
Since $g^{{}^{\prime}}(x)\geq 1$, for $x\geq b$,
$\displaystyle g(R_{s}^{\pi})-g(R_{s-}^{\pi})\leq-(L_{s}^{\pi}-L_{s-}^{\pi}),$
(7.13)
which, together with (7), implies that
$\displaystyle{\bf
E}\big{[}e^{-c(t\wedge\tau^{\pi}_{x})}g(R_{t\wedge\tau^{\pi}_{x}}^{\pi})\big{]}$
$\displaystyle+$ $\displaystyle{\bf
E}\big{[}\int_{0}^{t\wedge\tau^{\pi}_{x}}e^{-cs}dL_{s}^{\pi}\big{]}\leq g(x).$
(7.14)
By the definition of $\tau^{\pi}_{x}$ and $g(0)=0$, letting
$t\rightarrow\infty$ in (7.14), we get
$\displaystyle\liminf\limits_{t\rightarrow\infty}e^{-c(t\wedge\tau^{\pi}_{x})}g(R_{t\wedge\tau^{\pi}_{x}}^{\pi})$
$\displaystyle=$ $\displaystyle e^{-c\tau}g(0)I_{\\{\tau^{\pi}_{x}<\infty\\}}$
(7.15)
$\displaystyle+\liminf\limits_{t\rightarrow\infty}e^{-ct}g(R_{t})I_{\\{\tau^{\pi}_{x}=\infty\\}}$
$\displaystyle\geq$ $\displaystyle 0.$
We deduce from (7.14) and (7.15) that
$\displaystyle J(x,\pi)={\bf
E}\big{[}\int_{0}^{\tau^{\pi}_{x}}e^{-cs}dL_{s}^{\pi}\big{]}\leq g(x).$
So
$\displaystyle V(x,b)\leq g(x).$ (7.16)
If we choose the control policy
$\pi_{b}^{\ast}=\\{a^{*}_{b}(R^{\pi_{b}^{\ast}}_{\cdot}),L^{\pi_{b}^{\ast}}_{\cdot}\\}$,
which is uniquely determined by SDE (7.10), then all the inequalities above
become equalities. Hence
$\displaystyle V(x,b)=g(x).$
So we have
$\displaystyle V(x,b)=g(x,b).$ (7.17)
∎
Now we prove the main result of this paper. Proof of Theorem 3.1. If
$\mathbf{P}[\tau_{b_{0}}^{\pi^{*}_{b_{0}}}\leq T]\leq\varepsilon$, the
bankrupt probability constraint does not work and it turns to a usual optimal
control problem, thus the conclusion is obvious. If
$\mathbf{P}[\tau_{b_{0}}^{\pi^{*}_{b_{0}}}\leq T]>\varepsilon$, then by lemmas
6.4-6.6 there exists a unique $b^{\ast}$ solving the equation
$\mathbf{P}\\{\tau_{b}^{\pi^{*}_{b}}\leq T\\}=\varepsilon$. Moreover,
$b^{\ast}=\inf\\{b:b\in\mathfrak{B}\\}>b_{0}$. By Theorem 7.1 and Lemma 6.3,
$V(x,b)=g(x,b)$ for $b>b_{0}$ and $V(x,b)$ is decreasing w.r.t. $b$.
Therefore, we know that $b^{\ast}$ meets (3.6) and (3.7). So the optimal
policy associated with the optimal return function $V(x,b^{*})$ is
$\\{a^{\ast}_{b^{*}}(R^{\pi_{b^{*}}^{\ast}}_{t}),L^{\pi_{b^{*}}^{\ast}}_{t}\\}$,
which is uniquely determined by SDE (3.12). Due to the fact $b^{*}>b_{0}$, the
inequity (3.13) is a direct consequence of Lemma 6.3. Thus we complete the
proof. $\Box$
## 8\. Appendix
In this section, we first discuss some useful arguments, then we give the
proofs of the lemmas used in the previous sections. Due to the mathematical
model presented by (2.2), $a(t)$ is required to take values in the interval
$\alpha,\beta$, where $0<\alpha<\beta<+\infty$. Thus, $a(t)\mu-\delta$ may be
negative because $\delta>0$. If $\beta\mu\leq\delta$, there exists a trivial
solution to the corresponding HJB equations, which has been proved by Choulli,
Taksar, and Zhou[7]. In the next section, we always assume that
$\beta\mu\geq\delta$. Then the following statements are valid.
###### Lemma 8.1.
Let $\beta\mu>0$. Then
(i) $\frac{2\delta}{\mu}<\alpha$ if and only if $a(0)<\alpha$. In this case,
$\displaystyle a(0)=\frac{\mu\alpha^{2}}{2(\mu\alpha-\delta)}.$ (8.1)
(ii) $\alpha\leq\frac{2\delta}{\mu}<\beta$ if and only if $\alpha\leq
a(0)<\beta$. In this case,
$\displaystyle a(0)=\frac{2\delta}{\mu}.$ (8.2)
(iii) $\beta\leq\frac{2\delta}{\mu}$ if and only if $a(0)\geq\beta$. In this
case,
$\displaystyle a(0)=\frac{\mu\beta^{2}}{2(\mu\beta-\delta)}.$ (8.3)
###### Proof.
See Choulli, Taksar and Zhou[7] for details. ∎
Due to Lemma 8.1, there are three different cases to investigate. Since the
proof of each case is similar, we only give sketch proofs of lemmas in case
(i). Thus we suppose $\frac{2\delta}{\mu}<\alpha$ throughout the following
procedure to prove these lemmas. Proof of lemma 6.1. The complete proof is
given in Choulli, Taksar and Zhou[7](2003). $\Box$
Proof of lemma 6.2. Step 1. For each $x\geq 0$ and $a\geq 0$, define
$\displaystyle h(x,a)=\frac{1}{2}\sigma^{2}a^{2}g^{{}^{\prime\prime}}(x)+(\mu
a-\delta)g^{{}^{\prime}}(x)-cg(x).$ (8.4)
Then, by differentiating $h(x,a)$ w.r.t. $a$, we get the maximizing function
of $h(x,a)$
$\displaystyle a(x)=-\frac{\mu
g^{{}^{\prime}}(x)}{\sigma^{2}g^{{}^{\prime\prime}}(x)},\ \ x\geq 0.$ (8.5)
In view of Lemma 8.1 (i), $a(x)\leq\alpha$ for all $x$ in the right
neighborhood of 0. Substituting $a=\alpha$ into (6.11), and solving the
resulting second-order linear ODE, we get
$\displaystyle g(x)=k_{1}(e^{r_{+}(\alpha)}x-e^{r_{-}(\alpha)x}),\ \ 0\leq
x<x_{\alpha},$ (8.6)
where $k_{1}$ and $x_{\alpha}$ are to be determined and for $x>0$
$\displaystyle\left\\{\begin{array}[]{l l l}r_{+}(x)=\frac{-(\mu
x-\delta)+\sqrt{(\mu x-\delta)^{2}+2\sigma^{2}cx^{2}}}{\sigma^{2}x^{2}}\\\
r_{-}(x)=\frac{-(\mu x-\delta)-\sqrt{(\mu
x-\delta)^{2}+2\sigma^{2}cx^{2}}}{\sigma^{2}x^{2}}.\end{array}\right.$ (8.9)
Due to (8.5) and (8.6), for $x>0$
$\displaystyle a^{{}^{\prime}}(x)=\frac{-\mu
r_{+}(\alpha)r_{-}(\alpha)(r_{+}(\alpha)-r_{-}(\alpha))^{2}e^{(r_{+}(\alpha)+r_{-}(\alpha))x}}{\sigma^{2}(g^{{}^{\prime\prime}}(x))^{2}}>0.$
Therefore $a(x)$ increases to $\alpha$ at the point $x_{\alpha}$ given by
$\displaystyle
x_{\alpha}=\frac{1}{r_{+}(\alpha)-r_{-}(\alpha)}log(\frac{r_{-}(\alpha)(\mu+\alpha\sigma^{2}r_{-}(\alpha))}{r_{+}(\alpha)(\mu+\alpha\sigma^{2}r_{+}(\alpha))})>0.$
(8.10)
Step 2. In view of Proposition 8 in [7], $\alpha\leq a(x)\leq\beta$ in the
right neighborhood of $x_{\alpha}$. From (8.5), we get
$\displaystyle g^{{}^{\prime\prime}}(x)=-\frac{\mu
g^{{}^{\prime}}(x)}{\sigma^{2}a(x)}.$ (8.11)
Substituting (8.11) into (6.11), differentiating the resulting equation, and
using (8.11) again, we obtain
$\displaystyle
a^{{}^{\prime}}(x)=\frac{\mu^{2}+2c\sigma^{2}}{\mu\sigma^{2}}(1-\frac{u}{a(x)}),$
(8.12)
with
$\displaystyle u\equiv\frac{2\delta\mu}{\mu^{2}+2c^{2}\sigma^{2}}.$ (8.13)
Integrating (8.12), we get
$\displaystyle
G(a(x))=\frac{\mu^{2}+2c\sigma^{2}}{\mu\sigma^{2}}(x-x_{\alpha})+G(\alpha),$
(8.14)
where
$\displaystyle G(z)=z+ulog(z-u).$ (8.15)
Therefore
$\displaystyle
a(x)=G^{-1}(\frac{\mu^{2}+2c\sigma^{2}}{\mu\sigma^{2}}(x-x_{\alpha})+G(\alpha)).$
(8.16)
Obviously, $a(x)$ is increasing. Let $a(x_{\beta})=\beta$, we get
$\displaystyle x_{\beta}$ $\displaystyle=$
$\displaystyle\frac{\mu\sigma^{2}}{\mu^{2}+2c\sigma^{2}}[G(\beta)-G(\alpha)]+x_{\alpha}$
(8.17) $\displaystyle=$
$\displaystyle\frac{\mu\sigma^{2}}{\mu^{2}+2c\sigma^{2}}(\beta-\alpha)+\frac{\mu\sigma^{2}u}{\mu^{2}+2c\sigma^{2}}log(\frac{\beta-u}{\alpha-u})+x_{\alpha}.$
Solving (8.11),(8.15) and (8.16), we obtain
$\displaystyle
g(x)=g(x_{\alpha})+g^{{}^{\prime}}(x_{\alpha})\int_{x_{\alpha}}^{x}exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{y}\frac{dv}{a(v)})dy,\
\ x_{\alpha}\leq x<x_{\beta},$
where $g(x_{\alpha})$ and $g^{{}^{\prime}}(x_{\alpha})$ are free constants to
be determined. From (8.6) and (8.10), we deduce
$\displaystyle
g(x_{\alpha})=\frac{\alpha\mu-2\delta}{2c}g^{{}^{\prime}}(x_{\alpha})$ (8.19)
Let
$\displaystyle k_{2}\equiv g^{{}^{\prime}}(x_{\alpha}),$ (8.20)
Then (8.6) and (8.19) imply
$\displaystyle
k_{1}=\frac{\alpha\mu-2\delta}{2c(e^{r_{+}(\alpha)x_{\alpha}}-e^{r_{-}(\alpha)x_{\alpha}})}k_{2}$
(8.21)
Substituting (8.19) and (8.20) into (8), we get
$\displaystyle
g(x)=k_{2}[\frac{\alpha\mu-2\delta}{2c}+\int_{x_{\alpha}}^{x}exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{y}\frac{dv}{a(v)})dy],\
x_{\alpha}\leq x<x_{\beta}.$ (8.22)
Step 3. In view of Proposition 9 in [7], $a(x)\geq\beta$ holds for $x\geq
x_{\beta}$. Substituting $a=\beta$ into (6.11), and solving it, we get the
following solution
$\displaystyle
g(x)=k_{3}e^{r_{+}(\beta)(x-b_{0})}+k_{4}e^{r_{-}(\beta)(x-b_{0})},\
x_{\beta}\leq x<b,$ (8.23)
where $k_{3},k_{4}$ are free constants to be determined and $r_{\pm}(\beta)$
are given by (8.9). For $x\geq b$, the solution has the following form
$\displaystyle
g(x)=x-b+k_{3}e^{r_{+}(\beta)(b-b_{0})}+k_{4}e^{r_{-}(\beta)(b-b_{0})},\ x\geq
b.$ (8.24)
Next we apply the principle of smooth fit to determine the unknown constants
above. Note that
$\displaystyle\left\\{\begin{array}[]{l l l}g(x_{\beta}-)=g(x_{\beta}+)\\\
g^{{}^{\prime}}(x_{\beta}-)=g^{{}^{\prime}}(x_{\beta}+),\end{array}\right.$
(8.27)
we arrive at
$\displaystyle\left\\{\begin{array}[]{l l
l}k_{2}\xi=k_{3}e^{r_{+}(\beta)(x_{\beta}-b_{0})}+k_{4}e^{r_{-}(\beta)(x_{\beta}-b_{0})}\\\
k_{2}\eta=k_{3}r_{+}(\beta)e^{r_{+}(\beta)(x_{\beta}-b_{0})}+k_{4}r_{-}(\beta)e^{r_{-}(\beta)(x_{\beta}-b_{0})},\end{array}\right.$
(8.30)
where
$\displaystyle\left\\{\begin{array}[]{l l
l}\xi=\frac{\alpha\mu-2\delta}{2c}+\int_{x_{\alpha}}^{x_{\beta}}exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{y}\frac{dv}{a(v)})dy\\\
\eta=exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{x_{\beta}}\frac{dv}{a(v)}).\end{array}\right.$
(8.33)
Solving (8.30) for $k_{3}$ and $k_{4}$, we get
$\displaystyle\left\\{\begin{array}[]{l l l}k_{3}=\frac{\eta-\xi
r_{-}(\beta)}{(r_{+}(\beta)-r_{-}(\beta))e^{r_{+}(\beta)(x_{\beta}-b_{0})}}k_{2}\equiv
Ak_{2}\\\ k_{4}=\frac{\xi
r_{+}(\beta)-\eta}{(r_{+}(\beta)-r_{-}(\beta))e^{r_{-}(\beta)(x_{\beta}-b_{0})}}k_{2}\equiv
Bk_{2}.\end{array}\right.$ (8.36)
Substituting (8.36) into (8.23) and using $g^{{}^{\prime}}(b-)=1$, we obtain
$\displaystyle
k_{2}=\frac{1}{Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})}}.$
(8.37)
Thus, $k_{1},k_{2},k_{3},k_{4}$ are determined by (8.21),(8.36) and (8.37). We
claim that
$\displaystyle g^{{}^{\prime\prime}}(b_{-})\geq 0.$ (8.38)
In order to prove this statement, we consider $f(x)$ in Lemma 6.1 and notice
that $A$ and $B$ in (8.36) have the same expression both in $f(x)$ and $g(x)$.
Since $f^{{}^{\prime}}(b_{0})=1$ and $f^{{}^{\prime\prime}}(b_{0})=0$,
$\displaystyle\left\\{\begin{array}[]{l l
l}k^{f}_{2}(Ar_{+}(\beta)+Br_{-}(\beta))=1\\\
k^{f}_{2}(Ar^{2}_{+}(\beta)+Br^{2}_{-}(\beta))=0,\end{array}\right.$ (8.41)
where $k^{f}_{2}$ is the corresponding constant in Lemma 6.1. From (8.41), we
know that $A<0,B>0$ due to $r_{+}(\beta)>0,r_{-}(\beta)<0)$ and $k^{f}_{2}>0$
in $f(x)$. In addition, if we let
$\displaystyle l(b)\equiv
g^{{}^{\prime\prime}}_{3}(b-)=\frac{Ar^{2}_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br^{2}_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})}}{Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})}},$
(8.42)
then,
$\displaystyle\frac{\partial l}{\partial b}$ $\displaystyle=$
$\displaystyle\frac{AB(r^{3}_{+}(\beta)r_{-}(\beta)+r_{+}(\beta)r^{3}_{-}(\beta)-2r^{2}_{+}(\beta)r^{2}_{-}(\beta))e^{(r_{+}(\beta)+r_{(}\beta))(b-b_{0})}}{(Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})^{2}}$
$\displaystyle>$ $\displaystyle 0$
holds for $A>0,B<0,r_{+}(\beta)>0$ and $r_{-}(\beta)<0$. Since $b>b_{0}$, we
conclude that
$\displaystyle
g^{{}^{\prime\prime}}(b-)=l(b)>l(b_{0})=\tilde{g}^{{}^{\prime\prime}}(b_{0})=0,$
(8.44)
where $\tilde{g}(x)$ is the solution of (6.11) with replacing $b$ by $b_{0}$.
Step 4. Now we only need to prove the solution $g(x)$ satisfies (6.11). It
suffices to prove the following conditions,
$\displaystyle\max\limits_{a\in[\alpha,\beta]}[\frac{1}{2}\sigma^{2}a^{2}g^{{}^{\prime\prime}}(x)+(\mu
a-\delta)g^{{}^{\prime}}(x)-cg(x)]=0,\ \mbox{for}\ x\geq b.$
By (6.11), (8.44) and noticing that $g^{{}^{\prime}}(b-)=1$, we get
$\displaystyle\max\mathcal{L}g(x)$ $\displaystyle=$ $\displaystyle\mu
a-\delta-c(x-b+g(b))$ (8.45) $\displaystyle\leq$ $\displaystyle\mu a-\delta-
cg(b)$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sigma^{2}a^{2}g^{{}^{\prime\prime}}(b-)+(\mu
a-\delta)g^{{}^{\prime}}_{3}(b-)-cg(b)$ $\displaystyle\leq$ $\displaystyle 0.$
Thus, we complete the proof. We summarize the solution as follows. For
$\frac{2\delta}{\mu}<\alpha$,
$\displaystyle g(x)=\left\\{\begin{array}[]{l l
l}k_{1}(e^{r_{+}(\alpha)x}-e^{r_{-}(\alpha)x}),\ \ 0\leq x<x_{\alpha},\\\
k_{2}[\frac{\alpha\mu-2\delta}{2c}+\int_{x_{\alpha}}^{x}\exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{y}\frac{dv}{a(v)})dy],\
\ x_{\alpha}\leq x<x_{\beta},\\\
k_{3}e^{r_{+}(\beta)(x-b_{0})}+k_{4}e^{r_{-}(\beta)(x-b_{0})},\ \
x_{\beta}\leq x<b,\\\
x-b+k_{3}e^{r_{+}(\beta)(b-b_{0})}+k_{4}e^{r_{-}(\beta)(b-b_{0})},\ \ x\geq
b,\end{array}\right.$ (8.50)
where $r_{\pm}(x),x_{\alpha},x_{\beta},k_{1},k_{2},k_{3}$ and $k_{4}$ are
defined by (8.9), (8.10), (8.17), (8.21), (8.37) and (8.36), respectively.
$\Box$ Proof of lemma 6.3. For $b\geq b_{0},A<0,B>0$, together with (8.42) and
(8), we have
$\displaystyle\frac{\partial}{\partial b}g(b,x)$ $\displaystyle=$
$\displaystyle-\frac{(\alpha\mu-2\delta)}{2c(e^{r_{+}(\alpha)x_{\alpha}-r_{-}(\alpha)x_{\alpha}})}$
$\displaystyle\times\frac{Ar^{2}_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br^{2}_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})}}{(Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})^{2}}$
$\displaystyle\leq$ $\displaystyle 0,\ \ \ 0\leq x<x_{\alpha};$
$\displaystyle\frac{\partial}{\partial b}g(b,x)$ $\displaystyle=$
$\displaystyle-(\frac{\alpha\mu-2\delta}{2c}+\int_{x_{\alpha}}^{x}exp(-\frac{\mu}{\sigma^{2}}\int_{x_{\alpha}}^{y}\frac{dv}{a(v)})dy)$
$\displaystyle\times\frac{Ar^{2}_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br^{2}_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})}}{(Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})^{2}}$
$\displaystyle\leq$ $\displaystyle 0,\ \ \ x_{\alpha}\leq x<x_{\beta};$
$\displaystyle\frac{\partial}{\partial b}g(b,x)$ $\displaystyle=$
$\displaystyle(Ae^{r_{+}(\beta)(b-b_{0})}+Be^{r_{-}(\beta)(b-b_{0})})$
$\displaystyle\times\frac{(Ar^{2}_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br^{2}_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})}{(Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})^{2}}$
$\displaystyle\leq$ $\displaystyle 0,\ \ \ x_{\beta}\leq x<b;$
$\displaystyle\frac{\partial}{\partial b}g(b,x)$ $\displaystyle=$
$\displaystyle(Ae^{r_{+}(\beta)(b-b_{0})}+Be^{r_{-}(\beta)(b-b_{0})})$
$\displaystyle\times\frac{(Ar^{2}_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br^{2}_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})}{(Ar_{+}(\beta)e^{r_{+}(\beta)(b-b_{0})}+Br_{-}(\beta)e^{r_{-}(\beta)(b-b_{0})})^{2}}$
$\displaystyle\leq$ $\displaystyle 0,\ \ \ x\geq b.$
Thus, the proof is completed. $\Box$ Proof of lemma 6.4. We can prove that
$\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq T]$ is decreasing in $b$ along the
lines of Theorem 3.1 in [17](2008). Here we only need to prove that
$\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq T]$ is strictly decreasing in $b$ on
$[x_{\beta},D)$. We denote $\mathbf{P}[\tau_{b}^{\pi^{*}_{b}}\leq T]$ by
$\mathbf{P}[\tau_{b}^{b}\leq T]$ for simplicity. For any $b_{1},b_{2}$
satisfying $D\geq b_{2}\geq b_{1}\geq x_{\beta}$, we need to prove that
$\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{1}}\leq
T]-\mathbf{P}[\tau_{b_{2}}^{b_{2}}\leq T]>0$
By comparison theorem, we have
$\mathbf{P}[\tau_{b_{1}}^{b_{1}}\leq T]-\mathbf{P}[\tau_{b_{2}}^{b_{2}}\leq
T]\geq\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq
T]-\mathbf{P}[\tau_{b_{2}}^{b_{2}}\leq T].$
So we the proof can be reduced to proving that
$\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq
T]-\mathbf{P}[\tau_{b_{2}}^{b_{2}}\leq T]>0.$ (8.52)
Define stochastic processes $R_{t}^{(1)}$, $R_{t}^{(2)}$, $R_{t}^{(3)}$,
$R_{t}^{(4)}$ by the following SDEs:
$\displaystyle dR_{t}^{(1)}$ $\displaystyle=$ $\displaystyle[\mu
a_{b_{2}}^{*}(R_{t}^{(1)})-\delta]dt+a_{b_{2}}^{*}(R_{t}^{(1)})\sigma
d\mathcal{W}_{t}-dL_{t}^{b_{2}},R_{0}^{(1)}=b_{1};$ $\displaystyle
dR_{t}^{(2)}$ $\displaystyle=$ $\displaystyle[\mu
a_{b_{2}}^{*}(R_{t}^{(2)})-\delta]dt+a_{b_{2}}^{*}(R_{t}^{(2)})\sigma
d\mathcal{W}_{t}-dL_{t}^{b_{2}},R_{0}^{(2)}=b_{2};$ $\displaystyle
dR_{t}^{(3)}$ $\displaystyle=$ $\displaystyle[\mu
a_{b_{2}}^{*}(R_{t}^{(3)})-\delta]dt+a_{b_{2}}^{*}(R_{t}^{(3)})\sigma
d\mathcal{W}_{t}-dL_{t}^{b_{2}},R_{0}^{(3)}=\frac{b_{1}+b_{2}}{2};$
$\displaystyle dR_{t}^{(4)}$ $\displaystyle=$ $\displaystyle[\mu
a_{b_{2}}^{*}(R_{t}^{[4]})-\delta]dt+a_{b_{2}}^{*}(R_{t}^{(4)})\sigma
d\mathcal{W}_{t},R_{0}^{(4)}=\frac{b_{1}+b_{2}}{2},$
respectively, where $D\geq b_{2}\geq b_{1}\geq x_{\beta}$ and $a^{*}(\cdot)$
is defined by (6.25). First, let $\tau^{b_{1}}=\inf\limits_{t\geq
0}\\{t:R_{t}^{(2)}=b_{1}\\}$, $A=\\{\tau^{b_{1}}\leq T\\}$, $B=\big{\\{}\sl
R_{t}^{(2)}$ will go to bankruptcy in a time interval
$[\tau^{b_{1}},\tau^{b_{1}}+T]$ and $\tau^{b_{1}}\leq T\big{\\}}$,
$D=\\{\inf\limits_{0\leq t\leq T}R_{t}^{(3)}>b_{1}\\}$ and
$E=\\{\inf\limits_{0\leq t\leq T}R_{t}^{(4)}>b_{1},\sup\limits_{0\leq t\leq
T}R_{t}^{(4)}<b_{2}\\}$. Then
$\displaystyle\\{\tau_{b_{2}}^{b_{2}}\leq T\\}\subset B\subset A.$ (8.53)
Moreover, by using strong Markov property of $R_{t}^{[2]}$, we have
$\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq T]=\mathbf{P}[B|A].$ (8.54)
By comparison theorem on SDE, we have
$\displaystyle\mathbf{P}(A^{c})\geq\mathbf{P}(D)\geq\mathbf{P}(E).$ (8.55)
Since $a_{b_{2}}^{*}(x)=\beta$ we have
$\displaystyle
R_{t}^{(4)}=\frac{b_{1}+b_{2}}{2}+[\mu\beta-\delta]t+\sigma\mathcal{W}_{t}\
\mbox{ on $E$}.$ (8.56)
We deduce from (8.56) and properties of Brownian motion with drift (cf.
Andrei,Borodin,Paavo,Salminen [1] (2002)) that
$\displaystyle\mathbf{P}(E)=\frac{e^{-\mu^{\prime 2}T/2}}{\sqrt{2\pi
T}}\sum_{k=-\infty}^{\infty}\int_{b_{1}/\sigma\beta}^{b_{2}/\sigma\beta}e^{\mu^{\prime}(z-x)}[(e^{-(z-x+\frac{2k(b_{2}-b_{1})}{\sigma\beta})^{2}/2T})$
$\displaystyle-(e^{-(z+x-\frac{2b_{1}-2k(b_{2}-b_{1})}{\sigma\beta})^{2}/2T})]dz>0,$
where $\mu^{\prime}=(\beta\mu-\delta)/\sigma$ and
$x=\frac{b_{1}+b_{2}}{2\sigma\beta}$. Thus we get
$\displaystyle\mathbf{P}(A^{c})>0.$ (8.57)
We also know from Theorem 4.1 that $\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq
T]\geq\mathbf{P}[\tau_{b_{2}}^{b_{2}}\leq T]>0$, which together with (8.53),
(8.54) and (8.57), implies that
$\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq
T]-\mathbf{P}[\tau_{b_{2}}^{b_{2}}\leq T]$ $\displaystyle\geq$
$\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq T]-\mathbf{P}(B)$
$\displaystyle=$ $\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq
T]-\mathbf{P}(A)\mathbf{P}(B|A)$ $\displaystyle=$
$\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq T](1-\mathbf{P}(A))$
$\displaystyle=$ $\displaystyle\mathbf{P}[\tau_{b_{1}}^{b_{2}}\leq
T]\mathbf{P}(A^{c})$ $\displaystyle>$ $\displaystyle 0.$
Thus the proof is completed. $\Box$
Proof of lemma 6.5. Let $\phi(t,x)\equiv\phi^{b}(t,x)$. Setting
$\tau_{x}^{b}:=\tau_{x}^{\pi^{*}_{b}}$ and applying the generalized Itô
formula to $(R^{\pi_{b}^{\ast},x}_{t},L_{t}^{\pi_{b}^{\ast}})$ and $\phi(t,x)$
we have for $0<x\leq b$,
$\displaystyle\phi(T-(t\wedge\tau_{x}^{b}),R^{\pi_{b}^{\ast},x}_{t\wedge\tau_{x}^{b}})$
$\displaystyle=$ $\displaystyle\phi(T,x)$ $\displaystyle+$
$\displaystyle\int_{0}^{t\wedge\tau_{x}^{b}}(\frac{1}{2}a^{*2}(R^{\pi_{b}^{\ast},x}_{s})\sigma^{2}\phi_{yy}(T-s,R^{\pi_{b}^{\ast},x}_{s})$
$\displaystyle+$
$\displaystyle(a^{*}_{b}(R^{\pi_{b}^{\ast},x}_{s})\mu-\delta)\phi_{x}(T-s,R^{\pi_{b}^{\ast},x}_{s})$
$\displaystyle-$
$\displaystyle\phi_{t}(T-s,R^{\pi_{b}^{\ast},x}_{s}))ds-\int_{0}^{t\wedge\tau_{x}^{b}}\phi_{y}(T-s,R^{\pi_{b}^{\ast},x}_{s})dL_{s}^{b}$
$\displaystyle+$
$\displaystyle\int_{0}^{t\wedge\tau_{x}^{b}}a^{*}(R^{\pi_{b}^{\ast},x}_{s})\sigma\phi_{x}(T-s,R^{\pi_{b}^{\ast},x}_{s})dW_{s}.$
Setting $t=T$ and taking mathematical expectation at both sides of (8) yield
that
$\displaystyle\phi(T,x)$ $\displaystyle=$
$\displaystyle\mathbf{E}[\phi(T-(T\wedge\tau_{x}^{b}),R^{\pi_{b}^{\ast},x}_{T\wedge\tau_{x}^{b}})]$
$\displaystyle=$
$\displaystyle\mathbf{E}[\phi(0,R^{\pi_{b}^{\ast},x}_{T})1_{T<\tau_{x}^{b}}]+\mathbf{E}[\phi(T-\tau_{x}^{b},0)1_{T\geq\tau_{x}^{b}})]$
$\displaystyle=$ $\displaystyle\mathbf{E}[1_{T<\tau_{x}^{b}}]$
$\displaystyle=$ $\displaystyle 1-\psi(T,x).$
Thus we complete the proof. $\Box$
Define $u(x):=\frac{1}{2}a^{*2}(x)\sigma^{2}$, $v(x):=a^{*2}(x)\mu-\delta$,
the equation (4.13) becomes
$\displaystyle\phi_{t}^{b}(t,x)=u(x)\phi_{xx}^{b}(t,x)+v(x)\phi_{x}^{b}(t,x).$
(8.59)
Obviously, $u(x)$ and $v(x)$ are continuous in $[0,b]$ due to the fact that
$a^{*}(x)$ is continuous w.r.t $x$. Thus there exists a unique solution in
$C^{1}(0,\infty)\cap C^{2}(0,b)$ for (6.29). Moreover, $u^{{}^{\prime}}(x)$,
$v^{{}^{\prime}}(x)$, $u^{{}^{\prime\prime}}(x)$ are bounded in
$(0,x_{\alpha})$, $(x_{\alpha},x_{\beta})$, $(x_{\beta},b)$, respectively. Now
we are ready to prove that the bankrupt probability $\psi^{b}(T,b)$ is
continuous with respect to the dividends barrier $b(b\geq b_{0})$. Proof of
lemma 6.6. It suffices to prove that $\phi^{b}(t,x)$ is continuous in b. Let
$x=by$ and $\theta^{b}(t,y)=\phi^{b}(t,by)$, the equation (6.29) becomes
$\displaystyle\left\\{\begin{array}[]{l l
l}\theta_{t}^{b}(t,y)=[u(by)/b^{2}]\theta_{yy}^{b}(t,z)+[v(by)/b]\theta_{y}^{b}(t,y),\\\
\theta^{b}(0,y)=1,\ \mbox{for}\ 0<y\leq 1,\\\
\theta^{b}(t,0)=0,\theta_{y}^{b}(t,1)=0,\ \mbox{for}\ t>0.\end{array}\right.$
(8.63)
So the proof of Lemma 6.6 can be reduced to proving
$\lim\limits_{b_{2}\rightarrow b_{1}}\theta^{b_{2}}(t,1)=\theta^{b_{1}}(t,1)$
for fixed $b_{1}>b_{0}$. Setting
$w(t,y)=\theta^{b_{2}}(t,y)-\theta^{b_{1}}(t,y)$ and noticing that
$\theta^{b}(t,y)$ is continuous at $y=1$ for any $b\geq b_{0}$, it suffices to
show that
$\displaystyle\int_{0}^{t}\int_{0}^{1}w^{2}(s,y)dyds\rightarrow 0,\ \mbox{as}\
b_{2}\rightarrow b_{1}.$ (8.64)
Thus we have
$\displaystyle\left\\{\begin{array}[]{l l
l}w_{t}(t,y)&=&[u(b_{2}y)/b_{2}^{2}]w_{yy}(t,y)+[v(b_{2}y)/b_{2}]w_{y}(t,y)\\\
&+&\\{u(b_{2}y)/b_{2}^{2}-u(b_{1}z)/b_{1}^{2}\\}\theta_{yy}^{b_{1}}(t,y)\\\
&+&\\{u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}\\}\theta_{y}^{b_{1}}(t,y),\\\
w(0,y)&=&0,\ \mbox{for}\ 0<y\leq 1,\\\ w(t,0)&=&0,\ w_{y}(t,1)=0,\ \mbox{for}\
t>0.\end{array}\right.$ (8.70)
Multiplying the first equation in (8.70) by $w(t,y)$ and then integrating on
$[0,1]\times[0,t]$,
$\displaystyle\int_{0}^{t}\int_{0}^{1}w(s,y)w_{t}(s,y)dyds$ (8.71)
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}\big{\\{}[u(b_{2}y)/b_{2}^{2}]w(s,y)w_{yy}(s,y)$
$\displaystyle+\int_{0}^{t}\int_{0}^{1}[v(b_{2}y)/b_{2}]w(s,y)w_{y}(s,y)$
$\displaystyle+\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]w(s,y)\theta_{yy}^{b_{1}}(t,y)$
$\displaystyle+\int_{0}^{t}\int_{0}^{1}w(s,y)[v(b_{2}y)/b_{2}-v(b_{1}y)/b_{1}]w(s,y)\theta_{y}^{b_{1}}(t,y)\big{\\}}dyds$
$\displaystyle\equiv$ $\displaystyle E_{1}+E_{2}+E_{3}+E_{4}.$
We now estimate each terms at both sides of (8.71) as follows.
$\displaystyle\int_{0}^{t}\int_{0}^{1}w(s,y)w_{t}(s,y)dyds$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{1}{2}w^{2}(t,y)dy.$ (8.72)
By property of $a^{*}(x)$ and the definition of $u(x)$ and $v(x)$ there exit
positive constants $D_{1}$, $D_{2}$ and $D_{3}$ such that
$[v(b_{2}y)/b_{2}]^{2}\leq D_{1}$, $[u(by)/b^{2}]^{\prime}\geq 0$,
$[u(b_{2}y)/b_{2}^{2}]\geq D_{2}$ and $[a(b_{2}y)/b_{2}^{2}]^{\prime}\leq
D_{3}$. For any $\lambda_{1}>0$ and $\lambda_{2}>0$, by these facts and
Young’s inequality, we estimate $E_{1}$ and $E_{2}$ as follows,
$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}]w(s,y)w_{yy}(s,y)dyds$
(8.73) $\displaystyle=$
$\displaystyle-\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}]w_{y}^{2}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\alpha}/b_{2}}^{x_{\beta}/b_{2}}[u(b_{2}y)/b_{2}^{2}]^{{}^{\prime}}w_{y}(s,y)w(s,y)dyds$
$\displaystyle\leq$ $\displaystyle-
D_{2}\int_{0}^{t}\int_{0}^{1}w_{y}^{2}(s,y)dyds$
$\displaystyle+D_{3}\int_{0}^{t}\int_{0}^{1}[\lambda_{1}w_{y}^{2}(s,y)+\frac{1}{4\lambda_{1}}w^{2}(s,y)]dyds,$
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}[v(b_{2}y)/b_{2}]w(s,y)w_{y}(s,y)dyds$
(8.74) $\displaystyle\leq$
$\displaystyle\lambda_{2}\int_{0}^{t}\int_{0}^{1}w_{y}^{2}(s,y)dyds$
$\displaystyle+\frac{D_{1}}{4\lambda_{2}}\int_{0}^{t}\int_{0}^{1}w^{2}(s,y)dyds.$
It is easy to see that $[u(by)/b^{2}]$, $[u(by)/b^{2}]^{\prime}$and
$[v(by)/b]$ are Lipschitz continuous for all
$y\in(x_{\alpha}/b_{2},x_{\beta}/b_{1})$, that is, there exists $L>0$ such
that
$\displaystyle|[u(b_{2}y)/b_{2}^{2}]-[u(b_{1}y)/b_{1}^{2}]|\leq
L|b_{2}-b_{1}|,$
$\displaystyle|[u(b_{2}y)/b_{2}^{2}]^{\prime}-[u(b_{1}y)/b_{1}^{2}]^{\prime}|\leq
L|b_{2}-b_{1}|,$ $\displaystyle|[v(b_{2}y)/b_{2}]-[v(b_{1}y)/b_{1}]|\leq
L|b_{2}-b_{1}|.$ (8.75)
$E_{3}$ has the following expressions,
$\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}\\{u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}\\}w(s,y)\theta_{yy}^{b_{1}}(s,y)dyds$
(8.76) $\displaystyle=$
$\displaystyle-\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}u(b_{1}y)/b_{1}^{2}]w_{y}(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{0}^{x_{\alpha}/b_{2}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\alpha}/b_{2}}^{x_{\alpha}/b_{1}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\alpha}/b_{1}}^{x_{\beta}/b_{2}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\beta}/b_{2}}^{x_{\beta}/b_{1}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\beta}/b_{1}}^{1}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle=$
$\displaystyle-\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}u(b_{1}y)/b_{1}^{2}]w_{y}(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\alpha}/b_{2}}^{x_{\alpha}/b_{1}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\alpha}/b_{1}}^{x_{\beta}/b_{2}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{x_{\beta}/b_{2}}^{x_{\beta}/b_{1}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
$\displaystyle=$ $\displaystyle E_{31}+E_{32}+E_{33}+E_{34}.$
By (8) and (8.76)and Young’s inequality for any $\lambda_{3}>0$ and
$\lambda_{4}>0$,
$\displaystyle E_{31}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
(8.77) $\displaystyle\leq$
$\displaystyle\frac{L^{2}(b_{2}-b_{1})^{2}}{4\lambda_{3}}\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b_{1}}(s,y)]^{2}dyds$
$\displaystyle+\lambda_{3}\int_{0}^{t}\int_{0}^{1}[w_{y}^{2}(s,y)+w^{2}(s,y)]dyds$
and
$\displaystyle E_{33}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\int_{0}^{m/b_{2}}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]^{\prime}w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
(8.78) $\displaystyle\leq$
$\displaystyle\frac{L^{2}(b_{2}-b_{1})^{2}}{4\lambda_{4}}\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b_{1}}(s,y)]^{2}dyds$
$\displaystyle+\lambda_{4}\int_{0}^{t}\int_{0}^{1}[w_{y}^{2}(s,y)+w^{2}(s,y)]dyds.$
There exists a constant $D_{4}>0$ such that
$|[u(by)/b^{2}]^{\prime}-[v(by)/b]|\leq D_{4}$ and
$\lambda_{5}=\inf\limits_{b_{1}\leq b\leq b_{2}}\\{u(by)/b^{2}\\}>0$. Then by
the boundary conditions we estimate
$\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b}(s,y)]^{2}dyds$ for
$b\in[b_{1},b_{2}]$ as follows,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}\theta_{t}^{b}(s,y)\theta^{b}(s,y)$
(8.79)
$\displaystyle-[u(by)/b^{2}]\theta_{yy}^{b}(s,y)\theta^{b}(s,y)-[v(by)/b]\theta_{y}^{b}(s,y)\theta^{b}(s,y)dyds$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{1}[\theta^{b}(s,y)]^{2}dy+\int_{0}^{t}\int_{0}^{1}[u(by)/b^{2}][\theta_{y}^{b}(s,y)]^{2}dyds$
$\displaystyle+\int_{0}^{t}\int_{0}^{1}[u(by)/b^{2}]^{\prime}\theta_{y}^{b}(s,y)\theta^{b}(s,y)dyds$
$\displaystyle-\int_{0}^{t}\int_{0}^{1}[v(by)/b]\theta_{y}^{b}(s,y)\theta^{b}(s,y)dyds$
$\displaystyle\geq$
$\displaystyle\lambda_{5}\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b}(s,y)]^{2}dyds-\frac{\lambda_{5}}{2}\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b}(s,y)]^{2}dyds$
$\displaystyle-\frac{1}{2\lambda_{5}}\int_{0}^{t}\int_{0}^{1}[\theta^{b}(s,y)]^{2}dyds$
$\displaystyle\geq$
$\displaystyle\frac{\lambda_{5}}{2}\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b}(s,y)]^{2}dyds-\frac{D_{4}}{2\lambda_{5}}$
from which we deduce that
$\displaystyle\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b}(s,y)]^{2}dyds\leq\frac{D_{4}}{\lambda_{5}^{2}}.$
(8.80)
Therefore we conclude that
$\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b}(s,y)]^{2}dyds$ is bounded. Noticing
that $w(s,y)\leq 2$ and
$\displaystyle\lim\limits_{b_{2}\rightarrow b_{1}}\\{|E_{32}|+|E_{34}|\\}=0,$
(8.81)
as well as using the equalities (8.76)-(8.81), there exists a positive
function $B_{1}^{b_{1}}(b_{2})$ such that
$\displaystyle\lim\limits_{b_{2}\rightarrow b_{1}}B_{1}^{b_{1}}(b_{2})=0$
and for $0\leq t\leq T$
$\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}[u(b_{2}y)/b_{2}^{2}-u(b_{1}y)/b_{1}^{2}]w(s,y)\theta_{yy}^{b_{1}}(t,y)dyds$
$\displaystyle\leq$ $\displaystyle
B_{1}^{b_{1}}(b_{2})+(\lambda_{3}+\lambda_{4})\int_{0}^{t}\int_{0}^{1}[w_{y}^{2}(s,y)+w^{2}(s,y)]dyds.$
By the same way as in estimating $E_{3}$ we also find a positive function
$B_{2}^{b_{1}}(b_{2})$ such that
$\displaystyle\lim\limits_{b_{2}\rightarrow b_{1}}B_{2}^{b_{1}}(b_{2})=0$
and for any $\lambda_{6}>0$
$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{0}^{1}[v(b_{2}y)/b_{2}-v(b_{1}y)/b_{1}]w(s,y)\theta_{y}^{b_{1}}(s,y)dyds$
(8.83) $\displaystyle\leq$
$\displaystyle\frac{L^{2}(b_{2}-b_{1})^{2}}{4\lambda_{6}}\int_{0}^{t}\int_{0}^{1}[\theta_{y}^{b_{1}}(s,y)]^{2}dyds+\lambda_{6}\int_{0}^{t}\int_{0}^{1}w^{2}(s,y)dyds$
$\displaystyle\leq$ $\displaystyle
B_{2}^{b_{1}}(b_{2})+\lambda_{6}\int_{0}^{t}\int_{0}^{1}w^{2}(s,y)dyds.$
Choosing $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ and $\lambda_{4}$ small
enough such that $\lambda_{1}D_{3}+\lambda_{2}+\lambda_{3}+\lambda_{4}\leq
D_{2}$, we deduce from (8.71)-(8.74), (8) and (8.83) that there exist positive
constants $C_{1}$ and $C_{2}$ such that
$\displaystyle\int_{0}^{1}w^{2}(t,y)dy\leq
C_{1}\int_{0}^{t}\int_{0}^{1}w^{2}(s,y)dyds+C_{2}[B_{1}^{b_{1}}(b_{2})+B_{2}^{b_{1}}(b_{2})].$
Setting $F(t)=\int_{0}^{t}\int_{0}^{1}w^{2}(s,y)dyds$ and using the Gronwall
inequality,
$\displaystyle F(t)\leq
C_{2}[B_{1}^{b_{1}}(b_{2})+B_{2}^{b_{1}}(b_{2})]\exp\\{C_{1}t\\}.$
So
$\displaystyle\lim\limits_{b_{2}\rightarrow
b_{1}}\int_{0}^{t}\int_{0}^{1}[\theta^{b_{2}}(s,y)-\theta^{b_{1}}(s,y)]^{2}dyds=0.$
Thus we complete the proof. $\Box$ Acknowledgements. This work is supported by
Project 10771114 of NSFC, Project 20060003001 of SRFDP, the SRF for ROCS, SEM
and the Korea Foundation for Advanced Studies. We would like to thank the
institutions for the generous financial support. Special thanks also go to the
participants of the seminar stochastic analysis and finance at Tsinghua
University for their feedbacks and useful conversations.
## References
* [1] Andrei,N,Borodin.,Paavo,Salminen.,2002. Handbook of Brownian Motion. ISBN 3-7643-6705-9.
* [2] Asmussen, S., Taksar, M., 1997. Controlled Diffusion Models for Optimal Dividend Pay-out. Insurance: Math. Econ., Vol. 20, 1-15.
* [3] Asmussen, S., Højgaard, B., Taksar, M., 2000. Optimal Risk Control and Dividend Distribution Policies: Example of Excess-of-Loss Reinsurance for an insurance corporation, Finance Stochast.. Vol. 4, 199-324.
* [4] Avanzi, B., 2009. Strategies for Dividend Distribution: A Review. North American Actuarial Journal,Vol. 13, No. 2, pp. 217-251.
* [5] Cadenillas, A., Choulli, T., Taksar M., Zhang Lei, 2006. Classical and Impulse Stochastic Control for the Optimization of the Dividend and Risk Policies of an Insurance Firm. Mathematical Finance, Vol. 16, No. 1, 181-202.
* [6] Choulli, T., Taksar M., Xunyu Zhou, 2001. Excess-of-loss Reinsurance for a Company with Debt Liability and Constraints on Risk Reduction. Auantitative Finance, 1, pp. 573-596.
* [7] Choulli, T., Taksar M., Xunyu Zhou, 2003. An Optimal Diffusion Model of a Company with Contraints on Risk Control. SIAM J. Control Optim. 41 1946-1979. MR1972542
* [8] Choulli, T., Taksar M., Xunyu Zhou, 2004. Inerplay Between Dividend Rate and Business Constraints for a Financial Corporation. The Annals of Applied Probability, Vol. 14, No. 4, 1810-1837.
* [9] Emanuel D C,Harrison, J.M. and Taylor A. J., 1975. A diffusion approximation for the ruin probability with compounding assets. Scandinavian Acturial Journal 75, 240-247.
* [10] Grandell J., 1977. A class of approximations of ruin probabilities. Scandinavian Acturial Journal Suppl.77, 37-52.
* [11] Grandell J., 1978. A remark on a class of approximations of ruin probabilities. Scandinavian Acturial Journal78, 77-78.
* [12] Grandell J. 1990. Aspect of risk theory ( New York: Springer).
* [13] Guo,Xin , Liu,Jun , Zhou, Xunyu , 2004. A Constrained Nonlinear Regular-singular Stochastic Control Problem, with application. Stochastic Processes and Their Applications, Vol. 109, pp. 167-187.
* [14] Harrison, J.M.; Taksar, M.J.,1983. Instant control of Brownian motion, Mathematics of Operations Research. 8, 439-453.
* [15] Harrison, J.M., 1985. Brownian motion and stochastic flow systems( New York: Wiley).
* [16] Lin He, Ping Hou and Zongxia Liang, 2008. Optimal Financing and Dividend Control of the Insurance Company with Proportional Reinsurance Policy under solvency constraints. Insurance: Mathematics and Economics, Vol.43, 474-479.
* [17] Lin He, Zongxia Liang, 2008. Optimal Dividend Control of the Insurance Company with Proportional Reinsurance Policy under solvency constraints. Insurance: Mathematics and Economics, Vol.43, 474-479.
* [18] Lin He, Zongxia Liang, 2009. Optimal Financing and Dividend Control of the Insurance Company with fixed and Proportional transaction costs. Insurance: Mathematics and Economics 44 (2009) 88-94.
* [19] Højgaard, B., Taksar, M., 1999. Controlling Risk Exposure and Dividends Payout Schemes: Insurance company Example. Mathematical Finance, Vol. 9, No. 2, 153-182.
* [20] Højgaard, B., Taksar, M., 2001. Optimal Risk Control for a Large Corporation in the Presence of Returns on Investments. Finance Stochast. Vol. 5, 527-547.
* [21] Iglehart D.L.,1969. Diffusion approximations in collective risk theory. J.App. Probab. 6. 285-292.
* [22] Ikeda, N., Watanabe, S., 1981. Stochastic differentail Equ ations and Diffusion Processes. North-Holland, ISBN 0444-86172-6.
* [23] Ikeda, I. and Watanabe: A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. Vol.14, N.3, 619-633,1977.
* [24] Zongxia Liang, Jianping Huang: Optimal dividend and investing control of a insurance company with higher solvency constraints. arXiv:1005.1360.
* [25] Zongxia Liang, Jicheng Yao: Nonlinear optimal stochastic control of large insurance company with insolvency probability constraints. arXiv:1005.1361.
* [26] Zongxia Liang, Jicheng Yao: Optimal dividend policy of a large insurance company with positive transaction cost under higher solvency and security. arXiv:1005.1356.
* [27] Lions, P.-L.; Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Comm.Pure Appl. Math.37(1984)511-537.
* [28] Miller,M.H. and F. Modigliani: Dividend Policy, Growth, and the Valuation of Shares, J. Business 34, 411-433, 1961.
* [29] Paulsen, J., 2003. Optimal Dividend Payouts for Solvency Constraints. Finance Stochastics, Vol. 7, 457-473.
* [30] Schmidli H., 1994. Diffusion approximations for a risk process with the possibility of borrowing and interest. Commun. Stat. Stochast. Models. 10, 365-388.
* [31] Taksar, M., 2000. Optimal Risk/dividend Distribution Control Models: Applications to Insurance. Math. Methods Oper., Res. 1, 1-42.
* [32] Taksar, M., Xun Yu Zhou, 2008. Optimal Risk and Dividend Control for a Company with a Debt Liability. Insurance: Mathematics and Economics, Vol.22, 105-122.
|
arxiv-papers
| 2010-07-30T06:12:08 |
2024-09-04T02:49:11.955280
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zongxia Liang, Bin Sun",
"submitter": "Zongxia Liang",
"url": "https://arxiv.org/abs/1007.5376"
}
|
1008.0129
|
# Renormalization and quantum field theory
R. E. Borcherds
Department of Mathematics
University of California at Berkeley
CA 94720-3840 USA
Email: reb@math.berkeley.edu This research was supported by a Miller
professorship and an NSF grant. I thank the referees for suggesting many
improvements.
###### Abstract
The aim of this paper is to describe how to use regularization and
renormalization to construct a perturbative quantum field theory from a
Lagrangian. We first define renormalizations and Feynman measures, and show
that although there need not exist a canonical Feynman measure, there is a
canonical orbit of Feynman measures under renormalization. We then construct a
perturbative quantum field theory from a Lagrangian and a Feynman measure, and
show that it satisfies perturbative analogues of the Wightman axioms, extended
to allow time-ordered composite operators over curved spacetimes.
## 1 Introduction
We give an overview of the construction of a perturbative quantum field theory
from a Lagrangian. We start by translating some terms in physics into
mathematical terminology.
###### Definition 1
Spacetime is a smooth finite-dimensional metrizable manifold $M$, together
with a “causality” relation $\leqslant$ that is closed, reflexive, and
transitive. We say that two points are spacelike separated if they are not
comparable, in other words neither $x\leqslant y$ nor $y\leqslant x$.
The causality relation $a\leqslant b$ means informally that $a$ occurs before
$b$. The causality relation will often be constructed in the usual way from a
Lorentz metric with a time orientation, but since we do not use the Lorentz
metric for anything else we do not bother to give $M$ one. The Lorentz metric
will later appear implicitly in the choice of a cut propagator, which is often
constructed using a metric.
###### Definition 2
The sheaf of classical fields $\Phi$ is the sheaf of smooth sections of some
finite dimensional super vector bundle over spacetime.
When the sheaf of classical fields is a “super-sheaf”, one uses the usual
conventions of superalgebra: in particular the symmetric algebras used later
are understood to be symmetric algebras in the superalgebra sense, and the
usual superalgebra minus signs should be inserted into formulas whenever the
order of two terms is exchanged.
As usual, a global section of a sheaf of things is called a thing, so a
classical field $\varphi$ is a global section of the sheaf $\Phi$ of classical
fields, and so on. (A subtle point is sometimes things called classical fields
in the physics literature are better thought of as sections of the dual of the
sheaf of classical fields; in practice this distinction does not matter
because the sheaf of classical fields usually comes with a bilinear form
giving a canonical isomorphism with its dual.)
###### Definition 3
The sheaf of derivatives of classical fields or simple fields is the sheaf
$J\Phi=\operatorname{Hom}(J,\Phi)$, where $J$ is the sheaf of jets of $M$ and
the $\operatorname{Hom}$ is taken over the smooth functions on $M$, equal to
the inverse limit of the sheaves of jets of finite order of $M$, as in [8,
16.3].
###### Definition 4
The sheaf of (polynomial) Lagrangians or composite fields $SJ\Phi$ is the
symmetric algebra of the sheaf $J\Phi$ of derivatives of classical fields.
Its sections are (polynomial) Lagrangians, in other words polynomial in fields
and their derivations, so for example
$\lambda\varphi^{4}+m^{2}\varphi^{2}+\varphi\partial_{i}^{2}\varphi$ is a
Lagrangian, but $\sin(\varphi)$ is not.
Perturbative quantum field theories depend on the choice of a Lagrangian $L$,
which is the sum of a free Lagrangian $L_{F}$ that is quadratic in the fields,
and an interaction Lagrangian $L_{I}\in
SJ\Phi\otimes\boldsymbol{C}[[\lambda_{1},\ldots,\lambda_{n}]]$ whose
coefficients are infinitesimal, in other words elements of a formal power
series ring $\boldsymbol{C}[[\lambda_{1},\ldots,\lambda_{n}]]$ over the reals
with constant terms 0.
###### Definition 5
The sheaf of Lagrangian densities or local actions $\omega
SJ\Phi=\omega\otimes SJ\Phi$ is the tensor product of the sheaf $SJ\Phi$ of
Lagrangians and the sheaf $\omega$ of smooth densities (taken over smooth
functions on $M$).
For a smooth manifold, the (dualizing) sheaf $\omega$ of smooth densities (or
smooth measures) is the tensor product of the orientation sheaf with the sheaf
of differential forms of highest degree, and is non-canonically isomorphic to
the sheaf of smooth functions. Densities are roughly “things that can be
locally integrated”. For example, if $M$ is oriented, then
$(\lambda\varphi^{4}+m^{2}\varphi^{2}+\varphi\partial_{i}^{2}\varphi)d^{n}x$
is a Lagrangian density.
We use $\Gamma$ and $\Gamma_{c}$ to stand for spaces of global and compactly
supported sections of a sheaf. These will usually be spaces of smooth
functions (or compactly supported smooth functions) in which case they are
topologized in the usual way so that their duals are compactly supported
distributions (or distributions) taking values in some sheaf.
###### Definition 6
A (non-local) action is a polynomial in local actions, in other words an
element of the symmetric algebra $S\Gamma\omega SJ\Phi$ of the real vector
space $\Gamma\omega SJ\Phi$ of local actions.
We do not complete the symmetric algebra, so expressions such as $e^{i\lambda
L}$ are not in general non-local actions, unless we work over some base ring
in which $\lambda$ is nilpotent.
We will use $\ast$ for complex conjugation and for the antipode of a Hopf
algebra and for the adjoint of an operator and for the anti-involution of a
$*$-algebra. The use of the same symbol for all of these is deliberate and
indicates that they are all really special cases of a universal “adjoint” or
“antipode” operation that acts on everything: whenever two of these operations
are defined on something they are equal, so can all be denoted by the same
symbol.
The quantum field theories we construct depend on the choice of a cut
propagator $\Delta$ that is essentially the same as the 2-point Wightman
distribution
$\Delta(\varphi_{1},\varphi_{2})=\int_{x,y}\langle
0|\varphi_{1}(x)\varphi_{2}(y)|0\rangle dxdy$
###### Definition 7
A propagator $\Delta$ is a continuous bilinear map
$\Gamma_{c}\omega\Phi\times\Gamma_{c}\omega\Phi\rightarrow\boldsymbol{C}$.
* $\bullet$
$\Delta$ is called local if $\Delta(f,g)=\Delta(g,f)$ whenever the supports of
$f$ and $g$ are spacelike separated.
* $\bullet$
$\Delta$ is called Feynman if it is symmetric: $\Delta(f,g)=\Delta(g,f)$.
* $\bullet$
$\Delta$ is called Hermitian if $\Delta^{\ast}=\Delta$, where $\Delta^{\ast}$
is defined by $\Delta^{\ast}(f^{\ast},g^{\ast})=\Delta(g,f)^{\ast}$ (with a
change in order of $f$ and $g$).
* $\bullet$
$\Delta$ is called positive if $\Delta(f^{\ast},f)\geqslant 0$ for all $f$.
* $\bullet$
$\Delta$ is called cut if it satisfies the following “positive energy”
condition: at each point $x$ of $M$ there is a partial order on the cotangent
space defined by a proper closed convex cone $C_{x}$, such that if $(p,q)$ is
in the wave front set of $\Delta$ at some point $(x,y)\in M^{2}$ then
$p\leqslant 0$ and $q\geqslant 0$. Also, as a distribution, $\Delta$ can be
written in local coordinates as a boundary value of something in the algebra
generated by smooth functions and powers and logarithms of polynomials (the
boundary values taken so that the wave front sets lie in the regions specified
above). Moreover if $x=y$ then $p+q=0$.
A propagator can also be thought of as a complex distribution on $M\times M$
taking values in the dual of the external tensor product $J\Phi\boxtimes
J\Phi$. In particular it has a wave front set (see Hörmander [10]) at each
point of $M^{2}$, which is a cone in the imaginary cotangent space of that
point. If $A$ and $B$ are in $\Gamma_{c}\Phi$, then $\Delta(A,B)$ is defined
to be a compactly supported distribution on $M\times M$, defined by
$\Delta(A,B)(f,g)=\text{$\Delta(Af,Bf)$}$ for $f$ and $g$ in $\Gamma\omega$.
The key point in the definition of a cut propagator is the condition on the
wave front sets, which distinguishes the cut propagators from other
propagators such as Feynman propagators or advanced and retarded propagators
that can have more complicated wave front sets. For most common cut
propagators in Minkowski space, this follows from the fact that their Fourier
transforms have support in the positive cone. The condition about being
expressible in terms of smooth functions and powers and logs of polynomials is
a minor technical condition that is in practice satisfied by almost any
reasonable example, and is used in the proof that Feynman measures exist.
If $(p_{1},\ldots p_{n})$ is in the imaginary cotangent space of a point of
$M^{n}$, then we write $(p_{1},\ldots p_{n})\geqslant 0$ if $p_{j}\geqslant 0$
for all $j$, and call it positive if it is not zero.
###### Example 8
Over Minkowski space, most of the usual cut propagators are positive (except
for ghost fields), local, and Hermitian. Most of the ideas for the proof of
this can be seen for the simplest case of the propagator for massive Hermitian
scalar fields. Using translation invariance, we can write
$\Delta(x,y)=\Delta(x-y)$ for some distribution $\Delta$ on Minkowski
spacetime. Then the Fourier transform of this in momentum space is a
rotationally invariant measure supported on one of the two components of
vectors with $p^{2}=m^{2}$. This propagator is positive because the measure in
momentum space is positive. It satisfies the wave front set part of the cut
condition because the Fourier transform has support in the positive cone, and
explicit calculation shows that it can be written in terms of powers and logs
of polynomials. It satisfies locality because it is invariant under rotations
that preserve the direction of time, and under such rotations any space-like
vector is conjugate to its negative, so $\Delta(x)=\Delta(-x)$ whenever $x$ is
spacelike, in other words $\Delta(x,y)=\Delta(y,x)$ whenever $x$ and $y$ are
spacelike separated. The corresponding Feynman propagator is given by
$1/(p^{2}+m^{2}+i\varepsilon)$ where the $i\varepsilon$ indicates in which
direction one integrates around the poles, so the cut propagator is just the
residue of the Feynman propagator along one of the 2 components of the
2-sheeted hyperboloid $p^{2}=m^{2}$.
For other fields such as spinor fields in Minkowski space, the sheaf of
classical fields will usually be some sort of spin bundle. The propagators can
often be expressed in terms of the the propagator for a scalar field by acting
on it with polynomials in momentum multiplied by Dirac’s gamma matrices
$\gamma^{\mu}$, for example $i(\gamma^{\mu}p_{\mu}+m)/(p^{2}-m^{2})$.
Unfortunately there are a bewildering number of different notational and sign
conventions for gamma matrices.
Compactly supported actions give functions on the space $\Gamma\Phi$ of smooth
fields, by integrating over spacetime $M$. A Feynman measure is a sort of
analogue of Haar measure on a finite dimensional real vector space. We can
think of a Haar measure as an element of the dual of the space of continuous
compactly supported functions. For infinite dimensional vector spaces there
are usually not enough continuous compactly supported functions, but instead
we can define a measure to be an element of the dual of some other space of
functions. We will think of Feynman measures as something like elements of the
dual of all functions that are given by free field Gaussians times a compactly
supported action. In other words a Feynman measure should assign a complex
number to each compactly supported action, formally representing the integral
over all fields of this action times a Gaussian $e^{iL_{F}}$, where we think
of the action as a function of classical fields (or rather sections of the
dual of the space of classical fields, which can usually be identified with
classical fields). Moreover the Feynman measure should satisfy some sort of
analogue of translation invariance.
The space $e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ is a free rank 1 module over
$S\Gamma_{c}\omega SJ\Phi$ generated by the basis element $e^{iL_{F}}$, which
can be thought of either as a formal symbol or a formal power series. Its
elements can be thought of as representing functions of classical fields that
are given by a polynomial times the Gaussian $e^{iL_{F}}$, and will be the
functions that the Feynman measure is defined on. The symmetric algebra
$S\Gamma_{c}\omega SJ\Phi$ is topologized as the direct sum of the spaces
$S^{n}\Gamma_{c}\omega SJ\Phi$, each of which is toplogized by regarding it as
a space of smooth test functions over $M^{n}$.
For the definition of a Feynman measure we need to extend the propagator
$\Delta$ to a larger space as follows. We think of the propagator $\Delta$ as
a map taking $\Gamma_{c}J\Phi\otimes\Gamma_{c}J\Phi$ to distributions on
$M\times M$. We then extend it a map from
$\Gamma_{c}SJ\Phi\times\Gamma_{c}SJ\Phi$ to distributions on $M\times M$ by
putting $\Delta(a_{1}\cdots a_{n},b_{1}\cdots b_{n})=\sum_{\sigma\in
S_{n}}\Delta(a_{1},b_{\sigma(1)})\times\cdots\times\Delta(a_{1},b_{\sigma(n)})$
where the sum is over all elements of the symmetric group $S_{n}$ (and
defining it to be 0 for arguments of different degrees). Finally we extend it
to a map from $S^{m}\Gamma_{c}SJ\Phi\times S^{n}\Gamma_{c}SJ\Phi$ to
distributions on $M^{m}\times M^{n}$ using the “bicharacter” property: in
other words $\Delta(AB,C)=\sum\Delta(A,C^{\prime})\Delta(B,C^{\prime\prime})$
where the coproduct of $C$ is $\sum C^{\prime}\otimes C^{\prime\prime}$, and
similarly for $\Delta(A,BC)$.
###### Definition 9
A Feynman measure is a continuous linear map
$\omega:e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi\rightarrow\boldsymbol{C}$. The
Feynman measure is said to be associated with the propagator $\Delta$ if it
satisfies the following conditions:
* •
Smoothness on the diagonal: Whenever $(p_{1},\ldots,p_{n})$ is in the wave
front set of $\omega$ at the point $(x,\ldots,x)$ on the diagonal, then
$p_{1}+\ldots+p_{n}=0$
* •
Non-degeneracy: there is a smooth nowhere-vanishing function $g$ so that
$\omega(e^{iL_{F}}v)$ is $\int_{M}gv$ for $v$ in $\Gamma_{c}\omega
S^{0}J\Phi=\Gamma_{c}\omega$.
* •
Gaussian condition, or weak translation invariance: For $A\in
S^{m}\Gamma_{c}\omega SJ\Phi$, $B\in S^{n}\Gamma_{c}\omega SJ\Phi$, with both
sides interpreted as distributions on $M^{m+n}$,
$\omega(AB)=\sum\omega(A^{\prime})\Delta(A^{\prime\prime},B^{\prime\prime})\omega(B^{\prime})$
whenever there is no element in the support of $A$ that is $\leqslant$ some
element of the support of $B$. Here $\sum A^{\prime}\otimes
A^{\prime\prime}\in S\Gamma_{c}\omega SJ\Phi\otimes
S\Gamma_{c}\text{$SJ\Phi$}$ is the image of $A$ under the map
$S^{m}\Gamma_{c}\omega SJ\Phi\rightarrow S^{m}\Gamma_{c}\omega SJ\Phi\otimes
S^{m}\Gamma_{c}\text{$SJ\Phi$}$ induced by the coaction $\omega
SJ\Phi\rightarrow\omega SJ\Phi\otimes\text{$SJ\Phi$}$ of $SJ\Phi$ on $\omega
SJ\Phi$, and similarly for $B$. The product on the right is a product of
distributions, using the extended version of $\Delta$ defined just before this
definition.
We explain what is going on in this definition. We would like to define the
value of the Feynman measure to be a sum over Feynman diagrams, formed by
joining up pairs of fields in all possible ways by lines, and then assigning a
propagator to each line and taking the product of all propagators of a
diagram. This does not work because of ultraviolet divergences: products of
propagators need not be defined when points coincide. If these products were
defined then they would satisfy the Gaussian condition, which then says
roughly that if the vertices are divided into two disjoint subsets $a$ and
$b$, then a Feynman diagram can be divided into a subdiagram with vertices
$a$, a subdiagram with vertices $b$, and some lines between $a$ and $b$. The
value $\omega(AB)$ of the Feynman diagram would then be the product of its
value $\omega(A^{\prime})$ on $a$, the product
$\Delta(A^{\prime\prime},B^{\prime\prime})$ of all the propagators of lines
joining $a$ and $b$, and its value $\omega(B^{\prime})$ on $b$. The Gaussian
condition need not make sense if some point of $a$ is equal to some point of
$b$ because if these points are joined by a line then the corresponding
propator may have a bad singularity, but does make sense whenever all points
of $a$ are not $\leq$ all points of $b$. The definition above says that a
Feynman measure should at least satisfy the Gaussian condition in this case,
when the product is well defined.
Unfortunately the standard notation $\omega$ for a dualizing sheaf, such as
the sheaf of densities, is the same as the standard notation $\omega$ for a
state in the theory of operator algebras, which the Feynman measure will be a
special case of. It should be clear from the context which meaning of $\omega$
is intended.
If $\omega$ is a Feynman measure and $A\in e^{iL_{F}}S^{n}\Gamma_{c}\omega
SJ\Phi$ then $\omega(A)$ is a complex number, but can also be considered as
the compactly supported density on $M^{n}$ taking a smooth $f$ to
$\omega(A)(f)=\omega(Af)$. The integral of this density $\omega(A)$ over
spacetime is just the complex number $\omega(A)$.
Since $e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ is a coalgebra (where elements of
$\Gamma_{c}\omega SJ\Phi$ are primitive and $e^{iL_{F}}$ is group-like), the
space of Feynman measures is an algebra, whose product is called convolution.
The non-degeneracy condition just excludes some uninteresting degenerate
cases, such as the measure that is identically zero, and the function $g$
appearing in it is usually normalized to be 1. The condition about smoothness
on the diagonal implies that the product on the right in the Gaussian
condition is defined. This is because $\omega$ has the property that if an
element $(p_{1},\ldots p_{n})$ of the wave front set of some point is nonzero
then its components cannot all be positive and cannot all be negative. This
shows that the wave front sets are such that the product of distributions is
defined.
If $A$ is in $e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$, then $\omega(A)$ can be
thought of as a Feynman integral
$\omega(A)=\int A(\varphi)\mathcal{D}\varphi$
where $L_{F}$ is a quadratic action with cut propagator $\Delta$, and where
$A$ is considered to be a function of fields $\varphi$. The integral is
formally an integral over all classical fields. The Gaussian condition is a
weak form of translation invariance of this measure under addition of
classical fields. Formally, translation invariance is equivalent to the
Gaussian condition with the condition about supports omitted and cut
propagators replaced by Feynman propagators, but this is not well defined
because the Feynman propagators can have such bad singularities that their
products are sometimes not defined when two spacetime points coincide.
The Feynman propagator $\Delta_{F}$ of a Feynman measure $\omega$ is defined
to be the restriction of $\omega$ to
$\Gamma_{c}\omega\Phi\times\Gamma_{c}\omega\Phi$. It is equal to the cut
propagator at “time-ordered” points $(x,y)\in M^{2}$ where $x\nleqslant y$,
but will usually differ if $x\leqslant y$. As it is symmetric, it is
determined by the cut propagator except on the diagonal of $M\times M$. Unlike
cut propagators, Feynman propagators may have singularities on the diagonal
whose wave front sets are not contained in a proper cone, so that their
products need not be defined.
Any symmetric algebra $\operatorname{SX}$ over a module $X$ has a natural
structure of a commutative and cocommutative Hopf algebra, with the coproduct
defined by making all elements of $X$ primitive (in other words, $\Delta
x=x\otimes 1+1\otimes x$ for $x\in X$). In other words, $\operatorname{SX}$ is
the coordinate ring of a commutative affine group scheme whose points form the
dual of $X$ under addition. For general results about Hopf algebra see Abe
[1]. Similarly $SJ\Phi$ is a sheaf of commutative cocommutative Hopf algebras,
with a coaction on itself and the trivial coaction on $\omega$, and so has a
coaction on $S\omega SJ\Phi$, preserving the coproduct of $S\omega SJ\Phi$. It
corresponds to the sheaf of commutative affine algebraic groups whose points
correspond to the sheaf $J\Phi$ under addition.
###### Definition 10
A renormalization is an automorphism of $S\omega SJ\Phi$ preserving its
coproduct and the coaction of $SJ\Phi$. The group of renormalizations is
called the ultraviolet group.
The justification for this rather mysterious definition is theorem 15, which
shows that renormalizations act simply transitively on the Feynman measures
associated to a given local cut propagator. In other words, although there is
no canonical Feynman measure on the space of classical fields, there is a
canonical orbit of such measures under renormalization.
More generally, renormalizations are global sections of the sheaf of
renormalizations (defined in the obvious way), but we will make no use of this
point of view.
The (infinite dimensional) ultraviolet group really ought to be called the
“renormalization group”, but unfortunately this name is already used for a
quite different 1-dimensional group. The “renormalization group” is the group
of positive real numbers, together with an action on Lagrangians by
“renormalization group flow”. The relation between the renormalization group
and the ultraviolet group is that the renormalization group flow can be
thought of as a non-abelian 1-cocycle of the renormalization group with values
in the ultraviolet group, using the action of renormalizations on Lagrangians
that will be constructed later.
The ultraviolet group is indirectly related to the Hopf algebras of Feynman
diagrams introduced by Kreimer [11] and applied to renormalization by him and
Connes [5], though this relation is not that easy to describe. First of all
their Hopf algebras correspond to Lie algebras, and the ultraviolet group has
a Lie algebra, and these two Lie algebras are related. There is no direct
relation between Connes and Kreimer’s Lie agebras and the Lie algebra of the
ultraviolet group, in the sense that there seems to be no natural homomorphism
in either direction. However there seems to be a sort of intermediate Lie
algebra that has homomorphisms to both. This intermediate Lie algebra (or
group) can be defined using Feynman diagrams decorated with smooth test
functions rather than the sheaf $S\omega SJ\Phi$ used here. Unfortunately all
my attempts to explain the product of this Lie algebra explicitly have
resulted in an almost incomprehensible combinatorial mess so complicated that
it is unusable. Roughly speaking, the main differences between the ultraviolet
group and the intermediate Lie algebra is that the Lie algebra of the
ultraviolet group amalgamates all Feynman diagrams with the same vertices
while the intermediate Lie algebra algebra keeps track of individual Feynman
diagrams, and the main difference between the intermediate Lie algebra and
Kreimer’s algebra is that the intermediate Lie algebra is much fatter than
Kreimer’s algebra because it has infinite dimensioinal spaces of smooth
functions in it. In some sense Kreimer’s algebra could be thought of as a sort
of skeleton of the intermediate Lie algebra.
All reasonable Feynman measures for a given free field theory are equivalent
up to renormalization, but it is not easy to show that at least one exists. We
do this by following the usual method of constructing a perturbative quantum
field theory in physics. We first regularize the cut local propagator which
produces a meromorphic family of Feynman measures, following Etingof [6] in
using Bernstein’s theorem [3] on the analytic continuation of powers of a
polynomial to construct the regularization. We then use an infinite
renormalization to eliminate the poles of the regularized Feynman measure in
order of their complexity.
A quantum field theory satisfying the Wightman axioms [13, section 3.1] is
determined by its Wightman distributions, which are given by linear maps
$\omega_{n}:T^{n}\Gamma_{c}\omega\Phi\rightarrow\boldsymbol{C}$ from the
tensor powers of the space of test functions for each $n$. We will follow H.
J. Borchers [4] in combining the Wightman distributions into a Wightman
functional $\omega:\text{$T\Gamma_{c}\omega\Phi$}\rightarrow\boldsymbol{C}$ on
the tensor algebra $T\Gamma_{c}\omega\Phi$ of the space $\Gamma_{c}\omega\Phi$
of test functions (which is sometimes called a Borchers algebra or Borchers-
Uhlmann algebra or BU-algebra). In order to accommodate composite operators we
extend the algebra $T\Gamma_{c}\omega\Phi$ to the larger algebra
$T\Gamma_{c}\omega SJ\Phi$, and to accommodate time ordered operators we
extend it further to $\operatorname{TS}\Gamma_{c}\omega SJ\Phi$. In this set
up it is clear how to accommodate perturbative quantum field theories: we just
allow $\omega$ to take values in a space of formal power series
$\boldsymbol{C}[[\boldsymbol{\lambda}]]=\boldsymbol{C}[[\lambda_{1},\lambda_{2},\ldots]]$
rather than $\boldsymbol{C}$. For regularization $\omega$ sometimes takes
values in a ring of meromorphic functions. There is one additional change we
need: it turns out that the elements of $\Gamma_{c}\omega SJ\Phi$ do not
really represent operators on a space of physical states, but are better
thought of as operators that map a space of incoming states to a space of
outgoing states, and vice versa. If we identify the space of incoming states
with the space of physical states, this means that only products of an even
number of operators of $S\Gamma_{c}\omega SJ\Phi$ act on the space of physical
states. So the functional defining a quantum field theory is really defined on
the subalgebra $T_{0}S\Gamma_{c}\omega SJ\Phi$ of even degree elements.
So the main goal of this paper is to construct a linear map from
$T_{0}S\Gamma_{c}\omega SJ\Phi$ to $\boldsymbol{C}[\boldsymbol{\lambda}]$ from
a given Lagrangian, and to check that it satisfies analogues of the Wightman
axioms.
The space of physical states of the quantum field theory can be reconstructed
from $\omega$ as follows.
###### Definition 11
Let $\omega:T\rightarrow C$ be a $\boldsymbol{R}$-linear map between real
$*$-algebras.
* $\bullet$
$\omega$ is called Hermitian if $\omega^{\ast}=\omega$, where
$\omega^{\ast}(a^{\ast})=\omega(a)^{\ast}$
* $\bullet$
$\omega$ is called positive if it maps positive elements to positive elements,
where an element of a $*$-algebra is called positive if it is a finite sum of
elements of the form $a^{\ast}a$.
* $\bullet$
$\omega$ is called a state if it is positive and normalized by $\omega(1)=1$
* $\bullet$
The left, right, or 2-sided kernel of $\omega$ is the largest left, right or
2-sided ideal closed under * on which $\omega$ vanishes.
* $\bullet$
The space of physical states of $\omega$ is the quotient of $T$ by the left
kernel of $\omega$. Its sesquilinear form is $\left\langle
a,b\right\rangle=\omega(a^{\ast}b)$, and its vacuum vector is the image of
$1$.
* $\bullet$
The algebra of physical operators of $\omega$ is the quotient of $T$ by the
2-sided kernel of $\omega$.
The algebra of physical operators is a $*$-algebra of operators with a left
action on the physical states. If $\omega$ is positive or Hermitian then so is
the sesquilinear form $\left\langle,\right\rangle$. When $\omega$ is Hermitian
and positive and $C$ is the complex numbers the left kernel of $\omega$ is the
set of vectors $a$ with $\omega(a^{\ast}a)=0$, and the definition of the space
of physical states is essentially the GNS construction and is also the main
step of the Wightman reconstruction theorem. In this case the completion of
the space of physical states is a Hilbert space.
The maps $\omega$ we construct are defined on the real vector space
$T_{0}S\Gamma_{c}\omega SJ\Phi$ and will initially be $\boldsymbol{R}$-linear.
It is often convenient to extend them to be
$\boldsymbol{C}[[\boldsymbol{\lambda}]]$-linear maps defined on
$T_{0}S\Gamma_{c}\omega SJ\Phi\otimes\boldsymbol{C}[[\boldsymbol{\lambda}]]$,
in which case the corresponding space of physical states will be a module over
$\boldsymbol{C}[[\boldsymbol{\lambda}]]$ and its bilinear form will be
sesquilinear over $\boldsymbol{C}[[\boldsymbol{\lambda}]]$.
The machinery of renormalization and regularization has little to do with
perturbation theory or the choice of Lagrangian: instead, it is needed even
for the construction of free field theories if we want to include composite
operators. The payoff for all the extra work needed to construct the composite
operators in a free field theory comes when we construct interacting field
theories from free ones. The idea for constructing an interacting field theory
from a free one is simple: we just apply a suitable automorphism (or
endomorphism) of the algebra $T_{0}S\Gamma_{c}\omega SJ\Phi$ to the free field
state $\omega$ to get a state for an interacting field. For example, if we
apply an endomorphsim of the sheaf $\omega SJ\Phi$ then we get the usual field
theories of normal ordered products of operators, which are not regarded as
all that interesting. For any Lagrangian $L$ there is an infinitesimal
automorphism of $T_{0}S\Gamma_{c}\omega SJ\Phi$ that just multiplies elements
of $S\Gamma_{c}\omega SJ\Phi$ by $iL$, which we would like to lift to an
automorphism $e^{iL}$. The construction of an interacting quantum field theory
from a Feynman measure $\omega$ and a Lagrangian $L$ is then given by the
natural action $e^{-iL}\omega$ of the automorphism $e^{-iL}$ on the state
$\omega$. The problem is that $e^{iL_{I}}$ is only defined if the interaction
Lagrangian has infinitesimal coefficients, due to the fact that we only
defined $\omega$ on polynomials times a Gaussian, so this construction only
produces perturbative quantum field theories taking values in rings of formal
power series. This is essentially the problem of lifting a Lie algebra
elements $L_{I}$ to a group element $e^{iL_{I}}$, which is trivial for
operators on finite dimensional vector spaces, but a subtle and hard problem
for unbounded operators such as $L_{I}$ that are not self adjoint. This
construction works provided the interacting part of the Lagrangian not only
has infinitesimal coefficients but also has compact support. We show that the
more general case of Lagrangians without compact support can be reduced to the
case of compact support up to inner automorphisms, at least on globally
hyperbolic spacetimes, by showing that infra-red divergences cancel.
Up to isomorphism, the quantum field theory does not depend on the choice of
Feynman measure or Lagrangian, but only on the choice of propagator. In
particular, the interacting quantum field theory is isomorphic to a free one.
This does not mean that interacting quantum field theories are trivial,
because this isomorphism does not preserve the subspace of simple operators,
so if one only looks at the restriction to simple operators, as in the
Wightman axioms, one no longer gets an isomorphism between free and
interacting theories. The difference between interacting and free field
theories is that one chooses a different set of operators to be the “simple”
operators corresponding to physical fields.
The ultraviolet group also has a non-linear action on the space of
infinitesimal Lagrangians. A quantum field theory is determined by the choice
of a Lagrangian and a Feynman measure, and this quantum field theory is
unchanged if the Feynman measure and the Lagrangian are acted on by the same
renormalization. This shows why the choice of Feynman measure is not that
important: if one chooses a different Feynman measure, it is the image of the
first by a unique renormalization, and by applying this renormalization to the
Lagrangian one still gets the same quantum field theory.
Roughly speaking, we show that these quantum field theories $e^{iL_{I}}\omega$
satisfy the obvious generalizations of Wightman axioms whenever it is
reasonable to expect them to do so. For example, we will show that locality
holds by showing that the state vanishes on the “locality ideal” of definition
32, the quantum field theory is Hermitian if we start with Hermitian cut
propagators and Lagrangians, and we get a (positive) state if we start with a
positive (non-ghost) cut propagator. We cannot expect to get Lorentz invariant
theories in general as we are working over a curved spacetime, but if we work
over Minkowski space and choose Lorentz invariant cut propagators then we get
Lorentz invariant free quantum field theories. In the case of interacting
theories Lorentz invariance is more subtle, even if the Lagrangian is Lorentz
invariant. Lorentz invariance depends on the cancellation of infra-red
divergences as we have to approximate the Lorentz invariant Lagrangian by non
Lorentz invariant Lagrangians with compact support, and we can only show that
infra-red divergences cancel up to inner automorphisms. This allows for the
possibility that the vacuum is not Lorentz invariant, in other words Lorentz
invariance may be spontaneously broken by infra-red divergences, at least if
the theory has massless particles. (It seems likely that if there are no
massless particles then infra-red divergences cancel and we recover Lorentz
invariance, but I have been too lazy to check this in detail.)
In the final section we discuss anomalies. Fujikawa [7] observed that
anomalies arise from the lack of invariance of Feynman measures under a
symmetry group, and we translate his observation into mathematical language.
The definitions above generalize to the relative case where spacetime is
replaced by a morphism $X\rightarrow Y$, whose fibers can be thought of as
spacetimes parameterized by $Y$. For example, the sheaf of densities $\omega$
is replaced by the dualizing sheaf or complex $\omega_{X/Y}$. We will make no
serious use of this generalization, though the section on regularization could
be thought of as an example of this where $Y$ is the spectrum of a ring of
meromorphic functions.
## 2 The ultraviolet group
We describe the structure of the ultraviolet group, and show that it acts
simply transitively on the Feynman measures associated with a given
propagator.
###### Theorem 12
The map taking a renormalization $\rho:S\omega SJ\Phi\rightarrow S\omega
SJ\Phi$ to its composition with the natural map $S\omega SJ\Phi\rightarrow
S^{1}\omega S^{0}J\Phi=\omega$ identifies renormalizations with the elements
of $\operatorname{Hom}(S\omega SJ\Phi,\omega)$ that vanish on $S^{0}\omega
SJ\Phi$ and that are isomorphisms when restricted to $\omega=S^{1}\omega
S^{0}J\Phi$.
###### Proof.
This is a variation of the dual of the fact that endomorphisms $\rho$ of a
polynomial ring $R[x]$ correspond to polynomials $\rho(x)$, given by the image
of the polynomial $x$ under the endomorphism $\rho$. It is easier to
understand the dual result first, so suppose that $C$ is a cocommutative Hopf
algebra and $\omega$ is a vector space (with $C$ acting trivially on
$\omega$). Then the symmetric algebra $S\omega C=S(\omega\otimes C)$ is a
commutative algebra acted on by $C$, and its endomorphisms (as a commutative
algebra) correspond exactly to elements of $\operatorname{Hom}(\omega,S\omega
C)$ because any such map lifts uniquely to a $C$-invariant map from $\omega$
to $\omega C$, which in turn lifts to a unique algebra homomorphism from
$S\omega C$ to itself by the universal property of symmetric algebras. This
endomorphism is invertible if and only if the map from $\omega$ to
$\omega=S^{1}\omega C^{0}$ is invertible, where $C^{0}$ is the vector space
generated by the identity of $C$.
To prove the theorem, we just take the dual of this result, with $C$ now given
by $SJ\Phi$. There is one small modification we need to make in taking the
dual result: we need to add the condition that the element of
$\operatorname{Hom}(S\omega C,\omega)$ vanishes on $S^{0}\omega C$ in order to
get an endomorphism of $S\omega C$; this is related to the fact that
endomorphisms of the polynomial ring $R[x]$ correspond to polynomials, but
continuous endomorphisms of the power series ring $R[[x]]$ correspond to power
series with vanishing constant term. ∎
The ultraviolet group preserves the increasing filtration $S^{\leqslant
m}\omega SJ\Phi$ and so has a natural decreasing filtration by the groups
$G_{\geqslant n}$, consisting of the renormalizations that fix all elements of
$S^{\leqslant n}\omega SJ\Phi$. The group $G=G_{\geqslant 0}$ is the inverse
limit of the groups $G/G_{\geqslant n}$, and the commutator of $G_{\geqslant
m}$ and $G_{\geqslant n}$ is contained in $G_{\geqslant m+n}$, so in
particular $G_{\geqslant 1}$ is an inverse limit of nilpotent groups
$G_{\geqslant 1}/G_{\geqslant n}$. The group $G_{\geqslant n}$ is a semidirect
product $G_{\geqslant n+1}G_{n}$ of its normal subgroup $G_{\geqslant n+1}$
with the group $G_{n}$, consisting of elements represented by elements of
$\operatorname{Hom}(S\omega SJ\Phi,\omega)$ that are the identity on
$S^{1}\omega SJ\Phi$ if $n>0$, and vanish on $S^{m}\omega SJ\Phi$ for $m>1$,
$m\neq n+1$.
###### Lemma 13
The group $G$ is $\ldots G_{2}G_{1}G_{0}$ in the sense that any element of $G$
can be written uniquely as an infinite product $\ldots g_{2}g_{1}g_{0}$ with
$g_{i}\in G_{i}$, and conversely any such infinite product converges to an
element of $G$.
###### Proof.
The convergence of this product follows from the facts that all elements
$g_{i}$ preserve any space $S^{\leqslant m}\omega SJ\Phi$, and all but a
finite number act trivially on it. The fact that any element can be written
uniquely as such an infinite product follows from the fact that
$G/G_{\geqslant n}$ is essentially the product $G_{n-1}\ldots
G_{2}G_{1}G_{0}$. ∎
The natural map
$S\Gamma\omega SJ\Phi\rightarrow\Gamma S\omega SJ\Phi$
is not an isomorphism, because on the left the symmetric algebra is taken over
the reals, while on the right it is essentially taken over smooth functions on
$M$.
###### Lemma 14
The action of renormalizations on $\Gamma S\omega SJ\Phi$ lifts to an action
on $S\Gamma_{c}\omega SJ\Phi$ that preserves the coproduct, the coaction of
$\Gamma SJ\Phi$, and the product of elements with disjoint support.
###### Proof.
A renormalization is given by a linear map from $\Gamma_{c}S\omega SJ\Phi$ to
$\Gamma_{c}\omega$, which by composition with the map $S\Gamma_{c}\omega
SJ\Phi\rightarrow\Gamma S\omega SJ\Phi$ and the “integration over $M$” map
$\Gamma_{c}\omega\rightarrow\boldsymbol{R}$ lifts to a linear map from
$S\Gamma_{c}\omega SJ\Phi$ to $\boldsymbol{R}$. This linear map has the
special property that the product of any two elements with disjoint support
vanishes, because it is multilinear over the ring of smooth functions. As in
theorem 12, the linear map gives an automorphism of $S\Gamma_{c}\omega SJ\Phi$
preserving the coproduct and the coaction of $\Gamma SJ\Phi$. As the linear
map vanishes on products of disjoint support, the corresponding
renormalization preserves products of elements with disjoint support. ∎
In general, renormalizations do not preserve products of elements of
$S\Gamma_{c}\omega SJ\Phi$ that do not have disjoint support; the ones that do
are those in the subgroup $G_{0}$.
###### Theorem 15
The group of complex renormalizations acts simply transitively on the Feynman
measures associated with a given cut local propagator.
###### Proof.
We first show that renormalizations $\rho$ act on Feynman measures $\omega$
associated with a given local cut propagator. We have to show that
renormalizations preserve nondegeneracy, smoothness on the diagonal, and the
Gaussian property. The first two of these are easy to check, because the value
of $\rho(\omega)$ on any element is given by a finite sum of values of
$\omega$ on other elements, so is smooth along the diagonal.
To check that renormalizations preserve the Gaussian property
$\omega(AB)=\sum\omega(A^{\prime})\Delta(A^{\prime\prime},B^{\prime\prime})\omega(B^{\prime})$
we recall that renormalizations $\rho$ preserve products with disjoint support
and also commute with the coaction of $SJ\Phi$. Since $A$ and $B$ have
disjoint supports we have $\rho(AB)=\rho(A)\rho(B)$. Since $\rho$ commutes
with the coaction of $SJ\Phi$, the image of $\rho(A)$ under the coaction of
$SJ\Phi$ is $\sum\rho(A^{\prime})\otimes A^{\prime\prime}$, and similarly for
$B$. Combining these facts with the Gaussian property for $\rho(A)\rho(B)$
shows that
$\omega(\rho(AB))=\sum\omega(\rho(A^{\prime}))\Delta(A^{\prime\prime},B^{\prime\prime})\omega(\rho(B^{\prime}))$
or in other words the renormalization $\rho$ preserves the Gaussian property.
To finish the proof, we have to show that for any two normalized smooth
Feynman measures $\omega$ and $\omega^{\prime}$ with the same cut local
propagator, there is a unique complex renormalization $g$ taking $\omega$ to
$\omega^{\prime}$. We will construct $g=\ldots g_{2}g_{1}g_{0}$ as an infinite
product, with the property that $g_{n-1}\ldots g_{0}\omega$ coincides with
$\omega^{\prime}$ on $e^{iL_{F}}S^{\leqslant n}\Gamma_{c}\omega SJ\Phi$.
Suppose that $g_{0},\ldots,g_{n-1}$ have already been constructed. By changing
$\omega$ to $g_{n-1}\ldots g_{0}\omega$ we may as well assume that they are
all 1, and that $\omega$ and $\omega^{\prime}$ coincide on
$e^{iL_{F}}S^{\leqslant n}\Gamma_{c}\omega SJ\Phi$. We have to show that there
is a unique $g_{n}\in G_{n}$ such that $g_{n}\omega$ and $\omega^{\prime}$
coincide on $e^{L_{F}}S^{n+1}\Gamma_{c}\omega SJ\Phi$.
The difference $\omega-\omega^{\prime}$, restricted to
$e^{iL_{F}}S^{n+1}\Gamma_{c}\omega SJ\Phi$, is a continuous linear function on
$e^{iL_{F}}S^{n+1}\Gamma_{c}\omega SJ\Phi\text{}$, which we think of as a
distribution. Moreover, since both $\omega$ and $\omega^{\prime}$ are
determined off the diagonal by their values on elements of smaller degree by
the Gaussian property, this distribution has support on the diagonal of
$M^{n+1}$. Since $\omega$ and $\omega^{\prime}$ both have the property that
their wave front sets on the diagonal are orthogonal to the diagonal, the same
is true of their difference $\omega-\omega^{\prime}$, so the distribution is
given by a map $e^{iL_{F}}S^{n+1}\Gamma_{c}\omega SJ\Phi\rightarrow\omega$. By
theorem 12 this corresponds to some renormalization $g_{n}\in G_{n}$, which is
the unique element of $G_{n}$ such that
$g_{n}\omega\operatorname{and}\omega^{\prime}\operatorname{coincide}\operatorname{on}e^{iL_{F}}S^{n+1}\Gamma_{c}\omega
SJ\Phi$ . ∎
## 3 Existence of Feynman measures
We now prove theorem 21 showing the existence of at least one Feynman measure
associated to any cut local propagator, by using regularization and
renormalization. Regularization means that we construct a Feynman measure over
a field of meromorphic functions, which will usually have poles at the point
we are interested in, and renormalization means that we eliminate these poles
by acting with a suitable meromorphic renormalization.
###### Lemma 16
If $f_{1},\ldots,f_{m}$ are polynomials in several variables, then there are
non-zero (Bernstein-Sato) polynomials $b_{i}$ and differential operators
$D_{i}$ such that
$\text{ $b_{i}(s_{1},\ldots,s_{m})$}f_{1}(z)^{s_{1}}\ldots
f_{m}(z)^{s_{m}}=D_{i}(z)\left(f_{i}(z)f_{1}(z)^{s_{1}}\ldots
f_{m}(z)^{s_{m}})\right.$
###### Proof.
Bernstein’s proof [3] of this theorem for the case $m=1$ also works for any
$m$ after making the obvious minor changes, such as replacing the field of
rational functions in one variable $s_{1}$ by the field of rational functions
in $m$ variables. ∎
###### Corollary 17
If $f_{1},\ldots,f_{m}$ are polynomials in several variables then for any
choice of continuous branches of the multivalued functions,
$f_{1}(z)^{s_{1}}\ldots f_{m}(z)^{s_{m}}$ can be analytically continued from
the region where all $s_{j}$ have positive real part to a meromorphic
distribution-valued function for all complex values of $s_{1},\ldots,s_{m}$.
###### Proof.
This follows by using the functional equation of lemma 16 to repeatedly
decrease each $s_{j}$ by 1, just as in Bernstein’s proof [3] for the case
$m=1$. ∎
###### Theorem 18
Any cut local propagator $\Delta$ has a regularization, in other words a
Feynman measure with values in a ring of meromorphic functions whose cut
propagator at some point is $\Delta$.
###### Proof.
The following argument is inspired by the one in Etingof [6]. By using a
locally finite smooth partition of unity, which exists since we assume that
spacetime is metrizable, we can reduce to showing that a regularization exists
locally. If a local propagator is smooth, it is easy to construct a Feynman
measure for it, just by defining it as a sum of products of Feynman
propagators. Now suppose that we have a meromorphic family of local
propagators $\Delta_{d}$ depending on real numbers $d_{i}$, given in local
coordinates by a finite sum of boundary values of terms of the form
$s(x,y)p_{1}(x,y)^{d_{1}}\ldots p_{k}(x,y)^{d_{k}}\log(p_{k+1}(x,y))\ldots$
where $s$ is smooth in $x$ and $y$, and the $p_{i}$ are polynomials, and where
we choose some branch of the powers and logarithms in each region where they
are non-zero. In this case the Feynman measure can also be defined as a
meromorphic function of $d$ for all real $d$. To prove this, we can forget
about the smooth function $s$ as it is harmless, and we can eliminate the
logarithmic terms by writing $\log(p)$ as
$\frac{d}{dt}p^{t}\operatorname{at}t=0$. For any fixed number of fields with
derivatives of fixed order, the corresponding distribution is defined when all
variables $d_{i}$ have sufficiently large real part, because the product of
the propagators is smooth enough to be defined in this case. But this
distribution is given in local coordinates by the product the $d_{i}$’th
powers of polynomials of $x$ and $y$. By Bernstein’s corollary 17 these
products can be continued as a meromorphic distribution-valued function of the
$d_{i}$ to all complex $d_{i}$.
This gives a Feynman measure with values in the field of meromorphic functions
in several variables, and by restricting functions to the diagonal we get a
Feynman measure whose value are meromorphic functions in one variable. ∎
###### Example 19
Dimensional regularization. Over Minkowski space of dimension $d$, there is a
variation of the construction of a meromorphic Feynman measure, which is very
similar to dimensional regularization. In dimensional regularization, one
formally varies the dimension of spacetime, to get Feynman diagrams that are
meromorphic functions of the dimension of spacetime. One way to make sense out
of this is to keep the dimension of spacetime fixed, but vary the propagator
of the free field theory, by considering it to be a meromorphic function of a
complex number $d$. The propagator for a Hermitian scalar field, considered as
a distribution of $z$ in Minkowski space, can be written as a linear
combination of functions of the form
$K_{d/2-1}(c\sqrt{(z,z)})/\sqrt{(z,z)}^{d/2-1}$
where $K_{\nu}(z)$ is a multi-valued modified Bessel function of the third
kind, and where we take a suitable choice of branch (depending on whether we
are considering a cut or a Feynman propagator). A similar argument using
Bernstein’s theorem shows that this gives a Feynman measure that is analytic
in $d$ for $d$ with large real part and that can be analytically continued as
a meromorphic function to all complex $d$. This gives an explicit example of a
meromorphic Feynman measures for the usual propagators in Minkowski space.
###### Theorem 20
Any meromorphic Feynman measure can be made holomorphic by acting on it with a
meromorphic renormalization.
###### Proof.
This is essentially the result that a bare quantum field theory can be made
finite by an infinite renormalization. Suppose that $\omega$ is a meromorphic
Feynman measure. Using the same idea as in theorem 15 we will construct a
meromorphic renormalization $g=\text{$\ldots g_{2}g_{1}g_{0}$}$ as an infinite
product, but this time we choose $g_{n}\in G_{n}$ to kill the singularities of
order $n+1$. The key point is to prove that these lowest order singularities
are “local”, meaning that they have support on the diagonal. (In the special
case of translation-invariant theories on Minkowski spacetime this becomes the
usual condition that they are “polynomials in momentum”, or more precisely
that their Fourier transforms are essentially polynomials in momentum on the
subspace with total momentum zero). The locality follows from the Gaussian
property of $\omega$, which determines $\omega$ at each order in terms of
smaller orders except on the diagonal. In particular if $\omega$ is
nonsingular at all orders at most $n$, then the singular parts of the order
$n+1$ terms all have support on the diagonal. Since the difference is smooth
along the diagonal, we can find some $g_{n}\in G_{n}$ that kills off the order
$n+1$ singularities, as in theorem 15. Since renormalizations preserve the
Gaussian property we can keep on repeating this indefinitely, killing off the
singularities in order of their order. ∎
The famous problem of “overlapping divergences” is that the counter-terms for
individual Feynman diagrams used for renormalization sometimes contain non-
polynomial (logarithmic) terms in the momentum, which bring renormalization to
a halt unless they miraculously cancel when summed over all Feynman diagrams.
This problem is avoided in the proof above because by using the ultraviolet
group we only need to handle the divergences of lowest order at each step,
where it is easy to see that the logarithmic terms cancel.
###### Theorem 21
For any cut local propagator there is a Feynman measure associated to it.
###### Proof.
This follows from theorem 18, which uses regularization to show that there is
a meromorphic Feynman measure, and theorem 20 which uses renormalization to
show that the poles of this can be eliminated. ∎
## 4 Subgroups of the ultraviolet group
There are many additional desirable properties that one can impose on Feynman
measures, such as being Hermitian, or Lorentz invariant, or normal ordered,
and there is often a subgroup of the ultraviolet group that acts transitively
on the measures with the given property. We give several examples of this.
###### Example 22
A Feynman measure can be normalized so that on $S^{1}\Gamma_{c}\omega
S^{{}^{0}}J\Phi=\Gamma_{c}\omega$ its value is given by integrating over
spacetime (in other words $g=1$ in definition 9), by acting on it by a unique
element of the ultraviolet group consisting of renormalizations in $G_{0}$
that are trivial on $\omega S^{>0}J\Phi$. This group can be identified with
the group of nowhere-vanishing smooth complex functions on spacetime. The
complementary normal subgroup of the ultraviolet group consists of the
renormalizations that fix all elements of $\omega S^{0}J\Phi=\omega$, and this
acts simply transitively on the normalized Feynman measures. In practice
almost any natural Feynman measure one constructs is normalized.
###### Example 23
Normal ordering. In terms of Feynman diagrams, “normal ordering” means roughly
that Feynman diagrams with an edge from a vertex to itself are discarded. We
say that a Feynman measure is normally ordered if it vanishes on
$\Gamma_{c}\omega S^{>0}J\Phi$. Informally, $\omega S^{>0}J\Phi$ corresponds
to Feynman diagrams with just one point and edges from this point to itself.
We will say that a renormalization is normally ordered if it fixes all
elements of $\omega S^{>0}J\Phi$. The subgroup of normally ordered
renormalizations acts transitively on the normally ordered Feynman measures.
The group of all renormalizations is the semidirect product of its normal
subgroup $G_{>0}$ of normally-ordered renormalizations with the subgroup
$G_{0}$ preserving all products. For any renormalization, there is a unique
element of $G_{0}$ that takes it to a normally ordered renormalization. The
Feynman measures constructed by regularization (in particular those
constructed by dimensional regularization) are usually normally ordered if the
spacetime has positive dimension, but are usually not for 0-dimensional
spacetimes. This is because the propagators tend to contain a factor such as
$(x-y)^{-2d}$ which vanishes for large $-d$ when $x=y$, and so vanishes on
Feynman diagrams with just one point for all $d$ by analytic continuation. So
for most purposes we can restrict to normally-ordered Feynman measures and
normally-ordered renormalizations, at least for spacetimes of positive
dimension.
###### Example 24
Normalization of Feynman propagators. In general a renormalization fixes the
cut propagator but can change the Feynman propagator, by adding a distribution
with support on the diagonal. However there is often a canonical choice of
Feynman propagator: the one with a singularity on the diagonal of smallest
possible order, which will often also be a Green function for some
differential operator. We can add the condition that the Feynman propagator of
a Feynman measure should be this canonical choice; the subgroup of
renormalizations fixing the Feynman propagator, consisting of renormalizations
fixing $S^{2}\omega J\Phi$, acts simply transitively on these Feynman
measures.
###### Example 25
Simple operators. More generally, there is a subgroup consisting of
renormalizations $\rho$ such that $\rho(aB)=\rho(a)\rho(B)$ whenever $a$ is
simple (involving only one field), but where $B$ is arbitrary. This stronger
condition is useful because it says (roughly) that simple operators containing
only one field do not get renormalized; see the discussion in section 6. We
can find a set of Feynman measures acted on simply transitively by this group
by adding the condition that
$\omega(aB)=\sum\Delta_{F}(aB_{1})\omega(B_{2})$
whenever $a$ is simple and $\sum B_{1}\otimes B_{2}$ is the coproduct of $B$.
This relation holds whenever $a$ and $B$ have disjoint supports by definition
of a Feynman measure, so the extra condition says that it also holds even when
they have overlapping supports. The key point is that the product of
distributions above is always defined because any non-zero element of the wave
front set of $\Delta_{F}$ is of the form $(p,-p)$. This would not necessarily
be true if $a$ were not simple because we would get products of more than 1
Feynman propagator whose singularities might interfere with each other. In
terms of Feynman diagrams, this says that vertices with just one edge are
harmless: more precisely, with this normalization, adding a vertex with just
one edge to a Feynman diagram has the effect of multiplying its value by the
Feynman propagator of the edge. As this condition extends the Gaussian
property to more Feynman diagrams, it can also be thought of as a
strengthening of the translation invariance property of the Feynman measure.
###### Example 26
Dyson condition. Classically, Lagrangians were called renormalizable if all
their coupling constants have non-negative mass dimension. The filtration on
Lagrangian densities by mass dimension induces a similar filtration on Feynman
measures and renormalizations. The Feynman measures of mass dimension
$\leqslant 0$ are acted on simply transitively by the renormalizations of mass
dimension $\leqslant 0$. This is useful, because the renormalizations of mass
dimension at most 0 act on the spaces of Lagrangian densities of mass
dimension at most 0, and these often form finite dimensional spaces, at least
if some other symmetry conditions such as Lorentz invariance are added. For
example, in dimension 4 the density has dimension $-4$, so the (Lorentz-
invariant) terms of the Lagrangian density of mass dimension at most 0 are
given by (Lorentz invariant) terms of the Lagrangian of mass dimension at most
4, such as $\varphi^{4}$, $\varphi^{2}$, $\partial\varphi\partial\varphi$, and
so on: the usual Lorentz-invariant even terms whose coupling constants have
mass dimension at least 0. For example, we get a three-dimensional space of
theories of the form
$\lambda\varphi^{4}+m\varphi^{2}+z\partial\varphi\partial\varphi$ in this way,
giving the usual $\varphi^{4}$ theory in 4 dimensions.
###### Example 27
Boundary terms. The Feynman measures constructed in section 3 have the
property that they vanish on “boundary terms”. This means that we quotient the
space of local Lagrangians $\Gamma_{c}\omega SJ\Phi$ by its image under the
action of smooth vector fields such as $\partial/\partial x_{i}$, or in other
words we replace a spaces of $n$-forms by the corresponding de Rham cohomology
group. These measures are acted on simply transitively by renormalizations
corresponding to maps that vanish on boundary terms. This is useful in gauge
theory, because some symmetries such as the BRST symmetry are only symmetries
up to boundary terms.
###### Example 28
Symmetry invariance. Given a group (or Lie algebra) $G$ such as a gauge group
acting on the sheaf $\Phi$ of classical fields and preserving a given cut
propagator, the subgroup of $G$-invariant renormalizations acts simply
transitively on the $G$-invariant Feynman measures with given cut propagator.
In general there need not exist any $G$-invariant Feynman measure associated
with a given cut local propagator, though if there is then $G$-invariant
Lagrangians lead to $G$-invariant quantum field theories. The obstructions to
finding a $G$-invariant measure are cohomology classes called anomalies, and
are discussed further in section 7.
###### Example 29
Lorentz invariance. An important case of invariance under symmetry is that of
Poincare invariance for flat Minkowski space. In this case the spacetime $M$
is Minkowski space, the Lie algebra $G$ is that of the Poincare group of
spacetime translations and Lorentz rotations, and the cut propagator is one of
the standard ones for free field theories of fields of finite spin. Then
dimensional regularization is invariant under $G$, so we get a Feynman measure
invariant under the Poincare group, and in particular there are no anomalies
for the Poincare algebra. The elements of the ultraviolet group that are
Poincare invariant act simply transitively on the Feynman measures for this
propagator that are Poincare invariant. If we pick any such measure, then we
get a map from invariant Lagrangians to invariant quantum field theories.
###### Example 30
Hermitian conditions. The group of complex renormalizations has a real form,
consisting of the subgroup of (real) renormalizations. This acts simply
transitively on the Hermitian Feynman measures associated with a given cut
local propagator. The Hermitian Feynman measures (or propagators) are not the
real-valued ones, but satisfy a more complicated Hermitian condition described
in definition 36.
## 5 The free quantum field theory
We extend the Feynman measure $\omega:e^{iL_{F}}S\Gamma_{c}\omega
SJ\Phi\rightarrow\boldsymbol{C}$, which is something like a measure on
classical fields, to $\omega:Te^{iL_{F}}S\Gamma_{c}\omega
SJ\Phi\rightarrow\boldsymbol{C}$. This extension, restricted to the even
degree subalgebra $T_{0}e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$, is the free
quantum field theory. We check that it satisfies analogues of the Wightman
axioms.
Formulas involving coproducts can be confusing to write down and manipulate.
They are much simpler for the “group-like” elements $g$ satisfying
$\Delta(g)=g\otimes g$, $\eta(g)=1$, which form a group in any cocommutative
Hopf algebra. One problem is that most of the Hopf algebras we use do not have
enough group-like elements over fields: in fact for symmetric algebras the
only group-like element is the identity. However they have plenty of group-
like elements if we add some nilpotent elements to the base field, such as
$\exp(\lambda a)$ for any primitive $a$ and nilpotent $\lambda$ (in
characteristic 0). We will adopt the convention that when we talk about group-
like elements, we are tacitly allowing extensions of the base ring by
nilpotent elements.
Recall that $Te^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ is the tensor algebra of
$e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$, with the product denoted by $\otimes$ to
avoid confusing it with the product of $S\Gamma_{c}SJ\Phi$. We denote the
identity of $S\Gamma_{c}SJ\Phi$ by $1$, and the identity of
$\operatorname{TS}\Gamma_{c}\omega SJ\Phi$ by $1_{T}$. The involution $\ast$
is defined by $(A_{1}\otimes\ldots\otimes
A_{n})^{\ast}=A_{n}^{\ast}\otimes\ldots\otimes A_{1}^{\ast}$, and $\ast$ is
$-1$ on $\Gamma_{c}\omega SJ\Phi$.
###### Theorem 31
If $\omega:e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi\rightarrow C$ is a Feynman
measure then there is a unique extension of $\omega$ to
$Te^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ such that
* $\bullet$
Gaussian condition: if $A,B_{1},\ldots,B_{m}$ are group-like then
$\displaystyle e^{-iL_{F}}\omega(A\otimes B_{m}\otimes\ldots\otimes B_{1})$
$\displaystyle=$ $\displaystyle\sum e^{-iL_{F}}\omega(A\otimes
1\otimes\ldots\otimes 1)\Delta(A,B_{m}\ldots
B_{1})e^{-iL_{F}}\omega(B_{m}\otimes\ldots\otimes B_{1})$
Both sides are considered as densities, as in definition 9.
* $\bullet$
$e^{-iL_{F}}\omega(A\otimes A\otimes 1\otimes\ldots\otimes 1)=1$ for $A$
group-like (Cutkosky condition; see[9, section 6].)
###### Proof.
We first check that all the products of distributions are well defined by
examining their wave front sets. All the distributions appearing have the
property that their wave front sets have no positive or negative elements.
This follows by induction on the complexity of an element: if all smaller
elements have this property, it implies that the products defining it are well
defined, and also implies that it has the same property.
Existence and uniqueness of $\omega$ follows because the Cutkosky condition
defines it on elements of the form $A\otimes 1\otimes 1\otimes\ldots\otimes 1$
in terms of those of the form $A\otimes 1\otimes\ldots\otimes 1$, and the
Gaussian condition then determines it on all elements. ∎
We can also define $\omega$ directly as follows. When the propagator is
sufficiently regular then the Gaussian condition means that we can write
$\omega$ on $e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ as a sum over all ways of
joining up the fields of an element of $e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ in
pairs, where we take the propagator of each pair and multiplying these
together. This is of course essentially the usual sum over Feynman diagrams. A
minor difference is that we do not distinguish between “internal” vertices
associated with a Lagrangian and integrated over all spacetime, and “extenal”
vertices associated with a field and integrated over a compact set: all
vertices are associated with a composite operator that may be a Lagrangian or
a simple field or a more general composite operator, and all vertices are
integrated over compact sets as all coefficients are assumed to have compact
support.
Similarly we can define the extension of $\omega$ to
$Te^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ by writing the distributions defining
$\omega$ as a sum over more complicated Feynman diagrams whose vertices are in
addition labeled by non-negative integers, such that
* •
The propagators from $A_{i}$ to $A_{i}$ are Feynman propagators.
* •
The propagators from $A_{i}$ to $A_{j}$ for $i<j$ are cut propagators
$\Delta$, with positive wave front sets on $i$ and negative wave front sets on
$j$.
* •
The diagram is multiplied by a factor of $(-1)^{\deg(A_{2}A_{4}A_{6}\ldots)}$;
in other words we apply $\ast$ to $A_{2}$, $A_{4},\ldots$.
In general if the propagator is not sufficiently regular (so that products of
propagators might not be defined when some points coincide) we can construct
$\omega$ by regularization and renormalization as in section 3, which
preserves the conditions defining $\omega$.
Now we show that $\omega$ satisfies the locality property of quantum field
theories (operators with spacelike-separated supports commute) by showing that
it vanishes on the following locality ideal.
###### Definition 32
$T_{0}S\Gamma_{c}\omega SJ\Phi$ is the subalgebra of even degree elements of
$TS\Gamma_{c}\omega SJ\Phi$. The locality ideal is the 2-sided ideal of
$T_{0}S\Gamma_{c}\omega SJ\Phi$ spanned by the coefficients of elements of the
form
$\ldots\otimes Y_{1}\otimes ABD\otimes DBC\otimes X_{n}\otimes\ldots\otimes
X_{1}-\ldots\otimes Y_{1}\otimes AD\otimes DC\otimes X_{n}\otimes\ldots\otimes
X_{1}$
(for $A,C\in S\Gamma_{c}\omega SJ\Phi$ and $B,D\in S\Gamma_{c}\omega
SJ\Phi[[\boldsymbol{\lambda}]]$ with $B,D$ group-like) if $n$ is even and
there are no points in the support of $B$ that are $\leqslant$any points in
the support of $A$ or $C$, or if $n$ is odd and there are no points in the
support of $B$ that are $\geqslant$any points in the support of $A$ or $C$.
The algebra $T_{0}e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ and its locality ideal
are defined in the same way.
###### Remark 33
The map $\omega$ on $T_{0}e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ depends on the
choice of Feynman measure. We can define a canonical map independent of the
choice of Feynman measure by taking the underlying $*$-algebra to have
elements represented by pairs $(\omega,A)$ for a Gaussian measure $\omega$ and
$A\in\text{$T_{0}e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$}$, where we identify
$(\omega,A)$ with $(\rho\omega,\rho A)$ for any renormalization $\rho$. The
canonical state, also denoted by $\omega$, then takes an element represented
by $(\omega,A)$ to $\omega(A)$.
###### Theorem 34
$\omega$ vanishes on the locality ideal.
###### Proof.
We use the notation of definition 32. We prove this for elements with $n$
even; the case $n$ odd is similar. We can assume that the propagator $\Delta$
is sufficiently regular, as we can obtain the general case from this by
regularization and renormalization. We will first do the special case when
$D=1$. We can assume that $B=b_{1}\ldots b_{k}$ is homogeneous of some order
$k$ and write $B_{I}$ for $\prod_{j\in I}b_{j}$. If $k=0$ then the result is
obvious as $B$ is constant and both sides are the same, so we can assume that
$k>0$. We show that if $k>0$ then $\omega$ vanishes on
$\sum_{I\cup J=\\{1,\ldots k\\}}(-1)^{|I|}\ldots\otimes Y_{1}\otimes
AB_{I}\otimes B_{J}C\otimes X_{n}\otimes\ldots\otimes X_{1}$
by showing that the terms cancel out in pairs. This is because if $j$ is the
index for which the support of $b_{j}$ is maximal then $\omega$ has the same
value on
$\ldots\otimes Y_{1}\otimes AB_{I}b_{j}\otimes B_{J}C\otimes
X_{n}\otimes\ldots\otimes X_{1}$
and
$\ldots\otimes Y_{1}\otimes AB_{I}\otimes b_{j}B_{J}C\otimes
X_{n}\otimes\ldots\otimes X_{1}$
Now we do the case of general $D$. We can assume that the support of $D$ is
either $\leqslant$ all points of the support of $B$ or there are no points of
it that are $\leqslant$ any points in the support of $A$ or $C$. In the first
case the result follows from the special case $D=1$ by replacing $A$ and $C$
by $AD$ and $CD$. In the second case it follows from 2 applications of the
special case $D=1$, replacing $B$ by $D$ and $BD$, that both terms are equal
to $\ldots\otimes Y_{1}\otimes A\otimes C\otimes X_{n}\otimes\ldots\otimes
X_{1}$ and are therefore equal. ∎
This proof, in the special case that $\omega$ vanishes on $B\otimes B-1\otimes
1$ for $B$ group-like, is more or less the proof of unitarity of the S-matrix
using the “largest time equation” given in [9, section 6]. The locality ideal
is not the largest ideal on which $\omega$ vanishes, as $\omega$ also vanishes
on $A\otimes 1\otimes 1\otimes B-A\otimes B$; in other words we can cancel
pairs $1\otimes 1$ wherever they occur.
###### Theorem 35
Elements of $T_{0}S\Gamma_{c}SJ\Phi$ with spacelike-separated supports commute
modulo the locality ideal.
###### Proof.
It is sufficient to prove this for group-like degree 2 elements, as if two
even degree elements have spacelike-separated supports then they are
polynomials in degree 2 elements with spacelike separated supports. We will
work modulo the locality ideal. Suppose that the supports of the group-like
elements $W\otimes X\otimes Z$ and $Y$ are spacelike-separated. Then applying
theorem 34 twice gives
$W\otimes X\otimes YZ=WY\otimes XY\otimes YZ=WY\otimes X\otimes Z$
Applying this 4 times for various values of $W$, $X$, $Y$, and $Z$ shows that
if $A\otimes B$ and $C\otimes D$ are group-like and have spacelike separated
supports, then
$A\otimes B\otimes C\otimes D=AC\otimes B\otimes I\otimes D=AC\otimes I\otimes
I\otimes BD=AC\otimes D\otimes I\otimes B=C\otimes D\otimes A\otimes B$
$\operatorname{so}A\otimes B$ and $C\otimes D$ commute. ∎
Now we study when the quantum field theory $\omega$ is Hermitian, and show
that we can find a Hermitian quantum field theory associated to any Hermitian
local cut propagator, and show that the group of real renormalizations acts
transitively on them.
###### Definition 36
We say that a Feynman measure $\omega$ is Hermitian if its extension to
$TS\Gamma_{c}\omega SJ\Phi$ is Hermitian when restricted to the even
subalgebra $T_{0}S\Gamma_{c}\omega SJ\Phi$.
###### Lemma 37
If the local cut propagator $\Delta$ is Hermitian, then it has a Hermitian
Feynman measure associated with it.
###### Proof.
We can assume that the regularization of $\Delta$ is also Hermitian, by
replacing it by the average of itself and its Hermitian conjugate. We can
check directly that the meromorphic family of Feynman measures associated to
this Hermitian regularization is Hermitian on $T_{0}S\Gamma_{c}\omega SJ\Phi$
(but not on the whole of $TS\Gamma_{c}\omega SJ\Phi$); in other words
$\omega(A_{n}\otimes\ldots\otimes
A_{1})=\omega(A_{1}^{\ast}\otimes\ldots\otimes A_{n}^{\ast})^{\ast}$ if $n$ is
even. For example, we get a sign factor of
$-1^{\deg(A_{2})+\deg(A_{4})+\ldots}$ in the definition of $\omega$ on the
first term, a sign factor of $-1^{\deg(A_{1})+\deg(A_{3})+\ldots}$ form the
definition of $\omega$ for the second term, whose quotient is the factor
$-1^{\deg(A_{1})+\deg(A_{2})+\ldots}$ coming from the action of $\ast$ on
$A_{n}\otimes\ldots\otimes A_{1}$ because $n$ is even. We can then renormalize
using real renormalizations to eliminate the poles, and the resulting Feynman
measure will be Hermitian. ∎
###### Lemma 38
If a Feynman measure $\omega$ is Hermitian and $\rho$ is a complex
renormalization, then $\rho(\omega)$ is Hermitian if and only if $\rho$ is
real. In particular the subgroup of (real) renormalizations acts simply
transitively on the Hermitian Feynman measures associated with a given cut
local propagator.
###### Proof.
This follows from $\rho(\omega)^{\ast}=\rho^{\ast}(\omega^{\ast})$, and the
fact that complex renormalizations act simply transitively on Feynman measures
associated with a given cut local propagator. ∎
Next we show that $\omega$ is a state (in other words the space of physical
states is positive definite) when the cut propagator $\Delta$ is positive, by
using a representation of the physical states as a space of distributions. We
define the space $H_{n}$ of $n$-particle states to be the space of continuous
linear maps $S^{n}\Gamma\omega\Phi\rightarrow\boldsymbol{C}$ (considered as
compactly supported symmetric distributions on $M^{n}$) whose wave front sets
have no positive or negative elements, with a sesquilinear form given by
$\langle a,b\rangle=\int_{x,y\in
M^{n}}a(x_{1},\ldots)\prod_{j}\Delta(x_{j},y_{j})b(y_{j},\ldots)^{\ast}dxdy.$
This is similar to the usual definition of the inner product on the space of
states of a free field theory, except that we are using distributions rather
than smooth functions. We check this is well defined. To show the product of
distributions in the integral is defined we need to check that no sum of non-
zero elements of the wave front sets is zero, and this follows because nonzero
elements of the wave front set of the product of propagators are of the form
$(p,q)$ with $p>0$ and $q<0$, but $a$ and $b$ by assumption have no positive
or negative elements in their wave front sets. The integral over $M^{n}$ is
well defined because $a$ and $b$ have compact support.
###### Lemma 39
There is a map $f$ from $T_{0}S\Gamma_{c}\omega SJ\Phi$ to the orthogonal
direct sum$\oplus H_{{}_{n}}$ with
$\omega(AB)=\langle f(A^{\ast}),f(B)\rangle.$
###### Proof.
By theorem 31, $\omega(AB)$ is given by
$\sum\omega(A^{\prime})\Delta(A^{\prime\prime},B^{\prime\prime})\omega(B^{\prime})$
where $\sum A^{\prime}\otimes A^{\prime\prime}$ is the image of $A$ under the
coaction of $\Gamma_{c}SJ\Phi$. This is equal to $\langle
f(A^{\ast}),f(B)\rangle$ if we define $f(A)$ as follows. Suppose that
$A=A_{11}A_{12}\ldots\otimes A_{21}A_{22}\ldots$., and let the image of
$A_{jk}$ under the coaction of $\Gamma_{c}SJ\Phi$ be $\sum
A_{jk}^{\prime}\otimes A_{jk}^{\prime\prime}$. Then
$\omega(A^{\prime}_{11}A_{12}^{\prime}\ldots\otimes
A_{21}^{\prime}A_{22}^{\prime}\ldots$.) can be regarded as a distribution on
$M^{n}$, where $n$ is the total number of elements $A_{jk}$. On the other
hand, $A^{\prime\prime}_{11}A^{\prime\prime}_{12}\ldots
A^{\prime\prime}_{21}A^{\prime\prime}_{22}\ldots$ is a function on $M^{m}$,
where $m$ is the sum of the degree of the elements $A^{\prime\prime}_{jk}$, in
other words the number of fields occurring in them. There is also a map from
$m$ to $n$, which induces a map from $M^{n}$ to $M^{m}$, and so by push-
forward of densities a map from densities on $M^{n}$ to densities on $M^{m}$.
The image $f(A)$ is then given by taking the push-forward from $M^{n}$ to
$M^{m}$ of the compactly supported distribution
$\omega(A^{\prime}_{11}A_{12}^{\prime}\ldots\otimes
A_{21}^{\prime}A_{22}^{\prime}\ldots$.) on $M^{n}$, multiplying by the
function $A^{\prime\prime}_{11}A^{\prime\prime}_{12}\ldots
A^{\prime\prime}_{21}A^{\prime\prime}_{22}\ldots$ on $M^{m}$, symmetrizing the
result, and repeating this for each summand of $\sum A_{jk}^{\prime}\otimes
A_{jk}^{\prime\prime}$. ∎
###### Corollary 40
If the cut local propagator $\Delta$ is positive, then
$\omega:Te^{iL_{F}}S\Gamma_{c}\omega SJ\Phi\rightarrow\boldsymbol{C}$ is a
state.
###### Proof.
This follows from the previous lemma, because if $\Delta$ is positive then so
is the sesquilinear form $\langle,\rangle$ on $H_{n}$, and therefore
$\omega(A^{\ast}A)=\langle f(A),f(A)\rangle\geqslant 0$. ∎
## 6 Interacting quantum field theories
We construct the quantum field theory of a Feynman measure and a compactly
supported Lagrangian, by taking the image of the free field theory $\omega$
under an automorphism $e^{iL_{I}}$ where $L_{I}$ is the interaction part of
the Lagrangian. This automorphism is only well defined if the interaction
Lagrangian $L_{I}$ has infinitesimal coefficients, so the interacting quantum
field theories we construct are perturbative theories taking values in rings
of formal power series
$\boldsymbol{C}[\boldsymbol{\lambda}]=\boldsymbol{C}[\lambda_{1},\ldots]$ in
the coupling constants $\lambda_{1},\ldots$. (By “infinitesimal” we mean
elements of formal power series rings with vanishing constant term.) We then
lift the construction to all actions (possible without compact support) by
showing that infra-red divergences cancel up to inner automorphisms.
###### Lemma 41
The Hopf algebra $S\Gamma_{c}\omega SJ\Phi$ acts on the algebra
$T_{0}S\Gamma_{c}\omega SJ\Phi$, and maps the locality ideal to itself. Group-
like Hermitian elements of the Hopf algebra $S\Gamma_{c}\omega
SJ\Phi[[\boldsymbol{\lambda}]]$ preserve the subset of positive elements, and
therefore act on the space of states of $T_{0}S\Gamma_{c}\omega
SJ\Phi[[\boldsymbol{\lambda}]]$.
###### Proof.
Group-like elements are algebra automorphisms, and if they are also Hermitian
they commute with the involution $\ast$. In particular group-like Hermitian
elements preserve the set of positive elements (generated by positive linear
combinations of elements of the form $a^{\ast}a$), and so map positive linear
forms to positive linear forms. ∎
###### Definition 42
The quantum field theory of a Lagrangian $L=L_{F}+L_{I}$, where $L_{I}$ has
compact support and infinitesimal coefficients, is
$e^{-iL}\omega:T_{0}S\Gamma_{c}\omega
SJ\Phi\rightarrow\boldsymbol{C}[[\boldsymbol{\lambda}]]$.
The Hopf algebra $S\Gamma_{c}\omega SJ\Phi$ acts on the vector space
$S\Gamma_{c}\omega SJ\Phi$ by multiplication, so group-like elements of the
form $e^{iL_{F}+iL_{I}}$ take $S\Gamma_{c}\omega SJ\Phi$ to
$e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$ and $T_{0}S\Gamma_{c}\omega SJ\Phi$ to
$T_{0}e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$. Since $\omega$ is in the dual of
$T_{0}e^{iL_{F}}S\Gamma_{c}\omega SJ\Phi$, this shows that $e^{-iL}\omega$ is
in the dual of $T_{0}S\Gamma_{c}\omega SJ\Phi$.
###### Corollary 43
(Locality) Elements of $T_{0}S\Gamma_{c}\omega SJ\Phi$ with spacelike-
separated supports commute when acting on the space of physical states of
$e^{-iL}\omega$.
###### Proof.
By theorem 34 the operators of the locality ideal act trivially on the space
of physical states of $\omega$. Since $e^{-iL}$ preserves the locality ideal,
the locality ideal also acts trivially on the space of physical states of
$e^{-iL}\omega$. By lemma 35 this implies that operators with spacelike
separated supports commute on this space. ∎
This constructs the quantum field theory of a Lagrangian whose interaction
part has compact support (and is infinitesimal). We now extend this to the
case when the interaction part need not have compact support. We do this by
using a cutoff function to give the Lagrangian compact support, and then we
then try to show that the result is independent of the choice of cutoff
function, provided it is 1 in a sufficiently large region. To do this we need
to assume that spacetime is globally hyperbolic, and we also find that the
result is not quite independent of the choice of cutoff.
If $f$ is a smooth function on $M$ then multiplication by $f$ is a linear
transformation of $\Gamma\omega SJ\Phi$ and therefore induces a homomorphism
of $S\Gamma\omega SJ\Phi$, denoted by $A\rightarrow A^{f}$. If $A=e^{iL}$ is
group-like, then $A^{f}=e^{iLf}$. If $f$ has compact support then so does
$A^{f}$ so that $A^{f}\omega$ is defined. We try to extend the definition of
$A^{f}\omega$ to more general functions $f$ in the hope that we can take $f$
to be close to 1.
###### Lemma 44
Suppose that $f$ and $g$ are compactly supported smooth functions on $M$ and
$n$ is even. If $f=g$ on the past of $A_{1}\ldots A_{n}$ then (modulo the
locality ideal)
$\begin{array}[]{lll}e^{-iL_{F}}A^{f}\omega(A_{n}\otimes\ldots\otimes
A_{1})&=&e^{-iL_{F}}A^{g}\omega(A_{n}\otimes\ldots\otimes A_{1})\end{array}$
If $f=g$ on the future of $A_{1}\ldots A_{n}$ then
$\begin{array}[]{lll}e^{-iL_{F}}A^{f}\omega(A_{n}\otimes\ldots\otimes
A_{1})&=&e^{-iL_{F}}A^{g}\omega(A^{g-f}\otimes 1\otimes
A_{n}\otimes\ldots\otimes A_{1}\otimes 1\otimes A^{g-f})\end{array}$
###### Proof.
We work modulo the locality ideal. The first equality follows from
$A^{-f}A_{n}\otimes\ldots\otimes A^{-f}A_{1}=A^{-g}A_{n}\otimes\ldots\otimes
A^{-g}A_{1}$
which in turn follows from theorem 34 by repeatedly inserting $A^{f-g}\otimes
A^{f-g}$ (using the fact that $n$ is even). The second equality follows in the
same way from
$\displaystyle A^{-f}\otimes A^{-f}\otimes A^{-f}A_{n}\otimes\ldots\otimes
A^{-f}A_{1}\otimes A^{-f}\otimes A^{-f}$ $\displaystyle=$ $\displaystyle
A^{-f}\otimes A^{-g}\otimes A^{-g}A_{n}\otimes\ldots\otimes A^{-g}A_{1}\otimes
A^{-g}\otimes A^{-f}$
∎
This lemma shows that the restriction of $A^{f}\omega$ to arguments with
support in some fixed compact subset of $M$ is almost independent of the
choice of $f$ provided that $f$ is 1 on the convex hull of the argument:
different choices of $f$ are related by a locally inner automorphism of
$T_{0}S\Gamma_{c}\omega SJ\Phi$, given by conjugation by elements of the form
$1\otimes A^{h}$. If the spacetime is globally hyperbolic in the sense that
the convex hull of a compact set is contained in a compact set, then we can
always find a suitable $f$ that is 1 on the convex hull $X$ of the argument,
so we can construct the interacting quantum field theory. The result does not
depend on the choice of cutoff $f$ on the future of $X$, but does depend
slightly on the choice of cutoff in the past of $X$. The choice of cutoff in
the past corresponds to choices of the vacuum: roughly speaking, we turn off
the interaction in the distant past, which gives different vacuums. More
precisely, if we have two different cutoffs $f$ and $g$ then their vacuums,
which are the images of $e^{i(L_{F}+fL_{I})}$ and $e^{i(L_{F}+gL_{I})}$ will
differ by a factor of $e^{i(f-g)L_{I}}$. This does not change the observable
physics, beause all these choices of cutoffs give isomorphic quantum field
theories. However it does cause difficulties in constructing a Lorentz
invariant theory, because the choice of cutoff in the past is not Lorentz
invariant, so the vacuums are also not Lorentz invariant, or in other words
Lorentz invariance may be spontaneously broken. Presumably in theories with a
mass gap one can take the limit as the cutoff in the past tends to time
$-\infty$ and get a Lorentz invariant vacuum, but in theories with massless
particles such as QED there is an obstruction to constructing a Lorentz
invariant vacuum: Lorentz invariance might be spontaneously broken by infrared
divergences. This is a well known problem, which is not worth worrying about
too much, because the physical universe is not globally Lorentz invariant.
The time-ordered operator $T(A)$ of an element $A\in S\Gamma_{c}\omega SJ\Phi$
is defined to be $1\otimes A$. This has the property that
$T(A_{n}\ldots A_{1})=1\otimes A_{n}\ldots A_{1}=1\otimes
A_{n}\otimes\ldots\otimes 1\otimes A_{1}=T(A_{n})\ldots T(A_{1})$
whenever the composite fields $A_{i}\in\Gamma_{c}\omega SJ\Phi$ are in order
of increasing time of their supports. This formula is sometimes used as a
“definition” of the time-ordered product $T(A_{n}\ldots A_{1})$, though this
does not define it when some of the factors have overlapping supports, and in
general the time-ordered product depends on the choice of Feynman measure
$\omega$. The scattering matrix $S$ of the quantum field theory is
$S=T(e^{iL_{I}})=1\otimes e^{iL_{I}}$; this is essentially the LSZ reduction
formula of Lehmann, Symanzik, and Zimmermann [12].
We now show that if we change the Feynman measure, then we still get an
isomorphic quantum field theory provided we make a suitable change in the
Lagrangian. If we change $\omega$ to a different Feynman measure for the same
cut local propagator, these will differ by a unique renormalization $\rho$; in
other words the other Feynman measure will be $\rho\omega$. The quantum field
theory $e^{-iL}\omega$ changes under this renormalization of $\omega$ by
$\displaystyle e^{-iL}\omega(A_{1}\otimes\ldots)$ $\displaystyle=$
$\displaystyle\omega(e^{iL}A_{1}\otimes\ldots)$ $\displaystyle=$
$\displaystyle\rho(\omega)(\rho(e^{iL}A_{1})\otimes\ldots)$ $\displaystyle=$
$\displaystyle\rho(e^{-iL})\rho(\omega)(\rho(e^{-iL})\rho(e^{iL}A_{1})\otimes\ldots)$
so the quantum field theory stays the same under renormalization by $\rho$ if
we transform the Lagrangian by
$iL\rightarrow\log(\rho(\exp(iL)),$
which is a nonlinear transformation because renormalizations need not commute
with products or exponentiation, and change the operators $A_{n}$ by
$A_{n}\rightarrow\rho(e^{-iL})\rho(e^{iL}A_{n}).$
If $A_{n}$ is a simple operator and $\rho$ satisfies the condition of example
25 then $\rho(e^{iL}A_{n})=\rho(e^{iL})\rho(A_{n})=\rho(e^{iL})A_{n}$, so in
this special case $A_{n}$ is unchanged, or in other words simple operators are
not renormalized. The behavior of composite operators under renormalization
can be quite complicated when expanded out in terms of fields. The usual
Wightman distributions used to construct a quantum field theory use only
simple operators, so the only effect of renormalization on Wightman
distributions comes from the nonlinear transformation of the Lagrangian. This
nonlinear transformation of Lagrangians is the usual action of
renormalizations on Lagrangians used in physics texts to convert an infinite
“bare” Lagrangian $L$ to a finite physical one $L_{0}$; the bare and physical
Lagrangians are related by $iL_{0}=\log(\rho(\exp(iL))$, where $\rho$ is an
infinite renormalization taking an infinite Feynman measure, such as the one
given by dimensional regularization, to a finite one.
The orbit of a Lagrangian under this nonlinear action of the ultraviolet group
is in general infinite dimensional. It can sometimes be cut down to a finite
dimensional space as follows. As in example 26, we cut down to the group of
renormalizations of mass dimension at most 0, which acts on the space of
Lagrangians whose coupling constants all have mass dimension at least 0\. If
we also add the condition that the Lagrangian is Lorentz invariant, then we
sometimes get finite dimensional spaces of Lagrangians. The point is that the
classical fields themselves tend to have positive mass dimension, so if the
coupling constants all have non-negative mass dimension then the fields
appearing in any term of the Lagrangian have total mass at most $d$
(cancelling out the $-d$ coming from the density) which severely limits the
possibilities. At one time the Lagrangians with all coupling constants of non-
negative mass dimension were called renormalizable Lagrangians, though now all
Lagrangians are regarded as renormalizable in a more general sense where one
allows an infinite number of terms in the Lagrangian.
## 7 Gauge invariance and anomalies
If a Lagrangian is invariant under some group, this does not imply that the
quantum field theories we construct from it are also invariant, because as
Fujikawa [7] pointed out we also need to choose a Feynman measure and there
may not be an invariant way of doing this. The obstructions to finding an
invariant quantum field theory lie inside certain cohomology groups and are
called anomalies. We show that if these anomalies vanish then we can construct
invariant quantum field theories.
Suppose that a group $G$ acts on $SJ\Phi$ and preserves the set of Feynman
measures with given cut local propagator, and suppose that we have chosen one
such Feynman measure $\omega$. In practice we often start with an action of a
Lie algebra or superalgebra, such as that generated by the BRST operator,
which can be turned into a group action in the usual way by working over a
ring with nilpotent elements. If $g\in G$ then $g\omega$ is another Feynman
measure with the same propagator, so
$\omega=\rho_{g}g\omega$
for a unique renormalization $\rho_{g}$. This defines a non-abelian 1-cocycle:
$\rho_{gh}=\rho_{g}g(\rho_{h})$, where $g(\rho_{h})=g\rho_{h}g^{-1}$. Since
$\omega$ is invariant under $\rho_{g}g$, we find that
$\omega(e^{iL}A_{1})=\omega(\rho_{g}g(e^{iL}A_{1}))=\omega(e^{iL}e^{-iL}\rho_{g}g(e^{iL}A_{1}))$
so that $e^{-L}\omega$ is invariant under the transformation taking arguments
$A_{1}$ to $e^{-iL}\rho_{g}g(e^{iL}A_{1})$. This transformation fixes $1$ if
$e^{iL}$ is fixed by $\rho_{g}g$. If in addition
$\rho_{g}g(e^{iL}A_{1})=\rho_{g}g(e^{iL})\rho_{g}g(A_{1})$ (which is not
automatic as $\rho_{g}$ need not preserve products) then $A_{1}$ is taken to
$\rho_{g}g(A_{1})$ by this transformation.
This shows that we really want a Lagrangian $L$ such that $e^{iL}$ is
invariant under the modified action $e^{iL}\rightarrow\rho_{g}g(e^{iL})$. This
is not the same as asking for $\rho_{g}g(iL)=iL$ because $\rho_{g}$ need not
preserve products (although $g$ usually does). In practice we usually have a
Lagrangian $L$ with $L$ (and $e^{iL}$) invariant under $G$, and the problem is
whether it can be modified to $L^{\prime}$ so that $e^{iL^{\prime}}$ is
invariant under the twisted action. The powers of $L$ span a coalgebra all of
whose elements are $G$-invariant. Conversely, given a coalgebra $C$ all of
whose elements are invariant under some group action, there is a canonical
$G$-invariant group-like element associated to this coalgebra with
coefficients in the dual algebra of $C$. So a fundamental question is whether
the maximal coalgebra in the space of $G$-invariant classical actions is
isomorphic to the maximal coalgebra in the space of actions invariant under
the twisted action of $G$.
The simplest case is when one can find a $G$-invariant Feynman measure, in
which case the cocycle is trivial and the twisted action of $G$ is the same as
the untwisted action. In terms of the cocycle above, $\rho\omega$ is invariant
for some renormalization $\omega$ if and only if $\rho_{g}=\rho^{-1}g(\rho)$
for all $g$ (where $g(\rho)=g\rho g^{-1}$), in other words there is an
invariant measure $\omega$ if and only if the cocycle is a coboundary. This
case happens, for example, when spacetime $M$ is Minkowski space and $G$ is
the Lorentz or Poincare group (or one of their double covers). Dimensional
regularization in this case is automatically $G$-invariant, and so gives a
$G$-invariant Feynman measure.
In the case of BRST operators, there need not be any $G$-invariant Feynman
measure. In this case the following theorem shows that one can find suitable
coalgebras provided that certain obstructions, called anomalies, all vanish.
The renormalizations $\rho_{g}$ need not preserve products in $S\Gamma\omega
SJ\Phi$, but do preserve the coproduct and also fix all elements of
$\Gamma\omega SJ\Phi$ if they are normalized as in example 25. So we have an
action of $G$ on the space $V=\Gamma\omega SJ\Phi$, which lifts to two
different actions of the coalgebra $\operatorname{SV}$, the first
$\sigma_{1}(g)$ preserving the product, and the second
$\sigma_{2}(g)=\rho_{g}\sigma_{1}(g)$ given by twisting the first by the
cocycle $\rho_{g}$.
###### Theorem 45
Suppose that $V$ is a real vector space acted on by a group $G$, and there are
two extensions $\sigma_{1}$. $\sigma_{2}$ of this action to the coalgebra
$SV$. If the cohomology group $H^{1}(G,V)$ vanishes then the maximal
coalgebras in $SV$ whose elements are fixed by these 2 actions of $G$ are
isomorphic under an isomorphism fixing the elements of $V$.
###### Proof.
We construct an isomorphism $f$ from the maximal coalgebra in the space of
$\sigma_{1}$-invariant elements to the maximal coalgebra in the space of
$\sigma_{2}$-invariant elements by induction on the degree of elements. We
start by taking $f$ to be the identity map on elements of degree at most 1. We
can assume that the 2 actions coincide on elements of degree less than $n$,
and have to find an isomorphism $f$ making them the same on elements of degree
$n$, which we will do by adding elements of $V$ to a basis of the elements of
degree $n$. Suppose that $a$ is an element of degree $n>1$ contained in a
coalgebra of $G$-invariant elements. We want to find $v\in V$ so that
$\sigma_{1}(g)(a+v)=\sigma_{2}(g)(a)+v$
or equivalently
$\sigma_{1}(g)(v)-v=\sigma_{2}(g)(a)-a.$
The right hand side, as a function of $g$, is a 1-coboundary of an element
$a\in SV$, and therefore a 1-cocycle. We show that the right hand side is in
$V$. We have
$\Delta(a)=a\otimes 1+1\otimes a+\sum_{i}b_{i}\otimes c_{i}$
for some elements $b_{i}$ and $c_{i}$ of degrees less than $n$ invariant under
$G$ (for both actions, which coincide on elements of degree less than $n$).
Applying $\sigma_{2}$ we find that
$\Delta(\sigma_{2}(g)a)=\sigma_{2}(g)a\otimes
1+1\otimes\sigma_{2}(g)a+\sum_{i}b_{i}\otimes c_{i}$, so subtracting these two
identities shows that $\sigma_{2}(g)(a)-a$ is a primitive element of
$\operatorname{SV}$ and therefore in $V$. Therefore the right hand side, as a
function of $g$, is a 1-cocycle with values in $V$. The solvability of the
condition for $v$ says exactly that this expression is the coboundary of some
element $v\in V$. In other words the obstruction to finding a suitable $v$ is
exactly an element of the cohomology group $H^{1}(G,V)$, so as we assume this
group vanishes we can always solve for $v$. ∎
###### Example 46
We take $V$ to be $\Gamma\omega SJ\Phi$, and $G$ to be some group acting on
$V$. Then the spaces of classical and quantum actions are coalgebras acted on
by $G$, whose primitive elements can be identified with $V$. If
$H^{1}(G,\Gamma\omega SJ\Phi)$ vanishes, then the maximal $G$-invariant
coalgebra in the coalgebra of classical actions is isomorphic to the maximal
$G$-invariant coalgebra in the coalgebra of quantum actions. So if $L$ is a
$G$-invariant classical Lagrangian, then $e^{L}$ is a $G$-invariant classical
action, so gives a $G$-invariant quantum action. One cannot get a
$G$-invariant quantum action by exponentiating a $G$-invariant quantum
Lagrangian because the space of quantum actions does not in general have a
$G$-invariant product.
###### Example 47
Sometimes the group $G$ only fixes classical Lagrangians up to boundary terms,
in other words the Lagrangian is a $G$-invariant element of $\Gamma\omega
SJ\Phi/D$. In this case one replaces the cohomology group
$H^{1}(G,\Gamma\omega SJ\Phi)$ by $H^{1}(G,\Gamma\omega SJ\Phi/D)$.
The element $e^{iL_{F}}$ lies in the completion of $S\Gamma\omega SJ\Phi$ and
is fixed by the zeroth order part of the BRST operator. So the BRST operator
acts on $e^{iL_{F}}S\Gamma\omega SJ\Phi$.
The groups $H^{1}(G,\Gamma\omega SJ\Phi)\operatorname{and}H^{1}(G,\Gamma\omega
SJ\Phi/D)$ (and their variations for Poincare invariant Lagrangians) for the
BRST operators of gauge theories have been calculated in many cases, at least
for the case of Minkowski space (see for example Barnich, Brandt, and Henneaux
[2]) and are sometimes zero, in which case corresponding invariant quantum
field theories exist.
## References
* [1] Eiichi Abe. Hopf algebras, volume 74 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1980.
* [2] Glenn Barnich, Friedemann Brandt, and Marc Henneaux. Local BRST cohomology in gauge theories. Phys. Rep., 338(5):439–569, 2000.
* [3] I. N. Bernstein. Analytic continuation of generalized functions with respect to a parameter. Funkcional. Anal. i Prilo en., 6(4):26–40, 1972.
* [4] H.-J. Borchers. On structure of the algebra of field operators. Nuovo Cimento (10), 24:214–236, 1962\.
* [5] Alain Connes and Dirk Kreimer. Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys., 210(1):249–273, 2000.
* [6] Pavel Etingof. Note on dimensional regularization. In Quantum fields and strings: a course for mathematicians, Vol. 1 (Princeton, NJ, 1996/1997), pages 597–607. Amer. Math. Soc., Providence, RI, 1999.
* [7] Kazuo Fujikawa. Path-integral measure for gauge-invariant fermion theories. Phys. Rev. Lett., 42(18):1195–1198, Apr 1979.
* [8] A. Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math., (32):361, 1967.
* [9] G. ’t Hooft and M. Veltman. Diagrammar. In Under the spell of the gauge principle, pages 28–173. World Scientific, 1994.
* [10] Lars Hormander. The analysis of linear partial differential operators. I. Classics in Mathematics. Springer-Verlag, Berlin, 2003.
* [11] Dirk Kreimer. On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys., 2(2):303–334, 1998.
* [12] H. Lehmann, K. Symanzik, and W. Zimmermann. On the formulation of quantized field theories. Nuovo Cimento, 1:1425, 1955.
* [13] R. F. Streater and A. S. Wightman. PCT, spin and statistics, and all that. Princeton Landmarks in Physics. Princeton University Press, Princeton, NJ, 2000. Corrected third printing of the 1978 edition.
|
arxiv-papers
| 2010-07-31T22:51:13 |
2024-09-04T02:49:11.974662
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. E. Borcherds",
"submitter": "Richard E. Borcherds",
"url": "https://arxiv.org/abs/1008.0129"
}
|
1008.0187
|
# The Cosmological Constant as a Function of Extrinsic Curvature and Spatial
Curvature
Jin-Zhang Tang111Electronic address:JinzhangTang@pku.edu.cn, Qiang
Xu222Electronic address:xuqiang@pku.edu.cn Department of Physics, and State
Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China
(August 27, 2024
)
###### Abstract
In this paper we suppose that the cosmological constant will change when the
universe expends. For a general consideration, the cosmological constant is
assumed to be a function of scale factor and Hubble constant. According to the
ADM formulation, to the FRW metric, the extrinsic curvature $I$ equals
$-6H^{2}$ and spatial curvature $R$ equals $6k/a^{2}$. Therefore we suppose
cosmological constant is a function of extrinsic curvature and spatial
curvature. We investigate the cosmological evolution of FRW universe in these
models. At last we investigate two particular models which could fit the
observation data about dark energy well. Actually a changeless cosmological
constant is not essential. If when the universe expands, the cosmological
constant changes slowly and gradually flows to a constant, the observation
data about dark energy could also be fitted well by this kind of theory.
###### pacs:
98.80.Cq
## I introduction
Eintein’s general relativity(GR) has been considered as a fundamental theory
of gravity. Even from an effective field theory point of view, it should
describe all large scale gravitational physics, in particular the evolution of
our universe. However, the discovery of dark matter and dark energy from
various observation posed great challenge to the theory. To address the dark
energy issues, the most simplest method is adding a cosmological constant
which is called $\Lambda$CDM model. The cosmological constant problem has been
investigated by many papers such as CC-Weinberg ; CC-carroll ; CC-Peebles and
so on. Many other theories also have been proposed for the cosmic
acceleration, for instance, Quintessencequintessence-01 ; quintessence-02 ;
quintessence-03 , K-essenceK-essence-01 ; K-essence-02 , PhantomPhantom01 ;
Phantom02 , etc. And numerous versions of modification or extension of GR have
been proposed in the last and this century. The $f(R^{(4)})$ theory fR01 ;
fR02 ; fR03 ; fR04 is one of the modifications to GR. In this theory, the
Lagrangian density $f$ is an arbitrary function of $R^{(4)}$. The theory can
explain the cosmic acceleration without introducing a cosmological constant,
even though the theory would be very much restricted by the solar system test.
In this paper, we suppose the cosmological constant will change when the
universe expends. For the scalar factor and Hubble constant are main variable
to characterize a universe, the cosmological constant is supposed to be a
function of them. In the ADM formulation ADM , the metric is
$ds^{2}_{4}=-N^{2}dt^{2}+g_{ij}(dx^{i}-N^{i}dt)(dx^{i}-N^{i}dt).$ (1)
the building block to construct the action is the gauge invariant quantity:
the extrinsic curvature tensor
$K_{ij}=\frac{1}{2N}\left(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}\right),$
(2)
where a dot denotes a derivative with respect $t$, and the 3-dimensional
spatial curvature $R$ comes from $g_{ij}$. In terms of them, the action of GR
with a cosmological constant is
$S_{EH}=\frac{1}{16\pi G}\int
dtd^{3}x\sqrt{g}N\left[K_{ij}K^{ij}-K^{2}+R-2\Lambda\right],$ (3)
where $K=g^{ij}K_{ij}$. And we define extrinsic curvature $I$ as
$I\equiv K_{ij}K^{ij}-K^{2}.$ (4)
The homogeneous and isotropic universe is described by the FRW metric,
$ds^{2}=-dt^{2}+a(t)^{2}\left\\{\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right)\right\\},$
(5)
where $k=0,\pm 1$. To this FRW metric, After some trivial calculations, it is
easy to get the extrinsic curvature $I=-6H^{2}$ and spatial curvature
$R=6k/a^{2}$. Obviously $I$ is proportional to the square of Hubble constant
and $R$ is proportional to the inverse square on scalar factor $a$. Therefore
we suppose the cosmological constant $\Lambda=\Lambda(I,R)$ as a function of
extrinsic curvature $I$ and spatial curvature $R$.
For the calculational convenience, we define $F(I,R)\equiv I+R-2\Lambda(I,R)$
and the action of this theory is
$S_{F}=\frac{1}{16\pi G}\int dtd^{3}x\sqrt{g}NF\left(I,R\right)$ (6)
This theory could be regard as an extension of $f(R^{(4)})$ gravity model. In
the next section, we exhibit the basic equations by varying the action with
respect to the function $N,N_{k},g_{ij}$. Then we investigate the cosmological
evolution of FRW universe in these models. At last, we investigate two
particular models which could fit the observation data about dark energy well.
Actually a changeless cosmological constant is not essential. If the
cosmological constant changes slowly and gradually flows to a constant, the
observation data about dark energy could also be fitted well by this kind of
theory.
## II Basic Equations
Let us start from the action
$S_{F}=\frac{1}{16\pi G}\int dtd^{3}x\sqrt{g}NF\left(I,R\right).$ (7)
Actually the action could be more general if $F$ is an arbitrary function of
“$\hskip 8.61108ptg_{ij},K,K_{ij},\hskip 17.22217pt$
$\nabla_{i}K_{jk},\cdots,\nabla_{i_{1}}\nabla_{i_{2}}\cdots\nabla_{i_{n}}K_{jk},\cdots,R,R_{ij},R_{ijkl},\nabla_{i}R_{jk},\cdots$”.
In this paper we would like to focus on the simplest case
$F=F\left(I,R\right)$. Now we consider the case that there are matters couple
to the gravity. The action should be
$S=S_{F}+S_{m}=\int dtd^{3}x\sqrt{g}N\left[\frac{1}{16\pi
G}F\left(I,R\right)+\mathcal{L}_{m}\right].$ (8)
By varying the action with respect to the function $N,N_{k},g_{ij}$, we get
the Hamiltonian constraint, super-momentum constraint and dynamical equations,
$\displaystyle J_{N}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G}\left[F-2F_{,I}I\right],$ (9) $\displaystyle J^{k}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi G}\nabla_{j}\left[F_{,I}\pi^{jk}\right],$ (10)
$\displaystyle-\frac{1}{2}p^{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
G}\left\\{-\frac{1}{\sqrt{g}N}\partial_{t}\left[\sqrt{g}F_{,I}\pi^{ij}\right]\right.$
(11) $\displaystyle+$
$\displaystyle\left.\frac{1}{N}\nabla_{k}\left[F_{,I}\left(\pi^{ij}N^{k}-\pi^{kj}N^{i}-\pi^{ik}N^{j}\right)\right]-2F_{,I}\left[K^{i}_{m}K^{mj}-\lambda
KK^{ij}\right]\right.$ $\displaystyle-$
$\displaystyle\left.F_{,R}R^{ij}+\left(\nabla^{i}\nabla^{j}-g^{ij}\nabla^{2}\right)\left(NF_{,R}\right)+\frac{1}{2}Fg^{ij}\right\\}.$
$\pi^{ij}$ in above equations is defined as $\pi^{ij}=K^{ij}-\lambda Kg^{ij}$.
And $J_{N},J^{k},p^{ij}$ are defined as
$J_{N}\equiv-\frac{\delta(N\mathcal{L}_{m})}{\delta N};\hskip
8.61108ptJ^{k}\equiv-N\frac{\delta\mathcal{L}_{m}}{\delta N_{k}};\hskip
8.61108ptp^{ij}\equiv\frac{2}{\sqrt{g}}\frac{\delta\left(\sqrt{g}\mathcal{L}_{m}\right)}{\delta
g_{ij}}.$ (12)
If the matter could be taken to be the perfect liquid, then
$\rho_{m}=J_{N},J^{k}=0,p^{ij}=p_{m}g^{ij}.$ (13)
The quantities $\left(J_{N},J^{i},p^{ij}\right)$ should satisfy the
conservation laws,
$\displaystyle\partial_{t}g_{jk}p^{jk}+\frac{2}{\sqrt{g}}\partial_{t}\left(\sqrt{g}J_{N}\right)+\frac{2N_{k}}{\sqrt{g}N}\partial_{t}\left(\sqrt{g}J^{k}\right)=0,$
(14)
$\displaystyle\nabla_{k}p^{ik}-\frac{1}{\sqrt{g}N}\partial_{t}\left(\sqrt{g}J^{i}\right)-\frac{N^{i}}{N}\nabla_{k}J^{k}-\frac{J^{k}}{N}\left(\nabla_{k}N^{i}-\nabla^{i}N_{k}\right)=0.$
(15)
## III Cosmological Models
The homogeneous and isotropic universe is described by the FRW metric,
$ds^{2}=-c^{2}dt^{2}+a(t)^{2}\left\\{\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right)\right\\},$
(16)
where $k=0,\pm 1$. The extrinsic curvature of the spatial slices is
$K_{ij}=\frac{1}{c}\frac{\dot{a}(t)}{a(t)}g_{ij};\hskip
17.22217ptK=\frac{1}{c}\frac{3\dot{a}(t)}{a(t)}.$ (17)
Then $I=-6H^{2}$ where $H\equiv\dot{a}/a$ which is defined as Hubble constant.
The spatial slices are of constant-curvature
$R_{ij}=\frac{2k}{a^{2}(t)}g_{ij};\hskip 17.22217ptR=\frac{6k}{a^{2}(t)}.$
(18)
It can be shown that the momentum constraint (10) is satisfied identically
when $J^{k}=0$ for the perfect fluid. From the Hamiltonian constraint (9) and
the dynamical equation (11), and “$c$” is rescaled to unity, we get two
cosmological equations as
$\displaystyle F+12F_{,I}\left(\frac{\dot{a}}{a}\right)^{2}=16\pi G\rho_{m},$
(19) $\displaystyle
2\left[F_{,I}\frac{\ddot{a}}{a}+\dot{F}_{,I}\frac{\dot{a}}{a}\right]+4F_{,I}\left(\frac{\dot{a}}{a}\right)^{2}-F_{,R}\frac{2k}{a^{2}}+\frac{1}{2}F=-8\pi
Gp_{m},$ (20)
with the energy conservation equation
$\dot{\rho}_{m}+3\frac{\dot{a}}{a}\left(\rho_{m}+p_{m}\right)=0.$ (21)
The FRW equations with the matter and the dark energy could be rewritten as
$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{DE}\right),$ (22) $\displaystyle
2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k}{a^{2}}=-8\pi
G\left(p_{m}+p_{DE}\right).$ (23)
Here we have regarded the nonlinear terms in the equations (19),(20) as the
dark energy effectively. Comparing (22) to (19), (23) to (20), it is easy to
get
$\displaystyle 8\pi
G\rho_{DE}=3\left(1-2F_{,I}\right)\left(\frac{\dot{a}}{a}\right)^{2}+\frac{3k}{a^{2}}-\frac{1}{2}F,$
(24) $\displaystyle 8\pi
Gp_{DE}=2\left(F_{,I}-1\right)\frac{\ddot{a}}{a}+2\dot{F}_{,I}\frac{\dot{a}}{a}+\left(4F_{,I}-1\right)\left(\frac{\dot{a}}{a}\right)^{2}-\left(F_{,R}+\frac{1}{2}\right)\frac{2k}{a^{2}}+\frac{1}{2}F.$
(25)
So the equation of state $\omega_{DE}$ is given by
$\omega_{DE}=-1+\frac{2\left(F_{,I}-1\right)\dot{H}+2\dot{F}_{,I}H-\left(F_{,R}-1\right)\left(2k/a^{2}\right)}{3\left(1-2F_{,I}\right)H^{2}+\left(3k/a^{2}\right)-\left(1/2\right)F}.$
(26)
From the data of $WMAP+BAO+SN^{b}$, the constraints on $\omega_{DE}$ are
$\omega_{DE0}=-0.999^{+0.057}_{-0.056}$ observation01 . We now define a
deceleration parameter
$q=-\frac{\ddot{a}a}{\dot{a}^{2}}.$ (27)
From equations (22)and(23), we get
$q_{0}=-\frac{\rho_{DE0}-(1/2)\rho_{m0}}{\rho_{m0}+\rho_{DE0}-3k/(8\pi
Ga_{0}^{2})}=-\Omega_{DE0}+\frac{\Omega_{m0}}{2}\approx-0.592,$ (28)
here we have used “$\Omega_{DE0}=0.728,\Omega_{m0}=0.272$” observation01 from
the data of $WMAP+BAO+H_{0}$. The critical density of the universe if defined
as $\rho_{c}\equiv 3H^{2}/8\pi G$. From (22) it is easy to get
$\rho_{c0}=\frac{3H_{0}^{2}}{8\pi G}=\rho_{m0}+\rho_{DE0}-\frac{3k}{8\pi
Ga_{0}^{2}}.$ (29)
From above, The curvature density is defined as $\rho_{k}\equiv-3k/(8\pi
Ga^{2})$. Now we could define
$\Omega_{k}=\frac{\rho_{k}}{\rho_{c}}=\frac{-k}{a^{2}H^{2}}.$ (30)
The data of $WMAP+BAO+SN^{b}$ give the constraints on curvature density,
$\Omega_{k0}=-0.0057^{+0.0067}_{-0.0068}$ observation01 . Obviously, It is
probable that $k>0$ and the universe is close. If the universe is accelerating
as power-law that $a(t)\propto t^{y}$ today, it is easy to get
$q_{0}=y(1-y)/y^{2}=-0.592$ and $y=2.451$.
We consider the simple case that
$-2\Lambda(I,R)=g(I)+f(R).$ (31)
From (24) and (25), it is easy to get
$\displaystyle 8\pi G\rho_{DE}=g_{,I}I-\frac{1}{2}g(I)-\frac{1}{2}f(R),$ (32)
$\displaystyle 8\pi
Gp_{DE}=2\left(g_{,I}+2g_{,II}I\right)\dot{H}-g_{,I}I-f_{,R}\frac{2k}{a^{2}}+\frac{1}{2}g(I)+\frac{1}{2}f(R).$
(33)
The equation of state is
$\omega_{DE}=-1+\frac{2\left(g_{,I}+2g_{,II}I\right)\dot{H}-f_{,R}(2k/a^{2})}{g_{,I}I-(1/2)g(I)-(1/2)f(R)}.$
(34)
Now we investigate a concrete model $-2\Lambda(I,R)=\alpha e^{\beta R}$, here
$R=6k/a^{2}$ and we assume $k>0$. The state equation of this model is
$\omega_{DE}=-1+\frac{2}{3}\beta R.$ (35)
It is easy to get $-0.083<\beta R_{0}<0.087$ from
$\omega_{DE0}=-0.999^{+0.057}_{-0.056}$ observation01 . The coefficient
$\alpha$ can be fixed from the equation $8\pi G\rho_{DE0}=(-1/2)\alpha
e^{\beta R_{0}}$. From (21) and (22), the evolution equation of $a$ is
$\displaystyle\left(\frac{H}{H_{0}}\right)^{2}$ $\displaystyle=$
$\displaystyle\Omega_{k0}\left(\frac{a_{0}}{a}\right)^{2}+\Omega_{DE0}e^{\beta
R_{0}\\{(a_{0}/a)^{2}-1\\}}+\Omega_{m0}\left(\frac{a_{0}}{a}\right)^{3}+\left(1-\Omega_{DE0}-\Omega_{m0}-\Omega_{k0}\right)\left(\frac{a_{0}}{a}\right)^{4},$
where “$\Omega_{DE0}=0.73,\Omega_{m0}=0.27,\Omega_{r0}=0.014$ ”observation01 .
The details about the universe evolution are shown in Fig. 1 and 2. Obviously
in Fig. 1, the evolutions of deceleration parameter $q$ for $\beta
R_{0}=-0.01$ or $0.01$ are nearly identical to the case $\beta R_{0}=0$ which
corresponds to the $\Lambda$CDM model. Fig. 2 shows that the state function
$\omega_{DE}$ flow to $-1$ quickly for the cases $\beta R_{0}=-0.01$ or $\beta
R_{0}=0.01$ when $\omega_{DE}$ identically equal to $-1$ for $\beta R_{0}=0$.
Figure 1: The deceleration parameter $q=-\ddot{a}a/\dot{a}^{2}$ as a function
of $a$. Figure 2: The parameter $\omega_{DE}$ as a function of $a$.
Another specific model is $-2\Lambda(I,R)=\gamma e^{\eta I}$. The
corresponding state equation is
$\omega_{DE}=-1+\frac{2\eta I}{3}\frac{1+2\eta I}{1-2\eta
I}\frac{\dot{H}}{H^{2}}.$ (36)
When $\omega_{DE0}=-1$, $\eta$ has two solutions $\eta=0$ and $\eta
I_{0}=-1/2$. When $\eta=0$, it just is a $\Lambda CDM$ model. The attention
will be put on the second case $\eta I_{0}=-1/2$. The coefficient $\gamma$
could be fixed from
$8\pi G\rho_{DE0}=\gamma\eta e^{\eta I_{0}}I_{0}-\frac{1}{2}\gamma e^{\eta
I_{0}},$ (37)
and $\gamma=-3e^{1/2}\Omega_{DE0}H_{0}^{2}$. The evolution equation of $a$ is
$\displaystyle\left(\frac{H}{H_{0}}\right)^{2}$ $\displaystyle=$
$\displaystyle\Omega_{k0}\left(\frac{a_{0}}{a}\right)^{2}+\frac{1}{2}\Omega_{DE0}\left[1+\left(\frac{H}{H_{0}}\right)^{2}\right]e^{\frac{1}{2}[1-(H/H_{0})^{2}]}+\Omega_{m0}\left(\frac{a_{0}}{a}\right)^{3}+\left(1-\Omega_{DE0}-\Omega_{m0}-\Omega_{k0}\right)\left(\frac{a_{0}}{a}\right)^{4}.$
Similar to the discussion above, Fig. 3 is the evolution of deceleration
parameter $q$ which is similar to $\Lambda$CDM model. Fig. 4 shows that the
state function $\omega_{DE}$ flow to $-1$ quickly.
Figure 3: The deceleration parameter $q=-\ddot{a}a/\dot{a}^{2}$ as a function
of $a$. Figure 4: The parameter $\omega_{DE}$ as a function of $a$.
The two models discussed above could be regard as the modification to the
$\Lambda$CDM model. To address the dark energy issues, a changeless
cosmological constant is not essential. If the cosmological constant changes
slowly and gradually flows to a constant, the observation data about dark
energy could also be fitted well by this kind of theory.
## Acknowledgments
The work was partially supported by NSFC Grant No. 10775002, 10975005 and
RFDP. We would like to thank Professor Bin Chen very much for some useful
suggestions and his works on the modification of this paper.
## References
* (1) R.L. Arnowitt, S. Deser and C.W. Misner,_The dynamics of general relativity_ ,“Gravitation:an introduction to current research”, Louis Witten ed.(Wilew 1962),chapter 7,pp 227-265, [arXiv:gr-qc/0405109].
* (2) S. Weinberg, “The cosmological constant problem”, Rev. Mod. Phys. 61, 1(1989).
* (3) Sean M. Carroll, “The Cosmological Constant”, LivingRev. Rel. 4:1,2001, [arXiv:astro-ph/0004075v2].
* (4) P. J. E. Peebles, Bharat Ratra, “The Cosmological Constant and Dark Energy ”, Rev. Mod. Phys. 75: 559-606(2003), [arXiv:astro-ph/0207347v2].
* (5) Ivaylo Zlatev, Limin Wang, Paul J. Steinhardt, “Quintessence, Cosmic Coincidence, and the Cosmological Constant ”, Phys. Rev. Lett. 82:896-899(1999),[arXiv:astro-ph/9807002v2].
* (6) A. Yu. Kamenshchik, U. Moschella, V. Pasquier, “ An alternative to quintessence”, Phys. Lett. B 511:265-268 (2001), [arXiv:gr-qc/0103004v2].
* (7) Sean M. Carroll, “Quintessence and the Rest of the World”, Phys. Rev. Lett. 81: 3067-3070(1998), [arXiv:astro-ph/9806099v2].
* (8) C. Armendariz-Picon, V. Mukhanov, Paul J. Steinhardt, “Essentials of k-essence”, Phys. Rev. D 63:103510(2001), [arXiv:astro-ph/0006373v1].
* (9) Takeshi Chiba, “Tracking K-essence”, Phys. Rev. D 66(2002), [arXiv:astro-ph/0206298v2].
* (10) R.R. Caldwell, Phys.Lett.B 545:23-29(2002), [arXiv:astro-ph/9908168v2].
* (11) Robert R. Caldwell, Marc Kamionkowski, Nevin N. Weinberg, “Phantom Energy and Cosmic Doomsday”, Phys. Rev. Lett. 91(2003) 071301, [arXiv:astro-ph/0302506v1].
* (12) S. Nojiri and S. D. Odintsov, eConf C0602061, 06 (2006)[Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)][arXiv:hep-th/0601213].
* (13) T. P. Sotiriou and V. Faraoni, arXiv:0805.1726 [gr-qc].
* (14) A. de Felice and S. Tsujikawa, arXiv:1002.4928[hep-th].
* (15) Sean M. Carroll, Vikram Duvvuri, Mark Trodden, Michael S. Turner, “Is Cosmic Speed-Up Due to New Gravitational Physics?”, Phys. Rev. D 70, 043528 (2004)[arXiv:astro-ph/0306438].
* (16) E. Komatsu _et al._ , “Seven-Year Wilkinson Microwave Anisotropy Probe(WMAP) Observations: Cosmological Interpretation” , [arXiv:astro-ph/1001.4538v2].
|
arxiv-papers
| 2010-08-01T16:47:13 |
2024-09-04T02:49:11.986442
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jin-Zhang Tang, Qiang Xu",
"submitter": "Jinzhang Tang",
"url": "https://arxiv.org/abs/1008.0187"
}
|
1008.0294
|
# Can one identify the intrinsic structure of the yrast states in 48Cr after
the backbending?
Zao-Chun Gao1 Mihai Horoi2 Y. S. Chen1 Y. J. Chen1 Tuya3 1China Institute of
Atomic Energy P.O. Box 275-10, Beijing 102413, China
2Department of Physics, Central Michigan University, Mount Pleasant, Michigan
48859, USA
3College of Physics Science and Technology, Shenyang Normal University,
ShenYang, 110034, China
###### Abstract
The backbending phenomenon in 48Cr has been investigated using the recently
developed Projected Configuration Interaction (PCI) method, in which the
deformed intrinsic states are directly associated with shell model (SM)
wavefunctions. Two previous explanations, (i) $K=0$ band crossing, and (ii)
$K=2$ band crossing have been reinvestigated using PCI, and it was found that
both explanations can successfully reproduce the experimental backbending. The
PCI wavefunctions in the pictures of $K=0$ band crossing and $K=2$ band
crossing are highly overlapped. We conclude that there are no unique intrinsic
states associated with the yrast states after backbending in 48Cr.
###### pacs:
21.60.Ev, 21.60.Cs, 21.10.Re, 27.40.+z
## I Introduction
The backbending of 48Cr has been observed more than 10 years ago Cameron94 ;
Cameron96 , but its interpretation remains controversial and challenges the
existing nuclear models. Shell Model (SM) calculations have reproduced very
well the yrast states of 48Cr Caurier94 ; Caurier95 , but it is difficult to
provide the physical insight because the laboratory frame wavefunction doesn’t
imply any information associated with the deformed intrinsic structure. The
cranked Hartree Fock Bogoliubov(CHFB) method is a complementary theory chfb
often used to analyze the deformed intrinsic states. According to one CHFB
analysisCaurier95 , 48Cr is an axial rotor up to the backbending, after that
the system changes to a spherical shape. An alternative and more detailed CHFB
analysis Tanaka98 shows that the backbending of the 48Cr is not associated
with the single particle level crossing and that the intrinsic configuration
remains unchanged.
The Projected Shell Model (PSM) Hara95 ; Chen01 is an alternative technique
that mixes the best intrinsic shell model configuration with other associated
particle-hole configurations. A PSM analysis Hara99 indicates that the
backbending in 48Cr is due to a band crossing involving an excited
4-quasiparticle (qp) band with $K=2$, which represents a configuration of
broken neutron and proton pairs. Unfortunately, such an explanation is not
confirmed by recent SM Caurier95 ; Bran05 and CHFB Caurier95 ; Tanaka98
analyses. In addition, the same PSM Ref. Hara99 , uses the Generator
Coordinate Method (GCM) that provides the picture of a spherical band
crossing, which in some sense is similar to the result of CHFB. Obviously, the
above apparently conflicting explanations coming from different models need to
be reinvestigated.
In the present paper, the backbending in 48Cr is investigated in the framework
of the newly developed method called Projected Configuration Interaction
(PCI)Gao09a ; Gao09b . The PCI basis is built from a set of Slater
determinants (SD). Those SDs may have different shapes, including the
spherical shape. Hence, the nuclear states with different intrinsic shapes can
be mixed by the residual interaction. By using the same SM Hamiltonian, the
PCI results can be directly compared with those of full Shell Model
calculations. Moreover, PCI uses deformed single particle bases and therefore,
the physics insight of the results can be clearly analyzed. Different PCI
bases can be built in such a way that they reflect the nature of the
intrinsics states found in previous studies, such as CHFB (or GCM) and PSM.
PCI wavefunctions were shown to be very good approximations to those of full
SM, and they can be obtained using different bases. Thereafter, overlaps among
PCI wavefunctions can be calculated and analyzed to determine the validity of
various explanations. These features suggest that PCI could shed new light on
the interesting phenomenon of backbending in 48Cr. Other models using similar
techniques includes the family of VAMPIR vampir , and the quantum Monte Carlo
diagonalization (QMCD) methodqmcd .
The paper is organized as follows: Section II provides a short introduction
into the basics concepts used by the PCI method that would be required for an
understanding of the arguments used later in the analysis. Section III is
devoted to the analysis of the contribution of different intrinsics set of
states to the backbending in 48Cr. Section IV summarizes the conclusion of our
study.
## II Basic concepts of the PCI method
For completeness we give here a brief introduction of the PCI method (see
Refs. Gao09a ; Gao09b for more details). The deformed single particle (s.p.)
states need to be generated either by HF calculation or from a deformed s.p.
Hamiltonian. For simplicity, here we take the latter approach, and the s.p.
Hamiltonian can be written as
$\displaystyle
H_{\text{s.p.}}=h_{\text{sph}}-\frac{2}{3}\epsilon_{2}\hbar\omega_{0}\rho^{2}P_{2}+\epsilon_{4}\hbar\omega_{0}\rho^{2}P_{4},$
(1)
where $h_{\text{sph}}=\sum_{i}e_{i}c^{\dagger}_{i}c_{i}$ is the spherical s.p.
Hamiltonian assumed to have the same eigenfunctions as the spherical harmonic
oscillator, $e_{i}$ energies are properly adjusted such that the SD with
lowest energy is close to the HF vacuum. For the $pf$-shell we use
$e_{f_{7/2}}=0.0$MeV, $e_{p_{3/2}}=4.5$MeV, $e_{f_{5/2}}=5.0$MeV and
$e_{p_{1/2}}=6.0$MeV. In Eq. (1) $\epsilon_{2}$, $\epsilon_{4}$ are the
quadrupole and hexadecapole deformation parameters, $P_{l}$ are Legendre
polynomials, $\rho=r/b$, and we take $b=1.01A^{1/6}$ for the harmonic
oscillator parameter Caurier94 ; Caurier95 .
The Slater determinants can be built with deformed s.p. states. Following our
previous papers Gao09a ; Gao09b , the general structure of the PCI basis can
be written as
$\displaystyle\left\\{\begin{matrix}0{\text{p}}-0{\text{h}},&\>n{\text{p}}-n{\text{h}}\\\
|\kappa_{1},0\rangle,&|\kappa_{1},j\rangle,\cdots,\\\
|\kappa_{2},0\rangle,&|\kappa_{2},j\rangle,\cdots,\\\ {2}\\\
|\kappa_{N},0\rangle,&|\kappa_{N},j\rangle,\cdots\end{matrix}\right\\},$ (2)
where $|\kappa_{i},0\rangle$ ($i=1,...N$) is an optimal Gao09b set of
starting states having different deformations. Assuming that these
$|\kappa,0\rangle$ are found (in what follows we skip the subscript $i$ to
keep notation short), a number of relative $n$p-$n$h SDs, $|\kappa,j\rangle$,
on top of each $|\kappa,0\rangle$ are added to the SD basis selected the
constraint Gao09a
$\displaystyle\Delta
E=\frac{1}{2}(E_{0}-E_{j}+\sqrt{(E_{0}-E_{j})^{2}+4|V|^{2}})\geq
E_{\text{cut}}.$ (3)
Here $E_{0}=\langle\kappa,0|H|\kappa,0\rangle$,
$E_{j}=\langle\kappa,j|H|\kappa,j\rangle$ and
$V=\langle\kappa,0|H|\kappa,j\rangle$. The PCI basis is then obtained by
projecting the selected SDs onto good angular momentum. The wavefunctions, as
well as the energy levels, are obtained by solving the following generalized
eigenvalue equation:
$\displaystyle\sum_{\kappa^{\prime}}(H_{\kappa\kappa^{\prime}}^{I}-E^{I}N_{\kappa\kappa^{\prime}}^{I})f^{I}_{\kappa^{\prime}}=0.$
(4)
Here $H_{\kappa\kappa^{\prime}}^{I}$ and $N_{\kappa\kappa^{\prime}}^{I}$ are
given by
$\displaystyle
H_{\kappa\kappa^{\prime}}^{I}=\langle\kappa|HP^{I}_{KK^{\prime}}|\kappa^{\prime}\rangle,~{}~{}N_{\kappa\kappa^{\prime}}^{I}$
$\displaystyle=$
$\displaystyle\langle\kappa|P^{I}_{KK^{\prime}}|\kappa^{\prime}\rangle,$ (5)
where $P^{I}_{KK^{\prime}}$ is the angular momentum projection operator, and
$H$ is the shell model Hamiltonian. In this study we take the KB3 interaction
Poves81 , which has been used by Caurier et al in their shell model
calculations of 48Cr Caurier95 . However, results similar to those reported
here are provided by other interactions, such as KB3G kb3g or GXPF1A gxpf1a .
## III The PCI analysis of the backbending in 48Cr
To get some insight into the structure of the states contributing to the
backbending in 48Cr we chose two basic $|\kappa,0\rangle$ SDs. The first
$|\kappa_{1},0\rangle$ is a $K=0$ configuration with all 8 valence nucleons
occuping the $|\Omega|=1/2(1f_{7/2})$ orbits and the $|\Omega|=3/2(1f_{7/2})$
orbits, as shown in Fig. 1a. The deformation of $|\kappa_{1},0\rangle$ is
given by $\epsilon_{2}=0.19$ and $\epsilon_{4}=-0.05$. This deformation was
obtained by determining the minimum of the energy surface of $\langle
K=0|H_{\text{KB3}}|K=0\rangle$ as a function of $\epsilon_{2}$ and
$\epsilon_{4}$, (See the dashed line in Fig. 1b). A minimum energy of
$\langle\kappa_{1},0|H_{\text{KB3}}|\kappa_{1},0\rangle=-28.296$ MeV was
found, which is close to the HF energy $-28.423$ MeV. $|\kappa_{1},0\rangle$
is believed to be responsible for the low-spin yrast states before the
backbending, and the corresponding PCI basis that includes the $n$p-$n$h
states selected by Eq. (3) is denoted as ‘g.s.’, and it is shown in Table 1.
Table 1: PCI bases used for the backbending study in 48Cr. $K=0$ and $K=2$ configurations are shown in Fig. 1. Basis | | g.s. | | | A | | | B | | | C |
---|---|---|---|---|---|---|---|---|---|---|---|---
| $K$ | $\epsilon_{2}$ | $\epsilon_{4}$ | $K$ | $\epsilon_{2}$ | $\epsilon_{4}$ | $K$ | $\epsilon_{2}$ | $\epsilon_{4}$ | $K$ | $\epsilon_{2}$ | $\epsilon_{4}$
$|\kappa,0\rangle$ | 0 | 0.19 | $-0.05$ | 0 | 0.00 | 0.00 | 2 | 0.00 | 0.00 | 2 | 0.19 | $-0.05$
$N_{\kappa}$ | | 143 | | | 192 | | | 186 | | | 149 |
Figure 1: (color online) (a) The $K=0$ and $K=2$ configurations used in the
present calculations of 48Cr. All 4 levels are from $1f_{7/2}$ subshell. (b)
Potential energy curves of the $K=0$ configuration in (a) as functions of
$\epsilon_{2}$. The solid lines show the projected energies and the dashed
line, unprojected energy. $\epsilon_{4}$ was chosen to minimize the potential
energy for each $\epsilon_{2}$. KB3 interaction was used.
The second basic SD, $|\kappa_{2},0\rangle$, is chosen to describe the high
spin states in 48Cr, after the backbending. According to the previous studies
mentioned in the introduction, there are at least two candidate configurations
for $|\kappa_{2},0\rangle$. The first, suggested by the CHFB calculations,
according to which the backbending in 48Cr can be explained without a band
crossing Tanaka98 , but by a shape changes from well deformation to spherical
Caurier95 . Our PCI calculations seem to be in agreement with this
interpretation. The projected energy curves for the $K=0$ configuration at
each even-spin are shown in Fig. 1b (solid lines). One can see that the
deformation at minimum decreases as the spin increases. Guided by this result,
one can establish a possible $|\kappa_{2},0\rangle$, which has the same
configuration as $|\kappa_{1},0\rangle$, but whose shape is almost spherical.
Such choice of $|\kappa_{2},0\rangle$ SD would be consistent with the GCM
interpretation, in which HF vacua with different deformations are included in
the basis. However, in PCI one can further include particle-hole excitations
on top of each $|\kappa,0\rangle$. On should recall that in the PSM
interpretation Hara99 the backbending in 48Cr is caused by a $K=2$ band
crossing. Therefore, PCI seem to be ideally suited to include and analyze
another possible $|\kappa_{2},0\rangle$ with $K=2$. Its configuration is also
shown in Fig. 1a, and its structure can be chosen either spherical (see column
B in Table 1) or of the same deformation as that of $|\kappa_{1},0\rangle$
favored by the PSM approach (see column C in Table 1). In an attempt to find
the optimal structure of yrast states in 48Cr after the backbending, we
considered all three possibilities of $|\kappa_{2},0\rangle$, labeled with
‘A’, ‘B’ and ‘C’ in Table 1.
The $n$p-$n$h $|\kappa,j\rangle$ SDs built on top of each $|\kappa,0\rangle$
are selected by setting $E_{\text{cut}}=0.5$ keV in Eq. (3). Consequently, the
number of selected $|\kappa_{1},j\rangle$ is 142, and those of
$|\kappa_{2},j\rangle$ for A, B and C are 191, 185 and 148, respectively.
Adding the $|\kappa,0\rangle$ itself, the total number of the selected SDs,
$N_{\kappa}$, for each $|\kappa,0\rangle$ is listed in Table 1.
Figure 2: (color online) (a) PCI energies $E(I)$ as functions of spin for the
bases listed in Table 1. (b) Overlaps among the PCI wavefunctions for bases A,
B, and C. Figure 3: (color online) (a) $E2$ transition energies $E(I)-E(I-2)$
vs spin $I$. Experimental data are taken from Ref. Cameron96 , and the result
of the Full SM from Ref. Caurier95 . (b) Overlaps among the PCI wavefunctions
with basis sets g.s.+A, g.s.+B and g.s.+C.
These PCI bases were used to calculate all even-spin states from $I=0$ to
$I=16$. The calculated lowest energies with basis g.s., A, B and C as
functions of spin are shown in Fig. 2(a). The energies calculated with basis
g.s. alone increases very smoothly with the spin, without exhibiting any
backbending. For the bases A, B and C, all the corresponding energy curves
cross that of the g.s. band after spin $I=10$. To get a better understanding
of the relation among those bases, we calculated the overlaps between
corresponding wavefunctions. Let’s denote the PCI wavefunctions as
$|\Psi_{\kappa}(I)\rangle$, where $\kappa$ refers to certain basis or
combination. For instance, $\kappa$ can be ‘g.s.’, ‘A’, or ‘g.s.+A’, etc.
Overlaps of $|\langle\Psi_{A}(I)|\Psi_{B}(I)\rangle|$,
$|\langle\Psi_{A}(I)|\Psi_{C}(I)\rangle|$ and
$|\langle\Psi_{B}(I)|\Psi_{C}(I)\rangle|$ as functions of $I$ are plotted in
Fig. 2(b). The surprising result is that these overlaps are unexpectedly very
large, which means quite different intrinsic bases would generate almost the
same wavefunctions after the angular momentum projection. In particular, one
should note the large overlap $|\langle\Psi_{A}(I)|\Psi_{B}(I)\rangle|\approx
1$ at $I>10$, while without the angular momentum projection bases A and B are
strictly orthogonal due to different $K$ values .
The $E_{\gamma}(I)=E(I)-E(I-2)$ energies calculated using combinations of
bases, g.s.+A, g.s.+B and g.s.+C are shown in Fig. 3(a). The backbending
phenomenon seems to be easily reproduced by all those bases. As already
discussed, the g.s.+A basis is qualitatively similar to that used in the CHFB
and GCM analyses, while the g.s.+C basis follows the scenario proposed by the
PSM analysis. Furthermore, as shown in Fig. 3(b), the overlaps
$|\langle\Psi_{g.s.+A(I)}|\Psi_{g.s.+C}(I)\rangle|$ are very large, at least
$92\%$. Therefore, we come to the conclusion that the apparently contradictory
CHFB and PSM explanations, actually look qualitatively equivalent one to each
other. One should note, however, that the backbending described with the
g.s.+C basis is quantitatively not as good as that descibed with the g.s.+A
basis. The reason could be the large deformation of C basis. Changing the
deformation of C basis to spherical, one gets the g.s.+B basis, and the result
is improved. This feature supports the idea that the shape of 48Cr reduces
after backbending, as has been pointed out in Ref. Caurier95 ; Tanaka98 .
Notice that the results obtained with bases g.s.+A and g.s.+B are almost
identical, although the intrinsic bases A and B have a completely different
structure. Our calculations show that there are also other intrinsic bases
with $K$ different of 0 and 2 that can reproduce the backbending, and whose
corresponding wavefunctions are almost equivalent to the basis g.s.+A or
g.s.+B when projected on good angular momenta. In other words, one can not
find a unique intrinsic state for the yrast states in 48Cr for $I=12-16$. One
can get some insight into the apparent irrelevance of the intrinsic structure
at high spins by analyzing the case of $I=16$, which is the band termination
state. In the space of $\pi 1f^{4}_{7/2}\nu 1f^{4}_{7/2}$, there is only one
SD that reaches the maximum $K=16$, showing that only one $I=16$ state can be
constructed from that space. On the other hand many $\pi 1f^{4}_{7/2}\nu
1f^{4}_{7/2}$ SDs with various $K$ values can be projected onto good $I$.
Therefore, we have many projected states with $I=16$, and our calculations
prove that they are exactly identical and the projected energy is $-18.342$
MeV with KB3 interaction. Therefore, one can use any one of the $\pi
1f^{4}_{7/2}\nu 1f^{4}_{7/2}$ SDs to reproduce the state of band termination.
Figure 4: (color online) (a) Yrast state energies vs spin obtained with PCI
(open symbols) and Full SM(solid symbols). (b) The BE2 values with the same
wavefunctions as in (a). Result of the Full SM are taken from Ref. Caurier94 .
Figure 5: (color online) Overlaps O(I) in Eq.6 between the projected
intrinsic configurations with $K=0$ and $K=2$ shown in Fig.1a for spherical
and deformed ($\epsilon_{2}=0.19$ and $\epsilon_{4}=-0.05$) cases.
Fig. 3 shows good comparison of the $E2$ transition energies obtained with the
particular choice of bases described above and those of the full SM
calculations. One would like to further compare and validate the structure of
different approaches with the full SM results. It is difficult to calculate
the overlap between PCI and full SM wavefunctions because the large number of
norm matrix elements, $N_{\text{PCI}}\times N_{\text{SM}}$ that has to be
calculated (see Eq. (5)). Here $N_{\text{PCI}}$ and $N_{\text{SM}}$ are
dimensions of the PCI and full (M-scheme) SM spaces, where $N_{\text{SM}}$ is
a very large number. However, the PCI approximation can be further checked by
comparing the energies as well as the BE2 transition probabilities with those
of the full SM calculations. The results are shown in Fig. 4. The PCI energies
lie 500 keV on average above the exact values, but they were calculated using
less than 400 SDs (see Table. 1), while in full SM calculations the number of
the SDs used is 1963461. For the B(E2) calculations we used the ‘canonical’
effective charges, i.e. $1.5e$ for protons and $0.5e$ for neutrons, which are
the same as those used in Refs. Caurier94 ; Caurier95 . The BE2 values
obtained with PCI are very close to the exact ones, except those for $I=6,8$
and 10, which are slightly larger. That is very likely due to the fixed
deformation we used for the selected ‘g.s’ basis. As shown in Fig. 1b, the
deformation reduces gradually as the spin increases, conclusion also supported
by the CHFB Caurier95 and the cranked Nilsson-Strutinsky (CNS) Juodagalvis06
calculations. Therefore we overestimated the BE2 values at $I=6,8$ and 10.
On a final note the intrinsic states play a key role in studying the physics
of the nuclear system. Even if they are orthogonal, their overlap can be quite
large after the angular momentum projection. Here is a simple example. The
$K=0$ and $K=2$ configurations in Fig. 1a were projected on to good angular
momentum assuming spherical and deformed ($\epsilon_{2}=0.19$ and
$\epsilon_{4}=-0.05$) shapes, respectively. The overlap
$\displaystyle O(I)=\frac{\langle K=0|P^{I}_{02}|K=2\rangle}{\sqrt{\langle
K=0|P^{I}_{00}|K=0\rangle\langle K=2|P^{I}_{22}|K=2\rangle}}$ (6)
was calculated and plotted in Fig. 5. One can see that the deformed overlap
$O(I)$ is always smaller than the spherical one, indicating that the deformed
intrinsic states can be more clearly defined than the spherical ones. In the
low-spin region the overlaps $O(I)$ are small, and the two intrinsic
configurations can be easily distinguished. But at high spin, $O(I)$ are much
larger for both spherical and deformed cases, which means that the intrinsic
states can not be clearly identified, at least for the case of 48Cr.
## IV Summary
In summary, the backbending in 48Cr has been studied with a recently developed
Projected Configuration Interaction (PCI) method. PCI uses the same realistic
Hamiltonians and valence spaces as the SM calculations, but only a set of
properly selected SDs with different deformations and associated $n$p-$n$h
configurations. The backbending in 48Cr has been reproduced by using various
PCI bases, carefully selected to reflect the physics of the intrinsic states
found by the CHFB (GCM) and PSM analyses of this case. Using the PCI
capabilities of mixing these bases we show for the first time that the
backbending pictures proposed by the CHFB and PSM methods are qualitatively
equivalent. Our analysis supports the conclusion that there is no unique
intrinsic state for spins larger than 10 in 48Cr.
This work is supported by the NSF of China Contract No. 10775182. Z.G. and
M.H. acknowledge support from the US DOE UNEDF Grant No. DE-FC02-09ER41584.
M.H. acknowledges support from NSF Grant No. PHY-0758099. Y.S.C. acknowledges
support from the MSBRDP of China under Contract No. 2007CB815003.
## References
* (1) J. A. Cameron et al., Phys. Rev. C 49, 1347 (1994).
* (2) J. A. Cameron et al., Phys. Lett. B 387, 266(1996).
* (3) E. Caurier et al., Phys. Rev. C 50 225(1994).
* (4) E. Caurier et al., Phys. Rev. Lett.75, 2466 (1995).
* (5) P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, Berlin,1980).
* (6) T. Tanaka, K. Iwasawa, and F. Sakata, Phys. Rev. C 58, 2765(1998).
* (7) K. Hara and Y. Sun, Int. J. Mod. Phys. E4,637(1995).
* (8) Y. S. Chen and Z. C. Gao, Phys. Rev. C 63 014314(2000).
* (9) K. Hara, Y. Sun, and T. Mizusaki, Phys. Rev. Lett. 83, 1922(1999).
* (10) F. Brandolini and C. A. Ur, Phys. Rev. C 71, 054316(2005).
* (11) Zao-Chun Gao and Mihai Horoi, Phys. Rev. C79, 014311 (2009).
* (12) Zao-Chun Gao, Mihai Horoi and Y. S. Chen. Phys. Rev. C80, 034325 (2009).
* (13) K. W. Schmid, Prog. Part. Nucl. Phys. 52, 565 (2004).
* (14) M. Honma, T. Mizusaki, and T. Otsuka, Phys. Rev. Lett. 77, 3315 (1996).
* (15) A. Poves and A. P. Zuker, Phys. Rep. 71, 141 (1981).
* (16) A. Poves et al., Nucl. Phys. A 694, 157 (2001).
* (17) M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Eur. Phys. J. A 25, 499 (2005).
* (18) A. Juodagalvis, I. Ragnarsson, and S.Åberg, Phys. Rev. C73 044327(2006)
|
arxiv-papers
| 2010-08-02T13:03:48 |
2024-09-04T02:49:11.992838
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zao-Chun Gao, Mihai Horoi, Y. S. Chen, Y. J. Chen and Tuya",
"submitter": "Zaochun Gao",
"url": "https://arxiv.org/abs/1008.0294"
}
|
1008.0359
|
# Many-body calculations of low-energy eigenstates in magnetic and periodic
systems with self-healing diffusion Monte Carlo: steps beyond the fixed phase
Fernando Agustín Reboredo Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA
###### Abstract
The self-healing diffusion Monte Carlo algorithm (SHDMC) [Reboredo, Hood and
Kent, Phys. Rev. B 79, 195117 (2009); Reboredo, ibid. 80, 125110 (2009)] is
extended to study the ground and excited states of magnetic and periodic
systems. The method converges to exact eigenstates as the statistical data
collected increases if the wave function is sufficiently flexible. It is shown
that the wave functions of complex anti-symmetric eigen-states can be written
as the product of an anti-symmetric real factor and a symmetric phase factor.
The dimensionality of the nodal surface is dependent on whether phase is a
scalar function or not. A recursive optimization algorithm is derived from the
time evolution of the mixed probability density, which is given by an ensemble
of electronic configurations (walkers) with complex weight. This complex
weight allows the amplitude of the fixed-node wave function to move away from
the trial wave function phase. This novel approach is both a generalization of
SHDMC and the fixed-phase approximation [Ortiz, Ceperley and Martin, Phys Rev.
Lett. 71, 2777 (1993)]. When used recursively it simultaneously improves the
node and the phase. The algorithm is demonstrated to converge to nearly exact
solutions of model systems with periodic boundary conditions or applied
magnetic fields. The computational cost is proportional to the number of
independent degrees of freedom of the phase. The method is applied to obtain
low-energy excitations of Hamiltonians with magnetic field or periodic
boundary conditions. The method is used to optimize wave functions with
twisted boundary conditions, which are included in a many-body Bloch phase.
The potential applications of this new method to study periodic, magnetic, and
complex Hamiltonians are discussed.
###### pacs:
02.70.Ss,02.70.Tt
## I Introduction
Following the basic prescriptions of quantum mechanics, one could potentially
calculate any physical quantity. Finding the solutions of the Schrödinger
equation, the wave functions and the associated energies is all that it is
required. However, the computational cost of obtaining the solutions of many-
body problems is well known to increase exponentially with the number of
particles. Minimizing this exponential cost by either improved algorithms or
an insightful approximation is the central paradigm of condensed matter
theory.
Many physical quantities (observables) are only functionals of the probability
density: the square of the modulus of the wave function. Some other
observables, like the ground-state energy, are only functionals of the
electronic density hohenberg . However, very important quantities, such as the
current density or excitonic transition matrix elements, depend critically on
the wave function.
In confined systems, if the Hamiltonian has time reversal symmetry, a real-
value wave function is well known to exists. However, as soon as periodic
boundary conditions are introduced in the Hamiltonian or a magnetic field is
applied, the amplitude of the wave function is, in general, complex. If a wave
function has a complex amplitude, both its modulus $\Phi({\bf R})$ and its
complex phase $e^{{\bf i}\phi({\bf R})}$ can depend on the many-body
coordinate ${\bf R}=\\{{\bf r}_{1},{\bf r}_{2},\cdots,{\bf r}_{N_{e}}\\}$,
where ${\bf r}_{j}$ is the position of electron $j$ and $N_{e}$ is the number
of electrons.
In addition, for fermions, the many-body wave function must change sign when
the coordinates of any pair ${\bf r}_{j}$, ${\bf r}_{k}$ are interchanged in
${\bf R}$. In principle, if the wave function is real-valued, one only needs
to determine the exact surface where the wave function is zero and changes
sign (the node) to find the ground-state energy with diffusion Monte Carlo
(DMC) methods. Any error in the determination of the node results in an
overestimation of the ground-state energy anderson79 ; reynolds82 ; HLRbook .
The standard DMC method ceperley80 and improvements ortiz93 related to it
require, as an input, a trial wave function $\Psi_{T}({\bf R})=\Phi_{T}({\bf
R})e^{{\bf i}\phi({\bf R})}$, where both the modulus $\Phi_{T}({\bf R})$ and
$\phi({\bf R})$ can be chosen to be real. The node is the surface $S_{T}({\bf
R})$ in the $3N_{e}$ dimensional many-body space where $\Psi_{T}({\bf R})=0$.
The cost of a single DMC step in the standard algorithm is polynomial in the
number of electrons $N_{e}$, and can be reduced to almost linear if localized
orbitals are used williamson ; alfe04 ; reboredo05 . As a consequence, the
existence of an algorithm that finds the required node with polynomial cost in
$N_{e}$ has been subject of controversy ceperley91 ; mtroyerprl2005 . It has
been argued that one of the most important problems in many-body electronic
structure theory is to accurately find representations of the fermion nodes
ceperley91 ; mtroyerprl2005 , which could help in solving the so-called
“fermion sign problem.”
In general, a guess of the node $S_{T}({\bf R})$ and the phase $\phi({\bf R})$
can be obtained from mean field or quantum chemistry methods [such as density
functional theory (DFT), Hartree-Fock, or configuration interaction (CI)].
This initial trial wave function is often improved using various methods
mfoulkesrmp2001 ; ortiz95 ; jones97 ; guclu05 ; umrigar07 within a
variational Monte Carlo (VMC) context. This standard approach depends on the
accidental accuracy of the mean field to find the node or the possibility to
perform accurate CI calculations to pre-select a multideterminant expansion
for $\Psi_{T}({\bf R})$. In addition, it can be claimed that a variational
optimization of the trial wave function energy or its energy variance only
improves the nodes indirectly luchow07 .
In the last two years, we have developed a method to circumvent the sign
problem for the ground and low-energy eigenstates of confined systems keystone
; rockandroll . This method was recently validated in real molecular systems
rollingstones . We called the method self-healing diffusion Monte Carlo
(SHDMC) since the nodes are corrected in a DMC context (as opposed to a VMC
optimization) and the wave function converges to nearly exact keystone or
state-of-the-art solutions rollingstones , even starting from random. This
approach is based on the proof keystone that by locally smoothing the
discontinuities in the gradient of the fixed-node ground state $\Psi_{FN}({\bf
R})$ at $S_{T}({\bf R})$, a new trial wave function can be obtained with
improved nodes. This proof enables an algorithm that systematically moves the
nodal surface fn:sampling $S_{T}({\bf R})$ towards that of an eigenstate. The
trial wave function is self-corrected within a recursive DMC approach. If the
form of trial wave function is sufficiently flexible and given sufficient
statistics, the process leads to an exact eigenstate many-body wave function
keystone ; rockandroll ; rollingstones .
The success of the fixed-node approximation anderson79 used in the standard
DMC algorithm for real wave functions is related to the quadratic dependence
of the error in the fixed-node energy with the distance between $S_{T}({\bf
R})$ and the exact node HLRbook $S({\bf R})$. Because the probability density
goes to zero quadratically at $S({\bf R})$, errors due to small and short
wave-length departures of $S_{T}({\bf R})$ from $S({\bf R})$ do not propagate
far into the nodal pocket. Since the DMC energy is dominated by the average
far from the node, DMC tolerates short wave-length departures of $S_{T}({\bf
R})$ around $S({\bf R})$.
However, if the amplitude of wave function is complex, one must also determine
its phase $\phi({\bf R})$. The ground-state energy of complex wave functions
can be calculated within the fixed-phase approximation ortiz93 of DMC
(FPDMC). But any error in the phase also results in an overestimation of the
ground-state energy even if the exact nodes are provided ortiz93 . For complex
wave functions, moreover, the error in the phase can be more dramatic than the
nodal error, since the gradient of phase ${\bf\nabla}\phi({\bf R})$ is sampled
everywhere, and in particular in the regions of large probability density (see
below and Ref. ortiz93, ).
Since (i) periodic or infinite systems are dominant in solid state physics,
(ii) the ability to calculate complex-valued wave functions (with current) is
crucial to understanding transport, (iii) the response of quantum systems to
magnetic fields is key for basic understanding of correlated phenomena and
even applications such quantum computation, (iv) most physical systems of
interest are not confined, and (v) the error in the phase affects the result
more than the error in the node, solving “the phase problem” is, perhaps, as
important as solving the sign problem.
In this paper a method is derived to simultaneously obtain not only the node
but also the complex amplitude of the trial wave function for lower energy
eigenstates of Hamiltonians with periodic boundary conditions or under applied
magnetic fields. It is shown that if the phase of the wave function is a
scalar function, there is a ‘special’ gauge transformation of the many-body
Hamiltonian where the wave functions is real. These wave functions have nodes
that are optimized as in original SHDMC method. If the phase can only be
expressed by multi-valuate function, the nodal surface may have a reduced
dimensionality but there is no constraint to update the wave-function in SHDMC
if the nodes are removed.
The method is applied and validated in a model system studied previously
rosetta where near-analytical solutions can be obtained. The scaling of the
cost of this new approach is linear in the number of independent degrees of
freedom of the phase. The method is a generalization of both the “fixed-phase”
approach ortiz93 and the self-healing DMC algorithms developed to circumvent
the sign problem keystone ; rockandroll ; rollingstones . The amplitude of the
wave function is free to adjust to the complex weight of the walkers in a
recursive approach.
A study of Refs. keystone, ; rockandroll, ; rollingstones, in reverse
chronological order (with increasing detail) is recommended before reading
this article. Studying again the seminal fixed-phase paper by Ortiz, Ceperley
and Martin (OCM) ortiz93 and the importance sampling method by Ceperley and
Alder ceperley80 is also highly encouraged.
The rest of the paper is organized as follows: In Section II, the SHDMC and
FPDMC methods are generalized and blended into a new algorithm that optimizes
the complex amplitude (and, if there is one, the node) of the trial wave
function within a DMC approach. As in the case of SHDMC, the trial wave
function is adjusted recursively within a generalized DMC approach. In Section
III, the generalization of SHDMC is applied to a model Hamiltonian with
periodic boundary conditions. The results are compared with converged CI
results for the same model. In Section IV, the Zeeman splittings of the ground
and excited states of a model system are calculated and compared with
converged CI results. Section V describes the results obtained with a
realistic Coulomb interaction. Finally, Section VI discusses the advantages,
perspectives, and possible applications of these methods for many-body
problems.
## II A free-amplitude recursive diffusion Monte Carlo method
This section shows how one can obtain an improved trial wave function
$\Psi_{T}({\bf R},\tau(\ell+1))=\langle{\bf R}|\Psi_{T}^{\ell+1}\rangle$ by
applying a smoothing operator $\hat{D}$ and an evolution operator
$e^{-\tau\hat{\mathcal{H}}_{FN}^{\ell}}$ (during a small imaginary time
$\tau$) to the trial wave function $\Psi_{T}({\bf R},\tau\ell)=\langle{\bf
R}|\Psi_{T}^{\ell}\rangle$ provided before. The limit
$\tau^{\prime}=\ell\tau\rightarrow\infty$ is reached recursively as the
iteration index $\ell\rightarrow\infty$.
Following the seminal ideas of OCM ortiz93 , $\Psi_{T}({\bf R},\tau^{\prime})$
can be written fn:taup as an explicit product of a complex phase and an
amplitude $\Psi_{T}({\bf R},\tau^{\prime})=\Phi_{T}({\bf R})e^{{\bf
i}\phi({\bf R})}$. OCM chose $\Phi_{T}({\bf R})$ to be symmetric (bosonic
like), real, and positive, while the phase factor $e^{{\bf i}\phi({\bf R})}$
was antisymmetric for particle exchanges. However, the symmetry of a the phase
factor is arbitrary: a symmetric phase factor can be obtained as
$e^{{\bf i}\phi({\bf R})}=\left[\frac{\Psi_{T}({\bf
R},\tau^{\prime})}{\Psi_{T}^{*}({\bf R},\tau^{\prime})}\right]^{1/2}\;,$ (1)
since both $\Psi_{T}({\bf R},\tau^{\prime})$ and its complex conjugate change
sign for particle exchanges. Therefore, any eigenstate can also be written as
the product of a complex-symmetric phase factor $e^{{\bf i}\phi({\bf R})}$
(like the Jastrow factor) and a real function $\Phi_{T}({\bf R})$ where the
symmetry of $\Phi_{T}({\bf R})$ depends on whether fermions or bosons are
considered .
In this work it is proved (see Subsection II.1) that if the phase of a
fernionic eigenstate is a scalar function, then $\Phi_{T}({\bf R})$ has the
same nodal structure than real functions. Otherwise $\Phi_{T}({\bf R})$ might
not be zero except for ${\bf r}_{i}={\bf r}_{j}$. Thus, the node of the trial
wave function $\Psi_{T}({\bf R},\tau^{\prime})$ is given in any case by
$\Phi_{T}({\bf R})$ but the dimensionality of the nodal surfaces depend on the
phase.
The evolution for an additional imaginary time $\tau$ of $\Psi_{T}({\bf
R},\tau^{\prime})$ is given by
$\displaystyle\Psi_{T}({\bf R},\tau^{\prime}+\tau)=$
$\displaystyle\;e^{-\tau\hat{\mathcal{H}}_{FN}^{\ell}}\Psi_{T}({\bf
R},\tau^{\prime})$ (2) $\displaystyle=$
$\displaystyle\;e^{-\tau\hat{\mathcal{H}}_{FN}^{\ell}}\left[\Phi_{T}({\bf
R})e^{{\bf i}\phi({\bf R})}\right]$ (3) $\displaystyle=$
$\displaystyle\;\Phi_{T}({\bf R},\tau)e^{{\bf i}\phi({\bf R})}.$ (4)
Equation (4) includes all the time dependence of the wave function in
$\Phi_{T}({\bf R},\tau)$, while the phase $\phi({\bf R})$ remains fixed
ortiz93 .
In Eq. (2), $e^{-\tau\hat{\mathcal{H}}_{FN}^{\ell}}$ is the fixed-node
evolution operator, which is a function of the fixed-node Hamiltonian operator
$\hat{\mathcal{H}}_{FN}^{\ell}$ given by
$\hat{\mathcal{H}}_{FN}^{\ell}=\hat{\mathcal{H}}+\\!\infty\
\lim_{\epsilon\rightarrow 0}\theta\left\\{\epsilon-d_{m}[S_{T}({\bf
R^{\prime}},\ell\tau)-{\bf R}]\right\\}\;.$ (5)
The second term on the right-hand side of Eq. (5) adds an infinite potential
ortiz93 at the points ${\bf R}$ with minimum distance to any point on the
nodal surface $d_{m}[S_{T}({\bf R^{\prime}},\tau^{\prime})-{\bf R}]$ smaller
than $\epsilon$. The fixed-node Hamiltonian is dependent on $\ell$ since the
nodes $S_{T}({\bf R^{\prime}},\tau^{\prime})$ change from one iteration to the
next.
In Eq. (5), the many-body Hamiltonian $\hat{\mathcal{H}}$ is given in atomic
units by
$\hat{\mathcal{H}}=\sum_{j}^{N_{e}}\frac{(\nabla_{j}+{\bf
A}_{j})^{2}}{2}+V({\bf R})-E_{T}$ (6)
where ${\bf A}_{j}={\bf A}({\bf r}_{j})$ is a vector potential at point ${\bf
r}_{j}$, $V({\bf R})$ includes the electron-electron interaction and any
external potential, and $E_{T}$ is a complex fn:gauge energy reference,
adjusted to normalize the projected wave function, that cancels out any phase
shift resulting from arbitrary gauge choices for ${\bf A}_{j}$ (see remarks
below).
Using Eq (2), one can easily obtain
$\displaystyle\frac{d}{d\tau}\left[\Phi_{T}({\bf R},\tau)\right]=$
$\displaystyle-e^{-{\bf i}\phi({\bf
R})}\hat{\mathcal{H}}_{FN}e^{-\tau\hat{\mathcal{H}}_{FN}}\Psi_{T}({\bf
R},\tau^{\prime})$ $\displaystyle=$ $\displaystyle-\left[E_{L}({\bf
R},\tau)-E_{T}\right]\Phi_{T}({\bf R},\tau)$ (7)
with
$\displaystyle E_{L}({\bf R},\tau)=$
$\displaystyle-\frac{1}{2}\\!\\!\sum_{j}^{N_{e}}\frac{\nabla_{j}^{2}\Phi_{T}({\bf
R},\tau)}{\Phi_{T}({\bf R},\tau)}$ (8)
$\displaystyle+\frac{1}{2}\sum_{j}^{N_{e}}\left|{\bf
A}_{j}+{\bf\nabla}_{j}\phi({\bf R})\right|^{2}+V({\bf R})$ $\displaystyle-{\bf
i}\sum_{j}^{N_{e}}\left\\{\frac{{\bf\nabla}_{j}\Phi_{T}({\bf
R},\tau)}{\Phi_{T}({\bf R},\tau)}.\left[{\bf
A}_{j}\\!+\\!{\bf\nabla}_{j}\phi({\bf R})\right]\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\frac{{\bf\nabla}_{j}\cdot\left[{\bf
A}_{j}+{\bf\nabla_{j}}\phi({\bf R})\right]}{2}\right\\}.$
In Eq. (II), $E_{L}({\bf R},\tau)$ can be a real constant only if
$\Phi_{T}({\bf R})e^{{\bf i}\phi({\bf R})}$ is an eigenstate of
$\hat{\mathcal{H}}_{FN}$. In general, for an arbitrary trial wave function,
$E_{L}({\bf R},\tau)$ is a complex function of ${\bf R}$. The real part of
$E_{L}({\bf R},\tau)$ is given by the first three terms in Eq. (8), while the
imaginary contribution is given by the last one. OCM’s fixed-phase
approximation results from considering only the real part fn:fixed_phase of
$E_{L}({\bf R},\tau)$. With little effort, one can obtain Eq. (3) of OCM’s
work assuming $Im[E_{L}({\bf R})]=0$, which leads to a continuity-like
equation for fluids.
Note that if $\phi({\bf R})$ is held fixed and $Im[E_{L}({\bf R})]\neq 0$ [see
Eqs. (2), (4), (II), and (8)], then $\Phi_{T}({\bf R},\tau)$ must not only
change its modulus but also must be free to drift away from the real values as
$\tau$ increases.
If at $\tau=0$ an initial distribution of $N_{w}$ walkers $f({\bf R},0)$ is
generated to be equal to $N_{w}|\Phi_{T}({\bf R})|^{2}$, within a
generalization of the importance sampling algorithm of Ceperley and Alder
ceperley80 (see below), $f({\bf R},\tau)$ should evolve in imaginary time as
$\displaystyle f({\bf R},\tau)$ $\displaystyle=\Phi_{T}({\bf R})\Phi_{T}({\bf
R},\tau).$ (9)
Clearly $f({\bf R},\tau)$ can be complex for $\tau>0$ if $Im[E_{L}({\bf
R},\tau)]~{}\neq~{}0$ [see Eq. (II)].
Replacing Eq. (II) into Eq. (9), and following a procedure almost identical to
the one used in Ref. ceperley80, , one obtains
$\displaystyle\left.\frac{\partial f({\bf
R},\tau)}{\partial\tau}\right|_{\tau\approx 0}=$
$\displaystyle\frac{1}{2}\sum_{j}^{N_{e}}\left\\{\nabla_{j}^{2}f({\bf
R},\tau)-{\bf\nabla}_{j}\cdot\left[f({\bf R},\tau){\bf
F}_{Q}^{j}\right]\right\\}$ $\displaystyle-\left[E_{L}({\bf
R})-E_{T}\right]f({\bf R},\tau),$ (10)
where
$\displaystyle{\bf F}_{Q}^{j}={\bf\nabla}_{j}ln\left|\Phi_{T}({\bf
R})\right|^{2}\;,$ (11)
and $E_{L}({\bf R})=E_{L}({\bf R},0)$ is the complex local energy constructed
using Eq. (II). To obtain Eq. (10) one must to assume that
$\frac{{\bf\nabla}_{j}\Phi_{T}({\bf R},\tau)}{\Phi_{T}({\bf
R},\tau)}\simeq\frac{{\bf\nabla}_{j}\Phi_{T}({\bf R})}{\Phi_{T}({\bf R})},$
(12)
which implies that unlike the standard DMC algorithm ceperley80 , there is an
error in Eq. (10) when $\tau\rightarrow\infty$ if $\Psi_{T}({\bf R},\tau)$ is
not an eigenstate. This is only an apparent limitation since (i) $\tau$ at
first can be made as small as required for Eq. (12) to be valid, (ii) $\tau$
can be increased later as the wave function improves and converges to an
eigenstate, (iii) the limit $\tau^{\prime}\rightarrow\infty$ is reached by
applying this free-amplitude method recursively (see below), and (iv) $\tau$
is already limited to be small in SHDMC with correlated sampling so that the
weights remain close to $1$ (see below).
Although Eq. (10) above for $f({\bf R},\tau)$ is identical to Eq. (1) in Ref.
ceperley80, , it now has a slightly more complex interpretation as a
stochastic process. Each member of an ensemble of systems (walker) undergoes
(i) a random diffusion caused by the zero-point motion and (ii) drifting by
the trial quantum force $ln\left|\Phi_{T}({\bf R})\right|^{2}$ [which depends
only on $\Phi_{T}({\bf R})$ and not on the phase], but in variance with Ref.
ceperley80, , (iii) each walker carries a complex phase. In a nonbranching
algorithm, the complex weight of the walkers is multiplied by
$\exp\\{-\left[E_{L}({\bf R})-E_{T}\right]\delta\tau\\}$ at every time step.
Similar to the case of the “simple” SHDMC algorithm (see Refs. keystone, ;
rockandroll, ; rollingstones, for details), the weighted distribution of the
walkers can be written as
$\displaystyle f({\bf R},\tau)$
$\displaystyle=\lim_{N_{c}\rightarrow\infty}\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}W_{i}^{j}(k)\delta\left({\bf
R-R}_{i}^{j}\right).$ (13)
In Eq. (13), ${\bf R}_{i}^{j}$ corresponds to the position of the walker $i$
at step $j$ of $N_{c}$ equilibrated configurations. The complex weights
$W_{i}^{j}(k)$ are given by
$\displaystyle W_{i}^{j}(k)=e^{-\left[E_{i}^{j}(k)-E_{T}\right]\tau}$ (14)
with
$\displaystyle E_{i}^{j}(k)=\frac{1}{k}\\!\sum_{\ell=0}^{k-1}E_{L}({\bf
R}_{i}^{j-\ell}),$ (15)
where $E_{T}$ in Eq. (14) is now a complex energy reference periodically
adjusted so that $\sum_{i}W_{i}^{j}(k)\approx N_{c}$ and $\tau$ is
$k\delta\tau$ ($k$ is a small number of steps and $\delta\tau$ is a standard
DMC time step).
The trial wave function $\Psi_{T}({\bf R},\tau^{\prime}+\tau)$ for the next
iteration can be obtained as follows: All wave functions can be expanded in a
basis as
$\displaystyle\Psi_{T}({\bf R},\tau^{\prime})$ $\displaystyle=e^{J({\bf
R})}\sum_{n}^{\sim}\lambda_{n}(\tau^{\prime})\Phi_{n}({\bf R}).$ (16)
In Eq. (16), $\sum_{n}^{\sim}$ represents a truncated sum, $\\{\Phi_{n}({\bf
R})\\}$ forms a complete orthonormal basis of the antisymmetric Hilbert space
fn:basis , and $e^{J({\bf R})}$ is a symmetric Jastrow factor. The
$\lambda_{n}(\tau^{\prime})$ are complex coefficients to be defined [see Eq.
(24)]. Note that the expressions fn:taup
$\displaystyle\Phi_{T}({\bf R})$ $\displaystyle=\pm\sqrt{\Psi_{T}({\bf
R},\tau^{\prime})\Psi_{T}^{*}({\bf R},\tau^{\prime})},\text{ and }$ (17)
$\displaystyle\phi({\bf R})$ $\displaystyle=\ln[\Psi_{T}({\bf
R},\tau^{\prime})/\Psi_{T}^{*}({\bf R},\tau^{\prime})]/(2i)+\pi n$ (18)
allow the computation of all the quantities involved in $E_{L}({\bf R})$ in
terms of gradients and Laplacians of $\Phi_{n}({\bf R})$ and $J({\bf R})$. In
Eq. (18) $n$ is an arbitrary integer that changes the Riemann branch of the
natural logarithm $\ln$, but does not contribute to the gradient within a
branch. The local energy is thus independent on the choice of $n$ but at the
Riemann cuts where, sometimes, $n$ has to change to make the phase continuous.
However, the probability of a walker to touch the Riemann cut is, in practice,
zero.
From Eqs. (2), (9) and (13), one can formally obtain
$\displaystyle\tilde{\Psi}_{T}({\bf R},\tau^{\prime}+\tau)=$
$\displaystyle\;e^{{\bf i}\phi({\bf R})}f({\bf
R},\tau^{\prime}+\tau)/\Phi_{T}({\bf R})$ (19) $\displaystyle=$
$\displaystyle\;\langle{\bf
R}|e^{-\tau\hat{\mathcal{H}}_{FN}}|\Psi_{T}(\tau^{\prime})\rangle\;.$ (20)
The local smoothing operator is defined as
$\displaystyle\langle{\bf R}^{\prime}|\hat{D}|{\bf R}\rangle$
$\displaystyle=\tilde{\delta}\left({\bf R^{\prime},R}\right)$ (21)
$\displaystyle=\sum_{n}^{\sim}e^{J({\bf R^{\prime}})}\Phi_{n}({\bf
R^{\prime}})\Phi_{n}^{*}({\bf R})e^{-J({\bf R})}.$
Applying Eq. (21) to both sides of Eq. (19), using Eq. (13), and integrating
over ${\bf R}$, one can easily obtain
$\displaystyle\Psi_{T}({\bf R},\tau^{\prime}+\tau)=$
$\displaystyle\;\langle{\bf
R}|\hat{D}e^{-\tau\hat{\mathcal{H}}_{FN}}|\Psi_{T}(\tau^{\prime})\rangle$ (22)
$\displaystyle=$ $\displaystyle\;e^{J({\bf
R})}\sum_{n}^{\sim}\langle\lambda_{n}(\tau^{\prime}+\tau)\rangle\Phi_{n}({\bf
R}),$ (23)
with
$\displaystyle\langle\lambda_{n}(\tau^{\prime}\\!+\\!\tau)\rangle=\frac{1}{\mathcal{N}}\sum_{i}^{N_{c}}W_{i}^{j}(k)e^{-J({\bf
R}_{i}^{j})}\frac{\Phi^{*}_{n}({\bf R}_{i}^{j})}{\Psi^{*}_{T}({\bf
R}_{i}^{j},\tau^{\prime})}\gamma({\bf R}_{i}^{j})$ (24)
where $\mathcal{N}=\sum_{i=1}^{N_{c}}e^{-2J({\bf R}_{i}^{j})}$ normalizes the
Jastrow factor. $\gamma({\bf R})$ is the standard time-step correction [Eq.
(33) in Ref. umrigar93, ]:
$\displaystyle\gamma({\bf R})=\frac{-1+\sqrt{1+2|{\bf v}|^{2}\tau}}{|{\bf
v}|^{2}\tau}\text{ with }{\bf v}=\frac{\nabla\Phi_{T}({\bf R})}{\Phi_{T}({\bf
R})}.$ (25)
Note that $\Phi_{T}({\bf R})$ includes a Jastrow factor, thus Eq. (26) reduces
to the one used in the original “simple” SHDMC algorithm keystone ;
rockandroll for $\phi({\bf R})=0$.
In addition, as suggested by Umrigar umrigar_private for the ground-state
SHDMC algorithm keystone , correlated sampling can be used also for walkers
with complex weight. One can sample
$\delta\lambda_{n}=\lambda_{n}(\tau^{\prime}+\tau)-\lambda_{n}(\tau^{\prime})$,
which results in
$\displaystyle\langle\lambda_{n}(\tau^{\prime}+\tau)\rangle=\lambda_{n}(\tau)+\langle\delta\lambda_{n}\rangle$
(26)
$\displaystyle\langle\delta\lambda_{n}\rangle=\frac{1}{\mathcal{N}}\sum_{i=1}^{N_{c}}e^{-J({\bf
R}_{i}^{j})}\frac{\Phi_{n}^{*}({\bf R}_{i}^{j})}{\Psi^{*}_{T}({\bf
R}_{i}^{j},\tau^{\prime})}[W_{i}^{j}(k)-1]\gamma({\bf R}_{i}^{j}).$
These new $\lambda_{n}(\tau^{\prime}+\tau)$ [Eq. (26)] are used to construct a
new trial wave function [Eq. (16)] recursively within DMC. Equation (24) can
be related to the maximum-overlap method used for bosonic wave functions
reatto82 .
The error of $\langle\delta\lambda_{n}\rangle$ is obtained by sampling
$\displaystyle\langle\delta\lambda_{n}^{2}\rangle=\frac{1}{\mathcal{N}}\sum_{i=1}^{N_{c}}\left|e^{-J({\bf
R}_{i}^{j})}\frac{\Phi_{n}^{*}({\bf R}_{i}^{j})}{\Psi^{*}_{T}({\bf
R}_{i}^{j})}[W_{i}^{j}(k)-1]\gamma({\bf R}_{i}^{j})\right|^{2}.$ (27)
The truncation of the expansion of the delta function [Eq. (21)] is a key
ingredient in SHDMC since it decides how local is the smoothing operator
$\hat{D}$ and prevents noise to ruin the quality of the trial wave function.
If the absolute value of the error of $\langle\delta\lambda_{n}\rangle$ is
larger than $|\lambda_{n}(\tau^{\prime}+\tau)|/4$, the algorithm sets
$\lambda_{n}(\tau^{\prime}+\tau)$ equal to zero [which defines the truncation
criterion used in the sums ($\sum_{n}^{\sim}$) involved in Eqs. (16), (21) and
(22)]. See Ref. keystone, for a detailed theoretical justification of the
truncation procedure and the algorithm used. Briefly here, the coefficients
$\lambda_{n}$ are sampled at the end of each sub-block of $k$ DMC steps.
Statistical data is collected for number of sub-blocks $M$ before a wave
function update. At first, $M$ is set to a small number and increased
according to the recipe given in Ref. rockandroll, . In short, the algorithm
detects automatically the dominance of noise when the projection of two
successive sets of $\langle\delta\lambda_{n}\rangle$ becomes small and
multiplies by a factor larger than 1 the number of sub-blocks $M$ (see Ref.
rockandroll, for more details). As a result, the total number of
configurations $N_{c}$ sampled increases as the algorithm progresses.
Therefore the statistical error is reduced, and the number of basis functions
retained in the expansion increases over time. Thus, the smoothing operator
$\hat{D}$ tends to the delta function as $M$ increases, which allows the SHDMC
method to sample the wave function with increasing detail. The first quarter
of the data in each block, following a wave function update, is discarded.
In practice, the only difference between this new approach and the original
SHDMC method is the complex weight and the limitation for propagation to small
$\tau$. Therefore, a nonbranching algorithm for small $\tau$ has been used
(see Ref. rockandroll, for details). However, there are some formal
differences on the justification of the convergence of the SHDMC method that
are discussed in the next subsections.
### II.1 Gauge transformations and nodal structure of complex eigenstates
The phase $\phi({\bf R})$ must be continuous at any point of ${\bf R}$ where
$\Psi_{T}({\bf R})\neq 0$. Otherwise, if $\phi({\bf R})$ is not continuous,
its gradient in the local energy will introduce an effective infinite
potential at the discontinuity that will force $\Psi_{T}({\bf R})=0$. In some
cases, however, these discontinuities in the phase are not physicalfn:discont
and they can be removed by changing the Riemann sheet index $n$ in Eq. (18).
As a consequence, the wave functions can be split into two classes. In the
first class different Riemann sheets of $\phi({\bf R})$ are not connected. In
that case, one can choose as phase a single sheet of the Riemann surface The
phase in this class can be a continuous scalar function of ${\bf R}$ for every
${\bf R}$. The real wave functions, with constant phase, are special case of
this class. In the second class, the Riemann surfaces of $\phi({\bf R})$ for
different $n$ are connected at the Riemann cuts. Thus a continuous $\phi({\bf
R})$ can only be described by a multi-valuate function of ${\bf R}$.
Eigenstates with a scalar phase: Since
${\bf\nabla}_{j}\times{\bf\nabla}_{j}\cdot\Lambda({\bf R})=0$ for any scalar
function $\Lambda({\bf R})$, the magnetic field ${\bf
B}={\bf\nabla}_{j}\times{\bf A}_{j}$ is invariant for the gauge
transformations jackson ${\bf A}^{\prime}_{j}={\bf
A}_{j}+{\bf\nabla}_{j}[\Lambda({\bf R})+c(\tau)]$, where $\Lambda({\bf R})$ is
an arbitrary symmetric scalar function of ${\bf R}$ and $c(\tau)$ is an
arbitrary function of $\tau$ (independent of every ${\bf r}_{j}$ in ${\bf
R}$). If $\Psi_{T}({\bf R})$ is selected to be an eigenstate of
$\hat{\mathcal{H}}$ for a given gauge and if $\phi({\bf R})$ is a scalar
function , then a change in gauge ${\bf
A}_{j}+{\bf\nabla}_{j}\delta\Lambda({\bf R})$ could be readily compensated in
the phase by
$\tilde{\phi}(\bf R)=\phi({\bf R})-\delta\Lambda({\bf R})+c(\tau),$ (28)
without affecting $\Phi_{T}({\bf R})$ [since ${\bf A}_{j}$ and
${\bf\nabla}_{j}\phi({\bf R})$ always appear added in Eq. (8)]. This property
is particularly important, since implies that for this class of eigenstates of
$\hat{\mathcal{H}}$ there is a ‘special’ gauge where the wave function is
real.
Note that if one sets $\delta\Lambda({\bf R})=\phi({\bf R})$ in Eq. (28) then
$\tilde{\phi}(\bf R)~{}=~{}0$. Therefore, the new phase is a constant that can
be chosen to be zero. The vector potential in this special gauge is a many-
body object which includes the gradient of the many-body phase of the wave
function in a single particle gauge.
The norm $\Phi({\bf R})$ is invariant since the effective potential in Eq. (8)
is invariant using Eq. (28). This is expected since the expectation value of
an arbitrary operator $\hat{\mathcal{O}}({\bf R})$ must be independent of the
gauge choice for non-degenerate eigenstates. In particular, the nodes, which
are given by $\Phi({\bf R})$, are also invariant to gauge transformations.
Since the new phase is a constant, it can be easily shown that in the special
gauge, the amplitude of those eigenstates has the same structure as the trial
wave function used in the fixed-node approximation for real wave functions.
SHDMC self adjusts to an arbitrary gauge change $\delta\Lambda({\bf R})$
because $\phi({\bf R})$ is modified recursively by a change in the local
energy in Eq. (8) of the form
$\displaystyle\delta E_{L}({\bf R},\tau)=$ (29)
$\displaystyle\sum_{j}^{N_{e}}\left\\{Re\left[\left({\bf
A}_{j}+{\bf\nabla}_{j}\phi({\bf
R})\right)\cdot{\bf\nabla_{j}}\delta\Lambda({\bf
R})\right]+\frac{1}{2}|{\bf\nabla}_{j}\delta\Lambda({\bf R})|^{2}\right\\}$
$\displaystyle-\;{\bf
i}\sum_{j}^{N_{e}}\left\\{\frac{{\bf\nabla}_{j}\Phi_{T}({\bf
R},\tau)}{\Phi_{T}({\bf R},\tau)}\cdot{\bf\nabla_{j}}\delta\Lambda({\bf
R})+\frac{1}{2}\nabla^{2}_{j}\delta\Lambda({\bf R})\right\\}.$
Eigenstates with a multi-valuate phase: On the other hand if $\phi({\bf R})$
is not a scalar function then,
${\bf\nabla}_{j}\times{\bf\nabla}_{j}\cdot\phi({\bf R})\neq 0$ and, therefore,
${\bf\nabla}_{j}\cdot\phi({\bf R})$ cannot be included in the vector potential
without introducing an artificial many-body magnetic field. In that case, as
pointed out timely by an anonymous referee, there might be nodes only where
${\bf r}_{j}={\bf r}_{k}$. Eigenstates with this type of nodes, with reduced
dimensionality, can be found in states with current, degeneracy or magnetic
fields.
Sumarizing, the norm of the complex wave-functions of the eigenstates that
have a scalar phase in ${\bf R}$ has the same structure as the real wave
function used in the fixed-node approach because there is a special gauge
transformation where the wave function is real valued. This property has a
formal importance since it allows extending theorems developed in the context
of the fixed-node approximation. Instead, if different Riemann sheets of
$\phi({\bf R})$ are continuously connected, a continuous phase cannot be
described by a single scalar function. Then dimensionality of the nodal
surface might be smaller and limited to the cases in ${\bf R}$ where ${\bf
r}_{j}={\bf r}_{k}$. The nodes are an obstacle for DMC; SHDMC, however,
converges to eigenstates regardless of the dimensionality of the nodes (see
below).
### II.2 Remarks on the free-amplitude SHDMC method
Convergence of SHDMC to eigenstates: In II.1 it is shown that the wave-
function of any fermionic eigenstate can be factorized into an anti-symmetric
real function $\Phi_{T}({\bf R})$, with nodes, and a symmetric phase factor
(See Eq. (1)). The dimensionality of the nodal surface depends on the phase
properties. Since the amplitude can not change at the node in DMC, nodes are
the obstacle to overcame by SHDMC. Convergence is not affected if the initial
trial wave function has no nodes because, in this case, the amplitude and the
phase can evolve with $\tau$ everywhere. Indeed, SHDMC can be started from a
linear combination of real and imaginary parts with different nodes. As noted
by a referee, in this class of functions two particles can exchange without
crossing a node. However, the convergence of SHDMC is not affected in theory
and it is not affected in practice. But, if the phase is a scalar function,
nodes will develop as the trial wave function converges to an eigenstate. In
that limit, a kink at the node should appearkeystone until the exact node is
found.
The convergence of the SHDMC approach when the trial wave function approaches
an eigenstate and shows nodes, is based on the proof keystone that locally
smoothing the kinks of the fixed-node wave function improves the nodes. This
proof can be trivially extended to a complex wave function (when
$Im[E_{L}({\bf R})]\neq 0$ ), breaking the time evolution into a sequence of
pure real evolution followed by an imaginary evolution. If one assumes
$Im[E_{L}({\bf R})]=0$, the present approach reduces to SHDMC method, but
using the effective potential of the fixed-phase Hamiltonian ortiz93 . Thus
the best nodes in $\Phi_{T}({\bf R},\tau)$ for a given phase $\phi({\bf R})$
can be obtained by running SHDMC in a fixed-phase stage. It is trivial to show
that the phase, in turn, improves if the imaginary contribution is allowed to
evolve a short time from a trial wave function with optimal nodes. In
principle, a SHDMC fixed-phase stage can be propagated to infinite imaginary
time $\tau$ within a branching algorithm. The evolution of the phase, instead,
is limited to short times [for Eq. (12) to be valid]. In this work, however,
the real and the imaginary parts of the wave function are allowed to evolve
simultaneously during a short time without any observed adverse effect on the
accuracy.
Phase and nodal errors: Note in Eq. (8) that as in the fixed-phase approach
ortiz93 , an effective potential $\left|{\bf\nabla}_{j}\phi({\bf R})+{\bf
A}_{j}\right|^{2}$ is added to $V({\bf R})$, which depends in turn on
${\bf\nabla}\phi({\bf R})$ and ${\bf A}_{j}$. Thus small errors in the phase
$\phi({\bf R})$ have a global impact [in particular far from the node, where
$|\Phi_{T}({\bf R})|^{2}>>0$]. In contrast, small errors in $\Phi_{T}({\bf
R})$ only slightly displace the node and have smaller impact on the energy
HLRbook [since the local energy is seldom sampled because $\lim\Phi_{T}({\bf
R})\Phi_{T}({\bf R},\tau)\rightarrow 0$ at $S_{T}({\bf R})$]. For eigenstates
without nodal surfaces the phase is the sole source of error. SHDMC provides a
method to correct both the phase and the nodal error.
Complex $E_{T}$: Note that when $c(\tau)$ is changed in Eq. (28), it just
changes the normalization of $\Psi_{T}({\bf R})$ and, if complex, introduces a
global phase shift; however, $c(\tau)$ does not affect the local energy or
other observables.
For a random trial wave function $\Psi_{T}({\bf R})$ and an arbitrary choice
of gauge for ${\bf A}_{j}$, $E_{L}({\bf R},\tau)$ will have both real and
imaginary components. The average over the walkers‘ positions will have a real
contribution, which affects the norm, and a complex contribution, which
introduces a global phase shift. The average of $E_{L}({\bf R},\tau)$ only
contributes to $c(\tau)$. The correlated sampling approach is obviously more
efficient when the distribution of complex weights is centered around 1 since
the error in the coefficients is minimized [See Eq. 27 and use the standard
expression for the variance]. $E_{T}$ in Eq. (14) is thus complex. The real
part of $E_{T}$ renormalizes the wave function (that is, keeps the population
of walkers constant). The imaginary part of $E_{T}$ removes the global phase
shift [the average complex contribution of $E_{L}({\bf R},\tau)$ that only
contributes to $c(\tau)$]. For a converged trial wave-function,
$Im(E_{T})\simeq 0$.
Upper bound properties: The present approach should not be considered as a
method to estimate the energy of the trial wave function but instead as a
method to optimize the trial wave function before a final FPDMC calculation.
However, since the real part of $E_{L}({\bf R})$ corresponds to the fixed-
phase approximation, the real part of $E_{T}$ also converges to an upper bound
of the ground-state energy. This upper bound can be higher than the fixed-
phase approximation (if the limit of $\tau^{\prime}\rightarrow\infty$ is not
reached or the basis $\\{\Phi_{n}({\bf R})\\}$ is too small). Therefore, a
standard FPDMC calculation ortiz93 (with branching) must be performed to
obtain final values for the ground-state energy.
Known limitations and solutions: The present approach is inefficient when the
energy of the first excited state $E^{FP}_{1}$ of the fixed-phase Hamiltonian
is too close to the ground-state energy $E^{FP}_{0}$, since the coefficient of
the excited state component of the trial wave function decays as
$exp[-\tau(E^{FP}_{1}-E^{FP}_{0})]$. In that regime, satisfactory results can
be obtained as follows: begining with a small value for $M$, keep $M$ fixed
for a number of iterations until the lower energy excitations decay and, then
release $M$, allowing the high energy components of the wave function to
converge.
Moreover, the approximate excited-state wave functions can be calculated (see
Ref. rockandroll, ) and the lowest energy linear combination can be determined
with correlated function Monte Carlo jones97 , VMC umrigar07 or directly in
FPDMC using a restricted basis of low energy states. A final alternative is to
run this free-amplitude SHDMC method with larger $\tau=k\delta\tau$ but using
a smaller basis given by a few approximated excited states. The excited states
can be found as described in the next section.
### II.3 Generalization to excited states
Earlier estimates of excited state energies in the presence of magnetic fields
have been made by diagonalizing a matrix of correlation functions in imaginary
time jones97 ; ceperley88 . In addition, calculations of excited states have
been reported with the auxiliary field approach purwanto09 . The present
algorithm, in contrast, is almost identical to the SHDMC method keystone ;
rockandroll developed for the ground state and lower excitations of real wave
functions. The only relevant difference with Ref. rockandroll, is the complex
weight of the walkers. Thus the free-amplitude SHDMC method described above
for the ground state can be generalized in a straightforward way to study
excited states as in Ref. rockandroll, .
Readers are encouraged to follow a detailed theoretical justification of the
excited state algorithm in Ref. rockandroll, . Here only some key steps are
described [in particular, note Eqs. (31) and (32) that were omitted in Ref.
rockandroll, and are relevant for a non-unitary Jastrow factor].
As in the importance sampling algorithm ceperley80 , the generalization given
by Eq. (10) requires that $\Psi_{T}({\bf R},\tau^{\prime}\;+\;\tau)$ be zero
only at the nodes $S_{T}({\bf R},\tau^{\prime})$ of $\Psi_{T}({\bf
R},\tau^{\prime})$, being free to change both its modulus and phase elsewhere.
Therefore, $\Psi_{T}({\bf R},\tau^{\prime}+\tau)$ can develop a projection
into any lower energy state consistent with $S_{T}({\bf R},\tau^{\prime})$. To
obtain an excited state, the wave function $\Psi_{T}({\bf
R},\tau^{\prime}+\tau)$ must be projected in the subspace orthogonal to the
ground state and any other excited state calculated before. In alternative
approaches such as correlation function Monte Carlo ceperley88 , the
orthogonality of excited states is achieved by diagonalizing a generalized
eigenvalue problem. One could argue that the excited states obtained with that
approach share nodal error of the ground state. One of the advantages of SHDMC
is that the diagonalization of a large matrix of excitations is avoided, which
makes possible the consideration of a larger number of degrees of freedom. In
addition, the nodes of each excitations are found independently. But in SHDMC,
unless special conditions are satisfied rockandroll , one must calculate
lowerlying energy states before attempting the calculation of higher excited
states.
A projector is constructed with approximated expressions of the $\nu$
eigenstates $\Psi_{\mu}({\bf R})=\langle{\bf
R}|e^{\hat{J}}\breve{\Phi}_{\mu}\rangle=e^{J({\bf R})}\breve{\Phi}_{\mu}({\bf
R})$ calculated earlier as
$\hat{P}_{\nu}=e^{\hat{J}}\left[1-\sum_{\mu}^{\nu}|\breve{\Phi}_{\mu}\rangle\langle\breve{\Phi}_{\mu}^{\dagger}|\right]e^{-\hat{J}}\;\;.$
(30)
The operator $e^{\hat{J}}$ in Eq. (30) is the multiplication by a Jastrow. For
a non-unitary $e^{\hat{J}}$ the set $\\{|\breve{\Phi}_{\mu}\rangle\\}$ is
nonorthogonal. However, the conjugate (dual) basis hoffman ; prugovecki that
satisfies $\langle\breve{\Phi}_{\mu}|\breve{\Phi}_{m}\rangle=\delta_{\mu,m}$
can be obtained statistically as
$\langle\breve{\Phi}_{\mu}|{\bf
R}\rangle=\sum_{n}^{\nu}\xi_{n}^{\mu}\Phi^{*}_{n}({\bf R})$ (31)
with
$\displaystyle\xi_{n}^{\mu}=\lim_{N_{c}\rightarrow\infty}\frac{1}{\mathcal{N}}\sum_{i}^{N_{c}}\frac{W_{i}^{j}(k)}{e^{-J({\bf
R}_{i}^{j})}}\frac{\Phi_{n}({\bf R}_{i}^{j})}{\Psi_{T}^{\mu}({\bf
R}_{i}^{j},\tau^{\prime})}\gamma({\bf R}_{i}^{j})\;,$ (32)
where $\Psi_{T}^{\mu}({\bf R},\tau^{\prime})$ is the trial wave function used
to evaluate earlier the state $\mu$ for $\tau^{\prime}\rightarrow\infty$. Note
that the exponential involving $-J({\bf R}_{i}^{j})$ moves to the denominator
in Eq. (32) as compared with Eq. (24). Since $J({\bf R})$ is real, the phase
of $\langle\breve{\Phi}_{\mu}|{\bf R}\rangle$ must be conjugated to the phase
of $\langle{\bf R}|\breve{\Phi}_{\mu}\rangle$. The coefficients
$\xi_{n}^{\mu}$ should be sampled during the final FPDMC step (i.e., when the
final excited energy is sampled).
The projection of the conjugate function $\langle\breve{\Phi}_{\mu}|$ onto
earlier conjugate states should also be removed to obtain
$\langle\breve{\Phi}_{\mu}|=\langle\breve{\Phi}_{\mu}|\hat{P}^{T}_{\mu-1}|$
where $\hat{P}^{T}_{\nu}$ is the transpose of $\hat{P}_{\nu}$. Furthermore,
statistical errors in $\langle\breve{\Phi}_{\mu}|$ can be partially filtered
by inverting the overlap matrix
$S_{\mu,m}=\langle\breve{\Phi}_{\mu}|\breve{\Phi}_{m}\rangle$ as
$\langle\breve{\Phi}_{\mu}^{\dagger}|=\sum_{m}S^{-1}_{\mu,m}\langle\breve{\Phi}_{m}|\;.$
(33)
The scalar products resulting from applying $\hat{P}_{\mu}$ in Eq. (30) are
given by
$\langle\breve{\Phi}_{\mu}^{\dagger}|\breve{\Phi}_{m}\rangle=\sum_{n}\bar{\xi}_{n}^{\mu}\lambda_{n}^{m},$
(34)
with $\bar{\xi}_{n}^{\mu}=\sum_{\nu}S^{-1}_{\mu,\nu}\xi_{n}^{\mu}$ since
$\int\Phi_{m}^{*}({\bf R})\Phi_{n}({\bf R}){\bf dR}=\delta_{n,m}.$
The extension of SHDMC to the next excited $|\Psi_{\nu+1}\rangle$ can be
thought of as the recursive application of the evolution operator
$e^{-k\delta\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}}$, the projector $\hat{P}$
[Eq (30)], and a smoothing operation $\hat{D}$ [see Eq. (21)] to a trial wave
function $|\Psi_{T,\nu+1}^{\ell-1}\rangle$ [see Eq. (II.3)]. This procedure
can be derived analytically rockandroll as follows:
$\displaystyle|\Psi_{\nu+1}\rangle$ $\displaystyle=$
$\displaystyle\lim_{\tau\rightarrow\infty}\hat{P}\;e^{-\tau\hat{\mathcal{H}}}\hat{P}|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\hat{P}\;\prod_{\ell}\left(e^{-(\delta\tau^{\prime}+k\delta\tau)\hat{\mathcal{H}}}\hat{P}\right)|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\hat{P}\;\prod_{\ell}\left(e^{-\delta\tau^{\prime}\hat{\mathcal{H}}}e^{-k\delta\tau\hat{\mathcal{H}}_{FN}^{(\ell-1)}}\hat{P}\right)|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle\simeq$
$\displaystyle\lim_{\ell\rightarrow\infty}\hat{P}\prod_{\ell}\left(\tilde{D}e^{-k\delta\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}}\hat{P}\right)|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle|\Psi_{T,\nu+1}^{\ell\rightarrow\infty}\rangle.$
Replacing $e^{k\delta\tau\hat{\mathcal{H}}}$ in the infinite product in the
second line of Eqs. (II.3) with
$e^{-k\delta\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}}$ in the third line
generates the same projector keystone ; rockandroll . In turn, we proved
keystone that replacing $e^{-\delta\tau^{\prime}\hat{\mathcal{H}}}$ with a
large class of local smoothing operators $D$ has the same effect on the nodes.
The fixed-node Hamiltonian depends on the iteration index $\ell$ because the
trial wave function, the node, $E_{T}$, and the phase are different at every
iteration. Finally, the norm of the projected function can be fixed by
adjusting $E_{T}$ in every iteration $\ell$.
For states with inequivalent nodal pockets, special care must be taken within
the algorithm to avoid systematic errors (see Ref. rockandroll for additional
details about the algorithm).
## III Calculations for Hamiltonians with periodic boundary conditions
Usually periodic boundary conditions in a supercell with dimensions $a_{x}$,
$a_{y}$ and $a_{z}$ are set when studying crystalline systems that simulate an
infinite solid. By using the Bloch Theorem ashcroft , the trial wave function
at $\tau^{\prime}=\ell\tau$ can be written as the product of a many-body
phaserajagopal95 times a periodic part fn:taup $\mathcal{U}({\bf R})$ as
$\Psi_{T}({\bf R},\tau^{\prime})=e^{{\bf i}\left(\sum_{j}^{N_{e}}{\bf k\cdot
r}_{j}\right)}\mathcal{U}({\bf R})$ (36)
with
$\displaystyle\mathcal{U}(\\{{\bf r_{1}},\cdots,{\bf r}_{j},\cdots,{\bf
r_{N_{e}}}\\})=\mathcal{U}(\\{{\bf r_{1}},\cdots,{\bf r}_{j}+{\bf
a},\cdots,{\bf r_{N_{e}}}\\})$
for any $j$, where ${\bf
a}=a_{x}n_{x}{\bf\hat{\i}}+a_{y}n_{y}{\bf\hat{\j}}+a_{z}n_{z}{\bf\hat{k}}$,
with $n_{\mu}$ being arbitrary integers. $\mathcal{U}({\bf R})$, in turn,
fn:taup can be written as a product of a multi-determinant expansion times a
Jastrow factor. Each orbital entering each determinant in $\mathcal{U}({\bf
R})$ can be expanded in plane waves that satisfy periodic boundary conditions.
The theory developed in Section II can then be applied to periodic systems by
setting ${\bf A}_{j}=0$. Note, however, that since $\mathcal{U}({\bf R})$ is
in general a complex function, the phase entering in $E_{L}({\bf R})$ [see Eq.
8] must include both the phase of $\mathcal{U}({\bf R})$ and the many-body
Bloch phase. The resulting wave function $\Psi_{T}({\bf
R},\tau^{\prime}\rightarrow\infty)$ corresponds, in general, to a state with
current, and its solution can facilitate the calculation of transport problems
in a many-body context krcmar08 . The many-body Bloch phase component on the
trial wave function is often referred to in the literature as twisted boundary
conditions tbc ; fn:tbc .
### III.1 The model periodic system
Until this new development, the author had promised himself to halt
calculations in small model systems rosetta ; keystone ; rockandroll . Those
small model systems, however, while not very realistic, allow comparisons to
be performed with fully converged CI calculations. In addition, they can be
handled with symbolic programs like Mathematica, which, while computationally
very slow, are an ideal environment for developing new methods and comparing
the results with nearly analytical values. Therefore, small model systems
provide an ideal “workbench” for testing new theories and algorithms. On the
other hand, no method that fails in the simplest case has hopes of succeeding
in a realistic calculation involving the more challenging Coulomb interaction
with a large number of electrons. Past experience has shown, in contrast, that
earlier SHDMC developments tested and developed in small models keystone
could be implemented easily in realistic cases without additional
complications rollingstones . Indeed, calculations using this method in QWALK
wagner09 reproduced the results obtained for the triplet state of He for low
magnetic fieldsjones99 starting from a random linear combination of
determinants constructed with the Hartree-Fock solutions without magnetic
field. All electron calculations of atomic systems with tens of electrons are
currently under progress and will be published elsewhereelsewhere .
DMC calculations with a realistic Coulomb interaction in periodic systems
require a supercell large enough to prevent unphysical image interactions
between periodic replicas of the electrons from dominating the result. For the
purpose of testing the method, however, a model electron-electron interaction
can be chosen, and the system can be made as small as required for numerical
convenience. For validating the method, the Hamiltonian does not need to be
strictly realistic; however, one must solve the same Hamiltonian with SHDMC
and an established benchmark method (CI in this case).
The model studied in this section is related to the one considered in Refs.
rosetta, ; keystone, , and rockandroll, and consists of two spinless
electrons in a square of side $1$. However, instead of the hard-wall boundary
conditions used earlier, periodic boundary conditions are set.
Basis expansion: The ground state of the noninteracting system is degenerate.
Two states with zero total momentum can be constructed by placing two
electrons with opposite momenta ${\bf k}=\pm\pi{\bf\hat{\i}}$ or ${\bf
k}=\pm\pi{\bf\hat{\j}}$. The basis chosen to expand the wave function is an
antisymmetric combination of free-particle solutions that satisfy periodic
boundary conditions, which are plane waves of the form
$\displaystyle e^{2\pi{\bf i}\left[(n+1/2)x+my\right]}$ (37)
where $|n+1/2|<6$ and $|m|<5$, which results in a two-body basis with 1516
functions.
The confining potential and the interaction potential selected do not mix the
directions ${\bf\hat{\i}}$ and ${\bf\hat{\j}}$. They are given by
$\displaystyle V({\bf R})=\;$ $\displaystyle 4\pi^{2}\left\\{\cos(2\pi
x_{1})\\!+\\!\cos(2\pi x_{2})\\!+\\!\cos(2\pi y_{1})\\!+\\!cos(2\pi
y_{2})\right.$
$\displaystyle+\left.\cos[2\pi(x_{2}-x_{1})]+\cos[2\pi(y_{2}-y_{1})])\right\\}.$
(38)
The first line of Eq. (III.1) corresponds to an external potential applied to
electrons $1$ and $2$. The second line plays the role of an interaction
potential that depends on the difference between the electronic coordinates.
The Jastrow factor is set to zero to facilitate the analytical calculation of
the matrix elements of $V({\bf R})$, while the kinetic energy is a diagonal
matrix. The exact diagonalization of the Hamiltonian matrix is the CI result.
For $\hbar=1$, the energy difference units between the noninteracting ground
state and first excited states is $4\pi^{2}$. Since the interaction energy in
Eq. (III.1) is of the same order of magnitude as the kinetic energy, the
system is in the correlated regime.
### III.2 Results and discussion
Figure 1 shows the logarithm projection
$L_{P}(n)=ln|\langle\Psi_{n}^{CI}|\Psi_{T}(\ell\tau)\rangle|$ of the trial
wave function $|\Psi_{T}(\ell\tau)\rangle$ onto the $n$ eigenstate of the full
CI solution $|\Psi_{n}^{CI}\rangle$ as a function of the recursive iteration
index $\ell$. The wave function is constrained by the basis to have a many-
body Bloch phase $\phi({\bf R})=\exp[{\bf i}({\bf k\cdot r}_{1}+{\bf k\cdot
r}_{2})]$ with ${\bf k}=0.9\pi({\bf\hat{\i}}+{\bf\hat{\j}})$ (that is a twist
angle of $1.8\pi$ tbc ; fn:tbc both in the ${\bf\hat{i}}$ and ${\bf\hat{j}}$
directions).
The initial trial wave function $|\Psi_{T}(0)\rangle$ was chosen intentionally
to be of poor quality to demonstrate the strength of the method. The
coefficients of $|\Psi_{T}(0)\rangle$ corresponded to a linear combination of
the first 16 full CI eigenstates:
$|\Psi_{T}(0)\rangle=\sum_{n}c_{n}|\Psi_{n}^{CI}\rangle$, where the
coefficients $c_{n}$ are complex numbers of modulus $1/4$ and a random phase.
Note that the initial trial wave function has no nodes but at the coincidental
points because is a linear combination with random phase of different
eigenstates of the non interacting Hamiltonian with different nodes. The
calculation was run for 200 walkers with $\delta\tau=0.0004$ and $\tau=0.02$.
The coefficients $\lambda_{n}$ were sampled at the end of each sub-block of
$k=50$ DMC steps. At first the number of sub-blocks $M$ sampled before a wave
function update was set to $20$ and increased according to the recipe given in
Ref. rockandroll, and briefly in Section II. Therefore, the statistical error
is reduced, and the number of basis functions retained in the expansion
increases over time. As a result both the statistical error and the truncation
error diminish, and the wave function continues to improve. The final
iteration included $M=600$ blocks. The total optimization run cost $\approx
1.5\times 10^{5}$ DMC steps.
Figure 1 shows in increasingly lighter shading the results $L_{P}(n)$
corresponding to higher excited states. All the projections to the first 16
states start from the same value [$-\ln(4)$] by construction. The algorithm,
at first, increases the projection of the lower energy states at the expense
of the higher ones (thus $L_{P}(n)$ approaches zero for low $n$), while the
projections with higher $n$ (in lighter gray) become smaller and their
$L_{P}(n)$ is increasingly negative. As the algorithm progresses further, the
projection on lower-energy excitations also starts to decay. Finally,
$L_{P}(n)$ becomes increasingly negative for all states except the ground
state, which approaches zero.
As the number of recursive iterations $\ell$ increases, the projection onto
highly excited states becomes negligible. The values obtained for $L_{p}(n>0)$
are, therefore, dominated by statistical noise in the sampling. On the right
side of Fig. 1, the convergence of the wave function is no longer limited by
the initial trial wave function but by the statistical noise. Statistical
noise introduces a projection into higher excited states by two mechanisms:
(i) the coefficients $\lambda_{n}$ of the trial wave function expansion
include random noise and (ii) the trial wave function develops a projection
into excited states because it is truncated depending on the relative error of
$\lambda_{n}$, which in turn depends on $N_{c}$ keystone ; rockandroll .
Accuracy can be increased only by improving the statistics (increasing $M$ and
$N_{c}$).
The residual projection of the trial wave function
$|\Psi_{T}(\ell\tau)\rangle$ for iteration $\ell$ on the CI eigenstate
$|\Psi_{n}^{CI}\rangle$ is defined as
$L_{rp}^{n}=\ln\left(1-|\langle\Psi_{n}^{CI}|\Psi_{T}(\ell\tau)\rangle|\right).$
(39)
The final value for the residual projection for the calculation in Fig. 1 is
below $-7$. The value obtained for the SHDMC energy is -31.842(13) as compared
with a CI value of $-31.9486$. However, the SHDMC wave function retains only
$70$ coefficients in the expansion, whereas the CI has $1516$. The FPDMC
energy obtained with this wave function was $-32.00(2)$.
The results shown in Figure 1 demonstrate that the SHDMC method with complex
weights is able to correct both the phase and the nodal structure of the trial
wave function. SHDMC converges to the ground-state even starting from a poor
quality wave function with a random phase.
Figure 1: Logarithm of the projection of the trial wave function into the
lowest 16 eigenstates obtained with CI
[$L_{P}(n)=ln|\langle\Psi_{n}^{CI}|\Psi_{T}(\ell\tau)\rangle|$] as a function
of the SHDMC iteration index $\ell$. The results correspond to two electrons
in the triplet state with periodic boundary conditions and a many-body Bloch
phase $\phi({\bf R})=\exp[{\bf i}({\bf k\cdot r}_{1}+{\bf k\cdot r}_{2})]$
with ${\bf k}=0.9\pi({\bf\hat{\i}}+{\bf\hat{\j}})$ (that is a twist angle of
$1.8\pi$ tbc ; fn:tbc ). Darker symbols correspond to the projection with
lower energy CI eigenstates. The initial trial wave function was a linear
combination of the lowest 16 CI eigenstates with coefficients having the same
modulus and a random complex phase.
### III.3 Many-body band structure
Common electronic structure methods are based on a single-particle picture,
and the band structure is given by the evolution of the energy as a function
of the single-particle crystalline momentum. In this case, in contrast, the
energy of many-body states is a function of the many-body Bloch phase $e^{{\bf
i}\left(\sum_{j}^{N_{e}}{\bf k\cdot r}_{j}\right)}$ or the twist angle tbc ;
fn:tbc .
Figure 2 shows the many-body band structure for the ground and first excited
states as a function of the global crystalline momentum ${\bf
k}=k_{x}{\bf\hat{\i}}$ obtained for the same system studied in Fig. 1. The
calculations were done using the same parameters as in Fig. 1 described above.
The trial wave function for the ground state with ${\bf k}=0$ started from a
linear combination of the ground and first excited states of the free-particle
system with $\lambda_{0}=\lambda_{1}=1/\sqrt{2}$. For ${\bf k}\neq 0$, the
initial trial wave function for the ground state was constructed using the
Bloch part of the converged wave function with smaller $|{\bf k}|$. The
initial trial wave function for the first excited state for ${\bf k}=0$ was
constructed using a linear combination including $\lambda_{0}$ and
$\lambda_{1}$ orthogonal to the ground state. The trial wave functions for the
first excited states for ${\bf k}\neq 0$ were constructed using the Bloch part
of a converged previous calculation with the closest value of ${\bf k}$ and
using the projector $\hat{P}$ to orthogonalize it with the ground state. CI
results are shown with lines for validation of the SHDMC results in dots.
There is a very good agreement between the values obtained with Quantum Monte
Carlo and CI. In general, however, the Monte Carlo values have a higher energy
than the CI values. This is due to both the error in the complex phase and the
nodal error since the SHDMC wave function only retains $\approx 70$ of the
1516 basis functions retained in the CI. The energy difference is reduced
systematically as the algorithm progresses and more coefficients $\lambda_{n}$
are retained in the trial wave function.
Figure 2: Many-body band structure of a periodic model system with two
electrons in the triplet state obtained with SHDMC (dots) and compared with CI
(lines). The figure shows the energy of the ground state and the first excited
state as a function of the global crystalline momentum ${\bf k}$ (i.e., the
many-body Bloch phase or twist angle).
## IV Ground and excited states with applied magnetic field
Figure 3: SHDMC results (dots) obtained for (a) the first excited state and
(b) the ground state of a model system compared with CI results (lines) as a
function of the magnetic field. The system consists of two electrons in a two-
dimensional square in the triplet state. The results shown correspond to the
ground and first excited states with $E$ symmetry that transform as $x+{\bf
i}y$. The solution that transforms as $x-{\bf i}y$ can be obtained by changing
the sign of $B$.
This section describes the results obtained with the generalization of SHDMC
(described in Section II) for the ground and the first excited state of a
model system with an applied magnetic field. The results are compared with CI
calculations in the same model used in Refs. rosetta, ; keystone, and
rockandroll, .
### IV.1 Model system with magnetic field
Briefly, the lower energy eigenstates are found for two spinless electrons
moving in a two-dimensional square with a side length $1$ and a repulsive
interaction potential of the form $V({\bf r},{\bf
r^{\prime}})=8\pi^{2}\gamma\cos{[\alpha\pi(x-x^{\prime})]}\cos{[\alpha\pi(y-y^{\prime})]}$
with $\alpha=1/\pi$ and $\gamma=4$. The many-body wave function is expanded in
functions $\Phi_{n}({\bf R})$ that are eigenstates of the noninteracting
system. The basis functions in $\\{\Phi_{n}({\bf R})\\}$ are linear
combinations of functions of the form $\prod_{\nu}\sin(m_{\nu}\pi x_{\nu})$
with $m_{\nu}\leq 7$. Converged CI calculations were performed to obtain a
nearly exact expression of the lower energy states of the system
$\Psi_{n}({\bf R})=\sum_{m}a_{m}^{n}\Phi_{m}({\bf R})$. The matrix elements
involving the magnetic vector potential ${\bf A}$ (in the symmetric gauge)
were calculated analytically using the symbolic program Mathematica and were
included in the CI Hamiltonian. The Jastrow factor was set to zero in the
SHDMC run to facilitate a direct comparison between CI and SHDMC results.
This paper reports results for the triplet case. In the absence of a magnetic
field, the triplet ground state is degenerate. Its orbital symmetry
corresponds to the E symmetry of the D4 group. One of the solutions with E
symmetry transforms as $x$ and the other as $y$. Under an applied magnetic
field, the time reversal symmetry is broken, and the $x$ and $y$ solutions are
mixed. Under a magnetic field, the ground state can be expanded in a basis of
functions that transform as $x\pm{\bf i}y$. The energy of the $x-{\bf i}y$
solution can be obtained from the energy of $x+{\bf i}y$ by changing $B$ to
$-B$.
### IV.2 Results and discussion
Figure 3 shows energies of the ground state and first excited state of the
model system as a function of the magnitude of the magnetic field $B$ (the
curl of the vector potential ${\bf A}$). The calculations were run using
$\delta\tau=0.00004$ and $\tau=0.002$ and a total number of DMC steps of
$10^{5}$ for each calculated point. The calculation for the ground state
started from the noninteracting ground-state solution as a trial wave
function. The result obtained for $B=0$ for the ground and first excited
states compared well with the ones obtained with the same Hamiltonian in the
triplet case reported fn:errorCI in Table I of Ref. rockandroll, . Note that
in this case, the wave function is complex and the coefficients have the
freedom to be complex. Thus, in contrast with Ref. rockandroll, , where a real
wave function was enforced, here the phase was found within statistical error.
For $B\neq 0$, the time reversal symmetry is broken and so is the degeneracy
of the $x\pm{\bf i}y$ solutions. For higher (lower) magnetic fields, the
calculation began by using as the initial trial wave function the one obtained
previously with a lower (higher) magnetic field.
The excited states were obtained using the method outlined in subsection II.3
and described in detail in Ref. rockandroll, . The lines show the CI results
for reference. The calculation for the first excited state with $B=0$ started
from a linear combination of the ground and first excited state of the
noninteracting system orthogonal to the interacting ground state calculated
earlier. The initial trial wave functions of the excited states for $B\neq 0$
were taken from the previous calculations with smaller $|B|$ (keeping the wave
function orthogonal to the lower energy states with the operator $\hat{P}$).
Clearly, Fig. 3b shows good agreement between SHDMC and CI results for the
first excited state.
Table 1 summarizes the values obtained to construct Fig. 3. There is an
excellent agreement in the calculations obtained for the ground state using
SHDMC and CI. The SHDMC energy values are, within error bars converged FPDMC
results indicating that the remaining convergence errors in the basis are
small. The agreement is less satisfactory for the excited states than in the
ground state (using the same computational time). It its clear that the
residual projections are much larger for the excited state than for the
ground.
An independent way to measure the quality of the wave function is the
logarithm of the variance of the modulus of the weights given by
$L_{var}=\ln{\sqrt{\frac{1}{N_{c}}\sum_{i,j}(|W_{i}^{kj}(k)|-1)^{2}}}\;.$ (40)
The variance of the weights does not deteriorate as much as the residual
projection for excited states, which might signal that the differences in the
wave functions originate because CI and SHDMC minimize different things using
a truncated basis. rockandroll
Table 1: Comparison of the excitation energies obtained for the ground and
the first excited state of a model system with two spinless electrons and an
applied magnetic field (see Fig. 3). $L_{rp}$ quantifies the overlap of the
wave functions obtained with CI and SHDMC [see Eq. (39)]. $L_{var}$ is the
variance of the modulus of the walkers‘ weights [see Eq. (40)].
—Ground State—
$B$ | $E_{0}$ (SHDMC) | $E_{0}$ FPDMX | $E_{0}$ (CI) | $L^{0}_{rp}$ | $L_{var}$
---|---|---|---|---|---
-3.2 $\pi$ | 337.823 (13) | 337.820(7) | 337.821 | -9.9 | -4.7
-1.6 $\pi$ | 338.877 (4) | 338.867(4) | 338.870 | -12.7 | -5.3
-0.8 $\pi$ | 340.261 (7) | 340.256(5) | 340.256 | -12.9 | -5.7
-0.4 $\pi$ | 341.143 (6) | 341.153(5) | 341.162 | -10.3 | -5.9
-0.2 $\pi$ | 341.646 (11) | 341.662(6) | 341.667 | -13.6 | -6.0
-0.1 $\pi$ | 341.931 (5) | 341.930(7) | 341.933 | -14.2 | -6.1
0.0 $\pi$ | 342.207 (7) | 342.206(5) | 342.208 | -11.9 | -6.1
0.2 $\pi$ | 342.771 (6) | 342.782(6) | 342.782 | -12.4 | -6.0
0.4 $\pi$ | 343.387 (5) | 343.392(4) | 343.390 | -10.8 | -6.0
0.8 $\pi$ | 344.696 (8) | 344.689(6) | 344.704 | -11.7 | -5.8
1.6 $\pi$ | 347.699 (8) | 347.684(5) | 347.697 | -9.4 | -5.2
—First Excited State— $B$ $E_{1}$ (SHDMC) $E_{1}$ (CI) $L^{1}_{rp}$ $L_{var}$
-1.6 $\pi$ 394.161 (19) 394.114 -7.4 -5.0 -0.8 $\pi$ 391.532 (12) 391.504 -7.9
-5.4 -0.4 $\pi$ 389.744 (12) 389.741 -9.1 -5.7 -0.2 $\pi$ 388.786 (10) 388.769
-9.9 -5.8 -0.1 $\pi$ 388.253 (13) 388.265 -9.8 -5.7 0.0 $\pi$ 388.205 (44)
387.750 -5.7 -5.0 0.2 $\pi$ 386.697 (17) 386.694 -8.9 -5.7 0.8 $\pi$ 383.415
(14) 383.407 -8.9 -5.4 1.6 $\pi$ 379.159 (28) 379.057 -7.9 -4.8
## V Test with Coulomb interactions
The calculations with Coulomb interactions were performed in the same system
studied for the ground state in Ref. keystone, and for excited states in Ref.
rockandroll, but now with the additional ingredient of an applied magnetic
field. The more challenging triplet (antisymmetric) state was chosen for this
study.
The calculations were run with the same parameters and basis as in Fig. 3 and
Table 1 but with a Coulomb interaction potential of the form $V({\bf r},{\bf
r^{\prime}})=20\pi^{2}/|{\bf r-r\prime}|$. Since the average of the Coulomb
interaction is much larger than the single-particle energy differences, the
system is in the highly correlated regime.
Table 2: SHDMC and FPDMC energies as a function of an applied magnetic field for a model system with two electrons in a triplet state in a square box with Coulomb interactions. The quality of the wave function is measured by $L_{var}$ [see Eq. (40)]. State | $B$ | SHDMC | PFDMC | $L_{var}$
---|---|---|---|---
0 | -1.60 $\pi$ | 401.65 (2) | 401.67(4) | -4.1
0 | -1.26 $\pi$ | 401.80 (3) | | -4.2
0 | -0.80 $\pi$ | 401.92 (3) | 401.87(4) | -4.2
0 | -0.40 $\pi$ | 403.50 (6) | 402.39(7) | -3.4
0 | -0.20 $\pi$ | 402.97 (4) | 402.60(5) | -4.1
0 | 0.00 | 402.76 (4) | 402.58(3) | -4.0
0 | 0.40 $\pi$ | 403.23 (2) | 403.20(3) | -4.6
0 | 0.80 $\pi$ | 403.87 (3) | 403.73(3) | -4.2
0 | 1.26 $\pi$ | 404.93 (6) | | -3.7
0 | 1.60 $\pi$ | 405.54 (9) | 405.16(4) | -3.8
1 | -0.40 $\pi$ | 465.37 (10) | | -3.0
1 | -0.20 $\pi$ | 468.55 (7) | | -3.5
1 | 0.00 | 454.39 (8) | | -3.4
1 | 0.40 $\pi$ | 451.76 (8) | | -3.2
2 | -0.40 $\pi$ | 486.89 (7) | | -3.2
Table 2 displays the values obtained for the model system with Coulomb
interactions for the ground state and some excitations as a function of the
magnetic field. The quality of the wave function is characterized by the
logarithm of the variance of the modulus of weights given by Eq. (40). Note
that the variance of the weights increased when Coulomb interactions are
considered when compared with the case of the model interaction. This is due
to the Coulomb singularity and the lack of a Jastrow factor. While the
variance of the weights is larger in the Coulomb case, the quality of the wave
function improves from one SHDMC recursive iteration to the next (see below).
### V.1 Improvement of the wave function’s node and phase with SHDMC
Figure 4 shows the evolution of the real (a) and the imaginary (b) parts of
the local energy $E_{L}({\bf R})$ as a function of the DMC step for the first
excited state of two electrons in a square box with an applied magnetic field
of $0.4\pi$. The calculations started with a trial wave function with two
nonzero coefficients chosen to be orthogonal to the ground state calculated
earlier. It can be clearly seen in Fig. 4(a) that as the SHDMC algorithm
progresses, the real part of the local energy is quickly reduced and
stabilized at the first excited state energy. The imaginary part of the
$E_{L}({\bf R})$ should be zero for an eigenstate; otherwise, the divergence
of the current is nonzero ortiz93 . In SHDMC this strong condition is
satisfied only as the number of recursive iterations, the number of
configuration sampled $N_{c}$, and the size of the basis $N_{b}$ retained in
the wave function tend to infinity. Figure 4(b), however, clearly shows that
the variance of the raw data obtained for $Im[E_{L}({\bf R})]$ is reduced as
the SHDMC algorithm progresses. This is a clear indication of improvement of
the phase of the wave function.
Figure 4: (Color online) (a) Average of the real part of $E_{L}({\bf R})$
obtained with 200 walkers as a function of the DMC step. The results
correspond to the first excited state that transforms as $x+{\bf i}y$ with $E$
symmetry of the group $D_{4}$ of two electrons with Coulomb interactions in a
square box with an applied magnetic field units $B=0.4\pi$. (b) Average of
the imaginary part of $E_{L}({\bf R})$ as a function of the DMC step for the
same case. The vertical lines mark the end of the SHDMC block when the wave
function is updated.
Figure 5 shows the evolution of logarithm of the variance of the weights [see
Eq. (40)] as a function of the SHDMC block index (the number of wave function
updates). The reduction in weight variance is a clear indication of
convergence of the trial wave function towards an eigenstate of the
Hamiltonian rockandroll .
Figure 5: (Color online) Logarithm of the variance of the modulus of weights
as a function of the SHDMC block index $\ell$. The results correspond to the
first excited state with $B=0.4\pi$ and Coulomb interactions (the run shown in
Fig. 4).
## VI Summary and perspectives
A method that allows the calculation of the complex amplitude of a many-body
wave function has been presented and validated with model calculations. An
algorithm that finds the complex wave function is essential for any study of
many-body Hamiltonians with periodic boundary conditions or under external
magnetic fields. The method converges to nearly analytical results obtained
for model systems under applied magnetic fields or periodic boundary
conditions, with an accuracy limited only by statistics and the flexibility of
the wave function sampled.
It is found that for some eigenstates, the ones where the phase is a scalar
function of ${\bf R}$, there is a special gauge transformation in which wave
function is real. For this class of eigenstates the original proof of
convergence of SHDMC applies. For complex wave functions some fermionic
eigenstates may not have nodes. In the latter case, as in the case of
bosonsrockandroll ; reatto82 , the convergence of SHDMC is not affected since
the wave function can evolve everywhere.
This new approach goes beyond both fixed-phase DMC ortiz93 and SHDMC keystone
; rockandroll ; rollingstones . As in the real wave function version of SHDMC,
the method is recursive and the propagation to infinite imaginary time is
achieved as the number of iterations increases. As in FPDMC, the walkers
evolve under an effective potential that incorporates the gradient of the
phase of the trial wave function and the vector potential of the magnetic
field. But in this new algorithm, in contrast to FPDMC, the complex amplitude
of the wave function is free to adjust both its modulus and its phase. After
each iteration, the trial wave function is improved following a short time
evolution of an ensemble of walkers. These walkers follow the equation of
motion of a generalized importance sampling approach. Unlike previous
attempts, the walkers carry a complex weight resulting from eliminating the
fixed-phase constraint in the time evolution of the mixed probability density.
The modulus of the weight can be used to calculate real observables, such as
the energy. The phase of the weight of the walkers is used to improve the
phase of the trial wave function in the following iteration. As in earlier
versions of SHDMC, the modulus of the weights is also used to improve,
simultaneously, the node if there is any and the phase of the trial wave
function.
This free amplitude SHDMC method can be used to calculate not only the ground
state but also low energy excitations rockandroll within a DMC context.
Comparisons with nearly analytical results in model systems demonstrate that
the new approach converges to the many-body wave function of systems with
applied magnetic fields or with periodic boundary conditions for low energy
excitations.
This recursive method finds a solution to “the phase problem” and, if there is
any, finds the node at the same time. The many-body wave function can be used,
in principle, to calculate any observable. However, in very large systems,
when convergence with the size of the wave function basis cannot be fully
achieved, a standard fixed-phase calculation should be performed as a final
step to obtain a more accurate energy.
Scaling and cost: An analysis of the minimum cost required to determine the
node and the phase has to take into account the number of independent degrees
of freedom of the Hilbert space. Arguably, no method could scale better than
linear in the number of independent degrees of freedom of the problem studied;
otherwise, some degrees of freedom would be dependent from each other. A real-
space expansion of the many-body wave function with fixed resolution $L_{R}$
is ideal for counting independent degrees of freedom. The resolution $L_{R}$
can be connected to the energy cutoff of the excitations in a multideterminant
expansion. For a complex wave function, each point in the many-body space
${\bf R}$ has two independent degrees of freedom (modulus and phase). If the
volume of a system is proportional to the number of electrons $N_{e}$, its
size scales as $L\approx\alpha N_{e}^{1/3}$ (where $\alpha$ is of the order of
the Bohr radius $a_{B}$). Taking into account the $N_{e}!$ permutations of
identical particles, one finds $(L/L_{R})^{3N_{e}}/N_{e}!$ independent degrees
of freedom for each spin channel to determine the phase $\phi({\bf R})$. Thus,
the number of independent degrees of freedom scales as
$\exp\\{N_{e}(3\log(\alpha/L_{R}))+1)\\}$. The node $S_{T}({\bf R})$, if there
is any, requires one less dimension (which, if the nodal surface is not too
convoluted, could reduce the number of degrees of freedom by only up to a
factor $(L/L_{R})$). Since the number of independent degrees of freedom of the
phase increases exponentially with $N_{e}$, for a fixed resolution $L_{R}$ one
cannot find the phase with an algorithm polynomial in $N_{e}$.
This generalization of the SHDMC method, though tested in small systems, is
targeted to be used in large systems. The numerical cost of SHDMC scales
linearly with the number of independent degrees of freedom of the phase per
recursive step. However, the number of independent degrees of freedom (i.e,
the size of the basis expansion) should increase exponentially with the number
of electrons $N_{e}$ for a fixed resolution. The accuracy of SHDMC is limited
by the size of the basis sampled, the statistical error, and the number of
recursive iterations. The number of recursive steps required increases if the
product between $\tau$ and the lowest energy excitations is small. The SHDMC
method can be used in combination with other optimization approaches to
accelerate convergence in that limit.
The scaling of the cost of exact diagonalization methods such as CI is at
least quadratic with the number of degrees of freedom. Often a CI calculation
is used to preselect a multideterminant expansion to be improved within a VMC
context before a final FPDMC run. An advantage of SHDMC is that it
incorporates the Jastrow in the sampling of the coefficients. Thus SHDMC might
be more efficient than a CI filtering for large systems. The linear scaling of
SHDMC suggests that it could be the method of choice to optimize the wave
function phase and nodes for calculations in periodic solids.
The optimization of many-body wave functions with current in periodic boundary
conditions is now possible. Therefore, the new method can be used as a tool to
perform transport calculations including many-body effects. The calculation of
systems with an applied magnetic field is challenging, even in the case of
small molecules and atoms and particularly so when the magnetic field, the
many-body interactions, and the kinetic energy are of the same order of
magnitude jones96 . The calculations reported in this paper, though in a
simple model, suggest that the method can be applied to the study of molecular
or atomic systems in that difficult regime.
Our recent successful application of the ground-state algorithm for real wave
functions keystone to molecular systems rollingstones supports the idea that
this generalization of SHDMC can also be useful for real ab-initio
calculations beyond model systems. The implementation of the algorithm in
state-of-the-art DMC codes has been done. Initial results in atomic systems
show that the many-body wave function improves, which is shown by a reduction
of the average local energy, the energy variance and the variance of the
imaginary contribution to the total energy.
###### Acknowledgements.
The author would like to thank J. McMinis for an introduction to the fixed-
phase approximation and P. R. C. Kent, and G. Ortiz for a critical reading of
the manuscript. The author also thanks M. Bajdich for sharing all electron
calculations in atomic systems using this method as supplemental material for
the referees prior publication. Research sponsored by the Materials Sciences &
Engineering Division of the Office of Basic Energy Sciences U.S. Department of
Energy.
## References
* (1) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
* (2) J. B. Anderson, Int. J. Quantum Chem. 15, 109 (1979).
* (3) P. J. Reynolds, D. M. Ceperley, B. J. Alder, and W. A. Lester, J. Chem. Phys. 77, 5593 (1982).
* (4) B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry (World Scientific, Singapore-New Jersey-London-Hong Kong, 1994).
* (5) D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
* (6) G. Ortiz, D. M. Ceperley, and R. M. Martin, Phys Rev. Lett. 71, 2777 (1993).
* (7) A. J. Williamson, R. Q. Hood, and J. C. Grossman, Phys. Rev. Lett. 87, 246406 (2001).
* (8) D. Alfe and M. J. Gillan, Phys. Rev. B 70, 161101(R) (2004).
* (9) F. A. Reboredo and A. J. Williamson, Phys. Rev. B 71, 121105(R) (2005).
* (10) D. M. Ceperley, J. Stat. Phys. 63, 1237 (1991).
* (11) M. Troyer and U. J. Wiese, Phys. Rev. Lett. 94, 170201 (2005).
* (12) W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001).
* (13) G. Ortiz, and D. M. Ceperley Phys. Rev. Lett. 75, 4642 (1995).
* (14) M. D. Jones, G. Ortiz, and D. M. Ceperley, Phys. Rev. E, 55, 6202, (1997).
* (15) A. D. Güçlü and C. J. Umrigar, Phys. Rev. B, 72, 045309 (2005); A. D. Güçlü, G. S.. Jeon, C. J. Umrigar and J. K. Jain, Phys. Rev. B 72, 205327 (2005); G. S. Jeon, A. D. Güçlü, C. J. Umrigar, and J. K. Jain, Phys. Rev. B 72, 245312, (2005).
* (16) C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Phys. Rev. Lett. 98, 110201 (2007).
* (17) A. Lüchow, et al., J. Chem. Phys. 126, 144110 (2007).
* (18) F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B 79, 195117 (2009).
* (19) F. A. Reboredo, Phys. Rev. B 80, 125110 (2009).
* (20) M. Bajdich, M. L. Tiago, R. Q. Hood, P. R. C. Kent, and F. A. Reboredo, Phys. Rev. Lett. 104, 193001 (2010).
* (21) $\Psi_{FN}({\bf R})$ is not obtained; the new $\Psi_{T}({\bf R})$ is sampled directly.
* (22) F. A. Reboredo and P. R. C. Kent, Phys. Rev. B 77, 245110 (2008).
* (23) A complex energy reference stabilizes the run for arbitrary gauge choices for the vector potential ${\bf A}$.
* (24) Equation (2) in OCM work is only strictly valid if $Im[E_{L}({\bf R})]=0$, otherwise it is the so-called fixed-phase approximation ortiz93 .
* (25) A complete basis in the symmetric space can be used to calculate bosons. Other symmetries of the wave function can be enforced with the selection of the basis rockandroll .
* (26) Since $\tau^{\prime}=\tau\ell$ is essentially an iteration index, it is omitted in the trial wave function phase and amplitude for clarity.
* (27) C. J. Umrigar, M. P. Nightingale, and K. J. Runge, J. Chem. Phys. 99, 2865 (1993).
* (28) C. Umrigar (private communication).
* (29) L. Reatto, Phys. Rev. B 26, 130 (1982).
* (30) The position of the cut in the complex plane of the Riemann surface into sheets is arbitrary. Therefore, the discontinuities in the gradient and the effective potential are no physical if they can be removed changing $n$.
* (31) J. D. Jackson, Classical Electrodynamics third edition (John Willey & Sons, Inc) New York, (1998).
* (32) D. M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988); B. Bernu, D. M. Ceperley, and W. A. Lester, Jr., J. Chem. Phys. 93, 552 (1990).
* (33) W. Purwanto, S. Zhang, and H. Krakauer, J. Chem. Phys. 130, 094107 (2009).
* (34) K. Hoffman and R. Kunze, Linear Algebra second edition (Prentice -Hall) New Jersey (1971).
* (35) E. Prugovec̆ki Quantum Mechanics in Hilbert Space (Academic Press) New York (1981).
* (36) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing Harcourt Brace College Publishers, 1976).
* (37) G. Rajagopal, R. J. Needs, A. James, S. D. Kenny, and W. M. C. Foulkes, Phys. Rev B 51, 10591 (1995).
* (38) R. Krc̆már, A. Gendiar, M. Mos̆ko, R. Németh, P. Vagner, L. Mitas Physica E 40,1507 (2008).
* (39) C. Lin, F. H. Zong, and D. M. Ceperley, Phys. Rev. E 64, 016702 (2001).
* (40) The term “twist angle” was introduced tbc to avoid confusion with other usages of the term “phase”. Here the term “many-body Bloch phase” is also mentioned since the Bloch theory is well understood outside the many-body field.
* (41) L.K. Wagner, M. Bajdich, and L. Mitas, J. Comp. Phys. 228, 3390 (2009).
* (42) M. D. Jones, G. Ortiz, and D. M. Ceperley, Phys Rev. A 59, 2875 (1999).
* (43) F. A. Reboredo, M. Bajdich and P. R. C. Kent (work in progrees).
* (44) The energy unit is $\hbar^{2}/(2mL^{2})$ and the magnetic unit is given by $e/(c\hbar m^{(1/2)})$.
* (45) There was a small error in the CI calculations reported in Ref. rockandroll, .
* (46) M. D. Jones, G. Ortiz, and D. M. Ceperley, Phys Rev. A 54, 219 (1996).
|
arxiv-papers
| 2010-07-19T21:50:14 |
2024-09-04T02:49:12.000014
|
{
"license": "Public Domain",
"authors": "Fernando Agust\\'in Reboredo",
"submitter": "Fernando Reboredo",
"url": "https://arxiv.org/abs/1008.0359"
}
|
1008.0385
|
# Research Announcement: Finite–time Blow Up and Long–wave Unstable Thin Film
Equations
Marina Chugunova, M.C. Pugh, Roman M. Taranets
###### Abstract
We study short–time existence, long–time existence, finite speed of
propagation, and finite–time blow–up of nonnegative solutions for long-wave
unstable thin film equations
$h_{t}=-a_{0}(h^{n}h_{xxx})_{x}-a_{1}(h^{m}h_{x})_{x}$ with $n>0$, $a_{0}>0$,
and $a_{1}>0$. The existence and finite speed of propagation results extend
those of [Comm Pure Appl Math 51:625–661, 1998]. For $0<n<2$ we prove the
existence of a nonnegative, compactly–supported, strong solution on the line
that blows up in finite time. The construction requires that the initial data
be nonnegative, compactly supported in $\mathbb{R}^{1}$, be in
$H^{1}(\mathbb{R}^{1})$, and have negative energy. The blow-up is proven for a
large range of $(n,m)$ exponents and extends the results of [Indiana Univ Math
J 49:1323–1366, 2000].
2000 MSC: 35K65, 35K35, 35Q35, 35B05, 35B45, 35G25, 35D05, 35D10, 76A20
keywords: fourth-order degenerate parabolic equations, thin liquid films,
long–wave unstable, finite–time blow–up, finite speed of propagation
## 1 Introduction
Numerous articles have been published since the early sixties concerning the
problem of developing finite-time singularities by solutions of nonlinear
parabolic equations (see the survey papers of Levine [31] and of Bandle and
Brunner [2]). Such problems arise in various applied fields such as combustion
theory, the theory of phase separation in binary alloys, population dynamics
and incompressible fluid flow.
Whether or not there is a finite–time singularity, such as
$\|u(\cdot,t)\|_{\infty}\to\infty$ as $t\to T^{*}<\infty$, is strongly
affected by the nonlinearity in the PDE. For example, consider the semilinear
heat equation on the line:
$u_{t}=u_{xx}+u^{p},$ (1.1)
where $u$ is real–valued. If $p\leqslant 1$ then a solution of an initial
value problem exists for all time. If $1<p\leqslant 3$, then any non-trivial
solution blows up in finite time. If $p>3$ then some initial data yield
solutions that exist for all time and other initial data result in solutions
that have finite–time singularities. The manner in which solutions blow up is
well understood computationally and analytically. The blow up is of a
focussing type: there are isolated points in space around which the graph of
the solution narrows and becomes taller as $t\uparrow T^{*}$, converging to
delta functions centered at the blow–up points. As $t\uparrow T^{*}$, the
behaviour of the solution near the blow–up point(s) becomes more and more
self–similar. Proving this convergence to a self–similar solution uses the
maximum principle, which doesn’t hold for fourth–order equations like the one
we study in this article.
We study the longwave-unstable generalized thin film equation,
$h_{t}=-a_{0}\,(|h|^{n}\,h_{xxx})_{x}-a_{1}\,(|h|^{m}\,h_{x})_{x},$ (1.2)
where $a_{0}>0$, $a_{1}>0$, and $h$ is real valued. Perturbing a constant
steady state slightly, $h_{0}(x)=\bar{h}+\epsilon
h_{1}(x,0)=\bar{h}+\epsilon\cos(\xi x+\phi)$, and linearizing the equation
about $\bar{h}$, the small perturbation $h_{1}(x,t)$ will (approximately)
satisfy $h_{t}=-a_{0}|\bar{h}|^{n}h_{xxxx}-a_{1}|\bar{h}|^{m}h_{xx}$. Hence
the constant steady state is linearly unstable to long wave perturbations:
$\xi^{2}<|\bar{h}|^{m-n}a_{1}/a_{0}\quad\Longrightarrow\quad h_{1}(x,t)\sim
e^{-a_{0}\xi^{2}|\bar{h}|^{n}\left(\xi^{2}-\tfrac{a_{1}}{a_{0}}|\bar{h}|^{m-n}\right)t}\cos(\xi
x+\phi)\quad\mbox{grows}.$ (1.3)
Such equations arise in the modelling of fluids and materials. For example,
equation (1.2) with $n=m=1$ describes a thin jet in a Hele-Shaw cell [16]
where $h$ represents the thickness of the jet and $x$ is the axial direction;
if $n=3$ and $m=-1$ equation (1.2) describes Van der Waals driven rupture of
thin films [44] where $h$ represents the thickness of the film; if $n=m=3$ the
equation models fluid droplets hanging from a ceiling [21] with $h$
representing the thickness of the film, and finally if $n=0$ and $m=1$ the
equation is a modified Kuramoto-Sivashinsky equation and describes the
solidification of a hyper-cooled melt [9] where $h$ desribes the deviation
from flatness of a near planar interface. We note that in the first three
cases the solution $h$ must be nonnegative for the model to make physical
sense.
Hocherman and Rosenau [28] considered whether or not equation (1.2) could have
solutions that blow up in finite time. They conjectured that if $n>m$ then
solutions might blow up in finite time but if $n<m$ they would exist for all
time. Indeed, this conjecture is natural if one considers the linear stability
of a constant steady state $\bar{h}$: if $n>m$ the unstable band (1.3) grows
as $\bar{h}\to\infty$ suggesting that $n=m$ is critical.
Hocherman and Rosenau considered general, real–valued solutions. However, if
$n>0$ equation (1.2) may have solutions that are nonnegative for all time.
Bertozzi and Pugh [12] proposed that111In fact, the article considers (1.2)
with general coefficients: $f(h)$ and $g(h)$ instead of $|h|^{n}$ and
$|h|^{m}$ respectively. In the following, for simplicity, their results are
discussed for the power–law case. if the boundary conditions are such that the
mass, $\int h(x,t)\;dx$, is conserved then mass conservation, combined with
the nonnegativity results in a different balance: $m=n+2$ instead of $m=n$.
For such cases they introduced the regimes
$\begin{cases}m<n+2\Longrightarrow\mbox{subcritical regime}\\\
m=n+2\Longrightarrow\mbox{critical regime}\\\
m>n+2\Longrightarrow\mbox{supercritical regime}\end{cases}$
In [12], Bertozzi and Pugh considered equation (1.2) on a finite interval with
periodic boundary conditions. For a subset of the subcritical ($m<n+2$) regime
they proved some global-in-time results. Specifically, they proved that given
positive initial data, $h_{0}>0$, there is a nonnegative weak solution of
(1.2) that exists for all time if $0<n\leqslant m<n+2$. By restricting $n$
further to $0<n<3$ they can consider nonnegative initial data, $h_{0}\geqslant
0$. They prove that there a nonnegative weak solution that exists for all time
and also prove the local entropy bound needed for the finite speed of
propagation proof for $0<n<2$. For the critical ($m=n+2$) regime, they prove
that the above results will hold if the mass $\int h_{0}\;dx$ is sufficiently
small. Also, they provided numerical simulations suggesting that other initial
data can result in solutions that blow up in finite time and conjectured that
this is also true for the supercritical ($m>n+2$) regime.
In [13], they considered equation (1.2) on the line and found some analytical
results for the critical and supercritical ($m\geqslant n+2$) regimes in the
special case of $n=1$. They introduced a large class of ‘‘negative energy’’
initial data and proved that given initial data $h_{0}$ with compact support
and negative energy there is a nonnegative weak solution $h$ that blows up in
finite time: there is a time $T^{*}<\infty$ such that the weak solution
$h(\cdot,t)$ exists on $[0,T^{*})$ and
$\limsup_{t\to
T^{*}}\|h(\cdot,t)\|_{L^{\infty}}=\infty\qquad\mbox{and}\qquad\limsup_{t\to
T^{*}}\|h(\cdot,t)\|_{H^{1}}=\infty.$
The blow–up time $T^{*}$ depends only on $m$ and $H^{1}$-norm of the initial
data. We note that uniqueness of nonnegative weak solutions of (1.2) is an
open problem. Indeed, there are simple counterexamples to uniqueness for the
simplest equation $h_{t}=-(h^{n}h_{xxx})_{x}$ (see, e.g. [4]) although it is
hypothesized that solutions are unique if one considers the question within a
sufficiently restrictive class of weak solutions. For this reason, one cannot
exclude the possibility that the initial data $h_{0}$ might also be achieved
by a different weak solution, one that exists for all time.
Their proof relied on a second moment argument, found formally by Bernoff [8]:
if $h$ is a smooth compactly-supported solution of (1.2) on $[0,T^{*})$ then
the second moment of $h$ satisfies
$\int\limits_{-\infty}^{\infty}x^{2}h(x,t)\;dx\leqslant\int\limits_{-\infty}^{\infty}x^{2}h_{0}(x)\;dx+6\mathcal{E}(0)\,t$
(1.4)
holds for all $t\in[0,T^{*})$. Here, $\mathcal{E}(0)$ is the energy of the
initial data:
$\mathcal{E}(0):=\int\limits_{-\infty}^{\infty}\left\\{\tfrac{a_{0}}{2}{h_{0}}_{x}^{2}(x)-\tfrac{a_{1}}{m(m+1)}h_{0}^{m+1}(x)\right\\}\;dx.$
(1.5)
As a result, there could never be a global-in-time nonnegative smooth solution
with negative-energy initial data: for such a solution the left–hand side of
(1.4) would always be nonnegative but the right–hand side would become
negative in finite time. (This argument is strictly formal because, to date,
no-one has constructed nonnegative, compactly-supported, smooth solutions on
the line.) The blow-up result is found by first proving the short-time
existence of a nonnegative, compactly-supported, weak solution: it exists on
$[0,T_{0}]$ where the larger $\|h_{0}\|_{H^{1}}$ is, the smaller $T_{0}$ will
be. Also, the constructed solution satisfies the second moment inequality
(1.4) at time $t=T_{0}$. By ‘‘time–stepping’’ the short-time existence result,
they construct a solution on $[0,T^{*})$ such that the second moment
inequality (1.4) holds at a sequence of times $T_{i}$ with $T_{i}\to T^{*}$.
It then follows immediately that $T^{*}$ must be finite and therefore the
$\limsup_{t\to T^{*}}\|h(\cdot,t)\|_{H^{1}}$ must be infinite.
### Outline of results
The main results of this paper are: short-time existence of nonnegative strong
solutions on $[-a,a]$ given nonnegative initial data, finite speed of
propagation for these solutions if their initial data had compact support
within $(-a,a)$, and finite-time blow-up for solutions of the Cauchy problem
that have initial data with negative energy.
First, we consider equation (1.2) on a bounded interval $[-a,a]$ with periodic
boundary conditions. Given nonnegative initial data that has finite
‘‘entropy’’, in Theorem 1 we prove the short-time existence of a nonnegative
weak solution if $n>0$ and $m\geqslant n/2$. The solution is a ‘‘generalized
weak solution’’ as described in Section 2 and the entropy is introduced in
Section 3. Additional regularity is proven in Theorem 2: there is a second
type of entropy such that if this ‘‘$\alpha$-entropy’’ is also finite for the
initial data then there is a ‘‘strong solution’’ which satisfies Theorem 1 and
also has some additional regularity. We note that the work [12] described
above was primarily concerned with long-time222 Throughout this article, we
use phrases like “long-time”, “global-in-time” and “exist for all time” as
shorthand for the types of large-time results that have been proven in the
thin film literature to date: given a time $T<\infty$ there is a solution
$h(\cdot,t)$ for $t\in[0,T]$. Specifically, $T$ can be taken arbitrarily
large. existence results: for this reason the authors only addressed the
existence theory for the subcritical ($m<n+2$) case. However, given finite-
entropy initial data their methods easily imply a short-time result for
general $n>0$ as long as $m\geqslant n$. Our advance is prove the results for
the larger range of $m\geqslant n/2$. The left plot in Figure 1 presents the
$(n,m)$ parameter range for which our short-time existence results hold. The
darker region represents the parameters for which the methods of [12] would
have yielded the results. The lighter region represents the extended area
where our methods also yield results.
Figure 1: Left: the darker region represents the exponents $(n,m)$ for which
the methods of [12] yield short-time existence results like Theorems 1 and 2.
The larger, lighter region represents the additional exponents for which we’ve
proved our theorems. Middle: For $(n,m)$ in the shaded region, we prove that
the strong solutions of Theorem 2 can have finite speed of propagation
(Theorem 3). Right: For $(n,m)$ in the shaded region, we construct solutions
on the line that blow up in finite time (Theorem 4). The heavy solid line
denotes the $(n,m)$ for which this had already been done by [13].
In Theorem 3 we prove that if the initial data $h_{0}$ has compact support
then the strong solution of Theorem 2 will have finite speed of propagation.
Specifically, if the support of the initial data satisfies
$\mbox{supp}(h_{0})\subseteq[-r_{0},r_{0}]\subset(-a,a)$ then there is a
nondecreasing function $\Gamma(t)$ and a time $T_{speed}$ such that
$\mbox{supp}(h(\cdot,t))\subseteq[-r_{0}-\Gamma(t),r_{0}+\Gamma(t)]\subset(-a,a)$
for every time $t\in[0,T_{speed}]$. For $0<n<2$, we further prove that there
is a constant $C$ such that $\Gamma(t)\leqslant Ct^{1/(n+4)}$. This power law
behaviour is the same as has been found for $h_{t}=-(|h|^{n}h_{xxx})_{x}$ for
$0<n<2$ [5] and $2\leqslant n<3$ [29]; it corresponds to the rate of expansion
of the self–similar source–type solution.
The middle plot in Figure 1 presents the $(n,m)$ parameter range for which we
were able to prove finite speed of propagation. While we were successful in
proving finite speed of propagation for the entire expected range of
$m\geqslant n/2$ if $1/2<n<3$, if $0<n\leqslant 1/2$ we could prove it for
only a subset of $m\geqslant n/2$. This technical obstruction is discussed
further in Section 7. The $n$ values are restricted to $n<3$ because for
$n\geq 3$ if initial data has compact support in $(-a,a)$ then it will have
infinite entropy and will not be admissible initial data for Theorems 1 and 2.
To prove Theorem 3 we start by proving a local entropy estimate similar to
that of [5] for $0<n<2$ and a local energy estimate similar to that of [6, 29]
for $1/2<n<3$. Using these and well-chosen ‘‘localization’’ (or ‘‘test’’)
functions, we find systems of functional inequalities. In [24], Giacomelli and
Shishkov proved an extension of Stampacchia’s lemma to systems of
inequalities, we use this to then finish the proof.
A Stampacchia-like lemma for a single inequality was used by [29] to prove
finite speed of propagation for $h_{t}=-(|h|^{n}h_{xxx})_{x}$ and [18] proved
an extension of Stampacchia’s lemma (also for a single inequality) to study
waiting time phenomena. Similar approaches were subsequently used to study
finite speed of propagation and waiting time phenomena for related equations,
see [1, 19, 25, 27, 42, 23, 39, 40, 41]. Further, there are finite speed of
propagation and waiting time results [42, 35] that use the extension of
Stampacchia’s lemma to systems of [24] as well as other types of extensions to
systems [41].
Having proven finite speed of propagation in Theorem 3, we use this to prove a
short-time existence result for the Cauchy problem. Specifically, for the
range of exponents $(n,m)$ of Theorem 3 given nonnegative initial data with
bounded support in $\mathbb{R}^{1}$ we construct a nonnegative, compactly
supported strong solution on $\mathbb{R}^{1}\times[0,T_{0}]$ that satisfies
the bounds and regularity of Theorems 1 and 2 (with those bounds taken over
$\mathbb{R}^{1}$ rather than $\Omega$). The larger the $H^{1}$ norm of the
initial data, the shorter the time $T_{0}$. In Lemma 8.1 we prove that for a
subset of these exponents (see Figure 1), the entropy of the solution
satisfies a second-moment inequality at time $T_{0}$
$e^{-\widetilde{B}(T_{0})}\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h(x,T_{0}))\,dx}\leqslant\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\int\limits_{0}^{T_{0}}{e^{-\widetilde{B}(t)}\left(k_{1}\mathcal{E}_{0}(0)+k_{2}\int\limits_{-\infty}^{\infty}{x^{2}h^{2}_{xx}\,dx}\right)\,dt}$
where $k_{1}=2(4-n)$, $k_{2}=3a_{0}(n-1)/2$, $\mathcal{E}_{0}(0)$ is the
energy of the initial data (1.5),
$\widetilde{G}_{0}(z)=\tfrac{1}{2-n}z^{2-n},\qquad\mbox{and}\qquad\widetilde{B}(t)=\tfrac{a_{1}^{2}|(1-n)(2-n)|}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{t}{\|h(.,\tau)\|_{L^{\infty}(\Omega)}^{2m-n}\,d\tau}.$
Note that if $n=1$ then the second-moment inequality for the entropy reduces
to the second moment inequality (1.4) used by [13].
‘‘Time-stepping’’ this existence result, we construct a nonnegative, compactly
supported, strong solution on $\mathbb{R}^{1}\times[0,T^{*})$ such that our
second-moment inequality for the entropy holds for a sequence of times
$T_{i}\to T^{*}$. The left-hand side of the inequality is nonnegative. If the
initial data has negative energy, then the second term on the right-hand side
has an integrand which has the possibility of becoming negative. In Theorem 4
we prove that for a range of $(n,m)$ values (see Figure 1) that if
$T^{*}=\infty$ then the constructed solution would yield a right-hand side
that becomes negative in finite time, an impossibility. Hence $T^{*}$ must be
finite and therefore the $\limsup_{t\to T^{*}}\|h(\cdot,t)\|_{H^{1}}$ must be
infinite.
Ideally, we would have proven finite speed of propagation for all $(n,m)$ with
$0<n<3$ and all $n/2\leqslant m$ and would have proven the finite time blow up
result for all $(n,m)$ with $0<n<3$ and all $m\geqslant n+2$. We believe the
obstructions are technical ones.
We close by noting that our blow–up result does not give qualitative
information such as proving that there’s a focussing singularity, as suggested
by numerical simulations [12]. However, there has been some detailed study of
the critical regime $m=n+2$. There, a critical mass $M_{c}$ has been
identified [43], there are self-similar, compactly-supported, source-type
solutions with masses in $[0,M_{c})$ [3], and there are self-similar,
compactly supported, solutions with masses in $(M_{c},M_{1})$ that blow up in
finite time [37]. Further, these self-similar blow-up solutions have been
shown to be linearly stable [38].
## 2 Generalized weak solution
We study the existence of a nonnegative weak solution, $h(x,t)$, of the
initial–boundary value problem
$\displaystyle(\textup{P})$ $\displaystyle
h_{t}+\left(f(h)\left(a_{0}h_{xxx}+a_{1}D^{\prime\prime}(h)h_{x}\right)\right)_{x}=0\text{
in }Q_{T},\hfill$ (2.1) $\displaystyle(\textup{P})$
$\displaystyle\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)\text{ for
}t>0,\,i=\overline{0,3},\hfill$ (2.2) $\displaystyle(\textup{P})$
$\displaystyle\qquad\qquad h(0,x)=h_{0}(x)\geqslant 0,\hfill$ (2.3)
where $a_{0}>0$, $a_{1}\geqslant 0$, $f(h)=|h|^{n}$,
$D^{\prime\prime}(h)=|h|^{m-n}$, $n>0$, $m>0$, $\Omega=(-a,a)$, and
$Q_{T}=\Omega\times(0,T)$. We consider a weak solution like that considered in
[4, 5]:
###### Definition 2.1 (generalized weak solution).
Let $n>0$, $m>0$, $a_{0}>0$, and $a_{1}\geqslant 0$. A function $h$ is a
generalized weak solution of the problem $(\textup{P})$ if
$\displaystyle h\in C^{1/2,1/8}_{x,t}(\overline{Q}_{T})\cap
L^{\infty}(0,T;H^{1}(\Omega)),$ (2.4) $\displaystyle h_{t}\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime}),$ (2.5) $\displaystyle h\in
C^{4,1}_{x,t}(\mathcal{P}),\,\,\,\sqrt{f(h)}\,\left(a_{0}h_{xxx}+a_{1}D^{\prime\prime}(h)h_{x}\right)\in
L^{2}(\mathcal{P}),\,\,$ (2.6)
where $\mathcal{P}=\overline{Q}_{T}\setminus(\\{h=0\\}\cup\\{t=0\\})$ and $h$
satisfies (2.1) in the following sense:
$\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt-\iint\limits_{\mathcal{P}}{f(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}(h)h_{x})\phi_{x}\,dxdt}\
=0$ (2.7)
for all $\phi\in C^{1}(Q_{T})$ with $\phi(-a,\cdot)=\phi(a,\cdot)$;
$\displaystyle h(\cdot,t)\to h(\cdot,0)=h_{0}\mbox{ pointwise \& strongly in
$L^{2}(\Omega)$ as $t\to 0$},$ (2.8) $\displaystyle h(-a,t)=h(a,t)\;\forall
t\in[0,T]\;\mbox{and}\;\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)$ (2.9)
$\displaystyle\mbox{for}\;i=\overline{1,3}\;\mbox{at all points of the lateral
boundary where $\\{h\neq 0\\}$.}$
Because the second term of (2.7) has an integral over $\mathcal{P}$ rather
than over $Q_{T}$, the generalized weak solution is ‘‘weaker’’ than a standard
weak solution. Also note that the first term of (2.7) uses $h_{t}\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime})$; this is different from the definition of
weak solution first introduced by Bernis and Friedman [7]; there, the first
term was the integral of $h\phi_{t}$ integrated over $Q_{T}$. We do not
require a test function $\phi$ to be zero at both ends $t=0$ and $t=T$ that is
crucial for our construction of a continuation of the weak solution.
## 3 Main results
Our main results are: the short-time existence of a nonnegative generalized
weak solution (Theorem 1), the short-time existence of a nonnegative strong
solution (Theorem 2), finite-speed of propagation and finite-time blow-up for
some of these strong solutions (Theorems 3 and 4 respectively).
The short-time existence of generalized weak solutions relies on an integral
quantity: $\int G_{0}(h(x,t))\;dx$. This ‘‘entropy’’ was introduced by Bernis
and Friedman [7], where
$G_{0}(z):=\begin{cases}\tfrac{z^{2-n}}{(2-n)(1-n)}&\text{if
}n\in(0,1)\cup(2,\infty)\\\ \tfrac{z^{2-n}}{(2-n)(1-n)}+z+b&\text{if
}n\in(1,2)\\\ z\ln z-z+1&\text{ if }n=1\\\ -\ln z+z/e&\text{ if
}n=2.\end{cases}\Longrightarrow\quad G_{0}^{\prime\prime}(z)=\tfrac{1}{z^{n}}$
(3.1)
with $b$ chosen to ensure that $G_{0}(z)\geqslant 0$ for all $z\geqslant 0$.
###### Theorem 1 (Existence).
Let $n>0,\ m\geqslant\tfrac{n}{2}$, $a_{0}>0$, and $a_{1}\geqslant 0$ in
equation (1.2). Assume that the nonnegative initial data $h_{0}\in
H^{1}(\Omega)$ has finite entropy
$\int\limits_{\Omega}{G_{0}(h_{0}(x))\,dx}<\infty,$ (3.2)
where $G_{0}$ is given by (3.1) and either 1) $h_{0}(-a)=h_{0}(a)=0$ or 2)
$h_{0}(-a)=h_{0}(a)\neq 0$ and $\tfrac{\partial^{i}h_{0}}{\partial
x^{i}}(-a)=\tfrac{\partial^{i}h_{0}}{\partial x^{i}}(a)\text{ holds for
}i=1,2,3$. Then for some time $T_{loc}>0$ there exists a nonnegative
generalized weak solution $h$ on $Q_{T_{loc}}$ in the sense of the definition
2.1. Furthermore,
$h\in L^{2}(0,T_{loc};H^{2}(\Omega)).$ (3.3)
If
$D_{0}(z)=\begin{cases}\tfrac{z^{m-n+2}}{(m-n+1)(m-n+2)}&\mbox{if}\;m-n\notin\\{-2,-1\\}\\\
-\log(z)&\mbox{if}\;m-n=-2\\\
z\log(z)&\mbox{if}\;m-n=-1\end{cases},\quad\Longrightarrow\quad
D_{0}^{\prime\prime}(z)=z^{m-n},$ (3.4)
$\mathcal{E}_{0}(T):=\int\limits_{\Omega}{\\{\tfrac{a_{0}}{2}h_{x}^{2}(x,T)-a_{1}D_{0}(h(x,T))\\}\,dx},\qquad
B_{1}(T):=\tfrac{a_{1}^{2}}{a_{0}}\int\limits_{0}^{T}{\|h(.,t)\|^{2m-n}_{L^{\infty}(\Omega)}\,dt},$
(3.5)
and
$B_{2}(T):=\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{4m-n}\;dt+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{2m-n}\;dt,$
then the weak solution satisfies
$\mathcal{E}_{0}(T_{loc})+\iint\limits_{\\{h>0\\}}{h^{n}(a_{0}h_{xxx}+a_{1}h^{m-n}h_{x})^{2}\,dxdt}\leqslant\mathcal{E}_{0}(0),$
(3.6) $\int\limits_{\Omega}{h_{x}^{2}(x,T_{loc})\,dx}\leqslant
e^{B_{1}(T_{loc})}\int\limits_{\Omega}{h_{0x}^{2}(x)\,dx},$ (3.7)
and
$\int\limits_{\Omega}{\\{h_{x}^{2}(x,T_{loc})+G_{0}(h(x,T_{loc})\\}\,dx}\leqslant
e^{B_{2}(T_{loc})}\int\limits_{\Omega}{\\{h_{0x}^{2}(x)+G_{0}(h_{0}(x))\,dx}.$
(3.8)
The time of existence, $T_{loc}$, is determined by $a_{0}$, $a_{1}$,
$|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, and $\int G_{0}(h_{0})$.
There is nothing special about the time $T_{loc}$ in the bounds (3.6), (3.7),
and (3.8); given a countable collection of times in $[0,T_{loc}]$, one can
construct a weak solution for which these bounds will hold at those times.
Also, we note that the analogue of Theorem 4.2 in [7] also holds: there exists
a nonnegative weak solution with the integral formulation
$\int\limits_{0}^{T}\langle
h_{t}(\cdot,t),\phi\rangle\;dt+a_{0}\iint\limits_{Q_{T}}(nh^{n-1}h_{x}h_{xx}\phi_{x}+h^{n}h_{xx}\phi_{xx})\;dxdt-
a_{1}\iint\limits_{Q_{T}}{h^{m}h_{x}\phi_{x}\;dxdt}=0.$ (3.9)
We note that the existence theory for the long-wave stable case, $a_{1}<0$,
has already been considered by [10, 20].
Theorem 2 states that if the initial data also has finite ‘‘$\alpha$-entropy’’
then a solution can be constructed which satisfies Theorem 1 and has
additional regularity. This solution is called a ‘‘strong’’ solution; see [11,
4, 10]. The $\alpha$-entropy, defined below, was discovered by Leo Kadanoff
[14]. Let
$G_{0}^{(\alpha)}(z)=\begin{cases}\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}&\mbox{if
}\alpha\in(-\infty,n-2)\cup(n-1,\infty)\\\
\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}+z+c&\mbox{if
}\alpha\in(n-2,n-1)\\\ z\ln z-z+1&\mbox{if }\alpha=n-1\\\ -\ln z+z/e&\mbox{if
}\alpha=n-2,\end{cases}G_{0}^{(\alpha)^{\prime\prime}}(z)=\tfrac{z^{\alpha}}{z^{n}}$
(3.10)
where $c$ is chosen to ensure that $G_{0}^{(\alpha)}(z)\geqslant 0$ for all
$z\geqslant 0$.
###### Theorem 2 (Regularity).
Assume the initial data $h_{0}$ satisfies the hypotheses of Theorem 1 and also
has finite $\alpha$-entropy for some $\alpha\in(-1/2,1)$, $\alpha\neq 0$,
$\int\limits_{\Omega}{G_{0}^{(\alpha)}(h_{0}(x))\,dx}<\infty.$ (3.11)
Then for some time time $T_{loc}^{(\alpha)}\in(0,T_{loc}]$ there exists a
nonnegative generalized weak solution on $Q_{T_{loc}^{(\alpha)}}$ that
satisfies Theorem 1 and has the additional regularity
$h^{\tfrac{\alpha+2}{2}}\in L^{2}(0,T_{loc}^{(\alpha)};H^{2}(\Omega))\text{
and }h^{\tfrac{\alpha+2}{4}}\in
L^{2}(0,T_{loc}^{(\alpha)};W^{1}_{4}(\Omega)).$ (3.12)
###### Theorem 3 (Finite Speed of Propagation).
Consider the range of exponents $0<n\leqslant\tfrac{1}{2}$ and $n/2<m<6-n$ or
$\tfrac{1}{2}<n<3$ and $m\geqslant n/2$ (see Figure 1). Assume the initial
data satisfies the hypotheses of Theorem 2 and also has compact support in
$(-a,a)$: $\mbox{supp}(h_{0})\subseteq[-r_{0},r_{0}]\subset(-a,a)$. Then there
exists a time $0<T_{speed}\leqslant T_{loc}^{(\alpha)}$ and a nondecreasing
function $\Gamma(t)\in C([0,T_{speed}])$ such that the strong solution from
Theorem 2, has finite speed propagation, i.e.
$\mbox{supp}(h(\cdot,t))\subseteq[-r_{0}-\Gamma(t),r_{0}+\Gamma(t)]\subset(-a,a)$
for all $t\in[0,T_{speed}]$. Furthermore, if $0<n<2$ there exists a constant
$C$ which depends on $n$, $m$, $\alpha$, and $\int h_{0}$ such that
$\Gamma(t)\leqslant Ct^{1/(n+4)}$.
###### Theorem 4 (Finite Time Blow Up).
Consider the range of exponents $0<n\leqslant\tfrac{1}{2}$ and $4-n\leqslant
m<6-n$ or $\tfrac{1}{2}<n\leqslant 1$ and $m\geqslant 4-n$ or $1<n<2$ and
$m\geqslant n+2$ (see Figure 1). Assume the nonnegative initial data $h_{0}$
has compact support, negative energy (3.5)
$\mathcal{E}_{0}(0)=\int\limits_{-\infty}^{\infty}{\\{\tfrac{a_{0}}{2}h_{0x}^{2}-a_{1}D_{0}(h_{0})\\}\,dx}<0,$
and satisfies the hypotheses of Theorem 2 with $\Omega=\mathbb{R}^{1}$. Then
there exists a finite time $T^{*}<\infty$ and a nonnegative, compactly
supported, strong solution $h$ on $\mathbb{R}^{1}\times[0,T^{*})$ such that
$\mathop{\limsup}\limits_{t\to
T^{*}}\|h(.,t)\|_{H^{1}(\mathbb{R}^{1})}=\mathop{\limsup}\limits_{t\to
T^{*}}\|h(.,t)\|_{L^{\infty}(\mathbb{R}^{1})}=+\infty.$ (3.13)
The solution satisfies the bounds of Theorems 1 and 2 with
$\Omega=\mathbb{R}^{1}$.
## 4 Proof of Existence of Generalized Solutions
Our proof of existence of generalized weak solutions defined in the section 2
follows the main concept of the proof from [7].
### 4.1 Regularized Problem
Given $\delta,\varepsilon>0$, a regularized parabolic problem, similar to that
of Bernis and Friedman [7], is considered:
$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle
h_{t}+\left({f_{\delta\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})}\right)_{x}=0,\qquad\hfill$
(4.1) $\displaystyle(\textup{P}_{\delta,\varepsilon})$
$\displaystyle\tfrac{\partial^{i}h}{\partial
x^{i}}(-a,t)=\tfrac{\partial^{i}h}{\partial x^{i}}(a,t)\text{ for
}t>0,\,i=\overline{0,3},\hfill$ (4.2)
$\displaystyle(\textup{P}_{\delta,\varepsilon})$ $\displaystyle\qquad\qquad
h(x,0)=h_{0,\varepsilon}(x)\hfill$ (4.3)
where
$f_{\delta\varepsilon}(z):=f_{\varepsilon}(z)+\delta=\tfrac{|z|^{4+n}}{|z|^{4}+\varepsilon|z|^{n}}+\delta,\
D^{\prime\prime}_{\varepsilon}(z):=\tfrac{|z|^{m-n}}{1+\varepsilon|z|^{m-n}},\varepsilon>0,\delta>0.$
(4.4)
We note that $f_{\delta\varepsilon}\in C^{1+\gamma}(\mathbb{R}^{1})$ and
$f_{\delta\varepsilon}D^{\prime\prime}_{\varepsilon}\in
C^{1+\gamma}(\mathbb{R}^{1})$ for some $\gamma\in(0,1)$. The $\delta>0$ in
(4.4) makes the problem (4.1) regular (i.e. uniformly parabolic). The
parameter $\varepsilon$ is an approximating parameter which has the effect of
increasing the degeneracy from $f(h)\sim|h|^{n}$ to $f_{\varepsilon}(h)\sim
h^{4}$. For $\varepsilon>0$, the nonnegative initial data, $h_{0}$, is
approximated via
$\begin{gathered}h_{0}+\varepsilon^{\theta}\leq h_{0,\varepsilon}\in
C^{4+\gamma}(\overline{\Omega})\text{ for some }0<\theta<\tfrac{2}{5},\\\
\tfrac{\partial^{i}h_{0,\varepsilon}}{\partial
x^{i}}(-a)=\tfrac{\partial^{i}h_{0,\varepsilon}}{\partial x^{i}}(a)\text{ for
}i=\overline{0,3},\\\ h_{0,\varepsilon}\to h_{0}\text{ strongly in
}H^{1}(\Omega)\text{ as }\varepsilon\to 0.\end{gathered}$ (4.5)
The effect of $\varepsilon>0$ in (4.5) is to both ‘‘lift’’ the initial data,
making it positive, and to smooth the initial data from $H^{1}(\Omega)$ to
$C^{4+\gamma}(\Omega)$.
By Eĭdelman [22, Theorem 6.3, p.302], the regularized problem has a unique
classical solution $h_{\delta\varepsilon}\in
C_{x,t}^{4+\gamma,1+\gamma/4}(\Omega\times[0,\tau_{\delta\varepsilon}])$ for
some time $\tau_{\delta\varepsilon}>0$. For any fixed value of $\delta$ and
$\varepsilon$, by Eĭdelman [22, Theorem 9.3, p.316] if one can prove a uniform
in time an a priori bound $|h_{\delta\varepsilon}(x,t)|\leqslant
A_{\delta\varepsilon}<\infty$ for some longer time interval
$[0,T_{loc,\delta\varepsilon}]\quad(T_{loc,\delta\varepsilon}>\tau_{\delta\varepsilon}$)
and for all $x\in\Omega$ then Schauder-type interior estimates [22, Corollary
2, p.213] imply that the solution $h_{\delta\varepsilon}$ can be continued in
time to be in
$C_{x,t}^{4+\gamma,1+\gamma/4}(\Omega\times[0,T_{loc,\delta\varepsilon}])$.
Although the solution $h_{\delta\varepsilon}$ is initially positive, there is
no guarantee that it will remain nonnegative. The goal is to take $\delta\to
0$, $\varepsilon\to 0$ in such a way that a) $T_{loc,\delta\varepsilon}\to
T_{loc}>0$, b) the solutions $h_{\delta\varepsilon}$ converge to a
(nonnegative) limit, $h$, which is a generalized weak solution, and c) $h$
inherits certain a priori bounds. This is done by proving various a priori
estimates for $h_{\delta\varepsilon}$ that are uniform in $\delta$ and
$\varepsilon$ and hold on a time interval $[0,T_{loc}]$ that is independent of
$\delta$ and $\varepsilon$. As a result, $\\{h_{\delta\varepsilon}\\}$ will be
a uniformly bounded and equicontinuous (in the $C_{x,t}^{1/2,1/8}$ norm)
family of functions in $\bar{\Omega}\times[0,T_{loc}]$. Taking $\delta\to 0$
will result in a family of functions $\\{h_{\varepsilon}\\}$ that are
classical, positive, unique solutions to the regularized problem with
$\delta=0$. Taking a subsequence of $\varepsilon$ going to zero will then
result in the desired generalized weak solution $h$. This last step is where
the possibility of nonunique weak solutions arise; see [4] for simple examples
of how such a construction applied to $h_{t}=-(|h|^{n}h_{xxx})_{x}$ can result
in there being two different solutions arising from the same initial data.
### 4.2 A priori estimates
We start by proving a priori estimates for classical solutions of the
regularized problem (4.1)–(4.5). Appendix A contains the proofs of the lemmas
in this section.
We introduce function $G_{\delta\varepsilon}$ chosen such that
$G^{\prime\prime}_{\delta\varepsilon}(z)=\tfrac{1}{f_{\delta\varepsilon}(z)}\quad\mbox{and}\quad
G_{\delta\varepsilon}(z)\geqslant 0\quad\forall z\in\mathbb{R}^{1}.$ (4.6)
This is analogous to the ‘‘entropy’’ function (3.1) introduced by Bernis and
Friedman [7].
###### Lemma 4.1.
Let $n>0$ and $m\geqslant n/2$. There exist $\delta_{0}>0$,
$\varepsilon_{0}>0$, and time $T_{loc}>0$ such that if
$\delta\in[0,\delta_{0})$, $\varepsilon\in(0,\varepsilon_{0})$ and
$h_{\delta\varepsilon}$ is a classical solution of the problem (4.1)–(4.5)
with initial data $h_{0,\delta\varepsilon}$ that is built from a nonnegative
function $h_{0}$ satisfying the hypotheses of Theorem 1 then for any
$T\in[0,T_{loc}]$ the following inequalities hold true:
$\displaystyle\int\limits_{\Omega}{\\{h_{\delta\varepsilon,x}^{2}(x,T)+\tfrac{a_{1}^{2}}{2a_{0}^{2}}(1+\delta)G_{\delta\varepsilon}(h_{\delta\varepsilon}(x,T))\\}\,dx}$
(4.7) $\displaystyle\hskip
72.26999pt+a_{0}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h_{\delta\varepsilon})h^{2}_{\delta\varepsilon,xxx}\,dxdt}\leqslant
K_{1}<\infty,$
$\int\limits_{\Omega}G_{\delta\varepsilon}(h_{\delta\varepsilon}(x,T))\;dx+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}h_{\delta\varepsilon,xx}^{2}\;dxdt\leqslant
K_{2}<\infty.$ (4.8)
The energy $\mathcal{E}_{\delta\varepsilon}(t)$ (see (3.5)) satisfies:
$\mathcal{E}_{\delta\varepsilon}(T)+\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})^{2}}\;dxdt=\mathcal{E}_{\delta\varepsilon}(0).$
(4.9)
The time $T_{loc}$ and the constants $K_{1}$, and $K_{2}$ are independent of
$\delta$ and $\varepsilon$.
The existence of $\delta_{0}$, $\varepsilon_{0}$, $T_{loc}$, $K_{1}$, $K_{2}$,
and $C_{0}$ is constructive; how to find them and what quantities determine
them is shown in Section A.
Lemma 4.1 yields uniform-in-$\delta$-and-$\varepsilon$ bounds for $\int
h_{\delta\varepsilon,x}^{2}$, $\int
G_{\delta\varepsilon}(h_{\delta\varepsilon})$, $\iint
f_{\delta\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,xxx}^{2}$,
and $\iint h_{\delta\varepsilon,xx}^{2}$ which will be key in constructing a
nonnegative generalized weak solution. However, these bounds are found in a
different manner than in earlier work for the equation
$h_{t}=-(|h|^{n}h_{xxx})_{x}$, for example. Although the inequality (4.8) is
unchanged, the inequality (4.7) has an extra term involving
$G_{\delta\varepsilon}$. In the proof, this term was introduced to control
additional, lower–order terms. This idea of a ‘‘blended’’
$\|h_{x}\|_{2}$–entropy bound was first introduced by Shishkov and Taranets
especially for long-wave stable thin film equations with convection [34].
###### Lemma 4.2.
Assume $\varepsilon_{0}$ and $T_{loc}$ are from Lemma 4.1, $\delta=0$, and
$\varepsilon\in(0,\varepsilon_{0})$. If $h_{\varepsilon}$ is a positive,
classical solution of the problem (4.1)–(4.5) with initial data
$h_{0,\varepsilon}$ satisfying Lemma 4.1 then
$\int\limits_{\Omega}{h_{\varepsilon,x}^{2}(x,T)\,dx}\leqslant
e^{B_{1,\varepsilon}(T)}\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx},$
(4.10)
$\int\limits_{\Omega}{\\{h_{\varepsilon,x}^{2}(x,T)+G_{\varepsilon}(h_{\varepsilon}(x,T))\\}\,dx}\leqslant
e^{B_{2,\varepsilon}(T)}\int\limits_{\Omega}{\\{h_{0\varepsilon,x}^{2}+G_{\varepsilon}(h_{0,\varepsilon})\\}\,dx}$
(4.11)
hold true for all $T\in[0,T_{loc}]$. Here
$B_{1,\varepsilon}(T):=\tfrac{a_{1}^{2}}{a_{0}}\int\limits_{0}^{T}\|h_{\varepsilon}(\cdot,t)\|_{\infty}^{2m-n}\;dt,$
$B_{2,\varepsilon}(T):=\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\int\limits_{0}^{T}\|h_{\varepsilon}(\cdot,t)\|_{\infty}^{4m-n}\;dt+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}\|h_{\varepsilon}(\cdot,t)\|_{\infty}^{2m-n}\;dt.$
The final a priori bound uses the following functions, parametrized by
$\alpha$, chosen such that $G_{\varepsilon}^{(\alpha)}\geqslant 0$ and
$G_{\varepsilon}^{(\alpha)\prime\prime}(z)=z^{\alpha}/f_{\varepsilon}(z)$:
$G_{\varepsilon}^{(\alpha)}(z)=\begin{cases}\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}&\mbox{if
}\alpha\in(-\infty,n-2)\cup(n-1,\infty)\\\
\tfrac{z^{2-n+\alpha}}{(2-n+\alpha)(1-n+\alpha)}+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}+z+c&\mbox{if
}\alpha\in(n-2,n-1)\\\ z\ln
z-z+1+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}&\mbox{if
}\alpha=n-1\\\ -\ln
z+z/e+\varepsilon\tfrac{z^{\alpha-2}}{(\alpha-3)(\alpha-2)}&\mbox{if
}\alpha=n-2,\end{cases}$ (4.12)
###### Lemma 4.3.
Assume $\varepsilon_{0}$ and $T_{loc}$ are from Lemma 4.1, $\delta=0$, and
$\varepsilon\in(0,\varepsilon_{0})$. Assume $h_{\varepsilon}$ is a positive,
classical solution of the problem (4.1)–(4.5) with initial data
$h_{0,\varepsilon}$ that is built from a nonnegative function $h_{0}$
satisfying the hypotheses of Theorem 2 then there exists $K_{3}$,
$\varepsilon_{0}^{(\alpha)}$ and $T_{loc}^{(\alpha)}$ with
$0<\varepsilon_{0}^{(\alpha)}\leqslant\varepsilon_{0}$ and
$0<T_{loc}^{(\alpha)}\leqslant T_{loc}$ such that
$\int\limits_{\Omega}{\\{h_{\varepsilon,x}^{2}(x,T)+G_{\varepsilon}^{(\alpha)}(h_{\varepsilon}(x,T))\\}\,dx}+\iint\limits_{Q_{T}}\left[\beta
h_{\varepsilon}^{\alpha}h_{\varepsilon,xx}^{2}+\gamma
h_{\varepsilon}^{\alpha-2}h_{\varepsilon,x}^{4}\right]\;dx\,dt\leqslant
K_{3}<\infty$ (4.13)
holds for all $T\in[0,T_{loc}^{(\alpha)}]$. $K_{3}$ is determined by $\alpha$,
$\varepsilon_{0}$, $a_{0}$, $a_{1}$, $\Omega$ and $h_{0}$. Here,
$\beta=\begin{cases}a_{0}&\mbox{if }0<\alpha<1,\\\
a_{0}\tfrac{1+2\alpha}{4(1-\alpha)}&\mbox{if }-1/2<\alpha<0\end{cases}$
and
$\gamma=\begin{cases}a_{0}\tfrac{\alpha(1-\alpha)}{6}&\mbox{if }0<\alpha<1,\\\
a_{0}\tfrac{(1+2\alpha)(1-\alpha)}{36}&\mbox{if }-1/2<\alpha<0.\end{cases}$
Furthermore,
$h_{\varepsilon}^{\tfrac{\alpha+2}{2}}\in B_{R_{1}}(0)\subset
L^{2}(0,T_{loc};H^{2}(\Omega))\quad\mbox{and}\quad
h_{\varepsilon}^{\tfrac{\alpha+2}{4}}\in B_{R_{2}}(0)\subset\in
L^{2}(0,T_{loc};W^{1}_{4}(\Omega))$ (4.14)
where the radii $R_{1}$ and $R_{2}$ are independent of $\varepsilon$.
The $\alpha$–entropy, $\int G_{0}^{(\alpha)}(h)\;dx$, was first introduced for
$\alpha=-1/2$ in [14] and an a priori bound like that of Lemma 4.3 and
regularity results like those of Theorem 2 were found simultaneously and
independently in [4] and [11].
### 4.3 Proof of existence and regularity of solutions
Bound (4.7) yields uniform $L^{\infty}$ control for classical solutions
$h_{\delta\varepsilon}$, allowing the time of existence
$T_{loc,\delta\varepsilon}$ to be taken as $T_{loc}$ for all
$\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0})$. The
existence theory starts by constructing a classical solution
$h_{\delta\varepsilon}$ on $[0,T_{loc}]$ that satisfy the hypotheses of Lemma
4.1 if $\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0})$. The
a priori bounds of Lemma 4.1 then allow one to take the regularizing
parameter, $\delta$, to zero and prove that there is a limit $h_{\varepsilon}$
and that $h_{\varepsilon}$ is a generalized weak solution. One then proves
additional regularity for $h_{\varepsilon}$; specifically that it is strictly
positive, classical, and unique. It then follows that the a priori bounds
given by Lemmas 4.1, 4.2, and 4.3 apply to $h_{\varepsilon}$. This allows one
to take the approximating parameter, $\varepsilon$, to zero and construct the
desired generalized weak solution of Theorems 1 and 2:
###### Lemma 4.4.
Assume that the initial data $h_{0,\varepsilon}$ satisfies (4.5) and is built
from a nonnegative function $h_{0}$ that satisfies the hypotheses of Theorem
1. Fix $\delta=0$ and $\varepsilon\in(0,\varepsilon_{0})$ where
$\varepsilon_{0}$ is from Lemma 4.1. Then there exists a unique, positive,
classical solution $h_{\varepsilon}$ on $[0,T_{loc}]$ of problem
($\mbox{P}_{0,\varepsilon}$), see (4.1)–(4.5), with initial data
$h_{0,\varepsilon}$ where $T_{loc}$ is the time from Lemma 4.1.
The proof uses a number of arguments like those presented by Bernis & Friedman
[7] and we refer to that article as much as possible.
###### Proof.
Fix $\varepsilon\in(0,\varepsilon_{0})$ and assume $\delta\in(0,\delta_{0})$.
Because $G_{\delta\varepsilon}(z)\geqslant 0$, the bound (4.7) yields a
uniform-in-$\delta$-and-$\varepsilon$ upper bound on
$|h_{\delta\varepsilon}(x,T)|$ for
$(x,T)\in\overline{\Omega}\times[0,T_{loc}]$. As discussed in Subsection 4.1,
this allows the classical solution $h_{\delta\varepsilon}$ to be extended from
$[0,\tau_{\delta\varepsilon}]$ to $[0,T_{loc}]$.
By Section 2 of [7], the a priori bound (4.7) on
$\|h_{x}(\cdot,T_{loc})\|_{2}$ implies that $h_{\delta\varepsilon}\in
C^{1/2,1/8}_{x,t}(\overline{Q_{T_{loc}}})$ and that
$\\{h_{\delta\varepsilon}\\}$ is a uniformly bounded, equicontinuous family in
$\overline{Q_{T_{loc}}}$. By the Arzela-Ascoli theorem, there is a subsequence
$\\{\delta_{k}\\}$, so that $h_{\delta_{k}\varepsilon}$ converges uniformly to
a limit $h_{\varepsilon}\in C^{1/2,1/8}_{x,t}(\overline{Q_{T_{loc}}})$.
We now argue that $h_{\varepsilon}$ is a generalized weak solution, using
methods similar to those of [7, Theorem 3.1].
By construction, $h_{\varepsilon}$ is in
$C^{1/2,1/8}_{x,t}(\overline{Q_{T_{loc}}})$, satisfying the first part of
(2.4). The strong convergence $h_{\varepsilon}(\cdot,t)\to
h_{\varepsilon}(\cdot,0)$ in $L^{2}(\Omega)$ follows immediately. The uniform
convergence of $h_{\delta_{k}\varepsilon}$ to $h_{\varepsilon}$ implies the
pointwise convergence $h(\cdot,t)\to h(\cdot,0)=h_{0}$, and so
$h_{\varepsilon}$ satisfies (2.8).
Because $h_{\delta\varepsilon}$ is a classical solution,
$\iint\limits_{Q_{T}}{h_{\delta\varepsilon,t}\phi\;dxdt}-\iint\limits_{Q_{T_{loc}}}{f_{\delta\varepsilon}(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})\phi_{x}\;dxdt}=0.$
(4.15)
The bound (4.7) yields a uniform bound on
$\delta\iint\limits_{Q_{T_{loc}}}h_{\delta\varepsilon,xxx}^{2}\;dxdt$
for $\delta\in(0,\delta_{0})$. It follows that
$\delta_{k}\iint\limits_{Q_{T_{loc}}}(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})\phi_{x}\,dxdt\to
0\qquad\mbox{as $\delta_{k}\to 0$.}$
Introducing the notation
$H_{\delta\varepsilon}:=f_{\delta\varepsilon}(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x})$
(4.16)
the integral formulation (4.15) can be written as
$\iint\limits_{Q_{T}}{h_{\delta\varepsilon,t}\phi
dxdt}=\iint\limits_{Q_{T_{loc}}}H_{\delta\varepsilon}(x,t)\phi_{x}(x,t)\;dxdt.$
(4.17)
By the $L^{\infty}$ control of $h_{\delta\varepsilon}$ and the energy bound
(4.9), $H_{\delta\varepsilon}$ is uniformly bounded in $L^{2}(Q_{T_{loc}})$.
Taking a further subsequence of $\\{\delta_{k}\\}$ yields
$H_{\delta_{k}\varepsilon}$ converging weakly to a function $H_{\varepsilon}$
in $L^{2}(Q_{T_{loc}})$. The regularity theory for uniformly parabolic
equations implies that $h_{\delta\varepsilon,t}$, $h_{\delta\varepsilon,x}$,
$h_{\delta\varepsilon,xx}$, $h_{\delta\varepsilon,xxx}$, and
$h_{\delta\varepsilon,xxxx}$ converge uniformly to $h_{\varepsilon,t}$, …,
$h_{\varepsilon,xxxx}$ on any compact subset of $\\{h_{\varepsilon}>0\\}$,
implying (2.9) and the first part of (2.6). Note that because the initial data
$h_{0,\delta\varepsilon}$ is in $C^{4}$ the regularity extends all the way to
$t=0$ which is excluded in the definition of $\mathcal{P}$ in (2.6).
The energy $\mathcal{E}_{\delta\varepsilon}(T_{loc})$ is not necessarily
positive. However, the a priori bound (4.7), combined with the $L^{\infty}$
control on $h_{\delta\varepsilon}$, ensures that
$\mathcal{E}_{\delta\varepsilon}(T_{loc})$ has a uniform lower bound. As a
result, the bound (4.9) yields a uniform bound on
$\iint\limits_{Q_{T_{loc}}}f_{\delta\varepsilon}\left(h_{\delta\varepsilon})(a_{0}h_{\delta\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\delta\varepsilon,x}\right)^{2}\;dxdt.$
Using this, one can argue that for any $\sigma>0$
$\iint\limits_{\\{h_{\varepsilon}<\sigma\\}}\left|H_{\delta_{k}\varepsilon}\phi_{x}\right|\;dxdt\leqslant
C\sigma^{n/2}$
for some $C$ independent of $\delta$, $\varepsilon$, and $\sigma$. Taking
$\delta_{k}\to 0$ and using that $\sigma$ is arbitrary, we conclude
$H_{\delta_{k}\varepsilon}\to
H_{\varepsilon}=f_{\varepsilon}\left(h_{\varepsilon})(a_{0}h_{\varepsilon,xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h_{\delta\varepsilon})h_{\varepsilon,x}\right)\,\chi_{\\{h_{\varepsilon}>0\\}}.$
As a result, taking $\delta_{k}\to 0$ in (4.17) implies $h_{\varepsilon}$
satisfies (2.7) and the second part of (2.6).
The bound (4.7) yields a uniform bound on $\int
h_{\delta\varepsilon,x}^{2}(x,T)\;dx$ for every $T\in[0,T_{loc}]$. As a
result, $\\{h_{\delta_{k}\varepsilon}\\}$ is uniformly bounded in
$L^{\infty}(0,T_{loc};H^{1}(\Omega)).$
Therefore, another refinement of the sequence $\\{\delta_{k}\\}$ yields
$\\{h_{\delta_{k}\varepsilon}\\}$ weakly convergent in this space. As a
result, $h_{\varepsilon}\in L^{\infty}(0,T_{loc};H^{1}(\Omega))$ and the
second part of (2.4) holds.
Having proven then $h_{\varepsilon}$ is a generalized weak solution, we now
prove that $h_{\varepsilon}$ is a strictly positive, classical, unique
solution. This uses the entropy $\int
G_{\delta\varepsilon}(h_{\delta\varepsilon})$ and the a priori bound (4.8).
This bound is, up to the coefficient $a_{0}$, identical to the a priori bound
(4.17) in [7]. By construction, the initial data $h_{0,\varepsilon}$ is
positive (see (4.5)), hence $\int
G_{\varepsilon}(h_{0,\varepsilon})\;dx<\infty$. Also, by construction
$f_{\varepsilon}(z)\sim z^{4}$ for $z\ll 1$. This implies that the generalized
weak solution $h_{\varepsilon}$ is strictly positive [7, Theorem 4.1]. Because
the initial data $h_{0,\varepsilon}$ is in $C^{4}(\bar{\Omega})$, it follows
that $h_{\varepsilon}$ is a classical solution in
$C^{4,1}_{x,t}(\overline{Q_{T_{loc}}})$. This implies that
$h_{\varepsilon}(\cdot,t)\to h_{\varepsilon}(\cdot,0)$ strongly333 Unlike the
definition of weak solution given in [7], Definition 2.1 does not include that
the solution converges to the initial data strongly in $H^{1}(\Omega)$. in
$C^{1}(\bar{\Omega})$. The proof of Theorem 4.1 in [7] then implies that
$h_{\varepsilon}$ is unique. ∎
###### Proof of Theorem 1.
As in the proof of Lemma 4.4, following [7], there is a subsequence
$\\{\varepsilon_{k}\\}$ such that $h_{\varepsilon_{k}}$ converges uniformly to
a function $h\in C^{1/2,1/8}_{x,t}$ which is a generalized weak solution in
the sense of Definition 2.1 with $f(h)=|h|^{n}$.
The initial data is assumed to have finite entropy: $\int G_{0}(h_{0})<\infty$
where $G_{0}$ is given by (3.1). This, combined with $f(h)=|h|^{n}$, implies
that the generalized weak solution $h$ is nonnegative and, if $n\in[2,4)$ the
set of points $\\{h=0\\}$ in $Q_{T_{loc}}$ has zero measure and $h$ is
positive, smooth, and unique if $n\geqslant 4$ [7, Theorem 4.1].
To prove (3.6), start by taking $T=T_{loc}$ in the a priori bound (4.9). As
$\varepsilon_{k}\to 0$, the right-hand side of (4.9) is unchanged. Now,
consider the $\varepsilon_{k}\to 0$ limit of
$\mathcal{E}_{\varepsilon_{k}}(T_{loc})=\int\limits_{\Omega}{(\tfrac{a_{0}}{2}h_{\varepsilon_{k},x}^{2}(x,T_{loc})-a_{1}D_{\varepsilon_{k}}(h_{\varepsilon_{k}}(x,T_{loc})))\,dx},$
where $D_{\varepsilon}$ is defined in (4.4). By the uniform convergence of
$h_{\varepsilon_{k}}$ to $h$, the second term in the energy converges strongly
as $\varepsilon_{k}\to 0$. Hence the bound (4.9) yields a uniform bound on
$\\{\int_{\Omega}h_{\varepsilon_{k},x}^{2}(x,T_{loc})\;dx\\}$. Taking a
further refinement of $\\{\varepsilon_{k}\\}$, yields
$h_{\varepsilon_{k},x}(\cdot,T_{loc})$ converging weakly in $L^{2}(\Omega)$.
In a Hilbert space, the norm of the weak limit is less than or equal to the
$\liminf$ of the norms of the functions in the sequence, hence
$\mathcal{E}_{0}(T_{loc})\leqslant\liminf_{\varepsilon_{k}\to
0}\mathcal{E}_{\varepsilon_{k}}(T_{loc}).$ A uniform bound on $\iint
f_{\varepsilon}(h_{\varepsilon})\left(a_{0}h_{\varepsilon,xxx}+\dots\right)^{2}\;dx$
also follows from (4.9). Hence
$\sqrt{f_{\varepsilon_{k}}(h_{\varepsilon_{k}})}\left(a_{0}h_{\varepsilon_{k},xxx}+\dots\right)$
converges weakly in $L^{2}(Q_{T_{loc}})$, after taking a further subsequence.
We write the weak limit as two integrals: one over $\\{h=0\\}$ and one over
$\\{h>0\\}$. We can determine the weak limit on $\\{h>0\\}$: as in the proof
of Lemma 4.4, the regularity theory for uniformly parabolic equations allows
one to argue that the weak limit is $h^{n/2}\left(a_{0}h_{xxx}+\dots\right)$
on $\\{h>0\\}$. Using that 1) the norm of the weak limit is less than or equal
to the $\liminf$ of the norms of the functions in the sequence and that 2) the
$\liminf$ of a sum is greater than or equal to the sum of the $\liminf$s and
dropping the nonnegative term arising from the integral over $\\{h=0\\}$
results in the desired bound (3.6).
It follows from (4.8) that $h_{\varepsilon_{k},xx}$ converges weakly to some
$v$ in $L^{2}(Q_{T_{loc}})$, combining with strong convergence in
$L^{2}(0,T;H^{1}(\Omega))$ of $h_{\varepsilon_{k}}$ to $h$ by Lemma D.1 and
with the definition of weak derivative, we obtain that $v=h_{xx}$ and $h\in
L^{2}(0,T_{loc};H^{2}(\Omega))$ that implies (3.3). Hence
$h_{\varepsilon,t}\to h_{t}\text{ weakly in
}L^{2}(0,T;(H^{1}(\Omega))^{\prime})$ that implies (2.5). By Lemma D.2 we also
have $h\in C([0,T_{loc}],L^{2}(\Omega))$. ∎
###### Proof of Theorem 2.
Fix $\alpha\in(-1/2,1)$. The initial data $h_{0}$ is assumed to have finite
entropy $\int G_{0}^{(\alpha)}(h_{0}(x))\;dx<\infty$, hence Lemma 4.3 holds
for the approximate solutions $\\{h_{\varepsilon_{k}}\\}$ where this sequence
of approximate solutions is assumed to be the one at the end of the proof of
Theorem 1. By (4.14),
$\left\\{h_{\varepsilon_{k}}^{\tfrac{\alpha+2}{2}}\right\\}\quad\mbox{is
uniformly bounded in $\varepsilon_{k}$ in $L^{2}(0,T_{loc};H^{2}(\Omega))$}$
and
$\left\\{h_{\varepsilon_{k}}^{\tfrac{\alpha+2}{4}}\right\\}\quad\mbox{is
uniformly bounded in $\varepsilon_{k}$ in
$L^{2}(0,T_{loc};W^{1}_{4}(\Omega))$}.$
Taking a further subsequence in $\\{\varepsilon_{k}\\}$, it follows from the
proof of [17, Lemma 2.5, p.330] that these sequences converge weakly in
$L^{2}(0,T_{loc};H^{2}(\Omega))$ and $L^{2}(0,T_{loc};W^{1}_{4}(\Omega))$, to
$h^{\tfrac{\alpha+2}{2}}$ and $h^{\tfrac{\alpha+2}{4}}$ respectively. ∎
## 5 The subcritical regime: long–time existence of solutions
###### Lemma 5.1.
Let $h\in H^{1}(\Omega)$ be a nonnegative function and let
$M=\int\limits_{\Omega}{h(x)\,dx}$. Then
$\|h\|_{L^{p}(\Omega)}^{p}\leqslant
k_{1}M^{\tfrac{p+2}{3}}\biggl{(}\int\limits_{\Omega}{h^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}+k_{2}M^{p},\
\ p\geqslant 1,$ (5.1)
where $k_{1}=2^{\tfrac{4-p}{3}}3^{\tfrac{2(p-1)}{3}}(1-\epsilon)^{-1}$,
$k_{2}=|\Omega|^{1-p}(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{1-p}$, and
$\epsilon\in(0,1)$.
Note that by taking $h$ to be a constant function, one finds that the constant
$k_{2}M^{p}$ in (5.1) is sharp when $\epsilon\to 1$.
###### Proof.
Let $v=h-M/|\Omega|$. By (A.3),
$\|v\|_{L^{p}(\Omega)}^{p}\leqslant(\tfrac{3}{2})^{\tfrac{2(p-1)}{3}}\biggl{(}\int\limits_{\Omega}{v^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}\biggl{(}\int\limits_{\Omega}{|v|\,dx}\biggr{)}^{\tfrac{p+2}{3}}.$
Hence, due to the inequality
$|a-b|^{p}\geqslant(1-\epsilon)a^{p}-c_{0}(\epsilon,p)b^{p}$
for any $a\geqslant 0,\ b\geqslant 0,\ p>1,\ \epsilon\in(0,1)$,
$c_{0}(\epsilon,p)\geqslant\tfrac{1-\epsilon}{(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{p-1}}$,
$\|h\|_{L^{p}(\Omega)}^{p}\leqslant\tfrac{1}{1-\epsilon}(\tfrac{3}{2})^{\tfrac{2(p-1)}{3}}\biggl{(}\int\limits_{\Omega}{h^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}\biggl{(}\int\limits_{\Omega}{\left|h-\tfrac{M}{|\Omega|}\right|\,dx}\biggr{)}^{\tfrac{p+2}{3}}+\\\
\tfrac{1}{(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{p-1}}\tfrac{M^{p}}{|\Omega|^{p-1}}\leqslant\tfrac{1}{1-\epsilon}(\tfrac{3}{2})^{\tfrac{2(p-1)}{3}}\biggl{(}\int\limits_{\Omega}{h^{2}_{x}\,dx}\biggr{)}^{\tfrac{p-1}{3}}(2M)^{\tfrac{p+2}{3}}+\\\
\tfrac{1}{(1-(1-\epsilon)^{\tfrac{1}{p-1}})^{p-1}}\tfrac{M^{p}}{|\Omega|^{p-1}}.$
∎
###### Lemma 5.2.
Let $0<n/2\leqslant m<n+2$. Let $h$ be the generalized solution of Theorem 1.
Then
$\tfrac{a_{0}}{4}\|h(.,T_{loc})\|^{2}_{H^{1}(\Omega)}\leqslant\mathcal{E}_{0}(0)+c_{1}M^{\tfrac{m-n+4}{2-m+n}}+c_{2}M^{m-n+2}+c_{3}M^{2},$
(5.2)
where $\mathcal{E}_{0}(0)$ is defined in (3.5), and $M=\int h_{0}$. Moreover,
if $m=n+2$ and $0<M<M_{c}$ then
$\tfrac{a_{0}}{4}\|h(.,T_{loc})\|^{2}_{H^{1}(\Omega)}\leqslant\tfrac{2}{3(1-c_{4}M^{2})}\mathcal{E}_{0}(0)+\tfrac{c_{5}}{1-c_{4}M^{2}}M^{4}+c_{6}M^{2}.$
(5.3)
###### Proof of Lemma 5.2.
We present the proof for the $m-n\neq-1,-2$ case in (3.4), leaving the
$m-n=-1,-2$ cases to the reader. The first step is to find a priori bound
(5.2) that is the analogue of Proposition 2.2 in [12]. From (3.6) we deduce
$\tfrac{a_{0}}{2}\int\limits_{\Omega}{h_{x}^{2}\,dx}\leqslant\mathcal{E}_{0}(0)+\tfrac{a_{1}}{(m-n+1)(m-n+2)}\int\limits_{\Omega}{h^{m-n+2}\,dx}.$
(5.4)
Due to (5.1), we have
$\int\limits_{\Omega}{h^{m-n+2}\,dx}\leqslant
k_{1}M^{\tfrac{m-n+4}{3}}\Bigl{(}\int\limits_{\Omega}{h_{x}^{2}\,dx}\Bigr{)}^{\tfrac{m-n+1}{3}}+k_{2}M^{m-n+2}.$
(5.5)
Thus, from (5.4), in view of (5.5), we find that
$\tfrac{a_{0}}{2}\int\limits_{\Omega}{h_{x}^{2}\,dx}-\tfrac{a_{1}k_{1}M^{\tfrac{m-n+4}{3}}}{(m-n+1)(m-n+2)}\Bigl{(}\int\limits_{\Omega}{h_{x}^{2}\,dx}\Bigr{)}^{\tfrac{m-n+1}{3}}\leqslant\mathcal{E}_{0}(0)+\tfrac{a_{1}k_{2}M^{m-n+2}}{(m-n+1)(m-n+2)}.$
(5.6)
Moreover, due to (5.1), in view of the Young inequality (A.6), we have
$\int\limits_{\Omega}{h^{2}\,dx}\leqslant
6^{2/3}M^{\tfrac{4}{3}}\Bigl{(}\int\limits_{\Omega}{h_{x}^{2}\,dx}\Bigr{)}^{\tfrac{1}{3}}+\tfrac{M^{2}}{|\Omega|}\leqslant\tfrac{1}{4}\int\limits_{\Omega}{h_{x}^{2}\,dx}+(\tfrac{8\sqrt{3}}{3}+|\Omega|^{-1})M^{2}.$
(5.7)
In the subcritical ($m<n+2$) case, $\tfrac{m-n+1}{3}<1$ and using (5.6) and
(5.7) we deduce (5.2) with
$\begin{gathered}c_{1}=\bigl{(}\tfrac{a_{1}k_{1}}{(m-n+1)(m-n+2)}\bigr{)}^{\tfrac{3}{2-m+n}}\bigl{(}\tfrac{8(m-n+1)}{3a_{0}}\bigr{)}^{\tfrac{m-n+1}{2-m+n}}\tfrac{2-m+n}{3},\\\
c_{2}=\tfrac{a_{1}k_{2}}{(m-n+1)(m-n+2)},\
c_{3}=\tfrac{a_{0}}{2}(\tfrac{8\sqrt{3}}{3}+|\Omega|^{-1}).\end{gathered}$
(5.8)
In the critical ($m=n+2$) case, $\tfrac{m-n+1}{3}=1$. If
$M<M_{c}=\bigl{(}\tfrac{6a_{0}}{a_{1}k_{1}}\bigr{)}^{\tfrac{1}{2}}$ then using
(5.6) we arrive at
$\int\limits_{\Omega}{h_{x}^{2}\,dx}\leqslant\tfrac{12}{6a_{0}-a_{1}k_{1}M^{2}}\bigl{(}\mathcal{E}_{0}(0)+\tfrac{a_{1}k_{2}}{12}M^{4}\bigr{)}$
(5.9)
Using (5.7), from (5.9) we obtain (5.3) with
$c_{4}=\tfrac{a_{1}k_{1}}{6a_{0}},c_{5}=\tfrac{a_{1}k_{2}}{18},\
c_{6}=\tfrac{a_{0}}{3}(\tfrac{8\sqrt{3}}{3}+|\Omega|^{-1}).$ (5.10)
∎
Under certain conditions, a bound closely related to (5.2) implies that if the
solution of Theorem 1 is initially constant then it will remain constant for
all time:
###### Theorem 5.
Assume $m=n$, the coefficient $a_{1}\geqslant 0$ in (2.1), and
$|\Omega|^{2}<a_{0}/|a_{1}|$. If the initial data is constant, $h_{0}\equiv
C>0$, then the solution of Theorem 1 satisfies $h(x,t)=C$ for all
$x\in\bar{\Omega}$ and all $t>0$.
For the long–wave unstable case ($a_{1}>0$) the hypotheses correspond to the
domain not being ‘‘too large’’. We note that it is not yet known whether or
not solutions from Theorem 1 are unique and so Theorem 5 does have content: it
ensures that the approximation method isn’t producing unexpected (nonconstant)
solutions from constant initial data.
###### Proof.
Consider the approximate solution $h_{\varepsilon}$. The definition of
$\mathcal{E}_{\varepsilon}(T)$ combined with the uniform-in-time bound (4.9)
implies
$\tfrac{a_{0}}{2}\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\leqslant\mathcal{E}_{\varepsilon}(0)+\tfrac{|a_{1}|}{2}\int\limits_{\Omega}h_{\varepsilon}^{2}(x,T)\;dx.$
(5.11)
Letting $M_{\varepsilon}=\int h_{0,\varepsilon}\,dx$ and applying Poincaré’s
inequality (A.2) to $v_{\varepsilon}=h_{\varepsilon}-M_{\varepsilon}/|\Omega|$
and using $\int h_{\varepsilon}^{2}\,dx=\int
v_{\varepsilon}^{2}\,dx+M_{\varepsilon}^{2}/|\Omega|$ yields
$\left(\tfrac{a_{0}}{2}-\tfrac{|a_{1}|\,|\Omega|^{2}}{2}\right)\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,t)\;dx\leq\mathcal{E}_{\varepsilon}(0)+\tfrac{|a_{1}|M_{\varepsilon}^{2}}{2|\Omega|}.$
If $h_{0,\varepsilon}\equiv C_{\varepsilon}=C+\varepsilon^{\theta}$ this
becomes
$\left(\tfrac{a_{0}}{2}-\tfrac{|a_{1}||\Omega|^{2}}{2}\right)\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\leqslant(|a_{1}|-a_{1})\tfrac{C_{\varepsilon}^{2}|\Omega|}{2}.$
If $a_{1}\geqslant 0$ and $|\Omega|^{2}<a_{0}/|a_{1}|$ then $\int
h_{\varepsilon,x}^{2}(x,T)\;dx=0$ for all $T\in[0,T_{\varepsilon,loc}]$ and
that this, combined with the continuity in space and time of
$h_{\varepsilon}$, implies that $h_{\varepsilon}\equiv C_{\varepsilon}$ on
$Q_{T_{\varepsilon,loc}}$.
Taking the sequence $\\{\varepsilon_{k}\\}$ that yields convergence to the
solution $h$ of Theorem 1, $h\equiv C$ on $Q_{T_{loc}}$. ∎
This $H^{1}$ control in time of the generalized solution given by Lemma 5.2 is
now used to extend the short–time existence result of Theorem 1 to a long–time
existence result:
###### Theorem 6.
Let $0<n/2\leqslant m<n+2$. Let $T_{g}$ be an arbitrary positive finite
number. The generalized weak solution $h$ of Theorem 1 can be continued in
time from $[0,T_{loc}]$ to $[0,T_{g}]$ in such a way that $h$ is also a
generalized weak solution and satisfies all the bounds of Theorem 1 (with
$T_{loc}$ replaced by $T_{g}$).
Similarly, the short–time existence of strong solutions (see Theorem 2) can be
extended to long–time existence.
###### Proof.
To construct a weak solution up to time $T_{g}$, one applies the local
existence theory iteratively, taking the solution at the final time of the
current time interval as initial data for the next time interval.
Introduce the times
$0=T_{0}<T_{1}<T_{2}<\dots<T_{N}<\dots\quad\mbox{where}\quad
T_{N}:=\sum_{n=0}^{N-1}T_{n,loc}$ (5.12)
and $T_{n,loc}$ is the interval of existence (A.21) for a solution with
initial data $h(\cdot,T_{n})$:
$T_{n,loc}:=\tfrac{9}{20c_{11}(\gamma_{1}-1)}\min\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,T_{n})+\tfrac{2c_{2}}{a_{0}}G_{0}(h(x,T_{n}))\;dx\right)^{-(\gamma_{1}-1)}\right\\}$
(5.13)
where $\gamma_{1}=\max\\{3,2m-n\\}$ and $c_{2}$ and $c_{11}$ are given in the
proof of Lemma 4.1.
The proof proceeds by contradiction. Assume there exists initial data $h_{0}$,
satisfying the hypotheses of Theorem 1, that results in a weak solution that
cannot be extended arbitrarily in time:
$\sum_{k=0}^{\infty}T_{n,loc}=T^{*}<\infty\quad\Longrightarrow\quad\lim_{n\to\infty}T_{n,loc}=0.$
From the definition (5.13) of $T_{n,loc}$, this implies
$\lim_{n\to\infty}\int\limits_{\Omega}(h_{x}^{2}(x,T_{n})+\tfrac{2c_{2}}{a_{0}}G_{0}(h(x,T_{n})))\;dx=\infty.$
(5.14)
By (5.2),
$\tfrac{a_{0}}{4}\int\limits_{\Omega}h_{x}^{2}(x,T_{n})\;dx\leq\mathcal{E}_{0}(T_{n-1})+K,$
where $K=c_{1}M^{\tfrac{m-n+4}{2-m+n}}+c_{2}M^{m-n+2}+c_{3}M^{2}$. By (3.6),
$\mathcal{E}_{0}(T_{n-1})\leqslant\mathcal{E}_{0}(T_{n-2})\leqslant\dots\mathcal{E}_{0}(0).$
Combining these,
$\tfrac{a_{0}}{4}\int\limits_{\Omega}h_{x}^{2}(x,T_{n})\;dx\leq\mathcal{E}_{0}(0)+K.$
(5.15)
By assumption, $T_{n}\to T^{*}<\infty$ as $n\to\infty$ hence $\int
h_{x}^{2}(x,T_{n})$ remains bounded. Assumption (5.14) then implies that $\int
G_{0}(h(x,T_{n}))\to\infty$ as $n\to\infty$.
To continue the argument, we step back to the approximate solutions
$h_{\varepsilon}$. Let $T_{n,\varepsilon}$ be the analogue of $T_{n}$ and
$T_{n,loc,\varepsilon}$, defined by (A.20), be the analogue of $T_{n,loc}$. By
(A.16),
$\displaystyle\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n,\varepsilon}))\;dx\leqslant\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n-1,\varepsilon}))\;dx$
(5.16) $\displaystyle\hskip
144.54pt+c_{10}\int\limits_{T_{n-1,\varepsilon}}^{T_{n,\varepsilon}}\max\left\\{1,\Bigl{(}\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\Bigr{)}^{\gamma_{2}}\right\\}\;dT$
with $\gamma_{2}=3$. Using the bound (4.9), one can prove the analogue of
Lemma 5.2 for the approximate solution $h_{\varepsilon}$. However the bound
(5.2) would be replaced by a bound on $\|h_{\varepsilon}(\cdot,T)\|_{H^{1}}$
which holds for all $T\in[0,T_{\varepsilon,loc}]$. This bound would then be
used to prove a bound like (5.15) to prove boundedness of $\int
h_{\varepsilon,x}^{2}(x,T)$ for all $T\in[0,T_{n,\varepsilon}]$. Using this
bound,
$\displaystyle\int\limits_{T_{n-1,\varepsilon}}^{T_{n,\varepsilon}}\Bigl{(}\int\limits_{\Omega}h_{\varepsilon,x}^{2}(x,T)\;dx\Bigr{)}^{\gamma_{2}}dT\leq\bigl{(}\tfrac{4}{a_{0}}\bigr{)}^{\gamma_{2}}\int\limits_{T_{n-1,\varepsilon}}^{T_{n,\varepsilon}}\left(\mathcal{E}_{\varepsilon}(0)+K\right)^{\gamma_{2}}\,dT$
$\displaystyle\hskip
72.26999pt=\bigl{(}\tfrac{4}{a_{0}}\bigr{)}^{\gamma_{2}}\left(\mathcal{E}_{\varepsilon}(0)+K\right)^{\gamma_{2}}\,T_{n-1,loc,\varepsilon}.$
(5.17)
If the initial data is such that $4/a_{0}\;(\mathcal{E}_{\varepsilon}(0)+K)<1$
then before using (5.17) in (5.16) we replace $K$ by a larger value so that
$4/a_{0}\;(\mathcal{E}_{\varepsilon}(0)+K)>1$. Using (5.17) in (5.16), it
follows that
$\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n,\varepsilon}))\;dx\leqslant\int\limits_{\Omega}G_{\varepsilon}(h_{\varepsilon}(x,T_{n-1,\varepsilon}))\;dx+\beta\,T_{n-1,loc,\varepsilon}$
(5.18)
for
$\beta=c_{10}\bigl{(}\tfrac{4}{a_{0}}\bigr{)}^{\gamma_{2}}\left(\mathcal{E}_{\varepsilon}(0)+K\right)^{\gamma_{2}}$.
Here $\beta$ depends on $|\Omega|$, the coefficients of the PDE, and on the
initial data $h_{0,\varepsilon}$.
One now takes the sequence $\\{\varepsilon_{k}\\}$ that was used to construct
the weak solution of Theorem 1 on the interval $[T_{n-1},T_{n}]$. Taking
$\varepsilon_{k}\to 0$, (5.18) yields
$\int\limits_{\Omega}G_{0}(h(x,T_{n}))\;dx\leq\int\limits_{\Omega}G_{0}(h(x,T_{n-1}))\,dx+\beta\,T_{n-1,loc}.$
(5.19)
Applying (5.19) iteratively,
$\displaystyle\int\limits_{\Omega}G_{0}(h(x,T_{n}))\;dx\leqslant\int\limits_{\Omega}G_{0}(h_{0}(x))\;dx+\beta\sum_{k=0}^{n-1}T_{k,loc}$
$\displaystyle\hskip
72.26999pt=\int\limits_{\Omega}G_{0}(h_{0}(x))\;dx+\beta\,T_{n}.$
This upper bound proves that $\int G_{0}(h(x,T_{n}))$ cannot diverge to
infinity as $n\to\infty$, finishing the proof. ∎
## 6 Strong positivity
###### Proposition 6.1.
Let $m\geqslant n/2$. Assume the initial data $h_{0}$ satisfies $h_{0}(x)>0$
for all $x\in\omega\subseteq\Omega$ where $\omega$ is an open interval. Then
1. 1.
if $n>3/2$ and $\alpha\in(-1/2,\min\\{1,n-2\\})$ then the strong solution from
Theorem 2 satisfies $h(x,T)>0$ for almost every $x\in\omega$, for all
$T\in[0,T_{loc}^{(\alpha)}]$;
2. 2.
if $n>2$ and $\alpha\in(-1/2,\min\\{1,3n/4-2\\})$ then the strong solution
from Theorem 2 satisfies $h(x,T)>0$ for all $x\in\omega$, for almost every
$T\in[0,T_{loc}^{(\alpha)}]$;
3. 3.
if $n\geqslant 7/2$ and $m\geqslant n-1/2$ then the strong solution from
Theorem 2 satisfies $h(x,T)>0$ for all
$(x,T)\in\overline{Q}_{T_{loc}^{(\alpha)}}$.
The proof of Proposition 6.1 depends on a local version of the a priori bound
(4.13) of Lemma 4.3:
###### Lemma 6.1.
Let $\omega\subseteq\Omega$ be an open interval and $\zeta\in
C^{2}(\bar{\Omega})$ such that $\zeta>0$ on $\omega$,
$\text{supp}\,\zeta=\overline{\omega}$, and $(\zeta^{4})^{\prime}=0$ on
$\partial\Omega$. If $\omega=\Omega$, choose $\zeta$ such that
$\zeta(-a)=\zeta(a)>0$. Let $\xi:=\zeta^{4}$.
If the initial data $h_{0}$ and the time $T_{loc}^{(\alpha)}$ are as in
Theorem 2 then for all $T\in[0,T_{loc}^{(\alpha)}]$ the strong solution $h$
from Theorem 2 satisfies
$\int\limits_{\Omega}{\xi(x)\;G_{0}^{(\alpha)}(h(x,T))}\;dx<\infty$ (6.1)
The proof of Lemma 6.1 is given in Appendix A. The proof of Proposition 6.1 is
essentially a combination of the proofs of Theorem 6.1 and Corollary 4.5 in
[7] and is provided here for the reader’s convenience.
###### Proof of Proposition 6.1.
Choose the test function $\zeta(x)$ to satisfy the hypotheses of Lemma 6.1.
Hence, (6.1) holds for every $T\in[0,T_{loc}^{(\alpha)}]$.
Proof of 1): Assume it is not true that $h(x,T)>0$ for almost every
$x\in\omega$, for all $T\in[0,T_{loc}^{(\alpha)}]$. Then there is a time
$T\in[0,T_{loc}^{(\alpha)}]$ such that the set
$\\{x\;:\;h(x,T)=0\\}\cap\omega$ has positive measure. Then because
$\alpha-n+2<0$,
$\infty>\int\limits_{\Omega}\xi(x)h^{\alpha-n+2}(x,T)\;dx\geqslant\int\limits_{\\{h(\cdot,T)=0\\}\cap\omega}\xi(x)h^{\alpha-n+2}(x,T)\;dx=\infty.$
This contradiction implies there can be no time at which $h$ vanishes on a set
of positive measure in $\omega$, as desired.
Proof of 2): We start by noting that by (3.12),
$(h^{\tfrac{\alpha+2}{2}})_{xx}(\cdot,T)\in L^{2}(\Omega)$ for almost all
$T\in[0,T_{loc}^{(\alpha)}]$ hence $h^{\tfrac{\alpha+2}{2}}(\cdot,T)\in
C^{3/2}(\Omega)$ for almost all $T\in[0,T_{loc}^{(\alpha)}]$. To prove 2), it
therefore suffices to show that if $T$ such that
$h^{\tfrac{\alpha+2}{2}}(\cdot,T)\in C^{3/2}(\Omega)$ then $h(x,T)>0$ on
$\omega$. Assume this is not true and there is an $x_{0}\in\omega$ and $T_{0}$
such that $h(x_{0},T_{0})=0$ and $h^{\tfrac{\alpha+2}{2}}(\cdot,T_{0})\in
C^{3/2}(\Omega)$. Then there is a $L$ such that
$h^{\tfrac{\alpha+2}{2}}(x,T_{0})=|h^{\tfrac{\alpha+2}{2}}(x,T_{0})-h^{\tfrac{\alpha+2}{2}}(x_{0},T_{0})|\leqslant
L|x-x_{0}|^{3/2}.$
Hence
$\infty>\int\limits_{\Omega}{\xi(x)h^{\alpha-n+2}(x,T_{0})}\;dx\geqslant
L^{\tfrac{2(\alpha-n+2)}{\alpha+2}}\int\limits_{\Omega}\xi(x)|x-x_{0}|^{-\tfrac{3(n-\alpha-2)}{\alpha+2}}\;dx=\infty.$
This contradiction implies there can be no point $x_{0}$ such that
$h(x_{0},T_{0})=0$, as desired. Note that we used $\xi>0$ on $\omega$ and
$x_{0}\in\omega$ to conclude that the integral diverges.
Proof of 3): Taking $\alpha=-\tfrac{1}{2}$ in (A.43), the approximate solution
$h_{\varepsilon}$ satisfies
$\int\limits_{\Omega}{G^{(-1/2)}_{\varepsilon}(h_{\varepsilon}(x,T))\,dx}\leqslant\int\limits_{\Omega}{G^{(-1/2)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n-\tfrac{1}{2}}h^{2}_{\varepsilon,x}\,dxdt}.$
(6.2)
Now we use the estimate
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n-\tfrac{1}{2}}h_{\varepsilon,x}^{2}\,dxdt}=-\tfrac{2}{2m-2n+1}\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n+\tfrac{1}{2}}h_{\varepsilon,xx}\,dxdt}\\\
\leqslant\tfrac{2}{2m-2n+1}\left(\iint\limits_{Q_{T}}{h^{2}_{\varepsilon,xx}\,dxdt}\right)^{\tfrac{1}{2}}\left(\iint\limits_{Q_{T}}{h_{\varepsilon}^{2m-2n+1}\,dxdt}\right)^{\tfrac{1}{2}}.$
Using $m\geqslant n-1/2$ and the $L^{\infty}$ bound on $h_{\varepsilon}$ from
(4.7) as well as (4.8), we obtain
$\iint\limits_{Q_{T}}{h_{\varepsilon}^{m-n-\tfrac{1}{2}}h_{\varepsilon,x}^{2}\,dxdt}\leqslant
C\sqrt{T},\text{ i.\,e.
}\iint\limits_{Q_{T}}{\left(h_{\varepsilon}^{\tfrac{2m-2n+3}{4}}\right)_{x}^{2}\,dxdt}\leqslant
C\sqrt{T}.$ (6.3)
Returning to (6.2) and taking $\varepsilon_{k}\to 0$ along the subsequence
that yielded the strong solution $h$, we find that
$\int\limits_{\Omega}{G^{(-1/2)}_{0}(h)\,dx}=\tfrac{1}{(n-3/2)(n-1/2)}\int\limits_{\Omega}{h^{3/2-n}(x,T)\,dx}\leqslant
K<\infty.$ (6.4)
We now prove $h(x,T)>0$ for all $x\in\bar{\Omega}$, and for every
$T\in[0,T_{loc}^{(\alpha)}]$. By (2.4), $h(\cdot,T)\in C^{1/2}(\bar{\Omega})$
for all $T\in[0,T_{loc}^{(\alpha)}]$. Assume $T_{0}$ is such that
$h(x_{0},T_{0})=0$ at some $x_{0}\in\Omega$. Then there is a $L$ such that
$h(x,T_{0})=|h(x,T_{0})-h(x_{0},T_{0})|\leqslant L|x-x_{0}|^{1/2}.$
Hence since $n\geqslant 7/2$
$\infty>\int\limits_{\Omega}{h^{\tfrac{3}{2}-n}(x,T_{0})}\;dx\geq
L^{\tfrac{3-2n}{2}}\int\limits_{\Omega}|x-x_{0}|^{-\tfrac{2n-3}{4}}\;dx=\infty.$
This contradiction implies there can be no point $x_{0}$ such that
$h(x_{0},T_{0})=0$, as desired. ∎
## 7 Finite speed of propagation
### 7.1 Local entropy estimate
###### Lemma 7.1.
Let $\zeta\in C^{1,2}_{t,x}(\bar{Q}_{T})$ such that
$\text{supp}\,\zeta\subset\Omega$, $(\zeta^{4})_{x}=0$ on $\partial\Omega$,
and $\zeta^{4}(-a,t)=\zeta^{4}(a,t)$. Assume that $-\tfrac{1}{2}<\alpha<1$,
and $\alpha\neq 0$. Then there exist a weak solution $h(x,t)$ in the sense of
the theorem Theorem 2, constants $C_{i}\ (i=1,2,3)$ dependent on
$n,\,m,\,\alpha,\,a_{0}$, and $a_{1}$, independent of $\Omega$, such that for
all $T\leqslant T_{loc}^{(\alpha)}$
$\int\limits_{\Omega}{\zeta^{4}(x,T)G^{(\alpha)}_{0}(h(x,T))\,dx}-\iint\limits_{Q_{T}}{(\zeta^{4})_{t}G^{(\alpha)}_{0}(h)\,dxdt}+C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant\\\
\int\limits_{\Omega}{\zeta^{4}(x,0)G^{(\alpha)}_{0}(h_{0})\,dx}+C_{2}\iint\limits_{Q_{T}}{h^{\alpha+2}(\zeta_{x}^{4}+\zeta^{2}\zeta_{xx}^{2})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{2(m-n+1)+\alpha}\zeta^{4}\,dxdt}.$
(7.1)
###### Sketch of Proof of Lemma 7.1.
In the following, we denote the positive, classical solution $h_{\varepsilon}$
constructed in Lemma 4.3 by $h$ (whenever there is no chance of confusion).
Let $\phi(x,t)=\zeta^{4}(x,t)$. Recall the entropy function
$G^{(\alpha)}_{\varepsilon}(z)$ defined by (4.12). Multiplying (4.1) by
$\phi(x,t)G^{\prime(\alpha)}_{\varepsilon}(h_{\varepsilon})$, and integrating
over $Q_{T}$ yields
$\int\limits_{\Omega}{\phi(x,T)G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}-\int\limits_{\Omega}{\phi(x,0)G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}-\\\
\iint\limits_{Q_{T}}{\phi_{t}(x,t)G^{(\alpha)}_{\varepsilon}(h)\,dxdt}=\iint\limits_{Q_{T}}{\phi_{x}f_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}+\\\
\iint\limits_{Q_{T}}{\phi
h^{\alpha}(a_{0}h_{x}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2})\,dxdt}=:I_{1}+I_{2}.$
(7.2)
We now bound the terms $I_{1}$ and $I_{2}$. First,
$I_{1}=-a_{0}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)h_{xx}\,dxdt}-a_{0}\iint\limits_{Q_{T}}{\phi_{x}(h^{\alpha}+f^{\prime}_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h))h_{x}h_{xx}\,dxdt}-\\\
a_{1}\iint\limits_{Q_{T}}{\phi_{xx}F^{(\alpha)}_{\varepsilon}(h)\,dxdt}=-a_{0}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)h_{xx}\,dxdt}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{\phi_{xx}h^{\alpha}h_{x}^{2}\,dxdt}+\\\
\tfrac{a_{0}\alpha}{2}\iint\limits_{Q_{T}}{\phi_{x}h^{\alpha-1}h_{x}^{3}\,dxdt}-a_{0}\iint\limits_{Q_{T}}{\phi_{x}f^{\prime}_{\varepsilon}(h)G^{\prime(\alpha)}_{\varepsilon}(h)h_{x}h_{xx}\,dxdt}-a_{1}\iint\limits_{Q_{T}}{\phi_{xx}F^{(\alpha)}_{\varepsilon}(h)\,dxdt},$
(7.3)
where
$F^{(\alpha)}_{\varepsilon}(z):=\int\limits_{0}^{z}{f_{\varepsilon}(s)G^{\prime(\alpha)}_{\varepsilon}(s)g^{\prime}_{\varepsilon}(s)\,ds}$,
$I_{2}=-a_{0}\iint\limits_{Q_{T}}{(\phi_{x}h^{\alpha}h_{x}h_{xx}+\alpha\phi
h^{\alpha-1}h^{2}_{x}h_{xx}+\phi h^{\alpha}h^{2}_{xx})\,dxdt}+\\\
a_{1}\iint\limits_{Q_{T}}{\phi
h^{\alpha}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{\phi_{xx}h^{\alpha}h_{x}^{2}\,dxdt}+\tfrac{5a_{0}\alpha}{6}\iint\limits_{Q_{T}}{\phi_{x}h^{\alpha-1}h_{x}^{3}\,dxdt}+\\\
\tfrac{a_{0}\alpha(\alpha-1)}{3}\iint\limits_{Q_{T}}{\phi
h^{\alpha-2}h_{x}^{4}\,dxdt}-a_{0}\iint\limits_{Q_{T}}{\phi
h^{\alpha}h^{2}_{xx}\,dxdt}+a_{1}\iint\limits_{Q_{T}}{\phi
h^{\alpha}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}.$ (7.4)
One easily finds that for all $\varepsilon>0$ and all $z\geqslant 0$
$|f_{\varepsilon}(z)G^{\prime(\alpha)}_{\varepsilon}(z)|\leqslant\tfrac{z^{\alpha+1}}{|\alpha-n+1|},\
|f_{\varepsilon}^{\prime}(z)G^{\prime(\alpha)}_{\varepsilon}(z)|\leqslant\tfrac{sz^{\alpha}}{|\alpha-n+1|},\
|F_{\varepsilon}(z)|\leqslant\tfrac{z^{m-n+\alpha+2}}{|\alpha-n+1|(m-n+\alpha+2)}+o(\varepsilon).$
Using these bounds, and the Cauchy inequality, we bound $I_{1}+I_{2}$:
$I_{1}+I_{2}\leqslant\tfrac{a_{0}\alpha(\alpha-1)}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h_{x}^{4}\phi\,dxdt}-a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\phi\,dxdt}+\\\
a_{1}\iint\limits_{Q_{T}}{h^{m-n+\alpha}h_{x}^{2}\phi\,dxdt}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h_{x}^{2}|\phi_{xx}|\,dxdt}+\\\
\tfrac{a_{0}}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha+1}|h_{xx}||\phi_{xx}|\,dxdt}+\tfrac{a_{0}s}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha}|h_{x}||h_{xx}||\phi_{x}|\,dxdt}+\\\
+\tfrac{4a_{0}\alpha}{3}\iint\limits_{Q_{T}}{h^{\alpha-1}h_{x}^{3}\phi_{x}\,dxdt}+\tfrac{a_{1}}{|\alpha-n+1|(m-n+\alpha+2)}\iint\limits_{Q_{T}}{h^{m-n+\alpha+2}|\phi_{xx}|\,dxdt}.$
(7.5)
Due to (7.5), we deduce from (7.2) that
$\int\limits_{\Omega}{\phi(x,T)G^{(\alpha)}_{\varepsilon}(h(T))\,dx}-\iint\limits_{Q_{T}}{\phi_{t}(x,t)G^{(\alpha)}_{\varepsilon}(h)\,dxdt}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\phi\,dxdt}+\\\
\tfrac{a_{0}\alpha(1-\alpha)}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h_{x}^{4}\phi\,dxdt}\leqslant\int\limits_{\Omega}{\phi(x,0)G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\\\
a_{1}\iint\limits_{Q_{T}}{h^{m-n+\alpha}h_{x}^{2}\phi\,dxdt}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h_{x}^{2}|\phi_{xx}|\,dxdt}+\\\
\tfrac{a_{0}}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha+1}|h_{xx}||\phi_{xx}|\,dxdt}+\tfrac{a_{0}s}{|\alpha-n+1|}\iint\limits_{Q_{T}}{h^{\alpha}|h_{x}||h_{xx}||\phi_{x}|\,dxdt}+\\\
\tfrac{4a_{0}\alpha}{3}\iint\limits_{Q_{T}}{h^{\alpha-1}h_{x}^{3}\phi_{x}\,dxdt}+\tfrac{a_{1}}{|\alpha-n+1|(m-n+\alpha+2)}\iint\limits_{Q_{T}}{h^{m-n+\alpha+2}|\phi_{xx}|\,dxdt},$
(7.6)
where $\alpha\neq n-1$. Recalling $\phi=\zeta^{4}$, and using the Young’s
inequality (A.6) and simple transformations, from (7.6) we find that
$\int\limits_{\Omega}{\zeta^{4}(x,T)G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}-\iint\limits_{Q_{T}}{(\zeta^{4})_{t}G^{(\alpha)}_{\varepsilon}(h)\,dxdt}+\\\
C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant\int\limits_{\Omega}{\zeta^{4}(x,0)G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon})\,dx}+\\\
C_{2}\iint\limits_{Q_{T}}{h^{\alpha+2}(\zeta_{x}^{4}+\zeta^{2}\zeta_{xx}^{2})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{2(m-n+1)+\alpha}\zeta^{4}\,dxdt}.$
(7.7)
We now argue that the $\varepsilon\to 0$ limit of the right-hand side of (7.7)
is finite and bounded by $K$, allowing us to apply Fatou’s lemma to the left-
hand side of (7.7), concluding
$\int\limits_{\Omega}\zeta^{4}(x,T)\;G^{(\alpha)}_{0}(h(x,T))\;dx-\iint\limits_{Q_{T}}{(\zeta^{4})_{t}G^{(\alpha)}_{0}(h)\,dxdt}+C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant
K<\infty$
for every $T\in[0,T_{loc}^{(\alpha)}]$, as desired. (Note that in taking
$\varepsilon\to 0$ we will choose the exact same sequence $\varepsilon_{k}$
that was used to construct the weak solution $h$ of Theorem 2. Also, in
applying Fatou’s lemma we used the fact that $\\{h=0\\}$ having measure zero
in $Q_{T_{loc}^{(\alpha)}}$ implies $\\{h(\cdot,T)\\}$ has measure zero in
$\Omega$.)
It suffices to show that
$\int\zeta^{4}(x,0)G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon})\to\int\zeta^{4}(x,0)G^{(\alpha)}_{0}(h_{0})<\infty$
as $\varepsilon\to 0$ (the rest of items is bonded as $h\in
L^{\infty}(0,T_{loc}^{(\alpha)};H^{1}(\Omega))$). This uses the Lebesgue
Dominated Convergence Theorem. First, note that
$G^{(\alpha)}_{\varepsilon}(z)=\tfrac{z^{\alpha-n+2}}{(\alpha-n+2)(\alpha-n+1)}+\tfrac{\varepsilon
z^{\alpha-2}}{(\alpha-2)(\alpha-3)}=G^{(\alpha)}_{0}(z)+\tfrac{\varepsilon
z^{\alpha-2}}{(\alpha-2)(\alpha-3)},$
hence if $h_{0}(x)>0$ then
$G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))=G^{(\alpha)}_{0}(h_{0}(x)+\varepsilon^{\theta})+\tfrac{\varepsilon(h_{0}(x)+\varepsilon^{\theta})^{\alpha-2}}{(\alpha-2)(\alpha-3)}\leqslant
G^{(\alpha)}_{0}(h_{0}(x)+\varepsilon^{\theta})+\tfrac{\varepsilon^{1-\theta(2-\alpha)}}{|(\alpha-2)(\alpha-3)|}.$
Because $h_{0}$ has finite entropy ($\int G^{(\alpha)}_{0}(h_{0})<\infty$) it
is positive almost everywhere in $\Omega$. Using this and the fact that
$\theta$ was chosen so that $\theta<1/(2-\alpha)<2/5$, we have
$|\phi(x,0)\,G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))|\leqslant\phi(x,0)(G^{(\alpha)}_{0}(h_{0}(x))+c)\leqslant
C(G_{0}(h_{0}(x))+c)$ almost everywhere in $x$ and for all
$\varepsilon<\varepsilon_{0}$. The dominating function is in $L^{1}$, because
$h_{0}$ has finite entropy.
It remains to show pointwise convergence
$\phi(x,0)G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))\to\phi(x,0)G_{0}(h_{0}(x))$
almost everywhere in $x$:
$\displaystyle\left|G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|\leq\left|G^{(\alpha)}_{\varepsilon}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))\right|$
$\displaystyle\hskip
28.90755pt+\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|=\tfrac{\varepsilon
h_{0,\varepsilon}^{\alpha-2}(x)}{(\alpha-2)(\alpha-3)}+\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|$
$\displaystyle\hskip
28.90755pt\leqslant\tfrac{\varepsilon^{1-\theta(2-\alpha)}}{|(\alpha-2)(\alpha-3)|}+\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|$
As before, $\tfrac{\varepsilon^{1-\theta(2-\alpha)}}{|(\alpha-2)(\alpha-3)|}$
goes to zero by the choice of $\theta$. The term
$\left|G^{(\alpha)}_{0}(h_{0,\varepsilon}(x))-G^{(\alpha)}_{0}(h_{0}(x))\right|$
goes to zero for almost every $x\in\Omega$ because $G^{(\alpha)}_{0}(z)$ is
continuous everywhere except at $z=0$.
The proof is similar for the case $\alpha=n-1$ and $\alpha=n-2$. ∎
### 7.2 Proof of Theorem 3 for the case $0<n<2$
Let $0<n<2$, and let $\text{supp}\,h_{0}\subseteq(-r_{0},r_{0})\Subset\Omega$.
For an arbitrary $s\in(0,a-r_{0})$ and $\delta>0$ we consider the families of
sets
$\Omega(s)=\Omega\setminus(-r_{0}-s,r_{0}+s),\ Q_{T}(s)=(0,T)\times\Omega(s).$
(7.8)
We introduce a nonnegative cutoff function $\eta(\tau)$ from the space
$C^{2}(\mathbb{R}^{1})$ with the following properties:
$\eta(\tau)=\left\\{\begin{aligned} \hfill 0\ &\ \tau\leqslant 0,\\\
\hfill\tau^{2}(3-2\tau)\ &\ 0<\tau<1,\\\ \hfill 1\ &\ \tau\geqslant
1.\end{aligned}\right.$ (7.9)
Next we introduce our main cut-off functions $\eta_{s,\delta}(x)\in
C^{2}(\bar{\Omega})$ such that $0\leqslant\eta_{s,\delta}(x)\leqslant 1\
\forall\,x\in\bar{\Omega}$ and possess the following properties:
$\eta_{s,\delta}(x)=\eta\bigl{(}\tfrac{|x|-(r_{0}+s)}{\delta}\bigr{)}=\left\\{\begin{aligned}
\hfill 1\;&,x\in\Omega(s+\delta),\\\ \hfill 0\;&,x\in\Omega(s),\\\
\end{aligned}\right.\ \ \ |(\eta_{s,\delta})_{x}|\leqslant\tfrac{3}{\delta},\
|(\eta_{s,\delta})_{xx}|\leqslant\tfrac{6}{\delta^{2}}$ (7.10)
for all $s>0,\ \delta>0:r_{0}+s+\delta<a$. Choosing
$\zeta^{4}(x,t)=\eta_{s,\delta}(x)e^{-\tfrac{t}{T}}$, from (7.1) we arrive at
$\int\limits_{\Omega(s+\delta)}{h^{\alpha-n+2}(T)\,dx}+\tfrac{1}{T}\iint\limits_{Q_{T}(s+\delta)}{h^{\alpha-n+2}\,dxdt}+C\iint\limits_{Q_{T}(s+\delta)}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dxdt}\leqslant\\\
\tfrac{C}{\delta^{4}}\iint\limits_{Q_{T}(s)}{h^{\alpha+2}\,dxdt}+C\iint\limits_{Q_{T}(s)}{h^{2(m-n+1)+\alpha}\,dxdt}=:C\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}$
(7.11)
for all $s>0$, where we consider $(n-1)_{+}<\alpha<1$. We apply the Gagliardo-
Nirenberg interpolation inequality (see Lemma D.4) in the region
$\Omega(s+\delta)$ to a function $v:=h^{\tfrac{\alpha+2}{2}}$ with
$a=\tfrac{2\xi_{i}}{\alpha+2},\ b=\tfrac{2(\alpha-n+2)}{\alpha+2},\ d=2,\
i=0,\ j=2$, and
$\theta_{i}=\tfrac{(\alpha+2)(\xi_{i}-\alpha+n-2)}{\xi_{i}(4\alpha-3n+8)}$
under the conditions:
$\alpha-n+2<\xi_{i}\text{ for }i=1,2.$ (7.12)
Integrating the resulted inequalities with respect to time and taking into
account (7.11), we arrive at the following relations:
$\iint\limits_{Q_{T}(s+\delta)}{h^{\xi_{i}}}\leqslant
C\,T^{1-\tfrac{\theta_{i}\xi_{i}}{\alpha+2}}\Biggl{(}\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{1+\nu_{i}}\\!\\!\\!\\!\\!+C\,T\Biggl{(}\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{\tfrac{\xi_{i}}{\alpha-n+2}}\\!\\!\\!\\!,$
(7.13)
where $\nu_{i}=\tfrac{4(\xi_{i}-\alpha+n-2)}{4\alpha-3n+8}$. These
inequalities are true provided that
$\tfrac{\theta_{i}\xi_{i}}{\alpha+2}<1\Leftrightarrow\xi_{i}<5\alpha-4n+10\text{
for }i=1,2.$ (7.14)
Simple calculations show that inequalities (7.12) and (7.14) hold with some
$(n-1)_{+}<\alpha<1$ if and only if
$0<n<2,\ \ \tfrac{n}{2}<m<6-n.$
Since all integrals on the right-hand sides of (7.13) vanish as $T\to 0$, the
finite speed of propagations follows from (7.13) by applying Lemma D.5 with
$s_{1}=0$ and sufficiently small $T$. Hence,
$\textnormal{supp}\,h(T,.)\subset(-r_{0}-\Gamma_{0}(T),r_{0}+\Gamma_{0}(T))\Subset\Omega\
\text{for all }T:0\leqslant T\leqslant T_{speed}.$ (7.15)
Due to (7.15), we can consider the solution $h(x,t)$ with compact support in
the whole space $\mathbb{R}^{1}$ and for $T\leqslant T_{speed}$. In this case,
we can repeat the previous procedure for
$\Omega(s)=\mathbb{R}^{1}\setminus(-r_{0}-s,r_{0}+s)$ and we obtain
$G_{i}(s+\delta):=\iint\limits_{Q_{T}(s+\delta)}{h^{\xi_{i}}}\leqslant
C\,T^{1-\tfrac{\theta_{i}\xi_{i}}{\alpha+2}}\Biggl{(}\sum\limits_{i=1}^{2}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{1+\nu_{i}},$
(7.16)
instead of (7.13), and
$\Gamma_{0}(T)=C\bigl{(}T^{(1-\tfrac{\theta_{1}\xi_{1}}{\alpha+2})(1+\nu_{2})}T^{(1-\tfrac{\theta_{2}\xi_{2}}{\alpha+2})\nu_{1}(1+\nu_{1})}(G(0))^{\nu_{1}}\bigr{)}^{\tfrac{1}{4(1+\nu_{1})(1+\nu_{2})}},$
(7.17)
$H(s)=CT^{(1-\tfrac{\theta_{2}\xi_{2}}{\alpha+2})(1+\nu_{1})}T^{(1-\tfrac{\theta_{1}\xi_{1}}{\alpha+2})\nu_{2}(1+\nu_{2})}(G_{2}(0))^{\nu_{2}},$
where
$G(0)=C(T^{(1-\tfrac{\theta_{1}\xi_{1}}{\alpha+2})(1+\nu_{2})}(G_{2}(0))^{1+\nu_{1}}+T^{(1-\tfrac{\theta_{2}\xi_{2}}{\alpha+2})(1+\nu_{1})}(G_{1}(0))^{1+\nu_{2}}).$
(7.18)
Now we need to estimate $G(0)$. With that end in view, we obtain the following
estimates:
$G_{i}(0)\leqslant
C_{1}\,(C_{2}+C_{3}T)^{\tfrac{\xi_{i}-1}{\alpha+5}}T^{1-\tfrac{\xi_{i}-1}{\alpha+5}},\
\ i=1,2.$ (7.19)
where $1<\xi_{i}<\alpha+6$, and $C_{i}$ depends on $h_{0}(x)$ only. Really,
applying the Gagliardo-Nirenberg interpolation inequality (see Lemma D.4) in
$\Omega=\mathbb{R}^{1}$ to a function $v:=h^{\tfrac{\alpha+2}{2}}$ with
$a=\tfrac{2\xi_{i}}{\alpha+2},\ b=\tfrac{2}{\alpha+2},\ d=2,\ i=0,\ j=2$, and
$\widetilde{\theta}_{i}=\tfrac{(\alpha+2)(\xi_{i}-1)}{\xi_{i}(\alpha+5)}$
under the condition $\xi_{i}>1$, we deduce that
$\int\limits_{\mathbb{R}^{1}}{h^{\xi_{i}}dx}\leqslant
c\,\|h_{0}\|_{1}^{\tfrac{2(3\xi_{i}+\alpha+2)}{(\alpha+2)(\alpha+5)}}\Bigl{(}\int\limits_{\mathbb{R}^{1}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dx}\Bigr{)}^{\tfrac{\xi_{i}-1}{\alpha+5}}.$
(7.20)
Integrating (7.20) with respect to time and taking into account the Hölder
inequality ($\tfrac{\xi_{i}-1}{\alpha+5}\leqslant
1\Rightarrow\xi_{i}<\alpha+6\Rightarrow m\leqslant n+2$), we arrive at the
following relations:
$G_{i}(0)\leqslant
c\,\|h_{0}\|_{1}^{\tfrac{2(3\xi_{i}+\alpha+2)}{(\alpha+2)(\alpha+5)}}T^{1-\tfrac{\xi_{i}-1}{\alpha+5}}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dx}\Bigr{)}^{\tfrac{\xi_{i}-1}{\alpha+5}},\
m\leqslant n+2.$ (7.21)
From (7.21), due to (A.32) and (5.2), we find (7.19).
Inserting (7.19) into (7.18), we obtain after straightforward computations
that
$\Gamma_{0}(T)\leqslant\Gamma(T)=C\,T^{\tfrac{1}{n+4}}$ (7.22)
for $T\leqslant T_{speed}$, where $\tfrac{n}{2}<m\leqslant n+2$, and
$T_{speed}$ finds from the condition $H(0)<1$.
In the case of $m>n+2$, we have (7.21) for $i=1$ only, i. e.
$G_{1}(0)\leqslant
c\,\|h_{0}\|_{1}^{\tfrac{2(3\xi_{1}+\alpha+2)}{(\alpha+2)(\alpha+5)}}T^{1-\tfrac{\xi_{1}-1}{\alpha+5}}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\,dx}\Bigr{)}^{\tfrac{\xi_{1}-1}{\alpha+5}}.$
(7.23)
Hence
$G_{1}(0)\leqslant
C(C+C(1-(1-\tfrac{k_{1}(2m-n)\|h_{0}\|_{H^{1}}^{2m-n}}{2}T)^{\tfrac{m+2}{2m-n}})T^{1-\tfrac{\xi_{1}-1}{\alpha+5}}\leqslant
C_{0}T^{1-\tfrac{\xi_{1}-1}{\alpha+5}}.$ (7.24)
Next we need estimate the term $G_{2}(0)$. Really, by using the inequality
(A.3), we obtain
$G_{2}(0)\leqslant
b_{2}\|h_{0}\|_{1}^{\tfrac{\xi_{2}+2}{3}}\int\limits_{0}^{T}{\|h_{x}\|_{2}^{\tfrac{2}{3}(\xi_{2}-1)}dt}=b_{2}\|h_{0}\|_{1}^{\tfrac{\xi_{2}+2}{3}}\|h_{0}\|_{H^{1}}^{\tfrac{2(\xi_{2}-1)}{3}}\times\\\
\int\limits_{0}^{T}{(1-\tfrac{k_{1}(2m-n)\|h_{0}\|_{H^{1}}^{2m-n}}{2}t)^{-\tfrac{2(\xi_{2}-1)}{3(2m-n)}}dt}=b_{2}\|h_{0}\|_{1}^{\tfrac{\xi_{2}+2}{3}}\|h_{0}\|_{H^{1}}^{-\tfrac{2m+n-2(\alpha+1))}{3}}\times\\\
\tfrac{6}{k_{1}(2m+n-2(\alpha+1))}\bigl{(}1-(1-\tfrac{k_{1}(2m-n)\|h_{0}\|_{H^{1}}^{2m-n}}{2}T)^{\tfrac{2m+n-2(\alpha+1)}{3(2m-n)}}\bigr{)}\leqslant
C_{0}$ (7.25)
for all $T\leqslant T_{loc}$, where $C_{0}$ depends on $h_{0}(x)$. Inserting
(7.19) for $i=1$ and (7.25) into (7.18), we obtain (7.22).
### 7.3 Local energy estimate
###### Lemma 7.2.
Let $n\in\bigl{(}\tfrac{1}{2},3\bigr{)}$, and $m>\tfrac{2(n-1)_{+}}{3}$, and
$\beta>\tfrac{1-n}{3}$. Let $\zeta\in C^{2}(\bar{\Omega})$ such that
$\text{supp}\,\zeta$ in $\Omega$, and $(\zeta^{6})^{\prime}=0$ on
$\partial\Omega$, and $\zeta(-a)=\zeta(a)$. Then there exist constants $C_{i}\
(i=1,2,3)$ dependent on $n,\,m,\,a_{0}$, and $a_{1}$, independent of $\Omega$
and $\varepsilon$, such that for all $T\leqslant T_{loc}$
$\int\limits_{\Omega}{\zeta^{6}h^{2}_{x}(x,T)\,dx}+\int\limits_{\Omega}{\zeta^{4}h^{\beta+1}(T)\,dx}+C_{1}\iint\limits_{Q_{T}}{\zeta^{6}(h^{\tfrac{n+2}{2}})_{xxx}^{2}\,dxdt}\leqslant\int\limits_{\Omega}{\zeta^{6}h_{0}^{2}(x)\,dx}+\\\
\int\limits_{\Omega}{\zeta^{4}h_{0}^{\beta+1}\,dx}+C_{2}\iint\limits_{Q_{T}}{h^{n+2}(\zeta_{x}^{6}+\zeta^{3}|\zeta_{xx}|^{3})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt}+\\\
C_{4}\iint\limits_{Q_{T}}{\\{h^{n+2\beta}\zeta_{x}^{2}+\chi_{\\{\zeta>0\\}}h^{n+3\beta-1}+h^{\tfrac{6m-n+6\beta+4}{5}}\zeta^{\tfrac{12}{5}}\zeta_{x}^{\tfrac{6}{5}}+h^{\tfrac{3m-n+3\beta+1}{2}}\zeta^{3}\\}\,dxdt}.$
(7.26)
###### Sketch of Proof of Lemma 7.2.
Let $\phi(x)=\zeta^{6}(x)$. Multiplying (4.1) by $-(\phi(x)h_{x})_{x}$, and
integrating on $Q_{T}$, yields
$\tfrac{1}{2}\int\limits_{\Omega}{\phi(x)h^{2}_{x}(x,T)\,dx}-\tfrac{1}{2}\int\limits_{\Omega}{\phi(x)h_{0\varepsilon,x}^{2}(x)\,dx}=\\\
-\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})(\phi_{xx}h_{x}+2\phi_{x}h_{xx}+\phi
h_{xxx})\,dxdt}=\\\
-\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\phi_{xx}h_{x}\,dxdt}-\\\
2\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\phi_{x}h_{xx}\,dxdt}-\\\
\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\phi
h_{xxx}\,dxdt}=:I_{1}+I_{2}+I_{3}.$ (7.27)
We now bound the terms $I_{1}$, $I_{2}$ and $I_{3}$. First,
$I_{1}=-a_{0}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)h_{xxx}h_{x}\,dxdt}-a_{1}\iint\limits_{Q_{T}}{\phi_{xx}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=\\\
-6a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h_{xxx}h_{x}\zeta^{4}(5\zeta_{x}^{2}+\zeta\zeta_{xx})\,dxdt}-6a_{1}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\zeta^{4}(5\zeta_{x}^{2}+\zeta\zeta_{xx})\,dxdt}\leqslant\\\
\epsilon_{1}\iint\limits_{Q_{T}}{\zeta^{6}\\{f_{\varepsilon}(h)h_{xxx}^{2}+h^{n-4}h_{x}^{6}\\}\,dxdt}+C(\epsilon_{1})\iint\limits_{Q_{T}}{h^{n+2}(\zeta_{x}^{6}+\zeta^{3}|\zeta_{xx}|^{3})\,dxdt}+\\\
C(\epsilon_{1})\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt},$ (7.28)
$I_{2}=-2a_{0}\iint\limits_{Q_{T}}{\phi_{x}f_{\varepsilon}(h)h_{xxx}h_{xx}\,dxdt}-2a_{1}\iint\limits_{Q_{T}}{\phi_{x}f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{xx}h_{x}\,dxdt}=\\\
-12a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h_{xxx}h_{xx}\zeta^{5}\zeta_{x}\,dxdt}-12a_{1}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{xx}h_{x}\zeta^{5}\zeta_{x}\,dxdt}\leqslant\\\
\epsilon_{2}\iint\limits_{Q_{T}}{\zeta^{6}\\{f_{\varepsilon}(h)h_{xxx}^{2}+h^{n-2}h_{x}^{2}h^{2}_{xx}+h^{n-1}|h_{xx}|^{3}\\}\,dxdt}+\\\
C(\epsilon_{2})\iint\limits_{Q_{T}}{h^{n+2}\zeta_{x}^{6}\,dxdt}+C(\epsilon_{2})\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt},$
(7.29) $I_{3}=-a_{0}\iint\limits_{Q_{T}}{\phi
f_{\varepsilon}(h)h_{xxx}^{2}\,dxdt}-a_{1}\iint\limits_{Q_{T}}{\phi
f_{\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{xxx}h_{x}\,dxdt}\leqslant\\\
-a_{0}\iint\limits_{Q_{T}}{\zeta^{6}f_{\varepsilon}(h)h_{xxx}^{2}\,dxdt}+\epsilon_{3}\iint\limits_{Q_{T}}{\zeta^{6}(f_{\varepsilon}(h)h^{2}_{xxx}+h^{n-4}h_{x}^{6})\,dxdt}+\\\
C(\epsilon_{3})\iint\limits_{Q_{T}}{h^{3m-2n+2}\zeta^{6}\,dxdt}.$ (7.30)
Now, multiplying (4.1) by $\zeta^{4}(h+\gamma)^{\beta}$,
$\beta>\tfrac{1-n}{3},\ \gamma>0$ and integrating on $Q_{T}$, using the
Young’s inequality (A.6), letting $\gamma\to 0$, we obtain the following
estimate
$\int\limits_{\Omega}{\zeta^{4}h^{\beta+1}(T)\,dx}\leqslant\int\limits_{\Omega}{\zeta^{4}h_{0\varepsilon}^{\beta+1}\,dx}+\epsilon_{4}\iint\limits_{Q_{T}}{\zeta^{6}\\{f_{\varepsilon}(h)h_{xxx}^{2}+h^{n-4}h_{x}^{6}\\}\,dxdt}+\\\
C(\epsilon_{4})\iint\limits_{Q_{T}}{\\{h^{n+2\beta}\zeta_{x}^{2}+\chi_{\\{\zeta>0\\}}h^{n+3\beta-1}+h^{\tfrac{6m-n+6\beta+4}{5}}\zeta^{\tfrac{12}{5}}\zeta_{x}^{\tfrac{6}{5}}+h^{\tfrac{3m-n+3\beta+1}{2}}\zeta^{3}\\}\,dxdt},$
(7.31)
where $\beta>\max\\{\tfrac{1-n}{3},-m+\tfrac{n-1}{3}\\}=\tfrac{1-n}{3}$ as
$m>\tfrac{2(n-1)_{+}}{3}$.
If we now add inequalities (7.27) and (7.31), in view of (7.28)–(7.30), then,
applying Lemma D.3, choosing $\epsilon_{i}>0$, and letting $\varepsilon\to 0$,
we obtain (7.26). ∎
### 7.4 Proof of Theorem 3 for the case $\tfrac{1}{2}<n<3$
Let $\eta_{s,\delta}(x)$ denote by (7.10). Setting
$\zeta^{6}(x)=\eta_{s,\delta}(x)$ into (7.26), after simple transformations,
we obtain
$\int\limits_{\Omega(s+\delta)}{h^{2}_{x}(x,T)\,dx}+\int\limits_{\Omega(s+\delta)}{h^{\beta+1}(T)\,dx}+C\iint\limits_{Q_{T}(s+\delta)}{(h^{\tfrac{n+2}{2}})_{xxx}^{2}\,dxdt}\leqslant\\\
\tfrac{C}{\delta^{6}}\iint\limits_{Q_{T}(s)}{h^{n+2}\,dxdt}+\tfrac{C}{\delta^{2}}\iint\limits_{Q_{T}(s)}{h^{n+2\beta}\,dxdt}+\tfrac{C}{\delta^{\tfrac{6}{5}}}\iint\limits_{Q_{T}(s)}{h^{\tfrac{6m-n+6\beta+4}{5}}\,dxdt}+\\\
C\iint\limits_{Q_{T}(s)}{\\{h^{3m-2n+2}+h^{n+3\beta-1}+h^{\tfrac{3m-n+3\beta+1}{2}}\\}\,dxdt}=:C\sum\limits_{i=1}^{6}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}$
(7.32)
for all for all $s>0,\ \delta>0:r_{0}+s+\delta<a$. We apply the Gagliardo-
Nirenberg interpolation inequality (Lemma D.4) in the region
$\Omega(s+\delta)$ to a function $v:=h^{\tfrac{n+2}{2}}$ with
$a=\tfrac{2\xi_{i}}{n+2},\ b=\tfrac{2(\beta+1)}{n+2},\ d=2,\ i=0,\ j=3$, and
$\theta_{i}=\tfrac{(n+2)(\xi_{i}-\beta-1)}{\xi_{i}(n+5\beta+7)}$ under the
conditions:
$\beta<\xi_{i}-1\text{ for }i=\overline{1,6}.$ (7.33)
Integrating the resulted inequalities with respect to time and taking into
account (7.32), we arrive at the following relations:
$\iint\limits_{Q_{T}(s+\delta)}{h^{\xi_{i}}}\leqslant
C\,T^{1-\tfrac{\theta_{i}\xi_{i}}{n+2}}\Biggl{(}\sum\limits_{i=1}^{6}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{1+\nu_{i}}\\!\\!\\!\\!\\!+C\,T\Biggl{(}\sum\limits_{i=1}^{5}{\delta^{-\alpha_{i}}\iint\limits_{Q_{T}(s)}{h^{\xi_{i}}}}\Biggr{)}^{\tfrac{\xi_{i}}{\beta+1}}\\!\\!\\!\\!,$
(7.34)
where $\nu_{i}=\tfrac{6(\xi_{i}-\beta-1)}{n+5\beta+7}$. These inequalities are
true provided that
$\tfrac{\theta_{i}\xi_{i}}{n+2}<1\Leftrightarrow\beta>\tfrac{\xi_{i}-n-8}{6}\text{
for }i=\overline{1,6}.$ (7.35)
Simple calculations show that inequalities (7.33) and (7.35) hold with some
$\beta>\tfrac{1-n}{3}$ if and only if
$\tfrac{1}{2}<n<3,\ \ m>\tfrac{n}{2}.$
Since all integrals on the right-hand sides of (7.34) vanish as $T\to 0$, the
finite speed of propagations follows from (7.34) by applying Lemma D.5 with
$s_{1}=0$ and sufficiently small $T$. Hence,
$\textnormal{supp}\,h(T,.)\subset(-r_{0}-\Gamma(T),r_{0}+\Gamma(T))\Subset\Omega\
\text{for all }T:0\leqslant T\leqslant T_{speed}.$ (7.36)
## 8 Finite time blow up
###### Lemma 8.1.
Let $0<n<2,\,m\geqslant\max\\{n+2,4-n\\}$. Then the weak solution $h$ from
Theorem 1 satisfies the second-moment inequality:
$e^{-\widetilde{B}(T)}\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h(x,T))\,dx}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\\\
\int\limits_{0}^{T}{\biggl{(}k_{1}\mathcal{E}_{0}(0)+W^{\prime}(t)+k_{2}\int\limits_{\Omega}{x^{2}h^{2}_{xx}\,dx}\biggr{)}e^{-\widetilde{B}(t)}dt}$
(8.1)
for all $T\in[0,T_{loc}]$, where $k_{1}=2(4-n),\
k_{2}=\tfrac{3a_{0}(n-1)}{2}$. Here
$\widetilde{G}_{0}(z)=\tfrac{1}{2-n}z^{2-n},\
\widetilde{B}(T):=\tfrac{a_{1}^{2}|1-n|(2-n)}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}{\|h(.,\tau)\|_{L^{\infty}(\Omega)}^{2m-n}\,d\tau},$
$W(T):=\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}mD_{0}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}.$
(8.2)
Moreover, if $h(\cdot,t)$ has a compact support in $\Omega$ then
$W^{\prime}(t)\equiv 0$.
The proof of Lemma 8.1 is given in Appendix B Proof of second moment estimate.
###### Proof of Theorem 4.
Assume $0<n<2$ and $m$ satisfies the hypotheses of Theorem 3. By Theorems 2,
3, there is a compactly supported strong solution $h$ for $t\in[0,T_{0}]$.
Hence, we can assume that $W^{\prime}(t)=0$ in Lemma 8.1.
First, we construct a sequence of times $T_{0}<T_{1}<...$ and extend the
strong solution $h$ from the time interval $[0,T_{i}]$ to the time interval
$[0,T_{i+1}]$. Taking it as an initial datum, we obtain a time interval of
existence
$T_{1}-T_{0}=\tfrac{1}{2d_{7}(\gamma_{3}-1)}\biggl{(}\int\limits_{\Omega}{\\{\tfrac{1}{2}h_{x}^{2}(x,T_{0})+G_{0}^{(\alpha)}(h(x,T_{0}))\\}\,dx}\biggr{)}^{-(\gamma_{3}-1)},$
by (A.39). Applying Theorem 3 to the time interval $[T_{0},T_{1}]$, we have a
strong solution with compact support that satisfies all a priori estimates
with the time interval $[0,T_{0}]$ replaced by $[0,T_{1}]$. In this way, we
construct a nonnegative, compactly supported, strong solution on
$\mathbb{R}^{1}\times[0,T^{*})$ where
$T^{*}=\lim_{i\to\infty}T_{i}.$
If $T^{*}<\infty$ then
$\tfrac{1}{2e_{7}(\gamma_{3}-1)}\biggl{(}\int\limits_{\Omega}{\\{\tfrac{1}{2}h_{x}^{2}(x,T_{i})+G_{0}^{(\alpha)}(h(x,T_{i}))\\}\,dx}\biggr{)}^{1-\gamma_{3}}=T_{i+1}-T_{i}\to
0.$
Hence, the $H^{1}$ norms at times $T_{i}$ must blow up. And, due to (3.8), the
$L^{\infty}$-norm of the solution at times $T_{i}$ must also blow up.
Therefore, to finish the proof it suffices to prove that $T^{*}<\infty$.
Let
$V(0)=\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}\quad\mbox{and}\quad
V(T_{i}):=e^{-\widetilde{B}(T_{i})}\int\limits_{-\infty}^{\infty}{x^{2}\widetilde{G}_{0}(h(T_{i}))\,dx}.$
(8.3)
Using that $k_{1}>0$ and $\mathcal{E}_{0}(T_{i})\leqslant\mathcal{E}_{0}(0)$,
we apply the inequality (8.1) iteratively to find
$V(T_{i})\leqslant
V(0)+k_{1}\mathcal{E}_{0}(0)\int\limits_{0}^{T_{i}}e^{-\widetilde{B}(t)}\;dt+k_{2}\int\limits_{0}^{T_{i}}\int\limits_{-\infty}^{\infty}x^{2}h_{xx}^{2}(x,t)\;dxe^{-\widetilde{B}(t)}\;dt$
(8.4)
Case $n=1$: [13] In this case, $\widetilde{B}(t)=0$ and $k_{2}=0$. Hence (8.4)
becomes
$V(T_{i})\leqslant V(0)+k_{1}\mathcal{E}_{0}(0)\,T_{i}.$
If $T^{*}=\infty$ then the right-hand side will become negative, contradicting
that the left-hand side is nonnegative. Hence $T^{*}<\infty$.
Case $0<n<1$: In this case, $k_{2}<0$ and (8.4) implies
$V(T_{i})\leqslant
V(0)+k_{1}\mathcal{E}_{0}(0)\int\limits_{0}^{T_{i}}e^{-\widetilde{B}(t)}\;dt$
(8.5)
We wish to argue that if $T^{*}=\infty$ then the right-hand side of (8.5) must
become negative, leading to a contradition.
Let $g(t)=\int\limits_{0}^{t}{e^{-\widetilde{B}(s)}ds}$. Using (A.19), we have
$\|h(x,t)\|_{\infty}\leqslant K_{1}(T_{i}-t)^{-\tfrac{1}{2(2m-n-1)}}$ for all
$t<T_{i}$.
$\widetilde{B}(s)\leqslant\tfrac{a_{1}^{2}(1-n)(2-n)}{2a_{0}(m-n+1)^{2}}K_{1}^{2m-n}\int\limits_{0}^{s}{(T_{i}-s)^{-\tfrac{2m-n}{2(2m-n-1)}}\,d\tau}=K_{2}\bigl{(}T_{i}^{\tfrac{2m-n-2}{2(2m-n-1)}}-(T_{i}-s)^{\tfrac{2m-n-2}{2(2m-n-1)}}\bigr{)},$
whence
$g(T_{i})\geqslant
e^{-K_{2}T_{i}^{\tfrac{2m-n-2}{2(2m-n-1)}}}\int\limits_{0}^{T_{i}}{e^{K_{2}(T_{i}-s)^{\tfrac{2m-n-2}{2(2m-n-1)}}}ds}=\\\
\tfrac{\int\limits_{0}^{T_{i}}{e^{K_{2}s^{\tfrac{2m-n-2}{2(2m-n-1)}}}ds}}{e^{K_{2}T_{i}^{\tfrac{2m-n-2}{2(2m-n-1)}}}}\sim
T_{i}^{\tfrac{2m-n}{2(2m-n-1)}}\text{ as }T_{i}\to+\infty.$ (8.6)
Since $\mathcal{E}_{0}(0)<0$, if $T_{i}\to+\infty$ then for large times the
right-hand side of (8.5) would be negative: an impossibility. Therefore
$\mathop{\lim}\limits_{i\to+\infty}T_{i}=T^{*}<\infty$.
Case $1<n<2$: In this case, we write inequality (8.4) in the form:
$V(T_{i})\leqslant V(0)+k_{1}\mathcal{E}_{0}(0)\,g(T_{i})+k_{2}\,f(T_{i})$
(8.7)
where
$f(T_{i})=\int\limits_{0}^{T_{i}}e^{-\widetilde{B}(t)}\int\limits_{-\infty}^{\infty}x^{2}h_{xx}^{2}(x,t)\;dx\;dt.$
Assume $T^{*}=\infty$ and that $f(T_{i})$ grows slower than $g(T_{i})$ as
$T_{i}\to\infty$. Then the right–hand side of (8.7) will become negative in
finite time, which is an impossibility. It therefore suffices to show that
$f(T_{i})$ grows more slowly than $g(T_{i})$.
Due to (A.16) and (A.19) as $\varepsilon\to 0$, we have
$\tfrac{a_{0}}{2}\int\limits_{0}^{T_{i}}\int\limits_{-\infty}^{\infty}{h^{2}_{xx}\,dxdt}\leqslant\int\limits_{-\infty}^{\infty}{G_{0}(h_{0})\,dx}+C\int\limits_{0}^{T_{i}}{\|h_{x}\|_{2}^{2(m-n+1)}dt}\leqslant\\\
\int\limits_{-\infty}^{\infty}{G_{0}(h_{0})\,dx}+C\int\limits_{0}^{T_{i}}{(T_{i}-t)^{-\tfrac{m-n+1}{2m-n-1}}dt}=\int\limits_{-\infty}^{\infty}{G_{0}(h_{0})\,dx}+\tfrac{C(2m-n-1)}{m-2}T_{i}^{\tfrac{m-2}{2m-n-1}}.$
(8.8)
In view of (7.22), we know that $|x|\leqslant C(h_{0})$ for all $t\leqslant
T_{i}$. Hence, due to (8.8), we deduce
$f(T_{i})\leqslant\int\limits_{-\infty}^{\infty}{x^{2}h^{2}_{xx}\,dxdt}\leqslant
C\int\limits_{0}^{T_{i}}\int\limits_{-\infty}^{\infty}{h^{2}_{xx}\,dxdt}\leqslant
C(1+T_{i}^{\tfrac{m-2}{2m-n-1}}).$
Comparing with the exponent in (8.6), we see that $f(T_{i})$ grows more slowly
than $g(T_{i})$, as desired.
∎
## Appendix A Proofs of A Priori Estimates
The first observation is that the periodic boundary conditions imply that
classical solutions of equation (4.1) conserve mass:
$\int\limits_{\Omega}{h_{\delta\varepsilon}(x,t)\,dx}=\int\limits_{\Omega}{h_{0,\delta\varepsilon}(x)\,dx}=M_{\delta\varepsilon}<\infty\text{
for all }t>0.$ (A.1)
Further, (4.5) implies $M_{\delta\varepsilon}\to M=\int h_{0}$ as
$\varepsilon,\delta\to 0$. The initial data in this article have $M>0$, hence
$M_{\delta\varepsilon}>0$ for $\delta$ and $\varepsilon$ sufficiently small.
Also, we will relate the $L^{p}$ norm of $h$ to the $L^{p}$ norm of its zero-
mean part as follows:
$|h(x)|\leqslant\left|h(x)-\tfrac{M}{|\Omega|}\right|+\tfrac{M}{|\Omega|}\Longrightarrow\|h\|_{p}^{p}\leqslant
2^{p-1}\;\|v\|_{p}^{p}+\left(\tfrac{2}{|\Omega|}\right)^{p-1}\;M^{p}$
where $v:=h-M/|\Omega|$ and we have assumed that $M\geqslant 0$. We will use
the Poincaré inequality which holds for any zero-mean function in
$H^{1}(\Omega)$
$\|v\|_{p}^{p}\leqslant b_{1}\|v_{x}\|_{p}^{p}\qquad 1\leqslant p<\infty$
(A.2)
with $b_{1}=|\Omega|^{p}$.
Also used will be an interpolation inequality [30, Theorem 2.2, p. 62] for
functions of zero mean in $H^{1}(\Omega)$:
$\|v\|_{p}^{p}\leqslant b_{2}\,\|v_{x}\|_{2}^{ap}\;\|v\|_{r}^{(1-a)p}$ (A.3)
where $r\geqslant 1$, $p\geqslant r$,
$a=\tfrac{1/r-1/p}{1/r+1/2},\qquad b_{2}=\left(1+r/2\right)^{ap}.$
It follows that for any zero-mean function $v$ in $H^{1}(\Omega)$
$\|v\|_{p}^{p}\leqslant
b_{3}\|v_{x}\|_{2}^{p},\quad\Longrightarrow\quad\|h\|_{p}^{p}\leqslant
b_{4}\|h_{x}\|_{2}^{p}+b_{5}M_{\delta\varepsilon}^{p}$ (A.4)
where
$b_{3}=\begin{cases}b_{1}\;|\Omega|^{(2-p)/p}&\mbox{if}\quad 1\leqslant
p\leqslant 2\\\ b_{1}^{(p+2)/2}\;b_{2}&\mbox{if}\quad
2<p<\infty\end{cases},\quad b_{4}=2^{p-1}\,b_{3},\quad
b_{5}=\left(\tfrac{2}{|\Omega|}\right)^{p-1}$
To see that (A.4) holds, consider two cases. If $1\leqslant p<2$, then by
(A.2), $\|v\|_{p}$ is controlled by $\|v_{x}\|_{p}$. By the Hölder inequality,
$\|v_{x}\|_{p}$ is then controlled by $\|v_{x}\|_{2}$. If $p>2$ then by (A.3),
$\|v\|_{p}$ is controlled by $\|v_{x}\|_{2}^{a}\|v\|_{2}^{1-a}$ where
$a=1/2-1/p$. By the Poincaré inequality, $\|v\|_{2}^{1-a}$ is controlled by
$\|v_{x}\|_{2}^{1-a}$.
If $0<p<1$ then, instead of (A.4), we obtain
$\|h\|_{p}^{p}\leqslant\widetilde{b}_{4}\|h_{x}\|_{2}^{p}+\widetilde{b}_{5}M_{\delta\varepsilon}^{p}$
(A.5)
where
$\widetilde{b}_{4}=|\Omega|^{1-\tfrac{p}{2}}\,b_{4}^{\tfrac{p}{2}},\quad\widetilde{b}_{5}=|\Omega|^{1-\tfrac{p}{2}}\,b_{5}^{\tfrac{p}{2}}.$
The Cauchy inequality $ab\leqslant\epsilon a^{2}+b^{2}/(4\epsilon)$ with
$\epsilon>0$ will be used often as will Young’s inequality
$ab\leqslant\epsilon a^{p}+\tfrac{b^{q}}{q\,(\epsilon
p)^{q/p}},\qquad\tfrac{1}{p}+\tfrac{1}{q}=1,\;\epsilon>0.$ (A.6)
###### Sketch of Proof of Lemma 4.1.
In the following, we denote the classical solution $h_{\delta\varepsilon}$ by
$h$ whenever there is no chance of confusion.
To prove the bound (4.7) one starts by multiplying (4.1) by $-h_{xx}$,
integrating over $Q_{T}$, and using the periodic boundary conditions (4.2)
yields
$\displaystyle\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
(A.7) $\displaystyle\hskip
14.45377pt=\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}(x)\,dx}-a_{1}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}h_{xxx}\,dxdt}.$
The Cauchy inequality is used to bound some terms on the right-hand of (A.7):
$a_{1}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)D^{\prime\prime}_{\varepsilon}(h)h_{x}h_{xxx}\,dxdt}\leqslant\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}+\\\
\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)(D^{\prime\prime}_{\varepsilon}(h))^{2}h_{x}^{2}\,dxdt}.$
(A.8)
Using (A.8) in (A.7) yields
$\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+\\\
\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)(D^{\prime\prime}_{\varepsilon}(h))^{2}h_{x}^{2}\,dxdt}\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+\\\
\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{|h|^{2m-n}h_{x}^{2}\,dxdt}+\tfrac{\delta}{\varepsilon^{2}}\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}.$
(A.9)
Above, we used the bounds $f_{\delta\varepsilon}(z)\leqslant|z|^{n}+\delta$,
$D^{\prime\prime}_{\varepsilon}(z)\leqslant|z|^{m-n}$, and
$D^{\prime\prime}_{\varepsilon}(z)\leqslant\varepsilon^{-1}$. By the Cauchy
inequality, bound (A.4), (A.3) and bound (A.5),
$\displaystyle\iint\limits_{Q_{T}}|h|^{2m-n}h_{x}^{2}\;dxdt\leqslant\tfrac{1}{2}\iint\limits_{Q_{T}}h^{2(2m-n)}\;dxdt+\tfrac{1}{2}\iint\limits_{Q_{T}}h_{x}^{4}\;dxdt$
$\displaystyle\leqslant\tfrac{b_{4}}{2}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n}\;dt+\tfrac{b_{5}}{2}\;M_{\delta\varepsilon}^{2(2m-n)}\;T+\tfrac{b_{2}}{2}\int\limits_{0}^{T}\|h_{xx}(\cdot,t)\|_{2}\;\|h_{x}(\cdot,t)\|_{2}^{3}\;dt$
$\displaystyle\hskip
28.90755pt\leqslant\tfrac{1}{2}\iint\limits_{Q_{T}}h_{xx}^{2}\;dxdt+\tfrac{b_{2}^{2}}{8}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{3}\;dt+\tfrac{b_{4}}{2}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n}+c_{1}\;T,$
(A.10)
where $c_{1}=M_{\delta\varepsilon}^{2(2m-n)}\;b_{5}/2$,
$m\geqslant\tfrac{n}{2}$. From (A.9), due to (A.10), we arrive at
$\displaystyle\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
$\displaystyle\hskip
14.45377pt\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+c_{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+c_{3}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{3}\;dt$
$\displaystyle\hskip
43.36243pt+c_{4}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n}\;dt+c_{5}\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}+c_{6}\;T$
$\displaystyle\hskip
14.45377pt\leqslant\tfrac{1}{2}\int\limits_{\Omega}{{h_{0,\delta\varepsilon,x}}^{2}\,dx}+c_{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{1}}\right\\}\;dt$
(A.11)
where $\gamma_{1}=\max\\{3,2m-n\\}$, $m\geqslant\tfrac{n}{2}$,
$\displaystyle c_{2}=\tfrac{a_{1}^{2}}{4a_{0}},\quad
c_{3}=\tfrac{a_{1}^{2}b_{2}^{2}}{16a_{0}},\quad
c_{4}=\tfrac{a_{1}^{2}b_{4}}{4a_{0}},\
c_{5}=\tfrac{a_{1}^{2}}{2a_{0}}\tfrac{\delta}{\varepsilon^{2}}$ $\displaystyle
c_{6}=\tfrac{a_{1}^{2}}{2a_{0}}c_{1},\quad c_{7}=c_{3}+c_{4}+c_{5}+c_{6}.$
Now, multiplying (4.1) by $G^{\prime}_{\delta\varepsilon}(h)$, integrating
over $Q_{T}$, and using the periodic boundary conditions (4.2), we obtain
$\int\limits_{\Omega}{G_{\delta\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}=\int\limits_{\Omega}{G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{D^{\prime\prime}_{\varepsilon}(h)h^{2}_{x}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\,dx}+\tfrac{a_{1}}{\varepsilon}\iint\limits_{Q_{T}}{h^{2}_{x}\,dxdt}$
(A.12)
for limiting process on $\delta\to 0$, and
$\int\limits_{\Omega}{G_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{m-n}h^{2}_{x}\,dxdt},\
\ h_{\varepsilon}>0$ (A.13)
for limiting process on $\varepsilon\to 0$. In the case of (A.13), we estimate
the right-hand using the following equality
$\int\limits_{\Omega}{v^{a}v^{2}_{x}\,dx}=\tfrac{1}{a+1}\int\limits_{\Omega}{v^{a+1}v_{xx}\,dx},\
\ v>0,\ a>-1.$ (A.14)
Really, for $h_{\varepsilon}>0$ we deduce by the Cauchy inequality and bound
(A.4) that
$a_{1}\iint\limits_{Q_{T}}{h^{m-n}h^{2}_{x}\,dxdt}=-\tfrac{a_{1}}{m-n+1}\iint\limits_{Q_{T}}{h^{m-n+1}h_{xx}\,dxdt}\\\
\leqslant\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\iint\limits_{Q_{T}}{h^{2(m-n+1)}\,dxdt}\\\
\leqslant\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+c_{8}\int\limits_{0}^{T}\Bigl{(}\int\limits_{\Omega}h_{x}^{2}\,dx\Bigr{)}^{m-n+1}\,dt+c_{9}T,$
(A.15)
where $c_{8}=\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}b_{4}$,
$c_{9}=\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}b_{5}M_{\varepsilon}^{2(m-n+1)}$,
and $m\geqslant n-1$. Thus, from (A.12) due to (A.15), we deduce
$\int\limits_{\Omega}{G_{\varepsilon}(h(x,T))\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+c_{8}\int\limits_{0}^{T}\Bigl{(}\int\limits_{\Omega}h_{x}^{2}\,dx\Bigr{)}^{m-n+1}dt+c_{9}T\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+c_{10}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{2}}\right\\}\;dt,$
(A.16)
where $\gamma_{2}=\max\\{3,m-n+1\\}$, $m\geqslant n-1$, $c_{10}=c_{8}+c_{9}$.
Further, from (A.11) and (A.12) we find for limiting process on $\delta\to 0$
$\displaystyle\int\limits_{\Omega}{h_{x}^{2}\,dx}+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\delta\varepsilon}(h(x,T))\;dx+a_{0}\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
$\displaystyle\hskip
7.22743pt\leqslant\int\limits_{\Omega}{h_{0,\delta\varepsilon,x}}^{2}\;dx+\tfrac{2c_{2}}{a_{0}}\left(\int\limits_{\Omega}G_{\delta\varepsilon}(h(x,T))\;dx+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}h_{xx}^{2}\;dxdt\right)$
$\displaystyle\hskip
7.22743pt+2c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt\leqslant\int\limits_{\Omega}{h_{0,\delta\varepsilon,x}}^{2}\;dx$
$\displaystyle\hskip
7.22743pt+\tfrac{2c_{2}}{a_{0}}\left(\int\limits_{\Omega}G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\;dx+\tfrac{a_{1}}{\varepsilon}\iint\limits_{Q_{T}}{h^{2}_{x}\,dxdt}\right)$
$\displaystyle\hskip
7.22743pt+2c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt\leqslant\int\limits_{\Omega}{h_{0,\delta\varepsilon,x}}^{2}\;dx$
(A.17) $\displaystyle\hskip
7.22743pt+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\delta\varepsilon}(h_{0,\delta\varepsilon})\;dx+c_{11}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt$
where $c_{11}=\tfrac{2c_{2}a_{1}}{\varepsilon a_{0}}+2c_{7}$,
$\gamma_{1}=\max\\{3,2m-n\\}$, $m\geqslant\tfrac{n}{2}$. Similarly, from
(A.11) and (A.16) we find for limiting process on $\varepsilon\to 0$
$\int\limits_{\Omega}{h_{x}^{2}\,dx}+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\varepsilon}(h(x,T))\;dx+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0,\varepsilon,x}}^{2}\;dx\\\
+\tfrac{2c_{2}}{a_{0}}\left(\int\limits_{\Omega}G_{\varepsilon}(h_{0,\varepsilon})\;dx+c_{10}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{2}}\right\\}\;dt\right)\\\
+2c_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt\leqslant\int\limits_{\Omega}{h_{0,\varepsilon,x}}^{2}\;dx\\\
+\tfrac{2c_{2}}{a_{0}}\int\limits_{\Omega}G_{\varepsilon}(h_{0,\varepsilon})\;dx+c_{11}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\right)^{\gamma_{1}}\right\\}\;dt,$
(A.18)
where $c_{11}=\tfrac{2c_{2}c_{10}}{\varepsilon a_{0}}+2c_{7}$,
$\gamma_{1}=\max\\{3,2m-n,m-n+1\\}$, $m\geqslant\max\\{\tfrac{n}{2},n-1\\}$.
Applying the nonlinear Grönwall lemma [15] to
$v(T)\leqslant
v(0)+c_{11}\int\limits_{0}^{T}\max\\{1,v^{\gamma_{1}}(t)\\}\;dt$
with
$v(t)=\int(h_{x}^{2}(x,t)+2c_{2}/a_{0}\>G_{\delta\varepsilon}(h(x,t)))\;dx$
yields
$v(t)\leqslant\begin{cases}\begin{cases}v(0)+c_{11}t&\mbox{if}\quad
t<t_{0}:=\tfrac{1-v(0)}{c_{11}}\\\
\left(1-c_{11}(\gamma_{1}-1)(t-t_{0})\right)^{-1/(\gamma_{1}-1)}&\mbox{if}\quad
t\geqslant t_{0}\end{cases}&\mbox{if}\quad v(0)<1\\\
\left(v(0)^{1-\gamma_{1}}-c_{11}(\gamma_{1}-1)t\right)^{-1/(\gamma_{1}-1)}&\mbox{if}\quad
v(0)\geqslant 1.\end{cases}$
From this,
$\displaystyle\int\limits_{\Omega}\\{h_{x}^{2}(x,t)+\tfrac{2c_{2}}{a_{0}}G_{\delta\varepsilon}(h(x,t))\\}\;dx$
(A.19) $\displaystyle\hskip 28.90755pt\leqslant
2^{\tfrac{1}{\gamma_{1}-1}}\max\left\\{1,\int\limits_{\Omega}({h_{0,\delta\varepsilon,x}}^{2}(x)+\tfrac{2c_{2}}{a_{0}}G_{\delta\varepsilon}(h_{0,\delta\varepsilon}(x)))\;dx\right\\}=K_{\delta\varepsilon}<\infty$
for all $t\in[0,T_{\delta\varepsilon,loc}]$ where
$T_{\delta\varepsilon,loc}:=\tfrac{1}{2c_{11}(\gamma_{1}-1)}\min\left\\{1,\left(\int\limits_{\Omega}({h_{0,\delta\varepsilon,x}}^{2}(x)+\tfrac{2c_{2}}{a_{0}}G_{\delta\varepsilon}(h_{0,\delta\varepsilon}(x)))\,dx\right)^{-(\gamma_{1}-1)}\right\\}.$
(A.20)
Using the $\delta\to 0,\varepsilon\to 0$ convergence of the initial data and
the choice of $\theta\in(0,2/5)$ (see (4.5)) as well as the assumption that
the initial data $h_{0}$ has finite entropy (3.2), the times
$T_{\delta\varepsilon,loc}$ converge to a positive limit and the upper bound
$K$ in (A.19) can be taken finite and independent of $\delta$ and
$\varepsilon$ for $\delta$ and $\varepsilon$ sufficiently small. (We refer the
reader to the end of the proof of Lemma 6.1 in this Appendix for a fuller
explanation of a similar case.) Therefore there exists $\delta_{0}>0$ and
$\varepsilon_{0}>0$ and $K$ such that the bound (A.19) holds for all
$0\leqslant\delta<\delta_{0}$ and $0<\varepsilon<\varepsilon_{0}$ with $K$
replacing $K_{\delta\varepsilon}$ and for all
$0\leqslant t\leqslant T_{loc}:=\tfrac{9}{10}\lim_{\varepsilon\to 0,\delta\to
0}T_{\delta\varepsilon,loc}.$ (A.21)
Using the uniform bound on $\int h_{x}^{2}$ that (A.19) provides, one can find
a uniform-in-$\delta$-and-$\varepsilon$ bound for the right-hand-side of
(A.17) yielding the desired a priori bound (4.7). Similarly, one can find a
uniform-in-$\delta$-and-$\varepsilon$ bound for the right-hand-side of (A.16)
yielding the desired a priori bound (4.8).
To prove the bound (4.9), multiply (4.1) by
$-a_{0}h_{xx}-a_{1}D^{\prime}_{\varepsilon}(h)$, integrate over $Q_{T}$,
integrate by parts, use the periodic boundary conditions (4.2), to find
$\mathcal{E}_{\delta\varepsilon}(T)+\iint\limits_{Q_{T}}{f_{\delta\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})^{2}\,dxdt}=\mathcal{E}_{\delta\varepsilon}(0).$
(A.22)
The parameters $\delta_{0}$ and $\varepsilon_{0}$ are determined by $a_{0}$,
$a_{1}$, $|\Omega|$, $\int h_{0}$, $\|h_{0x}\|_{2}$, $\int h_{0}^{2-n}$, by
how quickly $M_{\delta\varepsilon}$ converges to $M$, and by how quickly the
approximate initial data (4.5), $h_{0,\delta\varepsilon}$, converge to $h_{0}$
in $H^{1}(\Omega)$.
The time $T_{loc}$ and the constants $K_{1}$, and $K_{2}$ are determined by
$\delta_{0}$, $\varepsilon_{0}$, $a_{0}$, $a_{1}$, $|\Omega|$, $\int h_{0}$,
$\|h_{0x}\|_{2}$, and $\int h_{0}^{2-n}$. ∎
###### Sketch of Proof of Lemma 4.2.
In the following, we denote the positive, classical solution $h_{\varepsilon}$
by $h$ whenever there is no chance of confusion.
Taking $\delta\to 0$ in (A.9) yields
$\displaystyle\tfrac{1}{2}\int\limits_{\Omega}{h_{x}^{2}\,dx}+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
$\displaystyle\leqslant\tfrac{1}{2}\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{2a_{0}}\iint\limits_{Q_{T}}{|h|^{2m-n}h^{2}_{x}\,dxdt}$
$\displaystyle\leqslant\tfrac{1}{2}\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{2a_{0}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{2m-n}\;\int\limits_{\Omega}h_{x}^{2}(x,t)\;dx\,dt.$
(A.23)
Applying the nonlinear Grönwall lemma [15] to
$v(T)\leqslant v(0)+\int\limits_{0}^{T}A_{1}(t)\;v(t)\;dt$
with $v(t)=\int h_{x}^{2}(x,t)\;dx$,
$A_{1}(t)=a_{1}^{2}/a_{0}\|h(\cdot,t)\|_{\infty}^{2m-n}$ yields
$v(T)\leqslant v(0)\;e^{B_{1}(T)},\text{ where
}B_{1}(t)=\int\limits_{0}^{t}{A_{1}(s)\;ds}.$
Similarly, taking $\delta\to 0$ in (A.9) and (A.12), due to (A.14) and (A.6),
yield
$\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}{h^{2m-n}h^{2}_{x}\,dxdt}\\\
=\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}-\tfrac{a_{1}^{2}}{a_{0}(2m-n+1)}\iint\limits_{Q_{T}}{h^{2m-n+1}h_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\iint\limits_{Q_{T}}{h^{2(2m-n+1)}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{4m-n}\;\int\limits_{\Omega}G_{\varepsilon}(h(x,t))\;dx\,dt,$
(A.24)
$\int\limits_{\Omega}{G_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{m-n}h^{2}_{x}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}-\tfrac{a_{1}}{m-n+1}\iint\limits_{Q_{T}}{h^{m-n+1}h_{xx}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\iint\limits_{Q_{T}}{h^{2(m-n+1)}\,dxdt}\leqslant\int\limits_{\Omega}{G_{\varepsilon}(h_{0,\varepsilon})\,dx}\\\
+\tfrac{a_{0}}{2}\iint\limits_{Q_{T}}{h^{2}_{xx}\,dxdt}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\int\limits_{0}^{T}\|h(\cdot,t)\|_{\infty}^{2m-n}\;\int\limits_{\Omega}G_{\varepsilon}(h(x,t))\;dx\,dt.$
(A.25)
Summing (A.24) and (A.25), we find that
$\int\limits_{\Omega}{\\{h_{x}^{2}(x,T)+G_{\varepsilon}(h(x,T))\\}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\\\
\leqslant\int\limits_{\Omega}{\\{h_{0\varepsilon,x}^{2}+G_{\varepsilon}(h_{0,\varepsilon})\\}\,dx}+\int\limits_{0}^{T}A_{2}(t)\;\int\limits_{\Omega}\\{h_{x}^{2}(x,t)+G_{\varepsilon}(h(x,t))\\}\;dx\,dt,$
(A.26)
where
$A_{2}(t)=\tfrac{a_{1}^{4}}{2a_{0}^{3}(2m-n+1)^{2}}\|h(\cdot,t)\|_{\infty}^{4m-n}+\tfrac{a_{1}^{2}}{2a_{0}(m-n+1)^{2}}\|h(\cdot,t)\|_{\infty}^{2m-n}$.
Applying the nonlinear Grönwall lemma [15] to
$v(T)\leqslant v(0)+\int\limits_{0}^{T}A_{2}(t)\;v(t)\;dt$
with $v(t)=\int\\{h_{x}^{2}(x,t)+G_{\varepsilon}(h(x,t))\\}\;dx$, yields
$v(T)\leqslant v(0)\;e^{B_{2}(T)},\text{ where
}B_{2}(t)=\int\limits_{0}^{t}{A_{2}(s)\;ds}.$
∎
###### Sketch of Proof of Lemma 4.3.
In the following, we denote the positive, classical solution $h_{\varepsilon}$
by $h$ whenever there is no chance of confusion.
Multiplying (4.1) with $\delta=0$ by $G^{\prime(\alpha)}_{\varepsilon}(h)$,
integrating over $Q_{T}$, taking $\delta\to 0$, and using the periodic
boundary conditions (4.2), yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}+a_{0}\tfrac{\alpha(1-\alpha)}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
(A.27)
$\displaystyle=\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{\alpha}D^{\prime\prime}_{\varepsilon}(h)h^{2}_{x}\,dxdt}\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}.$
Case 1: $0<\alpha<1$. The coefficient multiplying $\iint
h^{\alpha-2}h_{x}^{4}$ in (A.27) is positive and can therefore be used to
control the term $\iint h^{\alpha+m-n}h_{x}^{2}$ on the right–hand side of
(A.27). Specifically, using the Cauchy-Schwartz inequality and the Cauchy
inequality,
$a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}\leqslant
a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}\leqslant\\\
\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}+\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha+2(m-n+1)}\,dxdt}.$
(A.28)
Using the bound (A.28) in (A.27) yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}+a_{0}\tfrac{\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
(A.29)
$\displaystyle\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha+2(m-n+1)}\,dxdt}.$
(A.30)
By (A.4),
$\iint\limits_{Q_{T}}{h^{\alpha+2(m-n+1)}\,dxdt}\leqslant
b_{4}\int\limits_{0}^{T}{\left(\int\limits_{\Omega}{h^{2}_{x}\,dx}\right)^{\tfrac{\alpha}{2}+m-n+1}dt}+b_{5}M_{\varepsilon}^{\alpha+2(m-n+1)}\;T.$
(A.31)
Using (A.31) in (A.29) yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}+a_{0}\tfrac{\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leqslant$
$\displaystyle\hskip
14.45377pt\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+d_{1}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\;dt+d_{2}\;T$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+d_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}\;dt$
(A.32)
where
$d_{1}=b_{4}\;\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)},\quad
d_{2}=b_{5}\;\tfrac{3a_{1}^{2}}{2a_{0}\alpha(1-\alpha)}\;M_{\varepsilon}^{\alpha+2(m-n+1)},\quad
d_{3}=d_{1}+d_{2}.$
Taking $\delta\to 0$ in (A.9) yields
$\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}{h^{2m-n}h^{2}_{x}\,dxdt}.$
(A.33)
Applying the Cauchy inequality,
$\displaystyle\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}h^{2m-n}h_{x}^{2}\;dxdt$
(A.34) $\displaystyle\hskip
14.45377pt\leqslant\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt+\tfrac{3a_{1}^{4}}{2a_{0}^{3}\alpha(1-\alpha)}\iint\limits_{Q_{T}}h^{2(2m-n+1)-\alpha}\;dxdt.$
By (A.4),
$\iint\limits_{Q_{T}}{h^{2(2m-n+1)-\alpha}\,dxdt}\leqslant
b_{4}\int\limits_{0}^{T}{\left(\int\limits_{\Omega}{h^{2}_{x}\,dx}\right)^{2m-n+1-\tfrac{\alpha}{2}}dt}+b_{5}M_{\varepsilon}^{2(2m-n+1)-\alpha}\;T.$
(A.35)
Using (A.34) and (A.35) in (A.33) yields
$\displaystyle\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
$\displaystyle\hskip
14.45377pt+d_{4}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\;dt+d_{5}\;T\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}$
(A.36) $\displaystyle\hskip
14.45377pt+\tfrac{a_{0}\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}+d_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$
where
$d_{4}=\tfrac{3a_{1}^{4}}{2a_{0}^{3}\alpha(1-\alpha)}\;b_{4},\qquad
d_{5}=b_{5}\tfrac{3a_{1}^{4}}{2a_{0}^{3}\alpha(1-\alpha)}\;M_{\varepsilon}^{2(2m-n+1)-\alpha},\qquad
d_{6}=d_{4}+d_{5}.$
Add
$\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}$
to both sides of (A.36) and add
$a_{0}\iint\limits_{Q_{T}}h^{\alpha}h_{xx}^{2}\;dxdt$
to the right–hand side of the resulting inequality. Using (A.32) yields
$\displaystyle\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
(A.37) $\displaystyle\hskip
14.45377pt\leq\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+d_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}$
$\displaystyle\hskip
28.90755pt+d_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+d_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{3}}\right\\}$
where $d_{7}=d_{3}+d_{6}$,
$\gamma_{3}=\max\\{\alpha/2+m-n+1,2m-n+1-\alpha/2\\}$.
Applying the nonlinear Grönwall lemma [15] to
$v(T)\leqslant v(0)+d_{7}\int\limits_{0}^{T}\max\\{1,v^{\gamma_{3}}(t)\\}\;dt$
with $v(T)=\int h_{x}^{2}(x,T)+\>G_{\varepsilon}^{(\alpha)}(h(x,T))\;dx$
yields
$v(t)\leq\begin{cases}\begin{cases}v(0)+d_{7}t&\mbox{if}\quad
t<t_{0}:=\tfrac{1-v(0)}{d_{9}}\\\
\left(1-d_{7}(\gamma_{3}-1)(t-t_{0})\right)^{-\tfrac{1}{\gamma_{3}-1}}&\mbox{if}\quad
t\geqslant t_{0}\end{cases}&\mbox{if}\quad v(0)<1\\\
\left(v(0)^{1-\gamma_{3}}-d_{7}(\gamma_{3}-1)t\right)^{-\tfrac{1}{\gamma_{3}-1}}&\mbox{if}\quad
v(0)\geqslant 1\end{cases}$
From this,
$\displaystyle\int\limits_{\Omega}(h_{x}^{2}(x,T)+G_{\varepsilon}^{(\alpha)}(h(x,T)))\;dx$
(A.38) $\displaystyle\hskip 28.90755pt\leqslant
2^{\tfrac{1}{\gamma_{3}-1}}\max\left\\{1,\int\limits_{\Omega}({h_{0,\varepsilon}}_{x}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right\\}=K_{\varepsilon}<\infty$
for all
$0\leqslant T\leqslant
T_{\varepsilon,loc}^{(\alpha)}:=\tfrac{1}{2d_{7}(\gamma_{3}-1)}\min\left\\{1,\left(\int\limits_{\Omega}({h_{0,\varepsilon}}_{x}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right)^{-(\gamma_{3}-1)}\right\\}.$
The bound (A.38) holds for all $0<\varepsilon<\varepsilon_{0}$ where
$\varepsilon_{0}$ is from Lemma 4.1 and for all
$t\leq\min\\{T_{loc},T_{\varepsilon,loc}^{(\alpha)}\\}$ where $T_{loc}$ is
from Lemma 4.1.
Using the $\delta\to 0,\varepsilon\to 0$ convergence of the initial data and
the choice of $\theta\in(0,2/5)$ (see (4.5)) as well as the assumption that
the initial data $h_{0}$ has finite $\alpha$-entropy (3.11), the times
$T_{\varepsilon,loc}^{(\alpha)}$ converge to a positive limit and the upper
bound $K_{\varepsilon}$ in (A.38) can be taken finite and independent of
$\varepsilon$. (We refer the reader to the end of the proof of Lemma 6.1 in
this Appendix for a fuller explanation of a similar case.) Therefore there
exists $\varepsilon_{0}^{(\alpha)}$ and $K$ such that the bound (A.38) holds
for all $0<\varepsilon<\varepsilon_{0}^{(\alpha)}$ with $K$ replacing
$K_{\varepsilon}$ and for all
$0\leqslant t\leqslant
T_{loc}^{(\alpha)}:=\min\left\\{T_{loc},\tfrac{9}{10}\lim_{\varepsilon\to
0}T_{\varepsilon,loc}^{(\alpha)}\right\\}$ (A.39)
where $T_{loc}$ is the time from Lemma 4.1. Also, without loss of generality,
$\varepsilon_{0}^{(\alpha)}$ can be taken to be less than or equal to the
$\varepsilon_{0}$ from Lemma 4.1.
Using the uniform bound on $\int h_{x}^{2}$ that (A.38) provides, one can find
a uniform-in-$\varepsilon$ bound for the right-hand-side of (A.32) yielding
the desired bound
$\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}}\,dxdt+a_{0}\tfrac{\alpha(1-\alpha)}{6}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leqslant
K_{1}$ (A.40)
which holds for all $0<\varepsilon<\varepsilon_{0}^{(\alpha)}$ and all
$0\leqslant T\leqslant T_{loc}^{(\alpha)}$.
It remains to argue that (A.40) implies that for all
$0<\varepsilon<\varepsilon_{0}^{(\alpha)}$ that $h_{\varepsilon}^{\alpha/2+1}$
and $h_{\varepsilon}^{\alpha/4+1/2}$ are contained in balls in
$L^{2}(0,T;H^{2}(\Omega))$ and $L^{2}(0,T;W^{1}_{4}(\Omega))$ respectively. It
suffices to show that
$\iint\limits_{Q_{T}}\left(h_{\varepsilon}^{\alpha/2+1}\right)^{2}_{xx}\;dxdt\leqslant
K,\qquad\iint\limits_{Q_{T}}\left(h_{\varepsilon}^{\alpha/4+1/2}\right)^{4}_{x}\;dxdt\leqslant
K$
for some $K$ that is independent of $\varepsilon$ and $T$. The integral
$\iint(h_{\varepsilon}^{\alpha/2+1})_{xx}^{2}$ is a linear combination of
$\iint h^{\alpha-2}h_{x}^{4}$, $\iint h^{\alpha-1}h_{x}^{2}h_{xx}$, and $\iint
h^{\alpha}h_{xx}^{2}$. Integration by parts and the periodic boundary
conditions imply
$\tfrac{1-\alpha}{3}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}=\iint\limits_{Q_{T}}{h^{\alpha-1}h^{2}_{x}h_{xx}\,dxdt}$
(A.41)
Hence $\iint(h_{\varepsilon}^{\alpha/2+1})_{xx}^{2}$ is a linear combination
of $\iint h^{\alpha-2}h_{x}^{4}$, and $\iint h^{\alpha}h_{xx}^{2}$. By (A.40),
the two integrals are uniformly bounded independent of $\varepsilon$ and $T$
hence $\iint(h_{\varepsilon}^{\alpha/2+1})_{xx}^{2}$ is as well, yielding the
first part of (4.14).
The uniform bound of $\iint(h_{\varepsilon}^{\alpha/4+1/2})_{x}^{4}$ follows
immediately from the uniform bound of $\iint h^{\alpha-2}h_{x}^{4}$, yielding
the second part of (4.14).
Case 2: $-\tfrac{1}{2}<\alpha<0$. For $\alpha<0$ the coefficient multiplying
$\iint h^{\alpha-2}h_{x}^{4}$ in (A.27) is negative. However, we will show
that if $\alpha>-1/2$ then one can replace this coefficient with a positive
coefficient while also controlling the term $\iint h^{\alpha}h_{x}^{2}$ on the
right-hand side of (A.27).
Applying the Cauchy-Schwartz inequality to the right–hand side of (A.41),
dividing by $\sqrt{\iint h^{\alpha-2}h_{x}^{4}}$, and squaring both sides of
the resulting inequality yields
$\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leq\tfrac{9}{(1-\alpha)^{2}}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}\qquad\forall\alpha<1.$
(A.42)
Using (A.42) in (A.27) yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\tfrac{1+2\alpha}{1-\alpha}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
(A.43) $\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+a_{1}\iint\limits_{Q_{T}}{h^{\alpha+m-n}h^{2}_{x}\,dxdt}.$
Note that if $\alpha>-1/2$ then all the terms on the left–hand side of (A.43)
are positive. We now control the term $\iint h^{\alpha}h_{x}^{2}$ on the
right-hand side of (A.43).
By integration by parts and the periodic boundary conditions
$\iint\limits_{Q_{T}}h^{\alpha+m-n}h_{x}^{2}\;dxdt=-\tfrac{1}{\alpha+m-n+1}\iint\limits_{Q_{T}}h^{\alpha+m-n+1}h_{xx}\;dxdt$
(A.44)
Applying the Cauchy-Schwartz inequality and the Cauchy inequality to (A.44)
yields
$a_{1}\iint\limits_{Q_{T}}h^{\alpha+m-n}h_{x}^{2}\;dxdt\leq\tfrac{a_{0}(1+2\alpha)}{2(1-\alpha)}\iint\limits_{Q_{T}}h^{\alpha}h_{xx}^{2}\
\;dxdt+\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\iint\limits_{Q_{T}}h^{\alpha+2(m-n+1)}\;dxdt$
(A.45)
Using inequality (A.45) in (A.43) yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\tfrac{1+2\alpha}{2(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
(A.46)
$\displaystyle\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\iint\limits_{Q_{T}}h^{\alpha+2(m-n+1)}\;dxdt.$
Adding
$\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt$
to both sides of (A.46) and using the inequality (A.42) yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\tfrac{(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
(A.47) $\displaystyle\hskip
14.45377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt\leq\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}$
$\displaystyle\hskip
14.45377pt+\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\iint\limits_{Q_{T}}h^{\alpha+2(m-n+1)}\;dxdt.$
Using (A.31) in (A.47) yields
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+\tfrac{a_{0}(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
$\displaystyle\hskip
14.45377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leq\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}$
$\displaystyle\hskip
14.45377pt+e_{1}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\;dt+e_{2}\;T$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx}+e_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}\;dt$
(A.48)
where
$e_{1}=\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\;b_{4},\
\
e_{2}=b_{5}\tfrac{a_{1}^{2}(1-\alpha)}{2a_{0}(1+2\alpha)(\alpha+m-n+1)^{2}}\;M_{\varepsilon}^{\alpha+2(m-n+1)},$
and $e_{3}=e_{1}+e_{2}$.
Recall the bound (A.33):
$\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}{h^{2m-n}h^{2}_{x}\,dxdt}.$
(A.49)
As before, by the Cauchy-Schwartz inequality and the Cauchy inequality,
$\displaystyle\tfrac{a_{1}^{2}}{a_{0}}\iint\limits_{Q_{T}}h^{2m-n}h_{x}^{2}\;dxdt\leqslant\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}h^{\alpha-2}h_{x}^{4}\;dxdt$
(A.50) $\displaystyle\hskip
101.17755pt+\tfrac{9a_{1}^{4}}{a_{0}^{3}(1+2\alpha)(1-\alpha)}\iint\limits_{Q_{T}}h^{2(2m-n+1)-\alpha}\;dxdt.$
Using (A.50), and (A.35) in (A.49) yields
$\displaystyle\int\limits_{\Omega}{h_{x}^{2}\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}$
$\displaystyle\hskip
14.45377pt+e_{4}\int\limits_{0}^{T}\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\;dt+e_{5}\;T\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}$
(A.51) $\displaystyle\hskip
14.45377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}+e_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$
where
$e_{4}=\tfrac{9a_{1}^{4}}{a_{0}^{3}(1+2\alpha)(1-\alpha)}\;b_{4},\
e_{5}=b_{5}\tfrac{9a_{1}^{4}}{a_{0}^{3}(1+2\alpha)(1-\alpha)}\;M_{\varepsilon}^{2(2m-n+1)-\alpha},\
e_{6}=e_{4}+e_{5}.$
Add
$\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}$
to both sides of (A.51) and add
$\tfrac{a_{0}(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}h^{\alpha}h_{xx}^{2}\;dxdt$
to the right–hand side of the resulting inequality. Just as (A.32) and (A.33)
yielded (A.37), (A.48) combined with the above inequality yields
$\displaystyle\int\limits_{\Omega}{h_{x}^{2}(x,T)\,dx}+\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+a_{0}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)h^{2}_{xxx}\,dxdt}$
(A.52) $\displaystyle\hskip
14.45377pt\leq\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+e_{3}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\tfrac{\alpha}{2}+m-n+1}\right\\}$
$\displaystyle\hskip
28.90755pt+e_{6}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{2m-n+1-\tfrac{\alpha}{2}}\right\\}\;dt$
$\displaystyle\hskip
14.45377pt\leqslant\int\limits_{\Omega}{h_{0\varepsilon,x}^{2}\,dx}+\int\limits_{\Omega}G^{(\alpha)}_{\varepsilon}(h_{0\varepsilon})\,dx+e_{7}\int\limits_{0}^{T}\max\left\\{1,\left(\int\limits_{\Omega}h_{x}^{2}\;dx\right)^{\gamma_{3}}\right\\}$
where $e_{7}=e_{3}+e_{6}$,
$\gamma_{3}=\max\\{\alpha/2+m-n+1,2m-n+1-\alpha/2\\}$.
The rest of the proof now continues as in the $0<\alpha<1$ case. Specifically,
one finds a bound
$\displaystyle\int\limits_{\Omega}(h_{x}^{2}(x,T)+G_{\varepsilon}^{(\alpha)}(h(x,T)))\;dx$
(A.53) $\displaystyle\hskip 28.90755pt\leqslant
2^{\tfrac{1}{\gamma_{3}-1}}\max\left\\{1,\int\limits_{\Omega}({h_{0,\varepsilon,x}}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right\\}=K_{\varepsilon}<\infty$
for all
$0\leqslant T\leqslant
T_{\varepsilon,loc}^{(\alpha)}:=\tfrac{1}{2e_{7}(\gamma_{3}-1)}\min\left\\{1,\left(\int\limits_{\Omega}({h_{0,\varepsilon,x}}^{2}(x)+G_{\varepsilon}^{(\alpha)}(h_{0,\varepsilon}(x)))\;dx\right)^{-(\gamma_{3}-1)}\right\\}.$
The time $T_{loc}^{(\alpha)}$ is defined as in (A.39) and the uniform bound
(A.53) used to bound the right hand side of (A.48) yields the desired bound
$\displaystyle\int\limits_{\Omega}{G^{(\alpha)}_{\varepsilon}(h(x,T))\,dx}+\tfrac{a_{0}(1+2\alpha)}{4(1-\alpha)}\iint\limits_{Q_{T}}{h^{\alpha}h^{2}_{xx}\,dxdt}$
$\displaystyle\hskip
86.72377pt+\tfrac{a_{0}(1+2\alpha)(1-\alpha)}{36}\iint\limits_{Q_{T}}{h^{\alpha-2}h^{4}_{x}\,dxdt}\leqslant
K_{2}.$ (A.54)
∎
###### Sketch of Proof of Lemma 6.1.
In the following, we denote the positive, classical solution $h_{\varepsilon}$
constructed in Lemma 4.3 by $h$ (whenever there is no chance of confusion).
Recall the entropy function $G_{0}^{(\alpha)}(z)$ defined by (4.12). Using the
local entropy inequality (7.1) with $\zeta=\zeta(x)$ from Lemma 7.1, we obtain
$\int\limits_{\Omega}{\zeta^{4}(x)G^{(\alpha)}_{0}(h(x,T))\,dx}+C_{1}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})^{2}_{xx}\zeta^{4}\,dxdt}\leqslant\int\limits_{\Omega}{\zeta^{4}(x)G^{(\alpha)}_{0}(h_{0})\,dx}+\\\
C_{2}\iint\limits_{Q_{T}}{h^{\alpha+2}(\zeta_{x}^{4}+\zeta^{2}\zeta_{xx}^{2})\,dxdt}+C_{3}\iint\limits_{Q_{T}}{h^{2(m-n+1)+\alpha}\zeta^{4}\,dxdt}.$
(A.55)
Due to $h\in L^{\infty}(0,T_{loc}^{(\alpha)};H^{1}(\Omega))$, we deduce from
(A.55) that
$\int\limits_{\Omega}{\zeta^{4}(x)G^{(\alpha)}_{0}(h(x,T))\,dx}\leqslant\int\limits_{\Omega}{\zeta^{4}(x)G_{0}(h_{0}(x))\,dx}+C_{4}T\leqslant
K<\infty$ (A.56)
for any $T\in[0,T_{loc}^{(\alpha)}]$. ∎
## B Proof of second moment estimate
###### Sketch of Proof of Lemma 8.1.
Let
$\widetilde{G}_{\varepsilon}(z)=\tfrac{z^{2-n}}{2-n}-\tfrac{\varepsilon\,z^{2-s}}{s-2},\
\widetilde{G}^{\prime}_{\varepsilon}(z)=\tfrac{z}{f_{\varepsilon}(z)},$ (B.1)
$\widetilde{G}^{\prime\prime}_{\varepsilon}(z)=\tfrac{1}{f_{\varepsilon}(z)}-\tfrac{zf^{\prime}_{\varepsilon}(z)}{f^{2}_{\varepsilon}(z)}=(1-n)\tfrac{1}{f_{\varepsilon}(z)}-\varepsilon(s-n)z^{-s}.$
Here we use the equality
$f^{\prime}_{\varepsilon}(z)=nz^{-1}f_{\varepsilon}(z)+\varepsilon(s-n)z^{-(s+1)}f_{\varepsilon}^{2}(z).$
Multiplying (4.1) for $\delta=0$ by
$x^{2}\widetilde{G}^{\prime}_{\varepsilon}(h)$, and integrating on $Q_{T}$,
yields
$\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h)\,dx}-\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h_{0\varepsilon})\,dx}=\\\
\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\bigl{(}2x\tfrac{h}{f_{\varepsilon}(h)}+x^{2}\widetilde{G}^{\prime\prime}_{\varepsilon}(h)h_{x}\bigr{)}\,dxdt}=\\\
2\iint\limits_{Q_{T}}{xh(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}+\\\
(1-n)\iint\limits_{Q_{T}}{x^{2}h_{x}(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}-\\\
\varepsilon(s-n)\iint\limits_{Q_{T}}{x^{2}h^{-s}f_{\varepsilon}(h)h_{x}(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})\,dxdt}=:I_{1}+I_{2}+I_{3}.$
(B.2)
We now bound the terms $I_{1}$, $I_{2}$ and $I_{3}$. First,
$I_{1}=2a_{0}\int\limits_{0}^{T}{xhh_{xx}\biggl{|}_{\partial\Omega}\,dx}-2a_{0}\iint\limits_{Q_{T}}{\\{hh_{xx}+x\,h_{x}h_{xx}\\}\,dxdt}+\\\
2a_{1}\iint\limits_{Q_{T}}{xhD^{\prime\prime}_{\varepsilon}(h)h_{x}\,dxdt}=\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}h_{x}^{2}+2a_{1}L_{\varepsilon}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+\\\
3a_{0}\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-2a_{1}\iint\limits_{Q_{T}}{L_{\varepsilon}(h)\,dxdt},\
L_{\varepsilon}(z):=\int\limits_{0}^{z}{\tau\,D^{\prime\prime}_{\varepsilon}(\tau)\,d\tau},$
(B.3)
and
$I_{2}=a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h_{x}h_{xxx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=\\\
-2a_{0}(1-n)\iint\limits_{Q_{T}}{xh_{x}h_{xx}\,dxdt}-a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+\\\
a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}=-a_{0}(1-n)\int\limits_{0}^{T}{xh_{x}^{2}\biggl{|}_{\partial\Omega}\,dt}+a_{0}(1-n)\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-\\\
a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}.$
(B.4)
Using the Hölder inequality and (4.9) with $\delta=0$, we find
$I_{3}\leqslant\varepsilon(s-n)\Bigl{(}\iint\limits_{Q_{T}}{f_{\varepsilon}(h)(a_{0}h_{xxx}+a_{1}D^{\prime\prime}_{\varepsilon}(h)h_{x})^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}\times\\\
\Bigl{(}\iint\limits_{Q_{T}}{x^{4}h^{-2s}f_{\varepsilon}(h)h_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}\leqslant\varepsilon\,C\Bigl{(}\iint\limits_{Q_{T}}{\tfrac{h_{x}^{2}}{(h^{s-n}+\varepsilon)^{2}}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}.$
Using the Young’s inequality $ab\leqslant\tfrac{a^{p}}{p}+\tfrac{b^{q}}{q},\
\tfrac{1}{p}+\tfrac{1}{q}=1$ ($\Rightarrow p\,ab\leqslant
a^{p}+\tfrac{p}{q}b^{q}=a^{p}+(p-1)b^{q}$) with $a=z^{\tfrac{s-n}{p}}$ and
$b=\bigl{(}\tfrac{\varepsilon}{p-1}\bigr{)}^{\tfrac{1}{q}}$, we deduce
$I_{3}\leqslant\varepsilon^{\tfrac{q-1}{q}}\,C\Bigl{(}\iint\limits_{Q_{T}}{h^{-\tfrac{2(s-n)}{p}}h_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}=\varepsilon^{\tfrac{q-1}{q}}\,\widetilde{C}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{p-s+n}{p}})_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}},$
choosing $p=-\tfrac{s-n}{\alpha}>1$ and $q=\tfrac{2(s-n)}{2(s-n)+\alpha}>1$
($\Rightarrow 0>\alpha>-2(s-n)$), due to (4.14), we find
$I_{3}\leqslant\varepsilon^{\tfrac{q-1}{q}}\,\widetilde{C}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{p-s+n}{p}})_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}=\\\
\varepsilon^{-\tfrac{\alpha}{s-n}}\,\widetilde{C}\Bigl{(}\iint\limits_{Q_{T}}{(h^{\tfrac{\alpha+2}{2}})_{x}^{2}\,dxdt}\Bigr{)}^{\tfrac{1}{2}}\leqslant
C_{1}\varepsilon^{-\tfrac{\alpha}{s-n}},$ (B.5)
where the constant $C_{1}>0$ is independent of $\varepsilon$. Hence,
$\lim_{\varepsilon\to 0}{I_{3}}\leqslant 0.$ (B.6)
From (B.2)–(B.4), we deduce that
$\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h)\,dx}=\int\limits_{\Omega}{x^{2}\widetilde{G}_{\varepsilon}(h_{0\varepsilon})\,dx}+\\\
\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}L_{\varepsilon}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+a_{0}(4-n)\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-\\\
a_{0}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}D^{\prime\prime}_{\varepsilon}(h)h_{x}^{2}\,dxdt}-\\\
2a_{1}\iint\limits_{Q_{T}}{L_{\varepsilon}(h)\,dxdt}+I_{3}.$ (B.7)
Letting $\varepsilon\to 0$ in (B.7), due to (B.6), we find
$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+a_{0}(4-n)\iint\limits_{Q_{T}}{h_{x}^{2}\,dxdt}-\\\
2a_{1}(m-n+1)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}+\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}L_{0}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+\\\
a_{0}(n-1)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{m-n}h_{x}^{2}\,dxdt}=\\\
\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+2(4-n)\int\limits_{0}^{T}{\mathcal{E}_{0}(t)\,dt}-2\,a_{1}(m-3)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}+\\\
\int\limits_{0}^{T}{x\bigl{(}2a_{0}hh_{xx}-a_{0}(2-n)h_{x}^{2}+2a_{1}L_{0}(h)\bigr{)}\biggl{|}_{\partial\Omega}\,dt}+\\\
a_{0}(n-1)\iint\limits_{Q_{T}}{x^{2}h^{2}_{xx}\,dxdt}+a_{1}(1-n)\iint\limits_{Q_{T}}{x^{2}h^{m-n}h_{x}^{2}\,dxdt}.$
(B.8)
Due to
$\iint\limits_{Q_{T}}{x^{2}h^{m-n}h_{x}^{2}\,dxdt}=-2\int\limits_{0}^{T}{xD_{0}(h)\biggl{|}_{\partial\Omega}\,dt}+2\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}-\\\
\iint\limits_{Q_{T}}{x^{2}D^{\prime}_{0}(h)h_{xx}\,dxdt},$
from (B.8), in view of (3.6), we deduce that
$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}+2a_{1}(m+n-4)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\\\
k_{1}\mathcal{E}_{0}(0)T-(1-n)\iint\limits_{Q_{T}}{x^{2}h_{xx}(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))\,dxdt}+W(T)=\\\
\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T-\tfrac{1-n}{a_{0}}\iint\limits_{Q_{T}}{x^{2}(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))^{2}\,dxdt}+\\\
\tfrac{a_{1}(1-n)}{a_{0}}\iint\limits_{Q_{T}}{x^{2}D^{\prime}_{0}(h)(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))\,dxdt}+W(T),$
(B.9)
where $k_{1}=2(4-n)$, and $W(T)$ is from (8.2). From (B.9) we find that
$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}+2a_{1}(m+n-4)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}+\\\
\tfrac{1-n}{2a_{0}}\iint\limits_{Q_{T}}{x^{2}(a_{0}h_{xx}+a_{1}D^{\prime}_{0}(h))^{2}\,dxdt}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T+\\\
\tfrac{a_{1}^{2}(1-n)}{2a_{0}}\iint\limits_{Q_{T}}{x^{2}(D^{\prime}_{0}(h))^{2}\,dxdt}+W(T)\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T+\\\
\int\limits_{0}^{T}{\Bigl{(}\widetilde{A}(t)\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}\Bigr{)}dt}+W(T),$
(B.10)
where $m\geqslant 4-n$, and
$\widetilde{A}(t):=\tfrac{a_{1}^{2}|1-n|(2-n)}{2a_{0}(m-n+1)^{2}}\|h(.,t)\|_{L^{\infty}(\Omega)}^{2m-n}.$
Applying the nonlinear Grönwall lemma [15] to
$v(T)\leqslant
v(0)+k_{1}\mathcal{E}_{0}(0)T+\int\limits_{0}^{T}{\widetilde{A}(s)v(s)\,dt}+W(T)$
with $v(T)=\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h(x,T))\,dx}$ yields
$v(T)\leqslant
e^{\widetilde{B}(T)}\Bigl{(}v(0)+\int\limits_{0}^{T}{(k_{1}\mathcal{E}_{0}(0)+W^{\prime}(t))e^{-\widetilde{B}(t)}dt}\Bigr{)}$
with $\widetilde{B}(T):=\int\limits_{0}^{T}{\widetilde{A}(\tau)\,d\tau}$. From
this we obtain (8.1) for $0<n\leqslant 1$.
In the case of $1<n<2$, from (B.9) we find that
$\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}+2a_{1}(m+n-4)\iint\limits_{Q_{T}}{D_{0}(h)\,dxdt}\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+k_{1}\mathcal{E}_{0}(0)T+\\\
\tfrac{3a_{0}(n-1)}{2}\iint\limits_{Q_{T}}{x^{2}h_{xx}^{2}\,dxdt}+\tfrac{a_{1}^{2}(n-1)}{2a_{0}}\iint\limits_{Q_{T}}{x^{2}(D^{\prime}_{0}(h))^{2}\,dxdt}+W(T)\leqslant\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h_{0})\,dx}+\\\
k_{1}\mathcal{E}_{0}(0)T+k_{2}\iint\limits_{Q_{T}}{x^{2}h_{xx}^{2}\,dxdt}+\int\limits_{0}^{T}{\Bigl{(}\widetilde{A}(t)\int\limits_{\Omega}{x^{2}\widetilde{G}_{0}(h)\,dx}\Bigr{)}dt}+W(T),$
(B.11)
where $m\geqslant 4-n$, $k_{2}=\tfrac{3a_{0}(n-1)}{2}$, and $W(t)$ is from
(8.2). Applying the Grönwall lemma [15] to (B.11), we obtain the estimate
(8.1) for $1<n<2$.
∎
## Appendix C
###### Lemma D.1.
([32]) Suppose that $X,\ Y,$ and $Z$ are Banach spaces,
$X\\!\Subset\\!Y\subset\\!Z$, and $X$ and $Z$ are reflexive. Then the
imbedding $\\{u\in\\!L^{p_{0}}(0,T;$ $X):$ $\partial_{t}u\in
L^{p_{1}}(0,T;Z),1<p_{i}<\infty,i=0,1\\}\Subset L^{p_{0}}(0,T;Y)$ is compact.
###### Lemma D.2.
([36]) Suppose that $X,\ Y,$ and $Z$ are Banach spaces and
$X\\!\Subset\\!Y\subset\\!Z$. Then the imbedding $\\{u\in
L^{\infty}(0,T;X):\partial_{t}u\in L^{p}(0,T;Z)$, $p>1\\}\Subset C(0,T;Y)$ is
compact.
###### Lemma D.3.
([26, 6]) Let $\Omega\subset\mathbb{R}^{N},\ N<6$, be a bounded convex domain
with smooth boundary, and let
$n\in\bigl{(}2-\sqrt{1-\tfrac{N}{N+8}},3\bigr{)}$ for $N>1$, and
$\tfrac{1}{2}<n<3$ for $N=1$. Then the following estimates hold for any
strictly positive functions $v\in H^{2}(\Omega)$ such that $\nabla
v\cdot\vec{n}=0$ on $\partial\Omega$ and
$\int\limits_{\Omega}{v^{n}|\nabla\Delta v|^{2}}<\infty$:
$\int\limits_{\Omega}{\varphi^{6}\\{v^{n-4}|\nabla
v|^{6}+v^{n-2}|D^{2}v|^{2}|\nabla v|^{2}\\}}\leqslant
c\Bigl{\\{}\int\limits_{\Omega}{\varphi^{6}v^{n}|\nabla\Delta
v|^{2}}+\int\limits_{\\{\varphi>0\\}}{v^{n+2}|\nabla\varphi|^{6}}\Bigr{\\}},$
$\int\limits_{\Omega}{\varphi^{6}|\nabla\Delta
v^{\tfrac{n+2}{2}}|^{2}}\leqslant
c\Bigl{\\{}\int\limits_{\Omega}{\varphi^{6}v^{n}|\nabla\Delta v|^{2}}+\\\
+\int\limits_{\\{\varphi>0\\}}{v^{n+2}\\{|\nabla\varphi|^{6}+\varphi^{2}|D^{2}\varphi|^{2}|\nabla\varphi|^{2}+\varphi^{3}|\Delta\varphi|^{3}\\}}\Bigr{\\}},$
where $\varphi\in C^{2}(\Omega)$ is an arbitrary nonnegative function such
that the tangential component of $\nabla\varphi$ is equal to zero on
$\partial\Omega$, and the constant $c>0$ is independent of $v$.
###### Lemma D.4.
([33]) If $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with piecewise-
smooth boundary, $a>1$, $b\in(0,a),\ d>1,$ and $0\leqslant i<j,\
i,j\in\mathbb{N}$, then there exist positive constants $d_{1}$ and $d_{2}$
$(d_{2}=0\text{ if }\Omega$ is unbounded$)$ depending only on $\Omega,\ d,\
j,\ b,$ and $N$ such that the following inequality is valid for every $v(x)\in
W^{j,d}(\Omega)\cap L^{b}(\Omega)$:
$\left\|{D^{i}v}\right\|_{L^{a}(\Omega)}\leqslant
d_{1}\left\|{D^{j}v}\right\|_{L^{d}(\Omega)}^{\theta}\left\|v\right\|_{L^{b}(\Omega)}^{1-\theta}+d_{2}\left\|v\right\|_{L^{b}(\Omega)},\
\theta=\tfrac{{\tfrac{1}{b}+\tfrac{i}{N}-\tfrac{1}{a}}}{{\tfrac{1}{b}+\tfrac{j}{N}-\tfrac{1}{d}}}\in\left[{\tfrac{i}{j},1}\right)\\!\\!.$
###### Lemma D.5.
([24]) Let $(\beta_{1},\ldots,\beta_{m})\in\mathbb{R}^{m},m\geqslant 1$, and
let $\beta=\prod\limits_{j=1}^{m}\beta_{j},\
\overline{\beta_{i}}=\tfrac{\beta}{\beta_{i}}=\prod\limits_{j=1,j\neq
i}^{m}\beta_{j}$. Assume that $G_{i}(s)$ are nonnegative nonincreasing
functions satisfying the conditions
$G_{i}(s+\delta)\leqslant
c_{i}\Bigl{(}{\sum\limits_{i=1}^{m}\tfrac{G_{i}(s)}{\delta^{\alpha_{i}}}}\Bigr{)}^{\beta_{i}}\
\forall\,s>0,\ \delta>0,\ i=\overline{1,m}$
with real constants $c_{i}>0,\ \beta_{i}>1$, and $\alpha_{i}\geqslant 0$ for
$i=\overline{1,m}$, and $\alpha_{i}>0$ for $i=\overline{1,\ell}$. Let
$G(s)\\!=\\!\sum\limits_{i=1}^{m}\\!{\overline{(c_{i}^{\overline{\beta}_{i}})}\left({G_{i}(s)}\right)^{\overline{\beta}_{i}}}$,
and let the function
$H(s)=m^{\beta}\sum\limits_{i=\ell+1}^{m}\\!{c_{i}^{\overline{\beta}_{i}}\overline{(c_{i}^{\overline{\beta}_{i}})}^{1-\beta_{i}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\left({G_{i}(s)}\right)^{\beta_{i}-1}}$
be such that $H(s_{1})<1$ at a some $s_{1}\geqslant 0$. Then there exists a
positive constant $c>1$ depending on $m,\ \alpha_{i},\ \beta_{i},\ \ell$, and
$H(s_{1})$ such that $G_{i}(s_{0})\equiv 0$ for all $i=\overline{1,\ell}$,
where
$s_{0}=s_{1}+c\sum\limits_{i=1}^{\ell}{\bigl{(}{c_{i}^{\overline{\beta}_{i}}{\overline{(c_{i}^{\overline{\beta}_{i}})}}^{1-\beta_{i}}\left({G(s_{1})}\right)^{\beta_{i}-1}}\bigr{)}^{\tfrac{1}{\alpha_{i}\beta}}}$.
## References
* [1] Lidia Ansini and Lorenzo Giacomelli. Doubly nonlinear thin-film equations in one space dimension. Arch. Ration. Mech. Anal., 173(1):89–131, 2004.
* [2] Catherine Bandle and Hermann Brunner. Blowup in diffusion equations: a survey. J. Comput. Appl. Math., 97(1-2):3–22, 1998.
* [3] Elena Beretta. Selfsimilar source solutions of a fourth order degenerate parabolic equation. Nonlinear Anal., 29(7):741–760, 1997.
* [4] Elena Beretta, Michiel Bertsch, and Roberta Dal Passo. Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal., 129(2):175–200, 1995.
* [5] Francisco Bernis. Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differential Equations, 1(3):337–368, 1996.
* [6] Francisco Bernis. Finite speed of propagation for thin viscous flows when $2\leq n<3$. C. R. Acad. Sci. Paris Sér. I Math., 322(12):1169–1174, 1996\.
* [7] Francisco Bernis and Avner Friedman. Higher order nonlinear degenerate parabolic equations. J. Differential Equations, 83(1):179–206, 1990.
* [8] A. J. Bernoff. personal communication, 1998.
* [9] Andrew J. Bernoff and Andrea L. Bertozzi. Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition. Phys. D, 85(3):375–404, 1995.
* [10] A. L. Bertozzi and M. Pugh. The lubrication approximation for thin viscous films: the moving contact line with a ‘‘porous media’’ cut-off of van der Waals interactions. Nonlinearity, 7(6):1535–1564, 1994.
* [11] A. L. Bertozzi and M. Pugh. The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions. Comm. Pure Appl. Math., 49(2):85–123, 1996.
* [12] A. L. Bertozzi and M. C. Pugh. Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math., 51(6):625–661, 1998.
* [13] A. L. Bertozzi and M. C. Pugh. Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J., 49(4):1323–1366, 2000.
* [14] Andrea L. Bertozzi, Michael P. Brenner, Todd F. Dupont, and Leo P. Kadanoff. Singularities and similarities in interface flows. In Trends and perspectives in applied mathematics, volume 100 of Appl. Math. Sci., pages 155–208. Springer, New York, 1994.
* [15] I. Bihari. A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hungar., 7:81–94, 1956.
* [16] P Constantin, TF Dupont, RE Goldstein, LP Kadanoff, MJ Shelley, and SM Zhou. Droplet Breakup in a Model of the Hele-Shaw Cell. Physical Review E, 47(6):4169–4181, Jun 1993.
* [17] Roberta Dal Passo, Harald Garcke, and Günther Grün. On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal., 29(2):321–342 (electronic), 1998.
* [18] Roberta Dal Passo, Lorenzo Giacomelli, and Günther Grün. A waiting time phenomenon for thin film equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30(2):437–463, 2001.
* [19] Roberta Dal Passo, Lorenzo Giacomelli, and Günther Grün. Waiting time phenomena for degenerate parabolic equations—a unifying approach. In Geometric analysis and nonlinear partial differential equations, pages 637–648. Springer, Berlin, 2003.
* [20] Roberta Dal Passo, Lorenzo Giacomelli, and Andrey Shishkov. The thin film equation with nonlinear diffusion. Comm. Partial Differential Equations, 26(9-10):1509–1557, 2001\.
* [21] P Ehrhard. The Spreading of Hanging Drops. Journal of Colloid and Interface Science, 168(1):242–246, NOV 1994.
* [22] S. D. Èĭdel′man. Parabolic systems. Translated from the Russian by Scripta Technica, London. North-Holland Publishing Co., Amsterdam, 1969.
* [23] Lorenzo Giacomelli and Günther Grün. Lower bounds on waiting times for degenerate parabolic equations and systems. Interfaces Free Bound., 8(1):111–129, 2006.
* [24] Lorenzo Giacomelli and Andrey Shishkov. Propagation of support in one-dimensional convected thin-film flow. Indiana Univ. Math. J., 54(4):1181–1215, 2005.
* [25] Günther Grün. Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case. Interfaces Free Bound., 4(3):309–323, 2002.
* [26] Günther Grün. Droplet spreading under weak slippage: a basic result on finite speed of propagation. SIAM J. Math. Anal., 34(4):992–1006 (electronic), 2003.
* [27] Günther Grün. Droplet spreading under weak slippage: the waiting time phenomenon. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(2):255–269, 2004\.
* [28] T Hocherman and P Rosenau. On KS-Type Equations Describing the Evolution and Rupture of a Liquid Interface. Physica D, 67(1-3):113–125, AUG 15 1993.
* [29] Josephus Hulshof and Andrey E. Shishkov. The thin film equation with $2\leq n<3$: finite speed of propagation in terms of the $L^{1}$-norm. Adv. Differential Equations, 3(5):625–642, 1998.
* [30] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1967.
* [31] Howard A. Levine. The role of critical exponents in blowup theorems. SIAM Rev., 32(2):262–288, 1990.
* [32] J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969.
* [33] L. Nirenberg. An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3), 20:733–737, 1966.
* [34] A. E. Shishkov and R. M. Taranets. On the equation of the flow of thin films with nonlinear convection in multidimensional domains. Ukr. Mat. Visn., 1(3):402–444, 447, 2004.
* [35] Andrey E. Shishkov. Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Discrete Contin. Dyn. Syst., (Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl.):938–945, 2007.
* [36] Jacques Simon. Compact sets in the space $L^{p}(0,T;B)$. Ann. Mat. Pura Appl. (4), 146:65–96, 1987.
* [37] D. Slepčev and M. C. Pugh. Selfsimilar blowup of unstable thin-film equations. Indiana Univ. Math. J., 54(6):1697–1738, 2005.
* [38] Dejan Slepčev. Linear stability of selfsimilar solutions of unstable thin-film equations. Interfaces Free Bound., 11(3):375–398, 2009.
* [39] R. M. Taranets. Solvability and global behavior of solutions of the equation of thin films with nonlinear dissipation and absorption. In Proceedings of the Institute of Applied Mathematics and Mechanics. Vol. 7 (Russian), volume 7 of Tr. Inst. Prikl. Mat. Mekh., pages 192–209. Nats. Akad. Nauk Ukrainy Inst. Prikl. Mat. Mekh., Donetsk, 2002.
* [40] R. M. Taranets. Propagation of perturbations in the equations of thin capillary films with nonlinear absorption. In Proceedings of the Institute of Applied Mathematics and Mechanics. Vol. 8 (Russian), volume 8 of Tr. Inst. Prikl. Mat. Mekh., pages 180–194. Nats. Akad. Nauk Ukrainy Inst. Prikl. Mat. Mekh., Donetsk, 2003.
* [41] R. M. Taranets. Propagation of perturbations in equations of thin capillary films with nonlinear diffusion and convection. Sibirsk. Mat. Zh., 47(4):914–931, 2006.
* [42] R. M. Taranets and A. E. Shishkov. A singular Cauchy problem for the equation of the flow of thin viscous films with nonlinear convection. Ukraïn. Mat. Zh., 58(2):250–271, 2006.
* [43] T. P. Witelski, A. J. Bernoff, and A. L. Bertozzi. Blowup and dissipation in a critical-case unstable thin film equation. European J. Appl. Math., 15(2):223–256, 2004.
* [44] Thomas P. Witelski and Andrew J. Bernoff. Stability of self-similar solutions for van der Waals driven thin film rupture. Phys. Fluids, 11(9):2443–2445, 1999.
|
arxiv-papers
| 2010-08-02T19:32:53 |
2024-09-04T02:49:12.013324
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marina Chugunova, M.C. Pugh, Roman M. Taranets",
"submitter": "Roman Taranets",
"url": "https://arxiv.org/abs/1008.0385"
}
|
1008.0458
|
# Topological Nature of the Phonon Hall Effect
Lifa Zhang zhanglifa@nus.edu.sg Department of Physics and Centre for
Computational Science and Engineering, National University of Singapore,
Singapore 117542, Republic of Singapore Jie Ren renjie@nus.edu.sg NUS
Graduate School for Integrative Sciences and Engineering, Singapore 117456,
Republic of Singapore Department of Physics and Centre for Computational
Science and Engineering, National University of Singapore, Singapore 117542,
Republic of Singapore Jian-Sheng Wang phywjs@nus.edu.sg Department of
Physics and Centre for Computational Science and Engineering, National
University of Singapore, Singapore 117542, Republic of Singapore Baowen Li
phylibw@nus.edu.sg NUS Graduate School for Integrative Sciences and
Engineering, Singapore 117456, Republic of Singapore Department of Physics
and Centre for Computational Science and Engineering, National University of
Singapore, Singapore 117542, Republic of Singapore
(02 Aug 2010, Revised 25 Nov 2010)
###### Abstract
We provide a topological understanding of the phonon Hall effect in
dielectrics with Raman spin-phonon coupling. A general expression for phonon
Hall conductivity is obtained in terms of the Berry curvature of band
structures. We find a nonmonotonic behavior of phonon Hall conductivity as a
function of the magnetic field. Moreover, we observe a phase transition in the
phonon Hall effect, which corresponds to the sudden change of band topology,
characterized by the altering of integer Chern numbers. This can be explained
by touching and splitting of phonon bands.
###### pacs:
66.70.-f, 72.10.Bg, 03.65.Vf, 72.15.Gd
Recent years have witnessed a rapid development of an emerging field –
phononics, the science and technology of controlling heat flow and processing
information with phonons Wang2008 . Indeed, in parallel with electronics,
various functional thermal devices such as thermal diode diode , thermal
transistor transistor , thermal logic gates logic and thermal memory memory ,
etc., have been proposed to manipulate and control phonons, the carrier of
heat energy and information. However, different from electrons, phonons as
neutral quasiparticles, cannot directly couple to the magnetic field through
the Lorentz force. Therefore, it is a surprise that Strohm, Rikken, and Wyder
observed the phonon Hall effect (PHE) - the appearance of a temperature
difference in the direction perpendicular to both the applied magnetic field
and the heat current flowing through an ionic paramagnetic dielectric sample
strohm05 . It was confirmed later by Inyushkin and Taldenkov inyushkin07 .
Since then, several theoretical explanations have been proposed shengl06 ;
kagan08 ; wang09 to understand this novel phenomenon.
For electronic transport properties in various quantum, spin, or anomalous
Hall effects TKNN ; QSHE ; AHE , topological Berry phase has been successfully
used to understand the underlying mechanism xiao10 . Such an elegant
connection between mathematics and physics provides a broad and deep
understanding of basic material properties. However, because of the very
different nature of electrons and phonons, a topological picture related to
the PHE is not straightforward and obvious, and therefore, is still lacking.
In this Letter, we explore the topology of phonon bands in a two-dimensional
honeycomb lattice with Raman type spin-phonon interaction. A general
expression for phonon Hall conductivity in terms of Berry curvature is
derived. The phonon Hall effect is not quantized, although the Chern numbers
are quantized to integers. We find that there exists a phase transition
associated with the PHE, due to the discontinuous jump of Chern numbers.
We start with a Hamiltonian for an ionic crystal lattice in a uniform external
magnetic field holz72 , which reads in a compact form as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}(p-{\tilde{A}}u)^{T}(p-{\tilde{A}}u)+\frac{1}{2}u^{T}Ku\qquad$
(1) $\displaystyle=$
$\displaystyle\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}(K-{\tilde{A}}^{2})u+u^{T}\\!{\tilde{A}}\,p.$
Here, $u$ is a column vector of displacements from lattice equilibrium
positions for all the degrees of freedom, multiplied by the square root of
mass, $p$ is the conjugate momentum vector, and $K$ is the force constant
matrix. The superscript $T$ stands for the matrix transpose. ${\tilde{A}}$ is
an antisymmetric real matrix, which is block diagonal with elements
$\Lambda=\left(\begin{array}[]{rr}0&h\\\ -h&0\\\ \end{array}\right)$ (in two
dimensions), where $h$ is proportional to the magnitude of the applied
magnetic field, and has the dimension of frequency. For simplicity, we will
call $h$ the magnetic field later. The on-site term, $u^{T}{\tilde{A}}p$, can
be interpreted as the Raman (or spin-phonon) interaction supp . The
Hamiltonian (1) is positive definite.
By applying Bloch’s theorem, we can describe the system by the polarization
vector $x=(\mu,\epsilon)^{T}$, where $\mu$ and $\epsilon$ are associated with
the momenta and coordinates, respectively. The equation of motion can be
expressed as
$i\frac{\partial}{{\partial t}}x=H_{\rm eff}x,\;\;H_{\rm
eff}=i\left(\begin{array}[]{cc}-A&-D\\\ I&-A\end{array}\right),$ (2)
where $D({\bf k})=-A^{2}+\sum_{l^{\prime}}K_{l,l^{\prime}}e^{i({\bf
R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k}}$ is the dynamic matrix as a function
of wave vector ${\bf k}$; $K_{l,l^{\prime}}$ is the submatrix between unit
cell $l$ and $l^{\prime}$ in the full spring constant matrix $K$; ${\bf
R}_{l}$ is the real-space lattice vector; $A$ is block diagonal with elements
$\Lambda$, and $I$ is an identity matrix. Here, $D,A,K_{l,l^{\prime}}$, and
$I$ are all $4\times 4$ matrices for the two-dimensional honeycomb lattice.
The eigenvalue problem of the equation of motion (2) reads:
$H_{\rm eff}\,x_{\sigma}=\omega_{\sigma}\,x_{\sigma},$ (3)
where $x_{\sigma}=(\mu_{\sigma},\epsilon_{\sigma})^{T}$ is the right
eigenvector of the $\sigma$-th branch and $\omega_{\sigma}$ is the
corresponding eigenfrequency. Because of the non-Hermitian nature of $H_{\rm
eff}$, the left eigenvector is different, and is given by
${\tilde{x}_{\sigma}}^{T}=(\tilde{\mu}_{\sigma},\tilde{\epsilon}_{\sigma})=(\epsilon^{\dagger}_{\sigma},-\mu^{\dagger}_{\sigma})/(-2i\omega_{\sigma})$.
The orthonormal condition is
$\epsilon_{\sigma}^{{\dagger}}\epsilon_{\sigma^{\prime}}+\frac{i}{\omega_{\sigma}}\epsilon_{\sigma}^{{\dagger}}A\epsilon_{\sigma^{\prime}}=\delta_{\sigma,\sigma^{\prime}}$
wang09 .
By taking into account only positive eigenfrequency modes, displacement and
momentum operators can be written in the second quantization form. From the
definition of energy current density ${\bf
J}=\frac{1}{2V}\sum_{l,l^{\prime}}({\bf R}_{l}-{\bf
R}_{l^{\prime}})u^{T}_{l}K_{l,l^{\prime}}\dot{u}_{l^{\prime}}$ hardy63 ;
shengl06 ; kagan08 , the current density vector can be expressed as
${\bf J}={\bf J_{1}}(a^{\dagger}a)+{\bf J_{2}}(a^{\dagger}a^{\dagger},aa).$
(4)
Here, ${\bf
J_{1}}=\frac{\hbar}{4V}\sum\limits_{k,k^{\prime}}{\frac{\omega_{k}+\omega_{k^{\prime}}}{\sqrt{\omega_{k}\omega_{k^{\prime}}}}\epsilon_{k}^{\dagger}\frac{\partial
D({\bf k})}{\partial{\bf
k}}\epsilon_{k^{\prime}}\,a_{k}^{\dagger}a_{k^{\prime}}e^{i(\omega_{k}-\omega_{k^{\prime}})t}}\delta_{{\bf
k},{\bf k}^{\prime}}$, and ${\bf
J_{2}}=\frac{\hbar}{{4V}}\sum\limits_{k,k^{\prime}}{\sqrt{\frac{{\omega_{k^{\prime}}}}{{\omega_{k}}}}}\,(\epsilon_{k}^{\dagger}\frac{{\partial
D({\bf k})}}{\partial{\bf
k}}\epsilon_{k^{\prime}}^{*}a_{k}^{\dagger}a_{k^{\prime}}^{\dagger}e^{i(\omega_{k}+\omega_{k^{\prime}})t}+\,\epsilon_{k}^{T}\frac{{\partial
D^{*}({\bf k})}}{\partial{\bf
k}}\epsilon_{k^{\prime}}a_{k}a_{k^{\prime}}e^{-i(\omega_{k}+\omega_{k^{\prime}})t})\delta_{{\bf
k},-{\bf k}^{\prime}}$, where $k=({\bf k},\sigma)$ considers both the
wavevector and the phonon branch. It should be noted that the
$a^{\dagger}a^{\dagger}$ and $aa$ terms also contribute to the off-diagonal
elements of thermal conductivity tensor, although they have no contribution to
the average heat flux. The diagonal term $\epsilon_{k}^{\dagger}\frac{\partial
D({\bf k})}{\partial{\bf k}}\epsilon_{k}$ in ${\bf J_{1}}$ corresponds to
$\omega_{\sigma}\frac{\partial\omega_{\sigma}}{\partial{\bf k}}$. Only the
off-diagonal terms in ${\bf J_{1}}$ and ${\bf J_{2}}$ contribute to the Hall
conductivity, which can be regarded as the contribution from anomalous
velocities similar to the one in the intrinsic anomalous Hall effect AHE .
Using the Green-Kubo formula $\kappa_{xy}=\frac{V}{\hbar
T}\int_{0}^{\beta\hbar}d\lambda\int_{0}^{\infty}dt\,\bigl{\langle}J^{x}(-i\lambda)J^{y}(t)\bigr{\rangle}_{\rm
eq}$ mahan00 , one can obtain phonon Hall conductivity as supp :
$\displaystyle\kappa_{xy}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{{8VT}}\sum\limits_{\sigma\neq\sigma^{\prime}}{f(\omega_{\sigma})(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}}\times\qquad$
(5)
$\displaystyle\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial k_{y}}}\epsilon_{\sigma}-(k_{x}\leftrightarrow
k_{y})}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}},$
where $f(\omega_{\sigma})=(e^{\hbar\omega_{\sigma}/(k_{B}T)}-1)^{-1}$ is the
Bose distribution function, $V$ is the total volume of the sample, and the
phonon branch index $\sigma$ here includes both the positive and negative
eigenvalues without restrictions. It can be proved that the phonon Hall
conductivity $\kappa_{xy}$ satisfies the Onsager reciprocal relations supp .
Figure 1: (color online) (a) Phonon Hall conductivity vs magnetic field for
different temperatures. The dotted, dashed, and solid lines correspond to
$T=50,100$, and $300$ K, respectively. The inset shows $h$-dependence of
$\kappa_{xy}$ at low temperatures: $T=10$ (solid line), $20$ (dashed line),
and $40$ K (dotted line). (b) $d\kappa_{xy}/dh$ as a function of $h$ at
different temperatures: $T=50$ (dotted line), $100$ (dashed line), and $300$ K
(solid line); here $N_{L}=400$. The inset in (b) shows the $h$-dependence of
$d\kappa_{xy}/dh$ for different size $N_{L}$ at $T=50$ K, around $h\approx
25.5$ rad/ps; from top to bottom, $N_{L}=80,320$, and $1280$, respectively.
In Fig. 1 we show the phonon Hall conductivity of honeycomb lattices
calculated from Eq. (5). The parameters used in our numerical calculations are
the same as in Ref. wang09 . The coupling matrix between two sites is
configured such that the longitudinal spring constant is
$K_{L}=0.144\,$eV/(uÅ2) and the transverse one $K_{T}$ is 4 times smaller. The
unit cell lattice vectors are $(a,0)$ and $(a/2,a\sqrt{3}/2)$ with $a=1\,$Å.
It is found that when $h$ is small, $\kappa_{xy}$ is proportional to $h$ supp
, while the dependence becomes nonlinear when $h$ is large. As $h$ is further
increased, $\kappa_{xy}$ increases before it reaches a maximum at certain
value of $h$. Then $\kappa_{xy}$ decreases and goes to zero at very large $h$.
This can be understood as follows: numerical calculation shows that
$\omega_{\sigma}\approx\alpha h$, which can also be obtained from the equation
$\bigl{[}(-i\omega_{\sigma}+A)^{2}+D\bigr{]}\epsilon_{\sigma}=0$ supp , thus
we can obtain approximately $\kappa_{xy}\sim h^{2}/(e^{\beta\hbar\alpha h}-1)$
from Eq. (5). In the weak magnetic field limit $\kappa_{xy}\propto h$, while
in the strong field limit, $\kappa_{xy}\rightarrow 0$. The on-site term
$\tilde{A}^{2}$ in the Hamiltonian (1) increases with $h$ quadratically so as
to blockade the phonon transport, which competes with the spin-phonon
interaction. Therefore, as $h$ increases, $\kappa_{xy}$ first increases, then
decreases and tends to zero at last. At low temperatures, $\kappa_{xy}$
oscillates around zero with the variation of $h$, as shown in the inset in
Fig. 1(a).
There is a subtle singularity near $h\simeq 25$ rad/ps in Fig. 1(a); we thus
plot the first derivative of $\kappa_{xy}$ with respect to $h$ at different
temperatures in Fig. 1(b). It shows that, at the relatively high temperatures,
the first derivative of phonon Hall conductivity has a minimum at the magnetic
field $h_{c}\simeq 25.4778$ rad/ps for the finite-size sample $N_{L}=400$ (the
sample has $N=N_{L}^{2}$ unit cells). The first derivative $d\kappa_{xy}/dh$
at the point $h_{c}$ diverges when the system size increases to infinity. The
inset in Fig. 1(b) shows the finite-size effect. At the point $h_{c}$, the
second derivative $d^{2}\kappa_{xy}/dh^{2}$ is discontinuous. Therefore,
$h_{c}$ is a critical point for the PHE, across which a phase transition
occurs. At low temperatures, the divergence of $d\kappa_{xy}/dh$ is not so
evident as that at high temperatures. However, if the sample size becomes
larger, the discontinuity of $d^{2}\kappa_{xy}/dh^{2}$ is more obvious, as
illustrated in Fig. 1(b). For different temperatures, the phase transition
occurs at exactly the same critical value $h_{c}$, which strongly suggests
that the phase transition of the PHE is related to the topology of the phonon
band structure.
In the following, we would like to connect the PHE with the Berry phase to
examine the underlying topological mechanism. As is wellknown, the band
structure of crystals provides a natural platform to investigate the geometric
phase effect. Since the wave-vector dependence of the polarization vectors is
inherent to the Hall problems, the Berry phase effects are intuitively
expected for the PHE in the momentum space. Following Berry’s approach xiao10
, we set
$x(t)=e^{i\gamma_{\sigma}(t)-i\int_{0}^{t}{dt^{\prime}\omega_{\sigma}({\bf
k}(t^{\prime}))}}x_{\sigma}({\bf k}(t))$, and then insert it into Eq. (2). The
Berry phase is obtained as
$\gamma_{\sigma}=\oint\limits\mathbf{A}^{\sigma}_{\mathbf{k}}\cdot d{\bf k}$,
with $\mathbf{A}^{\sigma}_{\mathbf{k}}=i\tilde{x}_{\sigma}^{T}\frac{\partial
x_{\sigma}}{{\partial{\bf k}}},$ and the Berry curvature emerges as
$\Omega_{k_{x}k_{y}}^{\sigma}=\frac{\partial}{{\partial{k_{x}}}}\mathbf{A}^{\sigma}_{k_{y}}-\frac{\partial}{{\partial{k_{y}}}}\mathbf{A}^{\sigma}_{k_{x}}=\sum\limits_{\sigma^{\prime},\sigma^{\prime}\neq\sigma}{\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}},$
(6)
where,
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}=\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}-\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma}}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}$ (7)
is the contribution to the Berry curvature of the band $\sigma$ from a
different band $\sigma^{\prime}$. The associated topological Chern number is
obtained through integrating the Berry curvature over the first Brillouin zone
as
$C^{\sigma}=\frac{1}{{2\pi}}\int_{{\rm
BZ}}{dk_{x}dk_{y}\Omega_{k_{x}k_{y}}^{\sigma}}=\frac{{2\pi}}{{L^{2}}}\sum\limits_{\bf
k}{\Omega_{k_{x}k_{y}}^{\sigma}},$ (8)
where, $L$ is the length of the sample. The phonon Hall conductivity formula,
Eq. (5), is recasted into
$\kappa_{xy}=\frac{\hbar}{{8VT}}\sum\limits_{{\bf
k},\sigma\neq\sigma^{\prime}}{f(\omega_{\sigma})(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}}\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}.$
(9)
Here $V=L^{2}a$. The term $(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$
relating to the phonon energy is an analog of the electrical charge term
$e^{2}$ in the electron Hall effect, thus the phonon Hall conductivity Eq. (9)
is similar to but different from the electron case because the phonon energy
term can not be moved out from the summation. Although the formula is derived
from the phonon transport in the crystal-lattice system, we note that the
thermal Hall conductivity for the magnon Hall effect katsura10 can also be
cast into the form of Eq. (9) with a different expression for the Berry
curvature. Therefore, the Hall conductivity formula can be universally
applicable to the thermal Hall effect in phonon and magnon systems without
restriction for special lattice structures.
Figure 2: (color online) (a)-(d) The contour map of Berry curvatures for
bands $1-4$ at $h_{c-}=h_{c}-10^{-2}$ rad/ps; (e)-(h) The contour map of Berry
curvatures for bands $1-4$ at $h_{c+}=h_{c}+10^{-2}$ rad/ps. For (a)-(h), the
horizontal and vertical axes correspond to wave vector $k_{x}$ and $k_{y}$,
respectively. (i) $\Omega$ at different magnetic fields. The solid and dashed
lines correspond to $\Omega^{2}$ and $\Omega^{3}$ at $h_{c-}$ respectively,
while dotted and dash-dotted lines correspond to those at $h_{c+}$. (j) Chern
numbers of four bands: $C^{1}$ (solid line), $C^{2}$ (dashed line), $C^{3}$
(dotted line), and $C^{4}$ (dash-dotted line). (k) The dispersion relation of
band $2$ and $3$ at different magnetic fields in the vicinity of $h_{c}$. The
dashed, solid and dotted lines correspond to the bands at $h_{c-}$, $h_{c}$,
and $h_{c+}$, respectively. The lower three and upper three correspond to
bands 2 and 3, respectively. $k_{y}=0$ in (i) and (k).
Without the Raman spin-phonon interaction, namely, $h=0$,
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}$ is zero everywhere and the
phonon Hall conductivity vanishes. When a magnetic field is applied, the Berry
curvature is nonzero, and consequently, the PHE appears. It is found that if
the system exhibits symmetry satisfying $SDS^{-1}=D,$ $SAS^{-1}=-A$ (e.g.,
mirror reflection symmetry), the phonon Hall conductivity is zero wang09 ;
supp . This symmetry principle can also be applied to the topological property
of the phonon bands: we find that
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}=0$ provided that such symmetry
exists, such as in the square lattice system. Whereas if such symmetry is
broken for the dynamic matrix, the system can possess nontrivial Berry
curvatures. In the system with the PHE, if the magnetic field changes, the
Berry curvatures are quite different. However, we find that the associated
topological Chern numbers remain constant integers with occasional jumps when
$h$ is varied. Therefore, the Chern numbers given by Eq. (8) are topological
invariant, which indeed illustrates the nontrivial topology of the phonon band
structures. Although the Chern numbers are quantized to integers, the phonon
Hall conductivity is not, due to the extra term
$f(\omega_{\sigma})(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$. Thus, the
analogy to the quantum Hall effect is incomplete.
In the vicinity of the critical magnetic field $h_{c}$, we find that the phase
transition is indeed related to the abrupt change of the topology of band
structures. The Berry curvatures for different bands near the critical
magnetic field are illustrated in Fig. 2(a-h). We find that with an
infinitesimal change of magnetic field around $h_{c}$, the Berry curvatures
around the $\Gamma$ (${\bf k}=0$) point of bands $2$ and $3$ are quite
different, whereas those of band $1$ and $4$ remain unchanged. To illustrate
the change of the Berry curvatures clearly, we plot the cross section of the
Berry curvatures along the $k_{x}$ direction for bands $2$ and $3$ in Fig.
2(i), which shows explicitly that the Berry curvatures change dramatically
above and below the critical magnetic field $h_{c}$. Below the critical point,
the Berry curvature for band $2$ in the vicinity of $\Gamma$ point contributes
Berry phase $2\pi$ ($-2\pi$ for band $3$), which cancels that from $K$,and
$K^{\prime}$ points, so that the Chern number is zero for bands $2$ and $3$,
as indicated in Fig. 2(j). However, above the critical point, the sum of Berry
curvature at $\Gamma$ point is zero, and only the monopole at $K$,and
$K^{\prime}$ points contributes to Berry phase ($-2\pi$ for band $2$ and
$2\pi$ for band $3$). Therefore, the Chern numbers jump from $0$ to $\pm 1$,
as shown in Fig. 2(j). This jump indicates that the topology of the two bands
suddenly changes at the critical magnetic field, which is responsible for the
phase transition. From a calculation on the kagome lattice, which has been
used to model many real materials kagome , we also find qualitatively similar
phase transitions due to the sudden change of topology, where the phonon Hall
conductivity has three singularities of divergent first derivatives
corresponding to three jumps of the Chern numbers.
To further investigate the mechanism of the abrupt change of the phonon band
topology, we study the dispersion relation near the critical magnetic field.
From Fig. 2(k), we can see that band $2$ and $3$ are going to touch with each
other at the $\Gamma$ point if the magnetic field increases to $h_{c}$; at the
critical magnetic field, the degeneracy occurs and the two bands possess the
cone shape; above the critical point $h_{c}$, the two bands split up.
Therefore, the difference between the two bands decreases below and increases
above the critical point $h_{c}$. The property of the dispersion relation in
the vicinity of the critical magnetic field directly affects the Berry
curvature of the corresponding bands.
In summary, we have studied the PHE from a topological point of view. By
looking at the phases of the polarization vectors of both the displacements
and conjugate momenta as a function of the wave vector, a Berry curvature can
be defined uniquely for each band. This Berry curvature can be used to
calculate the phonon Hall conductivity. Because of the nature of phonons, the
phonon Hall conductivity, which is not directly proportional to the Chern
number, is not quantized. However, the quantization effect, in the sense of
discontinuous jumps in Chern numbers, manifests itself in the phonon Hall
conductivity as a singularity of the first derivative with respect to the
magnetic field.
The topological approach for phonon Hall conductivity proposed here is general
and can be applied to the real materials in low temperatures where the thermal
transport is ballistic. It can also be applied to the magnon Hall effect
discovered recently katsura10 . Phase transition in the PHE, explained from
topological nature and dispersion relations, can also be generalized to study
the phase transition in other Hall effects and/or nonequilibrium transport. In
line with recently reported Berry-phase-induced heat pumping Ren10 and the
Berry-phase contribution of molecular vibrational instability Lv10 , we hope
our present results do invigorate the studies aimed at uncovering intriguing
Berry phase effects and topological properties in phonon transport, which will
enrich further the discipline of phononics.
L.Z. thanks Bijay Kumar Agarwalla and Jie Chen for fruitful discussions. This
project is supported in part by Grants No. R-144-000-257-112 and No.
R-144-000-222-646 of NUS.
## References
* (1) L. Wang and B. Li, Physics World 21, No.3, 27 (2008).
* (2) B. Li, L. Wang and G Casati, Phys. Rev. Lett 93 184301 (2004); C. W Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121 (2006).
* (3) B. Li, L. Wang and G. Casati, Appl. Phys. Lett. 88, 143501 (2006).
* (4) L. Wang and B. Li, Phys. Rev. Lett 99, 177208 (2007).
* (5) L. Wang and B. Li, Phys. Rev. Lett. 101, 267203 (2008).
* (6) C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phys. Rev. Lett. 95, 155901 (2005).
* (7) A. V. Inyushkin and A. N. Taldenkov, JETP Lett. 86, 379 (2007).
* (8) L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 96, 155901 (2006).
* (9) Y. Kagan and L. A. Maksimov, Phys. Rev. Lett. 100, 145902 (2008).
* (10) J.-S. Wang and L. Zhang, Phys. Rev. B. 80, 012301 (2009); L. Zhang, J.-S. Wang, and B. Li, New J. Phys. 11, 113038 (2009).
* (11) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
* (12) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
* (13) M. Koenig, et al. J. Phys. Soc. Jpn. 77, 031007 (2008).
* (14) D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
* (15) A. Holz, Il Nuovo Cimento B 9, 83 (1972).
* (16) See the detailed discussion in supplementary information.
* (17) R. J. Hardy, Phys. Rev. 132, 168 (1963).
* (18) G. D. Mahan, Many-Particle Physics 3rd ed. (Kluwer Academic, New York, 2000).
* (19) H. Katsura, N. Nagaosa, and P. A. Lee, Phys. Rev. Lett. 104, 066403 (2010); Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, Y. Tokura, Science 329, 297 (2010).
* (20) I. Syozi, Prog. Theor. Phys. 6, 306 (1951); M. Takano, T. Shinjo, M. Kiyama, T. Takada, J. Phys. Soc. Jpn. 25, 902 (1968); M. Wolf, K. D. Schotte, J. Phys. A 21, 2195 (1988); V. Elser, Phys. Rev. Lett. 62, 2405 (1989); C. Broholm, G. Appli, G. P. Espinosa, A. S. Cooper, Phys. Rev. Lett. 65, 3173 (1990).
* (21) J. Ren, P. Hänggi, and B. Li, Phys. Rev. Lett. 104, 170601 (2010).
* (22) J.-T. Lü, M. Brandbyge, and P. Hedegård, Nano Lett. 10, 1657 (2010).
## I Supplementary information for “Topological Nature of Phonon Hall Effect”
#### I.0.1 ABSTRACT
_In this supplementary material, we discuss the origination of the Hamiltonian
[Eq. (1) in the main text] in the first section; then we present the detailed
derivation of the general formula of phonon Hall conductivity in terms of
Berry curvature, in which we also give the explicit expression for the dynamic
matrix and give the proof for the symmetry principle. Finally, we discuss the
numerical calculation for Chern numbers._
### I.1 DISCUSSION ON THE HAMILTONIAN
In the presence of a magnetic field, according to Ref. holz72 , the kinetic
energy of each site in ionic crystal lattices without free charges is
expressed as:
$T_{\alpha}=\frac{1}{2}m_{\alpha}|{\dot{\bf
r}}_{\alpha}|^{2}=\frac{1}{2m_{\alpha}}|{\bf
p}_{\alpha}\sqrt{m_{\alpha}}-q_{\alpha}{\bf A}_{\alpha}|^{2},$ (S1)
where, ${\bf r}_{\alpha}={\bf R}_{\alpha}+{\bf u}_{\alpha}/\sqrt{m_{\alpha}}$,
${\bf R}_{\alpha}$ is the equilibrium coordinate of the ion at site $\alpha$,
and ${\bf u}_{\alpha}$ denotes the displacement multiplied by the square root
of the ion mass $m_{\alpha}$. ${\bf p}_{\alpha}$ is the corresponding momentum
divided by the square root of mass $m_{\alpha}$. $q_{\alpha}$ is the ionic
charge at site $\alpha$. ${\bf A_{\alpha}}$ denotes the electromagnetic vector
potential, which, using the Lorenz gauge condition, can be related to the
ionic displacement as holz72
${\bf A}_{\alpha}=\frac{1}{2}{\bf B}\times{\bf u}_{\alpha}/\sqrt{m_{\alpha}}.$
(S2)
Thus, Eq. (S1) is recasted as:
$T_{\alpha}=\frac{1}{2}|{\bf p}_{\alpha}-\frac{q_{\alpha}}{2m_{\alpha}}{\bf
B}\times{\bf u}_{\alpha}|^{2}.$ (S3)
If the magnetic field with magnitude $B$ is applied along $z$ direction and we
only consider the two-dimensional ($x$ and $y$ direction) motion of the
system, then the kinetic energy of ion $\alpha$ can be expressed (it is
straightforward to generalize to high dimentions) as:
$T_{\alpha}=\frac{1}{2}(p_{\alpha}-\Lambda_{\alpha}u_{\alpha})^{T}(p_{\alpha}-\Lambda_{\alpha}u_{\alpha}),$
(S4)
where $p_{\alpha}=(p_{\alpha x},p_{\alpha y})^{T}$, $u_{\alpha}=(u_{\alpha
x},u_{\alpha y})^{T}$, and
$\Lambda_{\alpha}=\left(\begin{array}[]{cc}0&h_{\alpha}\\\ -h_{\alpha}&0\\\
\end{array}\right)$, where $h_{\alpha}=-q_{\alpha}B/(2m_{\alpha})$. Note that
there are both positive and negative ions in one unit cell. For a general
ionic paramagnetic dielectric, mostly, the mass of the positive ion is larger
than that of the negative one. For instance, in the experimental sample ${\rm
Tb_{3}Ga_{5}O_{12}}$, the ratio $m(+q)/m(-q)$ is about $4.3$ in one unit cell.
Therefore the negative ions will dominate in the contribution to $h_{\alpha}$,
which makes $h_{\alpha}$ have the same sign as that of the applied magnetic
field $B$. Under the mean-field approximation, we can set $h_{\alpha}=h$,
which is site-independent and is proportional to the magnitude of the applied
magnetic field.
Combining the kinetic energy with the harmonic inter-potential energy, we can
write the whole Hamiltonian as
$H=\frac{1}{2}(p-{\tilde{A}}u)^{T}(p-{\tilde{A}}u)+\frac{1}{2}u^{T}Ku,\\\ $
(S5)
where ${\tilde{A}}$ is an antisymmetric real matrix with block-diagonal
elements $\Lambda_{\alpha}$. $u$ and $p$ are column vectors denoting
displacements and momenta respectively, for all the degrees of freedom. $K$
indicates the force constant matrix. Finally, after the rearrangement, we have
$H=\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}(K-{\tilde{A}}^{2})u+u^{T}\\!{\tilde{A}}\,p,$
(S6)
which is exactly the second row of Eq. (1) in the text.
The Hamiltonian Eq. (S6) [Eq.(1) in the main text] is essentially the same as
that used in Ref. shengl06 ; kagan08 ; wang09 ; zhang09 resulting from the
phenomenological Raman interaction. The only difference is the term
proportional to $\tilde{A}^{2}$ which makes the above Hamiltonian positive
definite. The Raman interaction, proposed to study spin-phonon interactions
(SPI) based on quantum theory and fundamental symmetries old-sp-papers ; sp-
book ; ioselevich95 , can be expressed as
$H_{I}=g{\bf s}\cdot({\bf u}\times{\bf p}).$ (S7)
Here, $g$ denotes a positive coupling constant, and ${\bf s}$ is the isospin
for the lowest quasidoublet. In the presence of a magnetic field ${\bf B}$,
each site has a magnetization ${\bf M}$. For isotropic SPI, the isospin ${\bf
s}$ is parallel to ${\bf M}$, and the ensemble average of the isospin is
proportional to the magnetization, which can be expressed as $\langle{\bf
s}\rangle$ = $c{\bf M}$ with $c$ the proportionality coefficient (Ref.
shengl06 ; kagan08 ; wang09 ; zhang09 ). In the mean-field approximation, the
Raman type SPI reduces to
$H_{I}={\bf h}\cdot({\bf u}\times{\bf p}),$ (S8)
where ${\bf h}=gc{\bf M}$, and ${\bf M}$ is proportional to the magnetic field
${\bf B}$. If the magnetic field is applied along the $z$ direction, then the
SPI can be written as
$H_{I}=u^{T}\,\tilde{A}\,p.$ (S9)
By treating the phonon system under harmonic approximation, the total
Hamiltonian for the whole lattice can be written as (Ref. shengl06 ; kagan08 ;
wang09 )
$H=\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}Ku+u^{T}\\!{\tilde{A}}\,p.$ (S10)
Note that this Hamiltonian Eq. (S10) is not positive definite. In Ref.wang09 ,
the authors added an arbitrary onsite potential in order to make the
Hamiltonian positive definite. However, in the calculation of phonon Hall
effect for the four-terminal junctions, such non-positive-definite Hamiltonian
does not cause any problem because the thermal junctions will stabilize the
system zhang09 .
From the first physical picture of spin-phonon interaction in ionic crystal
lattice with an applied magnetic field (Eq. LABEL:eq-kinetic$\sim$S6), the
additional term proportional to ${\tilde{A}}^{2}$ emerges naturally to make
the Hamiltonian positive definite. Therefore, in this work we choose the
positive definite Hamiltonian Eq. (S6) [Eq. (1) in the main text].
### I.2 PHONON HALL CONDUCTIVITY FROM GREEN-KUBO FORMULA
The Hamiltonian Eq. (S6) is quadratic in $u$ and $p$, and we can write the
equation of motion as
$\displaystyle\dot{p}$ $\displaystyle=$
$\displaystyle-(K-\tilde{A}^{2})u-\tilde{A}p,$ (S11) $\displaystyle\dot{u}$
$\displaystyle=$ $\displaystyle p-\tilde{A}u.$ (S12)
The equation of motion for the coordinate is,
$\ddot{u}+2\tilde{A}\dot{u}+\tilde{A}^{2}u+(K-\tilde{A}^{2})u=0.$ (S13)
Since the lattice is periodic, we can apply the Bloch’s theorem
$u_{l}=\epsilon e^{i({\bf R}_{l}\cdot{\bf k}-\omega t)}$. The polarization
vector $\epsilon$ satisfies
$\bigl{[}(-i\omega+A)^{2}+D\bigr{]}\epsilon=0,$ (S14)
where $D({\bf k})=-A^{2}+\sum_{l^{\prime}}K_{l,l^{\prime}}e^{i({\bf
R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k}}$ denotes the dynamic matrix and $A$
is block diagonal with elements $\Lambda$. $D,K_{l,l^{\prime}},$ and $A$ are
all $nd\times nd$ matrices, where $n$ is the number of particles in one unit
cell and $d$ is the dimension of the motion.
To calculate the dynamic matrix $D({\bf k})$, we give an example for the two-
dimensional honeycomb lattice, where $n=2,d=2$. We only consider the nearest
neighbor interaction. The spring constant matrix along $x$ direction is
$K_{x}=\left(\begin{array}[]{cc}K_{L}&0\\\ 0&K_{T}\\\ \end{array}\right).$
(S15)
$K_{L}=0.144\,$eV/(uÅ2) is the longitudinal spring constant and the transverse
one $K_{T}$ is 4 times smaller. The unit cell lattice vectors are $(a,0)$ and
$(a/2,a\sqrt{3}/2)$ with $a=1\,$Å.
To obtain the explicit formula for the dynamic matrix, we first define a
rotation operator in two dimensions as:
$U(\theta)=\left({\begin{array}[]{*{20}c}{\cos\theta}&{-\sin\theta}\\\
{\sin\theta}&{\cos\theta}\\\ \end{array}}\right).$
The three kinds of spring-constant matrices between two atoms are
$K_{01}=U(\pi/2)K_{x}U(-\pi/2)$, $K_{02}=U(\pi/6)K_{x}U(-\pi/6)$,
$K_{03}=U(-\pi/6)K_{x}U(\pi/6)$, which are $2\times 2$ matrices. Then we can
obtain the on-site spring-constant matrix and the four spring-constant
matrices between the unit cell and its four nearest neighbors as:
$K_{0}=\left({\begin{array}[]{*{20}c}{K_{01}+K_{02}+K_{03}}&{-K_{02}}\\\
{-K_{02}}&{K_{01}+K_{02}+K_{03}}\\\ \end{array}}\right),$ (S16)
$K_{1}=\left({\begin{array}[]{*{20}c}0&0\\\
{-K_{03}}&0\end{array}}\right),\quad
K_{2}=\left({\begin{array}[]{*{20}c}0&0\\\ {-K_{01}}&0\\\
\end{array}}\right),\quad K_{3}=\left({\begin{array}[]{*{20}c}0&{-K_{03}}\\\
0&0\\\ \end{array}}\right),\quad
K_{4}=\left({\begin{array}[]{*{20}c}0&{-K_{01}}\\\ 0&0\\\
\end{array}}\right),\\\ $ (S17)
which are $4\times 4$ matrices. Finally we can obtain the $4\times 4$ dynamic
matrix $D({\bf k})$ as
$D({\bf
k})=-A^{2}+K_{0}+K_{1}e^{ik_{x}}+K_{2}e^{i(k_{x}/2+\sqrt{3}k_{y}/2)}+K_{3}e^{-ik_{x}}+K_{4}e^{-i(k_{x}/2+\sqrt{3}k_{y}/2)},$
(S18)
where, $A^{2}=-h^{2}\cdot I$, and $I$ denotes the $4\times 4$ identity matrix.
Equation (S14) can be written in a form as a standard eigen-problem given in
Eq. (3) in the main text, if we rewrite the equations of motion. Using Bloch
theorem, Eqs. (S11) and (S12) can be recasted as:
$i\frac{\partial}{{\partial t}}x=H_{\rm eff}x,\;\;\;\;\;\;\;\;\;\;H_{\rm
eff}=i\left(\begin{array}[]{cc}-A&-D\\\ I&-A\end{array}\right).$ (S19)
$x=(\mu,\epsilon)^{T}$ is the polarization vector , where column vectors $\mu$
and $\epsilon$ are associated with the momenta and coordinates respectively.
Therefore, the right eigenvector and left eigenvector satisfy:
$H_{\rm
eff}x_{\sigma}=\omega_{\sigma}x_{\sigma},\;\;\;\;\;\;\;\;\;\;\tilde{x}_{\sigma}^{T}H_{\rm
eff}=\omega_{\sigma}\tilde{x}_{\sigma}^{T}.$ (S20)
where the right eigenvector $x_{\sigma}=(\mu_{\sigma},\epsilon_{\sigma})^{T}$,
the left eigenvector
${\tilde{x}_{\sigma}}^{T}=(\epsilon^{\dagger}_{\sigma},-\mu^{\dagger}_{\sigma})/(-2i\omega_{\sigma})$,
and $\sigma$ indicates the branch index. Because the effective Hamiltonian
$H_{\rm eff}$ is not hermitian, the orthonormal condition then holds between
the left and right eigenvectors. The eigenmodes can be normalized as
${\tilde{x}_{\sigma}}^{T}x_{\sigma}=1$, which is equivalent to wang09
$\epsilon_{\sigma}^{\dagger}\epsilon_{\sigma}+\frac{i}{\omega_{\sigma}}\epsilon_{\sigma}^{\dagger}\\!A\epsilon_{\sigma}=1.$
(S21)
To solve the eigensystem, we require the following relations:
$\epsilon_{{-\bf k},-\sigma}^{*}=\epsilon_{{\bf k},\sigma};\;\omega_{{-\bf
k},-\sigma}=-\omega_{{\bf k},\sigma}.$ (S22)
In the following, we use $k=({\bf k},\sigma)$ to specify both the wavevector
and the phonon branch. By taking into account only positive eigen-modes
($\omega>0$), displacement and momentum operators are taken in the second
quantization form:
$\displaystyle u_{l}$ $\displaystyle=$
$\displaystyle\sum_{k}\epsilon_{k}e^{i({\bf R}_{l}\cdot{\bf
k}-\omega_{k}t)}\sqrt{\frac{\hbar}{2\omega_{k}N}}\;a_{k}+{\rm h.c.},$ (S23)
$\displaystyle p_{l}$ $\displaystyle=$ $\displaystyle\sum_{k}\mu_{k}e^{i({\bf
R}_{l}\cdot{\bf k}-\omega_{k}t)}\sqrt{\frac{\hbar}{2\omega_{k}N}}\;a_{k}+{\rm
h.c.},$ (S24)
where $\sigma>0$, $a_{k}$ is the annihilation operator, and h.c. stands for
hermitian conjugate. The momentum and displacement polarization vectors are
related through $\mu_{k}=-i\omega_{k}\epsilon_{k}+A\epsilon_{k}$. We can
verify that the canonical commutation relations are satisfied:
$[u_{l},p_{l^{\prime}}^{T}]=i\hbar\delta_{l,l^{\prime}}I$, and
$H=\sum_{k}\hbar\omega_{k}(a_{k}^{\dagger}a_{k}+1/2)$.
The energy current density is defined as hardy63 :
${\bf J}=\frac{1}{2V}\sum_{l,l^{\prime}}({\bf R}_{l}\\!-\\!{\bf
R}_{l^{\prime}})u^{T}_{l}K_{l,l^{\prime}}\dot{u}_{l^{\prime}},$ (S25)
where $V$ is the total volume of $N$ unit cells. The current density vector
can be expressed in terms of the creation/annihilation operators as
$\begin{array}[]{ll}{\bf J}={\bf J}_{1}(a^{\dagger}a)+{\bf
J}_{2}(a^{\dagger}a^{\dagger},aa);\\\ {\bf
J}_{1}=\frac{\hbar}{4V}\sum_{k,k^{\prime}}\left(\sqrt{\frac{\omega_{k}}{\omega_{k^{\prime}}}}+\\!\sqrt{\frac{\omega_{k^{\prime}}}{\omega_{k}}}\right)\epsilon_{k}^{\dagger}\frac{\partial
D({\bf k})}{\partial{\bf
k}}\epsilon_{k^{\prime}}\,a_{k}^{\dagger}a_{k^{\prime}}\delta_{{\bf k},{\bf
k}^{\prime}}e^{i(\omega_{k}-\omega_{k^{\prime}})t};\\\ {\bf
J}_{2}=\frac{\hbar}{{4V}}\sum\limits_{k,k^{\prime}}{\sqrt{\frac{{\omega_{k^{\prime}}}}{{\omega_{k}}}}}\,(\epsilon_{k}^{\dagger}\frac{{\partial
D({\bf k})}}{\partial{\bf
k}}\epsilon_{k^{\prime}}^{*}a_{k}^{\dagger}a_{k^{\prime}}^{\dagger}e^{i(\omega_{k}+\omega_{k^{\prime}})t}+\,\epsilon_{k}^{T}\frac{{\partial
D^{*}({\bf k})}}{\partial{\bf
k}}\epsilon_{k^{\prime}}a_{k}a_{k^{\prime}}e^{-i(\omega_{k}+\omega_{k^{\prime}})t})\delta_{{\bf
k},-{\bf k}^{\prime}}.\end{array}$ (S26)
We note that the $a^{\dagger}a^{\dagger}$ and $aa$ terms also contribute to
the off-diagonal elements of the thermal conductivity tensor, although they
have no contribution to the average energy current. Based on the expression of
heat current, the phonon Hall conductivity can be obtained through the Green-
Kubo formula mahan00 :
$\kappa_{xy}=\frac{V}{\hbar
T}\int_{0}^{\hbar/(k_{B}T)}\\!\\!\\!\\!d\lambda\int_{0}^{\infty}\\!dt\,\bigl{\langle}J^{x}(-i\lambda)J^{y}(t)\bigr{\rangle}_{\rm
eq},$ (S27)
where the average is taken over the equilibrium ensemble with Hamiltonian $H$.
Substituting the expression ${\bf J}$ into Eq. (S27), the phonon Hall
conductivity is obtained as
$\begin{array}[]{ll}\kappa_{xy}=\kappa_{xy}^{(1)}+\kappa_{xy}^{(2)};\\\
\kappa_{xy}^{(1)}=\frac{V}{\hbar
T}\int_{0}^{\hbar/(k_{B}T)}\,d\lambda\int_{0}^{\infty}\\!dt\,\bigl{\langle}J_{1}^{x}(-i\lambda)J_{1}^{y}(t)\bigr{\rangle}_{\rm
eq};\\\ \kappa_{xy}^{(2)}=\frac{V}{\hbar
T}\int_{0}^{\hbar/(k_{B}T)}\,d\lambda\int_{0}^{\infty}\\!dt\,\bigl{\langle}J_{2}^{x}(-i\lambda)J_{2}^{y}(t)\bigr{\rangle}_{\rm
eq}.\end{array}$ (S28)
Note that the averages of the cross terms
$\bigl{\langle}J_{1}^{x}(-i\lambda)J_{2}^{y}(t)\bigr{\rangle}_{\rm eq}$ and
$\bigl{\langle}J_{2}^{x}(-i\lambda)J_{1}^{y}(t)\bigr{\rangle}_{\rm eq}$ are
zero.
First we calculate the term $\kappa_{ab}^{(1)}$. Combining the result
$\langle a_{i}^{\dagger}a_{j}a_{k}^{\dagger}a_{l}\rangle_{\rm
eq}=f_{i}f_{k}\delta_{ij}\delta_{kl}+f_{i}(f_{j}+1)\delta_{il}\delta_{jk},$
(S29)
where $f_{i}=(e^{\beta\hbar\omega_{i}}-1)^{-1}$ is the Bose distribution
function, with the result
$\sum\limits_{\bf k}{\epsilon_{\sigma}^{\dagger}\frac{{\partial D}}{{\partial
k_{\alpha}}}\epsilon_{\sigma}}=-2i\sum\limits_{\bf
k}{\omega_{\sigma}{\tilde{x}_{\sigma}}^{T}\frac{{\partial H_{\rm
eff}}}{{\partial k_{\alpha}}}x_{\sigma}}=-2i\sum\limits_{\bf
k}{\omega_{\sigma}\frac{{\partial\omega_{\sigma}}}{{\partial
k_{\alpha}}}=0},\;\;\;\;\;\;\;\;(\alpha=x,y)$
which is obtained by differentiating the Eq. (S20) (Eq. (3) in the main text)
and $\tilde{x}_{\sigma}^{T}x_{\sigma}=1$, we obtain
$\kappa_{xy}^{(1)}=\frac{\hbar}{{16VT}}\sum\limits_{{\bf
k},\sigma>0,\sigma^{\prime}>0}{[f(\omega_{\sigma})-f(\omega_{\sigma}^{\prime})](\omega_{\sigma}+\omega_{\sigma}^{\prime})^{2}\frac{i}{{\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}}.$
(S30)
Because of Eq. (S22) and the following property:
$D_{ab}(-{\bf k})=D_{ab}^{*}({\bf k})=D_{ba}({\bf k}),$ (S31)
we can transform from the positive-frequency bands to the negative-frequency
band, and obtain
$\kappa_{xy}^{(1)}=\frac{\hbar}{{8VT}}\sum\limits_{{\bf
k},\sigma\sigma^{\prime}>0}{[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}}.$
(S32)
Here, $\sigma,\sigma^{\prime}$ can be both positive or negative.
Second, we calculate $\kappa_{ab}^{(2)}$. Utilizing the results
$\begin{array}[]{ll}\langle
a_{i}^{\dagger}a_{j}^{\dagger}a_{k}a_{l}\rangle_{\rm
eq}=f_{i}f_{j}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk});\\\ \langle
a_{i}a_{j}a_{k}^{\dagger}a_{l}^{\dagger}\rangle_{\rm
eq}=(1+f_{i})(1+f_{j})(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),\end{array}$
(S33)
and the relation $f(-\omega)=-1-f(\omega)$, after some algebraic derivation
similar to the above, we obtain
$\kappa_{xy}^{(2)}=\frac{\hbar}{{8VT}}\sum\limits_{{\bf
k},\sigma\sigma^{\prime}<0}{[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}}.$
(S34)
Therefore, the total phonon Hall conductivity can be written as
$\kappa_{xy}=\frac{\hbar}{{8VT}}\sum\limits_{{\bf
k},\sigma\neq\sigma^{\prime}}{[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}}.$
(S35)
We can prove $\kappa_{xy}=-\kappa_{yx}$, such that
$\kappa_{xy}=\frac{\hbar}{{16VT}}\sum\limits_{{\bf
k},\sigma\neq\sigma^{\prime}}{[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}},$
(S36)
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}=\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma}-\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{y}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma}}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}.$
(S37)
Because of
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}=-\Omega_{k_{x}k_{y}}^{\sigma^{\prime}\sigma}$,
the phonon Hall conductivity can be written eventually as
$\kappa_{xy}=\frac{\hbar}{{8VT}}\sum\limits_{{\bf
k},\sigma\neq\sigma^{\prime}}{f(\omega_{\sigma})(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}},$
(S38)
where $V$ is the total volume of $N=N_{L}^{2}$ unit cells. In the above
formula, the phonon branch $\sigma$ includes both positive and negative values
without restriction. We start with the positive frequency bands to derive the
conductivity formula. Through some transformations, we finally obtain the
simplified formula for phonon Hall conductivity which combines the
contribution from all the frequency bands. The formula Eq. (S38) is different
from that given in Ref. wang09 . In Ref. wang09 the contribution for phonon
Hall conductivity from $J_{2}$ was omitted, which is incorrect.
From the Eq. (S14), we obtain
$\epsilon_{{-\bf k},\sigma}^{*}(-A)=\epsilon_{{\bf
k},\sigma}(A);\;\omega_{{-\bf k},\sigma}=\omega_{{\bf k},\sigma},$ (S39)
and because of $D({\bf k})=D^{*}(-{\bf
k}),\;\epsilon_{\sigma}^{T}\frac{{\partial D^{*}}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}^{*}=\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial k_{x}}}\epsilon_{\sigma}$, we have
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}({\bf
k},-A)=\Omega_{k_{y}k_{x}}^{\sigma\sigma^{\prime}}(-{\bf
k},A)=-\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}(-{\bf k},A).$ (S40)
So we obtain
$\kappa_{xy}(-A)=\kappa_{yx}(A)=-\kappa_{xy}(A).$ (S41)
The Onsager reciprocal relations are satisfied.
If the system possesses the symmetry which satisfies
$SDS^{-1}=D,\;SAS^{-1}=-A,$ (S42)
where $S$ represents any symmetric operation, and from Eq. (S14), we obtain
$S\epsilon(A)=\epsilon(-A).$ (S43)
Using the definition of the dynamic matrix
$D=-A^{2}+\sum_{l^{\prime}}K_{l,l^{\prime}}e^{i({\bf R}_{l^{\prime}}-{\bf
R}_{l})\cdot{\bf k}}$ and $SDS^{-1}=D$, we can obtain
$S\frac{\partial D}{\partial k_{\alpha}}S^{-1}=\frac{\partial D}{\partial
k_{\alpha}},\;\;\;\;\;\;\;\;(\alpha=x,y)$ (S44)
Inserting $S^{-1}S=I$ into Eq.(S37), we obtain
$\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}(-A)=\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}(A).$
(S45)
Then it is easy to obtain $\kappa_{xy}(-A)=\kappa_{xy}(A)$, and because of the
Onsager relation, one can easily obtain that
$\kappa_{xy}=0,\;\;{\rm if}\;SDS^{-1}=D,\;SAS^{-1}=-A.$ (S46)
Figure S1: (color online) (a) Phonon Hall conductivity vs applied magnetic
field for two-dimensional honeycomb lattice. (b) Phonon Hall conductivity vs
temperature at fixed magnetic field $h=1$ rad/ps. The inset of (b) shows the
product of phonon Hall conductivity and temperature $\kappa_{xy}T$ vs magnetic
field $h$ for different temperatures.
Fig. S1(a) shows the phonon Hall conductivity with magnetic field for
different temperatures. In the weak magnetic field range, the phonon Hall
conductivity $k_{xy}$ is proportional to the magnetic field, which is
consistent with all the experimental and theoretical results. We plot phonon
Hall conductivity with a large range of temperatures in Fig. S1(b). At very
low temperatures, the phonon Hall conductivity is proportional to $1/T$.
$k_{xy}T$ will be constant for different temperatures lower than $1K$. This is
due to the contribution from $\kappa_{xy}^{(2)}$: if $T\rightarrow 0$,
$1+f\rightarrow 1$, then the conductivity linear with $1/T$ tends infinity.
While the longitudinal thermal conductivity $\kappa_{xx}$ is infinite for any
temperature wang09 , thus when $T\rightarrow 0$, the transverse Hall
conductivity, $\kappa_{xy}\rightarrow\infty$, has the ballistic property
similar to the longitudinal one. If temperature is very high, all the modes
contribute to the thermal transport, and $f\simeq k_{B}T/(\hbar\omega)$, then
the phonon Hall conductivity becomes a constant, which can be seen in Fig.
S1(b).
### I.3 THE BERRY PHASE AND BERRY CURVATURE
Using the similar method proposed by Berryberry84 , we derive the Berry phase
and Berry curvature in the following. Starting from
$i\frac{\partial}{{\partial t}}x(t)=H_{\rm eff}x(t)$ (S47)
and substituting
$x(t)=e^{i\gamma_{\sigma}(t)-i\int_{0}^{t}{dt^{\prime}\omega_{\sigma}({\bf
k}(t^{\prime}))}}x_{\sigma}({\bf k}(t)),$
we can obtain the Berry phase across the Brillouin zone as
$\gamma_{\sigma}=\oint\limits{{\bf A}^{\sigma}({\bf k})d{\bf
k}},\;\;\;\;\;\;\;\;{\bf A}^{\sigma}({\bf
k})=i\tilde{x}_{\sigma}^{T}\frac{\partial}{{\partial{\bf k}}}x_{\sigma}.$
Here $x_{\sigma},\tilde{x}_{\sigma}^{T}$ correspond to the right and left
eigenvectors, and
$\tilde{x}_{\sigma}^{T}x_{\sigma^{\prime}}=\delta_{\sigma\sigma^{\prime}}$,
$\sum\limits_{\sigma}{x_{\sigma}\tilde{x}_{\sigma}^{T}=I}$. ${\bf
A}^{\sigma}({\bf k})$ is the so-called Berry vector potential. Therefore the
Berry curvature is obtained through the Stokes theorem as:
$\Omega_{k_{x}k_{y}}^{\sigma}=\frac{\partial}{{\partial k_{x}}}{\bf
A}_{k_{y}}^{\sigma}-\frac{\partial}{{\partial k_{y}}}{\bf
A}_{k_{x}}^{\sigma}=i\sum\limits_{\sigma^{\prime}\neq\sigma}{\frac{{\tilde{x}_{\sigma}^{T}\frac{{\partial
H_{\rm eff}}}{{\partial
k_{x}}}x_{\sigma^{\prime}}\tilde{x}_{\sigma^{\prime}}^{T}\frac{{\partial
H_{\rm eff}}}{{\partial k_{y}}}x_{\sigma}-(k_{x}\leftrightarrow
k_{y})}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}}$ (S48)
Inserting the vector $x$ and the expression of matrix $H_{\rm eff}$, we obtain
$\Omega_{k_{x}k_{y}}^{\sigma}=\sum\limits_{\sigma^{\prime}\neq\sigma}{\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial
D}}{{\partial
k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial
D}}{{\partial k_{y}}}\epsilon_{\sigma}-(k_{x}\leftrightarrow
k_{y})}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}}=\sum\limits_{\sigma^{\prime}\neq\sigma}{\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}}$
(S49)
where $\Omega_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}$ indicates the contribution
to the Berry curvature of the band $\sigma$ from a different band
$\sigma^{\prime}$. Therefore, the phonon Hall conductivity formula Eq. (S38)
can be interpreted in terms of the Berry curvature.
### I.4 THE CALCULATION OF THE CHERN NUMBER
The topological Chern number is obtained by integrating the Berry curvature
over the first Brillouin zone as
$C^{\sigma}=\frac{1}{{2\pi}}\int_{\bf
BZ}{dk_{x}dk_{y}\Omega_{k_{x}k_{y}}^{\sigma}}.$ (S50)
For numerical calculation, we use
$C^{\sigma}=\frac{{2\pi}}{{L^{2}}}\sum\limits_{\bf
k}{\Omega_{k_{x}k_{y}}^{\sigma}}.$ (S51)
where ${\frac{1}{L^{2}}\sum\limits_{\bf
k}=\int{\frac{{dk_{x}dk_{y}}}{{(2\pi)^{2}}}}}$ and $V=L^{2}a$, $L^{2}$ is the
area of the sample.
Figure S2: (color online) (a) The four Chern number vs onsite potential
$V_{\mathrm{onsite}}$. The unit for onsite potential is longitudinal spring
constant $K_{L}$. Here $N=N_{L}^{2}=160000$; (b) The Chern number of the
fourth band changes with $N_{L}$ for different onsite potentials. For both (a)
and (b), $h=1$ rad/ps.
To calculate the integer Chern numbers, large ${\bf k}$-sampling points $N$ is
needed. However there is always a zero eigenvalue at the $\Gamma$ point of the
dispersion relation, which corresponds to a singularity of the Berry
curvature. Therefore, we cannot sum up the Berry curvature very near this
point to obtain Chern number of this band, unless we add a negligible on-site
potential $\frac{1}{2}u^{T}V_{\mathrm{onsite}}u$ to the original Hamiltonian.
In Fig. S2(a), without the on-site potential, the Chern number of the fourth
band is not an integer, no matter how large the sample size $N=N_{L}^{2}$ is
(see Fig. S2(b)). If we add the external on-site potential, the Chern number
of the fourth band will become integer. In Fig. S2(a), the $C_{4}$ changes
gradually to $-1$ with increasing the on-site potential, while other Chern
numbers do not change. And from Fig. S2(b), we see that with larger on-site
potential, the Chern number of the fourth band could be an integer for smaller
sample sizes.
## References
* (1) A. Holz, Il Nuovo Cimento B 9, 83 (1972).
* (2) L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 96, 155901 (2006).
* (3) Y. Kagan and L. A. Maksimov, Phys. Rev. Lett. 100, 145902 (2008).
* (4) J.-S. Wang and L. Zhang, Phys. Rev. B. 80, 012301 (2009).
* (5) L. Zhang, J.-S. Wang, and B. Li, New Journal of Physics 11, 113038 (2009).
* (6) R. de L. Kronig, Physica (Amsterdam) 6, 33 (1939); J. H. Van Vleck, Phys. Rev. 57, 426 (1940); R. Orbach, Proc. R. Soc. A 264, 458 (1961).
* (7) Spin-Lattice Relaxation in Ionic Solids, edited by A. A. Manenkov and R. Orbach (Harper & Row, New York, 1966).
* (8) A. S. Ioselevich and H. Capellmann, Phys. Rev. B 51, 11446 (1995).
* (9) R. J. Hardy, Phys. Rev. 132, 168 (1963).
* (10) G. D. Mahan, Many-Particle Physics 3rd ed. (Kluwer Academic, New York, 2000).
* (11) M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984)
|
arxiv-papers
| 2010-08-03T04:47:00 |
2024-09-04T02:49:12.028898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lifa Zhang, Jie Ren, Jian-Sheng Wang, and Baowen Li",
"submitter": "Lifa Zhang",
"url": "https://arxiv.org/abs/1008.0458"
}
|
1008.0471
|
# Stress Orientation Confidence Intervals from Focal Mechanism Inversion
S. A. Revets School of Earth & Environment
University of Western Australia
35 Stirling Highway M004
Crawley WA6008, Australia stefan.revets@uwa.edu.au
###### Abstract.
The determination of confidence intervals of stress orientation is a crucial
element in the discussion of homogeneity or heterogeneity of the stress field
under study. The error estimates provided by the grid search method Focal
Mechanism Stress Inversion of Gephart and Forsyth (1984) have been shown to be
too wide but the reasons for this failure have escaped elucidation.
Through the use of directional statistics and synthetic focal mechanisms, I
show that the grid search methodology does yield appropriate uncertainty
estimates. The direct perturbation of the synthetic focal mechanisms
introduces bias which leads to confidence intervals which become increasingly
too wide as the amount of perturbation increases. The synthetic data also show
at what point the method fails to overcome this bias and when confidence
intervals will be too wide. The indirect perturbation of the focal mechanisms
by perturbing the generating deviatoric stress tensor generates synthetic data
devoid of bias. Inversion of these data sets yields correct confidence
intervals.
The Focal Mechanism Stress Inversion method is vindicated as a highly
effective method, and with the use of appropriate directional statistics, its
results can be assessed and homogeneity or heterogeneity of the stress field
can be discussed with confidence.
I thank the Australian Research Council, the Australian National University,
Woodside Petroleum and Geoscience Australia for supporting this research
through ARC Linkage Grant LP0560955.
## 1\. Introduction
Gephart and Forsyth (1984) proposed a grid search method of inverting focal
mechanisms to obtain the stress tensor (focal mechanisms stress inversion,
henceforward FMSI), in which stress field parameters are tried systematically
against the focal mechanism orientations and the misfit calculated. They
defined the misfit as the sum of the minimum angle needed to bring the slip
direction of each focal mechanism into line with the resolved shear stress on
the fault plane. Both planes of each focal mechanism are tried and the
smallest misfit serves to differentiate fault plane from auxiliary plane. They
incorporated the one-norm measure as misfit criterion, adopting the
methodology proposed by Parker and McNutt (1980). Different techniques for
such inversions had been proposed already, and Gephart (1990b) discussed their
relative merits, particularly in the context of focal mechanisms and the
problems associated with the presence of two nodal planes.
Michael (1987) proposed a different method, relying on a linearisation and
bootstrapping to invert focal mechanisms for stress tensor calculation. He
drew attention to some differences in the size of the confidence intervals
obtained by these two different methods.
It became quickly apparent that the discrepancies in confidence intervals had
major repercussions on the study of actual stress fields. Different methods
led different groups of researchers to different conclusions regarding the
spatial and temporal variation in the stress regime in Southern California.
This unsatisfactory state of affairs led Hardebeck and Hauksson (2001) to test
thoroughly the different methodologies using synthetic data. They showed that
the confidence intervals estimated by FMSI were much too large, but they did
not succeed in determining the reasons for such overestimates.
Here, I attempt to uncover the nature of the confidence intervals estimated by
FMSI by investigating the Hardebeck and Hauksson (2001) analysis. Initial
attempts to resolve the problem of these too wide confidence intervals brought
to light some inaccuracies in the derivation and application of the one-norm
measure as proposed by Parker and McNutt (1980) and applied by Gephart and
Forsyth (1984) and Hardebeck and Hauksson (2001) (Revets, 2009). These
corrections proved insufficient to lessen significantly the width of the
confidence intervals (Hardebeck, pers. comm.).
This left two avenues of investigation: the nature of the statistics used to
calculate the confidence intervals, and the methodology of calculating the
synthetic data used to test the method.
## 2\. Directional statistics
Fisher (1953) pointed out that the initial development of the theory of errors
by Gauss aimed to help astronomers and surveyors with their accurate angular
measurements. Because of the accuracy of their measurements, the linear
approximation to which Gauss resorted was both appropriate and effective.
However, when the errors become substantial, the linear approximation is no
longer valid and the topological framework has to be taken into account.
The spherical mean direction $R$ and variance $S$ are defined as
(1)
$R^{2}=\big{(}\sum_{i=1}^{N}l_{i}\big{)}^{2}+\big{(}\sum_{i=1}^{N}m_{i}\big{)}^{2}+\big{(}\sum_{i=1}^{N}n_{i}\big{)}^{2}$
and
(2) $S=(N-R)/N$
where $l_{i}$, $m_{i}$, $n_{i}$ are the direction cosines of the angular
measurements (Mardia, 1972).
The spherical equivalent of the normal distribution is the Fisher distribution
(Fisher, 1953), defined by
(3) $df=\frac{\kappa}{2\sinh\kappa}e^{\kappa\cos\theta}\sin\theta d\theta$
in which $\kappa$ is a measure of concentration. When $\kappa$ approaches
zero, the distribution becomes uniform over the entire sphere whereas for
$\kappa$ large, the distribution is confined to a small region of the sphere
around the maximum. In the latter case, the distribution comes close to a two-
dimensional (isotropic) normal distribution where $\kappa$ plays the role of
the inverse of the variance.
The maximum likelihood estimate of $\kappa$ for the distribution on a sphere
is for larger values of $\kappa$
(4) $\mathrm{bar}{R}=\frac{I_{1.5}(\kappa)}{I_{0.5}(\kappa)}$
where $I_{n}$ is the modified Bessel function of the first kind and the n-th
order (Watson, 1944) and $\mathrm{bar}{R}$ the mean of $R$ (Watson and
Williams, 1956; Mardia, 1972). For large values of $\kappa$, the most likely
value is given by
(5) $\kappa=\frac{N-1}{N-R}$
(Fisher, 1953; Watson, 1960).
Confidence intervals for $\kappa$ can be determined from the relation
(6) $2\kappa(N-R)=\chi^{2}_{2N-2}$
(Watson and Williams, 1956; Mardia, 1972).
## 3\. Application to stress tensor inversion
To calculate and test the confidence intervals of stress tensors inverted from
focal mechanisms, I adapted and modified the method used by Hardebeck and
Hauksson (2001), taking particular care to adhere to the strictures and
requirements of directional statistics.
### 3.1. Generating synthetic focal mechanisms
Sets of focal mechanisms are made up from spherically randomly oriented fault
planes with slip directions determined by a chosen stress tensor. The azimuth
of the fault planes is chosen from uniformly distributed random numbers in the
$[0,2\pi]$ interval, while the dip is randomly taken out of the $[0,1]$
interval through
(7) $d_{i}\in\arccos[0,1]$
to avoid the (spherical) bias that the direct selection from the $[0,\pi/2]$
interval would cause. Sets contain either 20, 50 or 100 fault planes.
The direction of slip on the fault plane given the stress tensor follows from
the formalism of Ramsey and Lisle (1983)
(8) $l(m^{2}\phi+n^{2}):m(n^{2}(1-\phi)-l^{2}\phi):-n(l^{2}+m^{2}(1-\phi))$
where $l$, $m$, $n$, are the direction cosines of the fault plane in the
stress tensor reference frame and $\phi$ is the stress shape ratio
(9) $\phi=\frac{\sigma_{1}-\sigma_{2}}{\sigma_{1}-\sigma_{3}}$
This direction of slip, as the normal to the auxiliary plane, completes the
definition of the focal mechanism.
### 3.2. Generating error
I used a Fisher distribution to generate random directions and angles to
perturb orientations (Mardia, 1972, p. 231), and used quaternions to carry out
the required rotations. Figure 1 shows an example of such Fisher distributed
error angles and orientations plotted on a polar Schmidt net. Values of
$\kappa$ for the Fisher distribution which correspond to perturbations of 1 °,
5 °, 10 °, 15 ° and 20 ° can be obtained through interpolation from equation
4.
Figure 1. Polar Schmidt plots of examples of Fisher distributed errors, as
used to perturb the generating stress tensor. The crosses are 10 ° apart
There are two ways in which to incorporate errors in the synthetic data set.
There is the natural way of perturbing each focal mechanism directly. It is
also possible to perturb the generating stress tensor before calculating the
slip direction on each fault plane in the data set. The assumptions behind
these two ways are very different, and I discuss their implications,
statistical as well as physical, following the simulation results.
### 3.3. Data inversion and statistics
Each synthetic data set is then processed by FMSI, which inverts the focal
mechanisms to obtain the stress tensor. Thanks to FMSIETAB, part of the FMSI
suite of programs, it is possible to generate a list of angular deviation
between each fault plane and any given stress tensor. Such lists, comparing
the deviations between focal mechanisms with the generating stress tensor as
well as with the inverted stress tensor allow the calculation of the
respective spherical mean $R$ and the concentration parameter $\kappa$, using
equations 1 and 5. Each combination of $N$ fault planes with the different
amounts of perturbation is replicated 50 times.
Hardebeck and Hauksson (2001) proposed that for appropriate confidence
intervals the correct stress tensor should fall within the P confidence region
for an approximate proportion P of all the data sets. A plot of (sorted)
probabilities versus proportion should be a diagonal. If the confidence
intervals are too wide, the plot will show a curve above the diagonal and too
narrow confidence intervals will result in a curve below the diagonal. These
graphs are an equivalent of P-P plots (Wilk and Gnanadesikan, 1968; Holmgren,
1995). P-P plots are scatter plots of $F_{1}(q_{i}),F_{2}(q_{i})$, where
$F_{1}(q_{i})$ is obtained by applying
(10) $F_{1}(F_{2}^{-1}(p_{i}))$
to the two empirical cumulative density functions ($F$) of the two data sets
being compared. Here, I show P-P plots which compare the cumulative density
function of the misfit against the inverted stress tensor with the cumulative
density function of the misfit against the generating stress tensor.
Gephart (1990b) proposed a modification of the method proposed by Zizicas
(1955) to represent the orientation of a fault plane with respect to the
principal stress directions. Gephart’s unscaled Mohr Sphere uses the stress
components $\sigma$, $\tau_{S}$ and $\tau_{b}$ as axes.
The maximum shear stress and mean normal stress are respectively
(11)
$\tau_{m}=\frac{\sigma_{1}-\sigma_{3}}{2},\quad\sigma_{m}=\frac{\sigma_{1}+\sigma_{3}}{2}$
The absolute values of the stress tensor are inaccessible, but the relative
magnitudes of the principal stress components can be calculated. Their
relation is given by
(12) $\phi=\frac{\sigma_{2}-\sigma_{1}}{\sigma_{3}-\sigma_{1}}$
The three normalized stress components acting on a fault plane can be
described as
(13)
$\sigma=\frac{\sigma^{\prime}_{11}-\sigma_{m}}{\tau_{m}},\quad\tau_{b}=\frac{\sigma^{\prime}_{12}}{\tau_{m}},\quad\tau_{s}=\frac{\sigma^{\prime}_{13}}{\tau_{m}}$
and are completely determined by the dimensionless quantities $\phi$ and
$\beta_{ij}$, where $\beta_{ij}$ is determined from
(14) $\sigma^{\prime}_{ij}=\sigma_{kl}\beta_{ik}\beta{jl}$
which relates the stress tensor components between the coordinate systems of
the fault plane and of the regional stress tensor. The shear stress will match
the slip direction on a plane when $\sigma^{\prime}_{12}$ is zero (Gephart,
1990b).
Mohr Sphere plots of fault planes are often illuminating and are of great
assistance with the interpretation of the inversions. This is also the case
with the differently treated synthetic data sets, and I include a number of
such Mohr Sphere plots for some of the data used.
## 4\. Results
### 4.1. P-P Plots
The P-P plots obtained from the new analyses presented here (Figure 2–4) show
a considerable improvement for the method compared to the results obtained by
Hardebeck and Hauksson (2001). It appears that the change from linear
statistics to directional statistics improves the appropriateness of the
confidence interval estimation significantly. The graphs show a number of
trends, including some unexpected ones.
(a) Focal mechanism perturbation (b) Stress tensor perturbation
Figure 2. P–P plot of $\kappa$ for $N$=20, with the generating tensor values
along the x-axis and the inverted tensor values along the y-axis. The amount
of perturbation in degrees is shown on each individual curve.
The most obvious, and expected, trend is a decrease in precision of the
confidence interval estimates as the amount of error increases. This trend is
present in each individual graph (Figures 2–4).
(a) Focal mechanism perturbation (b) Stress tensor perturbation
Figure 3. P–P plot of $\kappa$ for $N$=50, with the generating tensor values
along the x-axis and the inverted tensor values along the y-axis. The amount
of perturbation in degrees is shown on each individual curve.
What is unexpected is that if the estimate of the confidence intervals is
imprecise, it is systematically too large. In the face of error, the method
performs better than one would expect, something that was very pronounced in
the study by Hardebeck and Hauksson (2001).
(a) Focal mechanism perturbation (b) Stress tensor perturbation
Figure 4. P–P plot of $\kappa$ for $N$=100, with the generating tensor values
along the x-axis and the inverted tensor values along the y-axis. The amount
of perturbation in degrees is shown on each individual curve.
The other noticeable and expected trend is an increase in precision of the
confidence interval estimates as the number of focal mechanisms increases.
This trend is very pronounced in graphs of the stress tensor perturbation set
(Figures 2b, 3b and 4b). The graphs of the focal mechanism perturbation set
(Figures 2a, 3a and 4a) show an increase in precision for the smaller amounts
of error, but an unexpected decrease for the larger amounts of error.
There is also a clear tendency for an increased precision for the stress
tensor perturbation sets against their focal mechanism perturbation
equivalents (the (b) subfigure against the (a) subfigure for each of Figures
2, 3 and 4).
### 4.2. Mohr Sphere Projections
(a) Generating Stress Tensor (b) Inverted Stress Tensor
Figure 5. Mohr Sphere projections of the poles of the fault planes relative to
the stress components defined by the respective stress tensor. The fault
planes have been subjected to a 1 ° Fisher distributed perturbation.
(a) Generating Stress Tensor (b) Inverted Stress Tensor
Figure 6. Mohr Sphere projections of the poles of the fault planes relative to
the stress components defined by the respective stress tensor. The stress
tensor has been subjected to a 1 ° Fisher distributed perturbation.
The Mohr Sphere projections (Figures 5–8) are examples of 4 data set of 50
focal mechanisms which have undergone either a 1 ° (Figures 5 and 6) or a 15 °
(Figures 7 and 8) Fisher distributed amount of error. Each figure shows two
sets of Mohr Sphere plots: one illustrating the fault planes against the
generating stress tensor (or its spherical average when it was the generating
stress tensor that underwent perturbation), and one showing the same set of
fault planes against the inverted stress tensor calculated by FMSI.
(a) Generating Stress Tensor (b) Inverted Stress Tensor
Figure 7. Mohr Sphere projections of the poles of the fault planes relative to
the stress components defined by the respective stress tensor. The fault
planes have been subjected to a 15 ° Fisher distributed perturbation.
(a) Generating Stress Tensor (b) Inverted Stress Tensor
Figure 8. Mohr Sphere projections of the poles of the fault planes relative to
the stress components defined by the respective stress tensor. The stress
tensor has been subjected to a 15 ° Fisher distributed perturbation.
The first trend which stands out is the larger amount of scatter on the Mohr
Sphere projections for the fault plane perturbed data sets (Figures 5 and 7)
compared to the projections for the stress tensor perturbed data sets (Figures
6 and 8) which is particularly noticeable on the $\sigma-\tau_{b}$ graphs.
This increase in scatter includes the presence of fault planes with the wrong
direction of slip ($\tau_{S}<0$, seen as crosses which plot below the X-axis
in the $\sigma-\tau_{S}$ and $\tau_{b}-\tau_{S}$ graphs).
A second trend is the larger amount of change between the Mohr Sphere plots
(going from the plot of the fault planes in the generating stress tensor Mohr
Sphere to the Inverted tensor Mohr Sphere) for the data sets that have
undergone fault plane perturbations compared to the data sets that have
undergone stress tensor perturbation. There is more change visible going from
the (a) subfigures to the (b) subfigures in Figures 5 and 7 than in Figures 6
and 8.
## 5\. Discussion
Theoretical and statistical reasoning requires us to use directional
statistics instead of linear statistics when we deal with focal mechanisms and
deviatoric stress tensor inversion. The Fisher distribution and its properties
are well established and its use in the present, theoretical study has been
effective. The question does arise if this distribution is a good
representation of actual seismological data.
Discussions of the distribution of data misfit to stress tensors in the
literature (Gephart and Forsyth, 1984; Hardebeck and Hauksson, 2001; Wyss et
al., 1992) consistently mention the non-normality of the data and refer
qualitatively to exponential or Poisson distributions. Turning to spherical
statistics resolves this issue immediately. The Fisher distribution, as given
in equation 3, shows directly its exponential nature. The method adopted by
Hardebeck and Hauksson (2001) for their simulations, in which they combine
exponentially distributed dip angles with uniformly distributed azimuth
angles, generates in effect a set of Fisher distributed orientations.
Since we are trying to assess the confidence intervals of the stress tensor
inversion through simulations, we have to ensure that we use appropriate
models and calculations to compare the perturbation distributions. It is
highly instructive to compare the effects of injecting uncertainty into the
simulations at two different points. It is for this reason that I opted to
inject uncertainty or error in the focal mechanisms to test the inversion
process by perturbing the orientation of the generating stress tensor, as well
as the more intuitive approach of perturbing the orientation of the individual
focal mechanisms directly.
The fault planes are chosen from uniform spherical random orientations
(equation 7) and the direction of slip is calculated from the stress tensor
according to equation 8. It is clear that the relation between orientation of
any of the focal mechanisms and the stress tensor is highly non-linear. That
non-linearity necessarily also applies to any perturbation and in particular
to the way in which it propagates in any calculation or inversion. It is
incorrect to assume that a perturbation distribution applied to a set of focal
mechanisms would translate to the same perturbation distribution of the
inverted stress tensor. This effect can be clarly seen in the Mohr Sphere
plots where the scatter is much larger in the plots of the data sets subjected
to fault plane perturbation (Figures 5 and 7) than those subjected to tensor
perturbation (Figures 6 and 8).
Comparing the dispersions of the data misfit to the inverted stress tensor
with the dispersion of the perturbations in this case will be misleading. The
dispersion of the stress tensor orientations (consistent with the perturbed
focal mechanisms) is necessarily larger than the dispersion used perturb the
focal mechanisms. It is therefore not surprising that the inversion method
will converge to the correct stress tensor more often than one would expect
from the dispersion used. This problem can be avoided by letting the
perturbation occur on the generating tensor, before generating the focal
mechanisms. This effect is clearly shown by the contrast between the P-P plots
of the data sets subjected to fault plane perturbation and the data sets
subjected to stress tensor perturbation (respectively the (a) and (b) subplots
in Figures 2–4).
Perturbing the fault planes directly in effect introduces bias into the data
set because each of the perturbed fault planes appears to have been generated
by a potentially very different stress tensor, including incompatibly oriented
stress tensors. This bias contaminates the dispersion. Using this contaminated
dispersion to measure the accuracy of the confidence intervals then leads to
the erroneous conclusion that the inversion method overestimates the
confidence intervals.
As far as simulations are concerned, the statistics can only legitimately be
compared if they are commensurable. That is not the case when error is
injected at the level of the individual fault plane. Unfortunately, this
situation mirrors exactly what happens in the real world: the parameters
describing a calculated focal mechanism are subject to error and uncertainty,
not the generating stress tensor. Therefore, it may seem that the fact that
FMSI yields correct confidence intervals only when the generating stress
tensor is subjected to error, but fails to do so when focal mechanisms are
error prone, is of no practical benefit.
The paradox can be resolved as follows. I have demonstrated that FMSI yields
correct confidence intervals when the dispersions used to calculate the
confidence intervals are commensurable. I have also shown that error in focal
mechanisms introduces bias in the statistics of the population at hand. But
thanks to the side-by-side simulations, it is possible to estimate to what
extent this bias contaminates the calculations of FMSI. As one would expect
from the law of large numbers (Révész, 1968), a statistic converges to its
theoretical value as the population size increases. This can be seen very
clearly in the P-P plots: the results of the simulations converge to the
diagonal as the sample size increases. This is very obvious for the
simulations in which the generating stress tensor was subjected to error (the
(b) subplots in Figures 2–4). Closer scrutiny of the P-P plots of the
simulations in which the focal mechanisms were subjected to perturbation shows
a similar, albeit slower convergence (the (a) subplots in Figures 2–4). There
is a trade-off between amount of error and population size: FMSI is capable of
estimating confidence intervals correctly up to a certain amount of bias.
Figure 2a shows that for populations of 20 focal mechanisms, FMSI is capable
of dealing with an average of 5 ° of error on the focal mechanisms. This
increases to 10 ° for 50 focal mechanisms (Figure 3a) and to 12–13 ° for 100
focal mechanisms (Figure 4a).
Figure 9. Empirical relation between the $\mathrm{bar}{R}$ values and mean
absolute deviation ( °) for the Fisher Distribution
The calculations by FMSI do not use directional statistics and the questions
arises if a simple conversion is possible from the dispersion measure given by
FMSI to directional statistics and allow the application of the correct
calculation of confidence intervals as shown here. The mean deviation
calculated by FMSI is an average of angles while the circular mean is the
average of the cosines of angles. There is no analytical formula to go from
one to the other. Nevertheless, there is a clear relation between the mean
absolute deviation and $\mathrm{bar}{R}$ for the Fisher Distribution (see
Figure 9) and graphs of this nature could be used to estimate values which can
then be used to determine the confidence intervals. The correct procedure,
albeit more calculation intensive, is to use the list of individual focal
mechanism misfits (which can be obtained through FMSIETAB) and to apply
equation 6.
## 6\. Conclusions
The processing and discussion of data from focal mechanisms and deviatoric
stress tensors should use directional statistics.
The genesis of synthetic data in the context of stress tensor inversion
requires very careful scrutiny of how perturbation is incorporated.
The inversion of deviatoric stress tensors from focal mechanisms using the
grid search method of Gephart yields reliable confidence intervals if correct,
directional statistics are used. When calculated misfits are small for
moderate to large numbers of focal mechanisms, the method recovers the true
stress tensor and confidence intervals become superfluous.
Gephart’s FMSI is a highly effective and reliable method for stress tensor
inversion.
## 7\. Data and Resources
The synthetic data, data inversion and additional calculations relied on a
combination of Bash shell scripts and procedures written for the Octave
program (Eaton, 2002). The FMSI suite of programs is in the public domain and
made available by Gephart (1990a). The fortran source code is freely available
from www.geo.cornell.edu/pub/FMSI.
## References
* Eaton (2002) Eaton, J. W. (2002). GNU Octave Manual, Network Theory Limited.
* Fisher (1953) Fisher, R. A. (1953). Dispersion on a sphere, Proc. Roy. Soc. Lond., Ser. A 217, 295–305.
* Gephart (1990a) Gephart, J. W. (1990a). FMSI: a FORTRAN program for inverting fault/slickenslide and earthquake focal mechanism data to obtain the regional stress tensor, Comp. Geosci. 16, 953–989.
* Gephart (1990b) Gephart, J. W. (1990b). Stress and the direction of slip on fault planes, Tectonics 9, no. 4, 845–858.
* Gephart and Forsyth (1984) Gephart, J. W., and D. W. Forsyth (1984). An improved method for determining the regional stress tensor using earthquake focal mechanism data: application to the San Fernando earthquake sequence, J. Geophys. Res. 89, no. B11, 9305–9320.
* Hardebeck and Hauksson (2001) Hardebeck, J. L., and E. Hauksson (2001). Stress orientations obtained from earthquake focal mechanisms: what are appropriate uncertainty estimates?, Bull. Seismol. Soc. Am. 91, no. 2, 250–262.
* Holmgren (1995) Holmgren, E. B. (1995). The P-P plot as a method for comparing treatment effects, J. Amer. Stat. Assoc. 90, no. 429, 360–365.
* Mardia (1972) Mardia, K. V. (1972). Statistics of Directional Data, Academic Press, London.
* Michael (1987) Michael, A. J. (1987). Use of focal mechanisms to determine stress: a control study, J. Geophys. Res. 92, no. B1, 357–368.
* Parker and McNutt (1980) Parker, R. L., and M. K. McNutt (1980). Statistics for the one-norm misfit measure, J. Geophys. Res. 85, no. B8, 4429–4430.
* Ramsey and Lisle (1983) Ramsey, J. G., and R. Lisle (1983). The techniques of Modern Structural Geology, vol. 3. Applications of Continuum Mechanics in Structural Geology, Academic Press, London.
* Révész (1968) Révész, P. (1968). The Law of Large Numbers, Probability and Mathematical Statistics, Academic Press, New York.
* Revets (2009) Revets, S. A. (2009). One-norm misfit statistics, Geophys. Res. Lett. 36, no. L20302.
* Watson (1944) Watson, G. N. (1944). A treatise on the Theory of Bessel Functions, Second Ed., Cambridge University Press.
* Watson (1960) Watson, G. S. (1960). More significance test on the sphere, Biometrika 47, 87–91.
* Watson and Williams (1956) Watson, G. S., and E. J. Williams (1956). On the construction of significance tests on the circle and the sphere, Biometrika 43, 344–352.
* Wilk and Gnanadesikan (1968) Wilk, M. B., and R. Gnanadesikan (1968). Probability plotting methods for the analysis of data, Biometrika 55, no. 1, 1–17.
* Wyss et al. (1992) Wyss, M., B. Y. Liang, W. R. Tanigawa, and X. P. Wu (1992). Comparison of orientations of stress and strain tensors based on fault plane solutions in Kaioki, Hawaii, J. Geophys. Res. 97, no. B4, 4769–4790.
* Zizicas (1955) Zizicas, G. A. (1955). Representation of three-dimensional stress distributions by Mohr circles, J. Appl. Mech. 22, 273–274.
|
arxiv-papers
| 2010-08-03T07:25:08 |
2024-09-04T02:49:12.036197
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefan A. Revets",
"submitter": "Stefan Revets",
"url": "https://arxiv.org/abs/1008.0471"
}
|
1008.0481
|
# Analytic Theory of Edge Modes in Topological Insulators
Shijun Mao1,2 Yoshio Kuramoto1 Ken-Ichiro Imura1,3 and Ai Yamakage1 Department
of PhysicsDepartment of Physics Tohoku University Tohoku University Sendai
980-8578 Sendai 980-8578 Japan 1
Department of Physics Japan 1
Department of Physics Tsinghua University Tsinghua University Beijing 100084
Beijing 100084 P.R.China 2
Department of Quantum Matter P.R.China 2
Department of Quantum Matter AdSM AdSM Hiroshima University Hiroshima
University Higashi-Hiroshima 739-8530 Higashi-Hiroshima 739-8530 Japan 3 Japan
3
###### Abstract
Spectrum and wave function of gapless edge modes are derived analytically for
a tight-binding model of topological insulators on square lattice. Particular
attention is paid to dependence on edge geometries such as the straight (1,0)
and zigzag (1,1) edges in the thermodynamic limit. The key technique is to
identify operators that combine to annihilate the edge state in the effective
one-dimensional (1D) model with momentum along the edge. In the (1,0) edge,
the edge mode is present either around the center of 1D Brillouin zone or its
boundary, depending on location of the bulk excitation gap. In the (1,1) edge,
the edge mode is always present both at the center and near the boundary.
Depending on system parameters, however, the mode is absent in the middle of
the Brillouin zone. In this case the binding energy of the edge mode near the
boundary is extremely small; about $10^{-3}$ of the overall energy scale.
Origin of this minute energy scale is discussed.
zigzag lattice, effective Hamiltonian, flat edge spectrum, reentrant edge mode
## 1 Introduction
Recently, a new class of topological insulator (TI), also referred to as
quantum spin Hall (QSH) insulator, has been attracting much interest in both
theory [1, 2, 3, 4, 5, 6, 7] and experiments [8, 9, 10]. Different from
conventional insulators, QSH insulators have topologically protected helical
edge states with the spectrum lying in the bulk insulating gap.[1, 12, 13]
Two-dimensinal (2D) topological insulator has been realized in HgTe/CdTe
quantum wells[8] following theoretical suggestion of Bernevig, Hughes and
Zhang[6] who proposed a useful model, hereafter referred to as the BHZ Model.
Helical edge state has been further studied using the resultant continuum
model.[14, 15] The BHZ model is not realistic away from the zone center, since
it is based on the $\mib k\cdot\mib p$ perturbation theory and envelope-
function approximation. However, a regularization of the model using the
tight-binding scheme has the simple structure for the whole Brillouin zone
(BZ), and is suitable for studying the general property of QSH systems. Note
that characterization of the topological property of the system requires
information of wave functions over the whole BZ.
In this paper, we derive the spectrum of edge modes analytically taking the
tight-binding version of the BHZ model for the square lattice. In contrast to
previous study of edge modes,[11, 14] we take not only the straight (1,0) edge
but also the zigzag (1,1) edge, and make detailed comparison of respective
edge modes. This comparison is partially inspired by the remarkable difference
between the zigzag and armchair edges in graphene [16, 17]. In contrast to the
latter, however, the edges in the present model should not be taken as
representing the HgTe/CdTe quantum wells since the short-distance behavior of
the model is not realistic. Nevertheless, the intuition gained by the exact
solution of the simplified model should provide useful information for
understanding more complicated systems.
In a separate paper[18], we have already derived the spectrum of both straight
and zigzag edges relying partially on numerical method. In particular, we have
found for the zigzag edge a reentrant mode with tiny binding energy. Since the
relevant energy is so small as compared with the overall energy, it is
desirable to characterize its nature analytically. Then the origin of tiny
binding energy should be clarified. In the present paper, we derive fully
analytic expressions of not only the spectrum for both edges, but also
momentum regions allowed for the modes. Namely we derive critical momentum
where the edge mode merges with bulk excitations. Our analytical method is a
systematic generalization of previous ones [19, 11] so that the spectrum can
be obtained for general direction of the edge.
This paper is organized as follows: In §2, we review the lattice version of
the BHZ model with nearest-neighbor transfer, paying attention to its symmetry
property in the 2D BZ. Sections 3 and 4 are devoted to analytic derivation of
edge modes for straight and zigzag edges, respectively. We use a systematic
method to derive the spectrum in terms of the “annihilator”. Existent regions
of edge modes are derived for both straight and zigzag edges. In the zigzag
case, we find a novel reentrant edge mode with a tiny binding energy below the
bulk spectrum. Finally, we summarize the results and discuss their implication
in Sec. 5.
## 2 Model with particle-hole symmetry
We consider the BHZ model given by the following $4\times 4$ matrix:
$\displaystyle H(\mib k)=\left[\begin{array}[]{cc}h(\mib k)&0\\\ 0&h^{*}(-\mib
k)\\\ \end{array}\right],$ (3)
where $\mib k=(k_{x},k_{y})$ is a 2D crystal momentum, measured from
$\Gamma$-point. The lower-right block $h^{*}(-\mib k)$ is a $2\times 2$
matrix, and is deduced from the upper-left block $h(\mib k)$ by time reversal
transformation. $h(\mib k)$ is parametrized as,
$h(\mib k)=\mib d(\mib
k)\cdot\mib\sigma=\left[\begin{array}[]{cc}d_{z}&d_{x}-id_{y}\\\
d_{x}+id_{y}&-d_{z}\end{array}\right],$ (4)
where $d_{i}(\mib k)$ are given by
$\displaystyle d_{x}({\mib k})=A\sin k_{x},\quad d_{y}({\mib k})=A\sin k_{y},$
(5) $\displaystyle d_{z}({\mib k})=\Delta-2B(2-\cos k_{x}-\cos k_{y}),$ (6)
with the lattice constant $a$ set to unity[6]. We only consider the case
$A>0,\ B>0$ and $\Delta\geq 0$ in this paper. The same signs of $B$ and
$\Delta$ are necessary for topological insulator. The solution for $A<0$ is
trivially obtained from that for $H(-\mib k)$. Each row and column of eq. (4)
represent spin-orbital states associated with the $s$-type $\Gamma_{6}$ and
the $p$-type $\Gamma_{8}$ orbitals of the 3D band structure of HgTe and CdTe.
Other parameters which appear in Ref.BHZ, i.e., $C$ and $D$ have been set to
zero. As a result, the spectrum has the particle-hole symmetry that simplifies
the analysis.
The $2\times 2$-matrix Hamiltonian $h(\mib k)$ is equivalent to the following
tight-binding Hamiltonian on the square lattice:
$\displaystyle h_{\uparrow}$
$\displaystyle=\sum_{I,J}c_{I,J}^{\dagger}\hat{\cal E}c_{I,J}$
$\displaystyle+\left(c_{I,J}^{\dagger}\hat{t}_{x}c_{I+1,J}+c_{I,J}^{\dagger}\hat{t}_{y}c_{I,J+1}+h.c.\right),$
(7)
where we have introduced for each site $(I,J)$ the two-component field:
$\displaystyle c^{\dagger}=(c^{\dagger}_{s\uparrow},c^{\dagger}_{p\uparrow})$
(8)
with orbitals $s,p$. The tight-binding parameters are given by
$\displaystyle\hat{\cal E}$ $\displaystyle=(\Delta-4B)\sigma_{z},$ (9)
$\displaystyle\hat{t}_{x}$ $\displaystyle=-i{A\over 2}\sigma_{x}+B\sigma_{z},$
(10) $\displaystyle\hat{t}_{y}$ $\displaystyle=-i{A\over
2}\sigma_{y}+B\sigma_{z},$ (11)
each of which is a $2\times 2$ matrix. The down spin part $h_{\downarrow}$
corresponding to the lower-right block of eq.(3) is obtained by replacing
$\hat{t}_{i}$ with its complex conjugation $\hat{t}_{i}^{*}$.
We consider the BHZ model over the whole BZ of the square lattice. Bulk energy
$E_{\rm b}({\mib k})$ is written as
$\displaystyle E_{\rm b}({\mib k})$
$\displaystyle=\pm\left\\{A^{2}\left(\sin^{2}k_{x}+\sin^{2}k_{y}\right)\right.$
$\displaystyle+\left.\left[\Delta-4B+2B\left(\cos k_{x}+\cos
k_{y}\right)\right]^{2}\right\\}^{1/2},$ (12)
which is symmetric with respect to positive (conduction) and negative
(valence) energy bands, being the signature of the particle-hole symmetry. As
one varies mass parameter $\Delta$, there appear four gap closing points,
namely at $\Gamma=(0,0)$, $X=(\pi,0)$, $X^{\prime}=(0,\pi)$ and $M=(\pi,\pi)$.
Note that these points are invariant against time-reversal operation. Gap
closing at $\Gamma$ occurs when $\Delta=0$, whereas the gap closing at $X$ and
$X^{\prime}$ occurs simultaneously when $\Delta=4B$, at $M$ when $\Delta=8B$.
## 3 Straight edge
Figure 1: Straight edge lattice ribbon with two boundaries in (1,0) direction.
### 3.1 Effective one-dimensional form
Let us first consider the geometry (see Fig. 1), in which electrons are
confined to $N_{r}$ rows in a strip between $y=1$ and $y=N_{r}$, i.e., the two
edges are along $x$-axis. Translational invariance along $x$-axis allows for
constructing a 1D Bloch state with a crystal momentum $k=k_{x}$:
$|k,J\rangle=\frac{1}{\sqrt{N_{c}}}\sum_{I}e^{ikI}|I,J\rangle,$ (13)
where $I,J$ represents lattice site, $N_{c}$ is the number of sites along the
$x$-axis, and $|I,J\rangle=c_{I,J}^{\dagger}|0\rangle$. In order to introduce
the edges, it is convenient to rewrite eq. (7) in form of a hopping
Hamiltonian between neighboring rows. In terms of the two-component creation
and annihilation operators $c^{\dagger}_{J}(k)$ and $c_{J}(k)$ associated with
Bloch state (13), one can rewrite eq. (7) as
$\displaystyle h_{\uparrow}$ $\displaystyle=\sum_{k}h_{01}(k),$ (14)
$\displaystyle h_{01}(k)$ $\displaystyle=\sum_{J}c^{\dagger}_{J}(k)\hat{\cal
E}(k)c_{J}(k)$
$\displaystyle+\sum_{J}\left[c^{\dagger}_{J}(k)\hat{t}_{y}c_{J+1}(k)+h.c.\right]$
(15)
where $\hat{\cal E}(k)$ is given by
$\hat{\cal E}(k)=A\sin k\sigma_{x}+\left(\Delta_{B}+2\cos k\right)\sigma_{z},$
(16)
Here and in the following, we use the notation $\Delta_{B}\equiv\Delta-4B$,
and take the energy unit so that $B=1$. The corresponding Schrödinger equation
is given by
$\hat{\cal
E}(k)\Psi_{J}+\hat{t}_{y}\Psi_{J-1}+\hat{t}_{y}^{\dagger}\Psi_{J+1}=E_{\uparrow}(k)\Psi_{J}$
(17)
where $\Psi_{J}$ is the two-component amplitude with row index $J$. The
straight edges along the $J=1$ row and $J=N_{r}$ row can be implemented by
open boundary condition $\Psi_{0}=\Psi_{N_{r}+1}=0$.
It has been shown [14] in the continuum approximation of eq.(17) that there
appear modes, which are localized on both edges of the system, with the
particle-hole symmetric spectrum and a minimum gap at $k=0$. The origin of the
gap is the overlap of wave functions on different edges. As $N_{r}$ tends to
infinity, the gap disappears since the overlap becomes zero. In this limit one
can identify edge modes localized on either edge. In the thermodynamic limit,
the spectrum $E(k)$ of an edge mode near $J=1$ becomes an odd function of $k$,
and the other edge mode has the spectrum $-E(k)$. The continuum approximation
of Ref.cm has a limited validity, but emergence of the zero mode at $k=0$ in
the thermodynamic limit $N_{r}\rightarrow\infty$ is guaranteed by the time-
reversal and particle-hole symmetries [19, 11]. In the following, we deal with
the thermodynamic limit.
### 3.2 Separation into Hermitian and annihilating parts
Our strategy to obtain the edge spectrum is best illustrated in the straight
edge. Although the spectrum in this case has already been obtained in the
literature [19, 11], we here present our way of derivation that can be
extended to the zigzag edge. We try an edge state solution with property:
$\Psi_{J+1}=\lambda\Psi_{J}\equiv\lambda^{J+1}\Psi$ with $|\lambda|<1$[19,
20], and derive the vector $\Psi$. Then eq.(17) can be written in the
following form:
$\left[\hat{\cal
E}(k)+\lambda\hat{t}_{y}^{\dagger}+\frac{1}{\lambda}\hat{t}_{y}\right]\Psi\equiv
P_{01}(\lambda,k)\Psi=E_{\uparrow}(k)\Psi,$ (18)
where $P_{01}(\lambda,k)$ can be rearranged as
$\displaystyle P_{01}(\lambda,k)=\mib\gamma\cdot\mib\sigma,$ (19)
with components
$\displaystyle\gamma_{x}$ $\displaystyle=A\sin k$ (20)
$\displaystyle\gamma_{y}$
$\displaystyle=i\frac{A}{2}\left(\lambda-\lambda^{-1}\right)$ (21)
$\displaystyle\gamma_{z}$
$\displaystyle=\left(\lambda+\lambda^{-1}\right)+\Delta_{B}+2\cos k$ (22)
For a general complex vector $\mib\gamma$, the eigenvalues of $P_{01}$ are
also complex. However, we obtain real energy $E_{\uparrow}(k)$ in eq.(18) if
one of the following conditions is met:
(a) all components of $\mib\gamma$ are real (including zero);
(b) the nonzero complex components combine to give zero when acting on the
edge state. Such combination of operators is referred to as annihilator.
The condition (a) is not relevant here, because we would then have two
eigenvalues $\pm E_{\uparrow}(k)$ and corresponding two eigenfunctions for a
given $k$. Actually we should have only one edge mode for $h_{\uparrow}(k)$.
Hence we have to accept the condition (b), and separate $P_{01}$ into a
Hermitian part that gives the spectrum by diagonalization, and the rest that
makes up the annihilator.
From eq.(21) and (22), coefficients $\gamma_{y}$ and $\gamma_{z}$ are both
real only if $|\lambda|=1$. On the other hand, due to time-reversal symmetry
and particle-hole symmetry of the system, the eigenvalue $E_{\uparrow}(k)$ is
an odd function of $k$ in the thermodynamic limit [11, 14]. Therefore,
$\sigma_{y}$ and $\sigma_{z}$ must both belong to the annihilator, and the
only component to be diagonalized is $\sigma_{x}$. The annihilator corresponds
to either $\sigma_{y}+i\sigma_{z}$ or $\sigma_{y}-i\sigma_{z}$, depending on
the eigenvalue $\pm 1$ of $\sigma_{x}$.
Accordingly, we decompose $P_{01}(\lambda,k)$ as
$\displaystyle P_{01}(\lambda,k)=H_{01}+F_{01},$ (23) $\displaystyle
H_{01}=A\sin k\,\sigma_{x},$ (24) $\displaystyle
F_{01}=\gamma_{y}\sigma_{y}+\gamma_{z}\sigma_{z},$ (25)
where $F_{01}$ should form the annihilator. Namely, we impose the relation
$\gamma_{z}=si\gamma_{y}$, according to the eigenvalue $s=\pm 1$ of
$\sigma_{x}$. The relation is equivalent to
$(1+\frac{sA}{2})\lambda+(1-\frac{sA}{2})\lambda^{-1}+m_{k}=0,$ (26)
where we have introduced the notation:
$\displaystyle m_{k}=\Delta_{B}+2\cos k.$ (27)
By solving eq.(26) with $A\neq 2$, we obtain
$\lambda_{\pm}(s)=\frac{-m_{k}\pm\sqrt{m^{2}_{k}+A^{2}-4}}{2+sA},$ (28)
with the relation
$\displaystyle\lambda_{\pm}(-s)=1/\lambda_{\mp}(s).$ (29)
Provided the eigenvalue equation
$\displaystyle H_{01}\Psi$ $\displaystyle=E_{\uparrow}(k)\Psi.$ (30)
is satisfied, then equality $F_{01}\Psi=0$ follows with eq.(26). In this way
the edge spectrum is simply derived as
$E_{\uparrow}(k)=sA\sin k.$ (31)
with the eigenstate written as
$\displaystyle\Psi(s)=\begin{pmatrix}1\\\ s\end{pmatrix}.$ (32)
Note that only one of $s=\pm 1$ is relevant, as discussed below.
The boundary condition $\Psi_{J=0}=0$ requires the edge state to have the form
$\displaystyle\Psi_{J}(s)=\left[\lambda_{+}(s)^{J}-\lambda_{-}(s)^{J}\right]\Psi(s),$
(33)
apart from the normalization factor. Since the wave function should decay as
$J$ increases, the edge state can only be realized with
$|\lambda_{\pm}(s)|<1$. Because of the relation eq.(29), there is at most one
$s$ for given $k$ that describes the edge mode with the spectrum eq.(31).
Hence there is only one edge state per spin and per momentum.
In a similar manner, we obtain another edge mode for $h_{\downarrow}$. The
corresponding energy $E_{\downarrow}(k)$ is given by
$\displaystyle E_{\downarrow}(k)=E_{\uparrow}(-k)=-E_{\uparrow}(k),$ (34)
where the first equality corresponds to the time-reversal symmetry. The
particle-hole symmetry connecting the rightmost and leftmost sides with the
same $k$ involves different spins. Note that the Kramers pair has the same
$\lambda_{\pm}(s)$ for given $k$ and $-k$. In certain range of $k$, however,
the edge modes do not exist. This problem is studied in detail in the next
section.
### 3.3 Allowed momentum range for edge modes
A pair of gapless edge modes per edge are always present in the topological
insulator phase (TI) that appears for $0<\Delta<8$, i.e., $|\Delta_{B}|<4$ in
the BHZ model. However, the number of zero points of helical edge states
should be an odd multiple of two, including the degeneracy, in the 1D BZ [12].
If the edge modes with the spectrum $E=\pm A\sin k$ were present for the whole
BZ, the zero points amount to four (an even multiple of two) which violates
the topological stability [21]. Hence the edge modes must merge into bulk
excitations at finite $\kappa_{m}$, and degenerate zero points occur either at
$k=0$ or $k=\pi$, but not at both. Note that $k=0$ and $k=\pi$ are two time
reversal invariant momenta in 1D BZ.
With $\Delta_{B}=0$, the energy gap closes at X points $(\pi,0)$ and $(0,\pi)$
in the 2D BZ, as seen from eq.(12). Let us classify the case of
$-4<\Delta_{B}<0$ as TI-1, and the case of $0<\Delta_{B}<4$ as TI-2. Figure 2
shows the spectrum of edge modes in the TI-1 (upper panel) and TI-2 (lower
panel). In the TI-1, the edge modes intersect at $k=0$, whereas in TI-2 they
meet at $k=\pi$. Also shown is approximate bulk spectrum in the system that is
derived from $E_{\rm b}(k,k_{y})$ by fixing $k_{y}$ to $2\pi n/N_{r}$ with
$n=1,\ldots,N_{r}$. Each curve for the bulk spectrum corresponds to integer
$n$.
Figure 2: 1D energy bands in the straight edge: TI-1 (top) with $\Delta=1.2$
and TI-2 (bottom) with $\Delta=4.5$. In both cases, we set $A=1$. The edge
modes have the spectrum $\pm A\sin k$ in the existent range of $k$.
We now derive the pair $\pm k_{m}$ of momentum where the edge mode merges with
bulk excitations. In the following, we always assume $k\geq 0$ for simplicity.
The merging occurs when the larger of $|\lambda_{\pm}(s)|$ becomes unity. Note
that such $\lambda_{\pm}(s)$ is real, since otherwise $|\lambda_{\pm}(s)|$ is
independent of $m_{k}$, and hence of $k$. For complex $\lambda_{\pm}(s)$ we
obtain from eq.(28),
$\displaystyle|\lambda_{\pm}(s)|^{2}=\frac{2-sA}{2+sA},$ (35)
which is less than unity only with $s=1$. In this section we deal with the
case $0<A<2$ that allows complex $\lambda_{\pm}(s)$. Then, edge modes are
assured to be present for such $k$ with complex $\lambda_{\pm}(1)$, which is
simply written as $\lambda_{\pm}$ hereafter. Edge modes in the case of $A>2$
will be discussed in §5.
#### TI-1 regime
In this case gapless points are present at $k=0$. Let us assume
$m_{k}=\Delta_{B}+2\cos k<0$ when merging occurs at $k=k_{m1}$, Then we obtain
$|\lambda_{+}|>|\lambda_{-}|$ for real $\lambda_{\pm}$, and the condition for
merging is reduced to
$\displaystyle\lambda_{+}=\frac{-m_{k}+\sqrt{m^{2}_{k}+A^{2}-4}}{2+A}=1.$ (36)
from eq.(28). Then we obtain
$\cos k_{m1}=1-\Delta/2,$ (37)
which justifies the assumption $m_{k}<0$. One can check that there is no
solution if we assume $m_{k}>0$. We note that eq.(37) has already been
obtained by König et al.[11]
The condition for merging is also to have the same energy as the lowest bulk
excitation for given $k$. The minimum of $E_{b}(k,k_{y})$ may occur either at
$k_{y}=0$ or $k_{y}=\pm\arccos\left[2m_{k}/(A^{2}-4)\right]$ depending on the
value of $k$. If the threshold of bulk excitations with $k=k_{m1}$ occurs at
$k_{y}=0$, merging momentum $k_{m1}$ is simply obtained from eq.(12) as
$\Delta_{B}+2+2\cos k=m_{k}+2=0,$ (38)
which is consistent with eq.(37). Namely, the condition $\lambda_{+}=1$ is
equivalent to having the same energy for edge mode and for the minimum of bulk
excitations. Furthermore, it is easily seen that the group velocity $A\cos
k_{m1}$ at merging point is common to both edge mode and the lowest bulk
excitation. Namely, the edge mode vanishes at such $k$ that it has the common
tangent with the threshold of bulk excitations $E_{b}(k,k_{y})$ with
$k_{y}=0$.
The crossings of 1D energy bands in Fig.2 indicates the transition from the
lowest bulk excitation $E_{b}(k,k_{y})$ with $k_{y}=0$ to with
$k_{y}=\pm\arccos\left[2m_{k}/(A^{2}-4)\right]$. The critical momentum
$k_{c1}$ satisfies the condition
$\partial^{2}E_{\rm b}(k_{c1},k_{y})/\partial k_{y}^{2}|_{k_{y}=0}=0,$ (39)
which gives the solution
$\cos k_{c1}=1-\frac{\Delta}{2}+\frac{A^{2}}{4}.$ (40)
The threshold has $k_{y}=0$ for $k>k_{c1}>0$. By comparing with eq.(37), we
find $\cos k_{c1}>\cos k_{m1}$, which means $0\leq k_{c1}<k_{m1}$. Namely,
merging with bulk excitations indeed occurs in the range where $k_{y}=0$
corresponds to the threshold.
#### TI-2 regime
In this case edge modes exist around $k=\pi$. Let us assume $m_{k}>0$ when
merging occurs at $k=k_{m2}>0$. The condition for merging is now given by
$\displaystyle\lambda_{-}=\frac{-m_{k}-\sqrt{m^{2}_{k}+A^{2}-4}}{2+A}=-1,$
(41)
which gives $m_{k}=2$ as the solution, or
$\displaystyle\cos k_{m2}=1-\Delta_{B}/2=3-\Delta/2.$ (42)
In the TI-2 regime, the threshold of bulk excitations occurs at $k_{y}=\pi$ or
$k_{y}=\pm\arccos\left[2m_{k}/(A^{2}-4)\right]$. It can be checked that the
bulk energy at $\mib k=(k_{m2},\pi)$ becomes the same as the edge mode with
the condition (42). Hence merging occurs with bulk excitations
$E_{b}(k,k_{y})$ with $k_{y}=\pi$.
The critical momentum $k=k_{c2}$, below which the minimum of $E_{b}(k,k_{y})$
no longer occurs at $k_{y}=\pi$, can be obtained by condition
$\partial^{2}E_{\rm b}(k_{c2},k_{y})/\partial k_{y}^{2}|_{k_{y}=\pi}=0,$ (43)
which gives the solution
$\cos k_{c2}=1-\frac{\Delta_{B}}{2}-\frac{A^{2}}{4}.$ (44)
Thus we have the relation $\cos k_{c2}<\cos k_{2m}$, or $k_{c2}>k_{2m}$. Hence
merging indeed occurs in the range where $k_{y}=\pi$ corresponds to the
threshold. In this way, we have quantified important characteristics of the
edge modes shown in Fig.2.
## 4 Zigzag edge
### 4.1 Effective one-dimensional form
Let us now consider a zigzag edge geometry, as illustrated in Fig.3.
Figure 3: Zigzag edge lattice ribbon, with two boundaries in (1,1) direction.
Electrons in a zigzag edge geometry are confined in a diagonal strip: $1\leq
y-x\leq N_{r}$, provided the edges are placed at $y-x=1$ and $y-x=N_{r}$,
normal to the $(1,-1)$-direction. The translational invariance remains along
the $(1,1)$-direction, where the conserved momentum is given by
$p=(k_{x}+k_{y})/\sqrt{2}$. For notational convenience we introduce
$\kappa=p/\sqrt{2}$ with ${-\pi/2<\kappa\leq\pi/2}$ and define the new basis
set as
$|\kappa,j\rangle=\frac{1}{\sqrt{N_{c}}}\sum_{I}\exp\left[i\kappa\left(2I+j\right)\right]|I,I+j\rangle,$
(45)
where the site summation goes along the (1,1) direction. The phase factor is
so chosen that it becomes unity for the state $|-\\!J,J\rangle$. Then, the
amplitude $\Phi_{j}$ in this basis satisfies the Schrödinger equation
analogous to eq.(17):
$\hat{\cal
E}\Phi_{j}+\hat{t}_{11}(\kappa)\Phi_{j-1}+\hat{t}^{\dagger}_{11}(\kappa)\Phi_{j+1}=E_{\uparrow}(\kappa)\Phi_{j}$
(46)
where $\hat{\cal E}$ has been defined by eq.(9) and hopping matrix
$\hat{t}_{11}(\kappa)$ is given by
$\displaystyle\hat{t}_{11}(\kappa)$
$\displaystyle=\frac{i}{2}Ae^{-i\kappa}\sigma_{x}-\frac{i}{2}Ae^{i\kappa}\sigma_{y}+2\cos
k\sigma_{z}$ (47)
We impose the boundary condition: $\Phi_{0}=\Phi_{N_{r}+1}=0$, which is
consistent with the zigzag edge geometry. Assuming eigenstate of eq.(46) with
property $\Phi_{j}=\lambda\Phi_{j-1}=\lambda^{j}\Phi$, where
$|\lambda|<1$,[19, 20] we obtain
$\displaystyle\left(\hat{\cal
E}+\lambda\hat{t}^{\dagger}_{11}(\kappa)+\lambda^{-1}\hat{t}_{11}(\kappa)\right)\Phi$
$\displaystyle=P_{11}(\lambda,\kappa)\Phi=E_{\uparrow}(\kappa)\Phi$ (48)
For later reference purpose, we write the bulk energy $E_{\rm b}$ in terms of
variables $\kappa=(k_{x}+k_{y})/2$ and $\xi=(k_{x}-k_{y})/2$. From eq.(12) we
obtain
$\displaystyle E_{\rm b}(\kappa,\xi)$
$\displaystyle=\pm\left[2A^{2}\left(\sin^{2}\kappa\cos^{2}\xi+\cos^{2}\kappa\sin^{2}\xi\right)\right.$
$\displaystyle+\left.\left(\Delta_{B}+4\cos\kappa\cos\xi\right)^{2}\right]^{1/2}.$
(49)
### 4.2 Derivation of spectrum in thermodynamic limit
We will separate $P_{11}$ into the Hermitian part $H_{11}$ and the
corresponding annihilator $F_{11}$. The separation now is not straightforward
in contrast with the case of straight edge. As a preliminary, we introduce the
following matrices:
$\displaystyle\sigma_{X}$
$\displaystyle=\left(\sigma_{x}+\sigma_{y}\right)/\sqrt{2},$ (50)
$\displaystyle\sigma_{Y}$
$\displaystyle=\left(\sigma_{y}-\sigma_{x}\right)/\sqrt{2}.$ (51)
Then we obtain
$\displaystyle\hat{t}_{11}(\kappa)=\frac{A}{\sqrt{2}}\left(\sin{\kappa}\,\sigma_{X}-i\cos{\kappa}\,\sigma_{Y}\right)+2\cos{\kappa}\,\sigma_{z}.$
(52)
We note that the spectrum of each edge mode is an odd function of $\kappa$.
Then we introduce a variable $\theta$, which is an odd function of $\kappa$,
and make the following transformation:
$\displaystyle\sigma_{\theta x}$
$\displaystyle={\cos{\theta}}\,\sigma_{X}+{\sin{\theta}}\,\sigma_{z},$ (53)
$\displaystyle\sigma_{\theta z}$
$\displaystyle={\cos{\theta}}\,\sigma_{z}-{\sin{\theta}}\,\sigma_{X},$ (54)
and rewrite as $\sigma_{\theta y}=\sigma_{Y}$. They keep the commutation
property:
$\displaystyle\left[\sigma_{\theta x},\sigma_{\theta
y}\right]=2i\sigma_{\theta z},$ (55)
and analogous cyclic ones that are the same as the original Pauli matrices.
Then we obtain
$\displaystyle\hat{\cal E}$
$\displaystyle=\Delta_{B}\left(\sin{\theta\,}\sigma_{\theta
x}+\cos{\theta\,}\sigma_{\theta z}\right),$ (56)
$\displaystyle\lambda\hat{t}^{\dagger}_{11}+\lambda^{-1}\hat{t}_{11}$
$\displaystyle=\gamma_{\theta x}\sigma_{\theta x}+\gamma_{\theta
y}\sigma_{\theta y}+\gamma_{\theta z}\sigma_{\theta z},$ (57)
where
$\displaystyle\gamma_{\theta x}$
$\displaystyle=(\lambda+\lambda^{-1})\left(\frac{A}{\sqrt{2}}\sin{\kappa}\cos{\theta}+2\cos{\kappa}\sin{\theta}\right),$
(58) $\displaystyle\gamma_{\theta y}$
$\displaystyle=i(\lambda-\lambda^{-1})\frac{A}{\sqrt{2}}\cos{\kappa},$ (59)
$\displaystyle\gamma_{\theta z}$
$\displaystyle=(\lambda+\lambda^{-1})\left(2\cos{\kappa}\cos{\theta}-\frac{A}{\sqrt{2}}\sin{\kappa}\sin{\theta}\right).$
(60)
We choose $\Phi$ as eigenstate of $\sigma_{\theta x}$ since the coefficient
$\Delta_{B}\sin\theta$ in eq.(56) is an odd function of $\kappa$. Then terms
with $\sigma_{\theta y}$ and $\sigma_{\theta z}$ must combine to form the
annihilator. Furthermore, since the coefficient of $\sigma_{\theta x}$ must be
real, and be an odd function of $\kappa$, we require $\gamma_{\theta x}=0$.
This condition determines $\theta$ in terms of $\kappa$ as
$\displaystyle\sin{\theta}$
$\displaystyle=-\frac{\tan{\kappa}}{\sqrt{\tan^{2}{\kappa}+8/A^{2}}},$ (61)
$\displaystyle\cos{\theta}$
$\displaystyle=\frac{\sqrt{8}/A}{\sqrt{\tan^{2}{\kappa}+8/A^{2}}}.$ (62)
In this way, we decompose $P_{11}=H_{11}+F_{11}$ in eq.(48) as follows:
$\displaystyle H_{11}$ $\displaystyle=\Delta_{B}\sin{\theta}\,\sigma_{\theta
x},$ (63) $\displaystyle F_{11}$
$\displaystyle=(\Delta_{B}\cos{\theta}+\gamma_{\theta z})\sigma_{\theta
z}+\gamma_{\theta y}\sigma_{\theta y}.$ (64)
By diagonalizing $H_{11}$, the eigenenergy $E_{\uparrow}(\kappa)$ is derived
as
$\displaystyle E_{\uparrow}(\kappa)=s\Delta_{B}\sin{\theta},\quad(s=\pm 1),$
(65)
where only one of the signs $\pm$ is relevant, as derived shortly. Note that
the spectrum has a form analogous to the case of straight edge given by
eq.(31). The condition for $F_{11}$ to form the annihilator is given by
$\displaystyle\Delta_{B}\cos{\theta}+\gamma_{\theta z}=is\gamma_{\theta y}$
(66)
which determines $\lambda$ for the edge mode as
$\displaystyle\lambda_{\pm}(s)=\frac{1}{2\cos\kappa}\cdot\frac{-\Delta_{B}\cos{\theta}\pm\sqrt{R}}{2/\cos{\theta}+sA/\sqrt{2}},$
(67)
with
$\displaystyle
R=\Delta_{B}^{2}\cos^{2}{\theta}-2\cos^{2}{\kappa}\left(8\cos^{-2}{\theta}-A^{2}\right).$
(68)
In the case of $R<0$, we obtain complex $\lambda_{\pm}(s)$ with absolute value
$|\lambda_{\pm}(s)|^{2}=\frac{\sqrt{8}-sA\cos{\theta}}{\sqrt{8}+sA\cos{\theta}},$
(69)
which is less than unity only for $s=1$, and positive for $A^{2}\leq 8$.
Therefore, the edge mode must have $s=1$ in eq.(65), and the group velocity is
positive (negative) in TP-1 (TP-2) regime. A special case occurs with
$\theta=\pm\pi/2$ that corresponds to $\kappa=\mp\pi/2$ according to eqs.(61)
and (62). Actually eq.(67) gives $\lambda_{\pm}=\pm i$ at $\kappa=\pi/2$.
Thus, at the boundary of the 1D BZ, there are no edge modes since
$|\lambda_{\pm}|=1$. The neighborhood of this special point has
$|\lambda_{\pm}|<1$, and there should be an edge mode. We emphasize that this
property is independent of $\Delta$ and $A$, and is specific to the zigzag
edge.
Due to the time-reversal symmetry, we obtain the Kramers partner from
$h_{\downarrow}$ with the spectrum
$E_{\downarrow}(\kappa)=-\Delta_{B}\sin{\theta}$. Figure 4 shows the edge
modes together with bulk excitations for the TI-1 and TI-2, and the boundary
case $\Delta_{B}=0$. The bulk spectrum illustrated is obtained from $E_{\rm
b}(\kappa,\xi)$ as a function of $\kappa$ with fixed $\xi$. In both regimes,
the edge mode becomes gapless at $\kappa=0$. Hence only $\kappa=0$ is the
relevant point where a pair of edge modes are degenerate by time-reversal
invariance. We note that the spectrum becomes completely flat with
$\Delta_{B}=0$ [18].
In the low momentum region, the spectrum tends to the linear dispersion
$\displaystyle
E_{\uparrow,\downarrow}(\kappa)=\mp\frac{1}{\sqrt{8}}A\Delta_{B}\kappa=\mp\frac{1}{4}A\Delta_{B}p.$
(70)
Especially, with $\Delta_{B}=\pm 4$, the spectrum becomes the same as the
corresponding modes in the straight edge. Thus we find that the system
acquires the axial symmetry only in the case of $\Delta_{B}=\pm 4$ even in the
long-wavelength limit. This is not surprising since the difference in the edge
shape remains even for long wavelength.
Figure 4: Spectrum of zigzag edge modes. The top panel is for TI-1 state with
$\Delta=1.2$, the center is for TI-2 state with $\Delta=4.5$. The bottom panel
shows the boundary case $\Delta=4$. All cases have $A=1$.
### 4.3 Allowed momentum range for edge modes
Let us first consider the edge mode in TI-1 regime corresponding to right-
going edge mode (see Figure 2 top panel), and
$\lambda_{\pm}=\lambda_{\pm}(1)$. We restrict to the region of positive
$\kappa$. Since complex $\lambda$ always has a pair of solutions with
$|\lambda_{\pm}|\leq 1$, the merging momentum $\kappa_{m}<\pi/2$ can be
obtained from the condition
$\lambda_{+}=1,$ (71)
which is equivalent to
$\left(8-A^{2}\right)\cos^{2}{\kappa_{m}}+2\Delta_{B}\cos{\kappa_{m}}+A^{2}=0,$
(72)
according to eq.(67). Note that there is no solution for
$0\leq\cos\kappa_{m}\leq 1$ in the case of $A^{2}\geq 8$. The relevant
solution in the case of $A^{2}<8$ is given by
$\cos{\kappa_{m}}=\frac{-\Delta_{B}\pm\sqrt{\Delta_{B}^{2}-A^{2}\left(8-A^{2}\right)}}{8-A^{2}},$
(73)
The critical value of $\Delta_{B}$ beyond which no real $\kappa_{m}$ exists is
given by
$\Delta_{4c}=-A\sqrt{8-A^{2}}<0,$ (74)
where we consider only the case $0<A<2$ as in the the straight edge. According
to eq.(74), we have three cases:
(i) no solution for $\kappa_{m}<\pi/2$ with $\Delta_{B}>\Delta_{4c}$;
(ii) single $\kappa_{m}$ with $\Delta_{B}=\Delta_{4c}$;
(iii) two solutions $\kappa_{m1}<\kappa_{m2}$ with $\Delta_{B}<\Delta_{4c}$.
Edge modes in the case of $A>2$ will be discussed in §5.
The threshold of bulk excitations can occur either at $\xi=0$ or $\xi\neq 0$
depending on $\kappa$. The critical value $\kappa_{c}$ separating the two
cases is determined by the condition
$\displaystyle\partial^{2}E_{\rm
b}(\kappa_{c},\xi)/\partial\xi^{2}|_{\xi=0}=0,$ (75)
which can be reduced to
$\displaystyle
2(A^{2}-4)\cos^{2}\kappa_{c}-2\Delta_{B}\cos\kappa_{c}-A^{2}=0.$ (76)
Then we obtain
$\displaystyle\cos\kappa_{c}=\frac{\Delta_{B}\pm\sqrt{\Delta_{B}^{2}+2A^{2}(A^{2}-4)}}{2(A^{2}-4)}.$
(77)
In the case of $\xi\neq 0$, the momentum $\xi$ at the threshold satisfy the
condition
$\displaystyle\cos\xi=\frac{-2\Delta_{B}\cos\kappa}{4+(4-A^{2})\cos 2\kappa}.$
(78)
At the zone boundary $\kappa=\pi/2$, eq.(78) gives $\xi=\pm\pi/2$ as the
solution. Here the edge mode has the energy $E_{\uparrow}(\pi/2)=-\Delta_{B}$,
and the lowest bulk excitation has the same energy $E_{\rm
b}(\pi/2,\pm\pi/2)=-\Delta_{B}$. Namely, $\kappa=\pi/2$ is always a merging
point in the zigzag edge for any parameter settings.
In the following, we analyze the spectrum of the edge mode near the merging
momentum according to classification (i), (ii), (iii) given above.
#### Edge modes for the whole BZ
Let us first consider the case (i): $\Delta_{B}>\Delta_{4c}$. With $A=1$, we
obtain $\Delta_{4c}=-\sqrt{7}$, i.e., $\Delta_{c}\sim 1.354$ from eq.(74).
Figure 5 shows $|\lambda_{\pm}|$ and the energy of the edge mode relative to
the threshold of bulk excitations in this case with $\Delta=1.4$.
At the zone boundary, the edge mode merges with bulk excitations. The
difference of energies is expanded as
$\displaystyle E_{\uparrow}(\kappa)-E_{\rm
b}(\kappa,\frac{\pi}{2})=\frac{A^{2}}{\Delta_{B}}(\kappa-\frac{\pi}{2})^{2}+O\left((\kappa-\frac{\pi}{2})^{4}\right).$
(79)
Note that only even order terms appear in the expansion since both the lowest
bulk excitation $E_{\rm b}$ and edge mode energy $E_{\uparrow}$ are symmetric
around $\pi/2$.
Figure 5: The parameter $|\lambda_{\pm}|$ and energy difference
$E_{\uparrow}-E_{\rm b}$ between edge mode and the threshold of bulk
excitations as a function of $\kappa$ for $\Delta=1.4$ with $A=1$. For complex
$\lambda$, we obtain $|\lambda_{+}|=|\lambda_{-}|$. In the region of $\kappa$
with real $\lambda_{\pm}$, $E_{\uparrow}-E_{\rm b}$ becomes almost zero, but
is marginally negative.
#### Edge modes with critical momentum $\kappa_{m}$
Next we consider the critical case (ii) characterized by single $\kappa_{m}$
with $\Delta_{B}=\Delta_{4c}$. Figure 6 shows $|\lambda_{\pm}|$ and the energy
difference. By comparing eqs.(73) and (77), we obtain
$\displaystyle 0<\kappa_{c1}<\kappa_{m}<\kappa_{c2}<\pi/2.$ (80)
Hence we find $\xi=0$ for the bulk momentum at the merging point.
Figure 6: The same quantities as in Fig.5 but with
$\Delta=\Delta_{c}=4-\sqrt{7}\sim 1.354$. Note the scale of the ordinate in
the lower panel, showing the minute energy difference as compared with the
overall energy scale of the system.
#### Edge modes with reentrance
We finally consider the case (iii): two solutions $\kappa_{m1}<\kappa_{m2}$
with $\Delta_{B}<\Delta_{4c}$. From eqs.(73) and (77) we obtain the relation
$\displaystyle 0<\kappa_{c1}<\kappa_{m1}<\kappa_{m2}<\kappa_{c2}<\pi/2$ (81)
Hence we obtain the bulk momentum $\xi=0$ for both merging points. Then we
expand $E_{\uparrow}-E_{\rm b}$ around merging points $\kappa_{mi}\ (i=1,2)$
as follows:
$\displaystyle E_{\uparrow}(\kappa)-E_{\rm
b}(\kappa,0)=a_{i}(\kappa-\kappa_{mi})^{2}+O\left((\kappa-\kappa_{mi})^{3}\right),$
(82)
where the expansion coefficient is calculated as
$\displaystyle a_{i}$
$\displaystyle=\frac{4(A^{4}-8A^{2}+\Delta_{B}^{2})}{A^{2}\Delta_{B}}$
$\displaystyle\times\left[\frac{A^{2}-4}{A^{2}-8}+(-1)^{i}\frac{4\sqrt{A^{4}-8A^{2}+\Delta_{B}^{2}}}{\Delta_{4}(A^{2}-8)}\right]^{1/2}$
(83)
Hence we have proven that the threshold of bulk excitation shares the same
energy and velocity with edge mode, since the lowest order term of expansion
is of second order.
At critical value of $\Delta_{B}=\Delta_{4c}$, we obtain
$\kappa_{m}=\kappa_{m1}=\kappa_{m2}$, and both second and third order terms in
eq.(82) tend to zero. Then the expansion around $\kappa_{m}$ begins from
fourth order. This explains the nearly flat shape of $E_{\uparrow}-E_{\rm b}$
in Fig.6.
Figure 7: The same quantities as in Fig.5 but with $\Delta=1$. There is no
edge mode for $\kappa$ between $\kappa_{m1}$ and $\kappa_{m2}$ where one of
$|\lambda_{\pm}|$ exceeds unity.
Presence of two merging points causes novel phenomenon in zigzag edge mode.
Figure 7 shows the parameter $|\lambda_{\pm}|$ and the energy difference as in
previous cases. In the region $0<\kappa<\kappa_{m1}$ and
$\kappa_{m2}<\kappa<\frac{\pi}{2}$, we obtain $|\lambda_{\pm}|<1$ and
$E_{\uparrow}-E_{\rm b}<0$. Namely, the edge mode has two separate momentum
ranges of its existence. The normal edge state (I) starts from zone center
($\kappa=0$), and vanishes at an intermediate point $\kappa=\kappa_{m1}$. In
addition the reentrant part (II) appears near the zone boundary:
$\kappa_{m2}<\kappa<\frac{\pi}{2}$. The reentrant edge mode has only
marginally lower energy than lowest bulk excitation, which approximatively
obeys $E_{\uparrow}-E_{\rm b}\propto(\kappa-\kappa_{m2})^{2}$. In the momentum
range $\kappa_{m1}\leq\kappa\leq\kappa_{m2}$, the edge mode disappears since
one of $|\lambda_{\pm}|$ exceeds unity.
Let us summarize the results for different $\Delta$ with fixed $A$ shown in
Figs. 5, 6 and 7. As $\Delta$ increases from zero, the two separate regions
for the edge modes widen in momentum space simultaneously. Namely, both
$\kappa_{m1}$ and $\pi/2-\kappa_{m2}$ increase, while
$\kappa_{m2}-\kappa_{m1}$ decreases with increasing $\Delta$. At critical
$\Delta_{c}$, the two regions merge ($\kappa_{m1}=\kappa_{m2}$), and the
unbroken edge mode emerges that disappears only at the zone boundary.
### 4.4 Edge modes in TI-2 regime
Let us now consider the TI-2 regime with $\Delta_{B}>0$. Since the Hamiltonian
$H(\mib k)$ has only a linear term of $\Delta_{B}$, the solution in the TI-2
regime can be obtained from that in the TI-1 regime by changing the sign of
$\Delta_{B}$. This conversion was indeed made in §3 for the straight edge.
In the zigzag edge, the spectrum given by eq.(65) has a negative slope because
of $\Delta_{B}>0$. Corresponding to eq.(41), the merging momentum
$\kappa_{m}>0$ is obtained from the condition
$\displaystyle\lambda_{-}=-1,$ (84)
instead of eq.(71). Then we obtain
$\cos{\kappa_{m}}=\frac{\Delta_{B}\pm\sqrt{\Delta_{B}^{2}-A^{2}\left(8-A^{2}\right)}}{8-A^{2}},$
(85)
instead of eq.(73), and
$\Delta_{4c}=A\sqrt{8-A^{2}}>0,$ (86)
instead of eq.(74). It is clear that the resultant solution of $\kappa_{m}$ is
the same as that in TI-1 range with the same $|\Delta_{B}|$. Similarly, one
can check that all relevant quantities such as
$\kappa_{m1},\kappa_{m2},\kappa_{c1},\kappa_{c2}$ also have the same
correspondence. The expression given by eqs.(79) and (82) remains valid, which
means that the difference is now positive. This is naturally understood since
the edge mode $E_{\uparrow}(\kappa)$ has the negative slope.
## 5 Summary and discussion
We have analytically obtained spectrum and wave function of helical edge
states for the BHZ model by identifying the annihilator for each case of (1,0)
and (1,1) edges. The simplicity of the BHZ model has allowed us to obtain the
complete information of the edge modes. Let us finally consider the case
$A>2$. In the straight edge, $\lambda_{\pm}$ are always real in this case.
Except for this difference, the property of the edge mode spectrum remains the
same. In the zigzag edge, the case $A>\sqrt{8}$ allows no solution for merging
momentum $\kappa_{m}$. This means the edge mode is present up to the zone
boundary.
Edge spectrum shows different properties depending on edge geometry. In
(1,0)-edge case, edge spectrum is proportional to $\sin k$. As the sign of
$\Delta_{B}$ changes from negative to positive, the main location of edge mode
moves from the center of the 1D BZ to the boundary. This movement is
associated with the change of location of the bulk energy gap.
For the (1,1) edge, we have obtained the spectrum in the form of
$\pm\Delta_{B}\sin{\theta(\kappa)}$, where $\theta(\kappa)$ is an odd function
of momentum $\kappa$. Since $\theta(\kappa)$ does not depend on $\Delta_{B}$
as shown in eqs.(61) and (62), $\Delta_{B}$ appears only as the scale factor
in the spectrum. With $\Delta_{B}=0$, the edge modes become completely flat
for the whole Brillouin zone, and two-fold degenerate as a consequence of the
time-reversal symmetry.
The edge mode in the (1,1) geometry contains a novel reentrant part with
extremely small binding energy. As seen from Fig.7, the binding energy for the
reentrant part is only $10^{-3}$ of the overall energy. Except for many-body
phenomena such as superconductivity and Kondo effect, we have been unaware of
emergence of such extraordinary different energy scale. In spite of the tiny
binding energy, the decay of the wave function toward inside the system looks
quite normal, as judged by $|\lambda_{\pm}|$ which deviates clearly from
unity.
Mathematically speaking, the tiny binding energy stems from the following
factors:
(i) The zone boundary $\kappa=\pi/2$ is a special point where the edge mode
merges with bulk excitations.
(ii) Group velocity of the edge mode is the same as that of threshold
excitation in the bulk at the merging point.
Let us assume $\kappa\geq 0$ for simplicity. The condition (ii) requires the
energy difference $E_{\uparrow}-E_{\rm b}$ to be proportional to
$(\kappa-\pi/2)^{2}$ near the zone boundary, and simultaneously to
$(\kappa-\kappa_{m2})^{2}$ near the critical momentum. Hence in the
intervening region $\kappa_{m2}<\kappa<\pi/2$, the growth of
$E_{\uparrow}-E_{\rm b}$ is constrained from both ends. Provided
$\pi/2-\kappa_{m2}\sim 0.1\pi$, we obtain the scaling factor $\sim 10^{-2}$
from the quadratic dependence of $E_{\uparrow}-E_{\rm b}$. It is hoped that
more physical explanation can be provided in the near future why the binding
energy is so small.
## References
* [1] C.L.Kane and E.J. Mele: Phys. Rev. Lett. 95, 146802 (2005).
* [2] C.L. Kane and E.J. Mele: Phys. Rev. Lett. 95, 226801 (2005).
* [3] L. Fu, C.L. Kane and E.J. Mele: Phys. Rev. Lett. 98, 106803 (2007).
* [4] L. Fu and C.L. Kane: Phys. Rev. B 76, 045302 (2007).
* [5] B.A. Bernevig and S.C. Zhang: Phys. Rev. Lett. 96, 106802 (2006).
* [6] B. A. Bernevig, T. L. Hughes and S.-C. Zhang: Science 314, 1757 (2006).
* [7] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and S.C. Zhang: Nat. Phys. 5, 438 (2009).
* [8] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Qi, and S.C. Zhang: Science 318, 766 (2007).
* [9] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava and M.Z. Hasan: Nature 452, 970 (2008).
* [10] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J. Cava and M.Z. Hasan: Nat. Phys. 5, 398 (2009).
* [11] M. König, H. Buhmann, L.W. Molenkamp, T.L. Hughes, C.X Liu, X.-L Qi and S.C Zhang: J. Phys. Soc. Jpn 77, 031007 (2008).
* [12] C. Wu, B.A. Bernevig, and S.C. Zhang: Phys. Rev. Lett. 96, 106401 (2006).
* [13] C. Xu and J. Moore, Phys. Rev. B 73, 045322 (2006).
* [14] B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu: Phys. Rev. Lett. 101, 246807 (2008).
* [15] E.B. Sonin: arXiv:1006.5218.
* [16] M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe: J. Phys. Soc.Jpn. 65,1920 (1996).
* [17] K. Nakada and M. Fujita: Phys. Rev. B 54, 17954 (1996).
* [18] K. Imura, A. Yamakage, S. Mao, A. Hotta and Yoshio Kuramoto: arXiv:1004.5019; Phys. Rev. B in press.
* [19] M. Creutz and I. Horvath:Phys. Rev. D 50, 2297 (1994).
* [20] M. Creutz: Rev. Mod. Phys. 73, 119 (2001).
* [21] H.B. Nielsen and M. Ninomiya: Nucl. Phys. B 185 20 (1981); ibid. 193 173 (1981).
|
arxiv-papers
| 2010-08-03T08:25:31 |
2024-09-04T02:49:12.042568
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shijun Mao, Yoshio Kuramoto, Ken-Ichiro Imura, and Ai Yamakage",
"submitter": "Shijun Mao",
"url": "https://arxiv.org/abs/1008.0481"
}
|
1008.0772
|
# The algebraic hyperstructure of elementary particles in physical theory
A. Dehghan Nezhada, M. Nadjafikhahb and S.M. Moosavi Nejadc
aFaculty of Mathematics, Yazd University, Yazd, Iran
anezhad@yazduni.ac.ir
bSchool of Mathematics, Iran University of Science and Technology,
Narmak, Tehran, Iran
m_nadjafikhah@iust.ac.ir
cFaculty of Physics, Yazd University, Yazd, Iran
and School of particles and accelerators, Institute for research in
fundamental science (IPM), P.O.Box 19395-5531, Tehran, Iran
mmoosavi@yazduni.ac.ir
###### Abstract
Algebraic hyperstructures represent a natural extension of classical algebraic
structures. In a classical algebraic structure, the composition of two
elements is an element, while in an algebraic hyperstructure, the composition
of two elements is a set. Algebraic hyperstructure theory has a multiplicity
of applications to other disciplines. The main purpose of this paper is to
provide examples of hyperstructures associated with elementary particles in
physical theory.
## 1 Introduction
Algebraic hyperstructures represent a natural extension of classical algebraic
structures and they were introduced in 1934 by the French mathematician F.
Marty [11]. In a classical algebraic structure, the composition of two
elements is an element, while in an algebraic hyperstructure, the composition
of two elements is a set. Since then, hundreds of papers and several books
have been written on this topic. One of the first books, dedicated especially
to hypergroups, is “Prolegomena of Hypergroup Theory”, written by P. Corsini
in 1993 [3]. Another book on “Hyperstructures and Their Representations”, by
T. Vougiouklis, was published one year later [14].
On the other hand, algebraic hyperstructure theory has a multiplicity of
applications to other disciplines: geometry, graphs and hypergraphs, binary
relations, lattices, groups, fuzzy sets and rough sets, automata,
cryptography, codes, median algebras, relation algebras, $C^{*}$-algebras,
artificial intelligence, probabilities and so on. A recent book on these
topics is “Applications of Hyperstructure Theory”, by P.Corsini and V.
Leoreanu, published by Kluwer Academic Publishers in 2003 [4]. We mention here
another important book for the applications in Geometry and for the clearness
of the exposition, written by W. Prenowitz and J. Jantosciak [13]. Another
monograph is devoted especially to the study of Hyperring Theory, written by
Davvaz and Leoreanu-Fotea [5]. It begins with some basic results concerning
ring theory and algebraic hyperstructures, which represent the most general
algebraic context, in which the reality can be modelled. Several kinds of
hyperrings are introduced and analyzed in this book. The volume ends with an
outline of applications in Chemistry and Physics [6, 7], canalizing several
special kinds of hyperstructures: $e-$hyperstructures and transposition
hypergroups. The theory of suitable modified hyperstructures can serve as a
mathematical background in the field of quantum communication systems.
The main purpose of this paper is to provide examples of hyperstructures
associated with elementary particles in physical theory.
## 2 General n-ary hyperstuctures
Throughout this paper, the symbol $H_{1},H_{2},\ldots,H_{n}$ will denote $n$
nonempty sets. Where $P^{\ast}(\bigcup_{i=1}^{n}H_{i})$ denotes the set of all
non empty subsets of $\bigcup_{i=1}^{n}H_{i}$.
#### Definition 2.1.
A general $n-$ary hyperstructure is $n$ non empty sets
$H_{1},H_{2},\ldots,H_{n}$ together with a hyperoperation,
$f:H_{1}\times H_{2}\times\ldots\times H_{n}\longrightarrow
P^{\ast}(\bigcup_{i=1}^{n}H_{i})$ $(x_{1},\ldots\,x_{n})\mapsto
f(x_{1},\ldots\,x_{n})\subseteq(\bigcup_{i=1}^{n}H_{i})-\emptyset.$
Let $f$ be a general $n$-ary hyperoperation on $H_{1},H_{2},\ldots,H_{n}$ and
$A_{i},$ subsets of $H_{i}$ for all $i=1,\ldots,n$. We define
$f(A_{1},\ldots,A_{n})=\bigcup\\{f(x_{1},\ldots,x_{n})|\ x_{i}\in
A_{i},i=1,\ldots,n\\}.$
We denote by $H^{n}$ the cartesian product $H\times\ldots\times H$ where $H$
appears $n$ times. An element of $H^{n}$ will be denoted by
$(x_{1},\ldots,x_{n})$ where $x_{i}\in H$ for any $i$ with $1\leq i\leq n$. In
general, a mapping $f:H^{n}\longrightarrow{\cal P}^{*}(H)$ is called an
$n$-ary hyperoperation and $n$ is called the order of hyperoperation. A
hyperalgebra $(H,f)$ is a non-empty set $H$ with one $n$-ary hyperoperations
$f$.
#### Remark 2.2.
A general hyperoperation $\ast:X\times Y\longrightarrow P^{\ast}(X\cup Y)$
yields a general hyperoperation
$\otimes:P^{\ast}(X)\times P^{\ast}(Y)\longrightarrow P^{\ast}(X\cup Y)$
defined by $\displaystyle A\otimes B=\bigcup_{a\in A,b\in B}a\ast b$.
Conversely a general hyperoperation on $P^{\ast}(X)\times P^{\ast}(Y)$ yields
a general hyperoperation on $X\times Y$, defined by $x\ast
y=\\{x\\}\otimes\\{y\\}.$
In the above definition if $A\subseteq X,\ B\subseteq Y,\ x\in X,\ y\in Y,$
then we define,
$A\ast y=A\ast\\{y\\}=\bigcup_{a\in A}a\ast y,\ x\ast B=\\{x\\}\ast
B=\bigcup_{b\in B}x\ast b,$ $A\otimes B=\bigcup_{a\in A,b\in B}a\ast b.$
#### Remark 2.3.
If we let $X=Y=H$, then we obtain the hyperstructure theory.
## 3 Algebraic hyperstructures
In this subsection, we summarize the preliminary definitions and results
required in the sequel.
#### Definition 3.1.
Let $H$ be a non-empty set and let ${\cal P}^{*}(H)$ be the set of all non-
empty subsets of $H$.
* (i)
A hyperoperation on $H$ is a map $\otimes:H\times H\longrightarrow{\cal
P}^{*}(H)$ and the couple $(H,\otimes)$ is called a hypergroupoid. If $A$ and
$B$ are non-empty subsets of $H$, then we denote
$A\otimes B=\bigcup_{a\in A,\,b\in B}a\otimes b,\ \ x\otimes A=\\{x\\}\otimes
A\mbox{\ \ and\ \ }A\otimes x=A\otimes\\{x\\}.$
* (ii)
A hypergroupoid $(H,\otimes)$ is called a semihypergroup if for all $x,y,z$ of
$H$ we have $(x\otimes y)\otimes z=x\otimes(y\otimes z)$, which means that
$\displaystyle\bigcup_{u\in x\otimes y}u\otimes z=\bigcup_{v\in y\otimes
z}x\otimes v.$
* (iii)
We say that a semihypergroup $(H,\otimes)$ is a hypergroup if for all $x\in
H$, we have $x\otimes H=H\otimes x=H.$ A hypergroupoid $(H,\otimes)$ is an
$H_{v}$-group, if for all $x,y,z\in H,$ the following conditions hold:
* (1)
$x\otimes(y\otimes z)\cap(x\otimes y)\otimes z\not=\emptyset$ (weak
associativity),
* (2)
$x\otimes H=H\otimes x=H$ (reproduction).
#### Definition 3.2.
Let $(L,\oplus),$ be a $H_{v}$-group, and $K$ be a nonempty subset of $L$.
Then $K$ is called $H_{v}$-subgroup of $(L,\oplus)$ if $a\oplus b\in{\cal
P}^{\ast}(K)$ for all $a,b\in K$. That is to say, $K$ is an $H_{v}$-subgroup
of $(L,\oplus)$ if and only if $K$ is closed under the binary hyperoperation
on $L$.
The concept of $H_{v}$-structures constitute a generalization of the well-
known algebraic hyperstructures ( hypergroup, hyperring, hypermodule, and so
on) (for example you see [1]).
## 4 Physical example (elementary particles)
Since long time ago, one of the most important questions is that what is our
universe made of?. To answer this question many efforts have been yet done.
Since 1897 when the electron was discovered by J. J. Thomson, the elementary
particle physics was born. Nowadays, the biggest particle accelerator which is
called the Large Hadron Collider (LHC) at CERN in France is being applied to
find the last elementary particle(Higgs boson). The existence of this particle
is necessary to understand our universe.
In particle physics, an elementary particle or fundamental particle is a
particle which have no substructure, i.e. it is not known to be made up of
smaller particles. If an elementary particle truly has no substructure, then
it is one of the basic building blocks of the universe from which all other
particles are made. To describe the elementary particles and the interacting
forces between them, some different theories are proposed that their most
important is the Standard Model [12]. The Standard Model(SM) of elementary
particles has proved to be extremely successful during the past three decades.
It has shown to be a well established theory. All predictions based on the SM
have been experimentally verified and most of its parameters have been fixed.
The only part of the SM that has not been directly experimentally verified yet
is the Higgs sector. In the SM, the Quarks, Leptons and Gauge bosons are
introduced as the elementary particles. The SM of particle physics contains
six types of quarks, known as flavors: Up, Down, Charm, Strange, Bottom and
Top plus their corresponding antiparticles. For every particle there is a
corresponding type of antiparticle, for every quark it is known as antiquark,
that differs from the particle only in some of its properties(like electric
charge) which have equal magnitude but opposite sign. Since the quarks are
never found in isolation, therefore quarks combine to form composite particles
which are called Hadrons, see Ref.[1]. In particle physics, hadrons are
composite particles made of quarks which are categorized into two families:
baryons (made of three quarks) and mesons (made of one quark and one
antiquark).
In the SM, gauge bosons consist of the photons($\gamma$), gluons($g$),
$W^{\pm}$ and $Z$ bosons act as carriers of the fundamental forces of
nature[9]. In fact, interactions between the particles are described by the
exchange of gauge bosons. The third group of the elementary particles are
leptons. There are six type of leptons including the electron($e$), electron
neutrino($\nu_{e}$), muon($\mu$), muon neutrino($\nu_{\mu}$), tau($\tau$) and
tau neutrino($\nu_{\tau}$). Every lepton has a corresponding antiparticle-
these antiparticles are known as antileptons. Leptons are an important part of
the SM, especially the electrons which are one of the components of atoms.
Since the leptons can be found freely in the universe and they are one of the
important groups of the elementary particles, in this article we only
concentrate on this group of particles.
### 4.1 Leptons
In the Standard Model, leptons still appear to be structureless. In this
model, there are six flavors of leptons and six corresponding antiparticles.
They form three generations [9, 10]. The first generation is the electronic
leptons, comprising the electron($e$), electron neutrino($\nu_{e}$) and their
corresponding antiparticles, i.e. positron($e^{+}$) and electron antineutrino
($\overline{\nu}_{e}$). The second generation is the muonic leptons, including
muon($\mu$), muon neutrino($\nu_{\mu}$), antimuon($\mu^{+}$) and muon
antineutrino($\overline{\nu}_{\mu}$). The third is the tauonic leptons,
consist of tau($\tau$), tau neutrino($\nu_{\tau}$), antitau($\tau^{+}$) and
tau antineutrino $(\overline{\nu}_{\tau})$. Therefore, there are 12 particles
$\\{e,\nu_{e},\mu,\nu_{\mu},\tau,\nu_{\tau},e^{+},\overline{\nu}_{e},\mu^{+},\overline{\nu}_{\mu},\tau^{+},\overline{\nu}_{\tau}\\}$
in the leptons group. In the leptons group, the electron, muon and tau have
the electric charge $Q=-1$ (the charge of a particle is expressed in unit of
the electron charge) and the neutrinos are neutral. According to the
definition of antiparticle, the electric charge of positron, antimuon and
antitau is $Q=+1$ but the antineutrinos are neutral as well as neutrinos. The
main difference between the neutrinos and antineutrinos is in the other
quantum numbers such as leptonic numbers, see Refs. [9, 10]. In the SM,
leptonic numbers are assigned to the members of every generation of leptons.
Electron and electron neutrino have an electronic number of $Le=1$ while muon
and muon neutrino have a muonic number of $L_{\mu}=1$ and tau and tau neutrino
have a tauonic number of $L_{\tau}=1$. The antileptons have their respective
generation’s leptonic numbers of $-1$. These numbers are classified in Table
1.
Classify | First Generation | | | | Second Generation | | | | Third Generation | | |
---|---|---|---|---|---|---|---|---|---|---|---|---
Leptons | $e$ | $\nu_{e}$ | $e^{+}$ | $\overline{\nu}_{e}$ | $\mu$ | $\nu_{\mu}$ | $\mu^{+}$ | $\overline{\nu}_{\mu}$ | $\tau$ | $\nu_{\tau}$ | $\tau^{+}$ | $\overline{\nu}_{\tau}$
$Q$ | $-1$ | $0$ | $+1$ | $0$ | $-1$ | $0$ | $+1$ | $0$ | $-1$ | $0$ | $+1$ | $0$
$L_{e}$ | $1$ | $1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$L_{\mu}$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$
$L_{\tau}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $-1$ | $-1$
Table 1: Leptons classification to three generations. $Q$ stands for charge in
unit of the electron charge. $L_{e},L_{\mu}$ and $L_{\tau}$ stand for the
electronic, muonic and tauonic numbers, respectively.
In every interaction the leptonic numbers are conserved. Conservation of the
leptonic numbers means that the number of leptons of the same type remains the
same when particles interact. This implies that leptons and antileptons must
be created in pairs of a single generation. For example, the following
processes are allowed under conservation of the leptonic numbers:
$e+\nu_{e}\rightarrow\\{e+\nu_{e}\\}=\\{e,\nu_{e}\\}\ or\
\mu+\nu_{\mu}\rightarrow\\{\mu+\nu_{\mu}\\}=\\{\mu,\nu_{\mu}\\},$
where in the first interaction, the electronic numbers and in the second
interaction the munic numbers are conserved. In other interactions, outgoing
particles might be different, therefore all leptonic numbers must be checked.
In the following interactions both the electronic and muonic numbers are
conserved:
$e+\nu_{\mu}\rightarrow\\{e+\nu_{\mu}\ or\
\mu+\nu_{e}\\}=\\{e,\nu_{\mu},\mu,\nu_{e}\\}.$
The only conditions required to occur a leptonic interaction are the
conservation of the electric charges and the leptonic numbers. Conservation of
the leptonic numbers means that the electronic, muonic and tauonic numbers
must be conserved sepreately. Considering these conservation rules, for the
electron-positron interaction the interacting modes are:
$e+e^{+}\rightarrow\\{e+e^{+}\ or\ \mu+\mu^{+}\ \ or\ \tau+\tau^{+}\ or\
\nu_{e}+\overline{\nu}_{e}\ or\ \nu_{\mu}+\overline{\nu}_{\mu}\ or\
\nu_{\tau}+\overline{\nu}_{\tau}\\}=L.$
Other interactions between the members of the leptons group are shown in Table
2. To arrange this table we avoided writing the repeated symbols. For example
in the productions of the electron-electron interaction we only write $e$
instead of $e+e$. All the interactions shown in Table 2 are in the first
order. It means in the higher order other particles can be produced that we do
not consider them. For example in the electron-electron scattering (Muller
scattering) one or several photons might be appeared in productions of the
interaction, i.e. $e+e\rightarrow e+e+\gamma$ or in the electron-positron
scattering(Bhabha scattering [2]) we can have: $e+e^{+}\rightarrow
e+e^{+}+\gamma$. There also exist other processes that we do not consider them
in this work. For example: $e+e^{+}\rightarrow\gamma+\gamma$,
$e+e^{+}\rightarrow W^{-}+W^{+}$, $\tau+\tau^{+}\rightarrow Z^{0}+Z^{0}$ and
so on.
## 5 The algebraic hyperstructure of leptons
Contemporary investigations of hyperstructures and their applications yield
many relationships and connections between various fields of mathematics.
Besides the motivation for investigation of hyperstructures coming from
noncommutative algebra, geometrical structures and other mathematical fields
there exist such physical fenomena as the nuclear fission. Nuclear fission
occurs when a heavy nucleus, such as U23 5 , splits, or fissions, into two
smaller nuclei. As a result of this fission process we can get several dozens
of diffierent combinations of two medium-mass elements and several neutrons
(as barium $Ba^{141}$ and krypton $Kr^{92}$ and 3 neutrons; strontium
$Sr^{94}$ , xenon $Xe^{14}$ 0 and 2 neutrons; lanthanum $La^{147}$ , bromum
$Br^{87}$ and 2 neutrons; $Sn^{132}$ , $Mo^{101}$ and 3 neutrons and so on) .
More precisely, the input of this reaction is always the same—the heavy
uranium is bombarded with neutrons, but the result is in general
different—there are about 90 different daughter nuclei that can be formed. The
fission also results in the production of several neutrons, typically two or
three. On the average about 2.5 neutrons are released per event. In any
fission equation, there are many combinations of fission fragments, but they
always satisfy the requirements of conservation of energy and charge.
Another typical example of the situation when the result of interaction
between two particles is the whole set of particles is the interaction between
a foton with certain energy and an electron. The result of this interaction is
not deterministic. A photo-electric effect or Coulomb repulsion effect or
changeover of foton onto a pair electron and positron can arise.
It is to be noted that a similar situation which occurs during uranium fission
appears during several nuclear fission, too. The result depends on conditions.
Although the input 2 particles are the same, the output can be variant. It can
differ both in the number of arising particles and in their kind.
Another motivation for investigation of hyperstructures yields from technical
processes as a time sequence of military car repairs with respect to its
roadability consequences and its operational behavior. In this section we
describe a certain construction of hyperstructures belonging to the important
class of elementary particles. We find the algebraic hyperstructure of
leptons.
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr\
\par\otimes\\!\\!&\\!\\!\displaystyle
e\\!\\!&\\!\\!\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle
e^{+}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\mu\\!\\!&\\!\\!\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\mu^{+}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\tau\\!\\!&\\!\\!\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\tau^{+}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\tau}\\\
\hline\cr e\\!\\!&\\!\\!\displaystyle e\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\mu\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\mu\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\tau\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau^{+}\\\ \bar{\nu}_{\tau}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\bar{\nu}_{\tau}\\\\[2.84526pt] \hline\cr\
\nu_{e}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\tau^{+}\\\
\displaystyle\nu_{e}\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\mu\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\nu_{e}\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\nu_{e}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\tau^{+}\atop\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{e}\end{array}\\\\[2.84526pt] \hline\cr\
e^{+}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e^{+}\\\
\displaystyle\mu^{+}\\\ \displaystyle\tau^{+}\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
e^{+}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\mu^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\tau^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\\\[2.84526pt] \hline\cr\
\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\mu\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{e}\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\tau\atop\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{e}\atop\displaystyle\bar{\nu}_{\tau}\\\\[2.84526pt]
\hline\cr\ \mu\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\mu\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\mu\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\mu\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\tau\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\bar{\nu}_{\tau}\\\\[2.84526pt]
\hline\cr\ \nu_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\nu_{e}\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\tau^{+}\\\
\displaystyle\nu_{e}\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\nu_{\mu}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\tau^{+}\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{\mu}\end{array}\\\\[2.84526pt] \hline\cr\
\mu^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\mu^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e^{+}\\\
\displaystyle\mu^{+}\\\ \displaystyle\tau^{+}\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\mu^{+}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\tau^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\\\[2.84526pt] \hline\cr\
\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{e}\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\mu}\mu^{+}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\tau\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\mu}\atop\displaystyle\bar{\nu}_{\tau}\\\\[2.84526pt]
\hline\cr\ \tau\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\tau\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\tau\atop\displaystyle\bar{\nu}_{e}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\tau\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\tau\atop\displaystyle\bar{\nu}_{\mu}\\!\\!&\\!\\!\displaystyle\tau\\!\\!&\\!\\!\displaystyle\tau\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\mu\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\\\[2.84526pt] \hline\cr\
\nu_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e\\\
\displaystyle\tau\\\ \displaystyle\nu_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\nu_{e}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\nu_{\mu}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\tau\atop\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\nu_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\mu^{+}\\\ \displaystyle\tau^{+}\\\
\displaystyle\nu_{e}\atop\displaystyle\\\ \displaystyle\nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle L\\\\[2.84526pt]
\hline\cr\ \tau^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle\tau^{+}\atop\displaystyle\nu_{e}\\!\\!&\\!\\!\displaystyle
e^{+}\atop\displaystyle\tau^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\tau^{+}\atop\displaystyle\nu_{\mu}\\!\\!&\\!\\!\displaystyle\mu^{+}\atop\displaystyle\tau^{+}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle e^{+}\\\
\displaystyle\mu^{+}\\\ \displaystyle\tau^{+}\\\
\displaystyle\displaystyle\nu_{e}\\\ \nu_{\mu}\\\
\displaystyle\nu_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\tau^{+}\\!\\!&\\!\\!\displaystyle\tau^{+}\atop\displaystyle\bar{\nu}_{\tau}\\\\[2.84526pt]
\hline\cr\ \bar{\nu}_{\tau}\\!\\!&\\!\\!\displaystyle
e\atop\displaystyle\bar{\nu}_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{e}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e^{+}\\\ \displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{e}\atop\displaystyle\bar{\nu}_{\tau}\\!\\!&\\!\\!\displaystyle\mu\atop\displaystyle\bar{\nu}_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\tau}\\\
\displaystyle\nu_{\mu}\end{array}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle\mu^{+}\\\
\displaystyle\tau^{+}\\\ \displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\mu}\atop\displaystyle\bar{\nu}_{\tau}\\!\\!&\\!\\!\displaystyle\begin{array}[]{c}\displaystyle
e\\\ \displaystyle\mu\\\ \displaystyle\tau\\\ \displaystyle\bar{\nu}_{e}\\\
\displaystyle\bar{\nu}_{\mu}\\\
\displaystyle\bar{\nu}_{\tau}\end{array}\\!\\!&\\!\\!\displaystyle
L\\!\\!&\\!\\!\displaystyle\tau^{+}\atop\displaystyle\bar{\nu}_{\tau}\\!\\!&\\!\\!\displaystyle\bar{\nu}_{\tau}\\\
\hline\cr\end{array}$
Table 2: Interaction between leptons are shown.
#### Theorem 5.1.
Let
$L=\\{e,\nu_{e},e^{+},\bar{\nu}_{e},\mu,\nu_{\mu},\mu^{+},\bar{\nu}_{\mu},\tau,\nu_{\tau},\tau^{+},\bar{\nu}_{\tau}\\}$.
Then $(L,\otimes)$ is an abelian $H_{v}$-group.
We summarize some results on associativity, in following lemmas.
#### Lemma 5.2.
The list of $[A,B,C]$’s that $[A,[B,C]]\not\subseteq[[A,B],C]$ is:
$[e^{+},\mu,e^{+}]$, $[e^{+},\tau,e^{+}]$, $[e^{+},\bar{\nu}_{\tau},e^{+}]$,
$[\mu,e^{+},e^{+}]$, $[\mu^{+},\bar{\nu}_{e},e^{+}]$, $[\tau,e^{+},e^{+}]$,
$[\tau^{+},\bar{\nu}_{e},e^{+}]$,
$[\bar{\nu}_{e},e^{+},e^{+}]$, $[\bar{\nu}_{e},e^{+},\mu]$,
$[\bar{\nu}_{e},e^{+},\tau^{+}]$, $[\bar{\nu}_{e},e^{+},\nu_{\mu}]$,
$[\bar{\nu}_{e},e^{+},\nu_{\tau}]$, $[\bar{\nu}_{\tau},e^{+},e^{+}]$,
$[\nu_{e},\bar{\nu}_{e},e^{+}]$,
$[\nu_{\tau},\bar{\nu}_{e},e^{+}]$
#### Lemma 5.3.
The list of $[A,B,C]$’s that $[[A,B],C]\not\subseteq[A,[B,C]]$ is:
$[e^{+},\bar{\nu}_{e},e^{+}]$, $[\mu,\bar{\nu}_{e},e^{+}]$,
$[\tau^{+},\bar{\nu}_{e},e^{+}]$, $[\bar{\nu}_{e},e^{+},\mu^{+}]$,
$[\bar{\nu}_{e},e^{+},\tau^{+}]$, $[\bar{\nu}_{e},e^{+},\nu_{e}]$,
$[\bar{\nu}_{e},e^{+},\nu_{\tau}]$,
$[\bar{\nu}_{e},\tau^{+},\bar{\nu}_{e}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},e^{+}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},\tau^{+}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},\nu_{\mu}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},\nu_{\tau}]$,
$[\bar{\nu}_{e},\nu_{\mu},\bar{\nu}_{e}]$,
$[\bar{\nu}_{e},\nu_{\tau},\bar{\nu}_{e}]$,
$[\nu_{\mu},\bar{\nu}_{e},e^{+}]$, $[\nu_{\tau},\bar{\nu}_{e},e^{+}]$.
#### Lemma 5.4.
The list of $[A,B,C]$’s that $[[A,B],C]\neq[A,[B,C]]$ is :
$[e^{+},\mu,e^{+}]$, $[e^{+},\tau,e^{+}]$, $[e^{+},\bar{\nu}_{e},e^{+}]$,
$[e^{+},\bar{\nu}_{\tau},e^{+}]$, $[\mu,e^{+},e^{+}]$,
$[\mu,\bar{\nu}_{e},e^{+}]$, $[\mu^{+},\bar{\nu}_{e},e^{+}]$,
$[t,e^{+},e^{+}]$, $[\tau^{+},\bar{\nu}_{e},e^{+}]$,
$[\bar{\nu}_{e},e^{+},e^{+}]$, $[\bar{\nu}_{e},e^{+},\mu]$,
$[\bar{\nu}_{e},e^{+},\mu^{+}]$, $[\bar{\nu}_{e},e^{+},\tau^{+}]$,
$[\bar{\nu}_{e},e^{+},\nu_{e}]$,
$[\bar{\nu}_{e},e^{+},\nu_{\mu}]$, $[\bar{\nu}_{e},e^{+},\nu_{\tau}]$,
$[\bar{\nu}_{e},\tau^{+},\bar{\nu}_{e}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},e^{+}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},\tau^{+}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},\nu_{\mu}]$,
$[\bar{\nu}_{e},\bar{\nu}_{e},\nu_{\tau}]$,
$[\bar{\nu}_{e},\nu_{\mu},\bar{\nu}_{e}]$,
$[\bar{\nu}_{e},\nu_{\tau},\bar{\nu}_{e}]$, $[\bar{\nu}_{\tau},e^{+},e^{+}]$,
$[\nu_{e},\bar{\nu}_{e},e^{+}]$, $[\nu_{\mu},\bar{\nu}_{e},e^{+}]$,
$[\nu_{\tau},\bar{\nu}_{e},e^{+}]$.
#### Remark 5.5.
In general. the hyperoperation $"\otimes"$ is not associative.
#### Theorem 5.6.
If $(L,\otimes)$ is above $H_{v}$-group, then the following statements hold.
* 1)
There is not any $H_{v}$-subgroups of order $5$, $7$, $8$, $9$, $10$, and,
$11$ for $(L,\otimes)$;
* 2)
All the $1-$dimensional $H_{v}$-subgroups of $L$ are:
$\displaystyle\begin{array}[]{lclclclcl}L^{1}_{1}=\\{e\\},&&L^{1}_{2}=\\{e^{+}\\},&&L^{1}_{3}=\\{\mu\\},&&L^{1}_{4}=\\{\mu^{+}\\},&&L^{1}_{5}=\\{\tau\\},\\\
L^{1}_{6}=\\{\tau^{+}\\},&&L^{1}_{7}=\\{\bar{\nu}_{e}\\},&&L^{1}_{8}=\\{\bar{\nu}_{\mu}\\},&&L^{1}_{9}=\\{\bar{\nu}_{\tau}\\},&&L^{1}_{10}=\\{\nu_{e}\\},\\\
L^{1}_{11}=\\{\nu_{\mu}\\},&&L^{1}_{12}=\\{\nu_{\tau}\\}\end{array}$
* 3)
All the $2-$dimensional $H_{v}$-subgroups of $L$ are:
$\displaystyle\begin{array}[]{lclclcl}L^{2}_{1}=\\{e,\mu\\},&&L^{2}_{2}=\\{e,\tau\\},&&L^{2}_{3}=\\{e,\bar{\nu}_{\mu}\\},&&L^{2}_{4}=\\{e,\bar{\nu}_{\tau}\\},\\\
L^{2}_{5}=\\{e,\nu_{e}\\},&&L^{2}_{6}=\\{e^{+},\mu^{+}\\},&&L^{2}_{7}=\\{e^{+},\tau^{+}\\},&&L^{2}_{8}=\\{e^{+},\nu_{\mu}\\},\\\
L^{2}_{9}=\\{e^{+},\nu_{\tau}\\},&&L^{2}_{10}=\\{\mu,\tau\\},&&L^{2}_{11}=\\{\mu,\bar{\nu}_{e}\\},&&L^{2}_{12}=\\{\mu,\bar{\nu}_{\tau}\\},\\\
L^{2}_{13}=\\{\mu,\nu_{\mu}\\},&&L^{2}_{14}=\\{\mu^{+},\tau^{+}\\},&&L^{2}_{15}=\\{\mu^{+},\bar{\nu}_{\mu}\\},&&L^{2}_{16}=\\{\mu^{+},\nu_{e}\\},\\\
L^{2}_{17}=\\{\mu^{+},\nu_{\tau}\\},&&L^{2}_{18}=\\{\tau,\bar{\nu}_{e}\\},&&L^{2}_{19}=\\{\tau,\bar{\nu}_{\mu}\\},&&L^{2}_{20}=\\{\tau,\nu_{\tau}\\},\\\
L^{2}_{21}=\\{\tau^{+},\bar{\nu}_{\tau}\\},&&L^{2}_{22}=\\{\tau^{+},\nu_{e}\\},&&L^{2}_{23}=\\{\tau^{+},\nu_{\mu}\\},&&L^{2}_{24}=\\{\bar{\nu}_{e},\bar{\nu}_{\mu}\\},\\\
L^{2}_{25}=\\{\bar{\nu}_{e},\bar{\nu}_{\tau}\\},&&L^{2}_{26}=\\{\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\},&&L^{2}_{27}=\\{\nu_{e},\nu_{\mu}\\},&&L^{2}_{28}=\\{\nu_{e},\nu_{\tau}\\},\\\
L^{2}_{29}=\\{\nu_{\mu},\nu_{\tau}\\}.\end{array}$
* 4)
All the $3-$dimensional $H_{v}$-subgroups of $L$ are:
$\displaystyle\begin{array}[]{lclcl}L^{3}_{1}=\\{e,\mu,\tau\\},&&L^{3}_{2}=\\{e,\mu,\bar{\nu}_{\tau}\\},&&L^{3}_{3}=\\{e,\tau,\bar{\nu}_{\mu}\\},\\\
L^{3}_{4}=\\{e,\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\},&&L^{3}_{5}=\\{e^{+},\mu^{+},\tau^{+}\\},&&L^{3}_{6}=\\{e^{+},\mu^{+},{\it\nu_{\tau}}\\},\\\
L^{3}_{7}=\\{e^{+},\tau^{+},\nu_{\mu}\\},&&L^{3}_{8}=\\{e^{+},\nu_{\mu},\nu_{e}\\},&&L^{3}_{9}=\\{\mu,\tau,\bar{\nu}_{e}\\},\\\
L^{3}_{10}=\\{\mu,\bar{\nu}_{e},\bar{\nu}_{\tau}\\},&&L^{3}_{11}=\\{\mu^{+},\tau^{+},\nu_{e}\\},&&L^{3}_{12}=\\{\mu^{+},\nu_{e},{\it\nu_{\tau}}\\},\\\
L^{3}_{13}=\\{\tau,\bar{\nu}_{e},\bar{\nu}_{\mu}\\},&&L^{3}_{14}=\\{\tau^{+},\nu_{e},\nu_{\mu}\\},&&L^{3}_{15}=\\{\bar{\nu}_{e},\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\},\\\
L^{3}_{16}=\\{\nu_{e},\nu_{\mu},{\it\nu_{\tau}}\\}.\end{array}$
* 5)
All the $4-$dimensional $H_{v}$-subgroups of $L$ are:
$\displaystyle\begin{array}[]{lclcl}L^{4}_{1}=\\{e,\mu,\nu_{e},\nu_{\mu}\\},&&L^{4}_{2}=\\{e,\mu^{+},\bar{\nu}_{\mu},\nu_{e}\\},&&L^{4}_{3}=\\{e,\tau,\nu_{e},{\it\nu_{\tau}}\\},\\\
L^{4}_{4}=\\{e,\tau^{+},\bar{\nu}_{\tau},\nu_{e}\\},&&L^{4}_{5}=\\{e^{+},\mu^{+},\bar{\nu}_{e},\bar{\nu}_{\mu}\\},&&L^{4}_{6}=\\{\mu,\tau,\nu_{\mu},{\it\nu_{\tau}}\\},\\\
L^{4}_{7}=\\{\mu,\tau^{+},\bar{\nu}_{\tau},\nu_{\mu}\\},&&L^{4}_{8}=\\{\mu^{+},\tau,\bar{\nu}_{\mu},{\it\nu_{\tau}}\\},&&L^{4}_{9}=\\{\mu^{+},\tau^{+},\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\}.\end{array}$
* 6)
All the $6-$dimensional $H_{v}$-subgroups of $L$ are:
$\displaystyle\begin{array}[]{lcl}L^{6}_{1}=\\{e,\mu,\tau,\bar{\nu}_{e},\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\},&&L^{6}_{2}=\\{e,\mu,\tau,\nu_{e},\nu_{\mu},{\it\nu_{\tau}}\\},\\\
L^{6}_{3}=\\{e,\mu,\tau^{+},\bar{\nu}_{\tau},\nu_{e},\nu_{\mu}\\},&&L^{6}_{4}=\\{e,\mu^{+},\tau,\bar{\nu}_{\mu},\nu_{e},{\it\nu_{\tau}}\\},\\\
L^{6}_{5}=\\{e,\mu^{+},\tau^{+},\bar{\nu}_{\mu},\bar{\nu}_{\tau},\nu_{e}\\},&&L^{6}_{6}=\\{e^{+},\mu^{+},\tau,\bar{\nu}_{e},\bar{\nu}_{\mu},{\it\nu_{t}au}\\},\\\
L^{6}_{7}=\\{e^{+},\mu^{+},\tau^{+},\bar{\nu}_{e},\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\},&&L^{6}_{8}=\\{e^{+},\mu^{+},\tau^{+},\nu_{e},\nu_{\mu},{\it\nu_{\tau}}\\}.\end{array}$
The result follows immediately from Theorem 5.6 and Definitions 3.1, 3.2.
#### Conclusion 5.7.
If $(L,\otimes)$ is above $H_{v}$-group, then the following statements (the
inclusion $H_{v}$-subgroups ) hold:
* (i)
for $1$-dimensional:
$L^{1}_{1}\subset L^{2}_{1}$, $L^{1}_{1}\subset L^{2}_{2}$,
$L^{1}_{1}\subset L^{2}_{3}$, $L^{1}_{1}\subset L^{2}_{4}$,
$L^{1}_{1}\subset L^{2}_{5}$, $L^{1}_{1}\subset L^{3}_{1}$,
$L^{1}_{1}\subset L^{3}_{2}$, $L^{1}_{1}\subset L^{3}_{3}$,
$L^{1}_{1}\subset L^{3}_{4}$, $L^{1}_{1}\subset L^{4}_{1}$,
$L^{1}_{1}\subset L^{4}_{2}$, $L^{1}_{1}\subset L^{4}_{3}$,
$L^{1}_{1}\subset L^{4}_{4}$, $L^{1}_{1}\subset L^{6}_{1}$,
$L^{1}_{1}\subset L^{6}_{2}$, $L^{1}_{1}\subset L^{6}_{3}$,
$L^{1}_{1}\subset L^{6}_{4}$, $L^{1}_{1}\subset L^{6}_{5}$,
$L^{1}_{1}\subset L^{12}$, $L^{1}_{2}\subset L^{2}_{6}$, $L^{1}_{2}\subset
L^{2}_{7}$, $L^{1}_{2}\subset L^{2}_{8}$, $L^{1}_{2}\subset L^{2}_{9}$,
$L^{1}_{2}\subset L^{3}_{5}$, $L^{1}_{2}\subset L^{3}_{6}$,
$L^{1}_{2}\subset L^{3}_{7}$, $L^{1}_{2}\subset L^{3}_{8}$,
$L^{1}_{2}\subset L^{4}_{5}$, $L^{1}_{2}\subset L^{6}_{6}$,
$L^{1}_{2}\subset L^{6}_{7}$, $L^{1}_{2}\subset L^{6}_{8}$,
$L^{1}_{2}\subset L^{12}$, $L^{1}_{3}\subset L^{2}_{1}$, $L^{1}_{3}\subset
L^{2}_{10}$, $L^{1}_{3}\subset L^{2}_{11}$, $L^{1}_{3}\subset L^{2}_{12}$,
$L^{1}_{3}\subset L^{2}_{1}3$, $L^{1}_{3}\subset L^{3}_{1}$,
$L^{1}_{3}\subset L^{3}_{2}$, $L^{1}_{3}\subset L^{3}_{9}$,
$L^{1}_{3}\subset L^{3}_{10}$, $L^{1}_{3}\subset L^{4}_{1}$,
$L^{1}_{3}\subset L^{4}_{6}$, $L^{1}_{3}\subset L^{4}_{7}$,
$L^{1}_{3}\subset L^{6}_{1}$, $L^{1}_{3}\subset L^{6}_{2}$,
$L^{1}_{3}\subset L^{6}_{3}$, $L^{1}_{3}\subset L^{12}$, $L^{1}_{4}\subset
L^{2}_{6}$, $L^{1}_{4}\subset L^{2}_{14}$, $L^{1}_{4}\subset L^{2}_{15}$,
$L^{1}_{4}\subset L^{2}_{16}$, $L^{1}_{4}\subset L^{2}_{17}$,
$L^{1}_{4}\subset L^{3}_{5}$, $L^{1}_{4}\subset L^{3}_{6}$,
$L^{1}_{4}\subset L^{3}_{11}$, $L^{1}_{4}\subset L^{3}_{12}$,
$L^{1}_{4}\subset L^{4}_{2}$, $L^{1}_{4}\subset L^{4}_{5}$,
$L^{1}_{4}\subset L^{4}_{8}$, $L^{1}_{4}\subset L^{4}_{9}$,
$L^{1}_{4}\subset L^{6}_{4}$, $L^{1}_{4}\subset L^{6}_{5}$,
$L^{1}_{4}\subset L^{6}_{6}$, $L^{1}_{4}\subset L^{6}_{7}$,
$L^{1}_{4}\subset L^{6}_{8}$, $L^{1}_{4}\subset L^{12}$, $L^{1}_{5}\subset
L^{2}_{2}$, $L^{1}_{5}\subset L^{2}_{10}$, $L^{1}_{5}\subset L^{2}_{18}$,
$L^{1}_{5}\subset L^{2}_{19}$, $L^{1}_{5}\subset L^{2}_{20}$,
$L^{1}_{5}\subset L^{3}_{1}$, $L^{1}_{5}\subset L^{3}_{3}$,
$L^{1}_{5}\subset L^{3}_{9}$, $L^{1}_{5}\subset L^{3}_{13}$,
$L^{1}_{5}\subset L^{4}_{3}$, $L^{1}_{5}\subset L^{4}_{6}$,
$L^{1}_{5}\subset L^{4}_{8}$, $L^{1}_{5}\subset L^{6}_{1}$,
$L^{1}_{5}\subset L^{6}_{2}$, $L^{1}_{5}\subset L^{6}_{4}$,
$L^{1}_{5}\subset L^{6}_{6}$, $L^{1}_{5}\subset L^{12}$, $L^{1}_{6}\subset
L^{2}_{7}$, $L^{1}_{6}\subset L^{2}_{14}$, $L^{1}_{6}\subset L^{2}_{21}$,
$L^{1}_{6}\subset L^{2}_{22}$, $L^{1}_{6}\subset L^{2}_{23}$,
$L^{1}_{6}\subset L^{3}_{5}$, $L^{1}_{6}\subset L^{3}_{7}$,
$L^{1}_{6}\subset L^{3}_{11}$, $L^{1}_{6}\subset L^{3}_{14}$,
$L^{1}_{6}\subset L^{4}_{4}$, $L^{1}_{6}\subset L^{4}_{7}$,
$L^{1}_{6}\subset L^{4}_{9}$, $L^{1}_{6}\subset L^{6}_{3}$,
$L^{1}_{6}\subset L^{6}_{5}$, $L^{1}_{6}\subset L^{6}_{7}$,
$L^{1}_{6}\subset L^{6}_{8}$, $L^{1}_{6}\subset L^{12}$, $L^{1}_{7}\subset
L^{2}_{11}$, $L^{1}_{7}\subset L^{2}_{18}$, $L^{1}_{7}\subset L^{2}_{24}$,
$L^{1}_{7}\subset L^{2}_{25}$, $L^{1}_{7}\subset L^{3}_{9}$,
$L^{1}_{7}\subset L^{3}_{10}$, $L^{1}_{7}\subset L^{3}_{13}$,
$L^{1}_{7}\subset L^{3}_{15}$, $L^{1}_{7}\subset L^{4}_{5}$,
$L^{1}_{7}\subset L^{6}_{1}$, $L^{1}_{7}\subset L^{6}_{6}$,
$L^{1}_{7}\subset L^{6}_{7}$, $L^{1}_{7}\subset L^{12}$, $L^{1}_{8}\subset
L^{2}_{3}$, $L^{1}_{8}\subset L^{2}_{15}$, $L^{1}_{8}\subset L^{2}_{19}$,
$L^{1}_{8}\subset L^{2}_{24}$, $L^{1}_{8}\subset L^{2}_{26}$,
$L^{1}_{8}\subset L^{3}_{3}$, $L^{1}_{8}\subset L^{3}_{4}$,
$L^{1}_{8}\subset L^{3}_{13}$, $L^{1}_{8}\subset L^{3}_{15}$,
$L^{1}_{8}\subset L^{4}_{2}$, $L^{1}_{8}\subset L^{4}_{5}$,
$L^{1}_{8}\subset L^{4}_{8}$, $L^{1}_{8}\subset L^{4}_{9}$,
$L^{1}_{8}\subset L^{6}_{1}$, $L^{1}_{8}\subset L^{6}_{4}$,
$L^{1}_{8}\subset L^{6}_{5}$, $L^{1}_{8}\subset L^{6}_{6}$,
$L^{1}_{8}\subset L^{6}_{7}$, $L^{1}_{8}\subset L^{12}$, $L^{1}_{9}\subset
L^{2}_{4}$, $L^{1}_{9}\subset L^{2}_{12}$, $L^{1}_{9}\subset L^{2}_{21}$,
$L^{1}_{9}\subset L^{2}_{25}$, $L^{1}_{9}\subset L^{2}_{26}$,
$L^{1}_{9}\subset L^{3}_{2}$, $L^{1}_{9}\subset L^{3}_{4}$,
$L^{1}_{9}\subset L^{3}_{10}$, $L^{1}_{9}\subset L^{3}_{15}$,
$L^{1}_{9}\subset L^{4}_{4}$, $L^{1}_{9}\subset L^{4}_{7}$,
$L^{1}_{9}\subset L^{4}_{9}$, $L^{1}_{9}\subset L^{6}_{1}$,
$L^{1}_{9}\subset L^{6}_{3}$, $L^{1}_{9}\subset L^{6}_{5}$,
$L^{1}_{9}\subset L^{6}_{7}$, $L^{1}_{9}\subset L^{12}$, $L^{1}_{10}\subset
L^{2}_{5}$, $L^{1}_{10}\subset L^{2}_{16}$, $L^{1}_{10}\subset L^{2}_{22}$,
$L^{1}_{10}\subset L^{2}_{27}$, $L^{1}_{10}\subset L^{2}_{28}$,
$L^{1}_{10}\subset L^{3}_{11}$, $L^{1}_{10}\subset L^{3}_{12}$,
$L^{1}_{10}\subset L^{3}_{14}$, $L^{1}_{10}\subset L^{3}_{16}$,
$L^{1}_{10}\subset L^{4}_{1}$, $L^{1}_{10}\subset L^{4}_{2}$,
$L^{1}_{10}\subset L^{4}_{3}$, $L^{1}_{10}\subset L^{4}_{4}$,
$L^{1}_{10}\subset L^{6}_{2}$, $L^{1}_{10}\subset L^{6}_{3}$,
$L^{1}_{10}\subset L^{6}_{4}$, $L^{1}_{10}\subset L^{6}_{5}$,
$L^{1}_{10}\subset L^{6}_{8}$, $L^{1}_{10}\subset L^{12}$,
$L^{1}_{11}\subset L^{2}_{8}$, $L^{1}_{11}\subset L^{2}_{13}$,
$L^{1}_{11}\subset L^{2}_{23}$, $L^{1}_{11}\subset L^{2}_{27}$,
$L^{1}_{11}\subset L^{2}_{29}$, $L^{1}_{11}\subset L^{3}_{7}$,
$L^{1}_{11}\subset L^{3}_{8}$, $L^{1}_{11}\subset L^{3}_{14}$,
$L^{1}_{11}\subset L^{3}_{16}$, $L^{1}_{11}\subset L^{4}_{1}$,
$L^{1}_{11}\subset L^{4}_{6}$, $L^{1}_{11}\subset L^{4}_{7}$,
$L^{1}_{11}\subset L^{6}_{2}$, $L^{1}_{11}\subset L^{6}_{3}$,
$L^{1}_{11}\subset L^{6}_{8}$, $L^{1}_{11}\subset L^{12}$,
$L^{1}_{12}\subset L^{2}_{9}$, $L^{1}_{12}\subset L^{2}_{17}$,
$L^{1}_{12}\subset L^{2}_{20}$, $L^{1}_{12}\subset L^{2}_{28}$,
$L^{1}_{12}\subset L^{2}_{29}$, $L^{1}_{12}\subset L^{3}_{6}$,
$L^{1}_{12}\subset L^{3}_{8}$, $L^{1}_{12}\subset L^{3}_{12}$,
$L^{1}_{12}\subset L^{3}_{16}$, $L^{1}_{12}\subset L^{4}_{3}$,
$L^{1}_{12}\subset L^{4}_{6}$, $L^{1}_{12}\subset L^{4}_{8}$,
$L^{1}_{12}\subset L^{6}_{2}$, $L^{1}_{12}\subset L^{6}_{4}$,
$L^{1}_{12}\subset L^{6}_{6}$, $L^{1}_{12}\subset L^{6}_{8}$,
$L^{1}_{12}\subset L^{12}$.
* (i)
for $2$-dimensional:
$L^{2}_{1}\subset L^{3}_{1}$, $L^{2}_{1}\subset L^{3}_{2}$,
$L^{2}_{1}\subset L^{4}_{1}$, $L^{2}_{1}\subset L^{6}_{1}$,
$L^{2}_{1}\subset L^{6}_{2}$, $L^{2}_{1}\subset L^{6}_{3}$,
$L^{2}_{1}\subset L^{12}$, $L^{2}_{2}\subset L^{3}_{1}$, $L^{2}_{2}\subset
L^{3}_{3}$, $L^{2}_{2}\subset L^{4}_{3}$, $L^{2}_{2}\subset L^{6}_{1}$,
$L^{2}_{2}\subset L^{6}_{2}$, $L^{2}_{2}\subset L^{6}_{4}$,
$L^{2}_{2}\subset L^{12}$, $L^{2}_{3}\subset L^{3}_{3}$, $L^{2}_{3}\subset
L^{3}_{4}$, $L^{2}_{3}\subset L^{4}_{2}$, $L^{2}_{3}\subset L^{6}_{1}$,
$L^{2}_{3}\subset L^{6}_{4}$, $L^{2}_{3}\subset L^{6}_{5}$,
$L^{2}_{3}\subset L^{12}$, $L^{2}_{4}\subset L^{3}_{2}$, $L^{2}_{4}\subset
L^{3}_{4}$, $L^{2}_{4}\subset L^{4}_{4}$, $L^{2}_{4}\subset L^{6}_{1}$,
$L^{2}_{4}\subset L^{6}_{3}$, $L^{2}_{4}\subset L^{6}_{5}$,
$L^{2}_{4}\subset L^{12}$, $L^{2}_{5}\subset L^{4}_{1}$, $L^{2}_{5}\subset
L^{4}_{2}$, $L^{2}_{5}\subset L^{4}_{3}$, $L^{2}_{5}\subset L^{4}_{4}$,
$L^{2}_{5}\subset L^{6}_{2}$, $L^{2}_{5}\subset L^{6}_{3}$,
$L^{2}_{5}\subset L^{6}_{4}$, $L^{2}_{5}\subset L^{6}_{5}$,
$L^{2}_{5}\subset L^{12}$, $L^{2}_{6}\subset L^{3}_{5}$, $L^{2}_{6}\subset
L^{3}_{6}$, $L^{2}_{6}\subset L^{4}_{5}$, $L^{2}_{6}\subset L^{6}_{6}$,
$L^{2}_{6}\subset L^{6}_{7}$, $L^{2}_{6}\subset L^{6}_{8}$,
$L^{2}_{6}\subset L^{12}$, $L^{2}_{7}\subset L^{3}_{5}$, $L^{2}_{7}\subset
L^{3}_{7}$, $L^{2}_{7}\subset L^{6}_{7}$, $L^{2}_{7}\subset L^{6}_{8}$,
$L^{2}_{7}\subset L^{12}$, $L^{2}_{8}\subset L^{3}_{7}$, $L^{2}_{8}\subset
L^{3}_{8}$, $L^{2}_{8}\subset L^{6}_{8}$, $L^{2}_{8}\subset L^{12}$,
$L^{2}_{9}\subset L^{3}_{6}$, $L^{2}_{9}\subset L^{3}_{8}$,
$L^{2}_{9}\subset L^{6}_{6}$, $L^{2}_{9}\subset L^{6}_{8}$,
$L^{2}_{9}\subset L^{12}$, $L^{2}_{10}\subset L^{3}_{1}$, $L^{2}_{10}\subset
L^{3}_{9}$, $L^{2}_{10}\subset L^{4}_{6}$, $L^{2}_{10}\subset L^{6}_{1}$,
$L^{2}_{10}\subset L^{6}_{2}$, $L^{2}_{10}\subset L^{12}$,
$L^{2}_{11}\subset L^{3}_{9}$, $L^{2}_{11}\subset L^{3}_{10}$,
$L^{2}_{11}\subset L^{6}_{1}$, $L^{2}_{11}\subset L^{12}$,
$L^{2}_{12}\subset L^{3}_{2}$, $L^{2}_{12}\subset L^{3}_{10}$,
$L^{2}_{12}\subset L^{4}_{7}$, $L^{2}_{12}\subset L^{6}_{1}$,
$L^{2}_{12}\subset L^{6}_{3}$, $L^{2}_{12}\subset L^{12}$,
$L^{2}_{13}\subset L^{4}_{1}$, $L^{2}_{13}\subset L^{4}_{6}$,
$L^{2}_{13}\subset L^{4}_{7}$, $L^{2}_{13}\subset L^{6}_{2}$,
$L^{2}_{13}\subset L^{6}_{3}$, $L^{2}_{13}\subset L^{12}$,
$L^{2}_{14}\subset L^{3}_{5}$, $L^{2}_{14}\subset L^{3}_{11}$,
$L^{2}_{14}\subset L^{4}_{9}$, $L^{2}_{14}\subset L^{6}_{5}$,
$L^{2}_{14}\subset L^{6}_{7}$, $L^{2}_{14}\subset L^{6}_{8}$,
$L^{2}_{14}\subset L^{12}$, $L^{2}_{15}\subset L^{4}_{2}$,
$L^{2}_{15}\subset L^{4}_{5}$, $L^{2}_{15}\subset L^{4}_{8}$,
$L^{2}_{15}\subset L^{4}_{9}$, $L^{2}_{15}\subset L^{6}_{4}$,
$L^{2}_{15}\subset L^{6}_{5}$, $L^{2}_{15}\subset L^{6}_{6}$,
$L^{2}_{15}\subset L^{6}_{7}$, $L^{2}_{15}\subset L^{12}$,
$L^{2}_{16}\subset L^{3}_{11}$, $L^{2}_{16}\subset L^{3}_{12}$,
$L^{2}_{16}\subset L^{4}_{2}$, $L^{2}_{16}\subset L^{6}_{4}$,
$L^{2}_{16}\subset L^{6}_{5}$, $L^{2}_{16}\subset L^{6}_{8}$,
$L^{2}_{16}\subset L^{12}$, $L^{2}_{17}\subset L^{3}_{6}$,
$L^{2}_{17}\subset L^{3}_{12}$, $L^{2}_{17}\subset L^{4}_{8}$,
$L^{2}_{17}\subset L^{6}_{4}$, $L^{2}_{17}\subset L^{6}_{6}$,
$L^{2}_{17}\subset L^{6}_{8}$, $L^{2}_{17}\subset L^{12}$,
$L^{2}_{18}\subset L^{3}_{9}$, $L^{2}_{18}\subset L^{3}_{13}$,
$L^{2}_{18}\subset L^{6}_{1}$, $L^{2}_{18}\subset L^{6}_{6}$,
$L^{2}_{18}\subset L^{12}$, $L^{2}_{19}\subset L^{3}_{3}$,
$L^{2}_{19}\subset L^{3}_{13}$, $L^{2}_{19}\subset L^{4}_{8}$,
$L^{2}_{19}\subset L^{6}_{1}$, $L^{2}_{19}\subset L^{6}_{4}$,
$L^{2}_{19}\subset L^{6}_{6}$, $L^{2}_{19}\subset L^{12}$,
$L^{2}_{20}\subset L^{4}_{3}$, $L^{2}_{20}\subset L^{4}_{6}$,
$L^{2}_{20}\subset L^{4}_{8}$, $L^{2}_{20}\subset L^{6}_{2}$,
$L^{2}_{20}\subset L^{6}_{4}$, $L^{2}_{20}\subset L^{6}_{6}$,
$L^{2}_{20}\subset L^{12}$, $L^{2}_{21}\subset L^{4}_{4}$,
$L^{2}_{21}\subset L^{4}_{7}$, $L^{2}_{21}\subset L^{4}_{9}$,
$L^{2}_{21}\subset L^{6}_{3}$, $L^{2}_{21}\subset L^{6}_{5}$,
$L^{2}_{21}\subset L^{6}_{7}$, $L^{2}_{21}\subset L^{12}$,
$L^{2}_{22}\subset L^{3}_{11}$, $L^{2}_{22}\subset L^{3}_{14}$,
$L^{2}_{22}\subset L^{4}_{4}$, $L^{2}_{22}\subset L^{6}_{3}$,
$L^{2}_{22}\subset L^{6}_{5}$, $L^{2}_{22}\subset L^{6}_{8}$,
$L^{2}_{22}\subset L^{12}$, $L^{2}_{23}\subset L^{3}_{7}$,
$L^{2}_{23}\subset L^{3}_{14}$, $L^{2}_{23}\subset L^{4}_{7}$,
$L^{2}_{23}\subset L^{6}_{3}$, $L^{2}_{23}\subset L^{6}_{8}$,
$L^{2}_{23}\subset L^{12}$, $L^{2}_{24}\subset L^{3}_{13}$,
$L^{2}_{24}\subset L^{3}_{15}$, $L^{2}_{24}\subset L^{4}_{5}$,
$L^{2}_{24}\subset L^{6}_{1}$, $L^{2}_{24}\subset L^{6}_{6}$,
$L^{2}_{24}\subset L^{6}_{7}$, $L^{2}_{24}\subset L^{12}$,
$L^{2}_{25}\subset L^{3}_{10}$, $L^{2}_{25}\subset L^{3}_{15}$,
$L^{2}_{25}\subset L^{6}_{1}$, $L^{2}_{25}\subset L^{6}_{7}$,
$L^{2}_{25}\subset L^{12}$, $L^{2}_{26}\subset L^{3}_{4}$,
$L^{2}_{26}\subset L^{3}_{15}$, $L^{2}_{26}\subset L^{4}_{9}$,
$L^{2}_{26}\subset L^{6}_{1}$, $L^{2}_{26}\subset L^{6}_{5}$,
$L^{2}_{26}\subset L^{6}_{7}$, $L^{2}_{26}\subset L^{12}$,
$L^{2}_{27}\subset L^{3}_{14}$, $L^{2}_{27}\subset L^{3}_{16}$,
$L^{2}_{27}\subset L^{4}_{1}$, $L^{2}_{27}\subset L^{6}_{2}$,
$L^{2}_{27}\subset L^{6}_{3}$, $L^{2}_{27}\subset L^{6}_{8}$,
$L^{2}_{27}\subset L^{12}$, $L^{2}_{28}\subset L^{3}_{12}$,
$L^{2}_{28}\subset L^{3}_{16}$, $L^{2}_{28}\subset L^{4}_{3}$,
$L^{2}_{28}\subset L^{6}_{2}$, $L^{2}_{28}\subset L^{6}_{4}$,
$L^{2}_{28}\subset L^{6}_{8}$, $L^{2}_{28}\subset L^{12}$,
$L^{2}_{29}\subset L^{3}_{8}$, $L^{2}_{29}\subset L^{3}_{16}$,
$L^{2}_{29}\subset L^{4}_{6}$, $L^{2}_{29}\subset L^{6}_{2}$,
$L^{2}_{29}\subset L^{6}_{8}$, $L^{2}_{29}\subset L^{12}$.
* (i)
for $3$-dimensional:
$L^{3}_{1}\subset L^{6}_{1}$, $L^{3}_{1}\subset L^{6}_{2}$,
$L^{3}_{1}\subset L^{12}$, $L^{3}_{2}\subset L^{6}_{1}$, $L^{3}_{2}\subset
L^{6}_{3}$, $L^{3}_{2}\subset L^{12}$, $L^{3}_{3}\subset L^{6}_{1}$,
$L^{3}_{3}\subset L^{6}_{4}$, $L^{3}_{3}\subset L^{12}$, $L^{3}_{4}\subset
L^{6}_{1}$, $L^{3}_{4}\subset L^{6}_{5}$, $L^{3}_{4}\subset L^{12}$,
$L^{3}_{5}\subset L^{6}_{7}$, $L^{3}_{5}\subset L^{6}_{8}$,
$L^{3}_{5}\subset L^{12}$, $L^{3}_{6}\subset L^{6}_{6}$, $L^{3}_{6}\subset
L^{6}_{8}$, $L^{3}_{6}\subset L^{12}$, $L^{3}_{7}\subset L^{6}_{8}$,
$L^{3}_{7}\subset L^{12}$, $L^{3}_{8}\subset L^{6}_{8}$, $L^{3}_{8}\subset
L^{12}$, $L^{3}_{9}\subset L^{6}_{1}$, $L^{3}_{9}\subset L^{12}$,
$L^{3}_{10}\subset L^{6}_{1}$, $L^{3}_{10}\subset L^{12}$,
$L^{3}_{11}\subset L^{6}_{5}$, $L^{3}_{11}\subset L^{6}_{8}$,
$L^{3}_{11}\subset L^{12}$, $L^{3}_{12}\subset L^{6}_{4}$,
$L^{3}_{12}\subset L^{6}_{8}$, $L^{3}_{12}\subset L^{12}$,
$L^{3}_{13}\subset L^{6}_{1}$, $L^{3}_{13}\subset L^{6}_{6}$,
$L^{3}_{13}\subset L^{12}$, $L^{3}_{14}\subset L^{6}_{3}$,
$L^{3}_{14}\subset L^{6}_{8}$, $L^{3}_{14}\subset L^{12}$,
$L^{3}_{15}\subset L^{6}_{1}$, $L^{3}_{15}\subset L^{6}_{7}$,
$L^{3}_{15}\subset L^{12}$, $L^{3}_{16}\subset L^{6}_{2}$,
$L^{3}_{16}\subset L^{6}_{8}$, $L^{3}_{16}\subset L^{12}$.
* (i)
for $4$-dimensional:
$L^{4}_{1}\subset L^{6}_{2}$, $L^{4}_{1}\subset L^{6}_{3}$,
$L^{4}_{1}\subset L^{12}$, $L^{4}_{2}\subset L^{6}_{4}$, $L^{4}_{2}\subset
L^{6}_{5}$, $L^{4}_{2}\subset L^{12}$, $L^{4}_{3}\subset L^{6}_{2}$,
$L^{4}_{3}\subset L^{6}_{4}$, $L^{4}_{3}\subset L^{12}$, $L^{4}_{4}\subset
L^{6}_{3}$, $L^{4}_{4}\subset L^{6}_{5}$, $L^{4}_{4}\subset L^{12}$,
$L^{4}_{5}\subset L^{6}_{6}$, $L^{4}_{5}\subset L^{6}_{7}$,
$L^{4}_{5}\subset L^{12}$, $L^{4}_{6}\subset L^{6}_{2}$, $L^{4}_{6}\subset
L^{12}$, $L^{4}_{7}\subset L^{6}_{3}$, $L^{4}_{7}\subset L^{12}$,
$L^{4}_{8}\subset L^{6}_{4}$, $L^{4}_{8}\subset L^{6}_{6}$,
$L^{4}_{8}\subset L^{12}$, $L^{4}_{9}\subset L^{6}_{5}$, $L^{4}_{9}\subset
L^{6}_{7}$, $L^{4}_{9}\subset L^{12}$.
* (i)
for $6$-dimensional:
$L^{6}_{1}\subset L^{12}$, $L^{6}_{2}\subset L^{12}$, $L^{6}_{3}\subset
L^{12}$, $L^{6}_{4}\subset L^{12}$, $L^{6}_{5}\subset L^{12}$,
$L^{6}_{6}\subset L^{12}$, $L^{6}_{7}\subset L^{12}$, $L^{6}_{8}\subset
L^{12}$.
## References
* [1] M. Aguilar-Benitez et al., Particles Data Group , Phys. Lett. 170B (1986).
* [2] H. Bhabha, Proc, Roy. Soc., A154 (1936) 195.
* [3] P. Corsini, Prolegomena of hypergroup theory, Second edition, Aviani editor, (1993).
* [4] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, (2003).
* [5] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, (2007).
* [6] B. Davvaz and A. Dehghan Nezhad, Chemical examples in Hypergroups, Ratio Mathematica-Numero 14 (2003), 71-74.
* [7] B. Davvaz, A. Dehghan Nezhad, and A. Benvidi, Chain reactions as experimental examples of ternary algebraic hyperstructures, Communications in mathematical and in computer chemistry, (2011), pp. 00.
* [8] A. Dehghan Nezhad and B. Davvaz, Universal hyperdynamical systems, Bulletin of the Korean Mathematical Society, 47 (2010), No. 3, pp. 513-528.
* [9] D. Griffiths, Introduction to Elementary Particles, John Wiley $\&$ Sons, (1987).
* [10] F. Halzen and A. Martin, Quarks $\&$ Leptons: An Introductory Course in Modern Particle Physics, John Wiley $\&$ Sons, (1984).
* [11] F. Marty, Sur une generalization de la notion de groupe, $8^{th}$ Congress Math. Scandenaves, Stockholm (1934), 45-49.
* [12] T. Muta, Foundations of quantum chromodynamics. Second edition, World Sci. Lect. Notes Phys. 57 (1998) 1.
* [13] W. Prenowitz and J. Jantosciak, Join Geometries, Springer-Verlag, UTM., (1979).
* [14] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Florida, (1994).
|
arxiv-papers
| 2010-08-04T13:09:03 |
2024-09-04T02:49:12.053811
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Akbar Dehghan Nezhad and Mehdi Nadjafikhah and Seyed Mohammad Moosavi\n Nejad",
"submitter": "Mehdi Nadjafikhah",
"url": "https://arxiv.org/abs/1008.0772"
}
|
1008.0990
|
# Demonstration of a microfabricated surface electrode ion trap
D. Stick dlstick@sandia.gov K. M. Fortier R. Haltli C. Highstrete D. L.
Moehring C. Tigges M. G. Blain Sandia National Laboratories, Albuquerque,
New Mexico 87185, USA
###### Abstract
In this paper we present the design, modeling, and experimental testing of
surface electrode ion traps fabricated in a heterostructure configuration
comprising a silicon substrate, silicon dioxide insulators, and aluminum
electrodes. This linear trap has a geometry with symmetric RF leads, two
interior DC electrodes, and 40 individual lateral DC electrodes. Plasma
enhanced chemical vapor deposition (PECVD) was used to grow silicon dioxide
pillars to electrically separate overhung aluminum electrodes from an aluminum
ground plane. In addition to fabrication, we report techniques for modeling
the control voltage solutions and the successful demonstration of trapping and
shuttling ions in two identically constructed traps.
Trapped Ions, Microfabrication, Quantum Computing, Laser Cooling
## I Introduction
Individually trapped ions are a leading candidate for quantum information
processing blatt:2008 ; wineland:2009 , as most of the DiVincenzo requirements
have already been substantially realized divincenzo:2000 . Of these
requirements, the current limiting factor is whether trapped ions constitute a
“scalable physical system”, due to the difficulty of creating large trapping
structures capable of independently controlling tens or hundreds of ions.
There has been an increasing emphasis on creating scalable architectures
wineland:1998 ; kielpinski:2002 ; rowe:2002 ; madsen:2004 ; stick:2006 ;
hensinger:2006 ; reichle:2006 ; brownnut:2006 ; leibfried:2007 ; amini:2008 ;
blakestad:2009 , with surface traps being an especially promising approach due
to their compatibility with standard fabrication techniques like
photolithography, via technology, wire bonding, and metal evaporation
chiaverini:2005 ; britton:2006 ; seidelin:2007 ; brown:2007 ; allcock:2009 ;
britton:2009 ; leibrandt:2009 ; hellwig:2009 . Most importantly, junctions and
backside loading holes can be incorporated in a surface geometry, the latter
of which we discuss in this paper.
## II Fabrication
Surface electrode traps have been previously demonstrated by a number of
groups seidelin:2007 ; brown:2007 ; allcock:2009 ; leibrandt:2009 using a
variety of techniques (e.g. gold on quartz, printed circuit board, aluminum on
silicon oxide). The traps reported here are similar in spatial scale and
geometry to these traps; however, particular emphasis is placed on the design
principle of minimizing the line of sight access to the ion from exposed
dielectrics, thereby reducing the impact of stray electric charges. Numerous
observations have been made of shifting trapping potentials over long time
scales (seconds) due to changes in the location and magnitude of stray charges
harlander:2010 .
To realize this design principle, the top metal layer of these traps
(comprising electrodes, their leads, and outside grounded regions) overhang
their supporting oxide pillars by 5 $\mu$m. The oxide pillars are grown
through multiple layers of plasma enhanced chemical vapor deposition, and are
between 9 and 14 $\mu$m thick (the traps reported here have 9 $\mu$m pillars).
The overhang distance is a controllable value achieved by using vertical etch
stops around the pillars (Figure 1). The overhang allows for vertical
deposition of metal on top of the aluminum electrode layer without shorting DC
control or RF electrodes. The lateral separation between electrically isolated
top metal layers (such as between neighboring electrodes) is set to be 7
microns, and the lateral dimensions of the electrodes can be arbitrarily
determined (see Figures 2 and 3 for specific dimensions). A hole through the
Si substrate of the trap chip runs the entire length of the trapping region to
allow for loading of ions from the backside of the trap (preventing shorting
of the trap electrodes by the atoms, which can occur when loading ions from
the side). DC rails inside the RF rails allow for additional principle axis
rotation and compensation. The back side of the chip is evaporated with gold
at a small off-normal angle to coat the exposed vertical edges of the silicon
substrate and the platform which supports the electrodes. This prevents charge
buildup by pinning the backside of the chip to ground (Figure 2a).
Figure 1: SEM image showing the 5 micron overhang from the supporting oxide
pillar and the 7 micron gap between neighboring electrodes.
Once fabricated, the chip is mounted in a 100 pin ceramic pin grid array
(CPGA) package from Kyocera [21]. A conductive cyanate ester adhesive (Johnson
Matthey Electronics JM7000) is used to attach the chip to a 1.5 mm thick
ceramic spacer, which holds the surface of the trap above the surface of the
package. The package and chip back side surfaces are gold coated to
electrically connect and ground the back of the package, the chip, the epoxy,
and the vertical silicon sidewalls (Figure 2).
Gold ribbons (12.5 $\mu$m thick by 75 $\mu$m wide) are wedge bonded to each
electrode at I/O pads located on an electrical plane (M1) 9 $\mu$m beneath the
trapping electrode plane (M2) and pulled taut to the package bond pads to
minimize their projection above the plane of the trap surface. The lower metal
plane, M1, serves primarily as a ground plane to prevent RF coupling into the
lossy silicon substrate. Each trap is electrically tested for shorts between
any trap electrodes (DC, RF, and grounds, including the ground plane). For
results using both traps reported here, no shorts below 100 M$\Omega$ were
observed. The RF capacity of the trap is measured by applying an RF drive (30
MHz at 2 Watts of power) to the trap through a resonator with a Q of 150. The
total capacitance from RF to ground (trap and CPGA package) is 7 pF. The
control electrodes are capacitively grounded outside of the vacuum chamber,
and low pass filters (3 kHz or 200 kHz, depending on whether shuttling tests
are being performed) are used to attenuate electronic noise.
Figure 2: Cross-sectional (a) and overhead (b) schematic of the ion trap.
Figure 3: Electrode layout.
## III Modeling
A custom boundary element modeling approach was used to find the charge
solution for all 42 control electrodes and the RF electrode. The area of each
element was chosen such that each element had the same charge; this
corresponds to larger elements at points farther away from the electrode in
question, and serves to standardize the error per element. The RF null was
then determined by finding the minimum pseudo-potential along the linear
trapping region. Control solutions were generated in order to maximize a
weighted figure of merit that includes secular frequency, trap depth, and
principal axis rotation, such that the electric field at a particular position
along the RF null is zero. The solution was verified using a flight simulator
which determined the ion’s motion according to the electric field at its
position (including both the control solution and an oversampled RF drive).
The ion’s flight was verified for thousands of times longer than the period of
the RF drive voltage, and the secular frequencies determined by Fourier
transforming the ion’s motion.
The simulations were experimentally validated by applying a voltage to each
control electrode such that the ion moved a fixed distance of 2 $\mu$m
axially. This was repeated many times and an image was take for each offset.
The ion’s location was precisely determined using a Gaussian fit of the image,
and the simulations corresponded to the experimental measurements of the ion’s
motion to within the error of the position measurement (10%).
## IV Trap Performance
The ion is observed to be trapped 80 $\mu$m above the top electrode layer,
consistent with simulations. The uncooled ion lifetime in the first trap was
3-5 minutes, and was observed to be sensitive to DAC cable shielding
(lifetimes dropped to $\sim$10 s without twisted pair shielding). The cooled
ion lifetime remained on the order of several hours throughout this period.
Two ion traps were tested side-by-side in separate UHV chambers, using
identical control voltage sets for storage and shuttling. The traps were
operated at a wide range of RF drive frequencies, although trapping multiple
Ca+ ions was easier with a $43$ MHz RF drive frequency compared to a $27$ MHz
drive frequency.
The secular frequencies for a given voltage set (typical frequencies are 1 MHz
axial and 4 MHz radial) were measured by observing driven motion for resonant
tickling voltages jefferts:1995 and by measuring the separation between two
trapped ions wineland:1998 , and were consistent over time. The observed drift
of the ion was measured to be $\leq.5\mu$m (axially) over a 250 s period of
observation, and depended on the power of the UV laser beams (Doppler and
photoionization) and the extent to which they struck the surface. To prevent
charge buildup on the inside of the imaging viewport, a wire mesh (88% open)
was attached inside the vacuum chamber, 1.5 mm above the surface of the trap.
Through simulations it was determined that this would have minimal impact on
the trapping potential.
The RF voltage is delivered through a cavity resonator with a Q $\approx 100$,
and an amplitude between 50 V and 200 V. By measuring the change in radial and
axial secular frequencies when scaling a particular DC voltage set at a fixed
RF voltage, the geometric potential factors were determined for the control
electrodes. An applied DC offset to the RF electrodes changes both the radial
secular frequencies and the rotation of the principal axes. The RF voltage
amplitude and principal axis rotation can be determined to within a few
percent by fitting these frequencies to a numerical model (Figure 4), and
agreed with electrostatic simulations to 10%.
Figure 4: The secular frequency was measured as a function of varying voltage
offsets $U_{r}$ to the RF electrode. Fitting this curve allowed us to
determine the amplitude of the RF voltage and the principal axis rotation, for
a particular DC voltage set and RF power applied. For this particular case,
the RF amplitude was determined to be 140 V (amplitude) and the principal axis
rotation was 39 degrees from vertical.
Motional control was demonstrated by shuttling a single ion over half the
length of the trapping structure (10 electrodes, 770 microns) for $10^{6}$
times without loss. This is a total travel distance of just over 1.5 km, and
was performed at a maximum average velocity of .77 m/s. Ion chains have also
been split into two parts and recombined. Future work on this trap will
include measurements of the induced heating of the ion for these shuttling and
splitting operations.
## V Conclusion
For trapped ions to be a suitable platform for quantum computing, a scalable-
in-principle technique for trap fabrication has to be demonstrated. The
surface geometry is the most amenable to microfabrication, but it poses
challenges related to the low trap depth and difficulty in making a working
shuttling junction. Our demonstration of a microfabricated surface electrode
trap addresses this first challenge, and given its consistency of fabrication
and trap performance, can be used to create more sophisticated structures with
similarly repeatable performance.
## References
* (1) R. Blatt and D. Wineland. Nature 453, 1008 (2008).
* (2) D. J. Wineland. Phys. Scr. T137, 014007 (2009).
* (3) D. P. DiVincenzo. Fortschr. Phys. 48, 771 (2000).
* (4) D. J. Wineland, et al. Journal of Research of the National Institute of Standards and Technology 103, 259 (1998).
* (5) D. Kielpinski, C. Monroe, and D. Wineland. Nature 417, 709 (2002).
* (6) M. A. Rowe, et al. Quant. Inf. Comp. 2, 257 (2002).
* (7) M. J. Madsen, W. K. Hensinger, D. Stick, J. A. Rabchuk, and C. Monroe. Appl. Phys. B 78, 639 (2004).
* (8) D. Stick, et al. Nature Phys. 2, 36 (2006).
* (9) W. K. Hensinger, et al. Appl. Phys. Lett. 88, 034101 (2006).
* (10) R. Reichle, et al. Fortschr. Phys. 54, 666 (2006).
* (11) M. Brownnutt, G. Wilpers, P. Gill, R. C. Thompson, and A. G. Sinclair. New J. Phys. 8, 232 (2006).
* (12) D. Leibfried, et al. Hyperfine Interactions 174, 1 (2007).
* (13) J. M. Amini, J. Britton, D. Leibfried, and D. J. Wineland. arXiv:0812.3907v1 [quant-ph] (2008).
* (14) R. B. Blakestad, et al. Phys. Rev. Lett. 102, 153002 (2009).
* (15) J. Chiaverini, et al. Quant. Inf. Comp. 5, 419 (2005).
* (16) J. Britton, et al. arXiv:quant-ph/0605170v1 (2006).
* (17) S. Seidelin, et al. Phys. Rev. Lett. 96, 253003 (2006).
* (18) K. R. Brown, et al. Phys. Rev. A 75, 015401 (2007).
* (19) D. T. C. Allcock, et al. New J. Phys. 12, 053026 (2010).
* (20) J. Britton, et al. Appl. Phys. Lett. 95, 173102 (2009).
* (21) D. R. Leibrandt, et al. Quant. Inf. Comp. 9, 901 (2009).
* (22) M. Hellwig, A. Bautista-Salvador, K. Singer, G. Werth, and F. Schmidt-Kaler. New J. Phys. 12, 065019 (2010).
* (23) M. Harlander, M. Brownnutt, W. Hänsel, and R. Blatt. arXiv:1004.4842v1 [quant-ph] (2010).
* (24) S. R. Jefferts, C. Monroe, E. W. Bell, and D. J. Wineland. Phys. Rev. A 51, 3112 (1995).
|
arxiv-papers
| 2010-08-05T14:58:10 |
2024-09-04T02:49:12.064584
|
{
"license": "Public Domain",
"authors": "D Stick, K M Fortier, R Haltli, C Highstrete, D L Moehring, C Tigges,\n M G Blain",
"submitter": "Daniel Stick",
"url": "https://arxiv.org/abs/1008.0990"
}
|
1008.1004
|
# Identification of Overlapping Communities
by Locally Calculating Community-Changing Resolution Levels
Frank Havemann
Michael Heinz
Alexander Struck Institut für Bibliotheks- und Informationswissenschaft
Humboldt-Universität zu Berlin
Unter den Linden 6
10099 Berlin
Germany Jochen Gläser Zentrum für Technologie und Gesellschaft
Technische Universität Berlin
Hardenbergstr. 36A
10623 Berlin
Germany
###### Abstract
The identification of thematic structures in networks of bibliographically or
lexically coupled papers is hindered by the fact that most publications
address more than one theme, which in turn means that themes overlap in
publications. An algorithm for the detection of overlapping natural
communities in networks was proposed by Lancichinetti, Fortunato, and Kertesz
(LFK) last year [1]. The LFK algorithm constructs natural communities of (in
principle) all nodes of a graph by maximising the local fitness of
communities. The authors define fitness as the ratio of the number of internal
links to the number of all links of the nodes of a community but the
denominator of the ratio is raised to the power of $\alpha$. This parameter
can be interpreted as the resolution at which natural communities are
determined. The resulting communites can, and are due to the constructing
approach likely to, overlap. The generation of communities can easily be
repeated for many values of $\alpha$; thus allowing different views on the
network at different resolutions. We implemented the main idea of the LFK
algorithm—to search for natural communities of each node of a network—in a
different way. We start with a value of the resolution parameter that is high
enough for each node to be its own natural community. When the resolution is
reduced, each node acquires other nodes as members of its natural community,
i.e. natural communities grow. For each community found at a certain $\alpha$
value we calculate the next lower $\alpha$ where a node is added. After adding
a node to a community of seed node $k$ we check whether the natural community
of node $k$ is also the natural community of a node that we have already
analysed. If this is the case, we can stop analysing node $k$. We tested our
algorithm on a small benchmark graph and on a network of about 500 papers in
information science weighted with the Salton index of bibliographic coupling.
In our tests, this approach results in characteristic ranges of $\alpha$ where
a large resolution change does not lead to a growth of the natural community.
Such results were also obtained by applying the LFK algorithm but since we
determine communities for all resolution values in one run, our approach is
faster than the original LFK approach.111The results presented were also shown
on a poster with the title A local algorithm to get overlapping communities at
all resolution levels in one run at ASONAM conference, Odense, Denmark, August
2010.
## I Introduction
Many real-world networks consist of substructures that overlap because nodes
are members of more than one substructure. Networks of scientific papers are a
case in point. Thematic structures such as common topics, approaches, or
methods are not disjunct. It is the rule rather than the exception that a
paper addresses more than one topic.
Hard clustering is inadequate for the investigation of real-world networks
with such overlapping substructures. Instead, methods are required that allow
nodes to be members of more than one community in the network. During the last
years a number of algorithms for detecting overlapping communities (or
modules) in graphs have been developed and tested. One approach starts from
hard clusters obtained by any clustering method and assigns the nodes at the
borders of clusters to several neighbouring modules [2, 3]. In another
approach links are clustered into disjoint modules and nodes are members of
all modules their links belong to [4, 5]. Our paper is based on a third
approach that constructs natural communities of all nodes which can overlap
each other [1].
In our search for methods that model scientific specialties as networks of
journal papers and enable the identification of thematic structures in those
networks, we applied the algorithm developed by Lancichinetti, Fortunas, and
Kertesz [1]. This LFK algorithm is well suited to our problem because it
identifies not only overlapping communities but also a hierarchical structure
of a graph if there is any. Since we assume that thematic structures are of
varying scope and that some of the smaller themes might be completely
contained in larger ones, an algorithm that detects both overlaps and
hierarchies is essential.
The main assumption of the LFK algorithm is that every node has its own
natural community. In our context this approach can be interpreted as the
construction of a thematic environment from the ’scientific perspective‘ of
the seed paper. This idea is not only attractive from a conceptual point of
view—the borders of topics are explored by a local algorithm i.e.
independently from papers located far away from the seed paper—but also for
services leading users of bibliographic databases from one relevant paper to
thematically similar ones.
The essence of the LFK algorithm is that independently constructed natural
communities of nodes can overlap. In accordance with the locality of their
approach Lancichinetti, Fortunas, and Kertesz evaluate the fitness of modules
of nodes with a function that uses only local information. It is based on the
assumption that a community should have more internal than external links. The
fitness function is defined as the ratio of the sum of internal degrees to the
sum of all degrees of nodes in a module $G$. The denominator is taken to the
power of $\alpha$, the resolution parameter:
$f(G,\alpha)=\frac{k_{in}(G)}{(k_{in}(G)+k_{out}(G))^{\alpha}}.$ (1)
For each node a natural community $G$ is constructed by including the
neighbour that produces the highest fitness gain. Then the fitness gain of
each node in $G$ is recalculated. If it is negative remove this node from $G$.
The community is complete if including any neighbour brings no fitness gain.
The authors conclude [1, p. 6]: “By varying the resolution parameter one
explores the whole hierarchy of covers of the graph, from the entire network
down to the single nodes, leading to the most complete information on the
community structure of the network.”
Since the LFK algorithm constructs natural communities of all nodes of a graph
and has to be repeated for each value of the resolution parameter within the
interval of interest, applying it to larger networks is time-consuming.
Acknowledging this, the authors proposed several ways in which their algorithm
could be optimised. They tested an implementation that starts from a random
node and after construction of its community switches to the next random node
outside this community until the whole graph is covered (we denote this
version of the algorithm by random LFK). Lancichinetti, Fortunas, and Kertesz
also proposed to use communities found at one level of resolution as starting
points for the next lower level because at lower resolution a community cannot
be smaller than at higher level.
We implemented the main idea of the LFK algorithm—to search for natural
communities of each node of a network—in a different way. For some
sufficiently high value of the resolution parameter alpha each node is a
single, i.e. it is its own natural community. Lowering the resolution makes
the single nodes include ‘companions’ because this increases the community’s
fitness function. The inclusion of nodes makes the natural community of each
node grow. For each community found at some alpha we look for the next lower
alpha at which new members are acquired. Whenever a node is added to a natural
community of seed node $k$ we check whether the natural community of node $k$
is fully contained by the natural community of any other node. If this is the
case, we can stop analysing node $k$. This way, we merge (completely)
overlapping natural communities. Therefore we choose the acronym MONC for our
algorithm.
Since we determine communities for all resolution values in one run our
algorithm is faster than the original LFK algorithm. Both algorithms are
different implementations of the idea of growing natural communities of nodes,
i.e. they are not totally equivalent. We discuss the differences between the
two algorithms in the following section.222In the Further Work section of
reference [6, p. 9] Lee et al. mention that they are working on a version of
their algorithm which also expands all seeds in parallel.
## II Algorithm
We assume that each node is its own natural community $G$ at infinite
resolution. The next vertex $V$ from the neighbourhood of $G$ included to $G$
is the one that increases the fitness of $G$ at the largest value of
resolution denoted by $\alpha_{\mathrm{incl}}(G,V)$.
In pseudo code the growth of a natural community $G$ can be described as
follows ($N(G)$ denotes the neighbourhood of $G$):
1: while $N(G)$ is not empty do
2: for each node $V$ in $N(G)$ do
3: calculate $\alpha_{\mathrm{incl}}(G,V)$
4: end for
5: include the node with maximum $\alpha_{\mathrm{incl}}$ into $G$
6: end while
If two nodes have equal $\alpha_{\mathrm{incl}}$ MONC should include both
(which we did not implement for the experiments described below).
If we use the fitness function as defined by Lancichinetti et al. [1] a node
cannot remain a single because for any alpha the module fitness of a single is
always zero and the module fitness of two neighbours is always larger then
zero. We can avoid this drawback of the algorithm by adding self-links to all
nodes i.e. we assume that a node is a friend of itself or most similar to
itself. To get results closer to those of reference [1] we change the fitness
function $F(G)$ only slightly by adding 1 to the numerator:
$f(G,\alpha)=\frac{k_{\mathrm{in}}(G)+1}{(k_{\mathrm{in}}(G)+k_{\mathrm{out}}(G))^{\alpha}}.$
(2)
From this definition we can derive a formula for calculating the maximum value
of resolution $\alpha_{\mathrm{incl}}(G,V)$, where a node $V$ does not
diminish the fitness of a module $G$ when included in it by demanding that for
$\alpha<\alpha_{\mathrm{incl}}(G,V)$ we have $f(G\cup
V,\alpha)>f(G,\alpha)$:333cf. Supplementary Information
$\alpha_{\mathrm{incl}}(G,V)=\frac{\log(k_{\mathrm{in}}(G\cup
V)+1)-\log(k_{\mathrm{in}}(G)+1)}{\log k_{\mathrm{tot}}(G\cup V)-\log
k_{\mathrm{tot}}(G)},$ (3)
where $k_{\mathrm{tot}}=k_{\mathrm{in}}+k_{\mathrm{out}}$ denotes the sum of
the degrees of all nodes of a module.
We can calculate $k_{\mathrm{in}}(G\cup V)$ from $k_{\mathrm{in}}(G)$ and
$k_{\mathrm{tot}}(G\cup V)$ from $k_{\mathrm{tot}}(G)$ i.e. the current values
of the module from the preceding ones (which saves computing time). For this
we define the interaction of a module and a node as
$k_{\mathrm{inter}}(G,V)=\sum_{i\in G}A_{Vi},$ (4)
where $A$ denotes the adjacency matrix of the undirected (and in general)
weighted graph and calculate the degree of a node or its weight as the sum of
the weights of its edges
$A_{V+}=\sum_{i}A_{Vi}.$ (5)
The weight of edges of internal nodes $k_{\mathrm{in}}$ is increased by
$2\cdot k_{\mathrm{inter}}$ because both directions have to be taken into
account:
$k_{\mathrm{in}}(G\cup V)=k_{\mathrm{in}}(G)+2\cdot k_{\mathrm{inter}}(G,V).$
(6)
The total of all weights is increased by the weights of the edges of the new
node:
$k_{\mathrm{tot}}(G\cup V)=k_{\mathrm{tot}}(G)+A_{V+}.$ (7)
We first include the neighbour $V$ of each node that improves the community’s
fitness at highest resolution. Then we continue with the new neighbourhood of
$G\cup V$ until all nodes are included in the natural community. After each
step we compare the current communities of all nodes to find duplicates. Thus
we can reduce the number of communities treated by the inclusion algorithm and
save further computing time. We merge overlapping natural communities of
nodes.
In addition to the changed fitness function described above, we deviated from
LFK’s approach in two more points. First, we do not allow the removal of nodes
from a natural community. The LFK algorithm rechecks the fitness contribution
of all community nodes after a new node has been added and excludes nodes if
their removal increases the fitness. However, this possibility of exclusion
contradicts the principle of locality. It can even lead to the exclusion of a
seed node from its own natural community. In our networks of papers, removing
nodes that reduce the fitness of a grown community is equivalent to shifting
from the individual thematic perspective of the seed paper to a collective
perspective of all papers in the community. Therefore our algorithm does not
remove nodes from a community. Similarly, Lee, Reid, McDaid, and Hurley [6]
implemented the LFK algorithm without exclusion mechanism.
Another modification concerns the starting point of the algorithm. If a graph
is characterised by a strong variation of its local density and the seed node
is located in a high density region, the MONC algorithm immediately leaves
this region because it searches for nodes with low degree first. These outside
nodes only moderately increase the number of links leaving the community and
thus often provide the earliest increase in fitness. We surmise that the LFK
algorithm ‘repairs’ this unwanted behaviour by allowing the exclusion of nodes
with negative fitness. Since we suppressed the exclusion of nodes, we solved
this problem by starting from cliques (i.e. totally linked subgraphs) instead
of single nodes. Lee et al., who applied the LFK algorithm without the
exclusion mechanism, also found that cliques as seeds gave better results than
single nodes [6].
While Lee et al.[6] use maximal cliques (i.e. cliques which are not subgraphs
of other cliques), we optimise clique size by excluding nodes that are only
weakly integrated. Thus, for our starting points we apply an analogon of the
LFK exclusion mechanism. In detail, we exclude the node $V$ that diminishes
the module fitness at lowest resolution, i.e. has the weakest coupling to the
rest of the module $G$. Analogously to $\alpha_{\mathrm{incl}}$ we calculate
$\alpha_{\mathrm{excl}}$ with
$\alpha_{\mathrm{excl}}(G,V)=\frac{\log(k_{\mathrm{in}}(G)+1)-\log(k_{\mathrm{in}}(G\setminus
V)+1)}{\log k_{\mathrm{tot}}(G)-\log k_{\mathrm{tot}}(G\setminus V)}.$ (8)
This procedure is repeated until only two nodes remain in each clique. From
the set of shrinking cliques we select the one which is most resistant to
further reduction i.e. those with highest $\alpha_{\mathrm{excl}}$ of the next
node to be excluded. After its exclusion the rest of the clique would be less
strongly coupled (for details see section Experiments and cf. Figure 12 in
Supplementary Information). That means, we choose the most cohesive subgraph
of a clique as optimal.
After optimising all cliques larger than pairs we determine the optimal clique
belonging to a seed node by searching for the clique where the seed is member
and has its maximum $\alpha_{\mathrm{excl}}$. Nodes which are not member of
any optimal clique remain single seeds. Every other node is assigned to one
clique, some of them to the same one.
## III Data
To compare our algorithm to that of Lancichinetti et al. we first applied both
to the network of social relations of 34 members of the well-known karate club
observed by Zachary [7]. As Lancichinetti et al. [1] we used the unweighted
version of this network.444s.
http://networkx.lanl.gov/examples/graph/karate_club.html
We also applied random LFK and MONC to a network of about 500 papers in volume
2008 of six information-science journals with a high portion of bibliometrics
(see details in Supplementary Information).
In the network of information-science papers, two nodes (papers) are linked if
they both have at least one cited source in common. The number of shared
sources, which is normalised in order to account for different lengths of
reference lists, provides a measure of the thematic similarity of papers. We
start from the affiliation matrix $M$ of the bipartite network of papers and
their cited sources. To account for different lengths of reference lists we
normalise the paper vectors of $M$ to an Euclidean length of one. Then the
element $a_{ij}$ of matrix $A=MM^{\mathrm{T}}$ equals Salton’s cosine index of
bibliographic coupling between paper $i$ and $j$. The symmetric adjacency
matrix $A$ describes a weighted undirected network of bibliographically
coupled papers. The elements of the main diagonal all equal 1, which means
that a document is most similar to itself. We could proceed with this main
diagonal i.e. with self-links but we omit them in the experiments described
here (cf. Algorithm section).
The main component of the bibliographic-coupling network of information
science 2008 contains 492 papers. Two small components (three and two papers,
respectively) and 34 isolated papers are of no interest for our experiments.
## IV Experiments
### IV-A Karate Club
Figure 1: Growing natural community of node 1 of Karate Club Figure 2: Graph
of growing natural community of node 1 of Karate Club Figure 3: Growing
natural community of node 33 of Karate Club Figure 4: Graph of growing natural
community of node 33 of Karate Club Figure 5: Growing natural community of
node 3 of Karate Club Figure 6: Graph of growing natural community of node 3
of Karate Club
Since the network of 34 karate club members is sparse—there is no clique with
six or more fighters—we can apply MONC by starting from each node rather than
using seed cliques. Figures 1–6 show the growing natural communities of three
nodes. The step curve in the diagrams gives the growing number of nodes in the
community as a function of $1/\alpha$. Each node is its own community at
$1/\alpha=0$. In our approach, the resolution always decreases, i.e.
$1/\alpha$ cannot decrease.
For example (cf. Figure 1), even if nodes 11, 6, 7, and 17 enter the community
of node 1 at lower $1/\alpha$ than their predecessor node 5, we display the
same value of $1/\alpha$ for all five nodes because the higher resolution for
the other four nodes becomes possible only after node 5 has been included. In
other words, adding node 5 to the community changes the latter’s properties in
a way that would enable adding other nodes at a smaller value of $1/\alpha$.
The network graphs (Figures 2 and 4) visualise the growth of communities by
displaying the seed node in black, the last nodes joining in white, and the
intermediate nodes on a grey scale corresponding to the resolution at which
they come in. Lancichinetti et al. [1, Fig. 6(a), p. 10] display the cover of
the karate network they obtain in the resolution interval $.76<\alpha<.84$
(which roughly equals the inverse resolution interval $1.2<1/\alpha<1.3$). We
see from the diagrams and graphs of nodes 1 and 33 that the MONC algorithm
detects exactly the same cover in this interval, i.e. the same set of
overlapping communities which cover the whole graph.
Another cover in this resolution range is less frequently obtained using the
random LFK algorithm. It becomes visible in the diagram and graph of node 3, a
node in the overlap of the two communities of the cover displayed by
Lancichinetti et al. In this resolution range the community of node 3 contains
all nodes except the five nodes on the right end of the karate graph. The
communities of these five nodes are identical and contain no other node in the
resolution interval considered.
These examples indicate that for the karate club our MONC algorithm gives at
least approximately the same results as the LFK algorithm. A detailed
comparison reveals that 31 of the modules we found with our implementation of
random LFK were also detected by MONC. Table II in Supplementary Information
lists the corresponding resolution intervals for both algorithms. Small
differences are partly due to the different fitness functions (s. Algorithm
section) and partly due to the randomness of LFK. Further 22 LFK modules were
not found by MONC. Their resolution intervals are mostly small (maximum .2716,
median .045).
In addition MONC detected 23 modules which random LFK did not find. Each of
these modules is found as an intermediate state of a growing natural community
of only one seed node (cf. column number of seeds in Table II for the number
of seed nodes of modules). 14 of them have $\alpha_{min}>2$ and could not be
found by LFK because in our implementation it run down from $\alpha=2$ to
$\alpha=0.65$.
In summary, both algorithms are not equivalent but display similarities in
many of their results. The LFK algorithm finds some modules MONC does not
find. This is probably due to the exclusion mechanism of LFK that allows
shifting a module away from its seed node.
### IV-B Information Science
The 1812 maximal cliques of 492 bibliographically coupled information-science
papers published in 2008 differ strongly in size. There are many small maximal
cliques and some large ones. The density variation across the graph requires
starting the MONC algorithm with seed cliques.
The largest clique is formed by 46 papers which all cite the paper by J. E.
Hirsch in 2005 where he proposes the h-index: the Hirsch paper couples all
these 46 papers. Many h-clique papers also have the term h-index in their
titles but some of them discuss it only as a method among others. We reduce
the h-clique by the method described above to 21 papers which all have the
h-index or its derivatives as a central topic (cf. Figure 12 in Supplementary
Information; the distribution of clique sizes before and after reduction is
given by Table III in Supplementary Information section).
Most papers belong to more than one reduced clique. Each paper is assigned to
the clique where it has its maximum $\alpha_{\mathrm{excl}}$. This leads to
the selection of 357 reduced cliques. 16 papers have their highest
$\alpha_{\mathrm{excl}}$ in the h-clique. 275 cliques belong only to one node.
Three papers do not belong to any reduced clique and are therefore used as
single paper seeds.
Figure 7: Graph of growing natural community of h-index clique (nodes are
positioned by force directed placement) Figure 8: Growing natural community of
h-index clique up to 100 papers Figure 9: Graph of growing natural community
of information-retrieval papers Figure 10: Growing natural community of IR-
papers
As an example, Figure 7 shows the graph of the growing natural community of
one paper that has its highest $\alpha_{\mathrm{excl}}$ in the reduced
h-clique, whose 21 papers form the black core of the dark cloud in the figure.
The corresponding diagram in Figure 8 visualises the growing natural community
up to 100 papers. After collecting further 21 papers more or less related to
the topic (mostly citing the Hirsch paper) the community’s growth decelerates.
This slow development lasts till $1/\alpha\approx 1$ ending up with 51 papers.
We get the same succession of modules accumulating papers attached to the
h-community by applying the random LFK algorithm to information-science papers
published in 2008. Even the corresponding thresholds of $\alpha$ obtained by
both algorithms are nearly the same (Table IV in Supplementary Information
section). Small differences between thresholds can be explained. First, MONC
values are more precise because the LFK experiment was done in $\alpha$ steps
of 1/100. Second, the MONC experiment is based on the modified fitness formula
(with + 1 in the numerator).
Figure 9 shows a sequential graph displaying intermediate steps while growing
a community around a clique of information-retrieval (IR) papers (cf. Figure
10). It visualises the separation of IR papers (left) from papers in
bibliometrics (right hand side).
MONC detected 5091 different modules as intermediate states of growing natural
communities of nodes. Random LFK identified 1116 modules between $\alpha=2$
and $\alpha=0.1$ (in steps of 1/100). The $\alpha$ intervals of 3219 MONC
modules overlap with this $\alpha$ region and are larger or equal to 0.2. The
corresponding modules have therefore a realistic chance to be found by LFK,
too. All in all, 211 modules across the whole spectrum of sizes were detected
by both algorithms. LFK probably finds modules not found by MONC due to its
exclusion mechanism. In addition, some smaller modules cannot be found by MONC
because it here starts from cliques. MONC probably detects modules not found
by LFK due to the latter’s randomness.
The random LFK experiment started from $\alpha=2$ and went down in steps of
1/100 to $\alpha=0.1$. We implemented both algorithms as R-scripts.555R is an
interpreted language and runs slower than compiled implementations. LFK
reached $\alpha=0.83$ after four hours and fifty minutes.666Intel(R) Xeon(R)
CPU X5550@2.67 GHz with 72 GB RAM installed The next value 0.82 is minimum
$\alpha$ of 70 modules and took the algorithm more than three hours. All in
all our slow random LFK implementation as an R-script (without storing
community parameters, see Algorithm section) needed 41 hours.
A straightforward implementation of our MONC algorithm (also without storing
community parameters, see Algorithm section) reduced computation time to about
10 hours. The optimised version of MONC (with storing community parameters and
neighbourhoods) needed less than 12 minutes for the network of 492 nodes. In
addition, the resolution thresholds computed by the MONC algorithm are much
more accurate and the hierarchy of modules is detected automatically by MONC.
Figure 11: Number of active communities in Information-Science experiment as a
function of nodes included
To illustrate merging of communities, Figure 11 displays how the number of
active communities first rises to above 300 (of a maximum of 492 node
communities or of 360 different seed cliques) and then is falling rapidly,
thus making MONC faster. By active we denote growing communities which up to
the current number of nodes included have not been made inactive by merging of
communities. They will merge later. Only three communities survive before they
are merged into the whole set of all 492 nodes.
## V Summary and Conclusions
The LFK algorithm detects overlapping natural communities of all nodes by
maximising a local fitness function that enables the tuning of the procedure’s
resolution [1]. Below some minimum resolution all nodes have the whole
(connected) graph as their common natural community, while above some maximum
resolution all nodes remain singles. If the algorithm is repeated for
different resolution levels in an appropriate number of steps between maximum
and minimum the hierarchical structure of the graph can be determined by
comparing all communities found at all resolution levels considered.
To maximise the local fitness function LFK includes nodes into a community
that increase its fitness and excludes nodes reducing it. However, the
exclusion of nodes violates the locality of the algorithm because nodes coming
in later can ‘throw out’ nodes that came in earlier, among them even the seed
node. A variant of LFK without exclusion of nodes also gives reasonable
results if it starts from maximal cliques instead of single nodes [6].
Another problem of the LFK algorithm is that it is time-consuming. LFK has
been made faster by randomly choosing a new seed node that is not included in
any community detected so far [1]. Diminishing the effects of randomness can
and should be done by multiple runs at the same $\alpha$-level or by using
small $\alpha$-steps. The random procedure rests on the assumption that after
each node is assigned to at least one community no further community has to be
detected (cf. Lee et al. [6], p. 3). If this assumption is unrealistic for the
network considered the non-random LFK variant has to be applied.
We propose an algorithm (MONC) that also uses local fitness maximisation to
include nodes but which is faster than LFK because it identifies overlapping
natural communities of all nodes in one run. In our test on a weighted
bibliometric graph of about 500 information-science papers non-optimised MONC
was four times faster than our non-optimised random LFK implementation.
Optimisation of MONC by storing community data accelerated it by a factor 50.
MONC includes nodes into communities but does not exclude nodes which diminish
fitness. At each step MONC tests whether intermediate modules of growing
communities of different nodes are equal. If this is the case, the two
communities are merged. This not only makes MONC faster but automatically
reveals the hierarchy of the network’s modules that can be visualised as a
dendrogram of overlapping communities. Thus, MONC can be seen as a truly
hierarchical algorithm that clusters growing natural communities of a graph
instead of its nodes.
If we follow the reasoning of Lancichinetti et al. [1, pp. 6–7] we get
$O(n^{2}\log n)$ as the worst case complexity of random LFK algorithm. One
factor $n$ is due to the exclusion mechanism and $log(n)$ is the order of the
number of $\alpha$-levels needed to reveal the hierarchy of the network with
$n$ nodes. Hence, the computing time of non-random LFK variants should scale
with $n^{3}\log n$ and that of MONC with $n^{2}$ because MONC does not exclude
nodes and uncovers the whole hierarchy in one run. Furthermore, MONC saves
time due to merging of communities. The estimation of complexity should be
examined by applying MONC to benchmark graphs.
For each node MONC calculates the resolution thresholds at which its natural
community grows by including new nodes from the neighbourhood, thereby
identifying the (overlapping) natural communities of all nodes. Intervals of
resolution at which the community does not expand are detected. These
relatively stable intermediate modules of a community correspond to
communities found in many LFK runs for different levels of resolution. MONC
detects resolution intervals much more easily and more precisely than LFK.
In the bibliometric test graph papers about the h-index form an area that is
very much denser than the rest of the graph because they constitute a clique
(by citing the paper where Hirsch introduced the h-index). Starting MONC with
a node in a region of high density would (due to fitness maximization)
immediately lead to sparse regions of the graph. We therefore use cliques as
starting points. However, we do not use maximal cliques as Lee et al. [6] do
because in bibliographic-coupling networks this could mean starting with
papers that are only weakly related to the seed paper. We reduce the maximal
cliques by excluding nodes until the maximum resolution threshold of the
clique is obtained. This procedure results in cliques with maximum cohesion as
starting points of MONC.
Some intermediate modules obtained by MONC while expanding communities for two
test graphs coincide with (often important) LFK communities and also exist for
similar resolution intervals. We take this as a hint that MONC and LFK results
are of comparable validity. By inspection, the structure of both test graphs
obtained by MONC can be evaluated as reasonable and meaningful. This is why we
expect MONC to produce valid modules when applied to large benchmark graphs.
The local fitness function defined by Lancichinetti et al.[1] was selected by
these authors among several alternatives (not specified by them) after some
tests. We think that at least one alternative should be tested, namely the
function
$f(G,\beta)=\frac{k_{in}(G,\beta)}{k_{in}(G,\beta)+k_{out}(G)},$ (9)
with $k_{in}(G,\beta)=k_{in}(G)+\beta|G|.$ That means that we calculate
$k_{in}$ (the sum of internal degrees of nodes in $G$) but include self-links
of weight $\beta$. Using weighted self-links for tuning resolution of
modularity maximising methods was proposed and tested by Arenas, Fernández,
and Gómez in 2008[8]. They argue that the links between nodes are not changed
by adding self-links. Thus the topology of the graph is not altered.
## Acknowledgement
This work is part of a project in which we develop methods for measuring the
diversity of research. The project is funded by the German Ministry for
Education and Research (BMBF). We would like to thank all developers of
R.777http://www.r-project.org
## References
* [1] A. Lancichinetti, S. Fortunato, and J. Kertesz, “Detecting the overlapping and hierarchical community structure in complex networks,” _New J. Phys_ , vol. 11, p. 033015, 2009.
* [2] J. Baumes, M. Goldberg, and M. Magdon-Ismail, “Efficient identification of overlapping communities,” _Intelligence and Security Informatics_ , vol. 3495, pp. 27–36, 2005.
* [3] X. Wang, L. Jiao, and J. Wu, “Adjusting from disjoint to overlapping community detection of complex networks,” _Physica A: Statistical Mechanics and its Applications_ , vol. 388, no. 24, pp. 5045–5056, 2009.
* [4] Y.-Y. Ahn, J. P. Bagrow, and S. Lehmann, “Link communities reveal multi-scale complexity in networks,” _Nature_ , vol. 466, pp. 761–764, 2010, arXiv:0903.3178.
* [5] T. Evans and R. Lambiotte, “Edge Partitions and Overlapping Communities in Complex Networks,” _arXiv:0912.4389_ , 2009.
* [6] C. Lee, F. Reid, A. McDaid, and N. Hurley, “Detecting highly overlapping community structure by greedy clique expansion,” _arXiv:1002.1827v2_ , 2010\.
* [7] W. Zachary, “An information flow model for conflict and fission in small groups,” _Journal of Anthropological Research_ , vol. 33, no. 4, pp. 452–473, 1977.
* [8] A. Arenas, A. Fernández, and S. Gómez, “Analysis of the structure of complex networks at different resolution levels,” _New Journal of Physics_ , vol. 10, p. 053039, 2008.
## Supplementary Information
To all tables in this section there is a reference in the main text.
TABLE I: 533 Papers (528 articles and 5 letters) in volume 2008 of six information science journals (source: Web of Science) journal | papers
---|---
INFORMATION PROCESSING & MANAGEMENT | 111
JOURNAL OF DOCUMENTATION | 40
JOURNAL OF INFORMATION SCIENCE | 49
JOURNAL OF INFORMETRICS | 31
JOURNAL OF THE AMERICAN SOCIETY FOR |
INFORMATION SCIENCE AND TECHNOLOGY | 176
SCIENTOMETRICS | 126
sum | 533
TABLE II: 31 modules with at least two nodes in Karate-Club network found by MONC and by LFK (cf. section Results) number of | MONC | | number of | LFK |
---|---|---|---|---|---
nodes | $\alpha_{\mathrm{min}}$ | $\alpha_{\mathrm{max}}$ | seeds | $\alpha_{\mathrm{min}}$ | $\alpha_{\mathrm{max}}$
34 | 0.0000000 | 0.7563793 | 34 | 0.6500 | 0.7662
29 | 0.6835612 | 0.8952971 | 13 | 0.6887 | 0.8468
20 | 0.7535657 | 0.8915217 | 12 | 0.7630 | 0.9023
19 | 0.8915217 | 0.9823978 | 4 | 0.9024 | 1.1177
19 | 0.7563793 | 0.9056675 | 7 | 0.7663 | 0.8479
14 | 0.8332970 | 1.0117767 | 4 | 0.8480 | 1.0320
14 | 0.9823978 | 1.2892272 | 4 | 1.0000 | 1.2549
12 | 1.0117767 | 1.2542579 | 4 | 0.8650 | 1.2979
12 | 1.2892272 | 1.3175164 | 1 | 1.3186 | 1.3415
11 | 1.3175164 | 1.6524283 | 1 | 1.3524 | 1.3785
9 | 1.8726915 | 1.9478173 | 1 | 1.9518 | 2.0000
6 | 0.8119532 | 1.0716644 | 6 | 0.8663 | 1.1541
6 | 1.2883392 | 2.1054487 | 1 | 1.3772 | 2.0000
5 | 1.0716644 | 1.0928830 | 2 | 1.1551 | 1.2029
5 | 0.6918777 | 1.0000000 | 5 | 0.7370 | 1.3569
5 | 1.6367610 | 2.7625538 | 1 | 1.8201 | 2.0000
5 | 2.1054487 | 2.3852809 | 1 | 1.6005 | 1.7242
4 | 1.0928830 | 1.6040811 | 2 | 1.2075 | 1.3567
4 | 0.8489011 | 1.1262455 | 4 | 0.9443 | 1.2892
4 | 1.1262455 | 1.6204646 | 1 | 1.2893 | 1.9527
3 | 1.4233850 | 2.2892242 | 1 | 1.7153 | 2.0000
3 | 1.6204646 | 3.0578458 | 1 | 1.9528 | 2.0000
3 | 1.0503397 | 1.2598510 | 2 | 1.1664 | 1.7095
3 | 1.1262455 | 2.7095113 | 3 | 1.2893 | 1.5849
3 | 0.9578836 | 1.6586832 | 3 | 1.0966 | 2.0000
2 | 1.0000000 | 1.8690664 | 2 | 1.2969 | 2.0000
2 | 1.2598510 | 3.8188417 | 1 | 1.3570 | 2.0000
2 | 1.4321881 | 1.9631546 | 1 | 1.9434 | 2.0000
2 | 1.0000000 | 1.5849625 | 2 | 1.3570 | 2.0000
2 | 1.2223924 | 1.5849625 | 2 | 1.1446 | 2.0000
2 | 0.8427577 | 2.7095113 | 2 | 1.1437 | 2.0000
TABLE III: Reducing cliques of 492 bibliographically coupled information-
science papers 2008 ($S$ is original size, cf. section Experiments)
nr. of excluded nodes | | | | |
---|---|---|---|---|---
$S$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 13 | 25 | sum
---|---|---|---|---|---|---|---|---|---|---|---|---
2 | 161 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 161
3 | 271 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 311
4 | 253 | 68 | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 344
5 | 200 | 115 | 38 | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 377
6 | 147 | 91 | 40 | 15 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 301
7 | 54 | 52 | 25 | 18 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 152
8 | 22 | 29 | 20 | 8 | 7 | 1 | 1 | 0 | 0 | 0 | 0 | 88
9 | 8 | 5 | 10 | 5 | 2 | 1 | 0 | 1 | 0 | 0 | 0 | 32
10 | 1 | 2 | 5 | 4 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 15
11 | 1 | 3 | 0 | 2 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | 9
12 | 0 | 0 | 1 | 0 | 1 | 0 | 2 | 2 | 0 | 0 | 0 | 6
13 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 5
14 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 2
15 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 2
16 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 4
18 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1
24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1
$\Sigma$ | 1118 | 406 | 164 | 78 | 25 | 5 | 5 | 6 | 3 | 1 | 1 | 1812
Figure 12: Optimisation of a clique of 46 $h$-index papers to 21 core papers TABLE IV: Comparison of α thresholds obtained by random LFK and by MONC algorithm, respectively, in a succession of modules growing from h-clique | random | | MONC |
---|---|---|---|---
| LFK | | (rounded) |
nr. nodes | $\alpha_{\mathrm{min}}$ | $\alpha_{\mathrm{max}}$ | $\alpha_{\mathrm{min}}$ | $\alpha_{\mathrm{max}}$
38 | 1.71 | 1.71 | 1.7019 | 1.7353
39 | 1.70 | 1.70 | 1.6880 | 1.7019
40 | 1.66 | 1.69 | 1.6361 | 1.6880
42 | 1.43 | 1.65 | 1.4233 | 1.6453
43 | 1.40 | 1.42 | 1.3955 | 1.4233
44 | 1.39 | 1.39 | 1.3587 | 1.3955
45 | 1.35 | 1.38 | 1.2817 | 1.3587
46 | 1.29 | 1.34 | 1.2792 | 1.2817
48 | 1.21 | 1.28 | 1.1903 | 1.3103
50 | 1.05 | 1.20 | 1.0308 | 1.1910
51 | 1.00 | 1.04 | 0.9956 | 1.0308
Figure 12 illustrates the optimisation of maximal cliques by exclusion of
nodes. Nodes with minimum $\alpha_{\mathrm{excl}}$ are excluded one after the
other from the clique. From the set of shrinking cliques we select the one
before maximum $\alpha_{\mathrm{excl}}$ (marked by the vertical line) is
reached.
We now derive the formula for calculating the maximum value of $\alpha$, where
a node $V$ does not diminish the fitness of a module $G$ when included in it.
For $V$ in neighbourhood of $G$ we demand therefore
$f(G\cup V,\alpha)>f(G,\alpha).$ (10)
With definitions given in Algorithm section we then have
$\frac{k_{\mathrm{in}}(G\cup V)+1}{k_{\mathrm{tot}}(G\cup
V)^{\alpha}}>\frac{k_{\mathrm{in}}(G)+1}{k_{\mathrm{tot}}(G)^{\alpha}}$ (11)
and therefore
$\frac{k_{\mathrm{in}}(G\cup
V)+1}{k_{\mathrm{in}}(G)+1}>\left[\frac{k_{\mathrm{tot}}(G\cup
V)}{k_{\mathrm{tot}}(G)}\right]^{\alpha}.$ (12)
We take logarithm on both sides of this equation and get
$\log\frac{k_{\mathrm{in}}(G\cup
V)+1}{k_{\mathrm{in}}(G)+1}>\alpha\log\frac{k_{\mathrm{tot}}(G\cup
V)}{k_{\mathrm{tot}}(G)}.$ (13)
That means, if $\alpha<\alpha_{\mathrm{incl}}$ with
$\alpha_{\mathrm{incl}}=\frac{\log({k_{\mathrm{in}}(G\cup
V)+1)}-\log({k_{\mathrm{in}}(G)+1)}}{\log{k_{\mathrm{tot}}(G\cup
V)}-\log{k_{\mathrm{tot}}(G)}}$ (14)
we have $f(G\cup V,\alpha)>f(G,\alpha)$.
|
arxiv-papers
| 2010-08-05T15:39:15 |
2024-09-04T02:49:12.068902
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Frank Havemann, Michael Heinz, Alexander Struck, Jochen Gl\\\"aser",
"submitter": "Frank Havemann",
"url": "https://arxiv.org/abs/1008.1004"
}
|
1008.1056
|
11institutetext: Parks College, Department of Chemistry, Saint Louis
University - Saint Louis, MO 63103, USA
Department of Mathematics, University of Leicester - Leicester LE1 7RH, UK
Department of Mathematical Analysis, Ghent University - Galglaan 2, B-9000
Gent, Belgium
Laboratory for Chemical Technology, Ghent University - Krijgslaan 281 (S5),
B-9000 Gent, Belgium
Kinetic theory Nonequilibrium and irreversible thermodynamics
# Reciprocal Relations Between Kinetic Curves
G.S. Yablonsky Long Term Structural Methusalem Funding by the Flemish
Government - grant number BOF09/01M0040911 A.N. Gorban Support from the
University of Leicester and Isaac Newton Institute for Mathematical Sciences22
D. Constales Support from BOF/GOA 01GA0405 of Ghent University33 V.V. Galvita
44 G.B. Marin (a)4(a)411223344
###### Abstract
We study coupled irreversible processes. For linear or linearized kinetics
with microreversibility, $\dot{x}=Kx$, the kinetic operator $K$ is symmetric
in the entropic inner product. This form of Onsager’s reciprocal relations
implies that the shift in time, $\exp(Kt)$, is also a symmetric operator. This
generates the reciprocity relations between the kinetic curves. For example,
for the Master equation, if we start the process from the $i$th pure state and
measure the probability $p_{j}(t)$ of the $j$th state ($j\neq i$), and,
similarly, measure $p_{i}(t)$ for the process, which starts at the $j$th pure
state, then the ratio of these two probabilities $p_{j}(t)/p_{i}(t)$ is
constant in time and coincides with the ratio of the equilibrium
probabilities. We study similar and more general reciprocal relations between
the kinetic curves. The experimental evidence provided as an example is from
the reversible water gas shift reaction over iron oxide catalyst. The
experimental data are obtained using Temporal Analysis of Products (TAP)
pulse-response studies. These offer excellent confirmation within the
experimental error.
###### pacs:
05.20.Dd
###### pacs:
05.70.Ln
## 1 Introduction
### 1.1 A bit of history
In 1931, L. Onsager [1, 2] gave the backgrounds and generalizations to the
reciprocal relations introduced in 19th century by Lord Kelvin and H. v.
Helmholtz. In his seminal papers, L. Onsager mentioned also the close
connection between these relations and detailed balancing of elementary
processes: at equilibrium, each elementary transaction should be equilibrated
by its inverse transaction. This principle of detailed balance was known long
before for the Boltzmann equation [3]. A. Einstein used this principle for the
linear kinetics of emission and absorption of radiation [4]. In 1901, R.
Wegscheider published an analysis of detailed balance for chemical kinetics
[5].
The connections between the detailed balancing and Onsager’s reciprocal
relations were clarified in detail by N. G. v. Kampen [6]. They were also
extended for various types of coordinate transformations which may include
time derivatives and integration in time [7]. Recently, [8], the reciprocal
relations were derived for nonlinear coupled transport processes between
reservoirs coupled at mesoscopic contact points. Now, an elegant geometric
framework is elaborated for Onsager’s relations and their generalizations [9].
Onsager’s relations are widely used for extraction of kinetic information
about reciprocal processes from experiments and for the validation of such
information (see, for example, [10]): one can measure how process A affects
process B and extract the reciprocal information, how B affects A.
The reciprocal relations were tested experimentally for many systems. In 1960,
D.G. Miller wrote a remarkable review on experimental verification of the
Onsager reciprocal relations which is often referred to even now [11].
Analyzing many different cases of irreversible phenomena (thermoelectricity,
electrokinetics, isothermal diffusion, etc), Miller found that these
reciprocal relations are valid. However, regarding the chemical reactions,
Miller’s point was : “The experimental studies of this phenomenon … have been
inconclusive, and the question is still open from an experimental point.”
According to Onsager’s work [1], the fluxes in chemical kinetics are time
derivatives of the concentrations and potentials are expressed through the
chemical potentials. The fluxes (near equilibrium) are linear functions of
potentials and the reciprocal relations state that the coefficient matrix of
these functions is symmetric. It is impossible to measure these coefficients
directly. To find them one has to solve the inverse problem of chemical
kinetics. This problem is often ill-posed.
Such a difficulty, appearance of ill-posed problems in the verification of the
reciprocal relations, is typical because these relations connect the kinetic
coefficients. Sometimes it is possible to find them directly in separate
experiments but if it is impossible then the inverse problem arises with all
the typical difficulties.
In our work we, in particular, demonstrate how it is possible to verify the
reciprocal relations without the differentiation of the empiric kinetic curves
and solving the inverse problems, and present the experimental results which
demonstrate these relations for one reaction kinetic system. For this purpose,
we have to formulate the reciprocal relations directly between the measurable
quantities.
These reciprocal relations between kinetic curves use the symmetry of the
propagator in the special entropic inner product. A dual experiment is defined
for each ideal kinetic experiment. For this dual experiment, both the initial
data and the observables are different (they exchange their positions), but
the results of the measurement is essentially the same function of time.
### 1.2 The structure of the paper
We start from the classical Onsager relations and reformulate them as
conditions on the kinetic operator $K$ for linear or linearized kinetic
equations $\dot{x}=Kx$. This operator should be symmetric in the entropic
inner product, whereas the matrix $L$ that transforms forces into fluxes
should be symmetric in the standard inner product, i.e. $L_{ij}=L_{ji}$. The
form of reciprocal relations with special inner product is well known in
chemical and Boltzmann kinetics [13, 14]. They are usually proved directly
from the detailed balance conditions. Such relations are also universal just
as the classical relations are.
Real functions of symmetric operators are also symmetric. In particular, the
propagator $\exp(Kt)$ is symmetric. Therefore, we can formulate the reciprocal
relation between kinetic curves. These relations do not include fluxes and
time derivatives, hence, they are more robust. We formulate them as the
symmetry relations between the observables and initial data (the observables-
initial data symmetry).
A particular case of this symmetry for a network of monomolecular chemical
reactions or for the Master equation, which describe systems with detailed
balance, seems rather unexpected. Let us consider two situations for a linear
reaction network.
1. 1.
The process starts at the state “everything is in $A_{q}$”, and we measure the
concentration of $A_{r}$. The result is $c^{a}_{r}(t)$ (“how much $A_{r}$ is
produced from the initial $A_{q}$”).
2. 2.
The process starts at the state “everything is in $A_{r}$”, and we measure the
concentration of $A_{q}$. The result is $c^{b}_{q}(t)$ (“how much $A_{q}$ is
produced from the initial $A_{r}$”) (the dual experiment).
The results of the dual experiments are connected by the identity
$\frac{c^{a}_{r}(t)}{c^{\rm eq}_{r}}\equiv\frac{c^{b}_{q}(t)}{c^{\rm
eq}_{q}}\,,$
where $c$ are concentrations and $c^{\rm eq}$ are equilibrium concentrations.
The symmetry with respect to the observables-initial data exchange gives the
general rule for production of the reciprocal relations between kinetic
curves.
Many real processes in chemical engineering and biochemistry include
irreversible reactions, i.e. the reactions with a negligible (zero) rate of
the reverse reaction. For these processes, the micro-reversibility conditions
and the backgrounds of classical Onsager relations are not applicable
directly. Nevertheless, they may be considered as limits of systems with
micro-reversibility when some of the rate constants for inverse reactions tend
to zero. We introduce the correspondent weak form of detailed balance,
formulate the necessary and sufficient algebraic conditions for this form of
detailed balance and formulate the observables-initial data symmetry for these
systems.
The experimental evidence of the observables-initial data symmetry is
presented for the reversible water gas shift reaction over iron oxide
catalyst. The experimental data are obtained using Temporal Analysis of
Products (TAP) pulse-response studies. These offer excellent confirmation
within experimental error.
## 2 Two forms of the reciprocal relations: forces, fluxes and entropic inner
product
Let us consider linear kinetic equations or kinetic equations linearized near
an equilibrium $x^{\rm eq}$ (sometimes, it may be convenient to move the
origin to $x^{\rm eq}$):
$\dot{x}=Kx\,.$ (1)
In the original form of Onsager’s relations, the vector of fluxes $J$ and the
vector of thermodynamic forces $X$ are connected by a symmetric matrix,
$J=LX$, $L_{ij}=L_{ji}$. The vector $X$ is the gradient of the corresponding
thermodynamic potential: $X_{i}=\partial\Phi/\partial x_{i}$. For isolated
systems, $\Phi$ is the entropy. For other conditions, other thermodynamic
potentials are used. For example, for the constant volume $V$ and temperature
$T$ conditions, $\Phi$ is $-F/T$ and for the constant pressure $P$ and
temperature conditions, $\Phi$ is $-G/T$, where $F$ is the Helmholtz energy
(free energy) and $G$ is the Gibbs energy (free enthalpy). These free entropy
functions are also known as the Massieu–Planck potentials [12]. Usually, they
are concave.
For the finite-dimensional systems, like chemical kinetics or the Master
equation, the dynamics satisfy linear (linearized) kinetic equation
$\dot{x}=Kx$, where
$K_{ij}=\sum_{l}L_{il}\left.\frac{\partial^{2}\Phi}{\partial x_{l}\partial
x_{j}}\right|_{x^{\rm eq}}\mbox{ i.e. }K=L(D^{2}\Phi)_{x^{\rm eq}}.$
This matrix is not symmetric but the product $(D^{2}\Phi)_{x^{\rm eq}}K$
$=(D^{2}\Phi)_{x^{\rm eq}}L(D^{2}\Phi)_{x^{\rm eq}}$ is already symmetric,
hence, $K$ is symmetric (self-adjoint) in the entropic scalar product
$\langle a\,|\,Kb\rangle_{\Phi}\equiv\langle Ka\,|\,b\rangle_{\Phi}\,,$ (2)
where
$\langle
a\,|\,b\rangle_{\Phi}=-\sum_{ij}a_{i}\left.\frac{\partial^{2}\Phi}{\partial
x_{l}\partial x_{j}}\right|_{x^{\rm eq}}b_{j}\,.$ (3)
Further on, we use the angular brackets for the entropic inner product (3) and
its generalizations and omit the subscript $\Phi$.
For the spatially distributed systems with transport processes, the variables
$x_{i}$ are functions of the space coordinates $\xi$, the equations of
divergence form appear, $\partial_{t}x_{i}=-\nabla_{\xi}\cdot J_{i}$,
thermodynamic forces include also gradients in space variables,
$X_{i}=\nabla_{\xi}\partial\Phi/\partial x_{i}$ and the operator $K$ has the
form
$K_{ij}=\sum_{l}L_{il}\left.\frac{\partial^{2}\Phi}{\partial x_{l}\partial
x_{j}}\right|_{x^{\rm eq}}\Delta_{\xi}\,\mbox{ i.e. }K=L(D^{2}\Phi)_{x^{\rm
eq}}\Delta_{\xi}\,,$
where $\Delta_{\xi}$ is the Laplace operator. This operator $K$ is self-
adjoint in the inner product which is just the integral in space of (3). The
generalizations to inhomogeneous equilibria, non-isotropic and non-euclidian
spaces are also routine but lead to more cumbersome formulas.
Symmetric operators have many important properties. Their spectrum is real,
for a function of a real variable $f$ with real values it is possible to
define $f(K)$ through the spectral decomposition of $K$, and this $f(K)$ is
also symmetric in the same inner product. This property is the cornerstone for
further consideration.
## 3 Symmetry between observables and initial data
The exponential of a symmetric operator is also symmetric, hence, Onsager’s
relations (2) immediately imply
$\langle a\,|\,\exp(Kt)\,b\rangle\equiv\langle b\,|\,\exp(Kt)\,a\rangle\,.$
(4)
The expression $x(t)=\exp(Kt)\,b$ gives a solution to the kinetic equations
(1) with initial conditions $x(0)=b$. The expression $\langle
a\,|\,x(t)\rangle$ is the result of a measurement: formally, for each vector
$a$ we can introduce a “device” (an observer), which measures the scalar
product of vector $a$ on a current state $x$.
The left hand side of (4) represents the result of such an experiment: we
prepare an initial state $x(0)=b$, start the process from this state and
measure $\langle a\,|\,x(t)\rangle$. In the right hand side, the initial
condition $b$ and the observer $a$ exchange their positions and roles: we
start from the initial condition $x(0)=a$ and measure $\langle
b\,|\,x(t)\rangle$. The result is the same function of time $t$.
This exchange of the observer and the initial state transforms an ideal
experiment into another ideal experiment (we call them dual experiments). The
left and the right hand sides of (4) represent different experimental
situations but with the same results of the measurements.
This observation produces many consequences. As a first class of examples, we
present the time–reversible Markov chains [15], or the same class of kinetic
equations, the monomolecular reactions with detailed balance (see any detailed
textbook in chemical kinetics, for example, [13]).
Here a terminological comment is necessary. The term “reversible” has three
different senses in thermodynamics and kinetics.
* •
First of all, processes with entropy growth are irreversible. In this sense,
all processes under consideration are irreversible.
* •
Secondly, processes with microreversibility, which satisfy the detailed
balance and Onsager relations, are time–reversible (or, for short, one often
calls them “reversible”). We always call them time–reversible to avoid
confusion.
* •
In the third sense, reversibility is the existence of inverse processes: if
transition $A\to B$ exists then transition $B\to A$ exists too. This condition
is significantly weaker than microreversibility.
“Time–reversibility” of irreversible processes sounds paradoxical and requires
comments. The most direct interpretation of “time–reversing” is to go back in
time: we take a solution to dynamic equations $x(t)$ and check whether $x(-t)$
is also a solution. For the microscopic dynamics (the Newton or Schrödinger
equations) we expect that $x(-t)$ is also a solution to the dynamic equations.
Nonequilibrium statistical physics combines this idea with the description of
macroscopic or mesoscopic kinetics by an ensemble of elementary processes:
collisions, reactions or jumps. The microscopic “reversing of time” turns at
this level into the “reversing of arrows”: reaction
$\sum_{i}\alpha_{i}A\to\sum_{j}\beta_{j}B_{j}$ transforms into
$\sum_{j}\beta_{j}B_{j}\to\sum_{i}\alpha_{i}A$ and conversely. The equilibrium
ensemble should be invariant with respect to this transformation. This leads
us immediately to the concept of detailed balance: each process is
equilibrated by its reverse process. “Time–reversible kinetic process” stands
for “irreversible process with the time–reversible underlying microdynamics”.
We consider a general network of linear reactions. This network is represented
as a directed graph (digraph) [13]: vertices correspond to components $A_{i}$
($i=1,2,\ldots,n$), edges correspond to reactions $A_{i}\to A_{j}$ ($i\neq
j$). For each vertex, $A_{i}$, a positive real variable $c_{i}$
(concentration) is defined. For each reaction, $A_{i}\to A_{j}$ a nonnegative
continuous bounded function, the reaction rate constant $k_{ji}>0$ is given.
The kinetic equations have the standard Master equation form
$\frac{{\mathrm{d}}c_{i}}{{\mathrm{d}}t}=\sum_{j,\,j\neq
i}(k_{ij}c_{j}-k_{ji}c_{i})\,.$ (5)
The principle of detailed balance (“time–reversibility”) means that there
exists such a positive vector $c^{\rm eq}_{i}>0$ that for all $i,j$ ($j\neq
i$)
$k_{ij}c^{\rm eq}_{j}=k_{ji}c^{\rm eq}_{i}\,.$ (6)
The following conditions are necessary and sufficient for existence of such an
equilibrium $c^{\rm eq}_{i}>0$:
* •
Reversibility (in the third sense): if $k_{ji}>0$ then $k_{ij}>0$;
* •
For any cycle $A_{i_{1}}\to A_{i_{2}}\to\ldots\to A_{i_{q}}\to A_{i_{1}}$ the
product of constants of reactions is equal to the product of constants of
reverse reactions,
$\prod_{j=1}^{q}k_{i_{j+1}i_{j}}=\prod_{j=1}^{q}k_{i_{j}i_{j+1}}\,,$ (7)
where $i_{q+1}=i_{1}$. This is the Wegscheider identity [5].
It is sufficient to consider in conditions (7) a finite number of basic cycles
[13].
The free entropy function for the Master equation (5) is the (minus) relative
entropy
$Y=-\sum_{i}c_{i}\ln\left(\frac{c_{i}}{c^{\rm eq}_{i}}\right)\,.$ (8)
In this form, the function $-RTY$ was used already by L. Onsager [1] under the
name “free energy”. The entropic inner product for the free entropy (8) is
$\langle a\,|\,b\rangle=\sum_{i}\frac{a_{i}b_{i}}{c^{\rm eq}_{i}}\,.$ (9)
Let $c^{a}(t)$ be a solution of kinetic equations (5) with initial conditions
$c^{a}(0)=a$. Then the reciprocity relations (4) for linear systems with
detailed balance take the form
$\sum_{i}\frac{b_{i}c^{a}_{i}(t)}{c^{\rm
eq}_{i}}=\sum_{i}\frac{a_{i}c^{b}_{i}(t)}{c^{\rm eq}_{i}}\,.$ (10)
Let us use for $a$ and $b$ the vectors of the standard basis in
$\mathbb{R}^{n}$: $a_{i}=\delta_{iq}$, $b_{i}=\delta_{ir}$, $q\neq r$. This
choice results in the useful particular form of (10). We compare two
experimental situations, $c^{a}_{i}(0)=\delta_{iq}$ (the process starts at the
state “everything is in $A_{q}$”) and $c^{b}_{i}(0)=\delta_{ir}$ (the process
starts at the state “everything is in $A_{r}$”); for the first situation we
measure $c^{a}_{r}(t)$ (“how much $A_{r}$ is produced from the initial
$A_{q}$”), for the second one we measure $c^{b}_{q}(t)$ (“how much $A_{q}$ is
produced from the initial $A_{r}$”). The reciprocal relations (10) give
$\frac{c^{a}_{r}(t)}{c^{\rm eq}_{r}}=\frac{c^{b}_{q}(t)}{c^{\rm eq}_{q}}\,.$
(11)
More examples of such relations for chemical kinetics are presented in [16].
It is much more straightforward to check experimentally these relations
between kinetic curves than the initial Onsager relations between kinetic
coefficients. We give an example of such an experiment below. For processes
distributed in space, instead of concentrations of $A$ and $B$ some of their
Fourier or wavelet coefficients appear.
## 4 Weak form of detailed balance
For many real systems some of the elementary reactions are practically
irreversible. Hence the first condition of detailed balance, the reversibility
(if $k_{ji}>0$ then $k_{ij}>0$) may be violated. Nevertheless, these systems
may be considered as limits of systems with detailed balance when some of the
constants tend to zero. For such limits, the condition (7) persists, and for
any cycle the product of constants of direct reactions is equal to the product
of constants of reverse reactions.
This is a weak form of detailed balance without the obligatory existence of a
positive equilibrium. In this section, we consider the systems, which satisfy
this weak condition, the weakly time–reversible systems.
For a linear system, the following condition is necessary and sufficient for
its weak time–reversibility: In any cycle $A_{i_{1}}\to A_{i_{2}}\to\ldots\to
A_{i_{q}}\to A_{i_{1}}$ with strictly positive constants $k_{i_{j+1}i_{j}}>0$
(here $i_{q+1}=i_{1}$) all the reactions are reversible ($k_{i_{j}i_{j+1}}>0$)
and the identity (7) holds.
The components $A_{q}$ and $A_{r}$ ($q\neq r$) are strongly connected if there
exist oriented paths both from $A_{q}$ to $A_{r}$ and from $A_{r}$ to $A_{q}$
(each oriented edge corresponds to a reaction with nonzero reaction rate
constant). It is convenient to consider an empty path from $A_{i}$ to itself
as an oriented path.
For strongly connected components of a weakly time–reversible system, all
reactions in any directed path between them are reversible. This is a
structural condition of the weak time–reversibility.
Under this structural condition, the classes of strongly connected components
form a partition of the set of components: these classes either coincide or do
not intersect and each component belongs to one of them. Each cycle belongs to
one class.
Let $A_{q}$ and $A_{r}$ be strongly connected. Let us select an arbitrary
oriented path $p$ between $A_{q}$ and $A_{r}$: $A_{q}\leftrightarrow
A_{i_{1}}\leftrightarrow A_{i_{2}}\leftrightarrow\ldots\leftrightarrow
A_{i_{l}}\leftrightarrow A_{r}$. For the product of direct reaction rate
constants in this path we use $K_{p}^{+}$ and for the product of reverse
reaction rate constants we use $K_{p}^{-}$. The ratio
$K_{rq}=K_{p}^{+}/K_{p}^{-}$ does not depend on the path $p$ and characterizes
the pair $A_{r},A_{q}$, because of the Wegscheider identity (7). This is the
quantitative criterion of the weak time–reversibility.
The constant $K_{rq}$ is an analogue to the equilibrium constant. Indeed, for
the systems with positive equilibrium and detailed balance, $K_{rq}c^{\rm
eq}_{q}=c^{\rm eq}_{r}$ and $K_{rq}=c^{\rm eq}_{r}/c^{\rm eq}_{q}$.
For weakly time–reversible system, the reciprocal relations between kinetic
curves can be formulated for any strongly connected pair $A_{q}$ and $A_{r}$.
Exactly for the same pair of kinetic curves, as in (11), we obtain
$\frac{c^{a}_{r}(t)}{c^{b}_{q}(t)}=K_{rq}\,.$ (12)
This formula describes two experiments: (i) we start the system at $t=0$ from
the pure $A_{q}$ and measure $c_{r}(t)$, then (ii) we start at $t=0$ from the
pure $A_{r}$ and measure $c_{q}(t)$. The ratio of these two kinetic curves,
$c_{r}(t)/c_{q}(t)$ does not depend on $t$ and is equal to the generalized
equilibrium constant $K_{rq}$.
The weak form of the Wegscheider identity for general (nonlinear) kinetic
systems is also possible. Let us consider the reaction system:
$\alpha_{r1}A_{1}+\ldots+\alpha_{rn}A_{n}\to\beta_{r1}A_{1}+\ldots+\beta_{rn}A_{n}\,,$
(13)
which satisfies the mass action law:
$\dot{c}=\sum_{r}\gamma_{r}k_{r}\prod_{i}c_{i}^{\alpha_{i}}\,,$ where
$k_{r}>0$, $\gamma_{ri}=\beta_{ri}-\alpha_{ri}$ is the stoichiometric vector
of the $r$th reaction, and the reverse reactions with positive constants are
included in the list (13) separately.
Let us consider linear relations between vectors $\\{\gamma_{r}\\}$:
$\sum_{r}\lambda_{r}\gamma_{r}=0\mbox{ and }\lambda_{r}\neq 0\mbox{ for some
}r\,.$ (14)
If all the reactions are reversible then the principle of detailed balance
gives us the identity [13]:
$\prod_{r}(k_{r}^{+})^{\lambda_{r}}=\prod_{r}(k_{r}^{-})^{\lambda_{r}}$ (15)
for any linear relation (14). For reversible reactions, we can take
$\lambda_{r}\geq 0$ in (15) for all $r$: if we substitute the reactions with
$\lambda_{r}<0$ by their reverse reactions then $\gamma_{r}$ and $\lambda_{r}$
change signs. It is sufficient to consider only the cone $\Lambda_{+}$ of non-
negative relations (14) ($\lambda_{r}\geq 0$) and take in (15) the direction
vectors of its extreme rays. Let $k_{r}^{-}=0$ for some $r$. The weak form of
the identity (15) is:
For any extreme ray of $\Lambda_{+}$ with a direction vector $\lambda_{r}\geq
0$ the reactions which correspond to the positive coefficients $\lambda_{r}>0$
are reversible ($k_{r}^{-}>0$) and their constants satisfy the identity (15).
## 5 Nonlinear Examples
It seems impossible to find a general relation between kinetic curves for
general nonlinear kinetics far from equilibrium. Nevertheless, simple examples
encourage us to look for a nontrivial theory for some classes of nonlinear
systems. In this Section, we give two examples of nonlinear elementary
reactions which demonstrate the equilibrium relations between nonequilibrium
kinetic curves [16].
### 5.1 $2A\leftrightarrow B$
The linear conservation law is $c_{A}+2c_{B}=const$. Let us take two initial
states with the same value $c_{A}+2c_{B}=1$: ($a$) $c_{A}(0)=1,\,c_{B}(0)=0$
and ($b$) $c_{A}(0)=0,\,c_{B}(0)=1/2$. We will mark the corresponding
solutions by the upper indexes $a,b$. The mass action law gives:
$\dot{c}_{A}=-2k^{+}c_{A}^{2}+k^{-}(1-c_{A})\,,\;c_{B}=(1-c_{A})/2\,.$ (16)
The analytic solution easily gives
$\frac{c_{B}^{a}(t)}{c_{A}^{a}(t)c_{A}^{b}(t)}=\frac{k^{+}}{k^{-}}=K^{\mathrm{eq}}=\frac{c_{B}^{\mathrm{eq}}}{(c_{A}^{\mathrm{eq}})^{2}}\,,$
(17)
the denominator involves the $A$ concentrations of both trajectories, $c^{a}$
(started from $c_{A}(0)=1,\,c_{B}(0)=0$) and $c^{b}$ (started from
$c_{A}(0)=0,\,c_{B}(0)=1/2$). A ratio is equal to the equilibrium constant at
every time $t>0$. This identity between the non-stationary kinetic curves
reproduces the equilibrium ratio.
### 5.2 $2A\leftrightarrow 2B$
The linear conservation law is $c_{A}+c_{B}=const$. Let us take two initial
states with the same value $c_{A}+c_{B}=1$: ($a$) $c_{A}(0)=1,\,c_{B}(0)=0$
and ($b$) $c_{A}(0)=0,\,c_{B}(0)=1$. The kinetic equation is
$\dot{c}_{A}=-2k^{+}c_{A}^{2}+k^{-}(1-c_{A})^{2}\,,\;c_{B}={1-c_{A}}\,.$ (18)
It can be solved analytically. For this solution,
$\frac{c_{B}^{a}(t)c_{B}^{b}(t)}{c_{A}^{a}(t)c_{A}^{b}(t)}=\frac{k^{+}}{k^{-}}=K^{\mathrm{eq}}=\frac{(c_{B}^{\mathrm{eq}})^{2}}{(c_{A}^{\mathrm{eq}})^{2}}\,,$
(19)
both the numerator and denominator include trajectories for both initial
states, $a$ and $b$. This identity between the kinetic curves also reproduces
the equilibrium ratio.
## 6 Experimental evidences
In this work, we investigate the validity of the reciprocal relations using
the TAP (Temporal Analysis of Products) technique proposed by Gleaves in 1988
[17]. It has been successfully applied in many areas of chemical kinetics and
engineering for non-steady-state kinetic characterization [18]. The studied
reaction is a part of the reversible water gas shift reaction over iron oxide
catalyst. The overall reaction is
$\mbox{H}_{2}\mbox{O}+\mbox{CO}\leftrightarrow\mbox{H}_{2}+\mbox{CO}_{2}$.
### 6.1 Experimental set-up
The TAP reactor system used in this work is made of quartz and is of the size
33 mm bed-length and 4.75 mm inner diameter. The products and the unreacted
reactants coming out of the reactor are monitored by a UTI 100C quadrupole
mass spectrometer (QMS). The number of molecules admitted during pulse
experiments amounts to $10^{15}$ molecules/pulse.
To ensure uniformity of the catalyst along the bed, we use a thin–zone TAP
reactor (TZTR), the width of the catalyst zone being 2mm. Experiments were
performed over 40 mg of Fe2O3 catalyst. The catalyst was packed in between two
inert zones of quartz particles of the same size ($250<d_{p}<500\mu m$). The
temperature of the reactor was measured by a thermocouple positioned in the
center of the catalyst bed. Several single pulse experiments were performed by
pulsing CO or CO2 at the temperature of 780K. In all the experiments, the
reaction mixture was prepared with Ar as one of the components, so that the
inlet amount of the components can be determined from the Ar response.
### 6.2 Application to the measurements
Figure 1: Fourier domain result values for the “B from A/A from B” ratio (20),
vs. frequency $f$ in Hz (so that $\omega=2\pi f$, $s=i\omega$); real and
imaginary part. The error bars were obtained from 10,000 resampled
measurements.
In a thin–zone TAP-reactor, the diffusion occurring in the inert zones
flanking the thin reactive zone must be accounted for. The Knudsen regime in
these zones guarantees a linear behaviour, so that the resulting outlet fluxes
can be expressed in terms of convolutions. Switching to the Laplace domain
greatly facilitates the analysis, and we can prove in general that the fixed
proportion property is equivalent to the following equality in terms of the
exit fluxes $F_{B_{A}}$ of gas $B$ given a unit inlet pulse of gas $A$ and
$F_{A_{B}}$, of $A$ given a unit inlet pulse of $B$, see [19]:
$K_{\mathop{\rm
eq}}={\frac{(\cosh\sqrt{s\tau_{1,A}})(\sqrt{\tau_{3,B}}\sinh\sqrt{s\tau_{3,B}})}{(\cosh\sqrt{s\tau_{1,B}})(\sqrt{\tau_{3,A}}\sinh\sqrt{s\tau_{3,A}})}}{\frac{{\cal
L}F_{B_{A}}(s)}{{\cal L}F_{A_{B}}(s)}}$ (20)
identically in the Laplace variable $s$, where
$\tau_{i,G}=\epsilon_{i}L_{i}^{2}/D_{G}$, $\epsilon_{i}$ denoting the packing
density of the $i$-th zone, $L_{i}$ its length, and $D_{G}$ the diffusivity of
gas $G$. To apply this in practice, we set $s=i\omega$ and switch to the
Fourier domain.
Performing these corrections, with $A$ denoting CO and $B$ CO2, the results of
Fig. 1 in the Fourier domain are obtained. The real and imaginary parts of the
right-hand side in (20) are graphed, with error bars corresponding to three
times the standard deviation estimated from resampling 10,000 times the exit
flux measurements using their principal error components. Ideally, all
imaginary values should be zero; we see that zero does lie within all the
confidence intervals. We also see that the smallest error in the real parts
occurs for the second frequency, 2.2 Hz. This confidence interval lies snugly
within the others, offering confirmation that (within experimental error) the
same value for all frequencies is obtained.
## 7 Conclusion
The shift in time operator is symmetric in the entropic inner product. Its
symmetry allows us to formulate the symmetry relations between the observables
and initial data. These relations could be validated without differentiation
of empiric curves and are, in that sense, more robust and closer to the direct
measurements. For the Markov processes and chemical kinetics, the symmetry
relations between the observables and initial data have an elegant form of the
symmetry between “$A$ produced from $B$” and “$B$ produced from $A$”: their
ratio is equal to the equilibrium constant and does not change in time (11),
(12). For processes distributed in space, instead of concentrations of $A$ and
$B$ some of their Fourier or wavelet coefficients appear.
The symmetry relations between the observables and initial data have a rich
variety of realizations, which makes the direct experimental verification
possible. On the other hand, this symmetry provides the possibility to extract
information about the experimental data through the dual experiments. These
relations are applicable to all systems with microreversibility.
## References
* [1] Onsager L. Phys. Rev.37 1931 405–426.
* [2] Onsager L. Phys. Rev.38 1931 2265–2279.
* [3] Boltzmann L. Lectures on gas theoryU. of California Press, Berkeley, CA 1964.
* [4] Einstein A. Verhandlungen der Deutschen Physikalischen Gesellschaft1813/14(1916) 318–323.
* [5] Wegscheider R. Monatshefte für Chemie / Chemical Monthly32 8(1911), 849–906. DOI: 10.1007/BF01517735
* [6] Van Kampen N. G. Physica 671973 1–22.
* [7] Stöckel H. Fortschritte der Physik/Progress of Physics311983 165–184.
* [8] Astumian R. D. Phys. Rev. Lett.1012008046802.
* [9] Grmela M. Physica A 3092002 304–328.
* [10] Ozera M. Provaznik I. J. Theor. Biology2332005 237–243.
* [11] Miller D. G. Chem. Rev. 601960 15–37.
* [12] Callen H. B. Thermodynamics and an Introduction to Thermostatistics (2nd ed.). John Wiley and Sons, New York 1985.
* [13] Yablonskii G. S., Bykov V. I., Gorban A. N. Elokhin V. I. Kinetic Models of Catalytic Reactions, Series: Comprehensive Chemical Kinetics 32 Elsevier, Amsterdam, The Netherlands 1991.
* [14] Gorban A. N. Karlin I. V. Invariant manifolds for physical and chemical kinetics, vol. 660 of Lect. Notes Phys Springer, Berlin-Heidelberg-New York 2005.
* [15] Hasegawa H. Progr. Theor. Phys.551976 90–105.
* [16] Yablonsky G. S., Constales D. Marin G. B. Equilibrium relationships for non-equilibrium chemical dependencies, Chem. Eng. Sci.662011 111–114.
* [17] Gleaves J. T., Ebner J. R. Kuechler T. C. Catal. Rev.- Sci. Eng.301988 49–116.
* [18] Yablonsky G. S., Olea M. Marin G. B. J. Catal.2162003 120–134.
* [19] Constales D., Yablonsky G. S., Marin G. B. Gleaves J. T. Chem. Eng. Sci.592004 3725–3736.
|
arxiv-papers
| 2010-08-05T19:59:29 |
2024-09-04T02:49:12.077089
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G.S. Yablonsky, A.N. Gorban, D. Constales, V. Galvita, G.B. Marin",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1008.1056"
}
|
1008.1146
|
arxiv-papers
| 2010-08-06T09:24:42 |
2024-09-04T02:49:12.083442
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X.Y.Cui, H.Negishi, A. N. Titov, S. G. Titova, M. Shi, L. Patthey",
"submitter": "Xiaoyu Cui",
"url": "https://arxiv.org/abs/1008.1146"
}
|
|
1008.1340
|
# Properties of Quantum Graphity at Low Temperature
Francesco Caravelli1,2 fcaravelli@perimeterinstitute.ca Fotini
Markopoulou1,2,3 fmarkopoulou@perimeterinstitute.ca 1 Perimeter Institute for
Theoretical Physics,
Waterloo, Ontario N2L 2Y5 Canada,
and
2 University of Waterloo, Waterloo, Ontario N2L 3G1, Canada,
and
3 Max Planck Institute for Gravitational Physics, Albert Einstein Institute,
Am Mühlenberg 1, Golm, D-14476 Golm, Germany.
###### Abstract
We present a mapping of dynamical graphs and, in particular, the graphs used
in the Quantum Graphity models for emergent geometry, to an Ising hamiltonian
on the line graph of a complete graph with a fixed number of vertices. We use
this method to study the properties of Quantum Graphity models at low
temperature in the limit in which the valence coupling constant of the model
is much greater than the coupling constants of the loop terms. Using mean
field theory we find that an order parameter for the model is the average
valence of the graph. We calculate the equilibrium distribution for the
valence as an implicit function of the temperature. In the approximation in
which the temperature is low, we find the first two Taylor coefficients of the
valence in the temperature expansion. A discussion of the susceptibility
function and a generalization of the model are given in the end.
###### pacs:
04.60.Pp , 04.60.-m
## I Introduction
It is commonly agreed that at high spacetime curvatures, when the quantum
effects of the gravitational field become significant, General Relativity
needs to be replaced by a quantum theory of gravity. In spite of progress in
several directions, finding this new theory has proven a challenging problem
for several decades. Current research in the field is paying substantial
attention to the numerous indications that gravity may only be emergent,
meaning that it is a collective, or thermodynamical, description of
microscopic physics in which we do not encounter geometric or gravitational
degree of freedom. An analogy to illustrate this point of view is fluid
dynamics and the transition from from thermodynamics to the kinetic theory.
What we currently know is the low energy theory, the analogue of fluid
dynamics. We are looking for the microscopic theory, the analogue of the
quantum molecular dynamics. Just as there are no waves in the molecular
theory, we may not find geometric degree of freedom in the fundamental theory.
Not surprisingly, this significant shift in perspective opens up new routes
that may take us out of the old problems.
The emergent viewpoint amounts to treating quantum gravity as a problem in
statistical physics. A powerful set of methods in statistical physics involve
the use of lattice-based models, such as the Ising model for ferromagnetism,
the Hubbard model for the conductor/insulator transition, etc. Such methods
are starting to be introduced in quantum gravity. Examples are G. Volovik’s
work on emergent Lorentz invariance at the Fermi point GV , X.-G. Wen’s work
on emergent matter and gravitons from a bosonic spin system Wen , the
emergence of a Lorentzian metric and aspects of gravitation such as Hawking
radiation in analog models of gravity analog , as well as long-standing
approaches such as matrix models MM , and more radical formulations of
geometry in terms of information Llo ; FMKQI .
It is natural for the lattice in the lattice system to play the role of (a
primitive form of) geometry. Now, General Relativity is a background
independent theory, by which we mean that the geometry of spacetime is fully
dynamical. By analogy, we expect that the use of a fixed lattice is
inappropriate and one instead needs models on a dynamical lattice. While for
this reason desirable, dynamical lattices raise difficult technical problems
that have not been previously addressed in the field of statistical physics.
The present article is concerned with exactly this problem and presents a
method that deals with dynamical lattices in certain situations.
In previous work, we introduced Quantum Graphity, a background independent
model of spacetime in which time is an external parameter and space is
described by a relational theory based on graphs graphity1 . The idea is to
represent locality by the adjacencies of a dynamical graph on which the
diffeomorphism group is replaced, in the high energy phase, by the symmetric
group on the complete graph which breaks down to its subgraphs at lower
energies. These kind of models are sometimes called event-symmetricevents ).
The principle of event symmetry refers to the replacement, at high energies,
of the diffeomorphism group with the group of permutations of events in
spacetime. In the context of Quantum Graphity, however, the event symmetry is
only spatial, in the sense that at high energy the graph is complete and every
vertex of the graph is at distance one from each other. If the dynamics is
such that the system settles into a minimum energy subgraph that exhibits
geometric symmetries, for instance, a discrete version of flat space in low
dimension, we say that geometry emerges in that phase. In graphity1 was shown
that desired symmetric lattices, i.e., discrete 2d FRW, are stable local
minima under certain choices of parameters in the hamiltonian (when the effect
of the matter on the lattice is neglected). Following this work, in graphity2
, we used the same concept of locality in terms of a dynamical lattice, but
with a new type of matter that interacts non-linearly with the geometry, a
precursor of gravity, and initiated a study of the quantum properties of that
system.
In the present article, we return to the technical issues of spin systems on
dynamical lattices and we introduce a method to deal with a theory of
dynamical graphs on $N$ vertices. Such graphs are subgraphs of
$\mathscr{K}_{N}$, the complete graph on $N$ vertices. We show how, by
transforming $\mathscr{K}_{N}$ to its line graph, the theory can be
approximated by an Ising model on the line graph of a complete graph. We then
use this to study the low energy properties of the Quantum Graphity model in
graphity1 . Using mean field theory we calculate the average valence of the
graph at low temperature and we evaluate the first corrections due to the
presence of 3-loops.
The paper is organized as follows. In section II, we review the Quantum
Graphity model graphity1 with no matter. In section III, we define the line
graph derived from a generic graph and summarize its properties. In section
IV, we show how, in a certain reasonable approximation, the hamiltonian of
Quantum Graphity can be recast as an Ising hamiltonian on the line graph of a
complete graph. In section V, we calculate the corrections due to loops at low
temperature and describe, in this framework, the behavior of the correlation
function in mean field theory. Conclusions follow in Section VI.
## II Quantum Graphity
Let us briefly introduce the Quantum Graphity model graphity1 . As the name
graphity implies, Quantum Graphity is a model for a quantum theory of gravity
in which the fundamental microstates are dynamical graphs, postulated to
describe relational physics at Planckian energies. There is no notion of
geometry or quantum geometry at high energy, instead, geometry emerges as the
system cools down and away from the Planckian regime. The microstates live in
a Hilbert space on the complete graph ${\mathscr{K}}_{N}$ of $N$ vertices,
given by
$\mathscr{H}=\bigotimes^{\frac{N(N-1)}{2}}_{ij}\mathscr{H}^{e}_{ij}\bigotimes^{N}_{j}\mathscr{H}^{v}_{j},$
where $e_{ij}$ represents the edge of the graph $K_{N}$ between the $i$ and
$j$ vertices, while $\mathscr{H}^{e}_{ij}$ and $\mathscr{H}^{v}_{j}$ are the
Hilbert spaces associated with edges and vertices respectively. In particular,
the Hilbert space associated with an edge between vertex $i$ and $j$ is the
two-level state space:
$\mathscr{H}^{e}_{ij}=\text{span}\\{|0\rangle,|1\rangle\\}.$ (1)
The two states 1,0 in (1) are interpreted as the edge being on or off
respectively. This choice means that basis states in the Hilbert space of the
edges represent subgraphs $\mathscr{G}_{{\mathscr{K}}_{N}}$ of the complete
graph ${\mathscr{K}}_{N}$. A generic state in the Hilbert space of the edges
is a superposition of such subgraphs:
$|\psi\rangle=\sum c_{t}\mathscr{G}_{K_{N};t}.$
In the full model of graphity1 , extra degrees of freedom are assigned to the
on states:
$\mathscr{H}^{e}_{ij}=\text{span}\\{|0\rangle_{ij},|1_{1}\rangle_{ij},|1_{2}\rangle_{ij},|1_{3}\rangle_{ij}\\}.$
(2)
In graphity1 , and in the present work, there are no degrees of freedom
associated with the vertices and hence we ignore ${\mathscr{H}}^{v}$.
Let us focus now on the state space (1). On the Fock space of the edges we can
introduce the ladder operators $\hat{a}$ and $\hat{a}^{\dagger}$, with the
usual action:
$\hat{a}^{\dagger}_{ij}|1\rangle_{ij}=\hat{a}_{ij}|0\rangle_{ij}=0\ ,\
\hat{a}|1\rangle_{ij}=|0\rangle_{ij}.$
Dynamics in Quantum Graphity is given by a hamiltonian acting on the graph
states of the formgraphity1 :
$\widehat{H}=\widehat{H}_{V}+\widehat{H}_{L}+interaction\ terms,$ (3)
where $\widehat{H}_{V}$ keeps track of how many on edges are attached to a
single vertex, and $\widehat{H}_{L}$ counts closed paths in the graph. The
interaction term will not be used in the following, but in a generic model
these terms produce Alexander moves on the graph.
In more detail, the term $\widehat{H}_{V}$ is usually chosen to be of the
form:
$\widehat{H}_{V}=g_{V}\sum_{j}e^{p\left(v_{0}\widehat{1}-\sum_{i}\widehat{N}_{ij}\right)^{2}},$
(4)
where the indices $i,j=1,...,N$ enumerate vertices,
$\widehat{N}_{ij}=\widehat{a}^{\dagger}_{ij}\widehat{a}_{ij}$ is the usual
number operator, and $g_{V}$ and $p$ are free couplings that we assume to be
positive. The purpose of this term is to ensure that at low energies the
system has a (low temperature) phase in which the average vertex valence (i.e.
on edges attached to a vertex) is $v_{0}$. Later on in the paper we will show
that, at least in the mean field theory approximation, this is indeed the
case. The term $\widehat{H}_{L}$ is given by
$\widehat{H}_{L}=-g_{L}\sum_{i}\sum_{L}\frac{r^{L}}{L!}\widehat{P}(i,L),$ (5)
where $g_{L}$ and $r$ are couplings assumed to be positive. The operator
$\widehat{P}(i,L)$ counts the number of non-retracing paths of length L based
at the vertex $i$. This operator is related to the trace of the adjacency
matrix in the original model. We will build this operator in another way in
the following. For $r\leq 1$, so that higher length loops contribute less than
short length loops, this term is semi-local. The $L!$ comes from the expansion
of the exponential of the loop-path operatorgraphity1 . It was shown in
graphity1 that $r$ determines the length on which loop size is peaked at low
energies.
In what follows we introduce a new method to analyze dynamical lattices, by
transforming ${\mathscr{K}}_{N}$ to its line graph which we define in the next
section. The on/off edges of ${\mathscr{K}}_{N}$ will become Ising spins on
the fixed line graph, so that standard methods of statistical physics can be
used.
## III Graphs and Line Graphs
We start by defining line graphs. Let $G=(V,E)$ denote a graph with vertex set
$V=\\{v_{1},v_{2},...\\}$ and edge set $E=\\{e_{1},e_{2},...\\}$. The line
graph $\mathscr{L}(G)=(\widetilde{V},\widetilde{E})$ is the graph of the
adjacencies of $G$, containing information on the connectivity of the original
graph. Each vertex $\tilde{v}\in{\widetilde{V}}({\mathscr{L}}(G))$ corresponds
to an edge $e\in E(G)$. Two vertices $\tilde{v}_{1}$ and $\tilde{v}_{2}$ in
$\widetilde{V}(\mathscr{L}(G))$ are adjacent if and only if the edges in G
corresponding to $\tilde{v}_{1}$ and $\tilde{v}_{2}$ share a vertex. The
correspondence between $G$ and $\mathscr{L}(G)$ is not one to one. From a
given graph $G$ we can construct only one $\mathscr{L}(G)$ but it is not true
that any graph is a line graph. In fact, according to the Beineke
classification, there are 9 non-minimal graphs that are not line graphs of
another graph and each graph containing them is not a line graphbeineke . The
simplest example of a line graph is depicted in Fig. 1.
Figure 1: The simplest example of element of the Beineke classification of a
minimal graph that is not a line graph of any other one. This means that in
general there is not a one-one correspondence between a graphs and _line-
duals_.
Given a graph $G$, we can construct its line graph using the following
procedure:
1. 1.
Enumerate the vertices of $G$.
2. 2.
Enumerate the edges of $G$ with a fixed prescription (see example below) and
put a blob on them.
3. 3.
If two edges share a vertex, draw a bold line between them.
4. 4.
Remove $G$ and its enumeration.
What is left is the line graph of $G$ where the blobs represent its vertices.
Let us now introduce some useful quantities:
###### Definition.
(Kirchhoff matrix) Let $G$ be a generic graph,
$V=\\{v_{1},v_{2},\cdots,v_{n}\\}$ be the set of vertices of $G$ and
$E=\\{e_{1},e_{2},\cdots,e_{p}\\}$ be the set of edges of $G$. Let us define
the matrix $P$ of size $n\times p$ with entries $P_{i\beta}$, where $i$ is an
integer between $1$ and $n$ on the set of vertices and $\beta$ is an integer
between $1$ and $p$ on the set of edges, such that:
$P_{i\beta}=\left\\{\begin{array}[]{rl}1&\text{if the edge $\beta$ has an
vertex on the vertex $i$},\\\ 0&\text{otherwise }.\end{array}\right.$ (6)
The Kirchhoff matrix $K$ is the $p\times p$ matrix built from $P$, such that:
$K=P^{t}\ P,$ (7)
$P^{t}$ representing the transpose of $P$.
A well-known theorem now gives the incidence matrix of the line graph
$\mathscr{L}(G)$:
###### Theorem.
Let $G$ be a graph with p edges and n vertices and let $\mathscr{L}(G)$ be its
line graph. Then the matrix:
$J=K-2\ \textbf{I},$ (8)
where I is the $p\times p$ identity matrix, is the incidence matrix of
$\mathscr{L}(G)$.
In the next section we will show how the graphity hamiltonian can be recast on
the line graph using (6) and (8).
## IV The line graph representation
Since in the hamiltonian (3) we neglect the terms in which vertices are
interacting because we assume there are no degrees of freedom on them, one
could expect that it can be rewritten only in terms of the connectivity of the
graph. To carry out such a reformulation, let us expand the first term in (3)
for small values of the parameter $p$:
$\displaystyle\widehat{H}_{V}$ $\displaystyle=$ $\displaystyle
g_{V}\sum_{i}\widehat{1}+pg_{V}\sum_{i}\left(v_{0}-\sum_{j}\widehat{N}_{ij}\right)^{2}+{\mathscr{O}}(p^{2})$
(9) $\displaystyle=$ $\displaystyle
g_{V}\left(1+v_{0}^{2}p\right)\sum_{i}\widehat{1}+pg_{V}\sum_{ijk}\widehat{N}_{ij}\widehat{N}_{jk}-2g_{V}pv_{0}\sum_{ij}\widehat{N}_{ij}+{\mathscr{O}}(p^{2}).$
As we will see later, such an expansion does not modify the properties of the
model at low temperature. The first term in (9) is an energy shift and, for
what is to come, can be neglected. We should now be able to recognize some
particular terms in the expansion (9). The third term is proportional to the
operator $\sum_{ij}\widehat{N}_{ij}$. It is the sum over all the edges of the
graph, zero or not, of the number operator. We will change the notation to
$\sum_{ij}\widehat{N}_{ij}\rightarrow 2\sum_{\beta}\widehat{N}_{\beta},$
where $\beta$, as in the previous section, runs from $1$ to $N(N-1)/2$ and
labels the edges of ${\mathscr{K}}_{N}$ or, equivalently in what follows, the
vertices of its line graph.
To rewrite the second term in (9), we need the matrix $P_{i\beta}$ of (6) in
this context. This matrix maps the graph to its line graph, as we will see. We
first fix a prescription to label edges. Let $\mathscr{K}_{N}$ be the complete
graph of $N$ vertices. Let $\mathscr{I}$ be any enumeration of
$V(\mathscr{K}_{N})$, $i\in\mathscr{I}=1,...,N$. We identify edges by their
endpoint vertices $(i,j)$, with $i,j\in\mathscr{I}$. A labeling $S_{\beta}$,
$\beta\in\mathscr{B}=\\{1,\cdots,N(N-1)/2\\}$ is an enumeration of
$\widetilde{E}(\mathscr{L}(\mathscr{K}_{N}))$, according to the following
prescription:
$\displaystyle S_{1},\cdots,S_{N-1}\ \text{label the edges connecting the
vertices}\ \\{(1,2),\cdots,(1,N)\\};$ $\displaystyle S_{N},\cdots,S_{2(N-1)}\
\text{label the edges connecting the vertices}\ \\{(2,3),\cdots,(2,N)\\};$
$\displaystyle\vdots$ $\displaystyle S_{N(N-1)/2}\ \text{labels the edge
connecting the vertices}\ (N-1,N).$ (10)
Using this prescription it is easy to see that the matrix $P_{i\beta}$
introduced in (6), for the complete graph $\mathscr{K}_{N}$, has the simple
(recursive) form:
$P^{N}=\left(\begin{array}[]{cc}\vec{V}_{N-1}&\vec{0}\\\
\textbf{I}^{N-1}_{b^{\prime}c^{\prime}}&P^{N-1}_{a^{\prime}\alpha^{\prime}}\\\
\end{array}\right),$ (11)
where $\vec{V}_{N-1}$ is a row vector of length $N-1$, $\textbf{I}^{N-1}$ is
the identity matrix of size $(N-1)\times(N-1)$ and $\vec{0}$ represents a null
row vector of length $N(N-1)/2-(N-1)$. The indices
$\\{a^{\prime},b^{\prime},c^{\prime}\\}$ and $\alpha^{\prime}$ run from $1$ to
$N-1$ and $1$ to $(N-1)(N-2)/2$ respectively. As an example, for the graphs of
Fig. 2 the $P$ matrices are:
$P^{3}=\left(\begin{array}[]{ccc}1&1&0\\\ 1&0&1\\\ 0&1&1\\\
\end{array}\right),$ (12) $P^{4}=\left(\begin{array}[]{cccccc}1&1&1&0&0&0\\\
1&0&0&1&1&0\\\ 0&1&0&1&0&1\\\ 0&0&1&0&1&1\end{array}\right),$ (13)
$P^{5}=\left(\begin{array}[]{cccccccccc}1&1&1&1&0&0&0&0&0&0\\\
1&0&0&0&1&1&1&0&0&0\\\ 0&1&0&0&1&0&0&1&1&0\\\ 0&0&1&0&0&1&0&1&0&1\\\
0&0&0&1&0&0&1&0&1&1\end{array}\right),$ (14)
for (a), (b) and (c) respectively.
Figure 2: Three examples of complete graphs labeled according to the
prescription (10).
It is easy to see that two edges $\alpha$ and $\beta$ have a common vertex if
and only if we have:
$\sum_{i\in\mathscr{I}}{P^{t}}_{\beta i}P_{i\alpha}=c\neq 0,$ (15)
where $\phantom{c}{}^{t}$ is the transposition operation. By construction, $c$
can take the following values only:
$c=\left\\{\begin{array}[]{rl}2&\text{if}\ \alpha=\beta,\\\ 1&\text{if}\
\alpha\neq\beta\ \text{and}\ \alpha\ \text{and}\ \beta\ \text{have a common
vertex},\\\ 0&\text{if}\ \alpha\neq\beta\ \text{and}\ \alpha\ \text{and}\
\beta\ \text{do not have a common vertex.}\end{array}\right.$ (16)
In particular, for $N=4$, $K$ is given by:
$K^{4}=\left(\begin{array}[]{cccccc}2&1&1&1&1&0\\\ 1&2&1&0&0&1\\\
1&1&2&0&1&1\\\ 1&0&0&2&1&1\\\ 1&0&0&1&2&1\\\ 0&1&1&1&1&2\\\
\end{array}\right),$ (17)
Using now the matrix $P_{i\beta}$ just introduced, we want to construct
generic n-string matrices as composition of $n$ edges of the graph, thus a
_path_ on the graph. The $n$-string matrices will be needed both for the non-
retracing loop term and the 2-edge interaction term in equation (9). The
quantity:
$K^{i}_{\alpha\beta}=P^{t}_{\alpha i}P_{i\beta}$ (18)
is the definition of the Kirchhoff matrix of equation (8) if we sum over the
index $i$. From $K^{i}_{\alpha\beta}$ we can construct strings of $P$’s of the
form
$Q^{i_{1}\cdots
i_{n}}_{\alpha_{1}\cdots\alpha_{n+1}}=K^{i_{1}}_{\alpha_{1}\alpha_{2}}K^{i_{2}}_{\alpha_{2}\alpha_{3}}\cdots
K^{i_{n}}_{\alpha_{n}\alpha_{n+1}},$
that we call string matrices of nth order. These string matrices represent
paths through vertices $i_{1}\cdots i_{n}$ and they are zero unless the edges
corresponding to $\alpha_{1}\cdots\alpha_{n+1}$ are in the correct order, that
means, they represent an actual path on the graph. For instance, the number of
paths of length 2 on the complete graph is given by
$\\#2\mbox{-strings}=\sum_{\alpha\neq\beta\in\mathscr{B}}\ \
\sum_{i\in\mathscr{I}}Q^{i}_{\alpha\beta}=\sum_{\alpha\neq\beta\in\mathscr{B}}K_{\alpha\beta},$
(19)
or, equivalently, we can use (6) and rewrite (19) as
$\\#{\mbox{2-strings}}=\sum_{\alpha\beta\in\mathscr{B}}\ \
\sum_{i\in\mathscr{I}}Q^{i}_{\alpha\beta}=\sum_{\alpha\beta\in\mathscr{B}}\left(K_{\alpha\beta}-2\
\textbf{I}_{\alpha\beta}\right).$ (20)
The subtraction of twice the identity in (20) is the same as the subtraction
of the self-energy of each edge. We now clearly see that this matrix is
precisely the incidence matrix of the line graph of $\mathscr{K}_{N}$
introduced in (8), with
$\alpha,\beta\in\widetilde{V}(\mathscr{L}(\mathscr{K}_{N}))$.
So far we have dealt with the complete graph only. We wish to extend this
formalism to a dynamical graph. In order to do that we return to the Hilbert
space formulation of the graph with _on/off_ edges. Recall that any graph on
$N$ vertices is a subgraph of the complete graph $\mathscr{K}_{N}$, with some
edges _off_. Thus, since we can always map a graph on a complete graph, we can
count paths on any graph by modifying (20) so that it counts paths of only on
edges on the corresponding complete graph. To do so, we introduce in the sum
the number operators $\widehat{N}_{\beta}$ in the following way:
$\displaystyle\\#\mbox{2-strings}$ $\displaystyle=$
$\displaystyle\sum_{\alpha\beta\in\mathscr{B}}\left(K_{\alpha\beta}-2\
I_{\alpha\beta}\right)\widehat{N}_{\alpha}\widehat{N}_{\beta}$ (21)
$\displaystyle=$
$\displaystyle\sum_{\alpha\beta\in\mathscr{B}}J_{\alpha\beta}\
\widehat{N}_{\alpha}\widehat{N}_{\beta}.$
This term does not contribute if any of the two edges $\alpha,\beta$ is off.
It is easy to see that this term of the hamiltonian is an Ising interaction.
The important difference between these two hamiltonians is that in our case
the spin system is on the line graph a complete graph $\mathscr{K}_{N}$.
By extension of the above, we are now able to construct a generic path
operator out of $K^{i}_{\alpha\beta}$’s. We define
$\displaystyle\widehat{P}(n)$ $\displaystyle:=$
$\displaystyle\sum_{\mathscr{Q}}\sum_{\alpha_{1}\cdots\alpha_{n}}K^{i_{1}}_{\alpha_{1}\alpha_{2}}K^{i_{2}}_{\alpha_{2}\alpha_{3}}\cdots
K^{i_{n-1}}_{\alpha_{n-1}\alpha_{n}}\
\widehat{N}_{\alpha_{1}}\cdots\widehat{N}_{\alpha_{n}}$ (22) $\displaystyle=$
$\displaystyle\sum_{\mathscr{Q}}\sum_{\alpha_{1}\cdots\alpha_{n}}Q^{i_{1}\cdots
i_{n-1}}_{\alpha_{1}\cdots\alpha_{n}}\
\widehat{N}_{\alpha_{1}}\cdots\widehat{N}_{\alpha_{n}},$
where the set $\mathscr{Q}$ is
$\mathscr{Q}=\left\\{\begin{array}[]{l}i_{1}\neq\cdots\neq
i_{n-1}\in\mathscr{I}\text{ for non-retracing paths},\\\
i_{1},\dots,i_{n-1}\in\mathscr{I}\text{ for retracing paths}.\\\
\end{array}\right.$ (23)
It is easy to see that it counts the number of paths of length $n$ in the
graph, that is why we call the $Q$’s _string matrices_. Note that
$Q^{i_{1}\cdots i_{n-1}}_{\alpha_{1}\cdots\alpha_{n}}$ can take values $0$ and
$1$ only because it is a product of $0$’s and $1$’s. This string matrix is not
the matrix multiplication of the Kirchhoff matrices: it only reduces to matrix
multiplication for retracing paths where we sum over all possible vertices.
In the following, we will denote the two sets in (23) as $\mathscr{Q}^{r}$ and
$\mathscr{Q}^{nr}$ for the retracing and non retracing cases respectively;
moreover, we may explicitly show the indices on which we are doing the sum as
$\mathscr{Q}^{r/nr}(i_{b(j)})$. In order to count loops, we just need to
impose $\alpha_{1}=\alpha_{n}$:
$\displaystyle\underbrace{P_{\alpha_{1}i_{1}}P_{i_{1}\alpha_{2}}}\underbrace{P_{\alpha_{2}i_{2}}P_{i_{2}\alpha_{3}}}\underbrace{P_{\alpha_{3}i_{3}}P_{i_{3}\alpha_{1}}}.$
$\displaystyle\ \ K^{i_{1}}_{\alpha_{1}\alpha_{2}}\ \ \ \ \
K^{i_{2}}_{\alpha_{2}\alpha_{3}}\ \ \ \ \ \ K^{i_{3}}_{\alpha_{3}\alpha_{1}}$
Thus we have discovered that, when there are no degrees of freedom on the
vertices of the graph and we neglect the interaction terms, we can recast the
Quantum Graphity hamiltonian on the line graph
${\mathscr{L}}({\mathscr{K}}_{N})$ representation in the weak coupling regime
at finite $N$.
We end this section with two properties of the nth-order string matrices. Let
us define:
$\widetilde{Q}^{r/nr}_{\alpha_{1}\cdots\alpha_{n}}=\sum_{\mathscr{Q}^{r/nr}}Q^{i_{1}\cdots
i_{n-1}}_{\alpha_{1}\cdots\alpha_{n}}.$ (24)
The following properties of the sum of these string matrices on complete
graphs will be required next:
Property 1: Let $\mathscr{G}=\mathscr{K}_{N}$. Then, for a loop of $n$ edges:
$\sum_{\alpha_{1}\neq\alpha_{2}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{nr}_{\alpha_{1}\cdots\cdots\alpha_{L}\alpha_{1}}=N(N-1)\cdots(N-L)$
(25)
and
$\sum_{\alpha_{1}\neq\alpha_{2}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{r}_{\alpha_{1}\cdots\cdots\alpha_{L}\alpha_{1}}=N^{L}.$
(26)
Proof. These two facts follow trivially if we note that the equations (25) and
(26) count the number of retracing and non-retracing paths of length $L$ on
the complete graph respectively.
Property 2: Let $\mathscr{G}=\mathscr{K}_{N}$. Then, for a loop of $n$ edges,
and for $L\geq 4$, we have:
$\displaystyle\sum_{\alpha_{3}\neq\alpha_{4}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{nr}_{\alpha_{1}\cdots\cdots\alpha_{L}\alpha_{1}}=(N-3)\cdots(N-3-(L-4))K_{\alpha_{1}\alpha_{2}},$
(27)
while, for $L=3$:
$\displaystyle\sum_{\alpha_{3}}\widetilde{Q}^{nr}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{1}}=K_{\alpha_{1}\alpha_{2}},$
(28)
if $\alpha_{1}\neq\alpha_{2}\neq\alpha_{3}\cdots\neq\alpha_{n}$.
Proof. Note that
$\sum_{\alpha_{3}\neq\alpha_{4}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{nr}_{\alpha_{1}\cdots\cdots\alpha_{n}\alpha_{1}}$
is the number of non-retracing loops of length $L$ on the complete graph
$\mathscr{K}_{N}$ which pass by the edges $\alpha_{1}$ and $\alpha_{2}$. Now,
it is easy to see that if the edges $\alpha_{1}$ and $\alpha_{2}$ do not share
a link this quantity is zero. Also note that, by the symmetry of the complete
graph, the number of non-retracing loops based on two neighboring edges must
be the same for each pair of edges $\alpha_{j_{1}}$,$\alpha_{j_{2}}$ sharing a
node. Since the matrix $K_{\alpha_{j_{1}}\alpha_{j_{2}}}$ takes values $1$ or
$0$ depending on whether the edges $\alpha_{j_{1}}$,$\alpha_{j_{2}}$ are
neighbors or not,
$\sum_{\alpha_{3}\neq\alpha_{4}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{nr}_{\alpha_{1}\cdots\cdots\alpha_{L}\alpha_{1}}$
must be proportional to the matrix $K_{\alpha_{1}\alpha_{2}}$. In order to
evaluate the proportionality constant, let us note that each loop is weighed
by a factor of 1 because n-string matrices take values $1$ or $0$ only. The
combinatorial quantity $(N-3)\cdots(N-3-(L-4))$ is then the number of non-
retracing loops of length $L$ passing from two consecutive fixed edges on the
complete graph of $N$ vertices, as can be easily checked. The special case
(LABEL:property2b) follows from the fact that if we fix two edges there is
only one edge which closes the 3-loop.
Note that, for $N\gg L$, we have:
$\sum_{\alpha_{3}\neq\alpha_{4}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{nr}_{\alpha_{1}\cdots\cdots\alpha_{L}\alpha_{1}}\approx\sum_{\alpha_{3}\neq\alpha_{4}\neq\cdots\neq\alpha_{n}}\widetilde{Q}^{r}_{\alpha_{1}\cdots\cdots\alpha_{L}\alpha_{1}}=N^{L}.$
(29)
We can now collect the results of this section to write the hamiltonian (9) as
$\widehat{H}=A\sum_{\alpha,\beta\in\mathscr{B}}J_{\alpha\beta}\widehat{N}_{\alpha}\widehat{N}_{\beta}-B\sum_{\alpha\in\mathscr{B}}\widehat{N}_{\alpha}-C\sum_{\alpha\neq\beta\neq\gamma\in\mathscr{B}}\widetilde{Q}^{nr}_{\alpha,\beta,\gamma}\
\widehat{N}_{\alpha}\widehat{N}_{\beta}\widehat{N}_{\gamma},$ (30)
where
$\displaystyle A$ $\displaystyle=$ $\displaystyle p\ g_{V},$ $\displaystyle B$
$\displaystyle=$ $\displaystyle 2\ g_{V}\ p\ v_{0},$ $\displaystyle C$
$\displaystyle=$ $\displaystyle g_{L}\ \frac{r^{3}}{6},$ (31)
and neglecting higher order loop terms.
## V Mean field theory approximation and low temperature expansion
Having rewritten the hamiltonian in an Ising fashion, we now can approach the
problem of finding a graph observable and its equilibrium distribution using
mean field theory. As we will see, the natural graph observable to consider is
the average valence of the graph. We will assume that the system is at
equilibrium and we neglect the interaction terms. In this case, it is
straightforward to use mean field theory analysis Parisi . In what follows, we
assume units in which the Boltzmann constant $k_{B}=1$.
We start by replacing the number operators $\widehat{N}_{\alpha}$ with semi-
classical analogs, imposing that their expectation value must lie in the
interval $I=[0,1]$:
$\widehat{N}_{\beta}\rightarrow\langle\widehat{N}_{\beta}\rangle_{P}=m_{\beta},$
where $P$ is a probability measure of the following form:
$P(m_{\beta})=m_{\beta}\delta_{1,m_{\beta}}+(1-m_{\beta})\delta_{0,m_{\beta}}.$
(32)
It is easy to see that this probability distribution forces the spin-average
to lie in $I$. Recall that in order to obtain the mean field theory
distribution we have to extremize the Gibbs functional given by
$\Phi[m]=H[m]-\frac{1}{\beta}S[m],$ (33)
where $\beta=T^{-1}$, $H[m]$ is the energy and $S[m]$ is the entropy
functional. The latter can be written as:
$S[m]=-\sum_{m_{\beta}=\\{0,1\\}}n_{i}P(m_{\beta}(i))\log P(m_{\beta}(i)),$
(34)
where $n_{i}$ is the degeneracy of the state.
### Case I: Non-degenerate edge states
In this subsection we focus on the case in which the states on and off are not
degenerate, so that $n_{i}=1$. In the next subsection we will deal with non-
degenerate edge states and in particular with 3-degenerate on states.
In the process of extremizing the Gibbs functional we will see how the average
valence of the graph naturally emerges. We impose:
$\partial_{m_{\beta}}\Phi[m]=0.$
Using
$\partial_{m_{\beta}}S[m]=-\log(\frac{m_{\beta}}{1-m_{\beta}})$
and
$\partial_{m_{\beta}}H[m]=A\sum_{\alpha\in\mathscr{B}}J_{\alpha\beta}m_{\alpha}-B-C\sum_{\alpha\gamma\in\mathscr{B},\alpha\neq\gamma\neq\beta}\tilde{Q}_{\alpha\beta\gamma}m_{\alpha}m_{\gamma},$
we find that the distribution for the $m_{\alpha}$ is
$m_{\beta}=\frac{e^{-\beta\partial_{m_{\beta}}H[m]}}{1+e^{-\beta\partial_{m_{\beta}}H[m]}}=\frac{1}{1+e^{\beta\partial_{m_{\beta}}H[m]}}.$
(35)
The solution of this equation gives the equilibrium value of $m_{\beta}$ once
the value of the temperature is fixed.
We now want to write (35) as a function of an average quantity on the graph.
Let us first note that, in the mean field theory approximation, we have
$\sum_{\alpha}J_{\alpha\beta}m_{\alpha}=2\ d(T),$ (36)
where $d(T)$ is the mean valence of the graph. The valence $d(T)$ is a good
graph observable that we can use also as a double check for our procedure
since it appears explicitly in the original formulation of the hamiltonian and
in the low temperature regime must take the value $v_{0}$. First, we note
that:
$m_{\alpha}=\frac{N_{\text{on
edges}}}{N(N-1)/2}=\frac{\sum_{i\in\mathscr{I}}d(i)/2}{N(N-1)/2}=\frac{d(T)}{N-1}.$
In the first equality, $N_{\text{on edges}}$ is the number of edges of the
graph which are in an on state. In the second equality, the average valence
(the sum over all the local valencies divided by the number of vertices) is
explicitly written as a temperature dependent quantity. In the third equality
we used the graph property:
$\sum_{i\in\mathscr{I}}\frac{d(i)}{N-1}=\langle d\rangle\equiv d(T).$
The most complicated term in the hamiltonian is the 3-loop one. The simplest
way to deal with it is to use the Ansatz dictated by the mean field theory:
$\sum_{\alpha\gamma\in\mathscr{B}}\tilde{Q}_{\alpha\beta\gamma}m_{\alpha}m_{\beta}\approx\xi(T)d^{2}(T).$
(37)
Let us replace $m_{\beta}$ with its average value: $d(T)/{N-1}$. Using eq.
(25) for non-retracing paths and assuming $N\gg 1$ we obtain the dependence on
$d(T)$. $\xi(T)$ is a function of order $\sim 1$ at low temperature, which we
assume is dependent on $T$. Using these approximations we can see that $d(T)$
is a natural order parameter for our mean field theory since it is easily
recognized as implicitly defined in the stable distribution:
$d(T)=\frac{N-1}{1+e^{\beta[2d(T)A-\xi\frac{C}{2}d^{2}(T)-B]}}.$ (38)
Again, in order to double check our procedure, we can ask if such an order
parameter behaves as expected at low temperature. We must keep in mind that
the starting hamiltonian (3) was constructed in such a way that the average
valence at zero temperature was a fixed value of the parameter $v_{0}$ at
finite $N$. We can now use (38) to check if this is the case. To do so, we
Taylor expand both sides and match the zeroth and first order coefficients on
the left and right hand side of the equation. That is, we start with the
expansion
$d(T)=\tilde{\alpha}+\tilde{\beta}T+O(T^{2}),$ (39)
and, for the approximation to be consistent at $T=0$, we require analyticity
of the order parameter (this has to be the case for a finite volume system in
ordinary statistical mechanics, which is the case for finite $N$) . We then
require that inside the exponential of equation (38) the temperature
independent terms in the numerator cancel out so that at $T=0$ the exponent is
well defined. This gives the second order equation in $\alpha$:
$2\alpha A-\xi(0)C\alpha^{2}=B.$
Now note that, while this equation has two solutions, we need to only look for
the one which is analytical in the parameters of the model and tends smoothly
to the solution $\tilde{\alpha}=\frac{B}{2A}$ in the $C\rightarrow 0$ limit.
This fixes $\alpha$ to the value $\tilde{\alpha}$, given by
$\tilde{\alpha}=\frac{A}{\xi(0)C}\left(1-\sqrt{1-\frac{C\xi(0)B}{A^{2}}}\right).$
(40)
We can now plug $\tilde{\alpha}$ at $T=0$ into (38):
$\tilde{\alpha}=\frac{N-1}{1+e^{(2A-\xi(0)C\tilde{\alpha})\tilde{\beta}}},$
to obtain the value of $\tilde{\beta}$ in (39):
$\tilde{\beta}=\frac{1}{2A-\xi(0)C\tilde{\alpha}}\log\left(\frac{N-1}{\tilde{\alpha}}-1\right).$
(41)
It is easy to see that in the limit $N\rightarrow\infty$ we have
$\tilde{\beta}\rightarrow\infty$, indicating a second-order phase transition
(a discontinuity in the first derivative of the order parameter). In our case,
this happens at $T=0$, meaning that this transition is not possible because
there is no way to cool down the system to zero temperature with an external
bath. However, we have to remember that we are just approximating the real
system with a semi-classical analog. We then simply interpret the above result
as the fact that the system reaches the ground state very quickly when the
temperature approaches zero.
It is interesting now to plug in the couplings. Inserting equations (31) into
(40), we find that at $T=0$
$d(T=0)=\tilde{\alpha}=\frac{A}{\xi C}\left(1-\sqrt{1-\frac{C\xi
B}{A^{2}}}\right)=\frac{6pg_{V}}{\xi(0)g_{L}r^{3}}\left(1-\sqrt{1-\xi(0)\frac{g_{L}r^{3}v_{0}}{3pg_{V}}}\right).$
(42)
Note that, for small values of $r$, when $r^{3}\ll\frac{3pg_{V}}{g_{L}v_{0}}$,
we have $\tilde{\alpha}=v_{0}$, meaning that at low temperature the mean
degree is the one imposed by the degree term of the hamiltonian, as expected.
We can, however, see how the 3-loops term contributes to this quantity by a
Taylor expansion in $r$:
$d(T=0)=\tilde{\alpha}=v_{0}\left(1+\frac{2}{3}\xi(0)\frac{g_{L}r^{3}v_{0}}{pg_{V}}\right).$
(43)
From this expression it is clear that the loop terms are suppressed if
$g_{V}\gg g_{L}$. This is the main result derived in this paper using the line
graph representation. A plot of the function $d(T)$ is shown in Figure 3.
Figure 3: The behavior of log($d(T)$) ($y$-axis) against $T$ ($x$-axis), for
increasing $N$.
We now have the tools to calculate the susceptibility function for the theory
in the mean field theory approximation. Recall that the susceptibility
function tells us how the system reacts to a variation of the external
magnetic field. In our case, the magnetic field is the combination
$2g_{V}pv_{0}$ and we note that the parameter $v_{0}$ in the hamiltonian
appears only here. We have the following analogy: $v_{0}$ represents the
external magnetic field, while $2g_{V}p$ represents the spin-coupling
combination.
In order to calculate the susceptibility, we assume that the constant B is
site-dependent (i.e. a field). Thus, we have
$\langle G_{\alpha\beta}\rangle_{B}=-\frac{1}{\beta}\frac{\partial}{\partial
B_{\beta}}\frac{\partial}{\partial
B_{\alpha}}F[h]=\frac{1}{\beta}\frac{\partial m_{\beta}}{\partial
B_{\alpha}}.$ (44)
In particular, we are interested in the susceptibility function when $T\approx
0$. From the study of it we can gain some information about the low energy
behavior of the model. We expand the equilibrium distribution
$m_{\beta}=\frac{1}{2}\left[1-\beta\left(A\sum_{\alpha\in\mathscr{B}}J_{\alpha\beta}m_{\alpha}-B_{\beta}-C\sum_{\alpha\gamma\in\mathscr{B}}\tilde{Q}_{\alpha\beta\gamma}m_{\alpha}m_{\gamma}\right)\right].$
(45)
Using the notation
$\rho_{\alpha\beta}:=\sum_{\gamma\in\mathscr{B}}\tilde{Q}_{\alpha\beta\gamma}m_{\gamma}$,
we obtain
$\tilde{B}_{\beta}=B_{\beta}-\frac{1}{2}=\sum_{\alpha\in\mathscr{B}}\left(2\frac{\delta_{\alpha\beta}}{\beta}+AJ_{\alpha\beta}+C\tilde{\rho}_{\alpha\beta}\right)m_{\alpha}.$
(46)
To invert this equation, we approximate $\rho_{\alpha\beta}$ by replacing
$m_{\gamma}\rightarrow d(T)/(N-1)$:
$\sum_{\gamma\in\mathscr{B}}\tilde{Q}_{\alpha\beta\gamma}m_{\gamma}\rightarrow\frac{d(T)}{N-1}\sum_{\gamma\in\mathscr{B}}\tilde{Q}_{\alpha\beta\gamma}.$
We can now can use property (LABEL:property2b) of the $\tilde{Q}$ matrices to
find that the sum of the $\tilde{Q}$’s reduces to the incidence matrix of
$\mathscr{L}(K_{N})$. Hence, inverting equation (46), we obtain
$m_{\beta}=\sum_{\gamma}Q_{\gamma\beta}\tilde{B}_{\gamma},$
where
$Q_{\gamma\beta}={(2\frac{\delta}{\beta}+c_{0}\textbf{J})^{-1}}_{\gamma\beta}$
and $c_{0}$ is an effective constant in front of the Ising term of the
hamiltonian:
$c_{0}\approx pg_{V}+g_{L}\frac{r^{3}}{3!}\frac{d(T)}{N-1}.$ (47)
It is interesting to note that, thanks to property (LABEL:property2a), we can
sum all the loop terms up to a finite number $1\ll\tilde{L}\ll N$ in the
hamiltonian if we assume the mean field theory approximation. Inserting the
couplings, we find
$c_{0}\approx
pg_{V}+g_{L}\sum_{L=3}^{\tilde{L}}\frac{r^{2}}{L!}\left(r\frac{d(T)}{N}\right)^{L-2}N^{L-3}\approx
pg_{V}+\frac{g_{L}r^{2}}{Nd(T)^{2}}\left(e^{r\ d(T)}-1-r\
d(T)-\frac{r^{2}}{2}d(T)^{2}\right).$ (48)
It is interesting to note that in the limit $r\rightarrow 0$, or $T\rightarrow
0$ (where $d(T)$ tends to a finite number for $N\gg 1$), this effective
constant tends to $pg_{V}$. We interpret this as the fact that at low
temperature the loops become less and less important and the model is
dominated by the Ising term. In particular, since the external “magnetic”
field is given by $v_{0}$ and is assumed to be nonzero, it is not surprising
that at $T=0$ the average valence, the equivalent “magnetization”, approaches
this value. We note that the $N\rightarrow\infty$ limit does not behave well
unless $L=3$. Higher loops are highly non-local objects. For a given pair of
edges, all the $L$-loops based on these edges span the whole graph already at
$L=4$, while of course this is not the case for $3$-loops. As a result, in
formula (48) there is a factor proportional to $N^{L-3}$ which is not present
at $L=3$.
### Case II: Degenerate edge states
The Quantum Graphity model graphity1 allows for degenerate on states on the
edges or the vertices of the graph. Degeneracy of edge states is necessary,
for instance, in order to have emergent matter via the string-net condensation
mechanism of Levin and Wen. Degeneracy requires modifying our calculations
above and we will address it in this subsection.
The first possible generalization of the Quantum Graphity model is to
introduce a Hilbert space on the edges of the form (2):
$\mathscr{H}^{e}_{\beta}=\text{span}\\{|0\rangle_{\beta},|1_{1}\rangle_{\beta},|1_{2}\rangle_{\beta},|1_{3}\rangle_{\beta}\\}.$
This changes the degeneracy number in equation (34). With $n_{1}=3$ and
$n_{0}=1$, we obtain
$\partial_{m_{\beta}}S[m]=-\log(\frac{m_{\beta}^{3}}{1-m_{\beta}})-2.$
The equilibrium distribution solves this equation. If we put
$Q=\exp[\beta(\partial_{m_{\beta}}H[m]-2)]$, we have
$m_{\beta}=\left(\frac{2}{3}\right)^{\frac{2}{3}}\frac{Q}{\left(9+\sqrt{3}\sqrt{27+4Q^{3}}\right)^{\frac{2}{3}}}+\frac{\left(9+\sqrt{3}\sqrt{27+4Q^{3}}\right)^{\frac{1}{3}}}{2^{\frac{1}{3}}3^{\frac{2}{3}}},$
(49)
obtained from the only real solution of the third order polynomial equation
$m_{\beta}^{3}+Qm_{\beta}-1=0$.
Using the same procedure as before, it is easy to see that equation (40)
remains unchanged: the low energy average valence is the same in both cases.
However, the first derivative, that is, the coefficient of the $T$ term in the
Taylor expansion of the average valence in the temperature, changes, so that
$\tilde{\beta}_{3,1}\geq\tilde{\beta}_{1,1}$ (with the obvious notation for
the two coefficients). This phenomenon can be understood using the following
argument. At high temperature, the two models behave in the same way, forcing
the valence to be high. When the temperature drops, $d(T)$ also goes down.
While in the (1,1) case the phase space of the on edges is the same as that of
the off edges, in the (3,1) case the system prefers to stay in the on state.
Thus, when the temperature decreases the system (3,1) is, at first, slowly
converging to the ground state, but at $T=0$ it is forced to go to the ground
state. For this reason the function $d(T)$ has a greater derivative near $T=0$
in the (3,1) case.
## VI Conclusions
In this paper we introduced a technique to map the Quantum Graphity
hamiltonian on the line graph of a complete graph. This procedure requires the
introduction of the Kirchhoff matrix of a graph and the $n$-string matrices
related to these. This mapping is general and not specific to Quantum
Graphity. Using this mapping in a weak coupling approximation of the model,
the mean field theory approximation and the low temperature expansion, we
studied the properties of the model near zero temperature after having
identified the average degree as a order parameter. We found that the model is
dual to an Ising model with external nonzero magnetic field if we neglect the
interaction terms due to loops. In particular, we showed that the average
valence is naturally a good order parameter for the mean field theory
approximation and we found, implicitly, its average distribution using the
mapped hamiltonian and the mean field theory approximation for the 3-loop
term. We then studied the susceptibility function and showed how the duality
with the Ising model can help to interpret the results. In particular, the
parameter $v_{0}$ plays the role of the external magnetic field. Since $v_{0}$
is assumed to be never zero, the model has no phase transition and at $T=0$
the system goes to the ground state as expected. In fact, we found that at
zero temperature the mean valence is determined by the parameter $v_{0}$ and
we approximated the first order correction, showing the dependence on the
coupling constants of the model. While these results were expected on general
grounds, the mapping used here simplified the problem and allowed a
quantitative analysis.
What emerged from the study of the average distribution for the valence is
that, if the vertex valence term dominates ($g_{V}\gg g_{L}$), the loop term
corrections to the average valence of the ground state are suppressed at
$T=0$. We found the dependence on the coupling constants explicitly. This
result is confirmed by the study of the susceptibility. Thanks to the mean
field theory approximation, we found the contribution of all loops to the
susceptibility and showed that the susceptibility function tends to the Ising
one when $T$ approaches zero. We then applied this procedure to the degenerate
case, and showed that the degeneracy does not change the average valence at
low temperature but only the speed with which this ground state is reached. As
a final remark, we stress that the vertex valence is an important quantity in
the model. In fact, as shown in iaif using the Lieb-Robinson bound, the speed
with which information can propagate on graphs is bounded by a valence-
dependent quantity. For this reason, as the temperature drops, the speed of
the emergent light field must drop with the valence.
We would like to stress that we assumed that the system was at equilibrium
with an external bath. The problem of the required external bath in the model
has been studied in graphity2 , where additional degrees of freedom (bosonic
particles on vertices) were introduced to let the system thermalize and reach
an equilibrium distribution.
## VII Aknowledgements
The authors are indebted to Cohl Furey, Piero Porta Mana, Isabeau Prémont-
Schwarz, Simone Severini, Lee Smolin, and especially Alioscia Hamma for
reading the manuscript and providing useful advice and comments. This work was
supported by NSERC grant RGPIN-312738-2007 and the Humboldt Foundation.
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of
Research & Innovation.
## References
* (1) G. Volovik, The Universe in a Helium Droplet, Oxford University Press (2009).
* (2) See, for instance, Michael Levin, Xiao-Gang Wen, Rev. Mod. Phys. 77, 871-879 (2005), cond-mat/0407140
* (3) W. G. Unruh, Phys. Rev. D 51, 2827-2838 (1995), arXiv:gr-qc/9409008; M. Visser, S. Weinfurtner, PoSQG-Ph:042,2007, arXiv:0712.0427.
* (4) See, for instance, N. Seiberg, Rapporteur talk at the 23rd Solvay Conference in Physics, December, 2005, arXiv:hep-th/0601234.
* (5) S. Lloyd, arXiv:quant-ph/0501135.
* (6) F. Markopoulou, in Approaches to Quantum Gravity - towards a new understanding of space, time and matter, edited by D. Oriti, Cambridge Univ. Press 2009.
* (7) C.Rovelli, Quantum Gravity, Cambridge U. Press, New York (2004); T.Thiemann, “Introduction to modern canonical quantum general relativity,” arXiv:gr-qc/0110034.; A.Ashtekar and J.Lewandowski, “Background independent quantum gravity: A status report,” Class. Quant. Grav. 21, R53 (2004) [arXiv:gr-qc/0404018].
* (8) A. Perez, “Introduction to Loop Quantum Gravity and Spin Foams”, arXiv:gr-qc/0409061; D. Oriti, “Quantum Gravity as a quantum field theory of simplicial geometry”, arXiv:gr-qc/0512103; L. Freidel, R. Gurau , D. Oriti, “Group field theory renormalization - the 3d case: power counting of divergences”, arXiv:0905.3772
* (9) J. Ambjorn, J.Jurkiewicz and R.Loll, “Quantum gravity, or the art of building spacetime,” arXiv:hep-th/0604212.
* (10) R. Sorkin, “Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)”, proceedings of the Valdivia Summer School, edited by A. Gomberoff and D. Marolf, arXiv:gr-qc/0309009.
* (11) J. Makela, “Quantum-Mechanical Model of Spacetime“, arXiv:gr-qc/0701128
* (12) P. Gibbs, “The Principle of Event Symmetry.”, Int.J.Theor.Phys.35:1037–1062, (1996)
* (13) T. Konopka, F.Markopoulou and L. Smolin, “Quantum graphity,” arXiv:hep-th/0611197 ; T. Konopka, F. Markopoulou, S. Severini, “Quantum Graphity: a model of emergent locality”, arXiv:0801.0861
* (14) M.Levin and X. G. Wen, “Fermions, strings, and gauge fields in lattice spin models,” Phys. Rev. B 67, 245316 (2003) [arXiv:cond-mat/0302460]; M.A.Levin and X.G. Wen, “String-net condensation: A physical mechanism for topological phases,” Phys. Rev. B 71, 045110 (2005) [arXiv:cond-mat/0404617]; M.Levin and X.G.Wen, “Quantum ether: Photons and electrons from a rotor model,” arXiv:hep-th/0507118.
* (15) T. Konopka, “Matter in toy dynamical geometries”,J.Phys.Conf.Ser.174:012051,2009 ,arXiv:0903.4342
* (16) A. Hamma, F. Markopoulou, S. Lloyd, F. Caravelli, S. Severini, K. Markstrom, “A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime”, arXiv:0911.5075
* (17) Beineke, L. W. (1968), ”Derived graphs of digraphs”, in Sachs, H.; Voss, H.-J.; Walter, H.-J., Beiträge zur Graphentheorie, Leipzig: Teubner, pp. 17–33 .
* (18) G. Parisi, Statistical Field Theory (Addison-Wesley, Reading, Mass.), 1988
* (19) I. Prémont-Schwarz, A. Hamma, I. Klich, F. Markopoulou-Kalamara,Lieb-Robinson bounds for commutator-bounded operators, arXiv:0912.4544v1 [quant-ph]; A. Hamma, F. Markopoulou, I. Premont-Schwarz, S. Severini, Lieb-Robinson bounds and the speed of light from topological order, 10.1103/PhysRevLett.102.017204, arXiv:0808.2495v2 [quant-ph]
|
arxiv-papers
| 2010-08-07T13:09:55 |
2024-09-04T02:49:12.091745
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Francesco Caravelli and Fotini Markopoulou",
"submitter": "Francesco Caravelli",
"url": "https://arxiv.org/abs/1008.1340"
}
|
1008.1345
|
# Adaptive post-Dantzig estimation and prediction for non-sparse “large $p$
and small $n$” models
Lu Lin, Lixing Zhu111Lu Lin is a professor of the School of Mathematics at
Shandong University, Jinan, China. His research was supported by NNSF project
(10771123) of China, NBRP (973 Program 2007CB814901) of China, RFDP
(20070422034) of China, NSF projects (Y2006A13 and Q2007A05) of Shandong
Province of China. Lixing Zhu is a chair professor of Department of
Mathematics at Hong Kong Baptist University, Hong Kong, China. Email:
lzhu@hkbu.edu.hk. He was supported by a grant from the University Grants
Council of Hong Kong, Hong Kong, China. Yujie Gai is a PHD student of the
School of Mathematics at Shandong University, Jinan, China. The first two
authors are in charge of the methodology development and material
organization. and Yujie Gai
###### Abstract
For consistency (even oracle properties) of estimation and model prediction,
almost all existing methods of variable/feature selection critically depend on
sparsity of models. However, for “large $p$ and small $n$” models sparsity
assumption is hard to check and particularly, when this assumption is
violated, the consistency of all existing estimations is usually impossible
because working models selected by existing methods such as the LASSO and the
Dantzig selector are usually biased. To attack this problem, we in this paper
propose adaptive post-Dantzig estimation and model prediction. Here the
adaptability means that the consistency based on the newly proposed method is
adaptive to non-sparsity of model, choice of shrinkage tuning parameter and
dimension of predictor vector. The idea is that after a sub-model as a working
model is determined by the Dantzig selector, we construct a globally unbiased
sub-model by choosing suitable instrumental variables and nonparametric
adjustment. The new estimation of the parameters in the sub-model can be of
the asymptotic normality. The consistent estimator, together with the selected
sub-model and adjusted model, improves model predictions. Simulation studies
show that the new approach has the significant improvement of estimation and
prediction accuracies over the Gaussian Dantzig selector and other classical
methods have.
Keywords. Adaptability, bias correction, Dantzig selector, instrumental
variable, nonparametric adjustment, Ultra high-dimensional regression.
AMS 2001 subject classification: 62C05, 62F10, 62F12, 62G05.
Running head. Adaptive post-Dantzig inference.
1\. Introduction
Estimation consistency is a natural criterion for estimation accuracy. In
classical settings with small/moderate number of variables in models, this
criterion can be adopted. For high-dimensional models, particularly, when the
number $p$ of variables involved is even larger than the sample size $n$, are
called “large $p$, small $n$” models. However in these paradigm estimation
consistency becomes a very challenging issue. This is because what we can work
on is only working models rather than full models after active variables are
selected into working models. For variable selection, some classical and newly
proposed methods are available, such as the LASSO (including the adaptive
LASSO) and the SCAD. These methods provide consistent and asymptotically
normally distributed estimation for the parameters in working models, but
these properties heavily depend on sparse structure, proper choice of
shrinkage tuning parameter and the diverging rate of the dimension of
parameter vector. For the relevant references see Huber (1973), Portnoy
(1988), Bai and Saranadasa (1996), Fan and Peng (2004), Fan, Peng and Huang
(2005), Lam and Fan (2008), Huang et al. (2008), and Li, Zhu and Lin (2009),
among others. As such, for models without spare structure, how to construct
consistent estimation is a great challenge. It is required to develop new or
extended statistical methodologies and theories to handle this challenge; see
for example Donoho (2000), Kettenring, Lindsay and Siegmund (2003).
To this end, we further review existing methods to get motivation for new
methodology development. The following methods were developed also under
sparse structure. The Dantzig selector that was proposed by Candés and Tao
(2007) and was extended to handle the generalized linear models by James and
Radchenko (2009) has received much attention. The connection between the
Dantzig selector and the LASSO was investigated by James et al. (2009). Under
the uniform uncertainty principle, the resulting estimator achieves an ideal
risk of order $O(\sigma\sqrt{\log p})$ with a large probability. This implies
that for large $p$, such a risk can be however large and then even under
sparse structure the estimator may also be inconsistent. To reduce the risk
and improve the performance of relevant estimation, the Gaussian Dantzig
Selector, a two-stage estimation, was suggested in the literature (Candés and
Tao 2007). Such an improved estimation is still inconsistent when the
shrinkage tuning parameter is chosen to be large (for details see the next
section). Another method is the Double Dantzig Selector (James and Radchenko
2009), by which one may choose a more accurate model and, at the same time,
get a more accurate estimator. But it critically depends on the choice of
shrinkage tuning parameter. Motivated by these problems, Fan and Lv (2008)
introduced a sure independent screening method that is based on correlation
learning to reduce high dimensionality to a moderate scale below the sample
size. Afterwards, variable selection and parameter estimation can be
accomplished by sophisticated methods, such as the LASSO, the SCAD or the
Dantzig selector. The relevant references include Kosorok and Ma (2007), Van
Der Lanin and Bryan (2001), Chen and Qin (2010), James, Radchenko and Lv
(2009) and Kuelbs and Anand (2010), among others.
However, for any model with very large $p$, without model sparsity, all
existing methods cannot provide estimation consistency for working models, and
any further data analysis would be questionable unless we can correct biases
later or at most we can obtain an approximation rather than estimation
consistency as the sample size goes to infinity. To deal with this problem, we
focus our attention on working sub-model that is chosen by the Dantzig
selector. In this paper, we suggest a method to construct consistent and
asymptotically normal distributed estimation for the parameters in the sub-
model. To achieve this, a nonparametric adjustment is recommended to construct
a globally unbiased sub-model and to correct the bias in working model. Here
the nonparametric adjustment may depend on a low-dimensional nonparametric
estimation via using proper instrument variables. We will show the following
properties. The estimator $\hat{\theta}$ of the parameter vector $\theta$ in
the sub-model satisfies
$\|\hat{\theta}-\theta\|^{2}_{{\ell}_{2}}=O_{p}(n^{-1})$ and the asymptotic
normality if the dimension $q$ of $\theta$ is fixed. Even for the case where
$q$ tends to infinity, the consistent and asymptotic normality still hold when
$q$ diverges at a certain rate. We will briefly discuss the theoretical
results for the case with diverging $q$. Furthermore, the new consistent
estimator, together with the unbiased adjustment sub-model or the original
sub-model, can also improve model prediction accuracy. We will prove that our
method possesses the adaptability. That is, the above properties always hold
whether the sub-model is small or large, the dimension of the parameter in the
original model is high or not, and the original model is sparse or not.
The rest of the paper is organized as follows. In Section 2 the properties of
the Dantzig estimator for the high-dimensional linear model are re-examined.
In Section 3 a bias-corrected sub-model is proposed via introducing
instrumental variables and a nonparametric adjustment, and a method about
instrumental variable selection is introduced. Estimation and prediction
procedures for the new sub-model are suggested and the asymptotic properties
of the resulting estimator and prediction are obtained. In Section 4 the
algorithms for constructing instrumental variables are proposed. Simulation
studies are presented in Section 5 to examine the performance of the new
approach when compared with the classical Dantzig selector and other methods.
The technical proofs for the theoretical results are postponed to the
Appendix.
2\. A brief review for the Dantzig selector
Consider the model
$Y=\beta^{\prime}X+\varepsilon,$ $None$
where $Y$ is the scale response, $X$ is the $p$-dimensional covariate and
$\varepsilon$ is the random error satisfying $E(\varepsilon|X)=0$ and
$Cov(\varepsilon|X)=\sigma^{2}$. Here $p$ will be greater than $n$ when we can
collect a sample of size $n$. Throughout this paper, our primary interest is
to construct consistent estimators for significant components of the parameter
vector $\beta=(\beta_{1},\cdots,\beta_{p})^{\prime}\in\mathscr{B}\subset
R^{p}$. These significant components of $\beta$, together with the
corresponding covariates, composes a working model. Then the second interest
of our paper is to obtain reasonable model prediction via our estimation.
To introduce the new estimation, we first re-examine the Dantzig selector. Let
${\bf Y}=(Y_{1},\cdots,Y_{n})^{\prime}$ be the vector of the observed
responses and ${\bf X}=(X_{1},\cdots,X_{n})^{\prime}=({\bf x}_{1},\cdots,{\bf
x}_{p})$ be the $n\times p$ matrix of the observed covariates. The Dantzig
selector of $\beta$ is defined as
$\tilde{\beta}^{D}=\arg\min_{\beta\in\mathscr{B}}\|\beta\|_{{\ell}_{1}}\ \
\mbox{ subject to }\ \ \sup_{1\leq j\leq p}|{\bf
x}^{\prime}_{j}r|\leq\lambda_{p}\,\sigma$ $None$
for some $\lambda_{p}>0$, where
$\|\beta\|_{{\ell}_{1}}=\sum_{j=1}^{p}|\beta_{j}|$ and $r={\bf Y}-{\bf
X}\beta$. As was shown by Candés and Tao (2007), under some regularity
conditions, this estimator satisfies that, with large probability,
$\|\tilde{\beta}^{D}-\beta\|^{2}_{{\ell}_{2}}\leq C\sigma^{2}\log p,$ $None$
where $C$ is free of $p$ and
$\|\tilde{\beta}^{D}-\beta\|^{2}_{{\ell}_{2}}=\sum_{j=1}^{p}|\tilde{\beta}_{j}^{D}-\beta_{j}|^{2}$.
In fact this is an ideal risk and thus cannot be improved in a certain sense.
However, such a risk can become large and may not be negligible when the
dimension $p>n$.
To reduce the risk and enhance the performance in practical settings, one
often uses a two-stage selection procedure (e. g., the Gaussian Dantzig
Selector) to construct a risk-reduced estimator for the obtained sub-model
(Candés and Tao 2007). For example, we can first estimate
$I=\\{j:\beta_{j}\neq 0\\}$ with
$\tilde{I}=\\{j:|\tilde{\beta}_{j}^{D}|>\varsigma\sigma\\}$ for some
$\varsigma\geq 0$ and then construct an estimator
$\tilde{\beta}_{(\tilde{I})}=(({\bf X}^{(\tilde{I})})^{\prime}{\bf
X}^{(\tilde{I})})^{-1}({\bf X}^{(\tilde{I})})^{\prime}{\bf Y}$
for $\beta_{(\tilde{I})}$ and set the other components of $\beta$ to be zero,
where $\beta_{(\tilde{I})}$ is the restriction of $\beta$ to the set
${\tilde{I}}$, and ${\bf X}^{(\tilde{I})}$ is the matrix with the column
vectors according to $\tilde{I}$.
Denote $\beta_{(\tilde{I})}=\theta$, a $q$-dimensional vector of interest.
Without loss of generality, suppose that $\beta$ can be partitioned as
$\beta=(\theta^{\prime},\gamma^{\prime})^{\prime}$ and, correspondingly, $X$
is partitioned as $X=(Z^{\prime},U^{\prime})^{\prime}$. Then the above two-
stage procedure implies that we can use the sub-model
$Y=\theta^{\prime}Z+\eta$ $None$
to replace the full-model (2.1), where $\eta=\gamma^{\prime}U+\varepsilon$ is
regarded as error. Here the dimension $q$ of $\theta$ can be either fixed or
diverging with $n$ at certain rate. Since the above sub-model is a replacer of
the full model (2.1), we call $\theta$ and $Z$ the main parts of $\beta$ and
$X$, respectively. From (2.1) and (2.4) it follows that
$E(\eta|Z)=\gamma^{\prime}E(U|Z)$. When both $\gamma\neq 0$ and $E(U|Z)\neq
0$, the sub-model (2.4) is biased and thus the two-stage estimator
$\tilde{\theta}_{S}=\tilde{\beta}_{(\tilde{I})}$ is also biased. It shows that
the two-stage estimator $\tilde{\theta}_{S}$ of $\theta$ is also inconsistent.
Note that for any non-sparse model, the condition $\gamma\neq 0$ always holds.
Then the above method is not possible to obtain consistent estimation.
Another method for improving the Dantzig selector is the Double Dantzig
Selector. By which more accurate model and estimation can be expected. In the
first step, the Dantzig selector is used with a relatively large shrinkage
tuning parameter $\lambda_{p}$ defined above to get a relatively accurate sub-
model in the sense that more significant variables are contained. The Dantzig
selector is further used in the selected sub-model to obtain a relatively
accurate estimator of $\theta$ via a small $\lambda_{p}$ and data $(Y,Z)$.
However, such a method cannot handle non-sparse model because the sub-model
selected in the first step has already been biased. It is also noted that this
method critically depends on twice choices of shrinkage tuning parameter
$\lambda_{p}$; for details see James and Radchenko (2009). On the other hand,
when estimation consistency and normality, rather than variable selection,
heavily depend on the choice of $\lambda_{p}$, it is practically not
convenient, and more seriously, the consistency is in effect not judgeable
unless a criterion of tuning parameter selection can be defined to ensure
consistency. Then it is desirable to have a new estimation/inference method
with which consistency is free of the choice of $\lambda_{p}$.
3\. Adaptive post-Dantzig estimation and prediction
3.1 Bias-corrected model. As was shown above, the sub-model (2.4) is usually
biased. Furthermore, this model is regarded as a non-random model after the
variable selection given by the Dantzig selector, i.e., the estimate
$\tilde{I}$ for the index set $I$ defined in the previous section is fixed
after variable selection.
It is clear that a bias correction is needed for the selected sub-model (2.4)
when we want to have a consistent estimation of the sub-vector
$\theta=(\theta_{1},\cdots,\theta_{q})^{\prime}$. To this end, a new model
with an instrumental variable is established. Denote
$Z^{\star}=(Z^{\prime},U^{(1)},\cdots,U^{(d)})^{\prime}$ and $W=AZ^{\star}$,
where $A$ is $d\times(q+d)$ matrix satisfying that its row vectors have length
1. Without loss of generality, $U^{(1)},\cdots,U^{(d)}$ are supposed to be the
first $d$ components of $U$, although they may be chosen as another components
of $U$ or pseudo-variables (artificial vavriables). Denote by $\lambda_{M}$
the maximum eigenvalue of $UU^{\prime}$ and set
$V=(\alpha^{\prime}U/\rho,W^{\prime})^{\prime}$ for some $\alpha$ to be chosen
later, where $\rho$ is a nonrandom positive number satisfying the condition
$\rho=O(\|\alpha\|_{{\ell}_{2}}\sqrt{\lambda_{M}})$. Choose $A$ and
$U^{(1)},\cdots,U^{(d)}$ such that
$E\\{(Z-E(Z|V))(Z-E(Z|V))^{\prime}\\}>0.$ $None$
This condition on the matrix we need can trivially hold because $V$ contains
$W$ that is a weighted sum of $Z$ and $U^{(1)},\cdots,U^{(d)}$. The condition
(3.1) can be used to guarantee the identifiability of the following model.
Denote $g(V)=E(\eta|V)$. Now we introduce a bias-corrected version of (2.4) as
$Y_{i}=\theta^{\prime}Z_{i}+g(V_{i})+\xi(V_{i}),\ i=1,\cdots,n,$ $None$
where $\xi(V)=\eta-g(V)$. Obviously, if $\alpha$ in $V$ is identical to
$\gamma$ in $\eta$, this model is unbiased, i.e., $E(\xi|Z,V)=0$; otherwise it
may be biased. This model can be regarded as a partially linear model with a
linear component $\theta^{\prime}Z$ and a nonparametric component $g(V)$, and
is identifiable because of the condition (3.1). From this structure, we can
see that when $V$ does not contain the instrumental variable $W$ and
$\alpha=\gamma$, the model goes back to the original model (2.4) as $\xi$ is
zero and $g(V)$ becomes the error term $\eta$ (if $\varepsilon$ is ignored).
This structure motivates our method. By introducing an instrumental variable
$V$ so that $\xi$ has a zero conditional mean, and then we can estimate
$g(\cdot)$ to correct the bias occurred in the original model. Although a
nonparametric function $g(v)$ is involved, it will be verified that the
dimension $d+1$ of the variable $v$ is low. Note that for $V$, the key is to
properly select $\alpha$ and $W$. From the above description, we can see that
although $\alpha=\gamma$ should be a natural and good choice, it is unknown
and when the dimension is large, is cannot be estimated consistently. Taking
this into account, we first consider a general $\alpha$ and construct a bias-
corrected model with suitable $W$, or equivalently a suitable matrix $A$.
Denote by $l=p-q$ the dimension of $\gamma$ and let
$\lambda=(0,\gamma_{2}-\frac{\gamma_{1}}{\alpha_{1}}\alpha_{2},\cdots,\gamma_{l}-\frac{\gamma_{1}}{\alpha_{1}}\alpha_{l})^{\prime}/\rho,$
where $\alpha_{1},\cdots,\alpha_{l}$ are the components of $\alpha$ and
$\alpha_{1}$ is supposed to be nonzero. We can ensure that, when $Z_{i}$ and
$U_{i}$ satisfy
$\lambda^{\prime}E(U_{i}|Z_{i},W_{i})=\lambda^{\prime}E(U_{i}|W_{i}),$ $None$
the model (3.2) is unbiased, i.e.,
$E(\xi(V_{i})|Z_{i},V_{i})=0.$ $None$
The proof of (3.4) will be presented in the Appendix.
When $(Z,U)$ is elliptically symmetrically distributed, the condition (3.3)
can be rewritten at population level as the following form:
$\begin{array}[]{ll}\lambda^{\prime}\Sigma_{U,Z^{\star}}A^{\prime}(A\Sigma_{Z^{\star},Z^{\star}}A^{\prime})^{-1}A(Z^{\star}-E(Z^{\star}))\vspace{1ex}\\\
=\lambda^{\prime}\Sigma_{U,Z^{\star}}B^{\prime}(B\Sigma_{Z^{\star},Z^{\star}}B^{\prime})^{-1}B(Z^{\star}-E(Z^{\star})),\end{array}$
$None$
where $\Sigma_{Z^{\star},Z^{\star}}=Cov(Z^{\star},Z^{\star})$,
$\Sigma_{U,Z^{\star}}=Cov(U,Z^{\star})$ and
$B=\left(\begin{array}[]{llll}I&0&\cdots&0\\\
A_{1}&a_{q+1}&\cdots&a_{q+d}\end{array}\right),$
$A=(A_{1},a_{q+1},\cdots,a_{q+d})$, $A_{1}$ is a $d\times q$ matrix and
$a_{j},j=q+1,\cdots,a_{q+d},$ are $d$-dimensional column vectors. Further, the
ellipticity condition can be slightly weakened to be the following linearity
condition:
$E(U|C^{\prime}Z^{\star})=E(U)+\Sigma_{U,Z^{\star}}C(C^{\prime}\Sigma_{Z^{\star},Z^{\star}}C)^{-1}C^{\prime}(Z^{\star}-E(Z^{\star}))$
for some given matrix $C$. This linearity condition also results in (3.5). The
linearity condition has been widely assumed in the circumstance of high-
dimensional models. Hall and Li (1993) showed that it often holds
approximately when the dimension $p$ is high.
Under either the equation (3.3) or (3.5), the bias-corrected model (3.2) is
unbiased. Thus, we are now in the position to determine the matrix $A$ by
solving either the equation (3.3) or (3.5). A solution is not difficult to be
obtained. For example, if $\Sigma_{Z^{\star},Z^{\star}}=I_{q+d}$ and $B^{-1}$
exists, then we choose $A$ satisfying
$\Sigma_{U,Z^{\star}}A^{\prime}(AA^{\prime})^{-1}A(Z^{\star}-E(Z^{\star}))=\Sigma_{U,Z^{\star}}(Z^{\star}-E(Z^{\star})).$
$None$
It is known that, if we can choose variables $U^{(1)},\cdots,U^{(d)}$ such
that the rank of matrix $\Sigma_{U,Z^{\star}}$ is $d$, then
$\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}=Q\left(\begin{array}[]{ll}I_{d}&0\\\
0&0\end{array}\right)Q^{\prime}=Q_{1}Q_{1}^{\prime},$
where $\Sigma_{U,Z^{\star}}^{+}$ is the Moore-Penrose generalized inverse
matrix of $\Sigma_{U,Z^{\star}}$, $I_{d}$ is a $d\times d$ identify matrix,
$Q=(Q_{1},Q_{2})$ an orthogonal matrix satisfying $Q^{\prime}Q=I_{q+d}$ and
$Q_{1}^{\prime}Q_{1}=I_{d}$. In this case, we choose
$A=Q_{1}^{\prime}.$ $None$
Such a matrix $A$ is a solution of (3.6) and thus a solution of (3.5). With
such a choice of $A$, the model (3.2) is always unbiased whether the model
(2.1) is sparse or not, the dimension of $\beta$ is high or low, and the
choice of $\lambda_{p}$ is proper or not.
However, sometimes the matrix $\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}$
is unknown. Under this situation, we will present a detailed procedure in
Section 4 to calculate $\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}$ and $A$.
From the above choice of $A$, we can see that $g(v)$ is a $d+1$-dimensional
nonparametric function. If $d$ is large, we choose a row vector to replace $A$
and will give a method in Section 4 to find an approximate solution. With
which, $g(v)$ is a 2-dimensional nonparametric function.
The above deduction shows that the above bias-correction procedure is free of
the choice of $\alpha$. However, choosing a proper $\alpha$ is of importance.
It is clear that, combining (3.2) and (3.3), choosing an $\alpha$ as close to
$\gamma$ as possible should be a good way although optimal choice leaves an
unsolved and interesting problem. In the estimation procedure, a natural
choice is the value $\tilde{\gamma}^{D}$ for $\gamma$, which is obtained in
the Dantzig selection step. The details are presented in Subsection 3.2 below.
We will also discuss the asymptotic properties of an estimation when we use a
given $\alpha$ in the next subsection.
3.2 Asymptotic normality of estimation. Throughout this subsection we assume
that the matrix $A$ satisfying (3.5) or (3.6) has been obtained. Although the
obtained $A$ is sometimes an estimator rather than an exact solution, in this
section we still regard it as a nonrandom solution of (3.5) or (3.6) because
such an estimator is $\sqrt{n}$-consistent (see Section 4 below) and, as a
result, when $A$ is thought of a random vector, the theoretical conclusions
given below still hold.
Recall that the bias-corrected model (3.2) can be thought of as a partially
linear model. We therefore design an estimation procedure as follows. First of
all, as mentioned above, for any $\alpha$, the model (3.2) is unbiased. Then
we can design the estimation procedure after $\alpha$ is determined by any
empirical method. An empirical choice $\alpha$ is designed as the Dantzig
selector $\tilde{\gamma}^{D}$ of $\gamma$ determined by (2.2). Generally,
given $\theta$ and for any $\alpha$, the nonparametric function $g(v)$ is
estimated by
$g_{\theta}(v)=\frac{\sum_{k=1}^{n}(Y_{k}-\theta^{\prime}Z_{k})L_{H}(V_{k}-v)}{\sum_{k=1}^{n}L_{H}(V_{k}-v)},$
where $L_{H}(\cdot)$ is a $(d+1)$-dimensional kernel function. Then
$g_{\theta}(v)$ is a $(d+1)$-variate nonparametric estimator. As was shown
above, the dimension $d+1$ is low. A simple choice of $L_{H}(\cdot)$ is a
product kernel as
$L_{H}(V-v)=\frac{1}{h^{d+1}}K\Big{(}\frac{V^{(1)}-v^{(1)}}{h}\Big{)}\cdots
K\Big{(}\frac{V^{(d+1)}-v^{(d+1)}}{h}\Big{)},$
where $V^{(j)},j=1,\cdots,d+1$, are the components of $V$, $K(\cdot)$ is an
1-dimensional kernel function and $h$ is the bandwidth depending on $n$.
Particularly, when $\alpha$ is chosen as $\tilde{\gamma}^{D}$, we get an
estimator of $g(v)$ as
$\hat{g}_{\theta}(v)=\frac{\sum_{k=1}^{n}(Y_{k}-\theta^{\prime}Z_{k})L_{H}(\hat{V}_{k}-v)}{\sum_{k=1}^{n}L_{H}(\hat{V}_{k}-v)}$
where $\hat{V}=(U^{\prime}\tilde{\gamma}^{D}/\hat{\rho},W^{\prime})^{\prime}$
and $\hat{\rho}=O(\|\tilde{\gamma}^{D}\|_{{\ell}_{2}}\sqrt{\lambda_{M}})$.
With these two estimations of $g(v)$, the bias-corrected model (3.2) can be
approximately expressed by the following two models:
$Y_{i}\approx\theta^{\prime}Z_{i}+g_{\theta}(V_{i})+\xi(V_{i})\ \ \mbox{ and
}\ \
Y_{i}\approx\theta^{\prime}Z_{i}+\hat{g}_{\theta}(\hat{V}_{i})+\xi(\hat{V}_{i}),$
equivalently,
$\tilde{Y}_{i}\approx\theta^{\prime}\tilde{Z}_{i}+\xi(V_{i})\ \ \mbox{ and }\
\ \hat{Y}_{i}\approx\theta^{\prime}\hat{Z}_{i}+\xi(\hat{V}_{i}),$ $None$
where
$\tilde{Y}_{i}=Y_{i}-\frac{\sum_{k=1}^{n}Y_{k}L_{H}(V_{k}-V_{i})}{\sum_{k=1}^{n}L_{H}(V_{k}-V_{i})},\
\
\tilde{Z}_{i}=Z_{i}-\frac{\sum_{k=1}^{n}Z_{k}L_{H}(V_{k}-V_{i})}{\sum_{k=1}^{n}L_{H}(V_{k}-V_{i})},$
$\hat{Y}_{i}=Y_{i}-\frac{\sum_{k=1}^{n}Y_{k}L_{H}(\hat{V}_{k}-\hat{V}_{i})}{\sum_{k=1}^{n}L_{H}(\hat{V}_{k}-\hat{V}_{i})},\
\
\hat{Z}_{i}=Z_{i}-\frac{\sum_{k=1}^{n}Z_{k}L_{H}(\hat{V}_{k}-\hat{V}_{i})}{\sum_{k=1}^{n}L_{H}(\hat{V}_{k}-\hat{V}_{i})}.$
Thus, the sub-models in (3.8) result in the estimations for $\theta$ as
$\tilde{\theta}=S_{n}^{-1}\frac{1}{n}\sum_{i=1}^{n}\tilde{Z}_{i}\tilde{Y}_{i}\
\ \mbox{ and }\ \
\hat{\theta}=S_{n}^{-1}\frac{1}{n}\sum_{i=1}^{n}\hat{Z}_{i}\hat{Y}_{i},$
$None$
where $S_{n}=\frac{1}{n}\sum_{i=1}^{n}\tilde{Z}_{i}\tilde{Z}_{i}^{\prime}$ or
$S_{n}=\frac{1}{n}\sum_{i=1}^{n}\hat{Z}_{i}\hat{Z}_{i}^{\prime}$,
respectively. Here we assume that the bias-corrected model (3.2) is
homoscedastic, that is $Var(\xi(V_{i}))=\sigma_{V}^{2}$ or
$Var(\xi(\hat{V}_{i}))=\sigma_{V}^{2}$ for all $i=1,\cdots,n$. If the model is
heteroscedastic, we respectively modify the above estimators as
$\tilde{\theta}^{*}={S_{n}^{*}}^{-1}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}(V_{i})}\tilde{Z}_{i}\tilde{Y}_{i}\
\ \mbox{ and }\ \
\hat{\theta}^{*}={S_{n}^{*}}^{-1}\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}(\hat{V}_{i})}\hat{Z}_{i}\hat{Y}_{i},$
where
$S_{n}^{*}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}(V_{i})}\tilde{Z}_{i}\tilde{Z}_{i}^{\prime}$
or
$S_{n}^{*}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}(\hat{V}_{i})}\hat{Z}_{i}\hat{Z}_{i}^{\prime}$,
respectively, and $\sigma_{i}^{2}(V_{i})=Var(\xi(V_{i}))$ and
$\sigma_{i}^{2}(\hat{V}_{i})=Var(\xi(\hat{V}_{i}))$. Here
$\sigma_{i}^{2}(V_{i})$ and $\sigma_{i}^{2}(\hat{V}_{i})$ are supposed to be
known. If they are unknown, we can use their consistent estimators to replace
them; for details about how to estimate them see for example Härdle et al.
(2000). In the following we only consider the estimators defined in (3.9).
Finally, the estimators of $g(v)$ can be defined as either
$g_{\tilde{\theta}}(v)$ or $\hat{g}_{\hat{\theta}}(v)$.
To study the consistency of the estimations, the following conditions for the
model (3.2) are assumed:
* (C1)
The first two derivatives of $g(v)$ and $\xi(v)$ are continuous.
* (C2)
Kernel function $K(\cdot)$ satisfies
$\int K(u)du=1,\int u^{j}K(u)du=0,j=1,\cdots,k-1,0<\int u^{k}K(u)du<\infty.$
* (C3)
$nh^{2(d+1)}\rightarrow\infty$.
Obviously, the conditions (C1)-(C3) are commonly used for semiparametric
models. Under these conditions, the following theorem provides the consistency
of the bias-corrected estimator $\tilde{\theta}$.
Theorem 3.1 Assume that the conditions (C1)-(C3) hold, and for given $\alpha$,
(3.1) and (3.3) are satisfied. When $q$ is fixed, and $p$ may be larger than
$n$, then, as $n\rightarrow\infty$,
$\sqrt{n}(\tilde{\theta}-\theta)\stackrel{{\scriptstyle
D}}{{\longrightarrow}}N(0,\sigma_{V}^{2}S^{-1}),$
where $S=E\\{(Z-E(Z|V))(Z-E(Z|V))^{\prime}\\}$.
Remark 3.1 For simplicity of presentation, in this theorem we only give the
the asymptotic normality for the case with fixed $q$. In fact, when $q$ tends
to infinity at a certain rate, the asymptotic normality still holds for every
component of $\theta$ (see for example Lam and Fan, 2008). This is because,
after bias-correction, the model (3.2) is indeed a partially linear model and
then the proof can be similar with more technical and tedious details. The
proof of this theorem is postponed to the Appendix. The results in the theorem
show that the new estimator $\tilde{\theta}$ is $\sqrt{n}$-consistent
regardless of the choice of the shrinkage tuning parameter $\lambda_{p}$ and
thus it is convenient to be used in practice. Furthermore, by the theorem and
the commonly used nonparametric techniques, we can prove that
$g_{\tilde{\theta}}(v)$ is also consistent. In effect, we can obtain the
strong consistency and the consistency of the mean squared error under some
stronger conditions. The details are omitted in this paper.
To investigate the asymptotic properties for the second estimator
$\hat{\theta}$ in (3.9) that is based on the Dantzig selector
$\tilde{\gamma}^{D}$, we need the following more conditions:
* (C4)
The bandwidth $h$ is optimally chosen, i.e., $h=O(n^{-1/(2(k+d+1))})$.
* (C5)
Suppose that there exists a vector, say $\alpha$, such that
$\|\alpha\|_{{\ell}_{2}}\geq c$ for a positive constant $c$ and
$\|\tilde{\gamma}^{D}-\alpha\|_{{\ell}_{2}}/\|\tilde{\gamma}^{D}\|_{{\ell}_{2}}=O_{p}(n^{-\mu})$
for some $\mu$ satisfying
$1/2-k/(2(k+d+1))\leq\mu\leq 1/2.$
As was stated in the previous sections, $\alpha$ was an arbitrary vector. The
vector $\alpha$ in the condition (C5) is then different. But for the
simplicity of representation we still use the same notation $\alpha$ in
different appearance. The condition (C5) is the key for the following theorem
and corollary. This condition does not mean that the Dantzig selector
$\tilde{\gamma}^{D}$ is consistent. Note that
$\|\tilde{\gamma}^{D}\|_{{\ell}_{2}}$ is large in non-sparse case, and the
accuracy of the solution of linear programm can guarantee that
$\|\tilde{\gamma}^{D}-\alpha\|_{{\ell}_{2}}$ is relatively small for the true
value of linear programm (2.2) at population level (see for example Malgouyres
and Zeng, 2009). These show that the condition (C5) is reasonable. Both (C4)
and (C5) can actually be weakened, but for the simplicity of technical proof
and presentation, we still use the current conditions in this paper.
Theorem 3.2 Under the conditions (C1)-(C5), (3.1) and (3.3), we have the
following asymptotic representation for the second estimator in (3.9):
$\sqrt{n}(\hat{\theta}-\theta)=S^{-1}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\Big{(}\tilde{Z}_{i}\tilde{g}(V_{i})+\tilde{Z}_{i}\tilde{\xi}(V_{i})\Big{)}+o_{p}(1),$
where $S=E\\{(Z-E(Z|V))(Z-E(Z|V))^{\prime}\\}$ and
$\begin{array}[]{ll}\tilde{g}(V_{i})=g(V_{i})-\frac{\sum\limits_{k=1}^{n}g(V_{k})L_{H}(V_{k}-V_{i})}{\sum\limits_{k=1}^{n}L_{H}(V_{k}-V_{i})},\vspace{1ex}\\\
\tilde{\xi}(V_{i})=\xi(V_{i})-\frac{\sum\limits_{k=1}^{n}\xi(V_{k})L_{H}(V_{k}-V_{i})}{\sum\limits_{k=1}^{n}L_{H}(V_{k}-V_{i})},\vspace{1ex}\\\
\tilde{Z}_{i}=Z_{i}-\frac{\sum\limits_{k=1}^{n}Z_{k}L_{H}(V_{k}-V_{i})}{\sum\limits_{k=1}^{n}L_{H}(V_{k}-V_{i})}.\end{array}$
The proof of the theorem is given in the Appendix. From Theorem 3.2, and
Theorem 2.1.2 of Härdle et al (2000), the asymptotic normality follows
directly. The following corollary states the detail.
Corollary 3.3 Under the conditions of Theorem 3.1, when $q$ is fixed but $p$
may be larger than $n$, then, as $n\rightarrow\infty$,
$\sqrt{n}(\hat{\theta}-\theta)\stackrel{{\scriptstyle
D}}{{\longrightarrow}}N(0,\sigma_{V}^{2}S^{-1}).$
As aforementioned in Remark 3.1, for the sub-model with diverging $q$, the
asymptotic normality can still hold under some stronger conditions, the
details are omitted here.
3.3 Prediction. Together the estimation consistency with the adjusted sub-
model (3.2), we obtain an improved prediction as
$\hat{Y}=\hat{\theta}^{\prime}Z+\hat{g}_{\hat{\theta}}(V)$ $None$
and the corresponding prediction error is
$\begin{array}[]{lll}E(Y-\hat{Y})^{2}&=&E((\hat{\theta}-\theta)^{\prime}Z)^{2}+E(\hat{g}_{\hat{\theta}}(V)-g(V))^{2}+E(\xi^{2}(V))\vspace{1ex}\\\
&&+2E((\hat{\theta}-\theta)^{\prime}Z(\hat{g}_{\hat{\theta}}(V)-g(V)))+2E((\hat{\theta}-\theta)^{\prime}Z\xi(V))\vspace{1ex}\\\
&&+2E((\hat{g}_{\hat{\theta}}(V)-g(V))\xi(V))\vspace{1ex}\\\
&=&E(\xi^{2}(V))+o(1).\end{array}$
Such a prediction is of a smaller prediction error than the one by the
classical Dantzig selector, and interestingly it is no need with any high-
dimensional nonparametric estimation.
In contrast, if we use the new estimator $\hat{\theta}$ and the sub-model
(2.4), rather than the adjusted sub-model (3.2), to construct prediction, the
resulting prediction is defined as
$\hat{Y}_{S}=\hat{\theta}^{\prime}Z+\bar{\hat{g}}_{\hat{\theta}},$ $None$
where
$\bar{\hat{g}}_{\hat{\theta}}=\frac{1}{n}\sum_{i=1}^{n}\hat{g}_{\hat{\theta}}(V_{i}).$
For prediction, we need to add $\bar{\hat{g}}_{\hat{\theta}}$ in (3.11)
because the sub-model (2.4) has a bias $E(g(V))$, otherwise, the prediction
error would be even larger. In this case, $\bar{\hat{g}}_{\hat{\theta}}$ is
free of the predictor $U$ and the resultant prediction (3.11) only uses the
predictor $Z$ in the sub-model (2.4). This is different from the prediction
(3.10) that depends on both the low-dimensional predictor $Z$ and high-
dimensional predictor $U$. Thus (3.11) is a sub-model based prediction. The
corresponding prediction error is
$\begin{array}[]{lll}E(Y-\hat{Y}_{S})^{2}&=&E((\hat{\theta}-\theta)^{\prime}Z)^{2}+E(\bar{\hat{g}}_{\hat{\theta}}-g(V))^{2}+E(\xi^{2}(V))\vspace{1ex}\\\
&&+2E((\hat{\theta}-\theta)^{\prime}Z(\bar{\hat{g}}_{\hat{\theta}}-g(V)))+2E((\hat{\theta}-\theta)^{\prime}Z\xi(V))\vspace{1ex}\\\
&&+2E((\bar{\hat{g}}_{\hat{\theta}}-g(V))\xi(V))\vspace{1ex}\\\
&=&E(\xi^{2}(V))+Var(g(V))+2E(E(g(V))-g(V))\xi(V))+o(1).\end{array}$
This error is usually larger than that of the prediction (3.10). But,
$|E(E(g(V))-g(V))\xi(V))|\leq(Var(g(V))Var(\xi(V)))^{1/2}$
and usually the values of both $Var(g(V))$ and $Var(\xi(V))$ are small. Then
such a prediction still has a smaller prediction error than the one obtained
by the sub-model (2.4) and the common LS estimator $\tilde{\theta}_{S}=({\bf
Z}^{\prime}{\bf Z})^{-1}{\bf Z}^{\prime}{\bf Y}$ as:
$\tilde{Y}_{S}=\tilde{\theta}_{S}^{\prime}Z$ $None$
with the corresponding error as
$\begin{array}[]{lll}E(Y-\tilde{Y}_{S})^{2}&=&E((\tilde{\theta}_{S}-\theta)^{\prime}Z)^{2}+E(\gamma^{\prime}U)^{2}+\sigma^{2}+2E((\tilde{\theta}_{S}-\theta)^{\prime}Z\gamma^{\prime}U).\end{array}$
Because $\tilde{\theta}_{S}$ does not tend to $\theta$, the values of both
$E((\tilde{\theta}_{S}-\theta)^{\prime}Z)^{2}$ and
$2E((\tilde{\theta}_{S}-\theta)^{\prime}Z\gamma^{\prime}U)$ are large and as a
result the prediction error is large.
The above results show that in the scope of prediction, the new estimator can
reduce prediction error under both the adjusted sub-model (3.2) and the
original sub-model (2.4). We will see that the simulation results in Section 5
coincide with these conclusions.
4\. Calculation for $A$
4.1 Calculation of $A$ for the case with unknown
$\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}$. In the previous section, we
suggested a simple choice of $A$ for the case with known
$\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}$. We now introduce an approach
for choosing vector $A$ such that (3.6) holds for the case with unknown
$\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}$. For the convenience of
representation, we here suppose $E(Z)=0$, E(U)=0 and $Cov(Z^{\star})=I$. In
this case, (3.6) can be rewritten as
$\Sigma_{U,Z^{\star}}A^{\prime}(AA^{\prime})^{-1}AZ^{\star}=\Sigma_{U,Z^{\star}}Z^{\star}.$
$None$
We denote
$\Sigma_{U,Z^{\star}}^{\prime}\Sigma_{U,Z^{\star}}=\Omega=(\omega_{ij})$ with
$\begin{array}[]{ll}\omega_{ij}=\sum_{k=1}^{l}E(U^{(k)}Z^{(i)})E(U^{(k)}Z^{(j)}),i,j\leq
q,\vspace{1ex}\\\
\omega_{i,q+s}=\omega_{q+s,i}=\sum_{k=1}^{l}E(U^{(k)}Z^{(i)})E(U^{(k)}U^{(s)}),i=1,\cdots,q,s=1,\cdots,d,\vspace{1ex}\\\
\omega_{q+r,q+s}=\sum_{k=1}^{l}E(U^{(k)}U^{(r)})E(U^{(k)}U^{(s)}),r,s=1,\cdots,d,\end{array}$
where $Z^{(i)}$ and $U^{(k)}$ are the components of $Z$ and $U$, respectively.
It is known that $\Omega$ can be decomposed as
$\Omega=Q\,\mbox{diag}\\{\phi_{1},\cdots,\phi_{d},0,\cdots,0\\}Q^{\prime},$
where $\phi_{k},k=1,\cdots,d,$ are the positive eigenvalues of $\Omega$ and
$Q$ is the orthogonal matrix. Note that $l$ depends on $n$ and tends to
infinity as $n\rightarrow\infty$. To get consistent estimator of $Q$, we need
the following condition
$\\#\left\\{\begin{array}[]{ll}E(U^{(i)}Z^{(j)}),E(U^{(k)}U^{(s)}):E(U^{(i)}Z^{(j)})\neq
0,E(U^{(k)}U^{(s)})\neq 0,\\\ \hskip 130.88284pt\mbox{ for all
}i,j,k,s\end{array}\right\\}\leq C$ $None$
for a positive constant $C$, where $\\#\\{S\\}$ denotes the number of elements
in the set $S$. Also we can use some weaker conditions to replace (4.2). In
fact the conditions we need are similar to those required for high-dimensional
linear models, for example, the weak and strong irrepresentable conditions
(Zhao and Yu 2006) and the uniform uncertainty principle (Candés and Tao
2007). Note that $\Omega$ is a low-dimensional matrix. Then, under the
condition (4.2), $\Omega$ can be $\sqrt{n}$-consistently estimated; for
example, a naive estimator of $\hat{\omega}_{ij}$ for $i,j\leq q$ can be
chosen as
$\begin{array}[]{ll}\hat{\omega}_{ij}\\\
=\sum\limits_{k=1}^{l}\frac{1}{n}\sum\limits_{s=1}^{n}U_{s}^{(k)}Z_{s}^{(i)}{\bf
1}\Big{\\{}\frac{1}{n}\Big{|}\sum\limits_{s=1}^{n}U_{s}^{(k)}Z_{s}^{(i)}\Big{|}>\frac{1}{\sqrt{n}}\Big{\\}}\frac{1}{n}\sum\limits_{s=1}^{n}U_{s}^{(k)}Z_{s}^{(j)}{\bf
1}\Big{\\{}\frac{1}{n}\Big{|}\sum\limits_{s=1}^{n}U_{s}^{(k)}Z_{s}^{(j)}\Big{|}>\frac{1}{\sqrt{n}}\Big{\\}},\end{array}$
where ${\bf 1}\\{S\\}$ is the indicator function of the set $S$. As was shown
above, we can express $\hat{\Omega}$ as
$\hat{\Omega}=\hat{Q}\,\mbox{diag}\\{\hat{\phi}_{1},\cdots,\hat{\phi}_{d},0\cdots,0\\}\hat{Q}^{\prime}$
$None$
and $\hat{Q}$ as $\hat{Q}=(\hat{Q}_{1},\hat{Q}_{2})$. Finally, the estimator
of $A$ is obtained by
$\hat{A}=\hat{Q}_{1}.$
4.1 Calculation of $A$ for large $d$. As we mentioned before, when $d$ is
large, the solution $A$ of (4.1) has $d$ columns and then $(d+1)$-dimensional
nonparametric estimation will be involved, which leads an inefficient
estimation. Thus, we consider an approximation solution of (4.1), which is a
row vector. Without confusion, we still use the notation $A$ to denote this
row vector. That is, we choose a row vector $A$ such that
$A^{\prime}AZ^{\star}=\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}Z^{\star}.$
$None$
The approximation solution is identical to the solution of (4.1) in form as
when $A$ is a row vector, recalling that it is normalized to be norm one,
$AA^{\prime}=1$. In this case, to get a low-dimensional nonparametric function
$g(v)$, we choose $d=1$, i.e., $Z^{*}$ is a $q+1$-dimensional vector. Similar
to the above determination, when $A$ is unknown, we can also construct an
estimation as follows. Denote $A=(a_{1},\cdots,a_{q},a_{q+1})$, $A_{k}=a_{k}A$
and
$\Sigma_{U,Z^{\star}}^{+}\Sigma_{U,Z^{\star}}=(D_{1}^{\prime},\cdots,D_{q}^{\prime},D_{q+1}^{\prime})^{\prime}$,
where $D_{k},k=1,\cdots,q+1$, are $(q+1)$-dimensional row vectors. Then we
estimate $A$ via solving the following optimization problem:
$\inf\Big{\\{}Q(a_{1},\cdots,a_{q+1}):\sum_{k=1}^{q+1}a_{k}^{2}=1\Big{\\}},$
$None$
where
$Q(a_{1},\cdots,a_{q+1})=\frac{1}{n}\sum_{i=1}^{n}\sum_{k=1}^{q+1}\|(A_{k}-D_{k})Z_{i}^{\star}\|^{2}$.
By the Lagrange multiplier, we obtain the estimators of
$A_{k},k=1,\cdots,q+1,$ as
$\hat{A}_{k}=\Big{(}D_{k}\frac{1}{n}\sum_{i=1}^{n}Z_{i}^{\star}{Z_{i}^{\star}}^{\prime}+cc_{k}e_{k}/2\Big{)}\Big{(}\frac{1}{n}\sum_{i=1}^{n}Z_{i}^{\star}{Z_{i}^{\star}}^{\prime}+c_{k}I\Big{)}^{-1},$
$None$
where $c_{k}>0$, which is similar to a ridge parameter, depends on $n$ and
tends to zero as $n\rightarrow\infty$, and $e_{k}$ is a row vector with $k$-th
component 1 and the others being zero. Note that the constraint $\|A\|=1$
implies $\|A_{k}\|=\pm a_{k}$. Finally, by combining (4.6) and this constraint
we get an estimator of $a_{k}$ as
$\hat{a}_{k}=\pm\|\hat{A}_{k}\|$
and consequently the estimator of $A$ is obtained by
$\hat{A}=(\hat{a}_{1},\cdots,\hat{a}_{q},\hat{a}_{q+1}).$
5\. Simulation studies
In this section we examine the performance of the new method by simulations.
By mean squared error (MSE), model prediction error (PE) and their $std$ MSE
and $std$ PE as well, we compare the method with the Gaussian-dantzig selector
first. In ultra-high dimensional scenarios, the Dantzig selector cannot work
well, we use the sure independent screening (SIS) (Fan and Lv 2008) to bring
dimension down to a moderate size and then to make comparison with the
Gaussian-dantzig selector. As is well known, there are several factors that
are of great impact on the performance of variable selection methods:
dimensions $p$ of covariate $X$, correlation structure between the components
of covariate $X$, and variation of the error which can be measured by
theoretical model R-square defined by
$R^{2}=(Var(Y)-\sigma^{2}_{\varepsilon})/Var(Y)$. In order to comprehensively
illustrate the theoretical conclusions and performance, we design three
experiments. The main goal of the first experiment is to examine the effect of
$R^{2}$ as the smaller $R^{2}$ is, the more difficult correctly selecting
variables is. The second experiment is to investigate the impact from the
correlation between the components of covariate $X$, and the third is to check
whether the two-step procedure of the SIS and the Dantzig selector works or
not.
Experiment 1. This experiment is designed mainly for: (1) comparing the new
estimator $\hat{\theta}$ defined by (3.9) with the Gaussian-dantzig selector
$\tilde{\theta}_{S}$; (2) examining the effect of different choices of the
theoretical model $R^{2}$ of the full model (2.1); (3) checking the effect of
the correlation between the components of $X$ when $R^{2}$ is fixed. To
achieve these goals, we compare the MSEs, the PEs and their $std$ MSE and
$std$ PE of the two different estimators $\hat{\theta}$ and
$\tilde{\theta}_{S}$, and the two models (2.4) and (3.2). In the simulation,
to determine the regression coefficients in our simulation, we decompose the
coefficient vector $\beta$ as two parts: $\beta_{I}$ and $\beta_{-I},$ where
$I$ denotes the set of locations of significant components of $\beta_{I}$, and
let $S=|I|$ denote the number of elements contained in $I$. Three types of
$\beta_{I}$ are considered:
Type (I): $\beta_{I}=(1,0.4,0.3,0.5,0.3,0.3,0.3)^{\prime}$ and $I$=
{1,2,3,4,5,6,7};
Type (II): $\beta_{I}=(1,0.4,0.3,0.5,0.3,0.3,0.3)^{\prime}$ and
$I=\\{1,17,33,49,65,81,97\\}$;
Type (III): $\beta_{I}=(1,0.4,-0.3,-0.5,0.3,0.3,-0.3)^{\prime}$ and
$I=\\{1,2,3,4,5,6,7\\}$.
As it is very rare that all other coefficients are exactly zero, non-sparse
models are considered. To mimic practical scenarios, we set the values of the
components $\beta_{-Ii}$’s of $\beta_{-I}$ as follows. Before performing the
variable selection and estimation, we generate $\beta_{-Ii}$’s from uniform
distribution $\mathcal{U}(-0.5,0.15)$ and the negative values of them are then
set to be zero. After the coefficient vector $\beta$ is determined, we
consider it as a fixed value vector and regard $\beta_{I}$ as the main part of
the coefficient vector $\beta$. We use this way to set the values of
$\beta_{-Ii}$’s because in the simulations below, there are too many
insignificant variables with small/zero coefficients and it makes little sense
to give a common value for them. As too many values for these insignificant
coefficients, we do not list all of them here. We use $\hat{I}$ to denote the
set of subscript of coefficients $\theta$ in $\beta$, that is the
coefficients’ subscript of variables selected into sub-model. we assume
$X\thicksim N_{p}(\mu,\Sigma_{X})$, with $\mu$ the components corresponding to
$I$ are 0 and others are 2 and the $(i,j)$-th element
$\Sigma_{ij}=(-\rho)^{\mid i-j\mid}$, $0<\rho<1.$ Furthermore, the error term
$\varepsilon$ is assumed to be normally distributed as $\varepsilon\thicksim
N(0,\sigma^{2})$. In this experiment, we choose different $\sigma$ to obtain
different type of full model with different $R^{2}$. In the simulation
procedure and the kernel function is chosen to be Gaussian kernel
$K(u)=\frac{1}{\sqrt{2\pi}}\exp\\{-\frac{u^{2}}{2}\\}$. In this experiment,
the choice of parameter $\lambda_{p}$ in the Dantzig selector is just like
that given by Candés and Tao (2007), which is the empirical maximum of
$|X^{\prime}z|_{i}$ over several realizations of $z\sim N(0,I_{n}).$
The following Tables 1 and 2 report the MSEs and the corresponding PEs via 200
repetitions. In these tables, $\hat{Y}$ is the prediction via the adjusted
model (3.2) that is based on the full dataset, $\hat{Y}_{S}$ is the prediction
via the sub-model (2.4) with the new estimator $\hat{\theta}$ defined in
(3.9), $\tilde{Y}_{S}$ stands for the prediction via the sub-model (2.4) and
the Gaussian-dantzig selector $\tilde{\theta}_{S}$. For the definitions of
$\hat{Y}$, $\hat{Y}_{S}$ and $\tilde{Y}_{S}$ see (3.10), (3.11) and (3.12),
respectively. The purpose of such a comparison is to see whether the
adjustment works and whether we should use the sub-model (2.4) when the high-
dimensional data are not available (say, too expensive to collect), whether
the new estimator $\hat{\theta}$ together with the sub-model (2.4) is helpful
for prediction accuracy. The sample size is $50$, and for the prediction, we
perform the experiment with 200 repetitions to compute the proportion $\tau$
of which the prediction error of $\hat{Y}_{S}$ is less than that of
$\tilde{Y}_{S}$ in the 200 repetitions. The larger $\tau$ is, the better the
new estimator is. We have the following considerations in designing the
experiment: a). We will study models with the theoretical model $R^{2}$
ranging between 0.3 and 1.0, which can be determined by the value of the
variance of error term $\sigma^{2}$, here we choose $\sigma^{2}$=0.2, 0.6,
0.9, 1.3 and 1.9 respectively; b). The correlation between the components of
$X$ should have effect for the estimation, we then consider different
correlation coefficients $0.1$ and $0.7$.
1\. Let $n=50,p=100,S=7$ and $\rho=0.1$ . For each type of $\beta$, we choose
different $\sigma$ to control the theoretical $R^{2}$ and consider five cases.
For type (I), we have the following results:
Case 1. $R^{2}=0.98$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,2,3,4,6,7\\}$;
Case 2. $R^{2}=0.82$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,2,4,6,7,55\\}$;
Case 3. $R^{2}=0.67$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,2,3,4,15,22,28,81\\}$;
Case 4. $R^{2}=0.50$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,2,4,27,29,49,53,84\\}$;
Case 5. $R^{2}=0.31$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,4,5,24,25,42,43,62\\}$.
For type (II), we have the following results:
Case 1. $R^{2}=0.98$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,17,33,49,65,81,97\\}$;
Case 2. $R^{2}=0.84$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,17,33,43,49,81\\}$;
Case 3. $R^{2}=0.71$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,15,17,33,49,62,72\\}$;
Case 4. $R^{2}=0.53$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,5,26,29,33,43,49,53,65,74\\}$;
Case 5. $R^{2}=0.35$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,7,17,26,29,31,49,72,80,96,97,98\\}$.
For type (III), we have the following results:
Case 1. $R^{2}=0.98$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,2,3,4,5,6\\}$;
Case 2. $R^{2}=0.83$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,3,4,5,6,7,15\\}$;
Case 3. $R^{2}=0.69$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,2,4,5,7,92\\}$;
Case 4. $R^{2}=0.51$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,5,7,8,67,71\\}$;
Case 5. $R^{2}=0.33$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,4,6,7,21,23,38,50,75,83\\}$.
Table 1. MSE, PE and their standard errors with $n=50,p=100,S=7$ and
$\rho=0.1$
MSE($std$ MSE) PE($std$ PE) type $R^{2}$ $\hat{\theta}$ $\tilde{\theta}_{S}$
$\hat{Y}$ $\hat{Y}_{S}$ $\tilde{Y}_{S}$ $\tau$ 0.98 0.0032(0.0118)
0.0866(0.3519) 0.1630(0.0405) 0.2299(0.0535) 1.1587(0.5549) 200/200 0.82
0.0134(0.0544) 0.1197(0.1654) 0.6603(0.1497) 0.7249(0.1564) 1.4755(0.3475)
200/200 (I) 0.67 0.0273(0.1288) 0.0430(0.1283) 1.3038(0.2952) 1.3438(0.3018)
1.4821(0.3266) 166/200 0.50 0.0543(0.2387) 0.0694(0.2221) 2.5371(0.5500)
2.5919(0.5633) 2.7176(0.6020) 142/200 0.31 0.1028(0.4689) 0.1131(0.4876)
4.9199(1.1856) 4.9960(1.2070) 5.0708(1.1965) 126/200 0.98 0.0052(0.0202)
0.3540(1.4263) 0.2584(0.0569) 0.2744(0.0583) 1.1324(2.4262) 200/200 0.84
0.0162(0.0686) 0.4087(0.3730) 0.8310(0.1823) 0.8417(0.1834) 3.7996(0.7909)
200/200 (II) 0.70 0.0292(0.1112) 0.1770 (0.2559) 1.4761(0.3028) 1.4727(0.3018)
2.6389(0.5804) 199/200 0.53 0.0588(0.3024) 0.0942(0.2988) 2.8825(0.6534)
2.8700(0.6460) 3.2707(0.6758) 171/200 0.35 0.1107(0.6896) 0.1251(0.6368)
5.4055 (1.1809) 5.3896(1.1856) 5.6004(1.2280) 141/200 0.98 0.0028(0.0113)
0.0879(0.2938) 0.1643(0.0410) 0.2365(0.0537) 1.2282(0.5590) 200/200 0.83
0.0114(0.0531) 0.0873(0.1589) 0.5874 (0.1332) 0.6938(0.1533) 1.3483(0.3118)
200/200 (III) 0.69 0.0234(0.0934) 0.1294(0.1667) 1.1922(0.2857) 1.2445(0.2961)
1.9950(0.4379) 196/200 0.51 0.0529(0.1715) 0.0913(0.1775) 2.6373(0.5788)
2.7418(0.6098) 2.9601(0.6288) 164/200 0.33 0.1006(0.5013) 0.1083(0.5158)
5.0952(1.2099) 5.1720(1.2241) 5.2372(1.2594) 119/200
The simulation results are reported in Table 1. The results suggest that the
adjustment of (3.2) works very well, the corresponding estimation and
prediction are uniformly the best among the competitors. Further, as we
mentioned, when the full dataset is not available and we thus use the sub-
model of (2.4), the new estimator $\hat{\theta}$ is also useful for
prediction. It can be seen that $\hat{Y}_{S}$ is better than $\tilde{Y}_{S}$,
and the value of $\tau$ is larger than 0.7 in 13 cases out of 15 cases and in
the other 2 cases, it is larger than or about 0.6.
2\. To provide more information, we also consider the case with higher
correlation $\rho=0.7$: $n=50,p=100,S=7$. Also different $\sigma$’s are chosen
to control the theoretical $R^{2}$.
For type (I), we consider the following five cases.
Case 1. $R^{2}=0.96$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,2,4,5,6,7\\}$;
Case 2. $R^{2}=0.71$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,2,4,81\\}$;
Case 3. $R^{2}=0.53$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,4,8,9\\}$;
Case 4. $R^{2}=0.35$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,4,8,51\\}$;
Case 5. $R^{2}=0.20$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,2,6,84\\}$.
For type (II), we consider the following five cases.
Case 1. $R^{2}=0.98$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,17,33,49,65,97\\}$;
Case 2. $R^{2}=0.84$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,18,49,65,97\\}$;
Case 3. $R^{2}=0.69$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,2,49,52,65\\}$;
Case 4. $R^{2}=0.52$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,15,33,49,76,84,98\\}$;
Case 5. $R^{2}=0.34$, $I=\\{1,17,33,49,65,81,97\\}$ and
$\hat{I}=\\{1,2,24,48,49,55,87,97\\}$.
For type (III), we consider the following five cases.
Case 1. $R^{2}=0.96$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,2,4,6,7\\}$;
Case 2. $R^{2}=0.74$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,4,6,7\\}$;
Case 3. $R^{2}=0.56$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,6,7,33,56\\}$;
Case 4. $R^{2}=0.38$, $I=\\{1,2,3,4,5,6,7\\}$ and $\hat{I}=\\{1,4,7,51,93\\}$;
Case 5. $R^{2}=0.23$, $I=\\{1,2,3,4,5,6,7\\}$ and
$\hat{I}=\\{1,2,7,31,45,80,85,88\\}$.
Table 2. MSE, PE and their standard errors with $n=50,p=100,S=7$ and
$\rho=0.7$
MSE($std$ MSE) PE($std$ PE) type $R^{2}$ $\hat{\theta}$ $\tilde{\theta}_{S}$
$\hat{Y}$ $\hat{Y}_{S}$ $\tilde{Y}_{S}$ $\tau$ 0.96 0.0136(0.0504)
0.3285(0.4226) 0.2472(0.0517) 0.2706(0.0599) 1.7397(0.3804) 200/200 0.71
0.0253(0.1426) 0.0709(0.2401) 0.6530(0.1463) 0.6945(0.1557) 1.9892(0.2070)
197/200 (I) 0.53 0.0373(0.1621) 0.1108(0.2310) 1.2779(0.2744) 1.3235 (0.2861)
1.5985(0.3736) 177/200 0.35 0.0613(0.3122) 0.0999(0.3289) 2.3431(0.5342)
2.3694(0.5395) 2.6339(0.5799) 161/200 0.2 0.1198(0.6479) 0.1292(0.6619)
5.1184(1.2643) 5.1347(1.2729) 5.1764(1.2420) 129/200 0.98 0.0122(0.0484)
0.2730(0.3789) 0.2648(0.0730) 0.2809(0.0757) 1.1952(0.2440) 200/200 0.84
0.0201(0.0924) 0.1799(0.2037) 0.6567(0.1453) 0.6580(0.1452) 1.6477(0.3560)
200/200 (II) 0.69 0.0303(0.1338) 0.2899(0.4442) 1.2955(0.2992) 1.2996(0.3047)
2.7125(0.5861) 200/200 0.52 0.0644(0.3395) 0.1141l(0.4388) 2.5572(0.5558)
2.5633(0.5582) 3.2790(0.6834) 191/200 0.34 0.1245(0.5615) 0.1831(0.6787)
5.0731(1.1850) 5.0818(1.1743) 5.5988(1.2782) 161/200 0.96 0.0239(0.0626)
0.6020(2.1653) 0.2596(0.0560) 0.2897(0.0630) 1.6754(1.4970) 200/200 0.74
0.0315(0.1158) 0.4401(0.5248) 0.6435(0.1435) 0.6485(0.1442) 2.7859(0.6035)
200/200 (III) 0.56 0.0749(0.2373) 0.1736(0.2679) 1.3334(0.2947) 1.4367(0.3217)
1.8643(0.3965) 189/200 0.38 0.0687(0.3227) 0.1701(0.3809) 2.3637(0.4538)
2.4645(0.4818) 2.9415(0.5992) 178/200 0.23 0.1740(0.8078) 0.2446(0.8718)
4.8488(1.1812) 4.8887(1.1968) 5.1471(1.1499) 145/200
Table 2 shows that when $\rho$ is larger, the conclusions about the comparison
are almost identical to those presented in Table 1; Thus it concludes that no
matter $\rho$ is larger or not, our new method always works quite well.
We are now in the position to make another comparison. In Experiments 2 and 3
below, we do not use the data-driven approach as given in Experiment 1 to
select $\lambda_{p}$, while manually select several values to see whether our
method works or not. This is because in the two experiments, it is not our
goal to study shrinkage tuning parameter, but is our goal to see whether the
new method works after we have a sub-model.
Experiment 2. In this experiment, our focus is how the correlation between
variables affects the estimations. The distribution of $X$ is the same as that
in Experiment 1 except for the dimension. The coefficient vector $\beta$ is
designed as type (I) in Experiment 1\. Furthermore, the error term
$\varepsilon$ is assumed to be normally distributed as $\varepsilon\thicksim
N(0,0.2^{2})$.
As different choices of $\lambda_{p}$ will usually lead to different sub-
models, equivalently, to different estimators $\hat{I}$ of $I$, we are then
able to examine, when the numbers of significant variables that are included
into the submodels are different, the performance of the new estimation by MSE
and PE. In this experiment, we consider two cases with two values of
$\lambda$. The setting is as follows. For $n=50,p=100,S=7$,
$\rho=0.1,0.3,0.5,0.7.$ We consider two cases for each $\rho$:
$\rho=0.1:\\\ $ Case 1. $\lambda_{p}=3.97,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1,
2, 3, 4, 5, 6, 7 }
Case 2. $\lambda_{p}=6.53,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1, 3, 4, 6, 95 }
$\rho=0.3:\\\ $ Case 1. $\lambda_{p}=3.32,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1,
2, 3, 4, 5, 6 }
Case 2. $\lambda_{p}=6.77,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={ 1, 2, 4, 6, 23 }
$\rho=0.5:\\\ $ Case 1. $\lambda_{p}=3.72,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1,
2, 4, 5, 6, 7 }
Case 2. $\lambda_{p}=7.29,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1, 4, 5, 7, 41,
58, 72 }
$\rho=0.7:\\\ $ Case 1. $\lambda_{p}=3.50,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1,
3, 4, 7, 41, 75}
Case 2. $\lambda_{p}=7.22,$ $I$={1,2,3,4,5,6,7}, $\hat{I}$={1, 4, 7, 51, 64,
67, 68, 83 }
Table 3. MSE, PE and their standard errors with $n=50,p=100,S=7$
MSE($std$ MSE) PE($std$ PE) $\rho$ Case $\hat{\theta}$ $\tilde{\theta}_{S}$
$\hat{Y}$ $\hat{Y}_{S}$ $\tilde{Y}_{S}$ $\tau$ 1 0.0052(0.0242) 0.2929(0.3877)
0.2580(0.0528) 0.2612(0.0527) 3.0195(0.6691) 200/200 0.1 2 0.0104(0.0357)
0.2347(0.1784) 0.5135(0.1074) 0.6430(0.1282) 5.921(0.4172) 200 /200 1
0.0070(0.0289) 0.4067(1.6692) 0.2732(0.0590) 0.3324(0.0735) 5.6406(1.8289)
200/200 0.3 2 0.0163(0.0458) 0.5048(0.4107) 0.4048(0.0881) 0.5014(0.1078)
6.4471(0.7697) 200/200 1 0.0079(0.0336) 0.4826(1.9425) 0.2436(0.0551)
0.3053(0.0674) 5.8204(1.8152) 200/200 0.5 2 0.0136(0.0512) 0.1532(0.1835)
0.3655(0.0841) 0.4245(0.0914) 6.4357(0.3262) 200/200 1 0.0157(0.0602)
0.2296(0.2970) 0.2688(0.0580) 0.3198(0.0711) 6.6313(0.3560) 200/200 0.7 2
0.0149(0.0637) 0.1914(0.1420) 0.2974(0.0624) 0.3225(0.0672) 7.5435(0.1169)
197/200
From Table 3, we can see clearly that the correlation is of impact on the
performance of the variable selection methods: the estimation gets worse with
larger $\rho$. However, the new method uniformly works much better than the
Gaussian Dantzig selector, when we compare the performance of the methods with
different values of $\lambda$ and then with different sub-models. We can see
that in case I, the sub-models are more accurate than those in case II in the
sense that they can contain more significant variables we want to select.
Then, the estimation based on the Gaussian Dantzig selector can work better
and so can the new method. Note that $\rho$ is about 1 meaning that in all the
200 repetitions, $\hat{Y}\leq\hat{Y}_{S}.$
In the following, we consider ultra high-dimensional data.
Experiment 3. For very large $p$, the Dantzig selector method alone cannot
work well. Thus, we use the sure independent screening (SIS, Fan and Lv 2008)
to reduce the number of variables to a moderate scale that is below the sample
size, and then perform the variable selection and parameter estimation
afterwards by the Gaussian Dantzig selector and our adjustment method.
We first consider $n=100,p=1000$ and $S=10$ with $\rho$=0.1, 0.5 and 0.9
respectively, and for each $\rho$ two $\lambda_{p}$ are used to obtain two
$\hat{I}$ as follows.
For $\rho$=0.1,
$\beta_{I}=(1.0,-1.5,2.0,1.1,-3.0,1.2,1.8,-2.5,-2.0,1.0)^{\prime}$, consider
two cases:
Case 1. $\lambda_{p}$=4.50, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{1,3,5,6,7,8,9,318,514,723,760\\}$;
Case 2. $\lambda_{p}$=7.30, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{2,3,5,8,515,886\\}$.
For $\rho$=0.5,
$\beta_{I}=(1.0,-1.5,2.0,1.1,-3.0,1.2,1.8,-2.5,-2.0,1.0)^{\prime}$, consider
two cases:
Case 1. $\lambda_{p}$=3.56, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{1,2,5,7,8,9,846,878,976\\}$;
Case 2.$\lambda_{p}$=6.92, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{2,3,5,8,10,882,963\\}$.
For $\rho$=0.9,
$\beta_{I}=(1.0,-1.5,2.0,1.1,-3.0,1.2,1.8,-2.5,-2.0,1.0)^{\prime}$, consider
two cases:
Case 1. $\lambda_{p}$=1.80, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{3,5,8,10,415,432\\}$;
Case 2.$\lambda_{p}$=5.83, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{2,3,5,114,121,839,853,882,984\\}$.
With this design, the $\lambda$ in case 1 results in that more significant
variables are selected into the sub-model than those in case 2 so that we can
see the performance of the adjustment method.
Table 4. MSE, PE and their standard errors with $n=100,p=1000,S=10$
MSE($std$ MSE) PE($std$ PE) $\rho$ Case $\hat{\theta}$ $\tilde{\theta}_{S}$
$\hat{Y}$ $\hat{Y}_{S}$ $\tilde{Y}_{S}$ $\tau$ 1 0.7588(0.3497)
71.4031(7.5501) 6.8104(1.5485) 8.0107(1.6574) 94.7515(19.2968) 200/200 0.1 2
0.8523(0.5343) 122.8426(15.0952) 13.1274(2.7772) 16.0812(3.4160)
189.7134(34.8081) 200/200 1 3.6170(1.1823) 104.8420(13.5089) 9.9151(1.9902)
11.2352(2.2316) 133.4762(26.5058) 200/200 0.5 2 3.4771(1.2683)
92.3485(12.5122) 11.6643(2.6704) 12.7811(2.8941) 134.3821(24.4896) 200/200 1
5.9027(2.7039) 107.6118(23.4383) 8.2842(1.6181) 11.3518(2.1745)
148.3143(27.4828) 200/200 0.9 2 3.8963(2.1760) 59.1525(11.3152)
10.8033(2.1411) 12.9395(2.4835) 68.7272(13.4061) 200/200
From Table 4, we can see that the SIS does work to reduce the dimension so
that the Gaussian Dantzig selector and our method can be performed. Whether
the correlation coefficient is small or large (the values of $\rho$ change
from 0.1 to 0.9), the new method works better than the Gaussian Dantzig
selector. The conclusions are almost identical to those when $p$ is much
smaller in Experiments 1 and 2. Thus, we do not give more comments here.
Further, when comparing the results of case 1 and case 2, we can see that the
adjustment can work better when the submodel is not well selected. The value
of $\rho=1.$
In the following we check the effect of model size when the dimension is
larger. In doing so, we choose $n=150,p=2000$, $\rho=0.3$ with $S=5,10$. For
each $S$ we choose two $\lambda_{p}$ to obtain two $\hat{I}.$
For $S$=5, $\beta_{I}=(4.0,-1.5,6.0,-2.1,-3.0)^{\prime}$, we consider two
cases:
Case 1. $\lambda_{p}$=3.45, $I$={1,2,3,4,5},
$\hat{I}=\\{1,2,3,4,5,15,1099,1733\\}$;
Case 2. $\lambda_{p}$=8.36, $I$={1,2,3,4,5}, $\hat{I}=\\{1,3,554,908\\}$.
For $S$=10,
$\beta_{I}=(4.0,-1.5,6.0,-2.1,-3.0,1.2,3.8,-2.5,-2.0,7.0)^{\prime}$, consider
two cases:
Case 1. $\lambda_{p}$=3.02, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{1,2,3,5,7,8,9,10,1701\\}$;
Case 2. $\lambda_{p}$=9.08, $I$={1,2,3,4,5,6,7,8,9,10},
$\hat{I}=\\{1,3,5,7,8\\}$.
Table 5. MSE, PE and their standard errors with $n=150,p=2000,\rho=0.3$
MSE($std$ MSE) PE($std$ PE) S Case $\hat{\theta}$ $\tilde{\theta}_{S}$
$\hat{Y}$ $\hat{Y}_{S}$ $\tilde{Y}_{S}$ $\tau$ 1 0.4245(0.2102)
262.6392(21.2109) 6.4015(1.3038) 6.3439(1.2879) 322.9945(62.6228) 200/200 5 2
1.9510(1.0923) 359.5838(32.4150) 24.1959(4.8932) 24.8013(5.1629)
559.3584(98.1216) 200/200 1 0.8799(0.5108) 498.7862(59.0383) 10.6009(2.3903)
12.3505(2.6381) 946.3400(175.1009) 200/200 10 2 1.8524(0.7599)
68.1862(43.3612) 15.0471(2.8069) 16.9161 (3.1755) 1623.4936(111.5972) 200/200
The results in Table 5 show that the SIS is again useful for reducing the
dimension for the use of the Gaussian Dantzig selector and our method. When
the model size is smaller, estimation accuracy can be better with smaller MSE
and PE. In other words, when the model size is smaller, variable selection can
perform better and sub-model can be more accurate (case 1 with $S=5$), the
adjustment method does not have much help, and in contrast, it is useful for
improving estimation accuracy when the sub-model is very different from the
full model.
In summary, the results in the five tables above clearly show the superiority
of the new estimator $\hat{\theta}$ and the new sub-model (3.2)/the sub-model
(2.4) over the others in the sense with smaller MSEs, PEs and standard errors,
and large proportion $\tau$. The good performance holds for different
combinations of the sizes of selected sub-models (values of $\lambda_{p}$),
$n,p,S,I$, $R^{2}$ and the correlation between the components of $X$. The new
method is particularly useful when a submodel, as a working model, is very
different from underlying true model. Thus, the adjustment method is very
worth of recommendation. However, as a trade-off, the adjustment method
involves nonparametric estimation, although low-dimensional ones, it might not
be that helpful when the sub-model is accurate enough. Thus, we may consider
using it after a check whether the submodel is significantly biased. The
relevant research is ongoing.
Appendix
Proof of (3.4) Note that
$\begin{array}[]{ll}&E(\xi(V)|Z,V)\vspace{1ex}\\\
&=E(Y-\theta^{\prime}Z-g(V)|Z,V)\vspace{1ex}\\\
&=E(Y-\theta^{\prime}Z|Z,V)-E(g(V)|Z,V)\vspace{1ex}\\\
&=E(\gamma^{\prime}U+\varepsilon|Z,V)-E(E(\gamma^{\prime}U+\varepsilon|V)|Z,V)\vspace{1ex}\\\
&=\gamma^{\prime}E(U|Z,V)-\gamma^{\prime}E(U|V)\vspace{1ex}\\\
&=\gamma^{\prime}E(U|Z,\alpha^{\prime}U/\rho,W)-\gamma^{\prime}E(U|\alpha^{\prime}U/\rho,W).\end{array}$
Further,
$\begin{array}[]{ll}E(\gamma^{\prime}U/\rho|Z=z,\alpha^{\prime}U/\rho=t,W=w)\vspace{1ex}\\\
=E(\gamma^{\prime}U/\rho|Z=z,U^{(1)}=(t\rho-\sum_{j=2}^{l}U^{(j)}\alpha_{j})/\alpha_{1},W=w)\vspace{1ex}\\\
=E(\frac{\gamma_{1}}{\alpha_{1}\rho}(t\rho-\sum_{j=2}^{l}U^{(j)}\alpha_{j})+\sum_{j=2}^{l}U^{(j)}\gamma_{j}/\rho|Z=z,W=w)\vspace{1ex}\\\
=E(\frac{\gamma_{1}}{\alpha_{1}}t+\sum_{j=2}^{l}U^{(j)}(\gamma_{j}-\frac{\gamma_{1}}{\alpha_{1}}\alpha_{j})/\rho|Z=z,W=w)\vspace{1ex}\\\
=\frac{\gamma_{1}}{\alpha_{1}}t+\lambda^{\prime}E(U|Z=z,W=w).\end{array}$
Similarly,
$E(\gamma^{\prime}U/\rho|W=w,U^{\prime}\gamma/\rho=t)=\frac{\gamma_{1}}{\alpha_{1}}t+\lambda^{\prime}E(U|W=w)$.
Combining the above results leads to
$E(\xi(V)|Z,V)=\rho\lambda^{\prime}(E(U|Z,W)-E(U|W)),$
as required.
Proof of Theorem 3.1 The proof follows directly from the unbiasedness of the
model (3.2) for any $\alpha$ and Theorem 2.1.2 of Härdle et al (2000).
Proof of Theorem 3.2 Denote
$V^{*}=({\gamma^{*}}^{\prime}U/\rho^{*},W^{\prime})^{\prime}$ and
$\rho^{*}=O(\|\gamma^{*}\|_{{\ell}_{2}}\sqrt{\lambda_{M}})$, where
$\gamma^{*}$ is a $l$-dimensional vector between $\tilde{\gamma}^{D}$ and
$\alpha$. Then there exists a vector $\gamma^{*}$ such that
$\begin{array}[]{ll}\frac{1}{n}\sum\limits_{k=1}^{n}L_{H}(\hat{V}_{k}-v)=\frac{1}{n}\sum\limits_{k=1}^{n}L_{H}(V_{k}-v)+\frac{1}{n}\sum\limits_{k=1}^{n}\dot{L}_{H}(V^{*}_{k}-v)(\hat{V}_{k}-V_{k}),\end{array}$
where $\dot{L}_{H}(\cdot)$ is the derivative of $L_{H}(\cdot)$. By the
conditions (C1) and (C5), we have
$\hat{V}_{k}-V_{k}=O_{p}(n^{-\mu})$
and, consequently,
$\begin{array}[]{ll}\frac{1}{n}\sum\limits_{k=1}^{n}\dot{L}_{H}(V^{*}_{k}-v)(\hat{V}_{k}-V_{k})=O_{p}(n^{-\mu}).\end{array}$
By standard nonparametric technique, see Härdle, et al (2000), it is easy to
have
$\begin{array}[]{ll}\frac{1}{n}\sum\limits_{k=1}^{n}L_{H}(V_{k}-v)-f_{V}(v)=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)},\end{array}$
and then
$\begin{array}[]{ll}\frac{1}{n}\sum\limits_{k=1}^{n}L_{H}(\hat{V}_{k}-v)-f_{V}(v)=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}+O_{p}(n^{-\mu}),\end{array}$
where $f_{V}$ is the density function of $V$. Similarly, we can prove
$\frac{1}{n}\sum\limits_{k=1}^{n}Z_{k}L_{H}(\hat{V}_{k}-v)-\int
zf_{Z,V}(z,v)dz=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}+O_{p}(n^{-\mu}),$
where $f_{Z,V}$ is the joint density function of $(Z,V)$. Combining the
results above leads to
$\hat{Z}=Z-E(Z|V)+O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}+O_{p}(n^{-\mu})$
and, consequently,
$S_{n}-S=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}+O_{p}(n^{-\mu}).$
Further, by the definition of $\hat{\theta}$ and the above result, we have
$\hat{\theta}-\theta=S^{-1}\frac{1}{n}\sum_{i=1}^{n}\Big{(}\hat{Z}_{i}\hat{g}(\hat{V}_{i})+\hat{Z}_{i}\hat{\xi}(\hat{V}_{i})\Big{)}\Big{\\{}1+O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}+O_{p}(n^{-\mu})\Big{\\}},$
where
$\begin{array}[]{ll}\hat{g}(\hat{V}_{i})=g(\hat{V}_{i})-\frac{\frac{1}{n}\sum_{k=1}^{n}g(\hat{V}_{k})L_{H}(\hat{V}_{k}-\hat{V}_{i})}{\frac{1}{n}\sum_{k=1}^{n}L_{H}(\hat{V}_{k}-\hat{V}_{i})},\vspace{1ex}\\\
\hat{\xi}(\hat{V}_{i})=\xi(\hat{V}_{i})-\frac{\frac{1}{n}\sum_{k=1}^{n}\xi(\hat{V}_{k})L_{H}(\hat{V}_{k}-\hat{V}_{i})}{\frac{1}{n}\sum_{k=1}^{n}L_{H}(\hat{V}_{k}-\hat{V}_{i})}.\end{array}$
Again by the conditions (C1) and (C4), we have
$\begin{array}[]{ll}\hat{g}(\hat{V}_{i})=g(V_{i})-\frac{\frac{1}{n}\sum_{k=1}^{n}g(V_{k})L_{H}(V_{k}-V_{i})}{\frac{1}{n}\sum_{k=1}^{n}L_{H}(V_{k}-V_{i})}+O_{p}(n^{-\mu})=\tilde{g}(V_{i})+O_{p}(n^{-\mu}),\vspace{1ex}\\\
\hat{\xi}(\hat{V}_{i})=\xi(V_{i})-\frac{\frac{1}{n}\sum_{k=1}^{n}\xi(V_{k})L_{H}(V_{k}-V_{i})}{\frac{1}{n}\sum_{k=1}^{n}L_{H}(V_{k}-V_{i})}+O_{p}(n^{-\mu})=\tilde{\xi}(V_{i})+O_{p}(n^{-\mu}),\vspace{1ex}\\\
\hat{Z}_{i}=Z_{i}-\frac{\frac{1}{n}\sum_{k=1}^{n}Z_{k}L_{H}(V_{k}-V_{i})}{\frac{1}{n}\sum_{k=1}^{n}L_{H}(V_{k}-V_{i})}+O_{p}(n^{-\mu})=\tilde{Z}_{i}+O_{p}(n^{-\mu}).\end{array}$
Note that, under the condition (C4),
$\begin{array}[]{ll}\tilde{g}(V_{i})=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}=O_{p}(n^{-k/(2(k+d+1))}),\vspace{1ex}\\\
\tilde{\xi}(V_{i})=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}=O_{p}(n^{-k/(2(k+d+1))}),\vspace{1ex}\\\
\tilde{Z}_{i}=O_{p}\Big{(}h^{k}+\frac{1}{\sqrt{nh^{2(d+1)}}}\Big{)}=O_{p}(n^{-k/(2(k+d+1))}).\end{array}$
Therefore combining the above results can complete the proof of the theorem.
Proof of Corollary 3.3 The proof follows directly from the result of Theorem
3.2 and Theorem 2.12 of Härdle et al (2000).
References
Bai, Z. and Saranadasa, H. (1996). Effect of high dimension: by an example of
a two sample problem. Statistica Sinica, 6, 311-329.
Candés, E. J. and Tao, T. (2005). Decoding by linear programming. IEEE Trans.
Inform. Theory 51, 4203-4215.
Candés, E. J. and Tao, T. (2006). Near-optimal signal recovery from random
projections: Universal encoding strategies? IEEE Trans. Inform. Theory 52,
5406-5425.
Candés, E. J. and Tao, T. (2007). The Dantzig selector: statistical estimation
when $p$ is much larger than $n$. Ann. Statist. 35, 2313-2351.
Chen, S. X. and Qin, Y. L. (2010). A two sample test for high dimensional data
with applications to gene-set testing. Ann. Statist., 38, 808-835.
Diaconis, P. and Freedman, D. (1984). Asymptotics of graphical projection
pursuit. Ann. Statist., 12, 793-815.
Donoho, D. L. (2000). High-dimensional data analysis: The curses and blessing
of dimensionality, Lecture on August 8, 2000, to the American Mathematical
Society on “Math Challenges of the 21st Century”.
Fan, J. and Peng, H. (2004). Nonconcave penalized likelihood with a diverging
number of parameters. Ann. Statist., 32, 928-961.
Fan, J., Peng, H. and Huang, T. (2005). Semilinear high-dimensional model for
normalized of microarray data: a theoretical analysis and partial consistency.
J. Am. Statist. Ass., (with discussion), 100, 781-813.
Fan, J. and Lv, J. (2008). Sure independence screening for ultrahigh
dimensional feature space. J. R. Statist. Soc. B 70, 849-911.
Härdle, W., Liang, H. and Gao, T. (2000). Partially linear models. Physica
Verlag.
Hall, P. and Li, K. C. (1993). On almost linearity of low dimensional
projection from high dimensional data. Ann. Statist. 21, 867-889.
Huang, J., Horowitz, J. L. and Ma, S. (2008). Asymptotic properties of Bridge
estimators in sparse high-dimensional regression models. Ann. Statist., 36,
2587-2613.
Huber, P. J. (1973). Robust regression: Asymptotic, conjectures and Montes
Carlo. Ann. Statist., 1, 799-821.
James, G. M. and Radchenko, P. (2009). A generalized Dantzig selector with
shrinkage tuning. Biometrika, 96, 323-337.
James, G. M. , Radchenko, P. and Lv, J. C. (2009). Dasso: connections between
the Dantzig selector and lasso. Journal of the Royal Statistical Society,
Series B, 71, 127-142.
James, G. M. and Radchenko, P. (2009). A generalized Dantzig selector with
shrinkage tuning. Biometrika 96, 323-337.
Kettenring, J., Lindsay, B. and Siegmund, D., eds. (2003). Statistics:
Challenges and opportunities for the twenty-first century. NSF report.
Available at www.pnl.gov/scales/docs/nsf_report.pdf
Kosorok, M. and Ma, S. (2007). Marginal asymptotics for the large $p$, small
$n$ paradigm: With application to Microarray data. Ann. statist., 35,
1456-1486.
Kuelbs, J. and Anand, N. W. (2010). Asymptotic inference for high dimensional
data. Ann. Statist., 38, 836-869.
Lam, C and Fan, J. (2008). Profile-kernel likelihood inference with diverging
number of parameters. Ann. Statist., 36, 2232-2260.
Li, G. R., Zhu, L. X. and Lin, L. (2008). Empirical Likelihood for a varying
coefficient partially linear model with diverging number of parameters.
Manuscript.
Malgouyres, F. and Zeng, T. (2009). A predual proximal point algorithm solving
a non negative basis pursuit denoising model. Int. J. comput Vis., 83,
294-311.
Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential
families when the number of parameters tends to infinity. Ann. Statist., 16,
356-366.
Robins, J. and Vaart, A. V. D. (2006). Adaptive nonparametric confidence sets.
Ann. Statist., 34, 229-253.
Van Der Lanin, M. J. and Bryan, J. (2001). Gene expression analysis with
parametric bootstrap. Biostatistics, 2, 445-461.
Zhao, P. and and Yu, B. (2006). On model selection consistency of Lasso.
Journal of Machine Learning Research, 7, 2541-2563.
|
arxiv-papers
| 2010-08-07T14:28:06 |
2024-09-04T02:49:12.100022
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lu Lin, Lixing Zhu and Yujie Gai",
"submitter": "Lixing Zhu",
"url": "https://arxiv.org/abs/1008.1345"
}
|
1008.1481
|
# Quantitative determination of anisotropic magnetoelectric coupling in
BiFeO3-CoFe2O4 nanostructures
Yoon Seok Oh CeNSCMR, Department of Physics and Astronomy, Seoul National
University, Seoul 151-747, Republic of Korea S. Crane Department of
Materials Science and Engineering, University of California, Berkeley, CA
94720, USA H. Zheng Department of Materials Science and Engineering,
University of California, Berkeley, CA 94720, USA Y. H. Chu Department of
Materials Science and Engineering, National Chiao Tung University, HsinChu
30010, Taiwan R. Ramesh Department of Materials Science and Engineering,
University of California, Berkeley, CA 94720, USA Kee Hoon Kim CeNSCMR,
Department of Physics and Astronomy, Seoul National University, Seoul 151-747,
Republic of Korea
###### Abstract
The transverse and longitudinal magnetoelectric susceptibilities (MES) were
quantitatively determined for (001) heteroepitaxial BiFeO3-CoFe2O4
nanostructures. Both of these MES values were sharply enhanced at magnetic
fields below 6 kOe and revealed asymmetric lineshapes with respect to the dc
magnetic field, demonstrating the strain-induced magnetoelectric effect. The
maximum transverse MES, which reached as high as $\sim$60 mV/cm Oe, was about
five times larger than the longitudinal MES. This observation signifies that
transverse magnetostriction of the CoFe2O4 nanopillars is enhanced more than
the bulk value due to preferred magnetic domain alignment along the [001]
direction coming from compressive, heteroepitaxial strain.
The magnetoelectric (ME) effect is a physical phenomenon in which the electric
polarization $P$ (magnetization $M$) is modulated by the magnetic field $H$
(electric field $E$). There is a growing interest in the application of ME
effects toward various devices, including magnetic sensors nanjap and energy
harvesters. dong As such, numerous efforts have been made to obtain strong ME
couplings in novel ME composites made of ferroelectric (or piezoelectric) and
ferromagnetic materials. boomgaard ; jryu In these types of ME composites,
$P$ is varied with $M$ via the strain ($u$)-coupling at the interface between
the piezoelectric and magnetostrictive phases. Thus, the configuration of this
interface is a significant control parameter for determining the extent of ME
coupling. nanjap In this respect, a layered sandwich structure [i.e., (2-2)
structures] produces a larger ME coupling than for particulates that are
dispersed in a matrix [i.e., (0-3) structures] because the former has a larger
interface area. In contrast, this common practice cannot be applied to
multilayered thin films, where clamping of the nonmagnetic substrate can
prevent strain-coupling between the layers. nanprl
In 2004, an epitaxial thin film composed of CoFe2O4 (CFO) nanopillars embedded
in a BaTiO3 (BTO) matrix [i.e., (1-3) structures] was grown and suggested to
be an alternative to circumvent the substrate clamping effect. On the other
hand, it has been quite difficult to determine quantitatively the ME coupling
of such nanostructures and, more generally, numerous ME films, except
observing the existence of nontrivial ME coupling. zhengscience One dominant
reason for this difficulty is that the ME voltage signal is proportional to
the film thickness; it typically becomes smaller than 1 $\mu$V for film
thicknesses less than 1 $\mu$m. To overcome this difficulty, a large ac
magnetic field ($H_{\rm{ac}}$) of up to $\sim$1 kOe was recently used to
obtain the effective MES as a function of $H_{\rm{ac}}$. yanapl ; yanjap
However, direct measurements of the MES based on a conventional scheme that
employs small $H_{\rm{ac}}$ values with variations in the dc magnetic field
($H_{\rm{dc}}$) would be useful for understanding the ME coupling of numerous
multiferroic films or nanostructures at the quantitative level. In this work,
by use of the conventional scheme, we provide experimental evidences that a
300 nm thick BiFeO3-CoFe2O4 (BFO-CFO) nanostructure has a peculiar MES
anisotropy that is not expected in bulk forms of those materials.
The self-assembled epitaxial BFO-CFO film with thickness of 300 nm was grown
on a (001) SrRuO3/SrTiO3 substrate by pulsed laser deposition. zhengadvm The
film had the CFO nanopillars embedded in a BFO matrix with a volume fraction
of 1:1. Top electrodes Pt/SrRuO3 were deposited on the film surface (Fig.
1(a)). For magnetic hysteresis measurements, a vibrating sample magnetometer
was utilized. For ferroelectric hysteresis loops, the displacement current was
measured using a fast digitizer and a high voltage amplifier.
Figure 1: (a) Schematic picture, (b) AFM image, and (c) XRD pattern of the
BFO-CFO film with nanopillar structures.
To investigate the MES (${\alpha}{\equiv}{\delta}P/{\delta}H_{\rm{ac}}$),
especially for thin films, we developed a highly sensitive ME susceptometer
that operates inside the PPMS (Quantum Design). A pair of solenoids was
designed to induce a $H_{\rm{ac}}$ of $\sim$4 Oe inside the solenoid pair and
a voltage pick-up coil was used to determine the phase of $H_{\rm{ac}}$. In
particular, modulated charges, instead of voltages, were measured using a
high-impedance charge amplifier with a gain factor of $10^{12}$ V/C. This
makes the amplified signal independent of film thickness but proportional to
electrode area so that the signal-to-noise ratio was improved. Based on this
scheme, we were able to detect small ME charges in a thin film with a
thickness $\sim$40 nm and a circular electrode of diameter $\sim$100 $\mu$m.
The lowest charge noise ($\Delta Q$) was ${\sim}10^{-17}$ C, which
corresponded to a voltage noise of $\Delta Q/C_{\rm{s}}$ ($C_{\rm{s}}$ =
sample capacitance). The determined $\alpha$ could be converted into the MES
expressed as a voltage unit $\alpha_{\rm{E}}{=}{\delta}E/{\delta}H_{\rm{ac}}$
using the relationship ${\alpha}{=}{\varepsilon}{\alpha}_{\rm{E}}$, where
$\varepsilon$ is the absolute permittivity of a specimen.
Figure 2: (a) Magnetic hysteresis loops for the in-plane and out-of-plane
directions measured at 300 K. $M$ is normalized to the volume fraction of CFO.
(b) $J_{\rm{PE}}(E)$ (solid circle) was integrated to estimate the $P$-$E$
hysteresis loop (solid line).
Fig. 1(b) shows an atomic force microscopy (AFM) image of the BFO-CFO film, in
which the CFO phase appears as rectangles embedded in the BFO matrix.
zhengadvm Fig. 1(c) further shows an x-ray diffraction (XRD) pattern of the
film obtained through a $\theta-2\theta$ scan around the (002) SrTiO3 peak.
The distinct (00$l$) peaks of CFO, BFO, and SrRuO3 are consistent with the
previous result zhengadvm that each phase was epitaxially grown on the SrTiO3
substrate. The $d$-spacing of the CFO nanopillars was estimated as 2.0867
$\rm{{\AA}}$ from the (004) peak. This indicates a compressive strain along
the [001] direction, $u_{001}{=}-0.33\%$, compared with the bulk CFO. Fig.
2(a) shows the magnetic hysteresis loops measured along the in-plane (i.e.,
$H{\parallel}[100]$) and out-of-plane (i.e., $H{\parallel}[001]$) directions.
The saturated moment of $\sim$3.4 $\mu_{B}$/f.u. for both directions was in
good agreement with the reported CFO value. pauthenet Moreover, there existed
a large uniaxial magnetic anisotropy with an easy axis along the [001]
direction. A linear extrapolation of the in-plane loop yielded a magnetic
anisotropy field of $\sim$25 kOe. In a previous study on BTO-CFO
nanostructures, the compressive strain of CFO caused by heteroepitaxial growth
was found to be a primary contribution to the uniaxial magnetic anisotropy;
zhengapl a large magnetic anisotropy field of 51 kOe was observed for
$u_{001}{=}-1.1\%$, while the anisotropy field decreased for smaller
$u_{001}$. Therefore, even in the BFO-CFO film studied here, the magnetic
anisotropy field of $\sim$25 kOe seems to originate from the presence of a
compressive, heteroepitaxial strain of $u_{001}{=}-0.33\%$ inside the CFO
nanopillars.
To determine the $P$-$E$ loop, displacement current $J(E)$ was measured while
negative, positive, and zero biases were applied successively in sequence
(i.e., $-E$ to $E$ to 0). The $J(E)$ curve showed two extremes at $-500$ kV/cm
and 200 kV/cm, at which a reversal of $P$ occurred. In addition to these
extremes, a nonlinear background was found in the $J(E)$ curve. This could
have been from either a Schottky barrier at the interface or a ferroelectric
diode effect. choi ; yang After subtracting the nonlinear background, the
remaining current density $J_{\rm{PE}}(E)$ was integrated. In the obtained
$P$-$E$ loop (Fig. 2(b)), the polarization value was normalized by the volume
fraction of BFO. The saturated $P$ ($P_{\rm{s}}$) of $\sim$62 $\mu$C/cm2 is
comparable to the previously obtained value in an epitaxial (001) BFO film.
wang After fully poling the sample along the positive $P$ direction, i.e. top
electrode direction as indicated in the inset of Fig. 2(b), all the MES
measurements were subsequently performed.
Figure 3: (a) Transverse ($\alpha_{31}$) and (b) longitudinal MES
($\alpha_{33}$) of the BFO-CFO nanostructure at 300 K. (c) Dashed (solid)
lines represent the transverse ($\lambda_{31}$) and longitudinal
($\lambda_{33}$) magnetostriction curves of a Co0.8Fe2.2O4 crystal with the
demagnetized (single) magnetic domain. (d) Demagnetized, (e) single, and (f)
preferred magnetic domain patterns of CFO are schematically drawn.
Fig. 3 summarizes transverse ($\alpha_{31}$) and longitudinal ($\alpha_{33}$)
MES curves as a function of $H_{\rm{dc}}$. We note that the similar
$\alpha_{31}$ and $\alpha_{33}$ curves were obtained at many different
electrode spots of the same film and at those of the same kind of
nanostructured film with a thickness $\sim$40 nm. The $\alpha_{31}$ clearly
shows a sign reversal with the direction of $H_{\rm{dc}}$ and develops extreme
points at $H_{\rm{dc}}{=}{\mp}6$ kOe. This is the archetypal lineshape
expected in the strain-coupled ME media composed of piezoelectric and
magnetostrictive materials, supporting that the measured MES data are
reliable. In addition, the $\alpha_{33}$ curve exhibited an asymmetric
lineshape with $H_{\rm{dc}}$, which was similar to the case of $\alpha_{31}$.
However, in this $\alpha_{33}$ curve, a small but non-negligible offset at
$H_{\rm{dc}}{=}0$ was observed, of which value was proportional to the
electrode area. Thus, this offset was attributed to a small contribution from
eddy currents generated inside the electrode due to ${\delta}H_{\rm{ac}}$.
Except for this offset, the $\alpha_{33}$ curve was almost an odd function of
$H_{\rm{dc}}$. This further supports that the strain-coupling is a dominant
source of the longitudinal ME effect as well.
As shown in Fig. 3, $\alpha_{31}{>}0$ for $H_{\rm{dc}}{>}0$ while
$\alpha_{33}{<}0$ for $H_{\rm{dc}}{>}0$. This experimental result reflects
that $P_{3}$ (i.e., $P{\parallel}[001]$) increases under
$H_{\rm{dc}}{\parallel}[100]$, while $P_{3}$ decreases under
$H_{\rm{dc}}{\parallel}[001]$. When the $H_{\rm{dc}}{\parallel}[100]$ was
applied to the CFO crystal, the transverse magnetostriction ($\lambda_{31}$)
was positive (dashed lines in Fig. 3(c)). Thus, the CFO nanopillars and the
BFO matrix, via strain-coupling, were expected to be elongated along the [001]
direction. bozorth It is known from an earlier study that $P_{3}$ increases
due to the rotation of $P$ with increasing length along the [001] direction.
ederer ; jang Therefore, the observation of ${\alpha}_{31}{>}0$ for
$H_{\rm{dc}}{>}0$ can be qualitatively understood as the result of an
elongation of CFO/BFO along the [001] direction under
$H_{\rm{dc}}{\parallel}[100]$ and the subsequent increase of $P_{3}$. The case
of decreasing $P_{3}$ under $H_{\rm{dc}}{\parallel}[001]$ can also be
understood in a similar way because the longitudinal magnetostriction
($\lambda_{33}$) in the CFO crystal was negative.
However, our MES data are seemingly inconsistent with the magnetostriction
behavior of a bulk CFO at the quantitative level. The maximum to minimum value
of $\alpha_{31}$ ($\Delta\alpha_{31}$) in Fig. 3(a), amounting to $\sim$260
ps/m ($\sim$120 mV/cm Oe), is about five times larger than that of
$\alpha_{33}$ ($\Delta\alpha_{33}$). According to the magnetostriction of a
Co0.8Fe2.2O4 crystal with demagnetized domains (Fig. 3(c)), a slope of the
$\lambda_{33}$ vs. $H$ curve is at least twice that of the $\lambda_{31}$ vs.
$H$ curve. bozorth Upon assuming that the magnetostriction of CFO nanopillars
follows a bulk behavior, these magnetostriction data predict that
$\Delta\alpha_{33}$ should be at least twice of $\Delta\alpha_{31}$, which is
in sharp contrast with the results in Fig. 3 (a) and (b).
Although there might exist several mechanisms to induce enhanced $\alpha_{31}$
as discussed in a recent anisotropic MES study using large $H_{\rm{ac}}$,
yanjap one most decisive factor could be the preferred magnetic domains
existing in the CFO nanopillars. The solid lines in Fig. 3(c) reproduce
published $\lambda^{\rm{s}}_{31}$ and $\lambda^{\rm{s}}_{33}$ for a
Co0.8Fe2.2O4 crystal with a single magnetic domain along the [001] direction.
The single magnetic domain was obtained through the magnetic annealing
process, i.e. cooling under $H$ from high to room temperature. bozorth In
this situation, applied $H{\parallel}[001]$ gave rise to the 180∘ domain wall
motion, which resulted in negligible $\lambda^{\rm{s}}_{33}$. In contrast,
$H{\parallel}[100]$ resulted in a 90∘ domain wall motion so that it produced
quite large $\lambda^{\rm{s}}_{31}$. As we discussed above, the compressive,
heteroepitaxial strain was a main source of enhanced magnetic anisotropy along
the [001] direction in the CFO nanopillars. It is thus likely that the
magnetic domains inside the CFO nanopillars have preferred alignment along the
[001] direction, as illustrated in Fig. 3(f). If so, similar to the case of
single magnetic domains, the CFO nanopillars are expected to have enhanced
$\lambda_{31}$ and suppressed $\lambda_{33}$. As a result, as observed in Fig.
3 (a) and (b), the BFO-CFO nanostructure will give rise to a bigger
$\alpha_{31}$ (smaller $\alpha_{33}$) than that expected based on the behavior
of bulk CFO magnetostriction.
These results point to the possibility that the strain-induced ME coupling in
the nanostructured film can be quite different from the macroscopic bulk
composite. Application of a compressive, heteroepitaxial strain to the CFO
nanopillars enables to achieve increased $\alpha_{31}$ to as high as $\sim$130
ps/m ($\sim$60 mV/cm Oe) at 6 kOe. Upon increasing $|u_{001}|$, as done in the
BTO-CFO nanostructures with growth temperatures, zhengapl the $\alpha_{31}$
can be further optimized. In comparison, our previous study on a thin film
made of NiFe2O4 nanoparticulates embedded in a PbZr0.52Ti0.48O3 matrix [i.e.,
the (0-3) structure] showed maximum $|\alpha_{31}|{=}4$ mV/cm Oe ($\sim$4
ps/m) and $|\alpha_{33}|{=}16$ mV/cm Oe ($\sim$14 ps/m) hryu , which was
clearly smaller than the maximum $\alpha_{31}{\sim}130$ ps/m found here.
Therefore, our results strongly support that thin films with the (1-3)
nanostructure have larger ME couplings than the other nanostructures [e.g.,
(0-3) structure].
In conclusion, we have determined the anisotropic MES of a 300 nm thick
BiFeO3-CoFe2O4 nanostructure. An enhancement was observed in the transverse
configuration, which can be explained by the preferred alignment of magnetic
domain resulting from the heteroepitaxial strain that is unique to the present
(1-3) nanostructure. This investigation offers quantitative evidences that the
nanoscale engineering of strain coupling is useful for the design of ME
devices.
We appreciate discussions with T. W. Noh. This study was supported by the NRF,
Korea through National Creative Research Initiatives, NRL (M10600000238) and
Basic Science Research (2009-0083512) programs and by MOKE through the
Fundamental R${\&}$D Program for Core Technology of Materials. YSO is
supported by Seoul R${\&}$BD (10543). Y.H.C. is supported by the National
Science Council, R.O.C. (NSC 98-2119-M-009-M016).
## References
* (1) C. W. Nan, M. I. Bichurin, S. X. Dong, D. Viehland, and G. Srinivasan, J. Appl. Phys. 103, 031101 (2008).
* (2) S. X. Dong, J. Y. Zhai, J. F. Li, D. Viehland, and S. Priya, Appl. Phys. Lett. 93, 103511 (2008).
* (3) J. van den Boomgaard, A. M. J. G. van Run, J. Van Suchtelen, Ferroelectrics 14, 727 (1976).
* (4) J. Ryu, A. V. Carazo, K. Uchino, and H. E. Kim, J. Electroceram. 7, 17 (2001).
* (5) C. W. Nan, G. Liu, Y. H. Lin, and H. D. Chen, Phys. Rev. Lett. 94, 197203 (2005).
* (6) H. Zheng, J. Wang, S. E. Lofland, Z. Ma, L. Mohaddes-Ardabili, T. Zhao, L. Salamanca-Riba, S. R. Shinde, S. B. Ogale, F. Bai, D. Viehland, Y. Jia, D. G. Schlom, M. Wuttig, A. Roytburd, and R. Ramesh, Science 303, 661 (2004).
* (7) L. Yan, Z. P. Xing, Z. G. Wang, T. Wang, G. Y. Lei, J. F. Li, and D. Viehland, Appl. Phys. Lett. 94, 192902 (2009).
* (8) L. Yan, Z. G. Wang, Z. P. Xing, J. F. Li, and D. Viehland, J. Appl. Phys. 107, 064106 (2010).
* (9) H. M. Zheng, F. Straub, Q. Zhan, P. L. Yang, W. K. Hsieh, F. Zavaliche, Y. H. Chu, U. Dahmen, and R. Ramesh, Adv. Mater. (Weinheim, Ger.) 18, 2747 (2006).
* (10) R. Pauthenet, Comptes Rendus 230, 1842 (1950).
* (11) H. M. Zheng, J. Kreisel, Y. H. Chu, R. Ramesh, and L. Salamanca-Riba, Appl. Phys. Lett. 90, 113113 (2007).
* (12) T. Choi, S. Lee, Y. J. Choi, V. Kiryukhin, and S. W. Cheong, Science 324, 63 (2009).
* (13) C. H. Yang, J. Seidel, S. Y. Kim, P. B. Rossen, P. Yu, M. Gajek, Y. H. Chu, L. W. Martin, M. B. Holcomb, Q. He, P. Maksymovych, N. Balke, S. V. Kalinin, A. P. Baddorf, S. R. Basu, M. L. Scullin, and R. Ramesh, Nat. Matter. 8, 485 (2009).
* (14) J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig, and R. Ramesh, Science 299, 1719 (2003).
* (15) R. M. Bozorth, E. F. Tilden, and A. J. Williams, Phys. Rev. 99, 1788 (1955).
* (16) C. Ederer and N. A. Spaldin, Phys. Rev. Lett. 95, 257601 (2005).
* (17) H. W. Jang, S. H. Baek, D. Ortiz, C. M. Folkman, R. R. Das, Y. H. Chu, P. Shafer, J. X. Zhang, S. Choudhury, V. Vaithyanathan, Y. B. Chen, D. A. Felker, M. D. Biegalski, M. S. Rzchowski, X. Q. Pan, D. G. Schlom, L. Q. Chen, R. Ramesh, and C. B. Eom, Phys. Rev. Lett. 101, 107602 (2008).
* (18) H. Ryu, P. Murugavel, J. H. Lee, S. C. Chae, T. W. Noh, Y. S. Oh, H. J. Kim, K. H. Kim, J. H. Jang, M. Kim, C. Bae, and J. G. Park, Appl. Phys. Lett. 89, 102907 (2006).
|
arxiv-papers
| 2010-08-09T10:23:23 |
2024-09-04T02:49:12.111492
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yoon Seok Oh, S. Crane, H. Zheng, Y. H. Chu, R. Ramesh, and Kee Hoon\n Kim",
"submitter": "Yoon Seok Oh",
"url": "https://arxiv.org/abs/1008.1481"
}
|
1008.1640
|
# Smooth double barriers in quantum mechanics
Avik Dutt quantumavik@gmail.com Department of Electronics and Electrical
Communication Engineering, Indian Institute of Technology, Kharagpur, 721302,
India Sayan Kar sayan@iitkgp.ac.in Department of Physics and Meteorology and
Center for Theoretical Studies, Indian Institute of Technology, Kharagpur,
721302, India
###### Abstract
Quantum mechanical tunneling across smooth double barrier potentials modeled
using Gaussian functions, is analyzed numerically and by using the WKB
approximation. The transmission probability, resonances as a function of
incident particle energy, and their dependence on the barrier parameters are
obtained for various cases. We also discuss the tunneling time, for which we
obtain generalizations of the known results for rectangular barriers.
## I Introduction
The usual example of quantum mechanical tunneling is the rectangular barrier
in one dimension.beiser ; cohen ; merzbacher Curious students might wonder
what happens if we consider smooth barriers. Do the tunneling results remain
the same? Are there quantitative or qualitative differences? Students might
also wonder what happens if there is more than one barrier. To answer these
questions (particularly for smooth barriers), numerical methods and
approximations are essential because little can be done analytically. In this
paper we discuss tunneling across smooth double barriers of various types.
Double barrier potentials arise in many diverse areas. We give a few examples
in the following.
Quantum heterostructures.heterostructures ; heterostructures2 Semiconductor
heterostructures are layered, thin (about 100 nanometers or less) sandwich
structures made with different semiconductor materials (for example, GaAs
between two AlAs layers). The existence of junctions of different materials is
the reason for the occurrence of sequences of wells and barriers in such
structures. Semiconductor heterostructures have made it possible to control
the parameters inside crystals and devices such as band gaps and the effective
masses of charge carriers and their mobilities, refractive indices, and
electron energy spectrum. Heterostructure electronics are widely used in many
areas, including laser-based telecommunication systems, light-emitting diodes,
bipolar transistors, and low-noise high-electron-mobility transistors for
high- frequency applications including satellite television.heterostructures2
High energy physics. A barrier penetration model has been developed for heavy
ion fusion,pddbphh taking into account the realistic features of the Coulomb
potential for the case of nuclei. The typical barrier shapes encountered for
nuclei conform to double barriers.
Extra dimensions. Another area where such potentials arise is in the physics
of extra dimensions. In particular, they appear in the context of localization
of particles and fields on a four-dimensional hypersurface, known as the
brane. In these brane-world modelsrs localized particles in four dimensions
are viewed as bound or quasi-bound (resonant) states in a double barrier
effective potential spread across the extra dimensional coordinate.
Nonlinear Schrödinger equation. The nonlinear Schrödinger equation has been
used in investigations of Bose-Einstein condensates to probe macroscopic
quantum tunnelingrapedius -paul and gravity surface waves in fluids, among
many other applications. The methods we will discuss here may be extended,
with minor modifications, to the Gross-Pitaevskii equation, a nonlinear
Schrödinger equation arising in the study of many body systems. Above barrier
reflection and tunneling in the nonlinear Schrödinger equation is discussed in
Ref. ishknayan, . The double Gaussian barrier, as well as the rectangular
barrier, has been used as model potentials in studiesrapedius ; paul of the
nonlinear Schrödinger equation. However, the definition of the tunneling
coefficient is ambiguous for weak nonlinearity, because the principle of
superposition does not strictly hold.
Apart from the applications we have discussed, we mention some recent
pedagogical articles on double barrier tunneling. Numerical solutions for
quantum tunneling through rectangular double barrier heterostructures have
been explored.mendez Further, a detailed analysis of the propagation of wave
packets across double barriers has been performed.double Three-dimensional
s-wave tunneling for double barrier potentials has also been
investigated.pramana
The paper is organized as follows. In Sec. II we introduce the various types
of double barriers that will be discussed. Section III discusses the theory of
double barrier tunneling, and the various methods we will use. Numerical and
semiclassical analyses of the smooth barriers mentioned in Sec. II are carried
out in Sec. IV. In Sec. V we determine the tunneling time across smooth double
barriers. Section VI summarizes our results and mentions some possibilities
for future studies.
## II Tunneling across double barriers
A simple way to model double barrier potentials is to add a term shifted in
position to the single barrier potential function, that is,
$V(x)=V_{\rm single}(x,V_{1},w_{1})+V_{\rm single}(x-a,V_{2},w_{2}),$ (1)
where $V_{1,2}$ and $w_{1,2}$ are the height and width of the two barriers.
The separation between the barrier heights is $a$.
Figure 1: Rectangular double barrier (solid line) of widths $w_{1}$ and
$w_{2}$ separated by the distance $a$. The heights of the two single barrier
potentials are $V_{1}$ and $V_{2}$. Also shown is a double Gaussian barrier
(dashed line).
The simplest and the easiest barrier to analyze is the rectangular double
potential barrier, whose parameters are defined in Fig. 1. We have
$V_{\rm single}(x,V_{1},w_{1})=\begin{cases}0&x<0\\\ V_{1}&0\leq x\leq
w_{1}\\\ 0&x>w_{1}\end{cases}.$ (2)
Similarly, the Gaussian double barrier (see Fig. 1) is written using
$V_{\rm single}(x,V_{1},\sigma_{1})=V_{1}e^{-x^{2}/2\sigma_{1}^{2}}$ (3)
and hence $V(x)=V_{\rm single}(x,V_{1},\sigma_{1})+V_{\rm
single}(x-a,V_{2},\sigma_{2})$.
The time-independent Schrödinger equation for a one-dimensional potential
$V(x)$ is
$\frac{d^{2}\psi}{dx^{2}}+\frac{2m}{\hbar^{2}}[E-V(x)]\psi=0,$ (4)
where $m$ is the mass of the particle, $E$ is its energy, and $\psi$ is the
wavefunction. The Schrödinger equation for a rectangular double barrier can be
readily solved for $\psi$, giving five solutions in the five regions. The
solutions and their derivatives can be matched at the boundaries between two
regions to obtain a relation between the incoming and outgoing waves. The
transmission amplitude isdoublebarrier
$\displaystyle T$ $\displaystyle=$ $\displaystyle
4[e^{ik_{1}(w_{1}+w_{2})}(2\cosh(k_{3}w_{2})-ik_{1-3}\sinh(k_{3}w_{2}))(2\cosh(k_{2}w_{1}))-ik_{1-2}\sinh(k_{2}w_{1})$
(5)
$\displaystyle{}+e^{ik_{1}(g+b-w_{1})}k_{1+3}k_{1+2}\sinh(k_{2}w_{1})\sinh(k_{3}w_{2})]^{-1},$
where $w_{1}$ and $w_{2}$ are given in Eq. (1),(2). The other parameters are
defined as
$\displaystyle g$ $\displaystyle=w_{1}+w_{2},$ $\displaystyle b=w_{1}+a+w_{2}$
(6a) $\displaystyle k_{1}$ $\displaystyle=\sqrt{2mE}/\hbar,$ $\displaystyle
k_{2}=\sqrt{2m(V_{1}-E)}/\hbar$ (6b) $\displaystyle k_{3}$
$\displaystyle=\sqrt{2m(V_{2}-E)}/\hbar),$ $\displaystyle k_{i\pm
j}=\dfrac{k_{i}}{k_{j}}\pm\dfrac{k_{j}}{k_{i}},\quad(i,j=1,2,3).$ (6c)
The transmission coefficient (probability) is $\mathcal{T}=T^{*}T$.
For the rectangular double barrier, for which we have analytical solutions, we
match $\psi$ and $d\psi/dx$ at the discontinuities to obtain the relations
between the wavefunction amplitudes, that is, $A$ and $C$. For a smooth
barrier, the ratio of $A$ and $C$ can be obtained by solving Eq. (4)
numerically. The ratio of the maxima on the right-hand side of the barrier to
the incident amplitude on the left-hand side of the barrier can be found from
the plot of $\psi$ versus $x$.
The analysis can be extended to the arbitrary potential functions.riccoazbel ;
cohen However, exact analytical solutions of Eq. (4) are usually difficult or
impossible to find. In particular, it is impossible to exactly solve the
Schrödinger equation for the Gaussian potential. To obtain useful results, we
either find numerical solutions or use the WKB approximation.
To solve the Schrödinger equation numerically, we write Eq. (4) as a system of
two first-order linear ordinary differential equations with $\phi_{1}=\psi$
and $\phi_{2}=d\psi/dx$. With these substitutions, Eq. (4) becomes
$\displaystyle\phi_{1}^{\prime}$ $\displaystyle=\phi_{2},$ (7a)
$\displaystyle\phi_{2}^{\prime}$
$\displaystyle=\frac{2m}{\hbar^{2}}(V(x)-E)\phi_{1}.$ (7b)
Equation (7) can be solved for $\phi_{1}$ to obtain $\psi(x)$. We used the
fourth order Runge-Kutta algorithm.
We compute the wavefunction on both sides of the barrier for different
energies in the required range (up to twice the maximum potential height of
the barriers). Because the potentials are rapidly decaying and we are mainly
interested in the barrier penetration properties, the use of asymptotic
solutions for $\psi$ to calculate $\mathcal{T}$ is justified. To the far right
and far left of the well, the forms of $\psi$ are,
$\psi_{\rm right}\rightarrow Ce^{ikx},\quad\psi_{\rm left}\rightarrow
Ae^{ikx}+Be^{-ikx}.$ (8)
The transmission coefficient is calculated from
$\mathcal{T}=\frac{|C|^{2}}{|A|^{2}}.$ (9)
The Wentzel-Kramers-Brillouin (WKB) approximationpowellcrasemann ; merzbacher
can be used to obtain partially analytic results for an arbitrary potential
well. In this approximation the wavefunction $\psi$ is expressed as a power
series in $\hbar$. We write the wavefunction in the form
$\psi=A(x)e^{i\phi(x)}.$ (10)
If we assume the potential is slowly varying and neglect the second
derivatives of $\phi$ and $A$, the solutions of the Schrödinger equation can
be expressed as
$\psi=\tilde{A}|p|^{-1/2}e^{\pm i\int|p|\,dx/\hbar},\quad
p=\sqrt{2m[E-V(x)]}.$ (11)
The limits of integration in Eq. (11) are determined by the classical turning
points at which $E=V(x)$. At these points the semiclassical approximation
fails because the wavefunction becomes infinite. To the far left and to the
far right of the barriers, the WKB method gives reasonably good approximations
for $\psi(x)$. The matching of $\psi$ near the classical turning points is
done by the use of special functions. The asymptotic forms of these functions
result in a net phase shift of $\pi/4$ when going from a classically forbidden
region to a classically allowed region or vice-versa.powellcrasemann ;
merzbacher
Figure 2: An arbitrary double barrier potential $V(x)$ with classical turning
points at $x_{1}$, $x_{2}$, $x_{3}$, and $x_{4}$. The regions $x_{1}<x<x_{2}$
and $x_{3}<x<x_{4}$ are classically forbidden, but the particle can tunnel
through these regions quantum mechanically.
For a general single barrier potential, the transmission probability
ispowellcrasemann
$\displaystyle\mathcal{T}$
$\displaystyle\approx\left(\frac{1}{T}+\frac{T}{4}\right)^{-2}=T^{2}/(1+T^{2}/4)^{2},$
(12) $\displaystyle T$
$\displaystyle=\exp\left[-\int_{x_{1}}^{x_{2}}\sqrt{2m(V(x)-E)}\,dx/\hbar\right],$
(13)
and $x_{1}$ and $x_{2}$ are the classical turning points. The WKB
approximation introduces errors which are consistent with $T$ being small for
the single barrier case. Thus $\mathcal{T}\simeq T^{2}$, or
$\mathcal{T}\simeq\exp\left[-\frac{2}{\hbar}\int_{x_{1}}^{x_{2}}\sqrt{2m(V(x)-E)}\,dx\right].$
(14)
For a double barrier we obtain four classical turning points instead of two as
evident from Fig. 2. Starting with the wave function on the far right of the
barrier (region V), we have
$\psi_{V}=Ap^{-1/2}\exp\left(i\\!\int_{x_{4}}^{x}p/\hbar\,dx+i\frac{\pi}{4}\right).\\\
$ (15)
Let $T_{1}$, $T_{2}$, and $T_{3}$ be defined as
$\displaystyle T_{1}$
$\displaystyle=\exp\left(-\int_{x_{3}}^{x_{4}}|p|/\hbar\,dx\right)$ (16a)
$\displaystyle T_{2}$ $\displaystyle=\int_{x_{2}}^{x_{3}}p/\hbar\,dx$ (16b)
$\displaystyle T_{3}$ $\displaystyle=\exp(-\int_{x_{1}}^{x_{2}}|p|/\hbar\,dx)$
(16c)
We expand the complex exponentials using Euler’s identity, and then match the
real and imaginary parts of $\psi$ near $x<x_{4}$ and $x>x_{4}$, and then near
$x<x_{3}$ and $x>x_{3}$ (by applying the connecting formulae of the WKB
method) and obtain
$\displaystyle\psi_{\rm III}$ $\displaystyle=$ $\displaystyle
iAp^{-1/2}\Bigg{[}\Big{(}\frac{T_{1}}{4}-\frac{1}{T_{1}}\Big{)}\exp\Big{(}i\\!\int_{x}^{x_{3}}p/\hbar\,dx+\frac{i\pi}{4}\Big{)}$
(17)
$\displaystyle{}+\Big{(}\frac{T_{1}}{4}+\frac{1}{T_{1}}\Big{)}\exp\Big{(}-i\\!\int_{x}^{x_{3}}p/\hbar\,dx-\frac{i\pi}{4}\Big{)}\Bigg{]}.$
Similarly, we find
$\displaystyle\psi_{I}$ $\displaystyle=$
$\displaystyle\frac{A}{\sqrt{|p|}}\Bigg{[}\Big{(}\frac{C_{3}+C_{4}}{iT_{3}}+\frac{iT_{3}C_{4}-iT_{3}C_{3}}{4}\Big{)}\exp\Big{(}i\\!\int_{x}^{x_{1}}p/\hbar\,dx+\frac{i\pi}{4}\Big{)}$
(18)
$\displaystyle{}+\Big{(}-\frac{C_{3}+C_{4}}{iT_{3}}+\frac{iT_{3}C_{4}-iT_{3}C_{3}}{4}\Big{)}\exp\Big{(}-i\\!\int_{x}^{x_{1}}p/\hbar\,dx-\frac{i\pi}{4}\Big{)}\Bigg{]}.$
The second term can be identified with the wave incident from the left, and
yields the transmission coefficient
$\mathcal{T}\approx\left|\frac{\psi_{V}}{\psi_{\rm
incident}}\right|^{2}=\left|\frac{T_{3}(C_{4}-C_{3})}{4}+\frac{C_{3}+C_{4}}{T_{3}}\right|^{-2}$
(19)
where
$\displaystyle C_{3}$ $\displaystyle=(1/T_{1}-T_{1}/4)\exp(iT_{2})$ (20a)
$\displaystyle C_{4}$ $\displaystyle=(T_{1}/4+1/T_{1})\exp(-iT_{2})$ (20b)
and $p(x)=\sqrt{2m[E-V(x)]}$.
Equation (19) can be applied to a Gaussian double barrier with the assumptions
that $E<V_{1},V_{2}$, $a\gg\sigma_{1},\sigma_{2}$, and
$a-3\sigma_{1}-3\sigma_{2}>0$. Hence, $x_{1,2}=\pm\sigma_{1}\sqrt{2(\log
V_{1}-\log E)}$, and $x_{3,4}=a\pm\sigma_{2}\sqrt{2(\log V_{2}-\log E)}$. The
integrations in the WKB method were performed using Simpson’s rule.
## III Numerical and semiclassical analysis for model smooth barriers
### III.1 Numerical Analysis
Figure 3: Plot of the potential (shaded) and wave function. The plots of the
wave function are for an incident energy of 4 eV. (a) $V_{1}=V_{2}=4$ eV,
$\sigma_{1}=\sigma_{2}=0.2$ nm, $a=1$ nm. (b) Transmission probability for the
barrier in (a). (c) $V_{1}=V_{2}=4$ eV, $\sigma_{1}=\sigma_{2}=0.2$ nm, $a$
increased to 3 nm. (d) Transmission probability for the barrier in (c) – the
number of resonances is greater due to the larger separation-to-width ratio.
(e) $V_{1}=V_{2}=4$ eV, $\sigma_{1}=\sigma_{2}=0.2$ nm, $a$ reduced to 0.4 nm
so that the two Gaussian potentials almost merge into a single barrier. (f)
Transmission probability for the barrier in (e)– the plot resembles the
transmission probability of a single Gaussian barrier. (g) $V_{1}=1,\,V_{2}=4$
eV, $\sigma_{1}=0.4$, $\sigma_{2}=0.1$ nm, $a=3$ nm. (h) Transmission
probability for the barrier in (g)– asymmetrical barriers have a marked effect
on the transmission probability.
Figure 4: Comparison of transmission probabilities for the rectangular and the
Gaussian double barrier. (a) Square : $V_{1}=4,V_{2}=4$ eV,
$w_{1}=0.6,w_{2}=0.6,a=0.75$ nm (b) Gaussian :
$V_{1}=4,V_{2}=4eV$,$\sigma_{1}=0.6,\sigma_{2}=0.6,a=4.35$ nm (c) Square :
$V_{1}=4,V_{2}=4$ eV, $w_{1}=0.2,w_{2}=0.8,a=1$ nm (d) Gaussian :
$V_{1}=4,V_{2}=4$ eV, $\sigma_{1}=0.2,\sigma_{2}=0.8,a=4$ nm (e) Square :
$V_{1}=4,V_{2}=8$ eV, $w_{1}=0.2,w_{2}=0.1,a=4$ nm (f) Gaussian :
$V_{1}=4,V_{2}=8$ eV, $\sigma_{1}=0.2,\sigma_{2}=0.1,a=4.9$ nm The
transmission probability asymptotically approaches unity for the Gaussian
barrier due to the smoothness of the Gaussian barrier potential.
The $\cal T$ versus $E$ plots in Fig. 3 show the variation of the transmission
probability with the energy of an electron incident on a one-dimensional
Gaussian double barrier. Figure 3 illustrates how the transmission coefficient
changes due to the presence of two barriers, a change in the barrier
separation, and a change in the height of one barrier. The energy of the
electron is assumed to be of the order of a few electron volts. In Fig. 3(a)
the two barriers are very close to each other and the number of resonances is
less (Fig. 3(b)) than obtained for larger $a$. In Fig. 3(c) the separation
between the barriers is increased,the well becomes wider than the well in Fig.
3(a), and consequently, the number of resonances increases (Fig. 3(d)). Figure
3(e) shows two barriers that are so close that they appear to be a single
continuous barrier, and hence no resonances (Fig. 3(f)) are seen. In Fig. 3(g)
we consider asymmetrical barriers for which $V_{1}\neq V_{2}$ and/or
$\sigma_{1}\neq\sigma_{2}$ with $\sigma_{1}V_{1}=\sigma_{2}V_{2}$. We note
that the effect is predominantly that of a single barrier though resonances do
appear (Fig. 3(h)), signaling the presence of a second barrier of much smaller
height.
Next, we compare the transmission coefficients for the Gaussian and the
rectangular double barrier (see Fig. 4). We choose
$a(\mbox{Gaussian})=3w_{1}+3w_{2}+a(\mbox{rect})$, $\sigma_{1}=w_{1}$, and
$\sigma_{2}=w_{2}$. We also vary the height of the barriers keeping the (width
$\times$ height) for both the barriers the same. Figure 4(a),(c),(e) shows the
transmission coefficient for a double rectangular barrier, and Fig.
4(b),(d),(f) shows the same for a double Gaussian. The qualitative nature of
the plots for the rectangular and the Gaussian barriers is similar. Tunneling
occurs for both barriers, though there are differences that may be used to
determine the barrier shape from plots of $\mathcal{T}$ versus $E$.
The number of peaks in $\mathcal{T}$ is large for both cases. However, the
peaks and valleys are more prominent for the rectangular barrier. The
transmission coefficient for the Gaussian does not reach unity (except
asymptotically) and the minima are also at lower values. The cause of this
quantitative difference is the smoothness of the Gaussian barrier.
The following observations are valid if $a/\sigma$ is large ($\geq 4$). The
${\mathcal{T}}$ versus $E$ plot (Fig. 4) depends strongly on the product of
width and height for any barrier. If we keep the width $\times$ height
constant, the graphs are qualitatively similar.
The plots show a large number of valleys and peaks, corresponding to resonant
states. Resonant states are a result of the destructive interference between
waves reflected from the two barriers. The valleys in the $\mathcal{T}$ versus
$E$ plots correspond to quasi-bound states. As the separation-to-width ratio
of the barriers is increased, the number of resonances increases, which means
that the well can accommodate a larger number of quasi-bound states. If the
well width is made indefinitely wide, the resonant states merge into a
continuum.
The peaks become sharper as the height of the barriers is increased; i.e. the
full width at half maximum of the peaks decreases. If $\Delta$ is the full
width at half maximum of the peak in ${\mathcal{T}}(E)$, then $\Delta$
decreases as $V_{0}$ and $V_{1}$ increase. The peaks also become sharper if
the widths of the barriers are increased, which makes it difficult to obtain
numerical solutions. The lower the energy at which the peak occurs, the
steeper is the peak. If $E\ll V_{0},\ V_{1}$, $\Delta$ is very small and vice
versa.
### III.2 WKB Analysis
The WKB method provides a way of obtaining analytical expressions for the
transmission coefficient. In Figure 5 we compare the results obtained for the
transmission probability using the semiclassical analysis method and the
numerical solutions Figures 5(a) and (b) differ because of the larger value of
$a$ in Fig. 5(b) (all other parameters are the same). In Fig. 5(c) the heights
are halved, and the widths are doubled compared to Fig. 5(b). In Fig. 5(d) the
Gaussian barrier is asymmetrical with $\sigma_{1}/\sigma_{2}=6$.
Figure 5: Comparison of the transmission probabilities calculated by numerical
solution of Schrödinger equation (gray) and the WKB approximation (black) for
a Gaussian double barrier with parameters: (a) $V_{1}=V_{2}=4$ eV,
$\sigma_{1}=\sigma_{2}=0.2$ nm, $a=1$ nm, (b) $V_{1}=V_{2}=4$ eV,
$\sigma_{1}=\sigma_{2}=0.2$ nm, $a$ is increased to $4$ nm. (c)
$V_{1}=V_{2}=2$ eV, $\sigma_{1}=\sigma_{2}=0.4$ nm, $a=4$ nm. The potential
barrier height is larger for this barrier. (d) $V_{1}=V_{2}=4$ eV,
$\sigma_{1}=0.6,\,\sigma_{2}=0.1$ nm, $a=4$ nm. In this case we consider an
asymmetrical barrier. The results obtained by the two methods are in good
agreement.
For $E\ll V_{1}$ the values of $\mathcal{T}$ are almost the same, or at least
of the same order of magnitude using either the numerical method or the WKB
approximation. The greatest deviations from the numerical solution are
observed for energies near $V_{0}$. This deviation is expected because the WKB
approximation is not valid near the classical turning points. The difference
between the WKB method and the numerical solution of Schrodinger’s equation
also becomes prominent when the separation between the barriers becomes
comparable to the width of the barriers. In the calculation of the turning
points we neglected the contribution of the first barrier to the turning
points of the second barrier and vice versa so that we could obtain analytic
expressions for the classical turning points; otherwise, we would have to
solve the transcendental equation
$V(x)=[V_{1}\exp(-x^{2}/2\sigma_{1}^{2})+V_{2}\exp(-(x-a)^{2}/2\sigma_{2}^{2})]=E$
for each value of $E$. Thus the difference between the semiclassical results
and the numerical solutions for the transmission probability becomes prominent
when the barriers are close to each other, but the assumption yields
acceptable results in return for a shorter computation time.
The WKB approximation becomes more accurate as the separation between the
barriers increases. The numerical values of ${\mathcal{T}}(E)$ versus $E$ in
Fig. 5 do not match the WKB approximation results for energies
$E>\min(V_{1},V_{2})$. For asymmetrical barriers with non-identical potential
heights, the WKB approximation breaks down for $E>\min(V_{1},V_{2})$ because
the formulation we have used involves four classical turning points. The
potential function reduces to a single barrier with two turning points for
$E>\min(V_{1},V_{2})$. However, the WKB results agree quite well for
$E<\min(V_{1},V_{2})$.
The advantages of the WKB method are that it requires about an order of
magnitude less computation time than the direct numerical solution of
Schrodinger’s equation. The main limitation of the WKB method is that it
relies heavily on the turning points, and hence is difficult to apply to
energies greater than the potential height of any of the barriers.
## IV Tunneling Time
With the advent of the fabrication of nanometer semi-conductor structures, the
calculation of tunneling time has acquired more importance.davies For one-
dimensional single barriers, the tunneling time becomes independent of the
barrier width for opaque barriers. This “Hartman effect”hartman predicts
superluminal and arbitrarily large group velocities inside sufficiently long
barriers. There is some controversy concerning the definition of the tunneling
time.haugestoveng The expression for the tunneling time we have used is
variously named the “group delay” or “phase time.”
If we assume a single rectangular barrier, with the potential
$V(x)=\begin{cases}V_{0}&(0<x<L)\\\ 0&\mbox{elsewhere},\\\ \end{cases}$ (21)
the wavefunction may be written ashaugestoveng
$\psi(x)=\begin{cases}\exp(ikx)+\sqrt{\mathcal{R}}\exp(-ikx+i\gamma)&(x<0)\\\
\psi_{1}(x)&(0\leq x\leq L)\\\ \sqrt{\mathcal{T}}\exp(ikx+i\alpha)&(x>L).\\\
\end{cases}$ (22)
where $\mathcal{T}$ and $\mathcal{R}$ are the transmission and reflection
probabilities, $k$ the wave vector, and $\alpha$ and $\gamma$ are phase
constants dependent on $k$. If we apply the stationary phase approximation to
the peak of the transmitted wave packet,
$\partial\\{\mbox{arg}(\psi(x,t))\\}/\partial k=0$, we find
$\frac{d\alpha}{dk}-\frac{1\,dE}{\hbar\ dk}t=0.$ (23)
Thus, the temporal delay may be defined as
$\tau=\hbar\frac{\partial\alpha}{\partial E}.$ (24)
From the transmission amplitude for a single barrier we find the phase shift
to be $\alpha=-\tan^{-1}((\kappa^{2}-k^{2})\tanh(\kappa L)/2\kappa k)$, where
$\kappa=\sqrt{2m(V_{0}-E)}/\hbar$ and $k=\sqrt{2mE}/\hbar$. By using Eq. (24)
we have
$\tau_{g}\stackrel{{\scriptstyle\kappa
L\rightarrow\infty}}{{\longrightarrow}}\frac{2m}{\hbar k\kappa},$ (25)
where the subscript $g$ denotes group delay.
For a rectangular double barrier the transmission amplitude is expressed by
Eq. (5), and the phase time isolkhrecami ; petrilloolkh
$\tau_{g}=\hbar\frac{\partial}{\partial E}\arg[T\exp(ik_{1}b)].$ (26)
In the limit of $k_{2}w_{1}$ and $k_{3}w_{2}\to\infty$, the phase time becomes
independent of the barrier width and separation:
$\tau_{g}=\frac{2m}{\hbar k_{1}k_{2}},$ (27)
except at resonance and anti-resonance.
To illustrate what happens at resonance and anti-resonance,hgwinful we take
$E=V_{1}/2$ and $V_{0}=V_{1}=V_{2}$. When the phase shift $2k_{1}a=2m\pi$
($m=1,2,3,\ldots$) (resonance), the group delay becomes
$\tau_{g}^{\rm res}=\frac{(1+R_{0})}{T_{0}}\frac{a}{v},$ (28)
where $v=p/m=\sqrt{2mE}/m$, and $R_{0}$ and $T_{0}$ denote the reflection and
transmission probabilities through a single barrier of width $w_{1}$. At anti-
resonance ($2k_{1}a=(2m+1)\pi$),
$\tau_{g}^{\mbox{\tiny anti-res}}=\frac{T_{0}}{(1+R_{0})}\frac{a}{v},$ (29)
indicating that the phase time increases linearly with the inter-barrier
separation, $a$, at both resonance and anti-resonance.
### IV.1 Tunneling times for general double barrier potentials
The wave function for $x>x_{4}$ (Fig. 2) is given in the WKB approximation by
$\psi_{V}=A|p|^{-1/2}\exp\left(\frac{i}{\hbar}\int_{x_{4}}^{x}|p|\,dx+i\frac{\pi}{4}\right),$
(30)
and the incident wavefunction is
$\psi_{\rm
incident}=A|p|^{-1/2}\left\\{i(C_{3}+C_{4})/T_{3}+iT_{3}(C_{4}-C_{3})/4\right\\}\exp\left(-i\\!\int_{x}^{x_{1}}p/\hbar\,dx-i\pi/4\right).$
(31)
In analogy with Eq. (26), the group delay is found to be
$\tau_{g}=\hbar\frac{\partial}{\partial E}\arg\left[(\psi_{V}/\psi_{\rm
incident})\exp\left(\int_{x_{1}}^{x_{4}}|p|/\hbar\,dx\right)\right].$ (32)
If we substitute $\psi_{V}$ and $\psi_{\rm incident}$, into Eq. 32 we obtain
$\tau_{g}=\hbar\frac{\partial}{\partial
E}\arg\left[\frac{T_{3}(C_{4}-C_{3})}{4}+\frac{C_{4}+C_{3}}{T_{3}}\right]^{-1},$
(33)
and after some algebra
$\displaystyle\tau_{g}$ $\displaystyle=\hbar\frac{\partial}{\partial
E}\tan^{-1}\frac{2\tan T_{2}}{T_{1}^{2}}$ (34a)
$\displaystyle=\frac{2}{T_{1}^{4}+4\tan^{2}T_{2}}T_{1}^{2}\left(\sec^{2}T_{2}\frac{\partial
T_{2}}{\partial E}-2\frac{\tan T_{2}}{T_{1}}\frac{\partial T_{1}}{\partial
E}\right).$ (34b)
Equations 27–29, which are applicable for a rectangular barrier, can be
recovered from Eq. (34) as follows. We have $T_{1}=\exp(-k_{2}w_{1})$ and
$T_{2}=k_{1}a$. We let $\phi=\tan^{-1}(2\tan(T_{2})/T_{1}^{2})$ and write
$\tau_{g}=\hbar\partial\phi/\partial E$. We find that the phase tunneling time
through a rectangular double barrier of width $w_{1}$ and separation $a$ is
$\tau_{g}=2\cos^{2}\phi\frac{m}{\hbar}\exp(2k_{2}w_{1})(a\sec^{2}(k_{1}a)/k_{1}-2w_{1}\tan(k_{1}a)/k_{2}).$
(35)
To illustrate the behavior of $\tau_{g}$ near a resonance, we assume
$E=V_{1}/2$ so that $k_{1}=k_{2}$. At resonance, $2k_{1}a=2m\pi$, where $m$ is
an integer. The group delay at resonance assumes the form
$\tau_{g}^{\rm res}=2e^{2k_{2}w_{1}}a/v\approx(1+R_{0})a/T_{0}v,$ (36)
because $\cos\phi=1$ and $\tan k_{1}a=0$. Here,
$R_{0}=(1/T_{1}-T_{1}/4)^{2}/(1/T_{1}+T_{1}/4)^{2}$ and
$T_{0}=(1/T_{1}+T_{1}/4)^{-2}$ are the reflection and transmission
probabilities through a single barrier
($(1+R_{0})/T_{0}=2(T_{1}^{-2}+T_{1}^{2}/16)\approx 2e^{k_{2}w_{1}}$).
Figure 6: Variation of tunneling time with barrier width $\sigma$ for a double
Gaussian barrier, with (a) $V_{0}=4$ eV; $a=8$ nm (b) $V_{0}=12$ eV; $a=8$ nm,
for various energies. The tunneling time increases up to a certain value of
$\sigma$ and then decreases rapidly. The maximum in the curve shifts to larger
values of the barrier width for higher energies.
Figure 7: Variation of the tunneling time through a double Gaussian barrier
with inter-barrier distance $a$ for (a) $V_{0}=6$ eV, $\sigma_{1}=0.2$ nm, and
(b) $V_{0}=1.50$ eV$,\sigma_{1}=0.8$ nm. Note that the tunneling time varies
over several orders of magnitude for various energies of the incident
particle.
Similarly at anti-resonance ($2k_{1}a=(2m+1)\pi)$, $m$ an integer, we have
$\tau_{g}^{\mbox{\tiny anti-res}}=T_{0}a/v(1+R_{0}).$ (37)
We next obtain $\tau_{g}$ for a Gaussian double barrier. The relevant
integrations for obtaining $T_{3}$ are performed numerically. Figure 6 shows
the variation of $\tau_{g}$ with $\sigma$ for fixed $a$. Figures 7(a) and (b)
show how $\tau_{g}$ varies with $a$ for fixed $\sigma$.
We see that the group delay increases with an increase in the barrier width up
to a certain value and then falls off rapidly on widening the barrier further.
The peak in the group delay versus barrier width shifts toward the greater
widths at higher energies. The maxima in $\tau_{g}$ shift to greater values of
$\sigma$ at higher energies. The group delay decreases with increasing
$V_{0}$, and for fixed $V_{0}$ $\tau_{g}$ increases with $E$ if other
parameters remain constant. The tunneling time is more or less independent of
the interbarrier distance as is evident from Fig. 7, which is consistent with
the Hartman effect.
## V DOUBLE BARRIERS IN HETEROSTRUCTURES
Double barriers of either type (smooth or square) appear in various physical
situations (as outlined in Sec. I). We now discuss in some detail, how they
may arise in the context of one such scenario-semiconductor heterostructures.
Figure 8: Semiconductor heterostructures and superlattices: (i) Energy band
diagram of an n-N semiconductor heterojunction; (ii) Heterojunctions can be of
three types classified according to the lineup of the conduction and valence
bands: straddled($E_{C2}>E_{C1},E_{V2}<E_{V_{1}}$),
staggered($E_{C2}>E_{C1},E_{C1}>E_{V2}>E_{V1}$), and
broken($E_{C2},E_{V2}>E_{C1}$); (iii) An unbiased GaAs/AlGaAs superlattice and
the corresponding potential energy diagram; (iv) A biased superlattice created
by the application of an electric field, which creates a gradient in the
potential function.
A heterojunction (Fig. 8(i),(ii)) is the interface between two dissimilar
solid state materials, including crystalline and amorphous structures of
metals, insulators, and semiconductors. A combination of one or more
heterojunctions in a device is a heterostructure. As discussed in Sec. I,
heterostructures having semiconductor substrates are widely used in the
fabrication of devices that have desired electron characteristics as functions
of applied potentials.heterostructures
Quantum structures may be classified as wells, wires, or dots, depending on
whether the carriers are confined in one, two or three dimensions.
Semiconductors can be used in all of these structures, but we consider one-
dimensional potential wells and barriers, which can be fabricated by growing a
single layer of one material (such as GaAs) between two layers of different
materials (such as AlGaAs), as in Fig. 8(iii). Other pairs of semiconductors
having profound importance in technological applications include the III-V
compounds GaInAs/InP, GaInAs/AlInAs, GaSb/InSb and the II-VI compounds
CdZnSe/ZnSe, ZnSTeSe/ZnSSe.
In a semiconductor the difference between the conduction and valence band
energies of the two materials at the heterojunction with respect to the vacuum
level and the Fermi level is responsible for the formation of quantum wells,
whose heights are of the order of a few hundred meV. This height is to be
compared with the thermal energies of carriers at room temperature ($\approx
26$ meV). Hence, the thermal motion of the electrons does not allow them to
frequently cross the barriers, and quantum mechanical tunneling is the main
transport phenomena at length scales less than the mean free path of the
electrons.
From Fig. 8(i) it is evident that the conduction band potential step is given
by $\Delta E_{c}=\chi_{1}-\chi_{2}$. Because the Fermi level must be
continuous in chemical and thermal equilibrium, the valence band potential
step is found to be $\Delta E_{v}=E_{G2}-E_{G1}-\Delta E_{c}$.
The I-V characteristics of a one-dimensional quantum heterostructure can be
related to the transmission probability according to the relationtsuesaki
$J=\frac{e}{4\pi^{3}\hbar}\\!\int_{0}^{\infty}dk_{l}\\!\int_{0}^{\infty}dk_{t}[f(E)-f(E^{\prime})]T^{*}T\frac{\partial
E}{\partial k},$ (38)
where $k_{l}$ and $k_{t}$ are the wave vectors in the longitudinal and
transverse directions respectively, and $f$ is the density of states given by
the Fermi-Dirac distribution. The theoretical results can be compared with
experimental data available from the current-voltage characteristics of the
device. The mass to be used in the Schrödinger equation is the effective mass
of the electron $m*$ in the longitudinal direction. As mentioned in Ref.
tsuesaki, the electrons lose coherence after tunneling through a distance of
the order of the mean free path of the carriers, resulting in a widening of
the peaks in the I-V characteristics. Calculations of the bound and quasi-
bound states of quantum heterostructures are available.ieee
There are two major approximations in the modeling of semiconductor
heterostructures by rectangular potentials. The effective mass $m*$ of the
carrier changes when the electrons pass through a heterojunction, but this
variation is not incorporated in the analysis. Also, the potential profile is
not steplike – the energy bands of the two materials at the heterojunction
change smoothly because of factors such as the inhomogeneities present at the
interface, space-dependent composition of the compounds, and possible
mechanical strain of the layers. The best known example is the SiO2/Si
heterojunction with a very small density of defects at the interface. Compound
semiconductors usually have a larger concentration of defects and
inhomogeneities, leading to a continuous barrier rather than a rectangular
one.chandkumar ; zeyrek The experimental data indicate a double Gaussian
distribution of heights for the potential barriers. Another major reason for
the smoothening of the heterojunction potential shapes is the formation of a
space charge.chebotarev
Our results may be used to compare models with experimental data which are
available for semiconductor heterostructures. This aspect requires more work –
we need to find out the theoretical I-V characteristics from the transmission
coefficients obtained by using Eq. (38). In principle, other functional forms
may also be chosen and the I-V characteristics obtained and compared with
experiments to determine the actual distribution of barrier heights. It is
possible that such investigations may help design device applications where
precise knowledge about fabricated structures is often very useful.
## VI REMARKS
We have carried out similar computations for the Lorentz barrier, and obtained
results closer to the rectangular barrier than to the Gaussian barrier. Other
functional forms of the barriers may be studied to learn their characteristic
features. A further challenging problem is to consider multiple barriers and
external, applied field effects, which are necessary to model realistic
systems in condensed matter/semiconductor physics.
###### Acknowledgements.
We express our gratitude to the anonymous reviewers for their valuable
suggestions in improving the article.
## References
* (1) A. Beiser, Concepts of Modern Physics (McGraw-Hill International, Singapore, 1984). See Chap. 5 for a discussion on the tunneling effect and Chap. 12 for its application to $\alpha$-decay.
* (2) E. Merzbacher, Quantum Mechanics (John Wiley & Sons, New York, 1986).
* (3) C. Cohen-Tannoudji, B. Liu, and F. Laloe, Quantum Mechanics, Vol. 1 (John Wiley & Sons, New York, 1977).
* (4) V. V. Mitin, V. A. Kochelap, and M. A. Stoscio, Quantum Heterostructures: Microelectronics and Optoelectronics (Cambridge University Press, Cambridge, UK, 1999), pp. 73–87.
* (5) Z. I. Alferov, “The double heterostructure concept and its applications to physics, electronics and technology,” Rev. Mod. Phys. 73(3), 767–782 (2001).
* (6) P. Descouvemont, D. Baye, and P.-H. Heenen, “A barrier penetration model for heavy ion fusion valid at all energies,” Z. Phys. A 306, 79–88 (1982).
* (7) L. Randall and R. Sundrum, “An alternative to compactification,” Phys. Rev. Lett. 83, 4690–4693 (1999) ibid. “A large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. 83, 3370–3373 (1999).
* (8) K. Rapedius and H. J. Korsch, “Barrier transmission for the one-dimensional nonlinear Schrodinger equation: Resonances and transmission profiles,” Phys. Rev. A 77, 063610 (2008).
* (9) H. A. Ishkhanyan and V. P. Krainov, “Above barrier reflection of cold atoms by resonant laser light within the Gross-Pitaevskii approximation,” Laser Physics 19, 1729–1734 (2009); H. A. Ishkhanyan and V. P. Krainov, “Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem,” Phys. Rev. A80, (045601-1–4) (2009); H. A. Ishkhanyan, V. P. Krainov, and A. M. Ishkhanyan, “Transmission resonances in above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential,” J. Phys. B: At. Mol. Opt. Phys. 43, 085306 (2010).
* (10) T. Paul, K. Richter, and P. Schlagheck, “Nonlinear resonant transport of Bose-Einstein condensates,” Phys. Rev. Lett. 94, 020404(2005).
* (11) B. Mendez and F. Dominguez-Adame,“Numerical study of electron tunneling through heterostructures,” Am. J. Phys. 62, 143–147 (1994).
* (12) A. P. Stamp and G. C. McIntosh, “A time-dependent study of resonant tunneling through a double barrier,” Am. J. Phys. 64, 264–276 (1996).
* (13) A. U. Maheswari, P. Prema, S. Mahadevan, and C.S. Shastry, “Quasi-bound states, resonance tunnelling, and tunnelling times generated by twin symmetric barriers,” Pramana J. Phys. 73(6), 969–988 (2009).
* (14) H. Yamamoto, “Resonant tunneling condition and transmission coefficient in a symmetrical one-dimensional rectangular double-barrier system,” Appl. Phys. A 42, 245–248 (1987).
* (15) B. Ricco, M. Ya. Azbel, “Physics of resonant tunneling: The one-dimensional double barrier case,” Phys. Rev. B 29, 1970–1981 (1984).
* (16) J. L. Powell and B. Crasemann, Quantum Mechanics (Addison-Wesley, Reading, MA, 1965).
* (17) P. C. W. Davies, “Quantum tunneling time,” Am. J. Phys. 73, 23–27 (2005).
* (18) T. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 33, 3427–3433 (1962).
* (19) E. H. Hauge and J. A. Stoveng, “Tunneling times: A critical review,” Rev. Mod. Phys. 61, 917–936 (1989)
* (20) V. S. Olkhovsky, E. Recami, and G. Salesi, “Superluminal tunneling through two succesive barriers,” Europhys. Lett., 57, 879-884 (2002).
* (21) V. Petrillo and V. S. Olkhovsky,“Time asymptotic expansion of the tunneled wave function for a double barrier potential,” Europhys. Lett. 74, 327–333 (2006).
* (22) H. G. Winful, “Tunneling time, the Hartman effect and superluminality: A proposed resolution of an old paradox,” Physics Reports 436, 1–69 (2006).
* (23) R. Tsu and L. Esaki, “Tunneling in a finite superlattice,” Appl. Phys. Lett. 22, 562–564 (1973).
* (24) E. Anemogiannis, E. N. Glytsis, amd T. K. Gaylord, “Bound and quasibound state calculations for biased/unbiased semiconductor quantum heterostructures,” IEEE J. Quantum Electronics 29, 2731–2740 (1993).
* (25) S. Chand and J. Kumar, “Evidence for the double distribution of barrier heights in $Pd_{2}Si/n-Si$ Schottky diodes from I-V-T measurements,” Semicond. Sci. Technol. 11, 1203–1208 (1996).
* (26) S. Zeyrek, M. M. Bulbul, S. Altindal, M. C. Baykul, and H. Yuzer, “The double Gaussian distribution of inhomogeneous barrier heights in Al/GaN/p-GaAs (MIS) Schottky diodes in wide temperature range,” Braz. J. Phys. 38(4), 591–597 (2008).
* (27) L. V. Chebotarev, “Resonant delay in tunnelling through smooth double and triple barriers,” Phys. Stat. Sol. (b) 208, 69–86 (1998).
|
arxiv-papers
| 2010-08-10T07:31:28 |
2024-09-04T02:49:12.119800
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Avik Dutt and Sayan Kar (IIT Kharagpur, India)",
"submitter": "Avik Dutt",
"url": "https://arxiv.org/abs/1008.1640"
}
|
1008.1686
|
11institutetext: 1 Department of Astronomy, Nanjing University, Nanjing
210093, China
2 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, China
11email: hyf@nju.edu.cn
# Early Rebrightenings of X-ray Afterglows from Ring-Shaped GRB Jets
M. Xu1,2 Y. F. Huang1,2
(Received 00 00, 0000; accepted 00 00, 0000)
###### Abstract
Aims. Early rebrightenings at a post-burst time of $10^{2}$ — $10^{4}$ s have
been observed in the afterglows of some gamma-ray bursts (GRBs). Unlike X-ray
flares, these rebrightenings usually last for a relatively long period. The
continuous energy injection mechanism usually can only produce a plateau in
the afterglow light curve, but not a rebrightening. Also, a sudden energy
injection can give birth to a rebrightening, but the rebrightening is a bit
too rapid.
Methods. Here we argued that the early rebrightenings can be produced by the
ring-shaped jet model. In this scenario, the GRB outflow is not a full cone,
but a centrally hollowed ring. Assuming that the line of sight is on the
central symmetry axis of the hollow cone, we calculate the overall dynamical
evolution of the outflows and educe the multiband afterglow light curves.
Results. It is found that the early rebrightenings observed in the afterglows
of a few GRBs, such as GRBs 051016B, 060109, 070103 and 070208 etc, can be
well explained in this framework.
Conclusions. It is suggested that these long-lasted early rebrightenings in
GRB afterglows should be resulted from ring-shaped jets.
###### Key Words.:
gamma rays: bursts - ISM: jets and outflows
## 1 Introduction
Gamma-ray bursts (GRBs) are powerful explosive events in the Universe. The
standard fireball model suggests that the prompt emission should result from
internal shocks and the afterglow should result from external shocks. Since
the launch of the $Swift$ satellite (Gehrels et al. 2004), enormous
improvements have been achieved in understanding the nature of GRBs (for
recent reviews, see: Zhang 2007, Gehrels et al. 2009). Generally, some
interesting components have been identified in the X-ray afterglow light
curves of GRBs, i.e., the steep decay phase, the shallow decay phase, the
normal decay phase, the post jet-break phase, and X-ray flares.
Rebrightening behavior is also a very interesting feature among a few GRBs.
Unlike X-ray flares which are usually characterized by a rapid rise and fall
of the flux (with the mean ratio of width to peak time $\langle\Delta
t/t\rangle=0.13\pm 0.1$, Chincarini et al. 2007), these rebrightenings are
generally very gentle, with $\Delta t/t>1$. Rebrightenings can occur either in
the early afterglow stage ($t\leq 10^{4}$ s) or in the late afterglow stage
($t\gg 10^{4}$ s). Rebrightening in the late stage was first discovered in the
X-ray and optical afterglow of GRB 970508, at a post-burst time of about
$t\sim 2$ day (Piro et al. 1998). More examples of late rebrightenings
(typically occuring at $t\geq 1$ — 2 day) can be found in GRBs 030329 (Lipkin
et al. 2004), 031203 (Ramirez-Ruiz et al. 2005), 050408 (de Ugarte Postigo et
al. 2007), and 081028 (Margutti et al. 2009). Interpretations for these late
rebrightenings include off-axis jet model, and microphysics variation
mechanism (Kong et al. 2010).
In this study, we will mainly concentrate on the early rebrightenings that
typically occur at a post-burst time of $t\leq 10^{4}$ s. Examples of such
early rebrightenings can be found in GRBs 051016B, 060109, 070103, 070208 etc.
Most GRBs, like GRB 061007 (Kocevski & Butler 2008), do not exhibit early
rebrightenings. Among more than 500 Swift GRBs (Gehrels et al. 2009), we note
that only less than 1% events show early X-ray rebrightenings. The energy-
injection mechanism is a natural explanation for rebrightenings, but it is
unlikely to take effect here. For example, the continuous energy-injection
mechanism usually is more likely to produce a plateau-like structure in the
afterglow light curve, but not an obvious rebrightening (Dai & Lu 1998; Zhang
& Mészáros 2001; Yu & Huang 2007). Also, although a sudden energy-injection
can give birth to a rebrightening, the rebrightening is generally a bit too
rapid (Huang et al. 2006) as compared with the early rebrightenings considered
here.
We suggest that this kind of early rebrightenings may be produced by ring-
shaped GRB jets. A ring-shaped jet is not a full cone, but a centrally
hollowed ring (Granot 2005; Fargion & Grossi 2006; Xu et al. 2008). In this
paper, we numerically investigate the afterglow features of such outflows and
use this mechanism to explain a few observed afterglows. The structure of our
paper is as follows. The geometry of ring-shaped jets is sketched in Section
2. The dynamical evolution and afterglow of ring-shaped jets are studied in
Section 3, together with a detailed comparison with a few observations.
Finally, Section 4 is our discussion and conclusion.
## 2 Ring-Shaped Jet
A ring-shaped jet can be produced by the central engine during the prompt GRB
phase. The most natural mechanism may be via precessing (Fargion & Grossi
2006; Zou & Dai 2006). In almost all GRB progenitor models, the central engine
is a rapidly rotating compact objects, such as a black hole or a neutron star.
The outflows accounting for the GRB are usually launched along the rotating
axis or the magnetic axis. If the nozzle of the central engine is precessing
while ejecting the outflow, then a ring-shaped jet will be naturally produced.
As an example, if GRBs are associated with the kick of neutron stars (Huang et
al. 2003), then the production of ring-shaped jet will be quite likely.
Alternatively, the apparent ring-shaped topology of the outflow can also arise
from a structured jet as suggested by van Putten & Levinson (2003) and van
Putten & Gupta (2009). This mechanism can explain the correlation between the
observed luminosity and variability in the light curves of long GRBs reported
by Reichert et al. (2001).
A detailed sketch of the ring-shaped jets has been presented in Xu et al.
(2008). Here we will adopt the notations there. The half opening angle of the
inner edge of the ring is denoted as $\theta_{\rm c}$, and the width of the
ring is $\Delta\theta$. Both the inner and outer edges may expand laterally,
so the inner edge will converge at the central axis and the ring-shaped jet
will finally become a conical jet after a period of time. In our calculations,
for simplicity, we assume that the line of sight is on the symmetry axis of
the centrally hollowed cone. Therefore, this scenario is more or less similar
to that of a normal off-axis jet at early stages, i.e. the line of sight is
not on the emitting outflow. A rebrightening is then expected to present in
the early afterglow phase, thanks to the deceleration and lateral expansion of
the ring-shaped jet.
Since our line of sight is “off-axis” initially, a natural problem is whether
the ring-shaped jet could produce the intensive $\gamma$-ray radiation in the
prompt GRB phase, similar to the problem confronted by a normal off-axis jet
model. Below, we show that the ring-shaped jet model has more advantages
against the off-axis jet model as long as this problem is concerned. The key
point is that all the material on a ring can contribute equally to the flux on
the central axis.
Assuming that the bulk Lorentz factor of a highly relativistic point-like jet
is $\gamma\gg 1$, and that it is viewed from an angle $\Theta$ relative to its
motion, then the observed flux will be amplified by a factor of $\sim D^{3}$,
due to the effect of relativistic beaming and Doppler boosting. Here the
Doppler factor is $D=[\gamma(1-\beta cos\Theta)]^{-1}$, with
$\beta=\sqrt{1-1/\gamma^{2}}$. For an extended outflow considered in our
framework, we should integrate over all the surface of the ring-shaped jet to
get the exact factor of amplification for the prompt GRB emission.
We first consider a normal conical jet with a full opening angle of
$\Delta\theta$ and a viewing angle of $\theta_{\rm c}+\frac{1}{2}\Delta\theta$
(similar to the viewing angle of our ring-shaped jet scenario). The observed
emission should be amplified by a factor of $f_{\rm j}=\int_{\theta_{\rm
c}}^{\theta_{\rm c}+\Delta\theta}D^{3}\phi(\Theta)d\Theta$ relative to that in
the jet rest frame, where $\phi(\Theta)$ is the toroidal angle given by
$\phi(\Theta)=2\arccos[\frac{\cos(\Delta\theta/2)-\cos(\theta_{\rm
c}+\Delta\theta/2)\cos\Theta}{\sin(\theta_{\rm
c}+\Delta\theta/2)\sin\Theta}].$ (1)
For simplicity, for a normal conical GRB jet with a full opening angle of
$\Delta\theta$ but with the viewing angle being $\sim 1/\gamma$, we define the
corresponding amplification factor as $f_{\rm o}$. Actually, $f_{\rm o}$ is a
measure of the amplification factor for an on-axis observer.
Now return to the ring-shaped jet considered in our framework. The
amplification factor for an observer on the central symmetry axis is $f_{\rm
r}=\int_{\theta_{\rm c}}^{\theta_{\rm
c}+\Delta\theta}D^{3}\phi(\Theta)d\Theta$, where $\phi(\Theta)=2\pi$. To get a
direct impression on this matter, let us simply take $\gamma=50,\theta_{\rm
c}=0.04$, $\Delta\theta=0.04$ as an example, and calculate the final
amplification factors. In this example, the viewing angle, i.e. the angle
between the jet axis and the line of sight, equals $3/\gamma$. We can easily
find that $f_{\rm o/r}\equiv f_{\rm o}/f_{\rm r}\approx 68.5$. It means that
the observed flux of our ring-shaped jet is only tens of times less than that
of an on-axis jet in the prompt GRB phase. At the same time, we can also find
that $f_{\rm r/j}\equiv f_{\rm r}/f_{\rm j}\approx 12.4$. It tells us that the
observed flux of our ring-shaped jet is more than 12 times larger than that of
an off-axis conical jet. So, the ring-shaped jet model is superior over the
normal off-axis conical jet model when considering the production of the
intensive $\gamma$-ray radiation in the prompt GRB phase.
## 3 Numerical Results
The overall dynamical evolution of a ring-shaped jet has been studied by Xu et
al. (2008) in detail. Generally, if the line of sight is on the central
symmetry axis of the hollow cone, a rebrightening phase should present in the
early afterglow light curve. Here, for completeness, we simply outline a few
important ingredients of the calculation. First, the dynamical equations
should be appropriate for both the ultra-relativistic and non-relativistic
stages. Recently, van Eerten et al. (2010) developed an accurate and also
correspondingly complex code for the dynamical evolution of GRB afterglows.
Here, we will use the simple and convenient equations proposed by Huang et al.
(1999). Secondly, for a ring-shaped jet, both the inner and outer edges may
expand laterally. When the inner edge converges at the central symmetry axis,
the ring-shaped jet will become a normal conical jet, and then we only need to
consider the sideways expansion of the outer edge. We suppose that the lateral
expansion is at the comoving sound speed ($c_{\rm s}$) approximately given by
$c_{\rm
s}^{2}=\hat{\gamma}(\hat{\gamma}-1)(\gamma-1)\frac{1}{1+\hat{\gamma}(\gamma-1)}c^{2},$
(2)
where $\hat{\gamma}\approx(4\gamma+1)/(3\gamma)$ is the adiabatic index (Dai
et al. 1999). We have $\hat{\gamma}\sim 4/3$ and $c_{s}=c/\sqrt{3}$ in
ultrarelativistic limit, and $\hat{\gamma}\sim 5/3$ and $c_{s}=\sqrt{5/9}\beta
c$ in nonrelativistic limit. The effect of $c_{\rm s}$ on the light curve of a
ring-shaped jet has been discussed by Xu et al. (2008) in detail. Generally
speaking, lateral expansion tends to make the light curve steeper and leads to
an earlier jet break. Thirdly, to calculate the afterglow flux, we integrate
the emission over the equal arrival time surface defined by
$\int\frac{1-\beta\cos\Theta}{\beta c}dR\equiv t,$ (3)
within the jet boundaries (Huang et al. 2000).
In this section, we will calculate the overall dynamical evolution and
multiband afterglow of ring-shaped jets, and compare the numerical results
with the observations of some $Swift$ GRBs with early rebrightenings, such as
GRBs 051016B, 060109, 070103, and 070208. In our calculations, we assume a
standard cosmology with $\Omega_{\rm M}=0.27$, $\Omega_{\Lambda}=0.73$, and
with the Hubble constant of $H_{0}=71\leavevmode\nobreak\ {\rm
km}\leavevmode\nobreak\ {\rm s}^{-1}{\rm Mpc}^{-3}$.
### 3.1 GRB 051016B
GRB 051016B was triggered and located as a soft burst by $Swift$-BAT at
18:28:09 UT on October 16, 2005 (Parsons et al. 2005). The light curve of its
prompt emission is shown in Figure 1a. Its duration is $T_{90}=4.0\pm 0.1$s.
The photon index in $15-150$ keV is $2.38\pm 0.23$, and the fluence is
$(1.7\pm 0.2)\times 10^{-7}{\rm erg\leavevmode\nobreak\ cm}^{-2}$ (Barbier et
al. 2005). Lying at a redshift of $z=0.9364$ (Soderberg et al 2005), the
isotropic-equivalent $\gamma$-ray energy release is $E_{\rm\gamma,iso}\sim
7.6\times 10^{50}$ erg. The light curve of early X-ray afterglow shows a
rising phase beginning at about 200s, and the rebrightening lasts for almost
one thousand seconds.
Using the ring-shaped jet model, we have fit the observed X-ray and optical
afterglow light curves of GRB 051016B numerically. The result is presented in
Figure 2, and the parameter values are given in Table 1. The observed X-ray
afterglow data are relatively abundant. We see that our model can reproduce
the rebrightening and the overall X-ray afterglow light curve satisfactorily.
Note that in the very early stage ($t\leq 200$ s), the afterglow may still be
in the steep decay phase and the flux should be dominant by the contribution
from the prompt tail emission. We thus have excluded these early data points
from our modeling. In Figure 2, the observed R-band optical afterglow light
curve is built up only by two data points, but they are in good agreement with
our prediction. From Table 1, we see that the ratio of $f_{\rm r/j}$ is about
5.8. It means that in the prompt GRB phase, the $\gamma$-ray flux of our model
is about six times more than that of a corresponding off-axis conical jet. So,
the ring-shaped jet model is better than the off-axis conical jet model when
considering the production of the intensive $\gamma$-ray radiation in the main
burst phase.
Figure 1: Prompt $\gamma$-ray light curves of GRBs 051016B, 060109, 070103,
and 070208 as observed by $Swift-BAT$ (15—350keV). The data are taken from the
$Swift$ website111$http://astro.berkeley.edu/\sim nat/swift/$ (Butler &
Kocevski 2007; Butler et al. 2007). Figure 2: Our best fit to the X-ray
(solid line) and R-band optical (dashed line) afterglow light curves of GRB
051016B by using the ring-shaped jet model. The square data points are
observed X-ray afterglow by $Swift$-XRT (see the $Swift$ website) and the
triangle data points are observed R-band afterglow (Chen et al. 2005; Sharapov
et al. 2005).
### 3.2 GRB 060109
At 16:54:41 UT on January 9, 2006, $Swift$-BAT triggered and located GRB
060109 (De Pasquale et al. 2006). The light curve of its prompt emission is
shown in Figure 1b. Its duration in 15 — 350 keV is $T_{90}=116\pm 3$ s, and
the fluence in 15 — 150 keV band is $(6.4\pm 1.0)\times 10^{-7}{\rm
erg\leavevmode\nobreak\ cm}^{-2}$ (Palmer et al. 2006). No redshift is
measured for this event. If assuming a typical redshift of $z=1$, then the
isotropic-equivalent $\gamma$-ray energy release is $E_{\rm\gamma,iso}\sim
3.74\times 10^{51}$ erg. In the early X-ray afterglow, the light curve shows a
rebrightening that begins in less than 1000 s and lasts for several thousand
seconds.
Figure 3 illustrates our fit to the observed X-ray afterglow of GRB 060109 by
using the the ring-shaped jet model. The parameters involved are given in
Table 1. In our modeling, again we have omitted the observed steep decaying
phase. We see that the theoretical light curve matches with the observational
data very well. The observed flux of the ring-shaped jet is about 10 times
larger than that of a corresponding off-axis conical jet ($f_{\rm r/j}\sim
10.4$, see Table 1). The ratio of $f_{\rm o/r}$ is $\sim 48.5$. It rougly
equals to $E_{\rm K,iso}/E_{\gamma,iso}$, which is about 18 as can be derived
from Table 1. This proves that our modeling is self-consistent.
Figure 3: Our best fit to the X-ray afterglow of GRB 060109 by using the ring-
shaped jet model. The square data points are observed by $Swift$-XRT (see the
$Swift$ website).
### 3.3 GRB 070103
At 20:46:39.41 UT on 2007 January 3, $Swift$-BAT triggered and located GRB
070103 (Sakamoto et al. 2007). The light curve of its prompt emission is shown
in Figure 1c. The duration in 15 — 350 keV is $T_{90}=19\pm 1$ s, and the
fluence in 15 — 150 keV is $(3.4\pm 0.5)\times 10^{-7}{\rm
erg\leavevmode\nobreak\ cm}^{-2}$ (Barbier et al. 2007). The isotropic-
equivalent $\gamma$-ray energy release is $E_{\rm\gamma,iso}\sim 1.8\times
10^{51}$ erg, assuming that the redshift is $z=1$. In the early X-ray
afterglow, the light curve shows a rebrightening that begins at about 200 s
and lasts for more than one thousand seconds.
In Figure 4, we illustrate our best fit to the X-ray afterglow of GRB 070103
by using the ring-shaped jet model, with the involving parameters given in
Table 1. Generally, the observed rebrightening and the overall light curve can
be well explained. According to Table 1, the value of $f_{\rm r/j}$ is about
5.3 . Note that the parameter of $f_{\rm o/r}$ is about 52.2, and the ratio of
$E_{\rm K,iso}/E_{\gamma,iso}$ is about 22. Again these two values are roughly
consistent with each other.
Figure 4: Our best fit to the X-ray afterglow of GRB 070103 by using the ring-
shaped jet model. The square data points are observed by $Swift$-XRT (see the
$Swift$ website).
### 3.4 GRB 070208
GRB 070208 was triggered and located by $Swift$-BAT at 09:10:34 UT on February
8, 2007 (Sato et al. 2007). The light curve of its prompt emission is shown in
Figure 1d. The duration in 15 — 350 keV is $T_{90}=48\pm 2$ s (Markwardt et
al. 2007), and the fluence in 15 — 150 keV is $(4.3\pm 1.0)\times 10^{-7}{\rm
erg\leavevmode\nobreak\ cm}^{-2}$. Lying at a redshift of $z=1.165$ (Cucchiara
et al. 2007), the isotropic-equivalent $\gamma$-ray energy release is
$E_{\rm\gamma,iso}\sim 3.32\times 10^{51}$ erg. The X-ray afterglow light
curve shows a rebrightening that begins at about 200 s and lasts for more than
one thousand seconds (Conciatore et al. 2007).
Using the ring-shaped jet model, we have tried to fit the multiband afterglow
of GRB 070208. The result is presented in Figure 5, with the parameters given
in Table 1. The observed early rebrightening in the X-ray afterglow can be
well reproduced. The optical afterglow can also be simultaneously explained.
Table 1 indicates that the radiation intensity of the ring-shaped jet is about
8 times more than that of a corresponding off-axis conical jet (i.e., $f_{\rm
r/j}\sim 7.7$). Note that the value of $f_{\rm o/r}$ is about 61. It again
agrees well with the ratio of $E_{K,iso}/E_{\rm\gamma,iso}$, which is about 24
in our modeling.
Figure 5: Our best fit to the X-ray (solid line) and R-band optical (dashed
line) afterglow light curves of GRB 070208 by using the ring-shaped jet model.
The square data points are observed X-ray afterglow by $Swift$-XRT (see the
$Swift$ website) and the triangle data points are observed R-band afterglow
(Cenko et al. 2009; Halpern & Miraba 2007).
## 4 Conclusion and Discussion
It is possible that GRBs may be produced by ring-shaped jets. Interestingly, a
few recent hydrodynamical and magnetohydrodymical simulations also give some
supports to this idea (Aloy & Rezzolla 2006; Mizuno et al. 2008). A notable
feature of the afterglow of a ring-shaped jet is that a rebrightening can be
observed in the early afterglow stage, assuming that the line of sight is
within the central hollow cone. In this study, we have clearly shown that the
early rebrightenings observed in a few GRBs, such as GRBs 051016B, 060109,
070103, and 070208, can be well explained by the ring-shaped jet model. In the
case of GRBs 060109 and 070103, of which only X-ray afterglow data are
available, the fit result is very good. For GRBs 051016B and 070208, of which
both the X-ray and optical afterglow data are available, the interpretation is
also satisfactory. We propose that these GRBs with long-lasting rebrightening
in the early afterglow should be produced by ring-shaped jets.
Comparing with a normal off-axis conical jet model, the advantage of our ring-
shaped jet model is obvious. For a normal conical jet, when the observer is
off-axis, the observed brightness will be notably reduced (as compared to the
on-axis case). It then has difficulty in explaining the intensive $\gamma$-ray
emission observed in the prompt GRB phase. On the contrary, in the case of a
ring-shaped jet, all the material on the whole ring can contribute to the
emission on the line of sight. The observed intensity then can be
significantly enhanced as compared to that of an off-axis conical jet. In
fact, for the 4 GRBs studied here, the amplification factor ($f_{\rm r/j}$) is
generally 5 — 10, comparing with the corresponding off-axis conical jet.
From Figure 1, we interestingly notice that all the four GRBs show similar
2-pulse behavior in the prompt $\gamma$-ray light curve. This behavior is also
observed in some other Swift GRBs (Butler et al. 2007). It is quite unclear
whether this 2-pulse behavior is intrinsic to the mechanism that produces the
ring-shaped jet or not. Huang et al. (2003) have argued that if the jet is
produced by rapid precessing process (with the precession period much less
than the GRB duration), then the prompt $\gamma$-ray light curve could be
highly variable and very complicated. It then seems that these four GRBs might
not be due to normal fast precession. However, a slow precession may still be
possible. The structured jet mechanism by van Putten & Gupta (2009) is of
course another choise. Other possibilities include the evolution of an extreme
Kerr Black Hole surrounded by a precessing disk (Lei et al. 2007), or
explanation of the second pulse as the tail emission from the first pulse.
From Table 1, we see that the initial angular radius of the inner edge of the
ring is generally very small ($\theta_{\rm c}\sim$ 0.01 — 0.02). The width of
the ring is also small ($\Delta\theta\sim$ 0.02 — 0.08). Taking typical
parameter values of $\theta_{c}=0.015$ and $\Delta\theta=0.04$, we can give a
rough estimate for the observed event rate of early rebrightenings. Among all
Swift GRBs, the fraction of well-monitored afterglows with potential jet-
breaks is around $60\%$ (Panaitescu 2007). For most of these jet candidates,
the outflows might be normal conical jets, but it is quite likely that a small
portion (here we take the ratio as 15%, see Xu et al. 2008) of the jets are
ring-shaped ones. For a ring-shaped jet with $\theta_{c}=0.015$ and
$\Delta\theta=0.04$, the possibility that our line of sight is within the
central hollow cone (as compared with the possibility that the line of sight
is just on the ring) is about 0.08 (calculated from
$(1-cos\theta_{c})/[(1-cos(\theta_{c}+\Delta\theta))-(1-cos\theta_{c})]$).
Final, we estimate that the fraction of Swift events predicted by our model to
display early rebrightening is $0.6\times 0.15\times 0.08\sim 0.7\%$. This
number is consistent with the observed fraction of Swift events that show
early X-ray rebrightening ($<1\%$).
The angular information may provide useful clues on the central engine of
GRBs. In many progenitor models of GRBs, the central engine is a rapidly
rotating compact star and the outflow is ejected along the magnetic axis. Our
study then indicates that the inclination angle of the magnetic axis with
respect to the rotating axis should be small ($\sim$ 0.02 — 0.06). In a few
other progenitor models, the central engine is also a rapidly rotating compact
star, but the outflow is ejected just along the rotating axis, then our
results suggest that the processing (angular) radius of the rotating axis
should be $\sim$ 0.02 — 0.06.
In our framework, since the rebrightening is mainly a geometric effect, it
should generally be achromatic, i.e. the rebrightening should appear
simultaneously in all the X-ray and optical bands. For the four GRBs studied
here, the X-ray afterglow data are relatively prolific. But the optical
afterglow data are generally so lacking that almost no rebrightening can be
discerned in the optical light curves. Anyway, we have shown that our model is
consistent with both the X-ray and optical observations. In the future, more
examples with abundant multiband afterglow data will be available, and tighter
constraints on the existence of ring-shaped jet could be derived.
In our model, we assume the ring-shaped jet is uniform. But note that the
actual structure of the GRB outflow may be very complicate. For example, it
may be two-component jet as predicted by some engine models (e.g. van Putten &
Levinson 2003). If these ingredients are included, then the afterglow behavior
of ring-shaped jets will be correspondingly much more complicated.
###### Acknowledgements.
We thank Fayin Wang and Yang Guo for helpful discussion. We also would like to
thank the anonymous referee for useful comments and suggestions that led to an
overall improvement of this manuscript. This work was supported by the
National Natural Science Foundation of China (Grant No. 10625313) and the
National Basic Research Program of China (973 Program, Grant No.
2009CB824800).
## References
* Aloy & Rezzolla (2006) Aloy, M. A., & Rezzolla, L., 2006, ApJ, 640, L115
* Barbieret al. (2005) Barbier, L., et al. 2005, GCN 4104
* Barbieret al. (2007) Barbier, L., et al. 2007, GCN 5991
* Butler & Kocevski (2007) Butler, N., & Kocevski, D., 2007, ApJ, 663, 407
* Butler (2007) Butler, N., Kocevski, D., et al. 2007, ApJ, 671, 656
* Cenkoet al. (2009) Cenko, S. B., Kelemen, J. F., Harrison, A., et al. 2009, ApJ, 693, 1484
* Chenet al. (2005) Chen, Y. T., Wu, Y. L., Huang, K-Y., et al. 2005, GCN 4105
* Chincariniet al. (2007) Chincarini, G., et al. 2007, ApJ, 671, 1903
* Conciatoreet al. (2007) Conciatore, M. L., et al. 2007, GCN 6085
* Cucchiaraet al. (2007) Cucchiara, A., et al. 2007, GCN 6083
* Daiet al. (1999) Dai, Z. G., Huang, Y. F., & Lu, T., 1999, ApJ, 520, 634
* Dai & Lu (1998) Dai, Z. G., & Lu, T., 1998, A&A, 333, L87
* De Pasqualeet al. (2006) De Pasquale, M., et al. 2006, GCN 4455
* de Ugarte Postigoet al. (2007) de Ugarte Postigo, A., et al. 2007, A&A, 462, L57
* Fargion & Grossi (2006) Fargion, D., & Grossi, M., 2006, Chin. J. Astron. Astrophys. (ChJAA), 6, 342
* Gehrelset al. (2004) Gehrels, N., et al. 2004, ApJ, 611, 1005
* Gehrelset al. (2009) Gehrels, N., Ramirez-Ruiz, E., & Fox, D. B., 2009, ARA&A, 47, 567
* Granot (2005) Granot, J., 2005, ApJ, 631, 1022
* Halpernet al. (2007) Halpern, J. P., & Miraba, N., 2007, GCN 6082
* (20) Huang, Y. F., & Cheng, K. S., 2003, MNRAS, 341, 263
* Huanget al. (2006) Huang, Y. F., Cheng, K. S., & Gao, T. T., 2006, ApJ, 637, 873
* Huanget al. (1999) Huang, Y. F., Dai, Z. G., & Lu, T., 1999, MNRAS, 309, 513
* Huanget al. (2003) Huang, Y. F., Dai, Z. G., Lu, T., Cheng, K. S., & Wu, X. F., 2003, ApJ, 594, 919
* Huanget al. (2000) Huang, Y. F., Gou, L. J., Dai, Z. G., & Lu, T., 2000, ApJ, 543, 90
* Kocevski & Butler (2008) Kocevski, D., & Butler, N., 2008, ApJ, 680, 531
* Konget al. (2010) Kong, S. W., et al. 2010, MNRAS, 402, 409
* Leiet al. (2007) Lei, W. H., Wang, D. X., Gong, B. P., & Huang, C. Y., 2007, A&A, 468, 563
* Lipkinet al. (2004) Lipkin, Y. M., et al. 2004, ApJ, 606, 381
* Marguttiet al. (2010) Margutti, R., et al. 2009, MNRAS, 402, 46
* Markwardtet al. (2007) Markwardt, C., et al. 2007, GCN 6081
* Mizunoet al. (2008) Mizuno, Y., Hardee, P., Hartmann, D., et al. 2008, ApJ, 672, 72
* Palmeret al. (2006) Palmer, D., et al. 2006, GCN 4476
* Panaitescu (2007) Panaitescu, A., 2007, MNRAS, 380, 374
* Parsonset al. (2005) Parsons, A., Barthelmy, S., et al. 2005, GCN 4103
* Piro et al. (1998) Piro, L., et al. 1998, A & A, 331, L4
* Ramirez-Ruizet al. (2005) Ramirez-Ruiz, E., et al. 2005, ApJ, 625, L91
* Reichertet al. (2001) Reichert, D. E., Lamb, D. Q., Fenimore E. E., et al. 2001, ApJ, 552, 57
* Sakamotoet al. (2007) Sakamoto, T., et al. 2007, GCN 5988
* Satoet al. (2007) Sato, G., et al. 2007, GCN 6074
* Sharapovet al. (2005) Sharapov, D., Ibrahimov, M., Pozanenko, A., et al. 2005, GCN 4185
* Soderberget al. (2005) Soderberg, A. M., et al. 2005, GCN 4186
* van Eertenet al. (2010) van Eerten, H. J., Leventis, K., Meliani, Z., Wijers, R. A. M. J., Keppens, R. 2010, MNRAS, 403, 300
* van Putten & Levinson (2003) van Putten, M. H. P. M., & Levinson, A., 2003, ApJ, 584, 937
* van Putten & Gupta (2009) van Putten, M. H. P. M., & Gupta, A. C., 2009, MNRAS, 394, 2238
* Xuet al. (2008) Xu, M., Huang, Y. F., & Kong, S. W., 2008, Chin. J. Astron. Astrophys. (ChJAA), 8, 411
* Yu & Huang (2007) Yu, Y., & Huang, Y. F., 2007, Chin. J. Astron. Astrophys. (ChJAA), 7, 669
* Zhang (2007) Zhang, B., 2007, Chin. J. Astron. Astrophys. (ChJAA), 7, 1
* Zhang & Mészáros (2001) Zhang, B., & Mészáros, P., 2001, ApJ, 552, L35
* (49) Zou, Y. C., & Dai, Z. G., 2006, Chin. J. Astron. Astrophys. (ChJAA), 6, 551
Table 1: Parameters of the 4 GRBs
GRB Name | GRB 051016B | GRB 060109 | GRB 070103 | GRB 070208
---|---|---|---|---
z | $0.9364$ | $1$ | $1$ | $1.165$
$E_{\rm\gamma,iso}$ (erg) | $7.6\times 10^{50}$ | $3.37\times 10^{51}$ | $1.79\times 10^{51}$ | $3.32\times 10^{51}$
$E_{\rm K,iso}$ (erg) | $1.3\times 10^{52}$ | $6.0\times 10^{52}$ | $4.0\times 10^{52}$ | $8.0\times 10^{52}$
$\theta_{\rm c}$ (rad) | $0.01$ | $0.018$ | $0.01$ | $0.009$
$\Delta\theta$ (rad) | $0.07$ | $0.024$ | $0.08$ | $0.022$
$\gamma$ | $180$ | $90$ | $120$ | $150$
n $(cm^{-3})$ | $0.05$ | $0.08$ | $0.1$ | $0.1$
$\epsilon_{\rm e}$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
$\epsilon_{\rm B}$ | $5.0\times 10^{-4}$ | $1.0\times 10^{-4}$ | $1.0\times 10^{-4}$ | $1.0\times 10^{-4}$
$p$ | $2.05$ | $2.2$ | $2.75$ | $2.02$
$f_{\rm r/j}$ | $5.8$ | $10.4$ | $5.3$ | $7.7$
$f_{\rm o/r}$ | $228.7$ | $48.5$ | $52.2$ | $61.0$
* Notes: z: redshift; $E_{\rm\gamma,iso}$: the isotropic-equivalent $\gamma$-ray energy release observed by $Swift$; $E_{\rm K,iso}$: the isotropic-equivalent kinetic energy of the outflow used in our model; $\theta_{\rm c}$: the half opening angle of the inner edge of the ring; $\Delta\theta$: the width of the ring; $\gamma$: the bulk Lorentz factor; $n$: number density of the circum-burst medium; $\epsilon_{\rm e}$: the electron energy fraction; $\epsilon_{\rm B}$: the magnetic energy fraction; $p$: the power-law index of the energy distribution function of electrons; $f_{\rm r/j}$: $f_{\rm r}/f_{\rm j}$ (see Section 2 for details); $f_{\rm o/j}$: $f_{\rm o}/f_{\rm r}$ (see Section 2 for details).
|
arxiv-papers
| 2010-08-10T10:49:26 |
2024-09-04T02:49:12.126565
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Xu and Y.-F. Huang",
"submitter": "Ming Xu",
"url": "https://arxiv.org/abs/1008.1686"
}
|
1008.1916
|
# Local convergence analysis of inexact Gauss-Newton like
methods under majorant condition
O. P. Ferreira IME/UFG, Campus II- Caixa Postal 131, 74001-970 - Goiânia, GO,
Brazil (E-mail:orizon@mat.ufg.br). The author was partly supported by CNPq
Grant 473756/2009-9, CNPq Grant 302024/2008-5, PRONEX-
Optimization(FAPERJ/CNPq) and FUNAPE/UFG. M. L. N. Gonçalves COPPE-Sistemas,
Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, RJ, BR
(E-mail:maxlng@cos.ufrj.br). The author was partly supported by CNPq Grant
473756/2009-9. P. R. Oliveira COPPE-Sistemas, Universidade Federal do Rio de
Janeiro, 21945-970 Rio de Janeiro, RJ, BR (Email: poliveir@cos.ufrj.br). This
author was partly supported by CNPq.
###### Abstract
In this paper, we present a local convergence analysis of inexact Gauss-Newton
like methods for solving nonlinear least squares problems. Under the
hypothesis that the derivative of the function associated with the least
square problem satisfies a majorant condition, we obtain that the method is
well-defined and converges. Our analysis provides a clear relationship between
the majorant function and the function associated with the least square
problem. It also allows us to obtain an estimate of convergence ball for
inexact Gauss-Newton like methods and some important, special cases.
Keywords: Nonlinear least squares problems; inexact Gauss-Newton like methods;
Majorant condition; Local convergence.
## 1 Introduction
Let $\mathbb{X}$ and $\mathbb{Y}$ be real or complex Hilbert spaces. Let
$\Omega\subseteq\mathbb{X}$ be an open set, and $F:\Omega\to\mathbb{Y}$ a
continuously differentiable nonlinear function. Consider the following
nonlinear least squares problems
$\min_{x\in\Omega}\;\|F(x)\|^{2}.$ (1)
The interest in this problem arises in data fitting, when
$\mathbb{X}=\mathbb{R}^{n}$ and $\mathbb{Y}=\mathbb{R}^{m}$ and $m$ is the
number of observations and $n$ is the number of parameters, see for example
[13]. A solution $x_{*}\in\Omega$ of (1) is also called a least-squares
solution of nonlinear equation $F(x)=0.$
When $F^{\prime}(x)$ is injective and has closed image for all $x\in\Omega$,
the Gauss-Newton’s method finds stationary points of the above problem.
Formally, the Gauss-Newton’s method is described as follows: Given an initial
point $x_{0}\in\Omega$, define
$x_{k+1}={x_{k}}+S_{k},\qquad
F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k}),\qquad
k=0,1,\ldots,$
where $A^{*}$ denotes the adjoint of the operator $A$. It is worth pointing
out that if $x_{*}$ is solution of (1), $F(x_{*})=0$ and $F^{\prime}(x_{*})$
is invertible, then the theories of the Gauss-Newton’s method merge into the
theories of Newton’s method. Early works dealing with the convergence of
Newton’s and Gauss-Newton’s methods include [1, 2, 3, 6, 8, 9, 10, 11, 14, 16,
17, 19, 20, 21, 24, 25, 28].
The inexact Gauss-Newton process is described as follows: Given an initial
point $x_{0}\in\Omega$, define
$x_{k+1}=x_{k}+S_{k},\qquad{k=0,1,...,}$
where $B_{k}:\mathbb{X}\to\mathbb{Y}$ is a linear operator and $S_{k}$ is any
approximated solution of the linear system
${B_{k}S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k})+r_{k},}$
for a suitable residual $r_{k}\in\mathbb{Y}$. In particular, the above process
is inexact Gauss-Newton method if
$B_{k}=F^{\prime}(x_{k})^{T}F^{\prime}(x_{k}),$ the process is inexact
modified Gauss-Newton method if
$B_{k}=F^{\prime}(x_{0})^{T}F^{\prime}(x_{0}),$ and it represents a inexact
Gauss-Newton like method if $B_{k}$ is an approximation of
$F^{\prime}(x_{k})^{T}F^{\prime}(x_{k}).$
For inexact Newton methods, as shown in [12], if
$\|r_{k}\|\leq\theta_{k}\|F(x_{k})\|$ for $k=0,1,\ldots$ and
$\\{\theta_{k}\\}$ is a sequence of forcing terms such that
$0\leq\theta_{k}<1$ then there exists $\epsilon>0$ such that the sequence
$\\{x_{k}\\}$, for any initial point $x_{0}\in
B(x_{*},\epsilon)=\\{x\in\mathbb{R}^{n}:\;\|x_{*}-x\|<\epsilon\\}$, is well
defined and converges linearly to $x_{*}$ in the norm
$\|y\|_{*}=\|F^{\prime}(x_{*})y\|$, where $\|\;\|$ is any norm in
$\mathbb{R}^{n}$. As pointed out by [22] (see also [23]) the result of [12] is
difficult to apply due to a dependence of the norm $\|\;\|_{*}$, which is not
computable.
Formally, the inexact Gauss-Newton like methods for solving (1), which we will
consider, are described as follows: Given an initial point $x_{0}\in{\Omega}$,
define
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k})+r_{k},\qquad k=0,1,\ldots,$
where $B(x_{k})$ is a suitable invertible approximation of the derivative
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ and the residual tolerance $r_{k}$
and the preconditioning invertible matrix $P_{k}$ (considered for the first
time in [23]) for the linear system defining the step $S_{k}$ satisfy
$\|P_{k}r_{k}\|\leq\theta_{k}\|P_{k}F^{\prime}(x_{k})^{*}F(x_{k})\|,$
for suitable forcing number $\theta_{k}$. Note that, if the forcing sequence
vanishes, i.e., $\theta_{k}=0$ for all $k,$ the inexact Gauss-Newton methods
include the class of Gauss-Newton iterative methods. Hence, the theories of
inexact Gauss-Newton methods merge into the theories of Gauss-Newton methods.
The classical local convergence analysis for the inexact Newton’s methods (see
[12, 23]) requires, among other hypotheses, that $F^{\prime}$ satisfies the
Lipschitz condition. In the last years, there have been papers dealing with
the issue of convergence of the Newton method and inexact Newton’s method,
including the Gauss-Newton’s method and inexact Gauss-Newton’s method, by
relaxing the assumption of Lipschitz continuity of the derivative (see for
example: [5, 6, 7, 9, 14, 15, 16, 17, 21, 28]). One of the main conditions
that relaxes the condition of the Lipschitz continuity of the derivative is
the majorant condition, which we will use, and Wang’s condition, introduced in
[28] and used in [5, 6, 7, 8, 20, 21] to study the Gauss-Newton’s and Newton’s
methods. In fact, it can be shown that these conditions are equivalent. But
the formulation as a majorant condition is in some sense better than Wang’s
condition, as it provides a clear relationship between the majorant function
and the nonlinear function under consideration. Besides, the majorant
condition provides a simpler proof of convergence.
In the present paper, we are interested in the local convergence analysis,
i.e., based on the information in a neighbourhood of a stationary point of (1)
we determine the convergence ball of the method. Following the ideas of [14,
15, 16, 17], we will present a new local convergence analysis for inexact
Gauss-Newton like methods under majorant condition. The convergence analysis
presented provides a clear relationship between the majorant function, which
relaxes the Lipschitz continuity of the derivate, and the function associated
with the nonlinear least square problem (see for example: Lemmas 12, 13 and
14). Besides, the results presented here have the conditions and the proof of
convergence in quite a simple manner. Moreover, two unrelated previous results
pertaining to inexact Gauss-Newton like methods are unified, namely, the
result for analytical functions and the classical one for functions with
Lipschitz derivative.
The organization of the paper is as follows. In Section 1.1, we list some
notations and basic results used in our presentation. In Section 2 the main
result is stated, and in Section 2.1 some properties involving the majorant
function are established. In Section 2.2 we present the relationships between
the majorant function and the non-linear function $F$. In Section 2.3 the main
result is proven and some applications of this result are given in Section 3.
Some final remarks are offered in Section 4.
### 1.1 Notation and auxiliary results
The following notations and results are used throughout our presentation. Let
$\mathbb{X}$ and $\mathbb{Y}$ be Hilbert spaces. The open and closed ball at
$a\in\mathbb{X}$ and radius $\delta>0$ are denoted, respectively by
$B(a,\delta):=\\{x\in\mathbb{X};\;\|x-a\|<\delta\\},\qquad
B[a,\delta]:=\\{x\in\mathbb{X};\;\|x-a\|\leqslant\delta\\}.$
The set $\Omega\subseteq\mathbb{X}$ is an open set and the function
$F:\Omega\to\mathbb{Y}$ is continuously differentiable, and $F^{\prime}(x)$
has closed image in $\Omega$.
Let $A:\mathbb{X}\to\mathbb{Y}$ be a continuous and injective linear operator
with closed image. The Moore-Penrose inverse
$A^{\dagger}:\mathbb{Y}\to\mathbb{X}$ of $A$ is defined by
$A^{\dagger}:=(A^{*}A)^{-1}A^{*},$
where $A^{*}$ denotes the adjoint of the linear operator $A$.
###### Lemma 1.
(Banach’s Lemma) Let $B:\mathbb{X}\to\mathbb{X}$ be a continuous linear
operator, and $\mbox{I}:\mathbb{X}\to\mathbb{X}$ the identity operator. If
$\|B-I\|<1$, then $B$ is invertible and $\|B^{-1}\|\leq
1/\left(1-\|B-I\|\right).$
###### Proof.
See the proof of Lemma 1, p. 189 of Smale [26] with $A=I$ and $c=\|B-I\|$. ∎
###### Lemma 2.
Let $A,B:\mathbb{X}\to\mathbb{Y}$ be a continuous linear operator with closed
image. If $A$ is injective, $E=B-A$ and $\|EA^{\dagger}\|<1$, then $B$ is
injective.
###### Proof.
In fact, $B=A+E=(I+EA^{\dagger})A,$ from the condition $\|EA^{\dagger}\|<1,$
we have of Lemma 1 that $I+EA^{\dagger}$ is invertible. So, $B$ is injective.
∎
The next lemma is proven in Stewart [27] ( see also, Wedin [29] ) for $m\times
n$ matrix with $m\geq n$ and $rank(A)=rank(B)=n$, that proof holds in more
general context as we will state below.
###### Lemma 3.
Let $A,B:\mathbb{X}\to\mathbb{Y}$ be continuous and injective linear operators
with closed images. Assume that $E=B-A$ and $\|A^{\dagger}\|\|E\|<1$, then
$\|B^{\dagger}\|\leq\frac{\|A^{\dagger}\|}{1-\|A^{\dagger}\|\|E\|},\qquad\|B^{\dagger}-A^{\dagger}\|\leq\frac{\sqrt{2}\|A^{\dagger}\|^{2}\|E\|}{1-\|A^{\dagger}\|\|E\|}.$
###### Proposition 4.
If $0\leq t<1$, then $\sum_{i=0}^{\infty}(i+2)(i+1)t^{i}=2/(1-t)^{3}.$
###### Proof.
Take $k=2$ in Lemma 3, pp. 161 of Blum, et al. [4]. ∎
Also, the following auxiliary results of elementary convex analysis will be
needed:
###### Proposition 5.
Let $R>0$. If $\varphi:[0,R)\to\mathbb{R}$ is convex, then
$D^{+}\varphi(0)={\lim}_{u\to
0+}\;\frac{\varphi(u)-\varphi(0)}{u}={\inf}_{0<u}\;\frac{\varphi(u)-\varphi(0)}{u}.\\\
$
###### Proof.
See Theorem 4.1.1 on pp. 21 of Hiriart-Urruty and Lemaréchal [18]. ∎
###### Proposition 6.
Let $\epsilon>0$ and $\tau\in[0,1]$. If
$\varphi:[0,\epsilon)\rightarrow\mathbb{R}$ is convex, then
$l:(0,\epsilon)\to\mathbb{R}$ defined by
$l(t)=\frac{\varphi(t)-\varphi(\tau t)}{t},$
is increasing.
###### Proof.
See Theorem 4.1.1 and Remark 4.1.2 on pp. 21 of Hiriart-Urruty and Lemaréchal
[18]. ∎
## 2 Local analysis for inexact Gauss-Newton like methods
In this section, we will state and prove a local theorem for inexact Gauss-
Newton like methods. Assuming that the function
$\Omega\ni x\mapsto F(x)^{*}F(x),$
has a point stationary $x_{*}$, we will, under mild conditions, prove that the
inexact Gauss-Newton like methods is well defined and that the generated
sequence converges linearly to this point stationary. The statement of the
theorem is as follows:
###### Theorem 7.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and
$c:=\|F(x_{*})\|,\qquad\beta:=\left\|F^{\prime}(x_{*})^{\dagger}\right\|,\qquad\kappa:=\sup\left\\{t\in[0,R):B(x_{*},t)\subset\Omega\right\\}.$
Suppose that $F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is
injective and there exists a $f:[0,\;R)\to\mathbb{R}$ continuously
differentiable such that
$\left\|F^{\prime}(x)-F^{\prime}(x_{*}+\tau(x-x_{*}))\right\|\leq
f^{\prime}\left(\|x-x_{*}\|\right)-f^{\prime}\left(\tau\|x-x_{*}\|\right),$
(2)
for all $\tau\in[0,1]$, $x\in B(x_{*},\kappa)$ and
* h1)
$f(0)=0$ and $f^{\prime}(0)=-1$;
* h2)
$f^{\prime}$ is convex and strictly increasing;
* h3)
$\alpha:=\sqrt{2}\,c\,\beta^{2}D^{+}f^{\prime}(0)<1$.
Take $0\leq\vartheta<1$, $0\leq\omega_{2}<\omega_{1}$ such that
$\omega_{1}(\alpha+\alpha\vartheta+\vartheta)+\omega_{2}<1$. Let the positive
constants
$\nu:=\sup\left\\{t\in[0,R):\beta[f^{\prime}(t)+1]<1\right\\},$
$\rho:=\sup\bigg{\\{}t\in(0,\nu):(1+\vartheta)\omega_{1}\beta\frac{tf^{\prime}(t)-f(t)+\sqrt{2}c\beta[f^{\prime}(t)+1]}{t[1-\beta(f^{\prime}(t)+1)]}+\omega_{1}\vartheta+\omega_{2}<1\bigg{\\}},\;r:=\min\left\\{\kappa,\,\rho\right\\}.$
Then, the inexact Gauss-Newton like methods for solving (1), with initial
point $x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k})+r_{k},\qquad\;k=0,1,\ldots,$ (3)
for the forcing term $\theta_{k}$ and the following conditions for the
residual $r_{k}$ and the invertible matrix $P_{k}$ preconditioning the linear
system in (3)
$\|P_{k}r_{k}\|\leq\theta_{k}\|P_{k}F^{\prime}(x_{k})^{*}F(x_{k})\|,\qquad
0\leq\theta_{k}\mbox{cond}(P_{k}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k}))\leq\vartheta,\qquad\;k=0,1,\ldots,$
where $B(x_{k})$ is an invertible approximation of
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ satisfying the following conditions
$\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})\|\leq\omega_{1},\qquad\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})-I\|\leq\omega_{2},\qquad\;k=0,1,\ldots,$
is well defined, contained in $B(x_{*},r)$, converges to $x_{*}$ and there
holds
$\|x_{k+1}-x_{*}\|\leq(1+\vartheta)\omega_{1}\beta\frac{[f^{\prime}(\|x_{0}-x_{*}\|)\|x_{0}-x_{*}\|-f(\|x_{0}-x_{*}\|)]}{\|x_{0}-x_{*}\|^{2}[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}{\|x_{k}-x_{*}\|}^{2}\\\
+\left(\frac{(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}[f^{\prime}(\|x_{0}-x_{*}\|)+1]}{\|x_{0}-x_{*}\|[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}+\omega_{1}\vartheta+\omega_{2}\right)\|x_{k}-x_{*}\|,\qquad
k=0,1,\ldots.$ (4)
###### Remark 1.
In particular, if taking $\vartheta=0$ (in this case $\theta_{k}\equiv 0$ and
$r_{k}\equiv 0$) in Theorem 7, we obtain the convergence of Gauss-Newton’s
like method under majorant condition which, for $\omega_{1}=1$ and
$\omega_{2}=0$, i.e., $B(x_{k})=F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$, has
been obtained by Ferreira et al. [16] in Theorem 7. Now, if taking $c=0$ (the
so-called zero-residual case) and $F^{\prime}(x_{*})$ is invertible, we obtain
the convergence of inexact Newton-Like methods under majorant condition, which
has been obtained by Ferreira, Gonçalves [15] in Theorem 4. Finally, if
$c=\vartheta=\omega_{2}=0$, $\omega_{1}=1$ and $F^{\prime}(x_{*})$ is
invertible in Theorem 7, we obtain the convergence of Newton’s method under
majorant condition, which has been obtained by Ferreira [14] in Theorem 2.1.
For the important case $\vartheta=0$, namely, Gauss-Newton’s like method under
majorant condition, the Theorem 7 becomes:
###### Corollary 8.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and
$c:=\|F(x_{*})\|,\qquad\beta:=\left\|F^{\prime}(x_{*})^{\dagger}\right\|,\qquad\kappa:=\sup\left\\{t\in[0,R):B(x_{*},t)\subset\Omega\right\\}.$
Suppose that $F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is
injective and there exists a $f:[0,\;R)\to\mathbb{R}$ continuously
differentiable such that
$\left\|F^{\prime}(x)-F^{\prime}(x_{*}+\tau(x-x_{*}))\right\|\leq
f^{\prime}\left(\|x-x_{*}\|\right)-f^{\prime}\left(\tau\|x-x_{*}\|\right),$
for all $\tau\in[0,1]$, $x\in B(x_{*},\kappa)$ and
* h1)
$f(0)=0$ and $f^{\prime}(0)=-1$;
* h2)
$f^{\prime}$ is convex and strictly increasing;
* h3)
$\alpha:=\sqrt{2}\,c\,\beta^{2}D^{+}f^{\prime}(0)<1$.
Take $0\leq\omega_{2}<\omega_{1}$ such that $\omega_{1}\alpha+\omega_{2}<1$.
Let $\nu:=\sup\left\\{t\in[0,R):\beta[f^{\prime}(t)+1]<1\right\\},$
$\rho:=\sup\bigg{\\{}t\in(0,\nu):\omega_{1}\beta\frac{tf^{\prime}(t)-f(t)+\sqrt{2}c\beta[f^{\prime}(t)+1]}{t[1-\beta(f^{\prime}(t)+1)]}+\omega_{2}<1\bigg{\\}},\quad
r:=\min\left\\{\kappa,\,\rho\right\\}.$
Then, the Gauss-Newton’s like method for solving (1), with initial point
$x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k}),\qquad\;k=0,1,\ldots,$
where $B(x_{k})$ is an invertible approximation of
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ satisfying
$\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})\|\leq\omega_{1},\qquad\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})-I\|\leq\omega_{2},\qquad\;k=0,1,\ldots,$
is well defined, contained in $B(x_{*},r)$, converges to $x_{*}$ and there
holds
$\|x_{k+1}-x_{*}\|\leq\omega_{1}\beta\frac{[f^{\prime}(\|x_{0}-x_{*}\|)\|x_{0}-x_{*}\|-f(\|x_{0}-x_{*}\|)]}{\|x_{0}-x_{*}\|^{2}[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}{\|x_{k}-x_{*}\|}^{2}\\\
+\left(\frac{\omega_{1}\sqrt{2}c\beta^{2}[f^{\prime}(\|x_{0}-x_{*}\|)+1]}{\|x_{0}-x_{*}\|[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}+\omega_{2}\right)\|x_{k}-x_{*}\|,\qquad
k=0,1,\ldots.$ (5)
###### Remark 2.
Despite the fact that the above corollary is a special case of Theorem 7, the
results contained therein extend the results of Chen and Li in [8], as the
results obtained [8] are only for the case $c=0.$
###### Remark 3.
Assumption (2) is crucial for our analysis. It should be pointed that, under
appropriate regularity conditions in the nonlinear function $F$, assumption
(2) always holds on a suitable neighbourhood of $x_{*}$. For instance, if $F$
is two times continuously differentiable, then the majorant function
$f:[0,\kappa)\to\mathbb{R}$, as defined by $f(t)=Kt^{2}/2-t,$ where
$K=\sup\\{\|F^{\prime\prime}(x)\|:x\in B[x_{*},\kappa]\\}$ satisfies
assumption (2). Estimating the constant $K$ is a very difficult problem.
Therefore, the goal is to identify classes of nonlinear functions for which it
is possible to obtain a majorant function. We will give some examples of such
classes in Section 3.
To prove Theorem 7 we need some results. From here on, we assume that all
assumptions of Theorem 7 hold.
### 2.1 The majorant function
In this section, we will prove that the constant $\kappa$ associated with
$\Omega$ and the constants $\nu$, $\rho$ and $r$ associated with the majorant
function $f$ are positive. We will also prove some results related to the
function $f$.
We begin by noting that $\kappa>0$, because $\Omega$ is an open set and
$x_{*}\in\Omega$.
###### Proposition 9.
The constant $\nu$ is positive and there holds
$\beta[f^{\prime}(t)+1]<1,\qquad t\in(0,\nu).$
###### Proof.
As $f^{\prime}$ is continuous in $(0,R)$ and $f^{\prime}(0)=-1,$ it is easy to
conclude that
$\lim_{t\to 0}\beta[f^{\prime}(t)+1]=0.$
Thus, there exists a $\delta>0$ such that $\beta(f^{\prime}(t)+1)<1$ for all
$t\in(0,\delta)$. Hence, $\nu>0.$
Using h2 and definition of $\nu$ the last part of the proposition follows.
∎
###### Proposition 10.
The following functions are increasing:
* i)
$[0,\,R)\ni t\mapsto 1/[1-\beta(f^{\prime}(t)+1)];$
* ii)
$(0,\,R)\ni t\mapsto[tf^{\prime}(t)-f(t)]/t^{2};$
* iii)
$(0,\,R)\ni t\mapsto[f^{\prime}(t)+1]/t;$
As a consequence, there is an increase of the following functions
$(0,\,R)\ni
t\mapsto\frac{tf^{\prime}(t)-f(t)}{t^{2}[1-\beta(f^{\prime}(t)+1)]},\qquad\qquad(0,\,R)\ni
t\mapsto\frac{f^{\prime}(t)+1}{t[1-\beta(f^{\prime}(t)+1)]}.$
###### Proof.
The item i is immediate, because $f^{\prime}$ is strictly increasing in
$[0,R)$.
For proving item ii, note that after some simple algebraic manipulations we
have
$\frac{tf^{\prime}(t)-f(t)}{t^{2}}=\int_{0}^{1}\frac{f^{\prime}(t)-f^{\prime}(\tau
t)}{t}\;d\tau.$
So, applying Proposition 6 with $f^{\prime}=\varphi$ and $\epsilon=R$ the
statement follows.
For establishing item iii use $\bf h2$, $f^{\prime}(0)=-1$ and Proposition 6
with $f^{\prime}=\varphi,$ $\epsilon=R$ and $\tau=0.$
To prove that the functions in the last part are increasing, combine item i
with ii for the first function, and i with iii for the second function. ∎
###### Proposition 11.
The constant $\rho$ is positive and there holds
$(1+\vartheta)\omega_{1}\beta\frac{tf^{\prime}(t)-f(t)+\sqrt{2}c\beta[f^{\prime}(t)+1]}{t[1-\beta(f^{\prime}(t)+1)]}+\omega_{1}\vartheta+\omega_{2}<1,\qquad\forall\;t\in(0,\,\rho).$
###### Proof.
First of all, note that the assumption $\bf h1$ implies, after simple
calculation, that
$\lim_{t\to
0}\frac{tf^{\prime}(t)-f(t)}{t[1-\beta(f^{\prime}(t)+1)]}=\lim_{t\to
0}\frac{f^{\prime}(t)-(f(t)-f(0))/t}{1-\beta(f^{\prime}(t)+1)}=0.$
Again, using $\bf h1$, some algebraic manipulation and that $f^{\prime}$ is
convex, we have by Proposition 5
$\lim_{t\to 0}\frac{f^{\prime}(t)+1}{t[1-\beta(f^{\prime}(t)+1)]}=\lim_{t\to
0}\frac{(f^{\prime}(t)-f^{\prime}(0))/t}{1-\beta(f^{\prime}(t)+1)}=D^{+}f^{\prime}(0).$
Hence, by combining the two above equalities it is easy to conclude that
$\lim_{t\to
0}(1+\vartheta)\omega_{1}\beta\frac{tf^{\prime}(t)-f(t)+\sqrt{2}c\beta[f^{\prime}(t)+1]}{t[1-\beta(f^{\prime}(t)+1)]}+\omega_{1}\vartheta+\omega_{2}=(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}D^{+}f^{\prime}(0)+\omega_{1}\vartheta+\omega_{2}.$
As, $\alpha=\sqrt{2}c\beta^{2}D^{+}f^{\prime}(0)$ and
$\omega_{1}(\alpha+\alpha\vartheta+\vartheta)+\omega_{2}<1$, we obtain that
there exists a $\delta>0$ such that
$(1+\vartheta)\omega_{1}\beta\frac{tf^{\prime}(t)-f(t)+\sqrt{2}c\beta[f^{\prime}(t)+1]}{t[1-\beta(f^{\prime}(t)+1)]}+\omega_{1}\vartheta+\omega_{2}<1,\qquad
t\in(0,\delta),$
Hence, $\delta\leq\rho$, which proves the first statement. To conclude the
proof, we use the definition of $\rho$, the above inequality, and the last
part of Proposition 10. ∎
### 2.2 Relationship of the majorant function with the non-linear function
In this section we will present the main relationships between the majorant
function $f$ and the function $F$ associated with the nonlinear least square
problem.
###### Lemma 12.
Let $x\in\Omega$. If $\|x-x_{*}\|<\min\\{\nu,\kappa\\}$, then
$F^{\prime}(x)^{*}F^{\prime}(x)$ is invertible and the following inequalities
hold
$\left\|F^{\prime}(x)^{\dagger}\right\|\leq\frac{\beta}{1-\beta[f^{\prime}(\|x-x_{*}\|)+1]},\qquad\left\|F^{\prime}(x)^{\dagger}-F^{\prime}(x_{*})^{\dagger}\right\|<\frac{\sqrt{2}\beta^{2}[f^{\prime}(\|x-x_{*}\|)+1]}{1-\beta[f^{\prime}(\|x-x_{*}\|)+1]}.$
In particular, $F^{\prime}(x)^{*}F^{\prime}(x)$ is invertible in $B(x_{*},r)$.
###### Proof.
Let $x\in\Omega$ such that $\|x-x_{*}\|<\min\\{\nu,\kappa\\}$. Since
$\|x-x_{*}\|<\nu$, using the definition of $\beta$, the inequality (2) and
last part of Proposition 9 we have
$\|F^{\prime}(x)-F^{\prime}(x_{*})\|\|F^{\prime}(x_{*})^{\dagger}\|\leq\beta[f^{\prime}(\|x-x_{*}\|)-f^{\prime}(0)]<1.$
For the sake of simplicity, the notations define the following matrices
$A=F^{\prime}(x_{*}),\qquad B=F^{\prime}(x),\qquad
E=F^{\prime}(x)-F^{\prime}(x_{*}).$ (6)
The last definitions, together with the latter inequality, imply that
$\|EA^{\dagger}\|\leq\|E\|\|A^{\dagger}\|<1,$
which, using that $F^{\prime}(x_{*})$ is injective, implies in view of Lemma 2
that $F^{\prime}(x)$ is injective. So, $F^{\prime}(x)^{*}F^{\prime}(x)$ is
invertible and by definition of $r$ we obtain that
$F^{\prime}(x)^{*}F^{\prime}(x)$ is invertible for all $x\in B(x_{*},r)$.
We already know that $F^{\prime}(x_{*})$ and $F^{\prime}(x)$ are injective.
Hence, to conclude the lemma use definitions in (6) and then combine the above
inequality and Lemma 3. ∎
Now, it is convenient to study the linearization error of $F$ at point in
$\Omega$, for which we define
$E_{F}(x,y):=F(y)-\left[F(x)+F^{\prime}(x)(y-x)\right],\qquad y,\,x\in\Omega.$
(7)
We will bound this error by the error in the linearization on the majorant
function $f$
$e_{f}(t,u):=f(u)-\left[f(t)+f^{\prime}(t)(u-t)\right],\qquad t,\,u\in[0,R).$
(8)
###### Lemma 13.
If $\|x-x_{*}\|<\kappa$, then there holds $\|E_{F}(x,x_{*})\|\leq
e_{f}(\|x-x_{*}\|,0).$
###### Proof.
Since $B(x_{*},\kappa)$ is convex, we obtain that $x_{*}+\tau(x-x_{*})\in
B(x_{*},\kappa)$, for $0\leq\tau\leq 1$. Thus, as $F$ is continuously
differentiable in $\Omega$, definition of $E_{F}$ and some simple
manipulations yield
$\|E_{F}(x,x_{*})\|\leq\int_{0}^{1}\left\|[F^{\prime}(x)-F^{\prime}(x_{*}+\tau(x-x_{*}))]\right\|\,\left\|x_{*}-x\right\|\;d\tau.$
¿From the last inequality and the assumption (2), we obtain
$\|E_{F}(x,x_{*})\|\leq\int_{0}^{1}\left[f^{\prime}\left(\left\|x-x_{*}\right\|\right)-f^{\prime}\left(\tau\|x-x_{*}\|\right)\right]\|x-x_{*}\|\;d\tau.$
Evaluating the above integral and using definition of $e_{f}$, the statement
follows. ∎
Define the Gauss-Newton step to the functions $F$ by the following equality:
$S_{F}(x):=-F^{\prime}(x)^{\dagger}F(x).$ (9)
###### Lemma 14.
If $\|x-x_{*}\|<\min\\{\nu,\kappa\\}$, then
$\|S_{F}(x)\|\leq\frac{\beta
e_{f}(\|x-x_{*}\|,0)+\sqrt{2}c\beta^{2}[f^{\prime}(\|x-x_{*}\|)+1]}{1-\beta[f^{\prime}(\|x-x_{*}\|)+1]}+\|x-x_{*}\|.$
###### Proof.
Using (9), $F^{\prime}(x_{*})^{*}F(x_{*})=0$ and some algebraic manipulation,
it follows from (7) that
$\displaystyle\|S_{F}(x)\|$
$\displaystyle=\|F^{\prime}(x)^{\dagger}\left(F(x_{*})-[F(x)+F^{\prime}(x)(x_{*}-x)]\right)-(F^{\prime}(x)^{\dagger}-F^{\prime}(x_{*})^{\dagger})F(x_{*})+(x_{*}-x)\|$
$\displaystyle\leq\|F^{\prime}(x)^{\dagger}\|\|E_{F}(x,x_{*})\|+\|F^{\prime}(x)^{\dagger}-F^{\prime}(x_{*})^{\dagger}\|\|F(x_{*})\|+\|x-x_{*}\|.$
So, the last inequality together with the Lemma 12, Lemma 13 and definition of
$c,$ imply that
$\|S_{F}(x)\|\leq\frac{\beta
e_{f}(\|x-x_{*}\|,0)}{1-\beta[f^{\prime}(\|x-x_{*}\|)+1]}+\frac{\sqrt{2}c\beta^{2}[f^{\prime}(\|x-x_{*}\|)+1]}{1-\beta[f^{\prime}(\|x-x_{*}\|)+1]}+\|x-x_{*}\|,$
which is equivalent to the desired inequality. ∎
###### Lemma 15.
Let $\Omega\subseteq\mathbb{X}$ be an open set and $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and $c,$
$\beta,$ $\kappa$ as a definition in Theorem 7. Suppose that
$F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is injective and there
exists a $f:[0,\;R)\to\mathbb{R}$ continuously differentiable satisfying (2),
h1, h2 and h3. Let $\alpha$, $\vartheta$, $\omega_{1}$, $\omega_{2}$, $\nu$,
$\rho$ and $r$ as in Theorem 7. Assume that $x\in
B(x_{*},r)\backslash\\{x_{*}\\}$, i.e., $0<\|x-x_{*}\|<r$. Define
$x_{+}={x}+S,\qquad B(x)S=-F^{\prime}(x)^{*}F(x)+r,$ (10)
where $B(x)$ is an invertible approximation of
$F^{\prime}(x)^{*}F^{\prime}(x)$ satisfying
$\|B(x)^{-1}F^{\prime}(x)^{*}F^{\prime}(x)\|\leq\omega_{1},\qquad\|B(x)^{-1}F^{\prime}(x)^{*}F^{\prime}(x)-I\|\leq\omega_{2},$
(11)
and the forcing term $\theta$ and the residual $r$ satisfy
$\theta\mbox{cond}(PF^{\prime}(x)^{*}F^{\prime}(x))\leq\vartheta,\qquad\|Pr\|\leq\theta\|PF^{\prime}(x)^{*}F(x)\|,$
(12)
with $P$ an invertible matrix(preconditioner for the linear system in (10)).
Then $x_{+}$ is well defined and there holds
$\|x_{+}-x_{*}\|\leq(1+\vartheta)\omega_{1}\beta\frac{[f^{\prime}(\|x-x_{*}\|)\|x-x_{*}\|-f(\|x-x_{*}\|)]}{\|x-x_{*}\|^{2}[1-\beta(f^{\prime}(\|x-x_{*}\|)+1)]}{\|x-x_{*}\|}^{2}\\\
+\left(\frac{(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}[f^{\prime}(\|x-x_{*}\|)+1]}{\|x-x_{*}\|[1-\beta(f^{\prime}(\|x-x_{*}\|)+1)]}+\omega_{1}\vartheta+\omega_{2}\right)\|x-x_{*}\|,\qquad
k=0,1,\ldots.$ (13)
In particular,
$\|x_{+}-x_{*}\|<\|x-x_{*}\|.$
###### Proof.
First note that, as $\|x-x_{*}\|<r$, it follows from Lemma 12 that
$F^{\prime}(x)^{*}F^{\prime}(x)$ is invertible. Now, let $B(x)$ an invertible
approximation of it satisfying (11). Thus, $x_{+}$ is well defined. Now, as
$F^{\prime}(x_{*})^{*}F(x_{*})=0,$ some simple algebraic manipulation and (10)
yield
$x_{+}-x_{*}=x-x_{*}-B(x)^{-1}F^{\prime}(x)^{*}\big{(}F(x)-F(x_{*})\big{)}+B(x)^{-1}{r}\\\
+B(x)^{-1}F^{\prime}(x)^{*}F^{\prime}(x)\left[F^{\prime}(x_{*})^{\dagger}F(x_{*})-F^{\prime}(x)^{\dagger}F(x_{*})\right].$
Again, some algebraic manipulation in the above equation gives
$x_{+}-x_{*}=B(x)^{-1}F^{\prime}(x)^{*}F^{\prime}(x)F^{\prime}(x)^{\dagger}\big{(}F(x_{*})-[F(x)+F^{\prime}(x)(x_{*}-x)]\big{)}+B(x)^{-1}{r}\\\
+B(x)^{-1}\left(F^{\prime}(x)^{*}F^{\prime}(x)-B(x)\right)(x-x_{*})+B(x)^{-1}F^{\prime}(x)^{*}F^{\prime}(x)[F^{\prime}(x_{*})^{\dagger}F(x_{*})-F^{\prime}(x)^{\dagger}F(x_{*})].$
The last equation, together with (7) and (11), imply that
$\|x_{+}-x_{*}\|\leq\omega_{1}\|F^{\prime}(x)^{\dagger}\|\|E_{F}(x,x_{*})\|+\|B(x)^{-1}{r}\|+\omega_{2}\|x-x_{*}\|+\omega_{1}\|F^{\prime}(x)^{\dagger}-F^{\prime}(x_{*})^{\dagger}\|\|F(x_{*})\|.$
On the other hand, using (9), (11) and (12) we have, by simple calculus,
$\displaystyle\|B(x)^{-1}{r}\|$ $\displaystyle\leq\|B(x)^{-1}P^{-1}\|\|P{r}\|$
$\displaystyle\leq\theta\|B(x)^{-1}F^{\prime}(x)^{*}F^{\prime}(x)\|\|(PF^{\prime}(x)^{*}F^{\prime}(x))^{-1}\|\|PF^{\prime}(x)^{*}F^{\prime}(x)\|\|F^{\prime}(x)^{\dagger}F(x)\|$
$\displaystyle\leq\omega_{1}\vartheta\|S_{F}(x)\|.$
Hence, it follows from the two last equations that
$\|x_{+}-x_{*}\|\leq\omega_{1}\|F^{\prime}(x)^{\dagger}\|\|E_{F}(x,x_{*})\|+\omega_{1}\vartheta\|S_{F}(x)\|+\omega_{2}\|x-x_{*}\|+\omega_{1}\|F^{\prime}(x)^{\dagger}-F^{\prime}(x_{*})^{\dagger}\|\|F(x_{*})\|.$
Combining the last equation with the Lemmas 12, 13 and 14, we obtain that
$\|x_{+}-x_{*}\|\leq(1+\vartheta)\beta\omega_{1}\frac{e_{f}(\|x-x_{*}\|,0)+\sqrt{2}c\beta(f^{\prime}(\|x-x_{*}\|)+1)}{1-\beta(f^{\prime}(\|x-x_{*}\|)+1)}+\omega_{1}\vartheta\|x-x_{*}\|+\omega_{2}\|x-x_{*}\|.$
Now, using (8) and some algebraic manipulation, we conclude from the last
inequality that
$\|x_{+}-x_{*}\|\leq(1+\vartheta)\beta\omega_{1}\frac{f^{\prime}(\|x-x_{*}\|)\|x-x_{*}\|-f(\|x-x_{*}\|)+\sqrt{2}c\beta(f^{\prime}(\|x-x_{*}\|)+1)}{1-\beta(f^{\prime}(\|x-x_{*}\|)+1)}\\\
+\omega_{1}\vartheta\|x-x_{*}\|+\omega_{2}\|x-x_{*}\|,$
which is equivalent to (13). To end the proof, note that the right hand side
of (13) is equivalent to
$\Bigg{[}(1+\vartheta)\omega_{1}\beta\frac{f^{\prime}(\|x-x_{*}\|)\|x-x_{*}\|-f(\|x-x_{*}\|)+\sqrt{2}c\beta(f^{\prime}(\|x-x_{*}\|)+1)}{\|x-x_{*}\|[1-\beta(f^{\prime}(\|x-x_{*}\|)+1)]}+\omega_{1}\vartheta+\omega_{2}\Bigg{]}\|x-x_{*}\|.$
On the other hand, as $x\in B(x_{*},r)/\\{x_{*}\\}$, i.e.,
$0<\|x-x_{*}\|<r\leq\rho$ we apply the Proposition 11 with $t=\|x-x_{*}\|$ to
conclude that the quantity in the bracket above is less than one. So, the last
inequality of the lemma follows. ∎
### 2.3 Proof of Theorem 7
Now, we will produce the proof of Theorem 7.
###### Proof.
Since $x_{0}\in B(x_{*},r)/\\{x_{*}\\},$ i.e., $0<\|x_{0}-x_{*}\|<r,$ by
combination of Lemma 12, last inequality in Lemma 15 and induction argument,
it is easy to see that $\\{x_{k}\\}$ is well defined and remains in
$B(x_{*},r)$.
We are going to prove that $\\{x_{k}\\}$ converges towards $x_{*}$. As,
$\\{x_{k}\\}$ is well defined and contained in $B(x_{*},r)$, applying Lemma 15
with $x_{+}=x_{k+1},$ $x=x_{k},$ $r=r_{k},$ $B(x)=B(x_{k}),$ $P=P_{k},$ and
$\theta=\theta_{k}$ we obtain
$\|x_{k+1}-x_{*}\|\leq(1+\vartheta)\omega_{1}\beta\frac{[f^{\prime}(\|x_{k}-x_{*}\|)\|x_{k}-x_{*}\|-f(\|x_{k}-x_{*}\|)]}{\|x_{k}-x_{*}\|^{2}[1-\beta(f^{\prime}(\|x_{k}-x_{*}\|)+1)]}{\|x_{k}-x_{*}\|}^{2}\\\
+\left(\frac{(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}[f^{\prime}(\|x_{k}-x_{*}\|)+1]}{\|x_{k}-x_{*}\|[1-\beta(f^{\prime}(\|x_{k}-x_{*}\|)+1)]}+\omega_{1}\vartheta+\omega_{2}\right)\|x_{k}-x_{*}\|,\qquad
k=0,1,\ldots.$
Now, using the last inequality of Lemma 15, it is easy to conclude that
$\|x_{k}-x_{*}\|<\|x_{0}-x_{*}\|,\qquad\;k=1,2\ldots.$ (14)
Hence, combining the last two inequalities with the last part of Proposition
10 we obtain that
$\|x_{k+1}-x_{*}\|\leq(1+\vartheta)\omega_{1}\beta\frac{[f^{\prime}(\|x_{0}-x_{*}\|)\|x_{0}-x_{*}\|-f(\|x_{0}-x_{*}\|)]}{\|x_{0}-x_{*}\|^{2}[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}{\|x_{k}-x_{*}\|}^{2}\\\
+\left(\frac{(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}[f^{\prime}(\|x_{0}-x_{*}\|)+1]}{\|x_{0}-x_{*}\|[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}+\omega_{1}\vartheta+\omega_{2}\right)\|x_{k}-x_{*}\|,\qquad
k=0,1,\ldots,$
which is the inequality (5). Now, using (14) and the last inequality we have
$\|x_{k+1}-x_{*}\|\leq\\\
\Bigg{[}(1+\vartheta)\omega_{1}\beta\frac{f^{\prime}(\|x_{0}-x_{*}\|)\|x_{0}-x_{*}\|-f(\|x_{0}-x_{*}\|)+\sqrt{2}c\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)}{\|x_{0}-x_{*}\|[1-\beta(f^{\prime}(\|x_{0}-x_{*}\|)+1)]}+\omega_{1}\vartheta+\omega_{2}\Bigg{]}\|x_{k}-x_{*}\|,$
for all $k=0,1,\ldots$. Applying Proposition 11 with $t=\|x_{0}-x_{*}\|$ it is
straightforward to conclude from the latter inequality that
$\\{\|x_{k}-x_{*}\|\\}$ converges to zero. So, $\\{x_{k}\\}$ converges to
$x_{*}$. ∎
## 3 Special cases
In this section, we present two special cases of Theorem 7. They include the
classical convergence theorem on Gauss-Newton’s method under the Lipschitz
condition and Smale’s theorem on Gauss-Newton for analytical functions.
### 3.1 Convergence result for Lipschitz condition
In this section we show a correspondent theorem for Theorem 7 under the
Lipschitz condition, instead of the general assumption (2).
###### Theorem 16.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and
$c:=\|F(x_{*})\|,\qquad\beta:=\left\|F^{\prime}(x_{*})^{\dagger}\right\|,\qquad\kappa:=\sup\left\\{t\in[0,R):B(x_{*},t)\subset\Omega\right\\}.$
Suppose that $F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is
injective and there exists a $K>0$ such that
$\alpha:=\sqrt{2}c\beta^{2}K<1,\qquad\qquad\left\|F^{\prime}(x)-F^{\prime}(y)\right\|\leq
K\|x-y\|,\qquad\forall\;x,y\in B(x_{*},\kappa).$
Take $0\leq\vartheta<1$, $0\leq\omega_{2}<\omega_{1}$ such that
$\omega_{1}(\alpha+\alpha\vartheta+\vartheta)+\omega_{2}<1$. Let
$r:=\min\left\\{\kappa,\frac{2(1-\omega_{1}\vartheta-\omega_{2})-2\sqrt{2}cK\beta^{2}\omega_{1}(1+\vartheta)}{\beta
K\left(2+\omega_{1}-\vartheta\omega_{1}-2\omega_{2}\right)}\right\\}.$
Then, the inexact Gauss-Newton like methods for solving (1), with initial
point $x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k})+r_{k},\qquad\;k=0,1,\ldots,$ (15)
with the following conditions for the residual $r_{k},$ and the forcing term
$\theta_{k}$
$\|P_{k}r_{k}\|\leq\theta_{k}\|P_{k}F^{\prime}(x_{k})^{*}F(x_{k})\|,\qquad
0\leq\theta_{k}\mbox{cond}(P_{k}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k}))\leq\vartheta,\qquad\;k=0,1,\ldots,$
where $\\{P_{k}\\}$ is an invertible matrix sequence (preconditoners for the
linear system in (15)) and $B(x_{k})$ is an invertible approximation of
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ satisfying
$\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})\|\leq\omega_{1},\qquad\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})-I\|\leq\omega_{2},\qquad\;k=0,1,\ldots,$
is well defined, contained in $B(x_{*},r)$, converges to $x_{*}$ and there
holds
$\|x_{k+1}-x_{*}\|\leq\frac{(1+\vartheta)\beta\omega_{1}K}{2(1-\beta
K\|x_{0}-x_{*}\|)}\|x_{k}-x_{*}\|^{2}+\left(\frac{(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}K}{1-\beta
K\|x_{0}-x_{*}\|}+\omega_{1}\vartheta+\omega_{2}\right)\|x_{k}-x_{*}\|,$
for all $k=0,1,\ldots.$
###### Proof.
It is immediate to prove that $F$, $x_{*}$ and $f:[0,\kappa)\to\mathbb{R}$ as
defined by $f(t)=Kt^{2}/2-t,$ satisfy the inequality (2), conditions h1 and
h2. Since $\sqrt{2}c\beta^{2}K<1$ the condition h3 also holds. In this case,
it is easy to see that constants $\nu$ and $\rho$ as defined in Theorem 7,
satisfy
$0<\rho=\frac{2(1-\omega_{1}\vartheta-\omega_{2})-2\sqrt{2}cK\beta^{2}\omega_{1}(1+\vartheta)}{\beta
K\left(2+\omega_{1}-\vartheta\omega_{1}-2\omega_{2}\right)}\leq\nu=1/\beta K,$
as a consequence, $0<r=\min\\{\kappa,\,\rho\\}.$ Therefore, as $F$, $r$, $f$
and $x_{*}$ satisfy all of the hypotheses of Theorem 7, taking $x_{0}\in
B(x_{*},r)\backslash\\{x_{*}\\}$ the statements of the theorem follow from
Theorem 7. ∎
For the case $\vartheta=0$, the Theorem 16 becomes:
###### Corollary 17.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and
$c:=\|F(x_{*})\|,\qquad\beta:=\left\|F^{\prime}(x_{*})^{\dagger}\right\|,\qquad\kappa:=\sup\left\\{t\in[0,R):B(x_{*},t)\subset\Omega\right\\}.$
Suppose that $F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is
injective and there exists a $K>0$ such that
$\alpha:=\sqrt{2}c\beta^{2}K<1,\qquad\qquad\left\|F^{\prime}(x)-F^{\prime}(y)\right\|\leq
K\|x-y\|,\qquad\forall\;x,y\in B(x_{*},\kappa).$
Take $0\leq\omega_{2}<\omega_{1}$ such that $\omega_{1}\alpha+\omega_{2}<1$.
Let
$r:=\min\left\\{\kappa,\frac{2(1-\omega_{2})-2\sqrt{2}cK\beta^{2}\omega_{1}}{\beta
K\left(2+\omega_{1}-2\omega_{2}\right)}\right\\}.$
Then, the Gauss-Newton’s like method for solving (1), with initial point
$x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k}),\qquad\;k=0,1,\ldots,$ (16)
where $B(x_{k})$ is an invertible approximation of
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ satisfying
$\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})\|\leq\omega_{1},\qquad\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})-I\|\leq\omega_{2},\qquad\;k=0,1,\ldots,$
is well defined, contained in $B(x_{*},r)$, converges to $x_{*}$ and there
holds
$\|x_{k+1}-x_{*}\|\leq\frac{\beta\omega_{1}K}{2(1-\beta
K\|x_{0}-x_{*}\|)}\|x_{k}-x_{*}\|^{2}+\left(\frac{\omega_{1}\sqrt{2}c\beta^{2}K}{1-\beta
K\|x_{0}-x_{*}\|}+\omega_{2}\right)\|x_{k}-x_{*}\|,$
for all $k=0,1,\ldots.$
Note that letting $c=0$ in the above corollary, we obtain the Corollary 6.1 of
[8].
### 3.2 Convergence result under Smale’s condition
In this section we present a correspondent theorem to Theorem 7 under Smale’s
condition. For more details see Smale [26] and Dedieu and Shub [11].
###### Theorem 18.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and
$c:=\|F(x_{*})\|,\qquad\beta:=\left\|F^{\prime}(x_{*})^{\dagger}\right\|,\qquad\kappa:=\sup\left\\{t\in[0,R):B(x_{*},t)\subset\Omega\right\\}.$
Suppose that $F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is
injective and
$\qquad\gamma:=\sup_{n>1}\left\|\frac{F^{(n)}(x_{*})}{n!}\right\|^{1/(n-1)}<+\infty,\qquad\qquad\alpha:=2\sqrt{2}c\beta^{2}\gamma<1.$
(17)
Take $0\leq\vartheta<1$, $0\leq\omega_{2}<\omega_{1}$ such that
$\omega_{1}(\alpha+\alpha\vartheta+\vartheta)+\omega_{2}<1$. Let
$a:=(1-\vartheta\omega_{1}-\omega_{2})$, $b:=(1+\vartheta)\omega_{1}\beta,$
$\bar{a}:=b+2a(1+\beta)-\sqrt{2}\gamma\beta bc$ and
$r:=\min\left\\{\kappa,\frac{\bar{a}-\sqrt{\bar{a}^{2}-4a(1+\beta)(a-2\sqrt{2}c\beta
b\gamma)}}{2a\gamma(1+\beta)}\right\\}.$
Then, the inexact Gauss-Newton like methods for solving (1), with initial
point $x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k})+r_{k},\qquad\;k=0,1,\ldots,$ (18)
with the following conditions for the residual $r_{k},$ and the forcing term
$\theta_{k}$
$\|P_{k}r_{k}\|\leq\theta_{k}\|P_{k}F^{\prime}(x_{k})^{*}F(x_{k})\|,\qquad
0\leq\theta_{k}\mbox{cond}(P_{k}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k}))\leq\vartheta,\qquad\;k=0,1,\ldots,$
where $\\{P_{k}\\}$ is an invertible matrix sequence (preconditoners for the
linear system in (18)) and $B(x_{k})$ is an invertible approximation of
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ satisfying
$\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})\|\leq\omega_{1},\qquad\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})-I\|\leq\omega_{2},\qquad\;k=0,1,\ldots,$
is well defined, contained in $B(x_{*},r)$, converges to $x_{*}$ and there
holds
$\displaystyle\|x_{k+1}-x_{*}\|\leq$
$\displaystyle\frac{(1+\vartheta)\omega_{1}\beta\gamma}{(1-\gamma\|x_{0}-x_{*}\|)^{2}-\beta\gamma(2\|x_{0}-x_{*}\|-\gamma\|x_{0}-x_{*}\|^{2})}\|x_{k}-x_{*}\|^{2}$
$\displaystyle+$
$\displaystyle\left(\frac{(1+\vartheta)\omega_{1}\sqrt{2}c\beta^{2}\gamma(2-\gamma\|x_{0}-x_{*}\|)}{(1-\gamma\|x_{0}-x_{*}\|)^{2}-\beta\gamma(2\|x_{0}-x_{*}\|-\gamma\|x_{0}-x_{*}\|^{2})}+\omega_{1}\vartheta+\omega_{2}\right)\|x_{k}-x_{*}\|,$
for all $k=0,1,\ldots.$
We need the following result to prove the above theorem.
###### Lemma 19.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ an
analytic function. Suppose that $x_{*}\in\Omega$ and
$B(x_{*},1/\gamma)\subset\Omega$, where $\gamma$ is defined in (17). Then, for
all $x\in B(x_{*},1/\gamma)$ there holds
$\|F^{\prime\prime}(x)\|\leqslant 2\gamma/(1-\gamma\|x-x_{*}\|)^{3}.$
###### Proof.
See the proof of the Lemma 21 of [16]. ∎
The next result gives a condition that is easier to check than condition (2),
whenever the functions under consideration are twice continuously
differentiable.
###### Lemma 20.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $x_{*}\in\Omega$ and
$F:{\Omega}\to\mathbb{Y}$ be twice continuously on $\Omega$. If there exists a
$f:[0,R)\to\mathbb{R}$ twice continuously differentiable such that
$\|F^{\prime\prime}(x)\|\leqslant f^{\prime\prime}(\|x-x_{*}\|),$ (19)
for all $x\in\Omega$ such that $\|x-x_{*}\|<R$. Then $F$ and $f$ satisfy (2).
###### Proof.
See the proof of the Lemma 22 of [16]. ∎
[Proof of Theorem 18]. Consider the real function
$f:[0,1/\gamma)\to\mathbb{R}$ defined by
$f(t)=\frac{t}{1-\gamma t}-2t.$
It is straightforward to show that $f$ is analytic and that
$f(0)=0,\quad f^{\prime}(t)=1/(1-\gamma t)^{2}-2,\quad f^{\prime}(0)=-1,\quad
f^{\prime\prime}(t)=(2\gamma)/(1-\gamma t)^{3},\quad
f^{n}(0)=n!\,\gamma^{n-1},$
for $n\geq 2$. It follows from the last equalities that $f$ satisfies h1 and
h2. Since $2\sqrt{2}c\beta^{2}\gamma<1$ the condition h3 also holds. Now, as
$f^{\prime\prime}(t)=(2\gamma)/(1-\gamma t)^{3}$ combining Lemmas 20, 19 we
conclude that $F$ and $f$ satisfy (2) with $R=1/\gamma$. In this case, it is
easy to see that constants $\nu$ and $\rho$ as defined in Theorem 7, satisfy
$0<\rho=\frac{\bar{a}-\sqrt{\bar{a}^{2}-4a(1+\beta)(a-2\sqrt{2}c\beta
b\gamma)}}{2a\gamma(1+\beta)}<\nu=((1+\beta)-\sqrt{\beta(1+\beta)})/(\gamma(1+\beta))<1/\gamma,$
and as a consequence, $0<r=\min\\{\kappa,\rho\\}.$ Therefore, as $F$,
$\sigma$, $f$ and $x_{*}$ satisfy all hypotheses of Theorem 7, taking
$x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$, the statements of the theorem
follow from Theorem 7.∎
For the case $\vartheta=0$, the Theorem 18 becomes:
###### Corollary 21.
Let $\Omega\subseteq\mathbb{X}$ be an open set, $F:{\Omega}\to\mathbb{Y}$ a
continuously differentiable function. Let $x_{*}\in\Omega,$ $R>0$ and
$c:=\|F(x_{*})\|,\qquad\beta:=\left\|F^{\prime}(x_{*})^{\dagger}\right\|,\qquad\kappa:=\sup\left\\{t\in[0,R):B(x_{*},t)\subset\Omega\right\\}.$
Suppose that $F^{\prime}(x_{*})^{*}F(x_{*})=0$, $F^{\prime}(x_{*})$ is
injective and
$\qquad\gamma:=\sup_{n>1}\left\|\frac{F^{(n)}(x_{*})}{n!}\right\|^{1/(n-1)}<+\infty,\qquad\qquad\alpha:=2\sqrt{2}c\beta^{2}\gamma<1.$
Take $0\leq\omega_{2}<\omega_{1}$ such that $\omega_{1}\alpha+\omega_{2}<1$.
Let
$\bar{a}:=\omega_{1}\beta+2(1-\omega_{2})(1+\beta)-\sqrt{2}\gamma\beta^{2}\omega_{1}c$
and
$r:=\min\left\\{\kappa,\frac{\bar{a}-\sqrt{\bar{a}^{2}-4(1-\omega_{2})(1+\beta)(1-\omega_{2}-2\sqrt{2}c\beta^{2}\omega_{1}\gamma)}}{2(1-\omega_{2})\gamma(1+\beta)}\right\\}.$
Then, the Gauss-Newton’s like method for solving (1), with initial point
$x_{0}\in B(x_{*},r)\backslash\\{x_{*}\\}$
$x_{k+1}={x_{k}}+S_{k},\qquad
B(x_{k})S_{k}=-F^{\prime}(x_{k})^{*}F(x_{k}),\qquad\;k=0,1,\ldots,$
where $B(x_{k})$ is an invertible approximation of
$F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})$ satisfying
$\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})\|\leq\omega_{1},\qquad\|B(x_{k})^{-1}F^{\prime}(x_{k})^{*}F^{\prime}(x_{k})-I\|\leq\omega_{2},\qquad\;k=0,1,\ldots,$
is well defined, contained in $B(x_{*},r)$, converges to $x_{*}$ and there
holds
$\displaystyle\|x_{k+1}-x_{*}\|\leq$
$\displaystyle\frac{\omega_{1}\beta\gamma}{(1-\gamma\|x_{0}-x_{*}\|)^{2}-\beta\gamma(2\|x_{0}-x_{*}\|-\gamma\|x_{0}-x_{*}\|^{2})}\|x_{k}-x_{*}\|^{2}$
$\displaystyle+$
$\displaystyle\left(\frac{\omega_{1}\sqrt{2}c\beta^{2}\gamma(2-\gamma\|x_{0}-x_{*}\|)}{(1-\gamma\|x_{0}-x_{*}\|)^{2}-\beta\gamma(2\|x_{0}-x_{*}\|-\gamma\|x_{0}-x_{*}\|^{2})}+\omega_{2}\right)\|x_{k}-x_{*}\|,$
for all $k=0,1,\ldots.$
Note that letting $c=0$ in the above corollary, we obtain the Example 1 of
[8].
## 4 Final remark
The Theorem 7 gives an estimate of the convergence radius for inexact Gauss-
Newton like methods. In particular, for $\vartheta=\omega_{1}=0$ and
$\omega_{2}=1$ is shown in Ferreira et al. [16], that $r$ is the best possible
convergence radius.
Another detail is that, as pointed out by Morini in [23] if preconditioning
$P_{k}$, satisfying
$\|P_{k}r_{k}\|\leq\theta_{k}\|P_{k}F^{\prime}(x_{k})^{*}F(x_{k})\|,$ (20)
for some forcing sequence $\\{\theta_{k}\\}$, is applied to finding the
inexact Gauss-Newton steep, then the inverse proportionality between each
forcing term $\theta_{k}$ and
$\mbox{cond}(P_{k}F^{\prime}(x_{k})^{*}F(x_{k}))$ stated in the following
assumption:
$0<\theta_{k}\mbox{cond}(P_{k}F^{\prime}(x_{k})^{*}F(x_{k}))\leq\vartheta,\qquad\;k=0,1,\ldots,$
(21)
is sufficient to guarantee convergence, and may be overly restrictive to bound
the sequence $\\{\theta_{k}\\}$, always such that the matrices
$P_{k}F^{\prime}(x_{k})^{*}F(x_{k})$, for $k=0,1,\ldots,$ are badly
conditioned. Moreover, $\theta_{k}$ does not depend on
$\mbox{cond}(F^{\prime}(x_{k})^{*}F(x_{k}))$ but only on the
$\mbox{cond}(P_{k}F^{\prime}(x_{k})^{*}F(x_{k}))$ and a suitable choice of
scaling matrix $P_{k}$ leads to a relaxation of the forcing terms.
## References
* [1] R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens, and M. Shub. Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal., 22(3):359–390, 2002.
* [2] F. Alvarez, J. Bolte, and J. Munier. A unifying local convergence result for Newton’s method in Riemannian manifolds. Found. Comput. Math., 8(2):197–226, 2008.
* [3] I. K. Argyros and S. Hilout. Improved generalized differentiability conditions for Newton-like methods. J. Complexity, In Press, Corrected Proof:–, 2010.
* [4] L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and real computation. Springer-Verlag, New York, 1998. With a foreword by Richard M. Karp.
* [5] J. Chen. The convergence analysis of inexact Gauss-Newton methods for nonlinear problems. Comput. Optim. Appl., 40(1):97–118, 2008.
* [6] J. Chen and W. Li. Convergence of Gauss-Newton’s method and uniqueness of the solution. Appl. Math. Comput., 170(1):686–705, 2005.
* [7] J. Chen and W. Li. Convergence behaviour of inexact Newton methods under weak Lipschitz condition. J. Comput. Appl. Math., 191(1):143–164, 2006.
* [8] J. Chen and W. Li. Local convergence results of Gauss-Newton’s like method in weak conditions. J. Math. Anal. Appl., 324(2):1381 – 1394, 2006.
* [9] J. P. Dedieu and M. H. Kim. Newton’s method for analytic systems of equations with constant rank derivatives. J. Complexity, 18(1):187–209, 2002.
* [10] J. P. Dedieu, P. Priouret, and G. Malajovich. Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal., 23(3):395–419, 2003.
* [11] J. P. Dedieu and M. Shub. Newton’s method for overdetermined systems of equations. Math. Comp., 69(231):1099–1115, 2000.
* [12] R. S. Dembo, S. C. Eisenstat, and T. Steihaug. Inexact Newton methods. SIAM J. Numer. Anal., 19(2):400–408, 1982.
* [13] J. E. Dennis, Jr. and R. B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations, volume 16 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Corrected reprint of the 1983 original.
* [14] O. P. Ferreira. Local convergence of Newton’s method in Banach space from the viewpoint of the majorant principle. IMA J. Numer. Anal., 29(3):746–759, 2009.
* [15] O. P. Ferreira and M. L. N. Goncalves. Local convergence analysis of inexact Newton-like methods under majorant condition. Comput. Optim. Appl., Article in Press:1–21, 2009.
* [16] O. P. Ferreira, M. L. N. Goncalves, and P. R. Oliveira. Local convergence analysis of gauss-newton’s method under majorant condition. To appear in J. Complexity, 1–20, 2010.
* [17] O. P. Ferreira and B. F. Svaiter. Kantorovich’s majorants principle for Newton’s method. Comput. Optim. Appl., 42(2):213–229, 2009.
* [18] J.-B. Hiriart-Urruty and C. Lemaréchal. Convex analysis and minimization algorithms. I, volume 305 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1993. Fundamentals.
* [19] L. V. Kantorovich. The principle of the majorant and Newton’s method. Doklady Akad. Nauk SSSR (N.S.), 76:17–20, 1951.
* [20] C. Li, N. Hu, and J. Wang. Convergence behavior of Gauss-Newton’s method and extensions of the smale point estimate theory. J. Complexity, In Press, Corrected Proof:–, 2010.
* [21] C. Li, W. H. Zhang, and X. Q. Jin. Convergence and uniqueness properties of Gauss-Newton’s method. Comput. Math. Appl., 47(6-7):1057–1067, 2004.
* [22] J. M. Martinez and L. Qi. Inexact Newton methods for solving nonsmooth equations. J. Comput. Appl. Math., 60:127–145, 1999.
* [23] B. Morini. Convergence behaviour of inexact Newton methods. Math. Comp., 68:1605–1613, 1999.
* [24] P. D. Proinov. General local convergence theory for a class of iterative processes and its applications to Newton’s method. J. Complexity, 25(1):38 – 62, 2009.
* [25] P. D. Proinov. New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complexity, 26(1):3 – 42, 2010.
* [26] S. Smale. Newton’s method estimates from data at one point. In The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), pages 185–196. Springer, New York, 1986.
* [27] G. W. Stewart. On the continuity of the generalized inverse. SIAM J. Appl. Math., 17:33–45, 1969.
* [28] X. Wang. Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal., 20(1):123–134, 2000.
* [29] P. A. Wedin. Perturbation theory for pseudo-inverses. Nordisk Tidskr. Informationsbehandling (BIT), 13:217–232, 1973\.
|
arxiv-papers
| 2010-08-11T14:27:51 |
2024-09-04T02:49:12.136202
|
{
"license": "Public Domain",
"authors": "O.P.Ferreira, M.L.N.Goncalves and P.R.Oliveira",
"submitter": "Max Leandro Nobre Goncalves",
"url": "https://arxiv.org/abs/1008.1916"
}
|
1008.2055
|
# Elements with $r$–th roots in finite groups
E. Khamseh Department of Mathematics, Islamic Azad University,
Mashhad–Branch, 91756, Mashhad, Iran elahehkhamseh@gmail.com , M. R. R.
Moghaddam Department of Mathematics, Khayyam Higher Education Institute, and
Centre of Excellence in Analysis and Algebraic Structure of Ferdowsi
University of Mashhad, Mashhad, Iran mrrm5@yahoo.ca , F. G. Russo DIEETCAM,
University of Palermo, Viale delle Scienze, 90128, Palermo, Italy, and,
Department of Mathematics, Universiti Teknologi Malaysia, 81310, Skudai, Johor
Bahru, Malaysia francescog.russo@yahoo.com and F. Saeedi Department of
Mathematics, Islamic Azad University, Mashhad–Branch, 91756, Mashhad, Iran
saeedi@mshdiau.ac.ir
###### Abstract.
The probability that a randomly chosen element of a finite group is an $r$–th
root (for any integer $r\geq 2$) has been studied largely in case $r=2$.
Certain techniques may be generalized for $r>2$ and here we find the exact
value of this probability for projective special linear groups. A result of
density is placed at the end, in order to show an analogy with the case $r=2$.
###### Key words and phrases:
$r$–th roots, probability, equations over finite groups, linear groups.
###### 2010 Mathematics Subject Classification:
Primary 20D15; 20P05; Secondary 20D60.
## 1\. Introduction
In the present paper all the groups are finite. In a group $G$, if there
exists an element $y\in G$ for which $x=y^{r}$, we say that $x$ $has$ $an$
$r$–$th$ $root$. For $r=2$, J. Blum described in [1] the probability
$\mathrm{Prob}_{2}(S_{n})=\frac{|S^{2}_{n}|}{n!}$
that a randomly chosen permutation of length $n$ has a $2$–nd root (or square
root), where $S_{n}$ is the permutation group on $n$ letters. Successively his
work was generalized in [3, 5, 6, 8] to the case of an arbitrary group.
Already in [2, 7] it was studied the probability
$\mathrm{Prob}_{r}(S_{n})=\frac{|S^{r}_{n}|}{n!}$
that a randomly chosen permutation of length $n$ has an $r$–th root for $r\geq
2$. Therefore, many results in [1] can be found as special situations of [2,
7], but, so far as we have searched in the literature, [2, 7] have not been
extended in the sense of [3, 5, 6, 8] to the case of an arbitrary group. This
is the beginning of our investigations and the motivation of the present work.
We define the probability
$\mathrm{Prob}_{r}(G)=\frac{|G^{r}|}{|G|},$
where $r\geq 2$ and $G^{r}=\\{g^{r}\ |\ g\in G\\}$ is the set of all elements
of $G$ having at least one $r$–th root. Unfortunately, $G^{r}$ is not a
subgroup of $G$ but only a set and this can give difficulties from the general
point of view.
Even if $\mathrm{Prob}_{2}(G)$ is known by [3, 5, 6] and
$\mathrm{Prob}_{r}(S_{n})$ by [2, 7], we have not found whether it is possible
to obtain some structural information on $G$ from the bounds of
$\mathrm{Prob}_{r}(G)$ or not. In the present paper we will investigate such
aspects and provide some restrictions of numerical nature for
$\mathrm{Prob}_{r}(G)$.
## 2\. Basic properties
We recall some fundamental notions on abelian groups. If $A$ is an abelian
group, then
$A^{r}=\\{a^{r}\ |\ a\in A\\}$
is a subgroup of $A$ and $A$ is called r-divisible, if $A^{r}=A$. $A$ is
divisible, if it is $r$–divisible for all $r\geq 2$. It is easy to see that
$A$ is divisible if, and only if, it is $p$–divisible for all primes $p$.
$A[r]=\\{a\in A\ |\ a^{r}=1\\}$
is a subgroup of $A$ and $A$ is of exponent $r$, if $A[r]=A$. If $r=p$ is a
prime, $A[p]$ is called p-socle of $A$ and is isomorphic to the additive group
of a vector space over the field with $p$ elements: In other words, $A[p]$ is
an elementary $p$–group of rank $k\geq 1$, that is,
$A[p]=C_{p}\times\ldots\times C_{p}=C^{k}_{p}.$
In general, for an arbitrary $r\geq 2$, the subgroups $A^{r}$ and $A[r]$ are
related by the First Isomorphism’s Theorem: $\varphi:a\in A\mapsto a^{r}\in
A^{r}$ is a homomorphism of groups, inducing $A^{r}\simeq A/A[r].$ The
following remark gives a complete characterization for abelian groups.
###### Remark 2.1.
Assume that $G$ is a nontrivial abelian group.
* (i)
$\mathrm{Prob}_{r}(G)=\frac{1}{|G[r]|}$.
* (ii)
If $r$ is prime, then $\mathrm{Prob}_{r}(G)=\frac{1}{r^{k}}$ for some $k\geq
1$. Furthermore, the set
$X=\\{\mathrm{Prob}_{r}(G)\ |\ G\mathrm{\ is\ an\ abelian\ group}\\}$
coincides with the subset
$Y=\left\\{\frac{1}{r^{k}}\ \Big{|}\ k\geq 1\right\\}$
of the interval $[0,1]$.
From Remark 2.1 (i), a nontrivial abelian group $G$ of exponent $r$ has
$\mathrm{Prob}_{r}(G)=\frac{1}{|G|}$. Now we will summarize most of the above
considerations in the next result.
###### Proposition 2.2.
Let $G$ be a nontrivial abelian group.
* (i)
$\mathrm{Prob}_{r}(G)=\frac{1}{|G[r]|}$. Furthermore, if $r$ is prime,
$\mathrm{Prob}_{r}(G)=\frac{1}{|G|}$ if and only if $G\simeq C^{k}_{r}$ for
some $k\geq 1$.
* (ii)
The sets $X$ and $Y$ of Remark 2.1 coincide.
###### Proof.
(i). The first part is exactly Remark 2.1 (i). Now assume that $r$ is prime.
If $G\simeq C^{k}_{r}$, then $G$ is isomorphic to the additive group of a
vector space over the field with $p$ elements, that is, $G[r]\simeq G$. Then
$\mathrm{Prob}_{r}(G)=\frac{1}{|G|}$. Conversely,
$\mathrm{Prob}_{r}(G)=\frac{|rG|}{|G|}=\frac{|G|}{|G[r]|\
|G|}=\frac{1}{|G[r]|}=\frac{1}{|G|}$ implies $|G[r]|=|G|$, then
$1=|rG|=\frac{|G|}{|G[r]|}$ via the isomorphism induced by $\varphi$, and so
$G[r]\simeq G$, from which the result follows.
(ii). It is exactly Remark 2.1 (ii). ∎
Now we describe $\mathrm{Prob}_{r}(G)=1$ for $r\geq 2$ and recall notions in
[5, 6].
###### Remark 2.3.
Let $G$ be a nontrivial group.
* (i)
$\mathrm{Prob}_{r}(G)=1$ if and only if $|G|=|G^{r}|$, that is, the number of
the elements of $G$ having an $r$–th root is the same of the number of the
elements of $G$.
* (ii)
(See [6]) The number of solutions of the equation $x^{r}=a$ in $G$ is a
multiple of $\mathrm{gcd}(r,|C_{G}(a)|)$, where $r\geq 2$ and $a$, $x\in G$.
In particular, the number of solutions of the equation $x^{r}=1$ over $G$ is a
multiple of $\mathrm{gcd}(r,|G|)$ and when $r$ is prime, $|x|=|a|$ or
$|x|=r|a|$.
We reformulate Remark 2.3 as follows.
###### Proposition 2.4.
Let $G$ be an arbitrary group and $r\geq 2$. $\mathrm{Prob}_{r}(G)=1$ if and
only if some multiple of $gcd(r,|G|)$ is equal to $1$.
Propositions 2.2 and 2.4 agree with [5, Proposition 2.1], when $r=2$. Now we
will proceed to list further properties.
###### Remark 2.5.
For an arbitrary group $G$, we have
$0<\frac{1}{|G|}\leq\mathrm{Prob}_{r}(G)\leq 1$. Propositions 2.2 (i) shows a
condition in which we achieve the lower bound $\frac{1}{|G|}$ in the abelian
case. Proposition 2.4 shows a more general condition in which we achieve the
upper bound.
It is well–known that the probability of independent events is multiplicative.
Here we have as follows.
###### Proposition 2.6.
Given two groups $A$ and $B$, $\mathrm{Prob}_{r}(A\times
B)=\mathrm{Prob}_{r}(A)\ \mathrm{Prob}_{r}(B).$
###### Proof.
$\mathrm{Prob}_{r}(A\times B)=\frac{|(A\times B)^{r}|}{|A\times
B|}=\frac{|A^{r}\times
B^{r}|}{|A||B|}=\frac{|A^{r}||B^{r}|}{|A||B|}=\mathrm{Prob}_{r}(A)\
\mathrm{Prob}_{r}(B).$
∎
For products of groups we draw the following conclusion.
###### Proposition 2.7.
If $G=AB$, where $A$ and $B$ are subgroups of $G$ such that $[A,B]=1$, then
$\mathrm{Prob}_{r}(G)=\frac{1}{|A^{r}\cap
B^{r}|}\mathrm{Prob}_{r}(A)\mathrm{Prob}_{r}(B).$
In particular, if $A\cap B=1$, then
$\mathrm{Prob}_{r}(G)=\mathrm{Prob}_{r}(A)\mathrm{Prob}_{r}(B)$.
###### Proof.
Given $a\in A$ and $b\in B$, $(ab)^{r}=a^{r}b^{r}$ if and only if $[a,b]=1$.
Therefore $[A,B]=1$ implies $(AB)^{r}=A^{r}B^{r}$ and so
$|A^{r}B^{r}|=\frac{|A^{r}||B^{r}|}{|A^{r}\cap B^{r}|}$. Then
$\mathrm{Prob}_{r}(G)=\frac{|G^{r}|}{|G|}=\frac{|(AB)^{r}|}{|AB|}=\frac{|A^{r}B^{r}|}{|AB|}=\frac{1}{|A^{r}\cap
B^{r}|}\frac{|A^{r}|}{|A|}\frac{|B^{r}|}{|B|}$
$=\frac{\mathrm{Prob}_{r}(A)\mathrm{Prob}_{r}(B)}{|A^{r}\cap B^{r}|}.$
In particular, $A^{r}\cap B^{r}\subseteq A\cap B=1$ implies
$\mathrm{Prob}_{r}(G)=\mathrm{Prob}_{r}(A)\mathrm{Prob}_{r}(B)$. ∎
The next two results show bounds in terms of subgroups and quotients.
###### Proposition 2.8.
Let $N$ be a normal subgroup of a group $G$. Then
$\mathrm{Prob}_{r}(G)\leq\mathrm{Prob}_{r}(G/N).$
###### Proof.
Note that $gN\in G/N$ has an $r$–th root if and only if there is $xN\in G/N$
for which $gN=(xN)^{r}$, that is, $x^{r}\in gN$. Therefore $gN\in G/N$ does
not have an $r$–th root if and only if there is no element $x\in G$ with
$x^{r}\in gN$. Hence, if a coset in $G/N$ does not have an $r$–th root, then
no element of this coset has an $r$–th root in $G$, and therefore
$|G|-|G^{r}|\geq|N|(|G/N|-|(G/N)^{r}|)$ . By dividing both sides by $|G|$ we
obtain $1-\mathrm{Prob}_{r}(G)\geq 1-\mathrm{Prob}_{r}(G/N)$ and so
$\mathrm{Prob}_{r}(G)\leq\mathrm{Prob}_{r}(G/N)$, as required. ∎
###### Proposition 2.9.
Let $H$ be a subgroup of a group $G$. Then
$|G|^{-1}\mathrm{Prob}_{r}(H)\leq\mathrm{Prob}_{r}(G).$
###### Proof.
Obviously $H^{r}\subseteq G^{r}$ implies $|H^{r}|\leq|G^{r}|$. Therefore
$\mathrm{Prob}_{r}(H)\leq|H|\cdot\mathrm{Prob}_{r}(H)=\frac{|H|}{|H|}\cdot|H^{r}|\leq\frac{|G|}{|G|}\cdot|G^{r}|=|G|\cdot\mathrm{Prob}_{r}(G)$
implies $|G|^{-1}\mathrm{Prob}_{r}(H)\leq\mathrm{Prob}_{r}(G)$ and the lower
bound follows. ∎
The following result is a lower bound of general interest.
###### Corollary 2.10.
Let $G$ be a solvable group and $P$ be a Sylow $p$–subgroup of $G$ for some
prime $p$. Then $\frac{1}{|P|}\leq\mathrm{Prob}_{p}(G)$.
###### Proof.
Since $H$ is solvable, there exists a $p^{\prime}$–Hall subgroup $H$ of $G$
such that $|G|=|H||P|$ and $H=H^{p}\subseteq G^{p}$. Therefore,
$\mathrm{Prob}_{p}(G)=\frac{|G^{p}|}{|G|}\geq\frac{|H|}{|G|}=\frac{|H|}{|H||P|}=\frac{1}{|P|}.$
∎
## 3\. Projective special linear groups and density
###### Theorem 3.1.
Let $q$ be a prime power. If $q\equiv 1\mod 4$, $q$ is odd and
$r=\frac{q-1}{2}\geq 2$ is prime, then
$\mathrm{Prob}_{r}(\mathrm{PSL}(2,q))=\frac{r+1}{2r}.$
In particular, if $r=2$, then
$\mathrm{Prob}_{2}(\mathrm{PSL}(2,q))=\frac{3}{4}.$
###### Proof.
We recall that
$|\mathrm{PSL}(2,q)|=\frac{q(q-1)(q+1)}{\gcd(2,q-1)}=q(q-1)(q+1).$
Let $\nu$ be a generator of the multiplicative group of the field of $q$
elements. Denote
$1=\left(\begin{array}[]{cccccccc}1&0\\\ 0&1\\\ \end{array}\right),\
c=\left(\begin{array}[]{cccccccc}1&0\\\ 1&1\\\ \end{array}\right),\
d=\left(\begin{array}[]{cccccccc}1&0\\\ \nu&1\\\ \end{array}\right),\
a=\left(\begin{array}[]{cccccccc}\nu&0\\\ 0&\nu^{-1}\\\ \end{array}\right).$
and $b$ an element of order $q+1$ (Singer cycle) in $\mathrm{SL}(2,q)$. By
abuse of notation, we use the same symbols for the corresponding elements in
$\mathrm{PSL}(2,q)=\mathrm{SL}(2,q)/Z(\mathrm{SL}(2,q))$. From the character
table of $\mathrm{SL}(2,q)$ (see [4, Theorem 38.1]), one gets easily the
character table of $\mathrm{PSL}(2,q)$. We reproduce it below for the
convenience of the reader. The elements $1,c,d,a^{l}$ and $b^{m}$ for $1\leq
l\leq\frac{q-1}{4}$ and $1\leq m\leq\frac{q-1}{4}$ form a set of
representatives for the conjugacy classes of $\mathrm{PSL}(2,q)$. For $1\leq
l\leq\frac{q-1}{4}$ and $1\leq m\leq\frac{q-1}{4}$, one can see from [4,
Theorem 38.1] that
$|C_{\mathrm{PSL}(2,q)}(1)|=|G|,\ \
|C_{\mathrm{PSL}(2,q)}(c)|=|C_{\mathrm{PSL}(2,q)}(d)|=p,$
$|C_{\mathrm{PSL}(2,q)}(a^{l})|=\frac{q-1}{2},\ \
|C_{\mathrm{PSL}(2,q)}(a^{\frac{q-1}{4}})|=q-1,\ \
|C_{\mathrm{PSL}(2,q)}(b^{m})|=\frac{q+1}{2},$
where $p$ is the prime of which $q$ is power. Now we count the elements which
do not have $r$–th roots and will deduce the probability of having $r$–th
roots.
Since $\langle a\rangle\simeq C_{r}$, $\langle a\rangle[r]\simeq\langle
a\rangle$ and $\langle a\rangle^{r}=1$, Proposition 2.2 (i) implies
$\mathrm{Prob}_{r}(\langle a\rangle)=\frac{1}{r}$ so the elements not having
$r$–th roots in $\langle a\rangle$ are exactly
$|\langle a\rangle-\langle a\rangle^{r}|=|\langle a\rangle|-|\langle
a\rangle^{r}|=|\langle a\rangle|-1=r-1=\frac{q-1}{2}-1=\frac{q-3}{2}.$
On the other hand, the (distinct) conjugates of $\langle a\rangle$ have
trivial intersection with $\langle a\rangle$ so that the total number of
elements of $\mathrm{PSL}(2,q)$ which do not have $r$–th roots is obtained by
multiplying $\frac{q-3}{2}$ by the number of conjugates of $\langle a\rangle$
, which is $|\mathrm{PSL}(2,q):N_{\mathrm{PSL}(2,q)}(\langle a\rangle)|$. This
means that
$|\mathrm{PSL}(2,q)|-|\mathrm{PSL}(2,q)^{r}|=|\mathrm{PSL}(2,q)-\mathrm{PSL}(2,q)^{r}|$
$=|\langle a\rangle-\langle
a\rangle^{r}|\cdot|\mathrm{PSL}(2,q):N_{\mathrm{PSL}(2,q)}(\langle
a\rangle)|=\frac{q-3}{2}\cdot\frac{|\mathrm{PSL}(2,q)|}{|N_{\mathrm{PSL}(2,q)}(\langle
a\rangle)|}$ $=\frac{q-3}{2}\cdot\frac{|\mathrm{PSL}(2,q)|}{q-1}$
and, dividing both sides by $|\mathrm{PSL}(2,q)|$, we get
$1-\mathrm{Prob}_{r}(\mathrm{PSL}(2,q))=\frac{q-3}{2}\cdot\frac{1}{q-1},$
that is,
$\mathrm{Prob}_{r}(\mathrm{PSL}(2,q))=1-\frac{q-3}{2(q-1)}=\frac{q+1}{2(q-1)}=\frac{2(r+1)}{2(2r)}=\frac{r+1}{2r}.$
∎
We note that the case $r=2$, which appears in the previous theorem, was found
in [5, Proposition 3.1]. A consequence is the following.
###### Corollary 3.2.
Let $q$ be a prime power. If $q\equiv 1\mod 4$, $q$ is odd and
$r=\frac{q-1}{2}\geq 2$ is prime, then
$\lim_{r\rightarrow\infty}\mathrm{Prob}_{r}(\mathrm{PSL}(2,q))=\frac{1}{2}.$
The computations for the projective special linear groups are important in
order to get [3, Theorem 1.1] and to prove that the set
$Z=\\{\mathrm{Prob}_{2}(G)\ |\ G\ \mathrm{is}\ \mathrm{an}\
\mathrm{arbitrary}\ \mathrm{group}\\}$
is dense in [0,1]. We are going to generalize for any prime $r\geq 2$, and we
will not use projective special linear groups as done in [3, Theorem 1.1], but
will assume a priori the existence of a certain group with a prescribed value
of probability. This is justified by evidences of computational nature.
###### Corollary 3.3.
For any $\epsilon\in\mathbb{R}$ with $\epsilon>0$ and given a prime $r\geq 2$,
there exists an abelian group $A$ such that $0<\mathrm{Prob}_{r}(A)<\epsilon$.
###### Proof.
Let $k>1$ be such that $1/r^{k}<\epsilon$ and $A$ be an elementary $r$–group
of rank $k$. By Proposition 2.2 (ii), the result follows. ∎
A proof of Corollary 3.3 when $r=2$ can be found in [3]. Briefly, Corollary
3.3 shows that $0$ is an accumulation point for the set $X$ in Remark 2.1.
###### Corollary 3.4.
Assume that $r\geq 2$ is a prime and $S$ is a group such that
$\mathrm{Prob}_{r}(S)=1-\frac{1}{|R|}$ for an elementary abelian $r$–Sylow
subgroup $R$ of $S$. Then for any $\epsilon\in\mathbb{R}$ with $\epsilon>0$ we
have $1-\epsilon<\mathrm{Prob}_{r}(S)<1$.
###### Proof.
Since $R$ is a Sylow $r$–subgroup of $S$ which is elementary abelian of rank
$k$ for some $k\geq 1$, we have $1/r^{k}<\epsilon$ and
$\mathrm{Prob}_{r}(S)=1-\frac{1}{r^{k}}=\frac{r^{k}-1}{r^{k}}$. On the other
hand, $1-\epsilon<\frac{r^{k}-1}{r^{k}}<1$, therefore
$1-\epsilon<\mathrm{Prob}_{r}(S)<1$, as claimed. ∎
Corollary 3.4 when $r=2$ can be found in [3]. Also Corollary 3.3 is
illustrating that $1$ is an accumulation point for the set
$T=\\{\mathrm{Prob}_{r}(G)\ |\ G\ \mathrm{is}\ \mathrm{an}\
\mathrm{arbitrary}\ \mathrm{group}\\},$
where $r\geq 2$ is a given prime.
###### Theorem 3.5.
Let $r\geq 2$ be a prime and assume that there exists a group $H$ such that
$\mathrm{Prob}_{r}(H)=1-\frac{1}{|R|}$ for an elementary abelian $r$–Sylow
subgroup $R$ of $H$. Then the set $T$ is dense in $[0,1]$.
###### Proof.
By Corollaries 3.3, 3.4, there is no loss of generality in showing that, if
$0<x<1$, then $x$ is a limit point of $T$. There exists an integer $m$ such
that $1/r<r^{m}x<1$. Note that $(0,1)={\underset{m\geq
0}{\bigcup}}[1/r^{m+1},1/r^{m})$. Let $y=r^{m}x$. We can choose an integer
$n_{1}\geq 1$ such that
$(r^{n_{1}}-1)/r^{n_{1}}\leq y\leq(r^{n_{1}+1}-1)/r^{n_{1}+1},$
noting that $[1/r,1)={\underset{n\geq
1}{\bigcup}}[(r^{n}-1)/r^{n},(r^{n+1}-1)/r^{n+1})$. Let
$s_{1}=(r^{n_{1}}-1)/r^{n_{1}}$ and $r_{1}=(r^{n_{1}+1}-1)/r^{n_{1}+1}$. Again
we can choose an integer $n_{2}\geq 1$ such that
$(r^{n_{2}}-1)/r^{n_{2}}\leq y/r_{1}\leq(r^{n_{2}+1}-1)/r^{n_{2}+1},$
noting that $1/r\leq y/r_{1}<1$. As before, let
$s_{2}=(r^{n_{2}}-1)/r^{n_{2}}$ and $r_{2}=(r^{n_{2}+1}-1)/r^{n_{2}+1}$.
Iterating this process, there exist positive integers
$n_{1},n_{2},n_{3},\ldots$ and two sequences $\\{s_{i}\\}$ and $\\{r_{i}\\}$
such that $s_{i}=(r^{n_{i}}-1)/r^{n_{i}},r_{i}=(r^{n_{i}+1}-1)/r^{n_{i}+1}$
and $s_{i}\leq\frac{y}{r_{1}r_{2}\ldots r_{i-1}}<r_{i}$ for all $i\geq 1$. Of
course, $0<s_{i}<r_{i}<1$ for all $i\geq 1$. We have $n_{i}\leq n_{i+1}$ for
all $i\geq 1$, since
$s_{i}\leq\frac{y}{r_{1}r_{2}\ldots r_{i-1}}<\frac{y}{r_{1}r_{2}\ldots
r_{i-1}r_{i}}<r_{i+1}.$
Thus $\\{s_{i}\\}$ is a monotonically increasing sequence, bounded by 1, and
so convergent. Moreover, $\\{s_{i}\\}$ has infinitely many distinct terms;
otherwise $\\{s_{i}\\}$, and hence $\\{r_{i}\\}$, would be eventually
constant, and so, for some $j\geq 1$, we would have
$\frac{y}{r_{1}r_{2}\ldots r_{j-1}r^{k-1}_{j}}<r_{j}$
or $r_{1}r_{2}\ldots r_{j-1}r^{k}_{j}$ for $k\geq 1$. This is impossible,
since $y>0$ and ${\underset{k\rightarrow\infty}{\lim}}r^{k}_{j}=0$. Therefore,
$\\{s_{i}\\}$ converges to 1 (after omitting repeated terms), because it is a
subsequence of $\\{(r^{n}-1)/r^{n}\\}$. This allows us to note that the
sequence $\\{a_{i}\\}$ converges to 1, where $a_{i}=y/r_{1}r_{2}\ldots
r_{i-1}$. Consequently, the sequence $\\{b_{i}\\}$ converges to $y$, where
$b_{i}=r_{1}r_{2}\ldots r_{i-1}$. Thus we have
$\lim_{k\rightarrow\infty}\frac{r_{1}r_{2}\ldots
r_{i-1}}{r^{m}}=\frac{y}{r^{m}}=x.$
For each $i\geq 1$ we consider the group $G^{(i)}=G_{0}\times
G_{1}\times\ldots\times G_{i-1}$, where $G_{0}=C^{m}_{r}$ and $G_{k}$ is a
sequence of groups isomorphic for each $k$ to the group $H$, introduced in the
assumptions. Propositions 2.2 and 2.6 imply
$\mathrm{Prob}_{r}(G^{(i)})=\mathrm{Prob}_{r}(G_{0})\
\mathrm{Prob}_{r}(G_{1})\ldots\mathrm{Prob}_{r}(G_{i-1})=\frac{1}{r^{m}}r_{1}r_{2}\ldots
r_{i-1}.$
We have ${\underset{i\rightarrow\infty}{\lim}}\mathrm{Prob}_{r}(G^{(i)})=x$
and the result follows. ∎
## References
* [1] J. Blum, Enumeration of the square permutations in $S_{n}$, J. Comb. Theory Ser. A 17 (1974), 156-161.
* [2] M. B${\rm\acute{o}}$na, A. McLennan and D. White, Permutations with roots, Random structures and algorithms 17 (2) (2000), 157–167.
* [3] A.K. Das, On group elements having square roots, Bull. Iranian Math. Soc. 31 (2005), 33–36.
* [4] L. Dornhoff, Group Representation Theory, Part A, Marcel Dekker, New York, 1971.
* [5] M.S. Lucido and M.R. Pournaki, Elements with square roots in finite groups, Algebra Colloq. 12 (2005), 677–690.
* [6] M.S. Lucido and M.R. Pournaki, Probability that an element of a finite group has a square root, Colloq. Math. 112 (2008), 147–155.
* [7] N. Pouyanne, On the number of permutations admitting an m-th root, Electr. J. Comb. 9 (2002), 12 pp. (electronic).
* [8] F.G. Russo, Elements with square roots in compact groups, Asian Eur. J. Math. 3 (2010), 495–500.
|
arxiv-papers
| 2010-08-12T06:51:48 |
2024-09-04T02:49:12.144514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Elaheh Khamseh (Islamic Azad University, Mashhad, Iran), Mohammed Reza\n R. Moghaddam (Ferdowsi University of Mashhad, Mashhad, Iran), Francesco G.\n Russo (Universita' degli Studi di Palermo, Palermo, Italy) and Farshid Saeedi\n (Islamic Azad University, Mashhad, Iran)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1008.2055"
}
|
1008.2122
|
# Secret Key and Private Key Constructions for Simple Multiterminal Source
Models
Chunxuan Ye and Prakash Narayan Chunxuan Ye was with the Department of
Electrical and Computer Engineering, and the Institute for Systems Research,
University of Maryland, College Park, MD 20742. He is now with InterDigital
Communications, LLC, King of Prussia, PA 19406, USA. E-mail:
chunxuan.ye@interdigital.com.Prakash Narayan is with the Department of
Electrical and Computer Engineering, and the Institute for Systems Research,
University of Maryland, College Park, MD 20742, USA. E-mail:
prakash@eng.umd.edu.The work of C. Ye and P. Narayan was supported by the
National Science Foundation under Grants CCF0515124, CCF0635271 and
CCF0830697. The material in this paper was presented in part at the IEEE
International Symposium on Information Theory, Adelaide, Australia, Sept.
2005, and at the Information Theory and Applications Workshop, San Diego, CA,
Feb. 2006.
###### Abstract
We propose an approach for constructing secret and private keys based on the
long-known Slepian-Wolf code, due to Wyner, for correlated sources connected
by a virtual additive noise channel. Our work is motivated by results of
Csiszár and Narayan which highlight innate connections between secrecy
generation by multiple terminals that observe correlated source signals and
Slepian-Wolf near-lossless data compression. Explicit procedures for such
constructions and their substantiation are provided. The performance of low
density parity check channel codes in devising a new class of secret keys is
examined.
Index terms: Secret key construction, private key construction, secret key
capacity, private key capacity, Slepian-Wolf data compression, binary
symmetric channel, maximum likelihood decoding, LDPC codes.
## I Introduction
The problem of secrecy generation by multiple terminals, based on their
observations of separate but correlated signals followed by public
communication among themselves, has been investigated by several authors
([23], [2], [7], among others). It has been shown that these terminals can
generate secrecy, namely “common randomness” which is kept secret from an
eavesdropper that is privy to said public communication and perhaps also to
additional “wiretapped” side information.
Our work is motivated by [8] which studies secrecy generation for
multiterminal “source models” with an arbitrary number of terminals, each of
which observes a distinct component of a discrete memoryless multiple source
(DMMS). Specifically, suppose that $d\geq 2$ terminals observe, respectively,
$n$ independent and identically distributed (i.i.d.) repetitions of finite-
valued random variables (rvs) $X_{1},\ldots,X_{d}$, denoted by ${\bf
X}_{1},\ldots,{\bf X}_{d}$, where ${\bf
X}_{i}=\left(X_{i1},\ldots,X_{in}\right),\ i=1,\ldots,d$. Thereupon,
unrestricted and noiseless public communication is allowed among the
terminals. All such communication is observed by all the terminals and by the
eavesdropper. The eavesdropper is assumed to be passive, i.e., unable to
tamper with the public communication of the terminals. In this framework, two
models considered in [8] dealing with a secret key (SK) and a private key (PK)
are pertinent to our work.
(i) Secret key: Suppose that all the terminals in $\\{1,\ldots,d\\}$ wish to
generate a SK, i.e., common randomness which is concealed from the
eavesdropper with access to their public communication and which is nearly
uniformly distributed 111In [8], a general situation is studied in which a
subset of the terminals generate a SK with the cooperation of the remaining
terminals.. The largest (entropy) rate of such a SK, termed the SK capacity
and denoted by $C_{S}$, is shown in [8] to equal
$C_{S}=H(X_{1},\cdots,X_{d})-R_{min},$ (1)
where
$R_{min}=\min_{(R_{1},\cdots,R_{d})\in{\cal R}}\ \ \sum_{i=1}^{d}R_{i},$ (2)
with222Here, $\subset$ denotes a proper subset.
$\displaystyle{\cal R}$ $\displaystyle=$
$\displaystyle\\{(R_{1},\cdots,R_{d}):\sum_{i\in B}R_{i}\geq$ (3)
$\displaystyle H(\\{X_{j},\ j\in B\\}|\\{X_{j},\ j\in
B^{c}\\}),B\subset\\{1,\cdots,d\\}\\},$
where $B^{c}=\\{1,\cdots,d\\}\backslash B$.
(ii) Private key: For a given subset $A\subset\\{1,\cdots,d\\}$, a PK for the
terminals in $A$, private from the terminals in $A^{c}$, is a SK generated by
the terminals in $A$ with the cooperation of the terminals in $A^{c}$, which
is concealed from an eavesdropper with access to the public interterminal
communication and also from the cooperating terminals in $A^{c}$ (and, hence,
private) 333A general model is considered in [8] for privacy from a subset of
$A^{c}$ of the cooperating terminals.. The largest (entropy) rate of such a
PK, termed the PK capacity and denoted by $C_{P}(A)$, is shown in [8] to be
$\displaystyle C_{P}(A)$ $\displaystyle=$ $\displaystyle
H(X_{1},\cdots,X_{d})-H(\\{X_{i},\ i\in A^{c}\\})-R_{min}(A)$ (4)
$\displaystyle=$ $\displaystyle H(\\{X_{i},\ i\in A\\}|\\{X_{i},\ i\in
A^{c}\\})-R_{min}(A),$
where
$R_{min}(A)=\min_{\\{R_{i},\ i\in A\\}\in{\cal R}(A)}\ \ \sum_{i\in A}R_{i},$
(5)
with
$\displaystyle{\cal R}(A)$ $\displaystyle=$ $\displaystyle\\{\\{R_{i},\ i\in
A\\}:\sum_{i\in B}R_{i}\geq$ (6) $\displaystyle H(\\{X_{j},\ j\in
B\\}|\\{X_{j},\ j\in B^{c}\\}),\ B\subset A\\}.$
The expressions in (1)–(3) and (4)–(6) afford the following interpretation
[8]. The joint entropy $H\left(X_{1},\ldots,X_{d}\right)$ in (1) corresponds
to the maximum rate of shared common randomness – sans secrecy constraints –
that can ever be achieved by the terminals in $\\{1,\ldots,d\\}$ when each
terminal becomes omniscient, i.e., reconstructs all the components of the DMMS
with probability $\cong 1$ as the observation length $n$ becomes large.
Further, $R_{min}$ in (2), (3) corresponds to the smallest aggregate rate of
interterminal communication that enables every terminal to achieve omniscience
[8]. Thus, from (1), the SK capacity $C_{S}$, i.e., largest rate at which all
the terminals in $\\{1,\ldots,d\\}$ can generate a SK, is obtained by
subtracting from the maximum rate of shared common randomness achievable by
these terminals, viz. $H(X_{1},\cdots,X_{d})$, the smallest overall rate
$R_{min}$ of the (data-compressed) interterminal communication that enables
all the terminals to become omniscient. A similar interpretation holds for the
PK capacity $C_{P}(A)$ in (4) as well, with the difference that the terminals
in $A^{c}$, which cooperate in secrecy generation and yet must not be privy to
the secrecy they help generate, can be assumed – without loss of generality –
to simply “reveal” their observations [8]. Hence, the entropy terms in (1),
(3) are now replaced in (4), (6) with additional conditioning on $\\{X_{i},\
i\in A^{c}\\}$. It should be noted that $R_{min}$ and $R_{min}(A)$ are
obtained as solutions to multiterminal Slepian-Wolf (SW) (near-lossless) data
compression problems not involving any secrecy constraints.
The form of characterization of the SK and PK capacities in (1) and (4) also
suggests successive steps for generating the corresponding keys. For instance,
and loosely speaking, in order to generate a SK, the terminals in
$\\{1,\ldots,d\\}$ first generate common randomness (without any secrecy
restrictions) using SW-compressed interterminal communication denoted
collectively by, say, ${\bf F}$. Thus, the terminals generate rvs
$L_{i}=L_{i}({\bf X}_{i},{\bf F}),\ i\in\\{1,\ldots,d\\}$, with
$\frac{1}{n}H(L_{i})>0$, which agree with probability $\cong 1$ for $n$
suitably large; suppressing subscripts, let $L$ denote the resulting “common”
rv where $\frac{1}{n}H(L)>0$. The second step entails an extraction from $L$
of a SK $K=g(L)$ of entropy rate $\frac{1}{n}H(L|{\bf F})$ by means of a
suitable operation $g$ performed identically at each terminal on the acquired
common randomness $L$. In particular, when the common randomness acquired by
the terminals corresponds to omniscience, i.e., $L\cong\left({\bf
X}_{1},\dots,{\bf X}_{d}\right)$, and is achieved using interterminal
communication ${\bf F}$ of the most parsimonious rate $\cong R_{min}$ in (2),
then the corresponding SK $K=g(L)$ has the best rate $C_{S}$ given by (1). It
is important to note, however, that as mentioned in ([8], Section VI) and
already known from [23], [2], neither communication by every terminal nor
omniscience is essential for generating secrecy (SK or PK) at the best rate;
for instance, the rv $L$ above need not correspond to omniscience for the SK
$K=g(L)$ to have the best possible rate in (1).
A similar approach as above can be used to generate a PK of the largest rate
in (4).
The discussion above suggests that techniques for SW data compression could be
used to devise constructive schemes for obtaining SKs and PKs that achieve the
corresponding capacities. Further, in SW data compression, the existence of
linear encoders of rates arbitrarily close to the SW bound has been long known
[5]. In the special situation when the i.i.d. sequences observed at the
terminals are related to each other in probability law through virtual
discrete memoryless channels (DMCs) characterized by independent additive
noises, such linear SW encoders can be obtained in terms of cosets of linear
error correction codes for such virtual channels, a fact first illustrated in
[37] for the case of $d=2$ terminals connected by a virtual binary symmetric
channel (BSC), and later exploited in most known linear constructions of SW
encoders (cf. e.g., [1], [4], [11], [12], [15]-[17], [19], [20], [24], [29],
[33]). When the i.i.d. sequences observed by $d=2$ terminals are connected by
an arbitrary virtual DMC, the corresponding SW data compression can be viewed
in terms of coding for a “semisymmetric” channel, i.e., a channel with
independent additive noise that is defined over an enlarged alphabet [14]; the
case of stationary ergodic observations at the terminals is also considered
therein. These developments in SW data compression can translate into an
emergence of new constructive schemes for secrecy generation.
Motivated by these considerations, we seek to devise new constructive schemes
for secrecy generation in source models in which SW data compression plays a
central role. The main technical contribution of this work is the following:
Considering four simple models of secrecy generation, we show how a new class
of SKs and PKs can be devised for them at rates arbitrarily close to the
corresponding capacities, relying on the SW data compression code in [37].
Additionally, we examine the performance of low density parity check (LDPC)
codes in the SW data compression step of the procedure for secrecy generation.
Preliminary results of this work have been reported in [38], [39]. In
independent work [25] for the case of $d=2$ terminals which is akin to but
different from ours, extraction of a SK from previously acquired common
randomness by means of a linear transformation has been demonstrated.
In related work, SK generation for a source model with two terminals that
observe continuous-amplitude signals, has been studied in [40], [36], [26],
[27], [41]. Furthermore, in recent years, several secrecy generation schemes
have been reported, relying on capacity-achieving channel codes, for “wiretap”
secrecy models that differ from ours. For instance, it was shown in [35] that
such a channel code can attain the secrecy capacity for any wiretap channel.
See also [3], [18].
The paper is organized as follows. Preliminaries are contained in Section II.
In Section III, we consider four simple source models for which we provide
elementary constructive schemes for SK or PK generation which rely on suitable
SW data compression codes; the keys thereby generated are shown to satisfy the
requisite secrecy and rate-optimality conditions in Section IV.
Implementations of these constructions using LDPC codes are illustrated in
Section V which also reports simulation results. Section VI contains closing
remarks.
## II Preliminaries
### II-A Secret Key and Private Key Capacities
Consider a DMMS with $d\geq 2$ components, with corresponding generic rvs
$X_{1},\cdots,X_{d}$ taking values in finite alphabets ${\cal
X}_{1},\cdots,{\cal X}_{d}$, respectively. Let ${\bf
X}_{i}=(X_{i,1},\cdots,X_{i,n})$ be $n$ i.i.d. repetitions of rv $X_{i}$,
$i\in{\cal D}=\\{1,\cdots,d\\}$. Terminals $1,\cdots,d$, with respective
observations ${\bf X}_{1},\cdots,{\bf X}_{d}$, represent the $d$ users that
wish to generate a SK by means of public communication. These terminals can
communicate with each other through broadcasts over a noiseless public
channel, possibly interactively in many rounds. In general, a communication
from a terminal is allowed to be any function of its observations, and of all
previous communication. Let ${\bf F}$ denote collectively all the public
communication.
Given $\varepsilon>0$, the rv $K_{\cal S}$ represents an $\varepsilon$-secret
key ($\varepsilon$-SK) for the terminals in ${\cal D}$, achieved with
communication ${\bf F}$, if there exist rvs $K_{i}=K_{i}({\bf X}_{i},{\bf
F})$, $i\in{\cal D}$, with $K_{i}$ and $K_{\cal S}$ taking values in the same
finite set ${\cal K_{S}}$, such that $K_{\cal S}$ satisfies
$\bullet$ the common randomness condition
$\Pr\\{K_{i}=K_{\cal S},\ i\in{\cal D}\\}\geq 1-\varepsilon;$
$\bullet$ the secrecy condition
$\frac{1}{n}I(K_{\cal S}\wedge{\bf F})\leq\varepsilon;$
and
$\bullet$ the uniformity condition
$\frac{1}{n}H(K_{\cal S})\geq\frac{1}{n}\log|{\cal K}_{\cal S}|-\varepsilon.$
Let $A\subset{\cal D}$ be an arbitrary subset of the terminals. The rv
$K_{\cal P}(A)$ represents an $\varepsilon$-private key ($\varepsilon$-PK) for
the terminals in $A$, private from the terminals in $A^{c}={\cal D}\backslash
A$, achieved with communication ${\bf F}$, if there exist rvs
$K_{i}=K_{i}({\bf X}_{i},{\bf F})$, $i\in A$, with $K_{i}$ and $K_{\cal P}(A)$
taking values in the same finite set ${\cal K_{P}}(A)$, such that $K_{\cal
P}(A)$ satisfies
$\bullet$ the common randomness condition
$\Pr\\{K_{i}=K_{\cal P}(A),\ i\in A\\}\geq 1-\varepsilon;$
$\bullet$ the secrecy condition
$\frac{1}{n}I\left(K_{\cal P}(A)\wedge\\{{\bf X}_{i},\ i\in A^{c}\\},{\bf
F}\right)\leq\varepsilon;$
and
$\bullet$ the uniformity condition
$\frac{1}{n}H(K_{\cal P}(A))\geq\frac{1}{n}\log\left|{\cal K}_{\cal
P}(A)\right|-\varepsilon.$
Definition 1 [8]: A nonnegative number $R$ is called an achievable SK rate if
$\varepsilon_{n}$-SKs $K_{\cal S}^{(n)}$ are achievable with suitable
communication (with the number of rounds possibly depending on $n$), such that
$\varepsilon_{n}\rightarrow 0$ and $\frac{1}{n}H\left(K_{\cal
S}^{(n)}\right)\rightarrow R$. The largest achievable SK rate is called the SK
capacity, denoted by $C_{S}$. The PK capacity for the terminals in $A$,
denoted by $C_{P}(A)$, is similarly defined. An achievable SK rate (resp. PK
rate) will be called strongly achievable if $\varepsilon_{n}$ above can be
taken to vanish exponentially in $n$. The corresponding capacities are termed
strong capacities.
Single-letter characterizations have been obtained for $C_{S}$ in the case of
$d=2$ terminals in [2], [23] and for $d\geq 2$ terminals in [8], given by (1);
and for $C_{P}(A)$ in the case of $d=3$ terminals in [2] and for $d\geq 3$
terminals in [8], given by (4). The proofs of the achievability parts exploit
the close connection between secrecy generation and SW data compression.
Loosely speaking, common randomness sans any secrecy restrictions is first
generated through SW-compressed interterminal communication, whereby all the
$d$ terminals acquire a (common) rv with probability $\cong 1$. In the next
step, secrecy is then extracted by means of a suitable identical operation
performed at each terminal on the acquired common randomness. When the common
randomness initially acquired by the $d$ terminals is maximal, the
corresponding SK has the best rate $C_{S}$ given by (1).
In this work, we consider four simple models for which we illustrate the
constructions of appropriate strong SKs or PKs.
### II-B Linear Codes for the Binary Symmetric Channel
The SW codes of interest will rely on the following classic result concerning
the existence of “good” linear channel codes for a BSC. A BSC with crossover
probability $p$, $0<p<\frac{1}{2}$, will be denoted by BSC($p$). Let
$h(p)=-p\log_{2}p-(1-p)\log_{2}(1-p)$ denote the binary entropy function.
Lemma 1 [9]: For every $\varepsilon>0$, $0<p<\frac{1}{2}$, and for all $n$
sufficiently large, there exists a binary linear $(n,n-m)$ code for a
BSC($p$), with $m<n[h(p)+\varepsilon]$, such that the average error
probability of maximum likelihood decoding is less than $2^{-n\eta}$, for some
$\eta>0$.
### II-C Types and Typical Sequences
The following standard facts regarding “types” and “typical sequences” and
their pertinent properties (cf. e.g., [6]) are compiled here in brief for
ready reference.
Given finite sets ${\cal X}$, ${\cal Y}$, the type of a sequence ${\bf
x}=(x_{1},\cdots,x_{n})\in{\cal X}^{n}$, ${\cal X}$ a finite set, is the
probability mass function (pmf) $P_{\bf x}$ on ${\cal X}$ given by
$P_{\bf x}(a)=\frac{1}{n}|\\{i:x_{i}=a\\}|,\ \ \ a\in{\cal X},$
and the joint type of a pair of sequences $({\bf x},{\bf y})\in{\cal
X}^{n}\times{\cal Y}^{n}$ is the joint pmf $P_{\bf xy}$ on ${\cal
X}\times{\cal Y}$ given by
$P_{\bf xy}(a,b)=\frac{1}{n}|\\{i:x_{i}=a,y_{i}=b\\}|,\ \ \ a\in{\cal X},\
b\in{\cal Y}.$
The numbers of different types of sequences in ${\cal X}^{n}$ (resp. ${\cal
X}^{n}\times{\cal Y}^{n}$) do not exceed $(n+1)^{|{\cal X}|}$ (resp.
$(n+1)^{|{\cal X}||{\cal Y}|}$).
Given rvs $X$, $Y$ (taking values in ${\cal X}$, ${\cal Y}$, respectively),
with joint pmf $P_{XY}$ on ${\cal X}\times{\cal Y}$, the set of sequences in
${\cal X}^{n}$ which are $X$-typical with constant $\xi$, denoted by
$T_{X,\xi}^{n}$, is defined as
$T_{X,\xi}^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\left\\{{\bf x}\in{\cal
X}^{n}:2^{-n[H(X)+\xi]}\leq P_{X}^{n}({\bf x})\leq 2^{-n[H(X)-\xi]}\right\\},$
where $P_{X}^{n}({\bf x})\stackrel{{\scriptstyle\triangle}}{{=}}\Pr\\{{\bf
X}={\bf x}\\}$, ${\bf x}\in{\cal X}^{n}$; and the set of pairs of sequences in
${\cal X}^{n}\times{\cal Y}^{n}$ which are $XY$-typical with constant $\xi$,
denoted by $T_{XY,\xi}^{n}$, is defined as
$\displaystyle T_{XY,\xi}^{n}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\\{({\bf
x},{\bf y})\in{\cal X}^{n}\times{\cal Y}^{n}:{\bf x}\in T_{X,\xi}^{n},{\bf
y}\in T_{Y,\xi}^{n},$ $\displaystyle 2^{-n[H(X,Y)+\xi]}\leq P_{XY}^{n}({\bf
x},{\bf y})\leq 2^{-n[H(X,Y)-\xi]}\\},$
where $P_{XY}^{n}({\bf x},{\bf
y})\stackrel{{\scriptstyle\triangle}}{{=}}\Pr\\{{\bf X}={\bf x},{\bf Y}={\bf
y}\\}$, ${\bf x}\in{\cal X}^{n}$, ${\bf y}\in{\cal Y}^{n}$. It readily follows
that for every $({\bf x},{\bf y})\in T_{XY,\xi}^{n}$,
$2^{-n[H(X|Y)+2\xi]}\leq P_{X|Y}^{n}({\bf x}|{\bf y})\leq
2^{-n[H(X|Y)-2\xi]},$
where $P_{X|Y}^{n}({\bf x}|{\bf
y})\stackrel{{\scriptstyle\triangle}}{{=}}\Pr\\{{\bf X}={\bf x}|{\bf Y}={\bf
y}\\}$, ${\bf x}\in{\cal X}^{n}$, ${\bf y}\in{\cal Y}^{n}$.
For every ${\bf y}\in{\cal Y}^{n}$, the set of sequences in ${\cal X}^{n}$
which are $X|Y$-typical with respect to ${\bf y}$ with constant $\xi$, denoted
by $T_{X|Y,\xi}^{n}({\bf y})$, is defined as
$T_{X|Y,\xi}^{n}({\bf y})\stackrel{{\scriptstyle\triangle}}{{=}}\left\\{{\bf
x}\in{\cal X}^{n}:({\bf x},{\bf y})\in T_{XY,\xi}^{n}\right\\},$
with $T_{X|Y,\xi}^{n}({\bf y})=\phi$ if ${\bf y}\not\in T_{Y,\xi}^{n}$. The
following is an independent and explicit statement of the well-known fact that
the probability of a nontypical set decays to 0 exponentially rapidly in $n$
(cf. e.g., [42, Theorem 6.3]).
Proposition 1: Given a joint pmf $P_{XY}$ on ${\cal X}\times{\cal Y}$ with
$P_{XY}(x,y)>0$, $x\in{\cal X}$, $y\in{\cal Y}$, for every $\xi>0$,
$\sum_{{\bf x}\in T_{X,\xi}^{n}}P_{X}^{n}({\bf x})\geq 1-(n+1)^{|{\cal
X}|}\cdot 2^{-n\frac{\xi^{2}}{2\ln 2\left[\sum_{a\in{\cal
X}}\log\frac{1}{P_{X}(a)}\right]^{2}}},$ (7)
and
$\displaystyle\sum_{({\bf x},{\bf y})\in T_{XY,\xi}^{n}}P_{XY}^{n}({\bf
x},{\bf y})$ (8) $\displaystyle\geq$ $\displaystyle 1-(n+1)^{|{\cal X}||{\cal
Y}|}\cdot 2^{-n\frac{\xi^{2}}{2\ln 2\left[\sum_{(a,b)\in{\cal X}\times{\cal
Y}}\log\frac{1}{P_{XY}(a,b)}\right]^{2}}},$
for all $n\geq 1$.
Proof: See Appendix A.
## III Main Results
We now present our main results on SK generation for three specific models,
and PK generation for a fourth model. The proofs of the accompanying Theorems
1 - 4 are provided in Section IV.
Model 1: Let the terminals $1$ and $2$ observe, respectively, $n$ i.i.d.
repetitions of the $\\{0,1\\}$-valued rvs $X_{1}$ and $X_{2}$ with joint pmf
$\displaystyle P_{X_{1}X_{2}}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-p)\delta_{x_{1}x_{2}}+\frac{1}{2}p\
(1-\delta_{x_{1}x_{2}}),$ (9) $\displaystyle\hskip 72.26999pt0<p<\frac{1}{2},$
with $\delta$ being the Kronecker delta function. These terminals wish to
generate a strong SK of maximum rate.
The (strong) SK capacity for this model [2], [8], [23], given by (1), is
$C_{S}=I(X_{1}\wedge X_{2})=1-h(p).$
We show a simple scheme for the terminals to generate a SK with rate close to
$1-h(p)$, which relies on Wyner’s well-known method for SW data compression
[37]. The SW problem of interest entails terminal $2$ reconstructing the
observed sequence ${\bf x}_{1}$ at terminal $1$ from the SW codeword for ${\bf
x}_{1}$ and its own observed sequence ${\bf x}_{2}$.
Observe that under the given joint pmf (9), ${\bf X}_{2}$ can be considered as
an input to a virtual BSC($p$), with corresponding output ${\bf X}_{1}$, i.e.,
we can write
${\bf X}_{1}={\bf X}_{2}\oplus{\bf V},$ (10)
where ${\bf V}=(V_{1},\cdots,V_{n})$ is an i.i.d. sequence of
$\\{0,1\\}$-valued rvs, independent of ${\bf X}_{2}$, and with
$\Pr\\{V_{i}=1\\}=p$, $1\leq i\leq n$.
(i) SW data compression [37]: Let ${\cal C}$ be a linear $(n,n-m)$ code as in
Lemma 1 with parity check matrix ${\bf P}$. Both terminals know ${\cal C}$
(and ${\bf P}$). Terminal $1$ communicates the syndrome ${\bf P}{\bf
x}_{1}^{t}$ to terminal $2$. The maximum likelihood estimate of ${\bf x}_{1}$
at terminal 2 is:
${\hat{\bf x}_{2}}(1)={\bf x}_{2}\oplus f_{\bf P}({\bf P}{\bf
x}_{1}^{t}\oplus{\bf P}{\bf x}_{2}^{t}),$
where $f_{\bf P}({\bf P}{\bf x}_{1}^{t}\oplus{\bf P}{\bf x}_{2}^{t})$ is the
most likely sequence ${\bf v}\in\\{0,1\\}^{n}$ (under the pmf of ${\bf V}$ as
above) with syndrome ${\bf P}{\bf v}^{t}={\bf P}{\bf x}_{1}^{t}\oplus{\bf
P}{\bf x}_{2}^{t}$, with $\oplus$ denoting addition modulo 2 and $t$ denoting
transposition. Note that in a standard array corresponding to the code ${\cal
C}$ above, $f_{\bf P}({\bf P}{\bf x}_{1}^{t}\oplus{\bf P}{\bf x}_{2}^{t})$ is
simply the coset leader of the coset with syndrome ${\bf P}{\bf
x}_{1}^{t}\oplus{\bf P}{\bf x}_{2}^{t}$. Also, ${\bf x}_{1}$ and ${\hat{\bf
x}_{2}}(1)$ lie in the same coset.
The probability of decoding error at terminal $2$ is given by
$\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf X}_{1}\\}=\Pr\\{{\bf X}_{2}\oplus f_{\bf
P}({\bf P}{\bf X}_{1}^{t}\oplus{\bf P}{\bf X}_{2}^{t})\neq{\bf X}_{1}\\},$
and it readily follows from (10) that
$\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf X}_{1}\\}=\Pr\\{f_{\bf P}({\bf P}{\bf
V}^{t})\neq{\bf V}\\}.$
By Lemma 1, $\Pr\\{f_{\bf P}({\bf P}{\bf V}^{t})\neq{\bf V}\\}<2^{-n\eta}$ for
some $\eta>0$ and for all $n$ sufficiently large, so that
$\Pr\\{{\hat{\bf X}_{2}}(1)={\bf X}_{1}\\}\geq 1-2^{-n\eta}.$
(ii) SK construction: Consider a (common) standard array for ${\cal C}$ known
to both terminals. Denote by ${\bf a}_{i,j}$ the element of the $i^{th}$ row
and the $j^{th}$ column in the standard array, $1\leq i\leq 2^{m}$, $1\leq
j\leq 2^{n-m}$.
Terminal $1$ sets $K_{1}=j_{1}$ if ${\bf X}_{1}$ equals ${\bf a}_{i,j_{1}}$ in
its coset $i$ in the standard array. Terminal $2$ sets $K_{2}=j_{2}$ if
${\hat{\bf X}_{2}}(1)$ equals ${\bf a}_{i,j_{2}}$ in the coset $i$ of the same
standard array.
The following theorem asserts that $K_{1}$ constitutes a strong SK with rate
approaching SK capacity.
Theorem 1: Let $\varepsilon>0$ be given. Then for some $\eta>0$ and for all
$n$ sufficiently large, the pair of rvs $(K_{1},K_{2})$ generated above, with
(common) range ${\cal K}_{1}$ (say), satisfy
$\Pr\\{K_{1}=K_{2}\\}\geq 1-2^{-n\eta},$ (11) $I(K_{1}\wedge{\bf F})=0,$ (12)
$H(K_{1})=\log|{\cal K}_{1}|,$ (13)
and
$\frac{1}{n}H(K_{1})>1-h(p)-\varepsilon.$ (14)
Remark: The probability of $K_{1}$ differing from $K_{2}$ equals exactly the
average error probability of maximum likelihood decoding when ${\cal C}$ is
used on a BSC($p$). Furthermore, the gap between the rate of the generated SK
and SK capacity equals the gap between the rate of ${\cal C}$ and channel
capacity.
Model 2: Let the terminals $1$ and $2$ observe, respectively, $n$ i.i.d.
repetitions of the $\\{0,1\\}$-valued rvs with joint pmf
$\displaystyle P_{X_{1}X_{2}}(0,0)$ $\displaystyle=$
$\displaystyle(1-p)(1-q),$ $\displaystyle P_{X_{1}X_{2}}(0,1)$
$\displaystyle=$ $\displaystyle pq,$ $\displaystyle P_{X_{1}X_{2}}(1,0)$
$\displaystyle=$ $\displaystyle p(1-q),$ $\displaystyle P_{X_{1}X_{2}}(1,1)$
$\displaystyle=$ $\displaystyle q(1-p),$ (15)
with $0<p<\frac{1}{2}$ and $0<q<1$. These terminals wish to generate a strong
SK of maximum rate.
Note that Model 1 is a special case of Model 2 for $q=\frac{1}{2}$. We show
below a scheme for the terminals to generate a SK with rate close to the
(strong) SK capacity for this model [2], [8], [23], which is given by (1) as
$C_{S}=I(X_{1}\wedge X_{2})=h(p+q-2pq)-h(p).$
(i) SW data compression: This step is identical to step (i) for Model 1. Note
that under the given joint pmf (15), ${\bf X}_{1}$ and ${\bf X}_{2}$ can be
written as in (10). It follows in the same manner as for Model 1 that for some
$\eta>0$ and for all $n$ sufficiently large,
$\Pr\\{{\hat{\bf X}_{2}}(1)={\bf X}_{1}\\}\geq 1-2^{-n\eta}.$
(ii) SK construction: Both terminals know the linear $(n,n-m)$ code ${\cal C}$
as in Lemma 1, and a (common) standard array for ${\cal C}$. Let $\\{{\bf
e}_{i}:1\leq i\leq 2^{m}\\}$ denote the set of coset leaders for all the
cosets of ${\cal C}$.
Denote by $A_{i}$ the set of sequences from $T_{X_{1},\xi}^{n}$ in the coset
of ${\cal C}$ with coset leader ${\bf e}_{i}$, $1\leq i\leq 2^{m}$. If the
number of sequences of the same type in $A_{i}$ is more than
$2^{n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]}$, where
$\varepsilon^{\prime}>\xi+\varepsilon$ with $\varepsilon$ satisfying
$m<n[h(p)+\varepsilon]$ in Lemma 1, then collect arbitrarily
$2^{n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]}$ such sequences to compose a
subset, which we term a regular subset (as it consists of sequences of the
same type). Continue this procedure until the number of sequences of every
type in $A_{i}$ is less than $2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}$. Let $N_{i}$ denote the number of distinct
regular subsets of $A_{i}$.
Enumerate (in any way) the sequences in each regular subset. Let ${\bf
b}_{i,j,k}$, where $1\leq i\leq 2^{m}$, $1\leq j\leq N_{i}$, $1\leq k\leq
2^{n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]}$, denote the $k^{th}$
sequence of the $j^{th}$ regular subset in the $i^{th}$ coset (with coset
leader ${\bf e}_{i}$).
Terminal $1$ sets $K_{1}=k_{1}$ if ${\bf X}_{1}$ equals ${\bf
b}_{i,j_{1},k_{1}}$; else, $K_{1}$ is set to be uniformly distributed on
$\left\\{1,\cdots,2^{n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]}\right\\}$,
independent of $({\bf X}_{1},{\bf X}_{2})$. Terminal $2$ sets $K_{2}=k_{2}$ if
${\hat{\bf X}_{2}}(1)$ equals ${\bf b}_{i,j_{2},k_{2}}$; else, $K_{2}$ is set
to be uniformly distributed on $\left\\{1,\cdots,2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}\right\\}$, independent of $({\bf X}_{1},{\bf
X}_{2},K_{1})$.
The following theorem says that $K_{1}$ constitutes a strong SK with rate
approaching SK capacity.
Theorem 2: Let $\varepsilon>0$ be given. Then for some
$\eta^{\prime}=\eta^{\prime}(\eta,\xi,\varepsilon,\varepsilon^{\prime})>0$ and
for all $n$ sufficiently large, the pair of rvs $(K_{1},K_{2})$ generated
above, with range ${\cal K}_{1}$ (say), satisfy
$\Pr\\{K_{1}=K_{2}\\}\geq 1-2^{-n\eta^{\prime}},$ (16) $I(K_{1}\wedge{\bf
F})=0,$ (17) $H(K_{1})=\log|{\cal K}_{1}|,$ (18)
and
$\frac{1}{n}H(K_{1})=h(p+q-2pq)-h(p)-\varepsilon^{\prime}.$ (19)
The next model is an instance of a Markov chain on a tree (cf. [13], [8]).
Consider a tree ${\cal T}$ with vertex set $V({\cal T})=\\{1,\cdots,d\\}$ and
edge set $E({\cal T})$. For $(i,j)\in E({\cal T})$, let $B(i\leftarrow j)$
denote the set of all vertices connected with $j$ by a path containing the
edge $(i,j)$. The rvs $X_{1},\cdots,X_{d}$ form a Markov chain on the tree
${\cal T}$ if for each $(i,j)\in E({\cal T})$, the conditional pmf of $X_{j}$
given $\\{X_{l},l\in B(i\leftarrow j)\\}$ depends only on $X_{i}$ (i.e., is
conditionally independent of $\\{X_{l},l\in B(i\leftarrow
j)\\}\backslash\\{X_{i}\\}$, conditioned on $X_{i}$). Note that when ${\cal
T}$ is a chain, this concept reduces to that of a standard Markov chain.
Model 3: Let the terminals $1,\cdots,d$ observe, respectively, $n$ i.i.d.
repetitions of $\\{0,1\\}$-valued rvs $X_{1},\cdots,X_{d}$ that form a Markov
chain on the tree ${\cal T}$, with joint pmf $P_{X_{1}\cdots X_{d}}$ specified
as: for $(i,j)\in E({\cal T})$,
$\displaystyle P_{X_{i}X_{j}}(x_{i},x_{j})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-p_{(i,j)})\delta_{x_{i}x_{j}}+\frac{1}{2}p_{(i,j)}\
(1-\delta_{x_{i}x_{j}}),$ $\displaystyle\hskip
72.26999pt0<p_{(i,j)}<\frac{1}{2},$
for $x_{i},x_{j}\in\\{0,1\\}$. These $d$ terminals wish to generate a strong
SK of maximum rate.
Note that Model 1 is a special case of Model 3 for $d=2$. Without any loss of
generality, let
$p_{max}=p_{(i^{*},j^{*})}=\max_{(i,j)\in E({\cal T})}p_{(i,j)}.$
Then, the (strong) SK capacity for this model [8] is given by (1) as
$C_{S}=I(X_{i^{*}}\wedge X_{j^{*}})=1-h(p_{max}).$
We show how to extract a SK with rate close to $1-h(p_{max})$ by using an
extension of the SW data compression scheme of Model 1 for reconstructing
${\bf x}_{i^{*}}$ at all the terminals.
(i) SW data compression: Let ${\cal C}$ be the linear $(n,n-m)$ code as in
Lemma 1 for a BSC($p_{max}$), and with parity check matrix ${\bf P}$. Each
terminal $i$ communicates the syndrome ${\bf P}{\bf x}_{i}^{t}$, $1\leq i\leq
d$.
Let ${\hat{\bf x}}_{i}(j)$ denote the corresponding maximum likelihood
estimate of ${\bf x}_{j}$ at terminal $i$, $1\leq i\neq j\leq d$. For a
terminal $i\neq i^{*}$, denote by $(i_{0},i_{1},\cdots,i_{r})$ the (only) path
in the tree ${\cal T}$ from $i$ to $i^{*}$, where $i_{0}=i$ and $i_{r}=i^{*}$;
this terminal $i$, with the knowledge of (${\bf x}_{i}$, ${\bf P}{\bf
x}_{i_{1}}^{t},\cdots,{\bf P}{\bf x}_{i_{r-1}}^{t},{\bf P}{\bf
x}_{i^{*}}^{t}$), forms its estimate ${\hat{\bf x}}_{i}(i^{*})$ of ${\bf
x}_{i^{*}}$ through the following successive maximum likelihood estimates of
${\bf x}_{i_{1}},\cdots,{\bf x}_{i_{r-1}}$:
$\displaystyle{\hat{\bf x}}_{i}(i_{1})$ $\displaystyle=$ $\displaystyle{\bf
x}_{i}\oplus f_{\bf P}({\bf P}{\bf x}_{i}^{t}\oplus{\bf P}{\bf
x}_{i_{1}}^{t}),$ $\displaystyle{\hat{\bf x}}_{i}(i_{2})$ $\displaystyle=$
$\displaystyle{\hat{\bf x}}_{i}(i_{1})\oplus f_{\bf P}({\bf P}{\bf
x}_{i_{1}}^{t}\oplus{\bf P}{\bf x}_{i_{2}}^{t}),$ $\displaystyle\vdots$
$\displaystyle\vdots$ $\displaystyle\vdots$ $\displaystyle{\hat{\bf
x}}_{i}(i_{r-1})$ $\displaystyle=$ $\displaystyle{\hat{\bf
x}}_{i}(i_{r-2})\oplus f_{\bf P}({\bf P}{\bf x}_{i_{r-2}}^{t}\oplus{\bf P}{\bf
x}_{i_{r-1}}^{t}),$
and finally,
${\hat{\bf x}}{i}(i^{*})={\hat{\bf x}}_{i}(i_{r-1})\oplus f_{\bf P}({\bf
P}{\bf x}_{i_{r-1}}^{t}\oplus{\bf P}{\bf x}_{i^{*}}^{t}).$ (20)
Proposition 2: By the successive maximum likelihood estimation above, the
estimate ${\hat{\bf X}_{i}}(i^{*})$ at terminal $i\neq i^{*}$, satisfies
$\Pr\\{{\hat{\bf X}_{i}}(i^{*})={\bf X}_{i^{*}}\\}\geq 1-d\cdot 2^{-n\eta},$
(21)
for some $\eta>0$ and for all $n$ sufficiently large.
Proof: See Appendix B.
It follows directly from (21) that for some
$\eta^{\prime}=\eta^{\prime}(\eta,m)>0$ and for all $n$ sufficiently large,
$\Pr\\{{\hat{\bf X}_{i}}(i^{*})={\bf X}_{i^{*}},1\leq i\neq i^{*}\leq d\\}\geq
1-2^{-n\eta^{\prime}}.$
(ii) SK construction: Consider a (common) standard array for ${\cal C}$ known
to all the terminals. Denote by ${\bf a}_{l,k}$ the element of the $l^{th}$
row and the $k^{th}$ column in the standard array, $1\leq l\leq 2^{m}$, $1\leq
k\leq 2^{n-m}$. Terminal $i^{*}$ sets $K_{i^{*}}=k_{i^{*}}$ if ${\bf
X}_{i^{*}}$ equals ${\bf a}_{l,k_{i^{*}}}$ in the standard array. Terminal
$i$, $1\leq i\neq i^{*}\leq d$, sets $K_{i}=k_{i}$ if ${\hat{\bf
X}_{i}}(i^{*})$ equals ${\bf a}_{l,k_{i}}$ in the same standard array.
The following theorem states that $K_{i^{*}}$ constitutes a strong SK with
rate approaching SK capacity.
Theorem 3: Let $\varepsilon>0$ be given. Then for some
$\eta^{\prime}=\eta^{\prime}(\eta,d)>0$ and for all $n$ sufficiently large,
the rvs $K_{1},\cdots,K_{d}$ generated above, with range ${\cal K}_{i^{*}}$
(say), satisfy
$\Pr\\{K_{1}=\cdots=K_{d}\\}>1-2^{-n\eta^{\prime}},$ (22)
$I(K_{i^{*}}\wedge{\bf F})=0,$ (23) $H(K_{i^{*}})=\log|{\cal K}_{i^{*}}|,$
(24)
and
$\frac{1}{n}H(K_{i^{*}})>1-h(p_{max})-\varepsilon.$ (25)
Model 4: Let the terminals 1, 2 and 3 observe, respectively, $n$ i.i.d.
repetitions of the $\\{0,1\\}$-valued rvs $X_{1}$, $X_{2}$, $X_{3}$, with
joint pmf $P_{X_{1}X_{2}X_{3}}$ given by:
$\displaystyle
P_{X_{1}X_{2}X_{3}}(0,0,0)=P_{X_{1}X_{2}X_{3}}(0,1,1)=\frac{(1-p)(1-q)}{2},$
$\displaystyle
P_{X_{1}X_{2}X_{3}}(0,0,1)=P_{X_{1}X_{2}X_{3}}(0,1,0)=\frac{pq}{2},$
$\displaystyle
P_{X_{1}X_{2}X_{3}}(1,0,0)=P_{X_{1}X_{2}X_{3}}(1,1,1)=\frac{p(1-q)}{2},$
$\displaystyle
P_{X_{1}X_{2}X_{3}}(1,0,1)=P_{X_{1}X_{2}X_{3}}(1,1,0)=\frac{q(1-p)}{2},$ (26)
with $0<p<\frac{1}{2}$ and $0<q<1$. Terminals 1 and 2 wish to generate a
strong PK of maximum rate, which is concealed from the helper terminal 3.
Note that under the joint pmf of $X_{1}$, $X_{2}$, $X_{3}$ above, we can write
${\bf X}_{1}={\bf X}_{2}\oplus{\bf X}_{3}\oplus{\bf V},$ (27)
where ${\bf V}=(V_{1},\cdots,V_{n})$ is an i.i.d. sequence of
$\\{0,1\\}$-valued rvs, independent of $({\bf X}_{2},{\bf X}_{3})$, with
$\Pr\\{V_{i}=1\\}=p$, $1\leq i\leq n$. Further, $(X_{2},X_{3})$ plays the role
of $(X_{1},X_{2})$ in Model 1 with $q$ in lieu of $p$ in the latter.
We show below a scheme for terminals 1 and 2 to generate a PK with rate close
to (strong) PK capacity for this model [2], [7], [8], given by (4) as
$C_{P}(\\{1,2\\})=I(X_{1}\wedge X_{2}|X_{3})=h(p+q-2pq)-h(p).$
The first step of this scheme entails terminal 3 simply revealing its
observations ${\bf x}_{3}$ to both terminals 1 and 2. Then, Wyner’s SW data
compression scheme is used for reconstructing ${\bf x}_{1}$ at terminal 2 from
the SW codeword for ${\bf x}_{1}$ and its own knowledge of ${\bf
x}_{2}\oplus{\bf x}_{3}$.
(i) SW data compression: This step is identical to step (i) for Model 1, as
seen with the help of (27). Obviously,
$\Pr\\{{\hat{\bf X}_{2}}(1)={\bf X}_{1}\\}\geq 1-2^{-n\eta},$
for some $\eta>0$ and for all $n$ sufficiently large.
(ii) PK construction: Suppose that terminals 1 and 2 know a linear $(n,n-m)$
code ${\cal C}$ as in Lemma 1, and a (common) standard array for ${\cal C}$.
Let $\\{{\bf e}_{i}:1\leq i\leq 2^{m}\\}$ denote the set of coset leaders for
all the cosets of ${\cal C}$.
For a sequence ${\bf x}_{3}\in\\{0,1\\}^{n}$, denote by $A_{i}({\bf x}_{3})$
the set of sequences from $T_{X_{1}|X_{3},\xi}^{n}({\bf x}_{3})$ in the coset
of ${\cal C}$ with coset leader ${\bf e}_{i}$, $1\leq i\leq 2^{m}$. If the
number of sequences of the same joint type with ${\bf x}_{3}$ in $A_{i}({\bf
x}_{3})$ is more than $2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$, where
$\varepsilon^{\prime}>2\xi+\varepsilon$ and $\varepsilon$ satisfies
$m<n[h(p)+\varepsilon]$ (as in Lemma 1), then collect arbitrarily
$2^{n[I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime}]}$ such sequences to
compose a regular subset. Continue this procedure until the number of
sequences of every joint type with ${\bf x}_{3}$ in $A_{i}({\bf x}_{3})$ is
less than $2^{n[I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime}]}$. Let
$N_{i}({\bf x}_{3})$ denote the number of distinct regular subsets of
$A_{i}({\bf x}_{3})$.
For a given sequence ${\bf x}_{3}$, enumerate (in any way) the sequences in
each regular subset. Let ${\bf b}_{i,j,k}({\bf x}_{3})$, where $1\leq i\leq
2^{m}$, $1\leq j\leq N_{i}({\bf x}_{3})$, $1\leq k\leq 2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$, denote the $k^{th}$ sequence of the
$j^{th}$ regular subset in the $i^{th}$ coset.
Terminal $1$ sets $K_{1}=k_{1}$ if ${\bf X}_{1}$ equals ${\bf
b}_{i,j_{1},k_{1}}({\bf x}_{3})$; else, $K_{1}$ is set to be uniformly
distributed on $\left\\{1,\cdots,2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}\right\\}$, independent of $({\bf
X}_{1},{\bf X}_{2},{\bf X}_{3})$. Terminal 2 sets $K_{2}=k_{2}$ if ${\hat{\bf
X}_{2}}(1)$ equals ${\bf b}_{i,j_{2},k_{2}}({\bf x}_{3})$; else, $K_{2}$ is
set to be uniformly distributed on $\left\\{1,\cdots,2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}\right\\}$, independent of $({\bf
X}_{1},{\bf X}_{2},{\bf X}_{3},K_{1})$.
The following theorem establishes that $K_{1}$ constitutes a strong PK with
rate approaching PK capacity.
Theorem 4: Let $\varepsilon>0$ be given. Then for some
$\eta^{\prime}=\eta^{\prime}(\eta,\xi,\varepsilon,\varepsilon^{\prime})>0$ and
for all $n$ sufficiently large, the pair of rvs $(K_{1},K_{2})$ generated
above, with range ${\cal K}_{1}$ (say), satisfy
$\Pr\\{K_{1}\neq K_{2}\\}<2^{-n\eta^{\prime}},$ (28) $I(K_{1}\wedge{\bf
X}_{3},{\bf F})=0,$ (29) $H(K_{1})=\log|{\cal K}_{1}|,$ (30)
and
$\frac{1}{n}H(K_{1})=I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime}.$ (31)
Remark: The PK construction scheme above applies for any joint pmf
$P_{X_{1}X_{2}X_{3}}$ satisfying (27), and is not restricted to the given
joint pmf in (26).
## IV Proofs of Theorems 1–4
Proof of Theorem 1: It follows from the SK construction scheme for Model 1
that
$\Pr\\{K_{1}\neq K_{2}\\}=\Pr\\{{\hat{\bf X}_{2}(1)}\neq{\bf
X}_{1}\\}<2^{-n\eta},$
which is (11). Since $X_{1}$ is uniformly distributed on $\\{0,1\\}$, we have
for $1\leq i\leq 2^{m}$, $1\leq j\leq 2^{n-m}$, that
$\Pr\\{{\bf X}_{1}={\bf a}_{i,j}\\}=2^{-n}.$
Hence,
$\displaystyle\Pr\\{K_{1}=j\\}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2^{m}}\Pr\\{{\bf X}_{1}={\bf a}_{i,j}\\}$
$\displaystyle=$ $\displaystyle 2^{-(n-m)},\ \ \ 1\leq j\leq 2^{n-m},$
i.e., $K_{1}$ is uniformly distributed on ${\cal
K}_{1}=\left\\{1,\cdots,2^{n-m}\right\\}$, and so
$H(K_{1})=\log 2^{n-m}=n-m=\log|{\cal K}_{1}|,$
which is (13). Therefore, (14) holds since $m<n[h(p)+\varepsilon]$.
It remains to show that $K_{1}$ satisfies (12) with ${\bf F}={\bf P}{\bf
X}_{1}^{t}$. Let $\\{{\bf e}_{i},1\leq i\leq 2^{m}\\}$ be the set of coset
leaders for the cosets of ${\cal C}$. For $1\leq i\leq 2^{m}$, $1\leq j\leq
2^{n-m}$,
$\displaystyle\Pr\\{K_{1}=j|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t}\\}$
$\displaystyle=$ $\displaystyle\frac{\Pr\\{K_{1}=j,{\bf P}{\bf X}_{1}^{t}={\bf
P}{\bf e}_{i}^{t}\\}}{\Pr\\{{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t}\\}}$
$\displaystyle=$ $\displaystyle\frac{\Pr\\{{\bf X}_{1}={\bf
a}_{i,j}\\}}{\sum_{j^{\prime}=1}^{2^{n-m}}\Pr\\{{\bf X}_{1}={\bf
a}_{i,j^{\prime}}\\}}$ $\displaystyle=$ $\displaystyle 2^{-(n-m)}$
$\displaystyle=$ $\displaystyle\Pr\\{K_{1}=j\\},$
i.e., $K_{1}$ is independent of ${\bf F}$, and so $I(K_{1}\wedge{\bf F})=0$,
establishing (12).
Proof of Theorem 2: Let ${\cal F}$ denote the union of all regular subsets in
$\bigcup_{i=1}^{2^{m}}A_{i}$. Clearly ${\cal F}\subseteq T_{X_{1},\xi}^{n}$,
so that
$\displaystyle\Pr\\{{\bf X}_{1}\in{\cal F}\\}$ (32) $\displaystyle=$
$\displaystyle\Pr\\{{\bf X}_{1}\in T_{X_{1},\xi}^{n},{\bf X}_{1}\in{\cal
F}\\}$ $\displaystyle=$ $\displaystyle\Pr\\{{\bf X}_{1}\in
T_{X_{1},\xi}^{n}\\}-\Pr\\{{\bf X}_{1}\in T_{X_{1},\xi}^{n}\backslash{\cal
F}\\}.$
By Proposition 1, $\Pr\\{{\bf X}_{1}\in T_{X_{1},\xi}^{n}\\}$ goes to 1
exponentially rapidly in $n$. We show below that $\Pr\\{{\bf X}_{1}\in
T_{X_{1},\xi}^{n}\backslash{\cal F}\\}$ decays to 0 exponentially rapidly in
$n$.
Since the number of different types of sequences in $\\{0,1\\}^{n}$ does not
exceed $(n+1)^{2}$, we have that
$\displaystyle\left|\\{{\bf x}_{1}:{\bf x}_{1}\in
T_{X_{1},\xi}^{n}\backslash{\cal F}\\}\right|$ $\displaystyle\leq$
$\displaystyle 2^{m}\cdot(n+1)^{2}\cdot 2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}$ $\displaystyle<$ $\displaystyle(n+1)^{2}\cdot
2^{n[H(X_{1})+\varepsilon-\varepsilon^{\prime}]},$
where the previous inequality is from
$m<n[h(p)+\varepsilon]=n[H(X_{1}|X_{2})+\varepsilon]$.
Since $P_{X_{1}}^{n}({\bf x}_{1})\leq 2^{-n[H(X_{1})-\xi]}$, ${\bf x}_{1}\in
T_{X_{1},\xi}^{n}$, we get
$\Pr\\{{\bf X}_{1}\in T_{X_{1},\xi}^{n}\backslash{\cal F}\\}<(n+1)^{2}\cdot
2^{-n(\varepsilon^{\prime}-\xi-\varepsilon)}.$
Choosing $\varepsilon^{\prime}>\xi+\varepsilon$, $\Pr\\{{\bf X}_{1}\in
T_{X_{1},\xi}^{n}\backslash{\cal F}\\}$ goes to 0 exponentially rapidly.
Therefore, it follows from (32) that $\Pr\\{{\bf X}_{1}\in{\cal F}\\}$ goes to
1 exponentially rapidly in $n$, with exponent depending on
$(\xi,\varepsilon,\varepsilon^{\prime})$.
By the SK construction scheme for Model 2,
$\displaystyle\Pr\\{K_{1}\neq K_{2}\\}$ $\displaystyle=$
$\displaystyle\Pr\\{K_{1}\neq K_{2},{\bf X}_{1}\in{\cal F}\\}+\Pr\\{K_{1}\neq
K_{2},{\bf X}_{1}\not\in{\cal F}\\}$ $\displaystyle\leq$
$\displaystyle\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf X}_{1},{\bf X}_{1}\in{\cal
F}\\}+\Pr\\{{\bf X}_{1}\not\in{\cal F}\\}$ $\displaystyle\leq$
$\displaystyle\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf X}_{1}\\}+\Pr\\{{\bf
X}_{1}\not\in{\cal F}\\}.$
Since $\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf X}_{1}\\}<2^{-n\eta}$, by the
observation in the previous paragraph, we have
$\Pr\\{K_{1}\neq K_{2}\\}<2^{-n\eta^{\prime}}$
for some
$\eta^{\prime}=\eta^{\prime}(\eta,\xi,\varepsilon,\varepsilon^{\prime})>0$ and
for all $n$ sufficiently large, which is (16).
Next, we shall show that $K_{1}$ satisfies (18). For $1\leq k\leq
2^{n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]}$, it is clear by choice that
$\Pr\\{K_{1}=k|{\bf X}_{1}\not\in{\cal F}\\}=2^{-n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]},$ (33)
and that
$\displaystyle\Pr\\{K_{1}=k|{\bf X}_{1}\in{\cal F}\\}=\frac{\Pr\\{K_{1}=k,{\bf
X}_{1}\in{\cal F}\\}}{\Pr\\{{\bf X}_{1}\in{\cal F}\\}}$ (34) $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{2^{m}}\sum_{j=1}^{N_{i}}\Pr\\{{\bf X}_{1}={\bf
b}_{i,j,k}\\}}{\sum_{i=1}^{2^{m}}\sum_{j=1}^{N_{i}}2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}\Pr\\{{\bf X}_{1}={\bf b}_{i,j,k}\\}}$
$\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]},$ (35)
where (34) is due to every regular subset consisting of sequences of the same
type. From (33) and (35),
$\Pr\\{K_{1}=k\\}=2^{-n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]},$ (36)
i.e., $K_{1}$ is uniformly distributed on ${\cal
K}_{1}=\left\\{1,\cdots,2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}\right\\}$, with
$\frac{1}{n}H(K_{1})=I(X_{1}\wedge X_{2})-\varepsilon^{\prime},$
which is (19).
It remains to show that $K_{1}$ satisfies (17) with ${\bf F}={\bf P}{\bf
X}_{1}^{t}$. For $1\leq i\leq 2^{m}$, $1\leq k\leq 2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}$, we have
$\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf
X}_{1}\not\in{\cal F}\\}=2^{-n[I(X_{1}\wedge X_{2})-\varepsilon^{\prime}]}$
by choice, and
$\displaystyle\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf
X}_{1}\in{\cal F}\\}$ $\displaystyle=$ $\displaystyle\frac{\Pr\\{K_{1}=k,{\bf
P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf X}_{1}\in{\cal F}\\}}{\Pr\\{{\bf
P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf X}_{1}\in{\cal F}\\}}$
$\displaystyle=$ $\displaystyle\frac{\sum_{j=1}^{N_{i}}\Pr\\{{\bf X}_{1}={\bf
b}_{i,j,k}\\}}{\sum_{j=1}^{N_{i}}2^{n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}\Pr\\{{\bf X}_{1}={\bf b}_{i,j,k}\\}}$
$\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}.$
Hence,
$\displaystyle\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t}\\}$
$\displaystyle=$ $\displaystyle\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf
P}{\bf e}_{i}^{t},{\bf X}_{1}\in{\cal F}\\}\times$ $\displaystyle\Pr\\{{\bf
X}_{1}\in{\cal F}|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t}\\}$
$\displaystyle+\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf
e}_{i}^{t},{\bf X}_{1}\not\in{\cal F}\\}\times$ $\displaystyle\Pr\\{{\bf
X}_{1}\not\in{\cal F}|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t}\\}$
$\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2})-\varepsilon^{\prime}]}$ $\displaystyle=$
$\displaystyle\Pr\\{K_{1}=k\\},$
where the previous equality follows from (36). Thus, $K_{1}$ is independent of
${\bf F}$, establishing (17).
Proof of Theorem 3: Applying the same arguments used in Theorem 1, we see that
the rvs $K_{1},\cdots,K_{m}$ satisfy (22), (24) and (25). It then remains to
show that $K_{i^{*}}$ satisfies (23) with ${\bf F}=({\bf P}{\bf
X}_{1}^{t},\cdots,{\bf P}{\bf X}_{d}^{t})$.
Under the given joint pmf $P_{X_{1}\cdots X_{d}}$, for each $i\neq i^{*}$, we
can write
${\bf X}_{i}={\bf X}_{i^{*}}\oplus{\bf V}_{i},$
where ${\bf V}_{i}=(V_{i,1},\cdots,V_{i,n})$ is an i.i.d. sequence of
$\\{0,1\\}$-valued rvs. Further, ${\bf V}_{i}$, $1\leq i\neq i^{*}\leq d$, and
${\bf X}_{i^{*}}$ are mutually independent. Then,
$\displaystyle I(K_{i^{*}}\wedge{\bf F})$ (37) $\displaystyle=$ $\displaystyle
I(K_{i^{*}}\wedge\\{{\bf P}{\bf X}_{i}^{t},\ 1\leq i\leq d\\})$
$\displaystyle\leq$ $\displaystyle I(K_{i^{*}}\wedge{\bf P}{\bf
X}_{i^{*}}^{t},\\{{\bf P}{\bf V}_{i}^{t},\ 1\leq i\neq i^{*}\leq d\\})$
$\displaystyle\leq$ $\displaystyle I(K_{i^{*}}\wedge{\bf P}{\bf
X}_{i^{*}}^{t})$ $\displaystyle+I(K_{i^{*}},{\bf P}{\bf
X}_{i^{*}}^{t}\wedge\\{{\bf P}{\bf V}_{i}^{t},\ 1\leq i\neq i^{*}\leq d\\}).$
Clearly, the first term on the right hand side of (37) is zero. Since for a
fixed ${\bf P}$, $(K_{i^{*}},{\bf P}{\bf X}_{i^{*}}^{t})$ is a function of
${\bf X}_{i^{*}}$,
$\displaystyle I(K_{i^{*}},{\bf P}{\bf X}_{i^{*}}^{t}\wedge\\{{\bf P}{\bf
V}_{i}^{t},\ 1\leq i\neq i^{*}\leq d\\})$ $\displaystyle\leq$ $\displaystyle
I({\bf X}_{i^{*}}\wedge\\{{\bf V}_{i},\ 1\leq i\neq i^{*}\leq d\\})=0,$
i.e., $K_{i^{*}}$ is independent of ${\bf F}$, establishing (23).
Proof of Theorem 4: For every ${\bf x}_{3}\in\\{0,1\\}^{n}$, let ${\cal
F}({\bf x}_{3})$ denote the union of all regular subsets in
$\bigcup_{i=1}^{2^{m}}A_{i}({\bf x}_{3})$. Since ${\cal F}({\bf
x}_{3})\subseteq T_{X_{1}|X_{3},\xi}^{n}({\bf x}_{3})$,
$\displaystyle\Pr\\{{\bf X}_{1}\in{\cal F}({\bf X}_{3})\\}$ $\displaystyle=$
$\displaystyle\Pr\\{{\bf X}_{1}\in T_{X_{1}|X_{3},\xi}^{n}({\bf X}_{3})\\}$
(38) $\displaystyle-\Pr\\{{\bf X}_{1}\in T_{X_{1}|X_{3},\xi}^{n}({\bf
X}_{3})\backslash{\cal F}({\bf X}_{3})\\}.$
It follows from Proposition 1 that $\Pr\\{{\bf X}_{1}\in
T_{X_{1}|X_{3},\xi}^{n}({\bf X}_{3})\\}$ goes to 1 exponentially rapidly in
$n$. We show below that $\Pr\\{{\bf X}_{1}\in T_{X_{1}|X_{3},\xi}^{n}({\bf
X}_{3})\backslash{\cal F}({\bf X}_{3})\\}$ goes to 0 exponentially rapidly in
$n$.
Recall that the number of different joint types of pairs in
$\\{0,1\\}^{n}\times\\{0,1\\}^{n}$ does not exceed $(n+1)^{4}$. Thus,
$\displaystyle\left|\\{{\bf x}_{1}:{\bf x}_{1}\in T_{X_{1}|X_{3},\xi}^{n}({\bf
x}_{3})\backslash{\cal F}({\bf x}_{3})\\}\right|$ $\displaystyle\leq$
$\displaystyle 2^{m}\cdot(n+1)^{4}\cdot 2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$ $\displaystyle<$
$\displaystyle(n+1)^{4}\cdot
2^{n[H(X_{1}|X_{3})+\varepsilon-\varepsilon^{\prime}]},$
where the previous inequality is from
$m<n[h(p)+\varepsilon]=n[H(X_{1}|X_{2},X_{3})+\varepsilon]$.
Since $P_{X_{1}|X_{3}}^{n}({\bf x}_{1}|{\bf x}_{3})\leq
2^{-n[H(X_{1}|X_{3})-2\xi]}$, $({\bf x}_{1},{\bf x}_{3})\in
T_{X_{1}X_{3},\xi}^{n}$, we get
$\Pr\\{{\bf X}_{1}\in T_{X_{1}|X_{3},\xi}^{n}({\bf X}_{3})\backslash{\cal
F}({\bf X}_{3})\\}<(n+1)^{4}\cdot
2^{-n(\varepsilon^{\prime}-2\xi-\varepsilon)}.$
Choosing $\varepsilon^{\prime}>2\xi+\varepsilon$, $\Pr\\{{\bf X}_{1}\in
T_{X_{1}|X_{3},\xi}^{n}({\bf X}_{3})\backslash{\cal F}({\bf X}_{3})\\}$ goes
to 0 exponentially rapidly. Therefore, it follows from (38) that $\Pr\\{{\bf
X}_{1}\in{\cal F}({\bf X}_{3})\\}$ goes to 1 exponentially rapidly in $n$,
with an exponent depending on $(\xi,\varepsilon,\varepsilon^{\prime})$.
By the PK construction scheme for Model 4,
$\displaystyle\Pr\\{K_{1}\neq K_{2}\\}$ $\displaystyle=$
$\displaystyle\Pr\\{K_{1}\neq K_{2},{\bf X}_{1}\in{\cal F}({\bf
x}_{3})\\}+\Pr\\{K_{1}\neq K_{2},{\bf X}_{1}\not\in{\cal F}({\bf x}_{3})\\}$
$\displaystyle\leq$ $\displaystyle\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf
X}_{1},{\bf X}_{1}\in{\cal F}({\bf x}_{3})\\}+\Pr\\{{\bf X}_{1}\not\in{\cal
F}({\bf x}_{3})\\}$ $\displaystyle\leq$ $\displaystyle\Pr\\{{\hat{\bf
X}_{2}}(1)\neq{\bf X}_{1}\\}+\Pr\\{{\bf X}_{1}\not\in{\cal F}({\bf
X}_{3})\\}.$
Since $\Pr\\{{\hat{\bf X}_{2}}(1)\neq{\bf X}_{1}\\}<2^{-n\eta}$ by the
observation in the previous paragraph, we have
$\Pr\\{K_{1}\neq K_{2}\\}<2^{-n\eta^{\prime}},$
for some
$\eta^{\prime}=\eta^{\prime}(\eta,\xi,\varepsilon,\varepsilon^{\prime})>0$ and
for all $n$ sufficiently large, which is (28).
Next, we shall show that $K_{1}$ satisfies (30). For ${\bf
x}_{3}\in\\{0,1\\}^{n}$ and $1\leq k\leq 2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$, it is clear by choice that
$\Pr\\{K_{1}=k|{\bf X}_{1}\not\in{\cal F}({\bf x}_{3}),{\bf X}_{3}={\bf
x}_{3}\\}=2^{-n[I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime}]},$
and that
$\displaystyle\Pr\\{K_{1}=k|{\bf X}_{1}\in{\cal F}({\bf x}_{3}),{\bf
X}_{3}={\bf x}_{3}\\}$ $\displaystyle=$ $\displaystyle\frac{\Pr\\{K_{1}=k,{\bf
X}_{1}\in{\cal F}({\bf x}_{3})|{\bf X}_{3}={\bf x}_{3}\\}}{\Pr\\{{\bf
X}_{1}\in{\cal F}({\bf x}_{3})|{\bf X}_{3}={\bf x}_{3}\\}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{2^{m}}\sum_{j=1}^{N_{i}({\bf
x}_{3})}\Pr\\{{\bf X}_{1}={\bf b}_{i,j,k}({\bf x}_{3})|{\bf X}_{3}={\bf
x}_{3}\\}}{\sum_{i=1}^{2^{m}}\sum_{j=1}^{N_{i}({\bf x}_{3})}2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}\Pr\\{{\bf X}_{1}={\bf b}_{i,j,k}({\bf
x}_{3})|{\bf X}_{3}={\bf x}_{3}\\}}$ $\displaystyle=$ $\displaystyle
2^{-n[I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime}]},$
where the second equality is due to every regular subset consisting of
sequences of the same joint type with ${\bf x}_{3}$. Therefore,
$\displaystyle\Pr\\{K_{1}=k\\}=\sum_{{\bf
x}_{3}\in\\{0,1\\}^{n}}\Pr\\{K_{1}=k,{\bf X}_{3}={\bf x}_{3}\\}$ (39)
$\displaystyle=$ $\displaystyle\sum_{{\bf x}_{3}\in\\{0,1\\}^{n}}[\Pr\\{{\bf
X}_{1}\in{\cal F}({\bf x}_{3}),{\bf X}_{3}={\bf x}_{3}\\}\times$
$\displaystyle\Pr\\{K_{1}=k|{\bf X}_{1}\in{\cal F}({\bf x}_{3}),{\bf
X}_{3}={\bf x}_{3}\\}$ $\displaystyle+\Pr\\{{\bf X}_{1}\not\in{\cal F}({\bf
x}_{3}),{\bf X}_{3}={\bf x}_{3}\\}\times$ $\displaystyle\Pr\\{K_{1}=k|{\bf
X}_{1}\not\in{\cal F}({\bf x}_{3}),{\bf X}_{3}={\bf x}_{3}\\}]$
$\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]},$
i.e., $K_{1}$ is uniformly distributed on ${\cal
K}_{1}=\left\\{1,\cdots,2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}\right\\}$, with
$\frac{1}{n}H(K_{1})=I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime},$
which is (31).
It remains to show that $K_{1}$ satisfies (29) with $({\bf X}_{3},{\bf
F})=({\bf X}_{3},{\bf P}{\bf X}_{1}^{t})$. For ${\bf x}_{3}\in\\{0,1\\}^{n}$,
$1\leq i\leq 2^{m}$ and $1\leq k\leq 2^{n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$, we have
$\displaystyle\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf
X}_{1}\not\in{\cal F}({\bf x}_{3}),{\bf X}_{3}={\bf x}_{3}\\}$
$\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$
by choice, and
$\displaystyle\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf
X}_{1}\in{\cal F}({\bf x}_{3}),{\bf X}_{3}={\bf x}_{3}\\}$ $\displaystyle=$
$\displaystyle\frac{\Pr\\{K_{1}=k,{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf
e}_{i}^{t},{\bf X}_{1}\in{\cal F}({\bf x}_{3})|{\bf X}_{3}={\bf
x}_{3}\\}}{\Pr\\{{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf
X}_{1}\in{\cal F}({\bf x}_{3})|{\bf X}_{3}={\bf x}_{3}\\}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=1}^{N_{i}({\bf x}_{3})}\Pr\\{{\bf X}_{1}={\bf
b}_{i,j,k}({\bf x}_{3})|{\bf X}_{3}={\bf x}_{3}\\}}{\sum_{j=1}^{N_{i}({\bf
x}_{3})}2^{n[I(X_{1}\wedge X_{2}|X_{3})-\varepsilon^{\prime}]}\Pr\\{{\bf
X}_{1}={\bf b}_{i,j,k}({\bf x}_{3})|{\bf X}_{3}={\bf x}_{3}\\}}$
$\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}.$
Hence,
$\displaystyle\Pr\\{K_{1}=k|{\bf P}{\bf X}_{1}^{t}={\bf P}{\bf e}_{i}^{t},{\bf
X}_{3}={\bf x}_{3}\\}$ $\displaystyle=$ $\displaystyle 2^{-n[I(X_{1}\wedge
X_{2}|X_{3})-\varepsilon^{\prime}]}$ $\displaystyle=$
$\displaystyle\Pr\\{K_{1}=k\\},$
where the previous equality follows from (39). Thus, $K_{1}$ is independent of
$({\bf X}_{3},{\bf F})$, establishing (29).
## V Implementation with LDPC Codes
We outline an implementation using LDPC codes (cf. e.g., [21], [30], [34],
[31]) of the scheme for the construction of a SK for Model 1 in Section III.
As will be indicated below, similar implementations can be applied to Models
2–4 as well.
### V-A SK construction
Without any loss of generality, we consider a systematic $(n,n-m)$ LDPC code
${\cal C}$ with generator matrix ${\bf G}=[{\bf I}_{n-m}\ {\bf A}]$, where
${\bf I}_{n-m}$ is an $(n-m)\times(n-m)$-identity matrix and ${\bf A}$ is an
$(n-m)\times m$-matrix. Then, the parity check matrix for ${\cal C}$ is ${\bf
P}=[{\bf A}^{t}\ {\bf I}_{m}]$, where ${\bf I}_{m}$ is an $m\times m$-identity
matrix. The first $n-m$ bits of every codeword in ${\cal C}$, namely the
information bits, are pairwise distinct. Further, since the coset with coset
leader ${\bf e}_{i}$, $1\leq i\leq 2^{m}$, must contain the sequence ${\bf
b}_{i}=[{\bf 0}_{n-m}\ {\bf e}_{i}{\bf P}^{t}]$, with ${\bf 0}_{n-m}$ denoting
a sequence of $n-m$ zeros, the first ($n-m$)-bit-segments of the sequences in
the coset $\\{{\bf b}_{i}\oplus{\bf c},\ {\bf c}\in{\cal C}\\}$ are pairwise
distinct.
Terminal $1$ transmits the syndrome ${\bf P}{\bf x}_{1}^{t}$, whereupon
terminal $2$, knowing $({\bf x}_{2},{\bf P}{\bf x}_{1}^{t})$, applies the
belief-propagation algorithm described in [19] to estimate ${\hat{\bf
x}_{2}}(1)$. Since the first $n-m$ bits of the sequences in each coset are
pairwise distinct, these bits can serve as the index of a sequence in its
coset. Then, terminal $1$ (resp. 2) sets $K_{1}$ (resp. $K_{2}$) as the first
$n-m$ bits of ${\bf x}_{1}$ (resp. ${\hat{\bf x}_{2}}(1)$).
The same implementation of the SW data compression scheme above holds for
Models 2 and 4, too. It can be applied repeatedly also for the successive
estimates (20) in Model 3. In Model 3, $K_{i^{*}}$ (resp. $K_{i}$, $i\neq
i^{*}$) is set as the first $n-m$ bits of ${\bf x}_{i^{*}}$ (resp. ${\hat{\bf
x}_{i}}(i^{*})$). It should be noted that the current complexity of generating
regular subsets in Models 2 and 4 poses a hurdle for explicit efficient
constructions of a SK and a PK, respectively, for these models.
### V-B Simulation Results
We provide simulation results for the tradeoff between the relative secret key
rate (i.e., the difference between the SK capacity and the rate of the
generated SK) and the rate of generating unequal SKs at different terminals
(corresponding to the bit error rate in SK-matching), when LDPC codes are used
for SK construction in Model 1.
For the purpose of comparison, three different LDPC codes were used: (i) a
$(3,4)$-regular LDPC code; (ii) a $(3,6)$-regular LDPC code; and (iii) an
irregular LDPC code with degree distribution pair (cf. [19])
$\displaystyle\lambda(x)$ $\displaystyle=$ $\displaystyle
0.234029x+0.212425x^{2}+0.146898x^{5}$
$\displaystyle+0.102840x^{6}+0.303808x^{19},$ $\displaystyle\rho(x)$
$\displaystyle=$ $\displaystyle 0.71875x^{7}+0.28125x^{8},$
with a common codeword length of $10^{3}$ bits, and upto 60 iterations of the
belief-propagation algorithm were allowed. Over $10^{3}$ blocks were
transmitted from terminal 1.
Simulation results are shown in Figures 1 and 2, where conditional entropy
(i.e., $H(X_{1}|X_{2})=h(p)$) is plotted against key bit error rate (KBER). We
note that in this simulation SKs are generated at fixed rates that are equal
to the rates of the LDPC codes used. Since for Model 1, SK capacity equals
$1-h(p)$, the conditional entropy $h(p)$ serves as an indicator of the gap
between SK capacity and the rate of the generated SK.
Figure 1 shows the performance of the $(3,6)$-regular and the irregular LDPC
codes; Figure 2 shows the performance of the $(3,4)$-regular LDPC code. It is
seen in both figures that KBER increases with $h(p)$. Since SK capacity
decreases with increasing $h(p)$, an increase of $h(p)$ narrows the gap
between SK capacity and the rate of the generated SK, but raises the
likelihood of generating unequal SKs at the two terminals.
It is seen from Figure 1 that the irregular LDPC code outperforms the
$(3,6)$-regular LDPC code. For instance, for a fixed crossover probability
$p=0.068$, say, and $h(p)\approx 0.3584$, the KBER for the irregular LDPC code
is as low as $10^{-5}$, while the KBER for the $(3,6)$-regular LDPC code is
only about $4\times 10^{-3}$.
Figure 1: Simulation results for the $(3,6)$-regular and the irregular LDPC
codes.
Figure 2: Simulation results for the $(3,4)$-regular LDPC code.
## VI Discussion
We have considered four simple secrecy generation models involving multiple
terminals, and propose a new approach for constructing SKs and PKs. This
approach is based on Wyner’s well-known SW data compression code for sources
connected by virtual channels with additive independent noise.
In all the models considered in this paper, the i.i.d. sequences observed at
the different terminals possesses the following structure: They can be
described in terms of sequences at pairs of terminals where each terminal in a
pair is connected to the other terminal by a virtual communication channel
with additive independent noise.
There are two steps in the SK construction schemes. The first step constitutes
SW data compression for the purpose of common randomness generation at the
terminals. Although the existence of linear data compression codes with rate
arbitrarily close to the SW bound has been long known for arbitrarily
correlated sources [5], constructions of such linear data compression codes
are understood in terms of the cosets of linear error-correction codes for the
virtual channel, say $P_{X_{1}|X_{2}}$, only when this virtual channel is
characterized by (independent) additive noise [37]. For instance, when two
terminals are connected by a virtual BSC $P_{X_{1}|X_{2}}$, a linear data
compression code, which attains the SW rate $H(X_{1}|X_{2})$ for terminal 2 to
reconstruct the signal at terminal 1, is then provided by a linear channel
code which achieves the capacity of the BSC $P_{X_{1}|X_{2}}$.
When the i.i.d. sequences observed at terminals 1 and 2 are arbitrarily
correlated, the associated virtual communication channel $P_{X_{1}|X_{2}}$
connecting them is no longer symmetric and corresponds to a virtual channel
with input-dependent noise. In this case, while linear codes are no longer
rate-optimal for the given channel [10], linear code constructions for a
suitably enlarged “semisymmetric” channel that are used for SW data
compression [14] could pave the way for devising schemes for SK construction.
The second step in the SK construction schemes involves SK extraction from the
previously acquired CR. It has been shown [25] that for the special case of a
two-terminal source model, this extraction can be accomplished by means of a
linear transformation. However, it is unknown yet whether this holds also for
a general source model with more than two terminals.
## Appendix A: Proof of Proposition 1
We shall prove (7) here. The proof of (8), which is similar, is omitted. Fix
$\delta>0$ and consider the set $T_{[P_{X}]_{\delta}}^{n}$ of sequences in
${\cal X}^{n}$ which are $P_{X}$-typical with constant $\delta$ (cf. [6, p.
33]), i.e.,
$T_{[P_{X}]_{\delta}}^{n}=\\{{\bf x}\in{\cal X}^{n}:\max_{a\in{\cal X}}|P_{\bf
x}(a)-P_{X}(a)|\leq\delta\\}.$
Since $T_{[P]_{\delta}}^{n}$ is the union of the sets of those types
$\tilde{P}$ of sequences in ${\cal X}^{n}$ that satisfy
$\max_{a\in{\cal X}}|\tilde{P}(a)-P_{X}(a)|\leq\delta,$ (A.1)
we have
$\displaystyle\sum_{{\bf
x}\in\left(T_{[P_{X}]_{\delta}}^{n}\right)^{c}}P_{X}^{n}({\bf x})$ (A.2)
$\displaystyle=$ $\displaystyle\sum_{\tilde{P}:\max_{a\in{\cal
X}}|\tilde{P}(a)-P_{X}(a)|>\delta}P_{X}^{n}\left(\\{{\bf x}:P_{\bf
x}=\tilde{P}\\}\right)$ $\displaystyle\leq$ $\displaystyle(n+1)^{|{\cal
X}|}\cdot 2^{-n\min_{\tilde{P}:\min_{a\in{\cal
X}}|\tilde{P}(a)-P_{X}(a)|>\delta}D(\tilde{P}||P_{X})},$
using the fact that $P_{X}^{n}(\\{{\bf x}:P_{\bf x}=\tilde{P}\\})\leq
2^{-nD(\tilde{P}||P)}$ (cf. [6, Lemma 2.6]).
Next, by Pinsker’s inequality (cf. e.g., [6, p. 58]),
$\displaystyle D(\tilde{P}||P)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2ln2}\left(\min_{a\in{\cal
X}}|\tilde{P}(a)-P_{X}(a)|\right)^{2}$ (A.3) $\displaystyle\geq$
$\displaystyle\frac{\delta^{2}}{2ln2},$
with the previous inequality holding for every $\tilde{P}$ in (A.1). It
follows from (A.2) and (A.3) that
$\sum_{{\bf x}\in T_{[P]_{\delta}}^{n}}P_{X}^{n}({\bf x})\geq 1-(n+1)^{|{\cal
X}|}\cdot 2^{-n\frac{\delta^{2}}{2ln2}}$ (A.4)
for all $n\geq 1$.
Finally, observe that
$T_{[P_{X}]_{\delta}}^{n}\subseteq T_{X,\xi}^{n},\ \ {\rm if}\ \
\xi=\delta\left[\sum_{a\in{\cal X}}\log\frac{1}{P_{X}(a)}\right],$ (A.5)
which is readily seen from the fact that for each ${\bf x}\in{\cal X}^{n}$,
$\displaystyle-\frac{1}{n}\log P_{X}^{n}({\bf x})-H(P_{X})$ $\displaystyle=$
$\displaystyle-\frac{1}{n}\log\left(2^{-n[H(P_{\bf x})+D(P_{\bf
x}||P_{X})]}\right)-H(P_{X})$ $\displaystyle=$ $\displaystyle H(P_{\bf
x})+D(P_{\bf x}||P_{X})-H(P_{X})$ $\displaystyle=$ $\displaystyle H(P_{\bf
x})-H(P_{\bf x})+\sum_{a\in{\cal X}}P_{\bf
x}(a)\log\frac{1}{P_{X}(a)}-H(P_{X})$ $\displaystyle=$
$\displaystyle\sum_{a\in{\cal X}}[P_{\bf
x}(a)-P_{X}(a)]\log\frac{1}{P_{X}(a)}.$
Clearly, (A.4) and (A.5) imply (7).
## Appendix B: Proof of Proposition 2
The proof of Proposition 2 relies on the following lemma concerning the
average error probability of maximum likelihood decoding.
A sequence ${\bf u}\in\\{0,1\\}^{n}$ is called a descendent of a sequence
${\bf v}\in\\{0,1\\}^{n}$ if $u_{i}=1$ implies that $v_{i}=1$, $1\leq i\leq
n$. A subset ${\Omega}\subset\\{0,1\\}^{n}$ is called quasiadmissible if the
conditions that ${\bf u}\in\Omega$ and ${\bf u}$ is a descendent of ${\bf v}$
together imply that ${\bf v}\in\Omega$.
Lemma 2 [22]: If $\Omega$ is a quasiadmissible subset of $\\{0,1\\}^{n}$, then
for $0\leq p\leq 1$,
$\frac{d\mu_{p}(\Omega)}{dp}>0,$
where
$\mu_{p}({\Omega})=\sum_{{\bf x}\in\Omega}p^{w_{H}({\bf
x})}(1-p)^{n-w_{H}({\bf x})},$
with $w_{H}({\bf x})$ denoting the Hamming weight of ${\bf x}$.
For a binary linear code, let ${\bf E}$ denote the set of coset leaders. It is
known (cf. [28, Theorem 3.11]) that
$\Omega^{\prime}=\\{0,1\\}^{n}\backslash{\bf E}$ is a quasiadmissible subset
of $\\{0,1\\}^{n}$. If a binary linear code is used on BSC($p$), the average
error probability of maximum likelihood decoding is given by (cf. [32, Theorem
5.3.3])
$\mu_{p}(\Omega^{\prime})=\sum_{{\bf x}\in\Omega^{\prime}}p^{w_{H}({\bf
x})}(1-p)^{n-w_{H}({\bf x})}.$
Lemma 2 implies that if the same binary linear code is used on two binary
symmetric channels with different crossover probabilities, say,
$0<p_{1}<p_{2}<\frac{1}{2}$, then the average error probability of maximum
likelihood decoding for a BSC($p_{1}$) is strictly less than that for a
BSC($p_{2}$); note that a BSC($p_{2}$) is a degraded version of a
BSC($p_{1}$), being a cascade of the latter and a
BSC($\frac{p_{2}-p_{1}}{1-2p_{1}}$).
Returning to the proof of Proposition 2, it follows from Lemma 1 that for some
$\eta>0$ and for all $n$ sufficiently large,
$\Pr\\{{\hat{\bf X}_{j^{*}}}(i^{*})\neq{\bf X}_{i^{*}}\\}<2^{-n\eta}.$
Recall that $p_{(i^{*},j^{*})}=\max_{(i,j)\in E({\cal T})}p_{(i,j)}$ and
$(i=i_{0},i_{1},\cdots,i_{r}=i^{*})$ is the path from $i$ to $i^{*}$. It
follows by Lemma 2 that
$\Pr\\{{\hat{\bf X}_{i}}(i_{1})\neq{\bf X}_{i_{1}}\\}<\Pr\\{{\hat{\bf
X}_{j^{*}}}(i^{*})\neq{\bf X}_{i^{*}}\\}<2^{-n\eta}.$
Consequently,
$\displaystyle\Pr\\{{\hat{\bf X}_{i}}(i_{2})\neq{\bf X}_{i_{2}}\\}$
$\displaystyle\leq$ $\displaystyle\Pr\\{{\hat{\bf X}_{i}}(i_{2})\neq{\bf
X}_{i_{2}},{\hat{\bf X}_{i}}(i_{1})\neq{\bf X}_{i_{1}}\\}$
$\displaystyle+\Pr\\{{\hat{\bf X}_{i}}(i_{2})\neq{\bf X}_{i_{2}},{\hat{\bf
X}_{i}}(i_{1})={\bf X}_{i_{1}}\\}$ $\displaystyle<$ $\displaystyle 2\cdot
2^{-n\eta}.$
Continuing this procedure, we have finally that
$\Pr\\{{\hat{\bf X}_{i}}(i^{*})\neq{\bf X}_{i^{*}}\\}<r\cdot 2^{-n\eta}<d\cdot
2^{-n\eta}.$
## References
* [1] A. Aaron and B. Girod, “Compression with side information using turbo codes,” Proc. IEEE Data Compression Conference, pp. 252–261, Snowbird, UT, Apr. 2002.
* [2] R. Ahlswede and I. Csiszár, “Common randomness in information theory and cryptography, Part I: Secret sharing,” IEEE Trans. Inf. Theory, vol. 39, pp. 1121–1132, July 1993.
* [3] M. Bloch, J. Barros, M. Rodrigues, S.W.M. McLaughlin, “Wireless Information Theoretic Security,” IEEE Trans. on Inf. Theory, Vol. 54, No. 6, pp. 2515–2534, June, 2008.
* [4] T. P. Coleman, A. H. Lee, M. Médard, and M. Effros, “On some new approaches to practical Slepian-Wolf compression inspired by channel coding,” Proc. IEEE Data Compression Conference, pp. 282–291, Snowbird, UT, March 2004.
* [5] I. Csiszár, “Linear codes for sources and source networks: Error exponents, universal coding,” IEEE Trans. Inf. Theory, vol. 28, no. 4, pp. 585–592, July, 1982.
* [6] I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic, New York, N.Y., 1982.
* [7] I. Csiszár and P. Narayan, “Common randomness and secret key generation with a helper,” IEEE Trans. Inf. Theory, vol. 46, pp. 344–366, March 2000.
* [8] I. Csiszár and P. Narayan, “Secrecy capacities for multiple terminals,” IEEE Trans. Inf. Theory, vol. 50, pp. 3047–3061, Dec. 2004.
* [9] P. Elias, “Coding for noisy channels,” IRE Convention Record, Part 4, pp. 37–46, 1955.
* [10] E. M. Gabidulin, “Bounds for the probability of decoding error when using linear codes over memoryless channels,” Prob. Pered. Inf., vol. 3, pp. 55–62, 1967.
* [11] J. Garcia-Frias and Y Zhao, “Compression of correlated binary sources using turbo codes,” IEEE Commun. Lett., vol. 5, pp. 417–419, Oct. 2001.
* [12] J. Garcia-Frias and W. Zhong, “LDPC codes for compression of multi-terminal sources with hidden Markov correlation,” IEEE Commun. Lett., vol. 7, no. 3, pp. 115–117, March 2003.
* [13] H.O. Georgii, Gobbs Measures and Phase Transitions. de Gruyter, Berlin – New York, 1988.
* [14] D-K. He and E-H. Yang, “On the duality between Slepian-Wolf coding and channel coding,” Proc. Int. Symp. Inf. Theory, pp. 2546–2550, Seattle, WA, July 2006.
* [15] R. Hu, R. Viswanathan and J. Li, “A new coding scheme for the noisy-channel Slepian-Wolf problem: Separate design and joint decoding,” Proc. Global Commun. Conference, Dallas, TX, 2004.
* [16] J. Li, Z. Tu, and R. Blum, “Slepian-Wolf coding for nonuniform sources using turbo codes,” Proc. IEEE Data Compression Conference, pp. 312–321, Snowbird, UT, March 2004.
* [17] C. Lan, A. Liveris, K. Narayanan, Z. Xiong, and C. N. Georghiades, “Slepian-Wolf coding of multiple M-ary sources using LDPC codes”, Proc. IEEE Data Compression Conference, p. 549, Snowbird, UT, March 2004.
* [18] Y. Liang, H.V. Poor and S. Shamai (Shitz), “Information Theoretic Security,” Foundations and Trends in Communications and Information Theory, vol. 5, no. 4-5, pp. 355-580, Now Publishers, MA, USA, 2008.
* [19] A. D. Liveris, Z. Xiong, C. N. Georghiades, “Compression of binary sources with side information at the decoding using LDPC codes,” IEEE Commun. Lett., vol. 6, pp. 440–442, Oct. 2002.
* [20] A. D. Liveris, C. Lan, K. R. Narayanan, Z. Xiong and C. N. Georghiades, “Slepian-Wolf coding of three binary sources using LDPC codes,” Proc. Int. Symp. Turbo Codes and Related Topics, Brest, France, Sept. 2003.
* [21] D. J. C. Mackay, “Good error correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory, vol. 45, pp. 399–431, Mar. 1999.
* [22] G. A. Margulis, “Probabilistic characteristics of graphs with large connectivity,” Probl. Inf. Trans. vol. 10, pp. 174–179, Apr. 1974\.
* [23] U. M. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory, vol. 39, pp. 733–742, May 1993.
* [24] P. Mitran and J. Bajcsy, “Turbo source coding: A noise-robust approach to data compression,” IEEE Data Compression Conference, p. 465, Snowbird, UT, Apr. 2002.
* [25] J. Muramatsu, “Secret key agreement from correlated source outputs using LDPC matrices,” IEICE Trans. Fundamentals, vol. E89-A, pp. 2036–2046, July, 2006.
* [26] S. Nitinawarat, “Secret key generation for correlated Gaussian sources,” Proc. Allerton Conf. Commun., Control, and Computing, Monticello, Illinois, pp. 1054-1058, Sept. 2007.
* [27] S. Nitinawarat, “Secret key generation for correlated Gaussian sources”, Proc. IEEE Int. Symp. Inf. Theory, pp. 702–706, Toronto, Canada, July, 2008.
* [28] W. W. Peterson and E. J. Weldon, Error-Correcting Codes, 2nd edition, MIT Press: Cambridge, Mass. 1972.
* [29] S. S. Pradhan and K. Ramchandran, “Distributed source coding using syndromes (DISCUS): Design and construction,” IEEE Trans. Inf. Theory, vol. 49, pp. 626–643, March 2003.
* [30] T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol. 47, pp. 599–618, Feb. 2001.
* [31] T. Richardson and R. Urbanke, Modern Coding Theory, New York: Cambridge, 2008.
* [32] S. Roman, Introduction to Coding and Information Theory, New York: Springer, 1996.
* [33] D. Schonberg, K. Ramchandran and S. S. Pradhan, “Distributed code constructions for the entire Slepian-Wolf rate region for arbitrarily correlated sources,” Proc. IEEE Data Compression Conference, pp. 292–301 Snowbird, UT, March 2004.
* [34] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory, vol. 27, pp. 533–547, Sept. 1981.
* [35] A. Thangaraj, S. Dihidar, A. R. Calderbank, S. McLaughlin and J. M. Merolla, “Capacity achieving codes for the wiretap channel with applications to quantum key distribution,” e-print cs. IT/0411003, 2004\.
* [36] R. Wilson, D. Tse and R. Scholtz, “Channel identification: Secret sharing using reciprocity in ultrawideband channels,” IEEE Trans. Inf. Foren. and Security, vol. 2, pp. 364–375, Sept. 2007.
* [37] A. D. Wyner, “Recent results in the Shannon theory,” IEEE Trans. Inf. Theory, vol. 20, pp. 2–10, Jan. 1974.
* [38] C. Ye and P. Narayan, “Secret key and private key constructions for simple multiterminal source models,” Proc. IEEE Int. Symp. Inf. Theory, pp. 2133–2137, Adelaide, Australia, Sept. 2005.
* [39] C. Ye and P. Narayan, “Secret key constructions for simple multiterminal source models,” Proc. Inf. Theory and Applications Workshop, San Diego, California, Feb. 2006.
* [40] C. Ye, A. Reznik and Y. Shah, “Extracting secrecy from jointly Gaussian random variables,” _Proc. Int. Symp. Inf. Theory_ , pp. 2593–2597, July 2006.
* [41] C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe and N. Mandayam, “Information theoretic secret key generation from wireless channels,” IEEE Trans. Inf. Foren. and Security, vol. 5, pp. 240-254, June 2010.
* [42] R. W. Yeung, Information Theory and Network Coding, New York: Springer, 2008.
|
arxiv-papers
| 2010-08-12T14:42:52 |
2024-09-04T02:49:12.152211
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chunxuan Ye and Prakash Narayan",
"submitter": "Chunxuan Ye",
"url": "https://arxiv.org/abs/1008.2122"
}
|
1008.2205
|
# Viewpoints: A high-performance high-dimensional exploratory data analysis
tool
P.R. Gazis SETI Institute, Mountain View, CA, 94043 pgazis@sbcglobal.net C.
Levit NASA Ames Research Center, Moffett Field, CA 94035
Creon.Levit@nasa.gov M.J. Way11affiliation: NASA Ames Research Center, Space
Science Division, Moffett Field, CA 94035, USA 22affiliation: Department of
Astronomy and Space Physics, Uppsala, Sweden NASA Goddard Institute for Space
Studies, 2880 Broadway, New York, NY, 10025 Michael.J.Way@nasa.gov
###### Abstract
Scientific data sets continue to increase in both size and complexity. In the
past, dedicated graphics systems at supercomputing centers were required to
visualize large data sets, but as the price of commodity graphics hardware has
dropped and its capability has increased, it is now possible, in principle, to
view large complex data sets on a single workstation. To do this in practice,
an investigator will need software that is written to take advantage of the
relevant graphics hardware. The Viewpoints visualization package described
herein is an example of such software. Viewpoints is an interactive tool for
exploratory visual analysis of large, high-dimensional (multivariate) data. It
leverages the capabilities of modern graphics boards (GPUs) to run on a single
workstation or laptop. Viewpoints is minimalist: it attempts to do a small set
of useful things very well (or at least very quickly) in comparison with
similar packages today. Its basic feature set includes linked scatter plots
with brushing, dynamic histograms, normalization and outlier
detection/removal. Viewpoints was originally designed for astrophysicists, but
it has since been used in a variety of fields that range from astronomy,
quantum chemistry, fluid dynamics, machine learning, bioinformatics, and
finance to information technology server log mining. In this article, we
describe the Viewpoints package and show examples of its usage.
Multivariate Datasets, Visualization
## 1 Introduction
Analysis and visualization of extremely large and complex data sets has become
one of the more significant challenges facing the scientific community today.
Current instruments and simulations produce enormous volumes of data. These
data sets include hyperspectral images from spacecraft, multivariate data of
high dimensionality from sky surveys, time-varying three-dimensional flows
from supercomputer simulations, and complex interrelated time series from
vehicle telemetry, DNA microarrays or physiological monitoring, to name a few.
One representative example from astronomy is the Sloan Digital Sky (SDSS York
et al., 2000). The spectroscopic catalog from the SDSS Data Release 7 (DR7
Abazajian et al., 2009) contains information for over 1.5 million objects. For
each object in the joint spectroscopic and photometric catalog there are over
100 physical measurements. Properties in these catalogs include position on
the sky, spectroscopic redshift, photometric magnitude in five bands,
energy/width of up to 50 spectral lines, galaxy angular size, bulge-to-disk
ratio, etc. In addition, many other physical quantities of interest can be
derived from these data, such as absolute magnitudes, photometric redshifts,
galaxy type, average distance to nearest neighbors and other properties of the
galaxy’s local environment. This is the kind of catalog in which Viewpoints
excels on a relatively modern desktop computer.
On the other hand the DR7 photometric catalog has measured over 350 million
unique objects. This is an enormous amount of data by traditional standards
and these kinds of data volumes will continue to grow in size over the coming
decade. The actual data volume in the SDSS DR7, including pre- and
postprocessing products, is more than 50 TB, while the searchable database for
the photometric and redshift catalogs is over 3.5
TB.111http://www.sdss.org/dr7 We do not claim that Viewpoints is able to
explore a data set of this size all at once as it is beyond the memory
capabilities of a normal desktop computer. Still, Viewpoints can quickly
explore subsets of a large complex data set like this. When Viewpoints does
reach the limits of desktop hardware, one can reach for more capable, but
expensive and rare, custom-built hardware systems such as the hyperwall
(Sandstrom et al., 2003). The hyperwall can then be utilized efficiently using
the knowledge already obtained through exploration via Viewpoints.
Although advances in hardware speed and storage technology have kept up with
(and, indeed, have driven) increases in database size, the same is not true of
the tools we use to explore and understand these data (e.g., SubbaRao et al.,
2008). Modern data sets routinely outstrip the capabilities of contemporary
visualization and analysis software, and this problem can only get worse as
data volumes from future astronomy-related programs (e.g., Large Synoptic
Survey Telescope [LSST]222See http://www.lsst.org, Supernova Acceleration
Probe [SNAP]333See http://snap.lbl.gov, Panoramic Survey Telescope and Rapid
Response System [PanSTARRS]444See http://pan-starrs.ifa.hawaii.edu/public, and
Dark Energy Survey [DES]555See http://www.darkenergysurvey.org) and ever-
larger supercomputer simulations increase by orders of magnitude. See Borne
(2009) for a review. Data sets of this size and dimensionality defy
interactive analysis with standard tools such as MATLAB,666See
http://www.mathworks.com Octave,777See http://www.octave.org
Mathematica,888See http://www.wolfram.com R,999See http://www.r-project.org
S++,101010See http://www.splusplus.com xGobi,111111See
http://www.research.att.com/areas/stat/xgobi Mirage,121212See http://www.bell-
labs.com/project/mirage gnuplot,131313See http://www.gnuplot.info etc. To
address this problem, we have developed Viewpoints.
Viewpoints was inspired to a large extent by the hyperwall (Sandstrom et al.,
2003). This facility, developed by the NASA Advanced Supercomputing division
at NASA Ames Research Center, is a high-performance visualization cluster that
was specifically designed to address the problem of exploring, visualizing,
and analyzing large, complex, multidimensional data. See Wong & Bergeron
(1997) for a review on the history of multidimensional multivariate
visualization. In its original form, the hyperwall was a 50 node Linux cluster
with a 7$\times$7 array of tiled displays that could be used to display data
in a variety of formats that might range from simple plots to elaborate 3D
rendering. It has since been superseded by even larger facilities at NASA Ames
Research Center and several other institutions throughout the world.
Hyperwalls have been applied to a wide range of problems in astrophysics,
Earth science, aerodynamics, and life sciences (MacDougall & Henze, 2003;
Brown et al., 2004; Murman et al., 2004). Unfortunately, tools such as the
hyperwall require custom hardware and a team of onsite specialists at a
dedicated facility, which places them beyond the reach of most investigators
and research programs.
Viewpoints, shown in Figure 1, is intended to provide a small but powerful
subset of the hyperwall’s functionality on any working scientist’s desktop or
laptop computer.
Figure 1: Viewpoints displaying a 2$\times$2 array of scatter plots of SDSS
data. (Upper left) xy positions of $\sim$20,000 galaxies from the SDSS. Colors
indicate different classes identified by a clustering algorithm. (Upper right)
Density classes identified by a Bayesian partitioning scheme vs class
identified by a Self-Organizing Map. (Lower left) Local density vs local
gradient. (Lower right) Local density vs density classes identified by a
Bayesian partitioning scheme. (Far right) The viewpoints control panel.
Viewpoints provides a high-performance interactive multidimensional visual
exploratory data analysis environment that allows users to quickly explore and
visually analyze the most common forms of large multidimensional scientific
data. It can also be run as a “plug in” or external module for scientific
programming environments such as MATLAB.
In contrast to programs like Mirage (Ho, 2007) or VisIVO (Becciani et al.,
2010) which have a large number of supporting features, Viewpoints does a few
things very well. Its basic feature set includes linked scatter plots with
brushing (e.g., Becker & Cleveland, 1987; Davidson & Sardy, 2000; Chen, 2003;
Stump et al., 2004; Co et al., 2005; Jordan et al., 2008), dynamic histograms,
normalization, and outlier detection/removal. It also includes a limited
capability for Boolean operations with brushes analogous to those of Hurter et
al. (2009).
## 2 Overview and Capabilities
Viewpoints is a fast interactive tool for the analysis and visualization of
high-dimensional data primarily via linked scatter plots with brushing. It is
intended for general-purpose visual data analysis, interactive exploratory
data mining, and machine-assisted pattern discovery. The targeted application
is the exploration of large, complex, multivariate data sets, with
dimensionalities up to 100 or more and sample sizes up to 105–107. Viewpoints
takes advantage of hardware-accelerated graphics (graphics processing units
[GPUs]) and other capabilities of modern desktop workstations and laptops
(e.g., multithreading and memory-mapped input/output). As previously above, it
is highly interactive, and almost all of its parameters can be modified in
real time to provide immediate visual feedback.
Viewpoints can be invoked either from a desktop icon or from the command line
with a set of optional parameters. Once it is running, it is controlled by a
conventional GUI and/or configuration files saved from previous sessions.
Viewpoints reads multidimensional data in the form of ASCII files, binary
files, or simple Flexible Image Transport System (FITS) tables and displays
these data as an arbitrary number of simultaneously linked 2D and 3D scatter
plots (e.g., Comparato et al., 2007) with independently controllable viewing
transformations, normalization schemes, overplotting schemes (Zhang et al.,
2003), axis labels, and other formatting. These data can then be visualized
and explored interactively using a set of graphical, statistical, and
semiautomated pattern-recognition techniques. The user can select, flag,
and/or delete subsets of the data using brushes, and the modified data set can
be written back to disk in ASCII, binary, or FITS format.
The program’s capabilities were carefully selected using a minimalist
philosophy. We have chosen to provide a small core of essential features that
are universally useful and keep the program size small, execution speeds high,
and the learning curves shallow. Figure 1 shows an example of Viewpoints
displaying a small 2$\times$2 array of linked scatter plots. This figure is
described in detail in §5. In practice, one can simultaneously display an
arbitrary number of linked scatter plots, and we routinely use up to 30
windows to examine data sets of high dimensionality. Viewpoints has a
conventional GUI, shown in the right panel of Figure 1, with an assortment of
menus, buttons, and sliders that provide access to its features. The most
important of these capabilities can also be invoked by mouse gestures and/or
keyboard shortcuts. It has a full range of basic help facilities, such as
usage messages, help files, and tooltips. It also produces a diagnostic output
stream when invoked via the command line.
Viewpoints has a tightly integrated set of features. A partial list of these
features is provided next. Some were in the original release, others were
added in response to users requests, and a few were contributed to the open
source repository141414See http://www.assembla.com/wiki/show/viewpoints by a
small community of developers.
1. 1.
Multiple linked scatter plots with dynamic brushing rendered via OpenGL with
update rates of greater than $10^{7}$ points $s^{-1}$ for most modern graphics
cards.
2. 2.
The ability to brush separate selections using separate colors and/or symbols.
3. 3.
The ability to map densely overplotted regions smoothly to different hues
and/or intensities.
4. 4.
The ability to save the program’s complete state to an XML formatted
configuration file at any time.
5. 5.
The ability to initialize and/or restore the program’s state from a
configuration file.
6. 6.
The ability to append and merge additional data and create or destroy
additional plot windows on the fly.
7. 7.
Per-axis one-dimensional equiwidth marginal histograms with dynamically
adjustable bin widths and dynamic update of the selected fraction in each bin.
Histograms are individually selectable and adjustable for both the $x$ and $y$
axes of each plot separately.
8. 8.
Per-axis automatic normalizations based data limits, quantiles, moments, rank,
or simple Gaussianization.
9. 9.
Elementary outlier detection and removal.
10. 10.
The ability to change most parameters globally across all plots
simultaneously, or locally on a single plot.
11. 11.
The ability to lock (and unlock) individual plots so they are immune from
(susceptible to) global operations.
12. 12.
Rudimentary semi-automated browsing through high-dimensional data by permuting
all (unlocked) axes on all plots.
13. 13.
Real-time display of quantitative data for individual axes and for the current
selection.
14. 14.
Basic online help.
15. 15.
A comprehensive set of error tests along with graceful error recovery if a
user attempts to read corrupt and ill-formatted data or configuration files.
16. 16.
The ability to replace missing values with user-specified default values.
17. 17.
Basic ability to plot time series analogous to that of Akiba & Ma (2007).
18. 18.
Viewpoints also achieves smooth interactive performance on large data as long
as:
1. (a)
The (binary) array corresponding to the dataset fits into real system (CPU)
memory
2. (b)
The coordinates of all vertices being displayed fit into available graphics
(GPU) memory. For example, on a 512 Mbyte graphics card viewpoints can
simultaneously update and display (4
plots)$\times$(8x106points/plot)$\times$(3 vertices/point)$\times$ (4
bytes/vertex)= $\sim$400Megabytes
## 3 Software Design
Viewpoints uses hardware-accelerated OpenGL 1.5 on all platforms and performs
rendering via OpenGL151515http://www.opengl.org vertex buffer objects to
provide extremely high graphics performance. The package has cross-platform
availability and can be easily ported to other architectures. Table 1 has a
list of operating systems and architectures known to work to-date. Viewpoints
has a preliminary interface to MATLAB and Octave. It has the potential to work
with scripting languages such as Python161616See http://www.python.org or
Perl.171717See http://www.perl.org Most importantly, Viewpoints is
specifically engineered for use with large data sets. As noted previously, the
targeted application is the interactive analysis of large, complex,
multivariate data sets, with dimensionalities that may surpass 100 and sample
sizes that may exceed 105–107.
Table 1: Platforms Known To Run viewpoints Operating System | OS Revision | Architecture
---|---|---
Apple | Mac OS 10.3–10.6 | Intel
Apple | Mac OS 10.3–10.6 | PPC
Linux | Most Distributions with Kernels 2.4.x–2.6.x | Intel
Windows | 2000 | Intel
Windows | XP | Intel
Windows | Vista | Intel
Windows | 7 | Intel
In the interests of portability, extensibility and efficiency, Viewpoints is
written in C++. This allows us to program at a high level of abstraction and
still achieve excellent runtime performance on all target platforms. Most
importantly, C++ allows us to leverage vast amounts of code written by other
people. Identical code compiles and runs on Linux, Microsoft Windows, and
Apple OS X.
Viewpoints is a relatively small program. The compiled Linux binary is only
5.5 Mbytes.181818The compiled file contains a large number of statically
linked libraries Since it makes extensive use of libraries including some that
leverage C++ template metaprogramming (e.g.,
Blitz++191919http://www.oonumerics.org/blitz and
Boost202020http://www.boost.org), the Viewpoints source tree itself is also
rather small (approximately 40,000 lines). The libraries it depends on are
listed in Table 2.
Table 2: C++ libraries used in viewpoints Library | Description
---|---
libc++ | The standard runtime library for basic math, input/output, and memory management
STL | The standard template library for data structures and algorithms
OpenGL | The standard cross-platform library for hardware accelerated graphics
FLTK | A lightweight platform-independent open source graphical user interface (GUI) library
Blitz++ | A library of extensions that add FORTRAN 90-style array expressions (and fortran-90 style numerical performance) to c++
GSL | The GNU scientific library. Includes functionality such as statistics and linear algebra
Boost serialization | a library of extensions for saving and restoring the state of c++ objects.
## 4 GPU Usage and OpenGL Implementation
Graphics boards that were available at the time development began, such as
NVIDIA Geforce 7950, can transform and display over a billion floating-point
vertices per second and can drive two or more 4 Mpixel displays
simultaneously. This capability continues to increase. In its default mode,
Viewpoints uses GPUs explicitly to take advantage of this tremendous
processing and rendering power. This allows the package to provide smooth
high-frame-rate interactivity while manipulating multiple linked views, each
containing millions of data points. If for some reason a GPU is not available,
Viewpoints can also be instructed to use a system’s ordinary CPU, at some cost
in performance.
The current release uses the following advanced OpenGL features:
1. 1.
GPU-resident vertex buffer objects (VBOs) for point coordinates. This
conserves main memory (by offloading relevant data to GPU memory), increases
graphics performance (by caching frequently rendered vertex data locally on
the GPU), and frees up essentially all of the CPU bandwidth and bus bandwidth
for other uses.
2. 2.
GPU-resident VBOs for point indices. A single copy of the vertex index arrays
(which encode the selection status) is shared by all plots.
3. 3.
Squared alpha blending with separate blending control per plot. For data with
wide variations in projected density this provides fast, adjustable mapping of
overplot density to varying hue and/or intensity.
4. 4.
Point sprites for displaying symbols and/or smooth kernels instead of just
simple points (splatting).
5. 5.
Antialiased points and/or point sprites of variable color, brightness, and
size per brush and/or per plot.
6. 6.
Floating-point color and alpha values, allowing both more sensitivity and more
dynamic range for overplotting.
These design features provide dramatic graphics performance. When one of our
early beta releases was run on a 1997 MacBook Pro laptop, we measured
throughput of over $70\times 10^{6}$ vertices s-1. This consisted of six
linked plots, each with 2 million data points simultaneously updating at five
frames s-1.
Viewpoints has proved quite useful on small-sized (e.g., netbooks) and medium-
sized computer platforms with as little as 1Gbyte of RAM. We do most of our
demos on a laptop displaying up to 18-27 variables in 6-9 windows on a single
monitor. To take full advantage of this package we recommend using a high-end
workstation with high-end graphics card, two high-resolution displays, and
several gigabytes of RAM. Experience suggests that this is a good impedance
match to the visual perception of a scientist or a team of up to three
collaborators.
The best way to think about the limitations of Viewpoints with respect to
common desktop computer hardware is that the data should be able to literally
fit in RAM memory. Specifically Viewpoints can achieve smooth interactive
performance on data as long as:
1. 1.
The (binary) array corresponding to the data set fits into real system (CPU)
memory
2. 2.
The coordinates of all vertices being displayed fit into available graphics
(GPU) memory.
For those looking to the future please note that the relationship between the
amount of memory on a graphics card and the allowable size of the data set
appears to be linear. So the latest graphics cards with 1.5Gbyte GPU memory
should display four plots with $\sim$25$\times 10^{6}$ points per plot, with
an update rate of a few frames per second. The update rate is harder to
estimate, since “vertices per second” is no longer a common figure of merit in
GPU specs. However, this appears to scale roughly along with clock speed and
GPU memory.
When visible plots exhaust GPU memory, one can run Viewpoints with the
“–no_vbo” flag, and check the “defer redraws” box in the lower right-hand
control panel. We have run 100 million point data sets on a laptop this way
and it is quite usable.
## 5 Examples
Figure 1 shows a simple example of Viewpoints usage with a small 2$\times$2
array of linked scatter plots of data from the SDSS (York et al., 2000),
several methods of calculating local density, and a cluster detection scheme.
The upper left panel shows $xy$ coordinates of $\sim$20,000 galaxies from the
SDSS. When used this way, a scatter plot becomes a map of a region of space
that is approximately 30 Mpc across. The blue, yellow, and black colors
indicate cluster galaxies, halo galaxies, and field galaxies identified by a
partitioning scheme based on self-organizing maps (SOMs). It can be seen that
cluster galaxies (blue) are concentrated into filaments while the field
galaxies (black) are distributed more or less uniformly. The upper right panel
shows density classes identified by a Bayesian partitioning scheme versus
classes identified by the SOM-based algorithm. This panel is where the
selection capability of Viewpoints was used. The rectangle in the lower right
side of the panel is the selection box that was used to flag the halo galaxies
(yellow points). The lower left panel shows local density versus local
gradient to display the distribution of the three galaxy classes in parameter
space. The lower right panel shows local density versus density class
identified by a Bayesian partitioning scheme. This panel shows a simple
example of the histogram capability of Viewpoints. Figure 1 provides a real-
world example of the abilities of Viewpoints to quickly deduce scientifically
useful features for research purposes. The example provided here is directly
related to a recent publication by two of the authors of this article (Way,
Gazis & Scargle, 2010). All of the figures comparing the three different
clustering technologies described in (Way, Gazis & Scargle, 2010) were
originally devised using Viewpoints. Publication-quality plots were later
produced using MATLAB.
Figure 2 shows a more elaborate usage involving a 2$\times$2 array of linked
scatter plots of astrometric data from the Tycho mission (Perryman et al.,
1997; Høg et al., 2000). For clarity, the control panel has been eliminated
from this figure.
Figure 2: Viewpoints displaying a 2$\times$2 array of linked scatter plots of
Tycho data for 1 million stars. (Upper left) Right ascension vs declination.
(Upper right) Number of transits vs parallax error, with a selection box
outlined in white. (Lower left) Signal to noise ratio vs a quality flag.
(Lower right) V band magnitude plotted vs the B-V magnitude difference.
The upper left panel shows a plot of right ascension versus declination for 1
million stars. This panel illustrates the effect of overplotting with alpha
blending. The galactic plane, with its greater concentration of stars, is
plainly visible as a curving white band that meanders through the image. The
upper right panel shows a plot of the number of transits versus parallax
error. The selection box (outlined in blue) has been used to select points for
which more than 250 transits were available. The legend at the top of the
panel reports that these involved 32,503 points or 3.14% of the total. This
selection is repeated in the other panels. In the upper left panel described
previously, it appears as two parallel and roughly sinusoidal bands that mark
the part of the spacecraft’s orbit from which it was possible to obtain this
number of transits. The lower left panel shows a plot of signal-to-noise ratio
(S/N) versus a quality flag. This is included to illustrate use of a marginal
(conventional) and a conditional histogram. The latter peaks at a
significantly higher S/N ratio than the former, as would be expected, since
measurements that involve more transits should have a higher S/N. In the lower
right panel the $V$-band magnitude has been plotted against the $B--V$
magnitude difference to produce a typical color-magnitude diagram.
Figure 3 shows an example of Viewpoints used to visualize the set of solar
wind data that is provided with the Viewpoints package for demonstration
purposes.
Figure 3: Viewpoints displaying a 2$\times$3 array of linked scatter plots of
solar wind data. (Upper left) Solar wind speed versus time. (Upper middle)
Solar wind density vs time. (Upper right) Solar wind temperature vs time.
(Lower left) Density vs speed. (Lower middle) Temperature vs speed. (Lower
right) Temperature vs density.
In the top three panels, solar wind speed, density, and temperature have been
plotted versus day since 1950 to produce a conventional time series. These
show a typical structure in the solar wind, in which high-speed/low-density
plasma (visible on the left of each panel) overtakes low-speed/low-density
material (visible on the right) to produce an interaction region (visible in
the middle) that is characterized by intermediate speeds, high density, and
high temperature. These interaction regions are often subdivided into a
leading portion and trailing portion with different ranges of densities and
temperatures separated by a well-defined boundary. These are plainly visible
in this time-series plot. The selection capability of Viewpoints has been used
to select these regions as blue and yellow. The bottom three panels show
scatter plots of velocity versus density, velocity versus temperature, and
density versus temperature. It can be seen that the leading portion of this
interaction region (blue) was associated with a large range of densities and
moderate range of temperatures, while the trailing portion was associated with
a moderate range of densities and large range of temperatures.
As mentioned in the Abstract, Viewpoints has been used extensively in a
variety of fields to advance specific scientific goals. Further examples
include:
* •
Geophysics, Relationships between earthquake occurrence frequency, location,
season, and time of day.
* •
Information Technology, Relationships between query response time, query
source location, time of day, type of query, etc. These were causing slow
response and high system loads on commercial servers which viewpoints helped
resolve quickly.
* •
Finance, Currency arbitrage relationships and portfolio clustering in
statistical arbitrage.
## 6 Opening up to the open source community
Two significant concerns with any software tools are long-term support and
response to the user community. Viewpoints has been released as an open source
and is on a collaborative-oriented public wiki site called Assembla.212121See
http://www.assembla.com We now have a significant and growing user community
that is successfully and enthusiastically applying Viewpoints to a huge
variety of problem spread across multiple disciplines.
## 7 Conclusions and the future of viewpoints
Viewpoints was adapted from its alpha prototype to general use with seed
funding from the NASA Applied Information Systems Research Program.222222See
http://aisrp.nasa.gov It is now available on all major computing platforms
(see Table 1) and is in use by scientists and engineers across all NASA
mission directorates, as well as by some early adopters in industry and
academia. Its operation, interface, source code, and impressive performance
are the same on all platforms. Viewpoints is now directly callable from
MATLAB. There is a Viewpoints wiki and Web site232323See
http://www.assembla.com/wiki/show/viewpoints that allows the open source
community to take part in development and affords users access to the latest
information regarding bug fixes and program enhancements. In the near future
we expect more development to come from the open source community and plan to
modify the base code to take advantages of new advances in video chip
technology as it continues to evolve.
Many thanks go to Chris Henze and Jeff Scargle for their input and ideas. The
authors would also like to thank Joe Bredekamp and his NASA Applied
Information Systems Research Program for pilot funding of Viewpoints. M.J.W.
would like to thank the Astronomy Department of Uppsala University for their
kind hospitality. This research has made use of NASA’s Astrophysics Data
System. Viewpoints has been released under the NASA Open Source Agreement and
is considered an open source license. The source code and precompiled binaries
can be freely downloaded from the Viewpoints Web site. If you use Viewpoints
in your research, please consider referencing this work.
## References
* Abazajian et al. (2009) Abazajian, K. et al. 2009, ApJS, 182, 543
* Akiba & Ma (2007) Akiba, H. & Kwan-Liu, M. 2007, “An Interactive Interface for Visualizing Time-Varying Multivariate Volume Data”, Proceedings of the Joint Eurographics-IEEE VGTC Symposium on Visualization, May 2007
* Becciani et al. (2010) Becciani, U. et al. 2010, PASP, 122, 119
* Becker & Cleveland (1987) Becker, R.A. & Cleveland, W.S. 1987, Technometrics, 29, 127
* Borne (2009) Born, K. 2009, arXiv:0911.0505v1
* Brown et al. (2004) Brown, J., Tobak, M., Prabhu, D. & Sandstrom, T. “Flow Topology About an Orbiter Leading Edge Cavity at STS-107 Reentry Conditions”, AIAA-2004-2285 37th AIAA Thermophysics Conference, Portland, Oregon, 2004.
* Chen (2003) Chen, H. 2003, IEEE Symp. on Inf. Vis. (Washington: IEEE), 181
* Co et al. (2005) Co, C.S. et al. 2005, Eurographics/IEEE-VGTC Symp. Vis. (Washington: IEEE) 279
* Comparato et al. (2007) Comparato, M. et al. 2007, PASP, 119, 898
* Davidson & Sardy (2000) Davidson, A.C. & Sardy, S. 2000, J. Comput. and Graph. Stat., 9, 750
* Ho (2007) Ho, T.K. 2007, in Statistical Challenges in Modern Astronomy IV, ASP Conferences Series, Vol. 371, p.391. eds G. Jogesh Babu and Eric D. Feigelson
* Høg et al. (2000) Høg, E. et al. 2000, A&A, 355, L27
* Hurter et al. (2009) Hurter, C., Tissoires, B. & Conversy, S. 2009, “FromDaDy: Spreading Aircraft Trajectories Across Views to Support Iterative Queries”, IEEE Transactions on Visualization and Computer Graphics, 15, 1017
* Jordan et al. (2008) Jordan, D.D. et al. 2008, AAS 08-245, AAS/AIAA Space Flight Mechanics Meeting, (AAS 08-245; Springfield; AAS), 499
* MacDougall & Henze (2003) MacDougall, P. & Henze C. “Fleshing-out pharmacophores with volume rendering of molecular charge densities and hyperwall visualization technology, 226th ACS National Meeting New York, NY September 07-11, 2003.
* Murman et al. (2004) Murman, S.M., Aftosimis, M.J. & Nemec, M. “Automated Parameter Studies Using a Cartesian Method”, AIAA 2004-5076 22nd AIAA Applied Aerodynamics Conference, Providence, RI., 2004
* Perryman et al. (1997) Perryman, M.A.C., et al., 1997, The Hipparcos and Tycho Catalogues, (SP-1200, Noordwijk: ESA), 117
* Sandstrom et al. (2003) Sandstrom, T.A., Henze, C., & Levit, C. 2003, Coordinated & Multiple Views in Exploratory Visualization (Piscataway: IEEE), 124
* Stump et al. (2004) Stump, G.M. et al. 2004, Aerospace Conference (Piscataway: IEEE), 6, 3885
* SubbaRao et al. (2008) SubbaRao, M. U., Aragón-Calvo, M.A., Chen, H.W., Quashnock, J.M., Szalay, A.S. & York, D.G. 2008, NJP, 10, 125015
* Way, Gazis & Scargle (2010) Way, M.J., Gazis, P.R. & Scargle, J.S. 2010, ApJ, submitted (arXiv:1009.0387v1)
* Wong & Bergeron (1997) Wong, P.C. & Bergeron, R.D. 1997, Scientific Visualization Overviews Methodologies and Techniques (Washington: IEEE), 3
* York et al. (2000) York, D.G. et al. 2000, AJ, 120, 1579
* Zhang et al. (2003) Zhang, J. et al. 2003, IEEE International Conference on Data Mining (Washington: IEEE)
|
arxiv-papers
| 2010-08-12T20:00:28 |
2024-09-04T02:49:12.161924
|
{
"license": "Public Domain",
"authors": "P.R. Gazis (Seti), C. Levit (NASA/Ames), M.J.Way (NASA/GISS)",
"submitter": "Michael Way",
"url": "https://arxiv.org/abs/1008.2205"
}
|
1008.2247
|
# Symmetry and Uncountability of Computation
小林 弘二
(2010-09-22)
###### 概要
This paper talk about the complexity of computation by Turing Machine. I take
attention to the relation of symmetry and order structure of the data, and I
think about the limitation of computation time. First, I make general problem
named “testing problem”. And I get some condition of the NP complete by using
testing problem. Second, I make two problem “orderly problem” and “chaotic
problem”. Orderly problem have some order structure. And DTM can limit some
possible symbol effectly by using symmetry of each symbol. But chaotic problem
must treat some symbol as a set of symbol, so DTM cannot limit some possible
symbol. Orderly problem is P complete, and chaotic problem is NP complete.
Finally, I clear the computation time of orderly problem and chaotic problem.
And P != NP.
## 1 Introduction
本論文ではTMで扱う問題の複雑性について述べる。情報の対称性と順序構造の関係に着目し、計算時間の限界を考察する。
始めに、DTMやNTMで扱う問題を一般化した検定問題を定義する。検定問題によってNP完全性の条件や性質を明らかにし、計算量についての議論を容易にする。
次に、検定問題を秩序問題と混沌問題に分割する。秩序問題は、検定問題にある種の順序が存在し、問題の可能性を効率良く制約することのできる問題のことである。秩序問題はP完全になる。混沌問題は、秩序問題のような順序は存在せず、大きな塊を丸ごと扱う必要のある問題のことである。混沌問題はNP完全になる。
そして、秩序問題の持つ順序による計算の効率性と、混沌問題の非効率性を比較し、それぞれの複雑性の違いを明らかにする。結果として、$P\neq NP$を導く。
## 2 Define the testing problem
DTMやNTMで扱う問題を一般化した問題を定義する。
「検証機（verifier）」はある問題を判定可能なTMである。検証機はどのような入力であっても必ず停止状況に到達し、受理状況か拒否状況になる。「検証情報(verify
data)」は検証機が計算する情報である。「検証問題(verifying problem)」は検証機が判定できる問題である。「特定文字(specific
symbol)」は検証情報を構成する文字である。
「検査機(checker)」は「総称文字(generic symbol)」を扱うことのできるTMである。「総称文字(generic
symbol)」は特定文字のいずれか一つを意味する特別な文字である。さらに総称文字は、その総称文字の意味する特定文字のより少ない総称文字を意味することがある。「検査情報(check
data)」は総称文字を含む情報である。検査機は、検証機が総称文字を特定文字に置き換えた検証情報を全て拒否する検査情報のみを拒否し、それ以外の検査情報を受理する。つまり、検証情報のうちいずれか一つでも検証機が受理するものがある検査情報を受理する。「検査問題(checking
problem)」は検査機が判定できる問題である。
選択文字(selected
symbol)は、検査情報の総称文字において置き換えることのできる特定文字（あるいは意味する特定文字のより少ない総称文字）のことである。
検査機は検証機に総称文字を処理する遷移関数を追加したTMとなる。もしも入力に総称文字が含まれない場合、検査機は検証機と同様の計算を行い、同じ停止局面で計算を終了する。
検証問題と検査問題の両方を検定問題(testing
problem)、検証機と検査機の両方を検定機(tester)、検証情報と検査情報の両方を検定情報(test data)と呼ぶこともある。
「計算結果(computation
result)」は、ある検定情報を検定機で計算した停止状況（受理または拒否）のことである。また、計算結果を受理とも拒否とも決定していない状況を明示するために「残留状況(remaining
configuration)」と呼ぶこともある。
検定問題と検証問題には下記の関係がある。
###### Theorem 1.
検証問題がPの時、検定問題はNP完全となる。
###### 証明.
検定問題がNPに属することは容易に示すことができる。ある検定情報が受理可能な時、その検定情報には、総称文字以外が一致してなおかつ検証問題で真となる検証情報が存在する。その検証情報の検証問題はPであるとしているので、検定問題はNPに属する。
検定問題がNP困難であることは、SATを検定問題に多項式時間で帰着させることで行う。この検定情報は、SATの論理式を特定文字として、SATの真理値割当を総称文字として持つ。この検定情報への帰着は、論理式に対応する真理値割当は論理式の長さよりも短くなること、及び論理式の変数の抽出は多項式時間で可能なことを考えると、多項式時間で行うことができる。
以上より、検証問題がPの時、検定問題はNP完全となる。 ∎
## 3 Symmetry and order of the problem
問題の複雑さを考えるために2種類の検定問題を考える。
### 3.1 Orderly problem
始めに、総称文字をそれぞれ別々に扱うことのできる検定問題を考える。
###### Definition 2.
全ての検定情報に、選択文字への置換により計算結果が確定する総称文字が少なくとも一つ以上存在し、またその総称文字と選択文字を多項式時間で計算可能な検定問題を秩序問題(orderly
problem)と呼ぶ。また、この総称文字を焦点文字(focus symbol)、結果が確定する選択文字を対称文字(symmetry
symbol)と呼ぶ。また、焦点文字の計算結果を維持する選択文字を簡約文字(simplify symbol)と呼ぶ。
秩序問題の検定情報には、全ての計算結果が対称となる選択文字が少なくとも一つ存在する。簡約文字は、他の総称文字がどのような選択文字であったとしても計算結果が対称になる。この対称性を活用することにより検定問題の総称文字を順番に限定することができる。もしその計算結果が受理の場合は総称文字をその選択文字に置換することによって、計算結果が拒否の場合は総称文字をその選択文字以外の特定文字を意味する総称文字に置換することによって、検定情報の計算結果を維持しながらより選択肢の少ない検定情報に変更することができる。つまり、焦点文字は、計算結果の観点から他の総称文字と独立している部分を持つ。
なお、焦点文字を特定文字に置き換えた時に受理/拒否/残留のいずれかを判定することができれば、簡約文字がどの選択文字になるかを、全ての選択文字について判定を行うことで多項式時間で探すことができる。
また、秩序問題では、焦点文字を簡約文字に置き換えた検定情報も秩序問題の条件を満たす。よって下記が成立する。
###### Theorem 3.
秩序問題の検定情報に含まれる総称文字には、少なくとも一つの順序が存在する。
###### 証明.
2より、秩序問題の検定情報には対応する焦点文字が少なくとも一つ存在し、簡約文字に置き換えることができる。この置き換えた検定情報にもまた対応する焦点文字が少なくとも一つ存在し、簡約文字に置き換えることができる。この関係は検定情報に総称文字が含まれなくなるまで繰り返すことができる。このようにして得られた検定情報の列は、検定情報に含まれる総称文字を、計算結果を変更せずに選択文字に置き換える順序を表す。∎
###### Theorem 4.
検証問題がPの秩序問題はP完全となる。
###### 証明.
まず始めに、検証問題がPとなる秩序問題がPに属することを示す。
3より、秩序問題は多項式時間で順番に焦点文字を簡約文字に変更することが可能であり、最終的には検定情報を検証情報にすることができる。この変更は、総称文字が検定情報よりも少ない数しか存在しないことを考慮すると、高々多項式時間しかかからない。また検証情報の検証もまた多項式時間しかかからないということも考慮すると、この秩序問題の計算は多項式時間で行うことができる。よって、検証問題がPの秩序問題はPに属する。
条件を満たす秩序問題に検査問題がP困難であることは、HornSATを秩序問題に対数領域で帰着させることで行う。
秩序問題の検証問題をHornSAT検証とすると、1と同様にして論理式と真理値割当を意味する検査情報を作成することができる。この計算は、論理式から変数を取り出す計算であるから、作業領域には論理式の現在の位置を記録するだけで行うことができる。よって、HornSATは対数領域で帰着することができる。
HornSATを帰着させた検査情報が秩序問題の条件を満たすことを示す。HornSATの変数には順序があり、その順序に従って真にしかならない変数を特定していき、最終的に論理式が充足するかどうかを判定することができる。これはHornSATを帰着させた検査情報でも同様であり、順番に真理値割当を表す総称文字を特定文字に変更することができる。よってHornSATを検査情報にした検査問題は秩序問題の条件を満たす。
つまり、HornSATを多項式時間で秩序問題に帰着させることができる。
よって、検証問題がPの秩序問題はP完全となる。 ∎
### 3.2 Chaotic problem
次に、複数の総称文字を塊として扱う必要のある検定問題を考える。
###### Definition 5.
どの総称文字をどの選択文字に置換しても計算結果が確定せず、計算結果を確定するためには多数の総称文字を選択文字に置換する必要のある検定情報が無数に存在する検定問題を混沌問題(chaotic
problem)と呼ぶ。これらの総称文字の組合せを拡張文字(extended symbol)、拡張文字を構成する一つ一つの文字を単位文字(unit
symbol)と呼び、総称文字の数を拡張文字長(extended symbol
length)と呼ぶ。総称文字の一部あるいは全てを特定文字に変更した組合せも拡張文字と呼ぶことがある。そして混沌問題では拡張文字も無数に存在するものとする。また、混沌問題の補問題を普遍問題(universal
problem)と呼ぶ。
混沌問題は、単独の焦点文字の簡約文字を決定するちょうどいい制約が存在せず、ある総称文字の選択文字を決定しようとすると同時に複数の総称文字の選択文字も決定してしまう振る舞いをする。つまり、複数の総称文字を拡張文字として一塊に扱う必要があり、秩序問題のように扱うことができない。総称文字や特定文字が増加した形になり、総称文字や特定文字に対して拡張文字長で指数倍した数の拡張文字を扱う必要がある。つまり、拡張文字を構成するそれぞれの単位文字は検定機に対して対称となる。検定機が一部の単位文字を優先して対称性を対称性の破ったとしても、混沌問題全ての検定情報を集めた場合は全体として対称となる。
なお、拡張文字が有限数しか存在しない場合は、その拡張文字を総称文字や特定文字にした別のTMを用意することにより、容易に秩序問題に変更することができる。
混沌問題は検定問題から秩序問題を除いた集合となる。秩序問題が検定問題と一致すれば混沌問題は空集合となる。秩序問題に収まらない検定問題が存在すれば、それは混沌問題となる。
###### Theorem 6.
検証問題がPの時、混沌問題は空集合とならない検定問題が存在する。
###### 証明.
背理法で証明する。検証問題がPの時、混沌問題は空集合になると仮定する。
混沌問題が空集合ということは検定問題が全て秩序問題であることを意味する。つまり、全ての検定情報には焦点文字と対称文字が存在し、焦点文字を対称文字に置き換えた検定情報は結果が対称となる。
しかし、検定問題はSATを含むNP完全問題であり、検定情報と計算結果には秩序問題のような対称性のない問題が存在する。例えば、SATでは論理式の構成を変更することにより任意の真理値表を実現することが可能である。そのため、全ての変数について、その変数の値にかかわらず他の変数の値によって論理式の値が真偽どちらにもなりうる論理式もまた存在する。SATを還元させた検定問題でも同様に焦点文字や対称文字は存在せず、秩序問題の条件を満たさない。
つまり、検証問題がPの時に検定問題が全て秩序問題であるという仮定と矛盾し、混沌問題は空集合となるという仮定とも矛盾する。
よって背理法より、検証問題がPの時、混沌問題は空集合とならない検定問題が存在することがいえる。 ∎
## 4 Difference between Orderly problem and Chaotic problem
混沌問題と秩序問題の最も大きな違いは計算量の違いである。下記で混沌問題の計算量の違いを明らかにする。
###### Theorem 7.
検証問題がPの混沌問題はPに含まれない。
###### 証明.
背理法で証明する。検証問題がPの時、混沌問題がPに含まれると仮定する。
ある検定情報Aと、その検定問題の総称文字のいずれか1つを選択文字に変更した検定情報の集合Bを考えると、仮定より、それらの計算結果も多項式時間で求めることができる。また集合Bから検定情報Aの計算結果が同じ検定情報を選別することも多項式時間で可能である。つまり、ある検定情報Aの総称文字の選択文字を多項式時間で求めることができる。
しかし、混沌問題の条件より、この総称文字と選択文字は他の総称文字から独立しておらず、他の選択文字と同時にしか決定することができない。また、拡張文字を構成する全ての単位文字は検定機に対して対称であるため、検定機がいずれの単位文字を優先したとしても、混沌問題全体として見た場合は全て対称となる。よって、混沌問題全体としては結局拡張文字をそれぞれ独立した文字として扱う必要がある。この拡張文字の数は、特定文字の種類の数を拡張文字長で指数倍した規模であり、多項式時間で計算可能という条件と矛盾する。
また、もしもこの総称文字の選択文字を拡張文字と関係なく判定することができる場合は、この総称文字は他の総称文字から独立しているということであり、拡張文字の一部にはならない。そしてこの関係は全ての検定情報で成立するため、結果としてこの検定問題には拡張文字が存在しないということになり、混沌問題の条件と矛盾する。
よって混沌問題がPに含まれると仮定すると、混沌問題の条件と矛盾する。
従って、背理法より、検証問題がPの時、混沌問題はPに含まれないことがわかる。 ∎
ここで、検証問題がPの時、秩序問題はP完全、混沌問題を含む検定問題はNP完全であるから、下記を結論付けることができる。
###### Theorem 8.
$P\neq NP$
## 5 More detail about difference between Orderly problem and Chaotic problem
混沌問題の計算時間をより詳細に説明する。
秩序問題の限界を明らかにするために、下記の問題を定義する。
###### Definition 9.
秩序問題のうち、総称文字及び特定文字を組み合わせた情報で秩序問題の検定情報とならないものが無数に存在する問題を飽和秩序問題と呼ぶ。
###### Theorem 10.
飽和秩序問題は存在する。
###### 証明.
CNFSATを変形した検定問題を使用して証明する。
1と同様にして作成したCNFSATの検定問題を考える。論理式が特定文字であり、真理値割当が総称文字となる。ここで総称文字を特定文字に変更してCNFSATをHornSAT相当に変更することを考える。この変更は、CNFSATの論理式の各項に含まれる肯定リテラルが0または1になるように、真理値割当に偽値を割り当てることにより行うことができる。このようにして変更した検定情報はHornSAT相当であり、秩序問題の条件を満たす。
しかし、このようにして作成した秩序問題には必ず制約された真理値割当が存在し、全ての総称文字と特定文字の組み合わせにはならない。全ての総称文字と特定文字を組み合わせた検定情報にはCNFSATが必ず含まれており、HornSATの条件を満たさない。また、条件を満たさないCNFSATは無数に存在することを考えると、条件を満たさない検定情報も無数に存在する。
よって、このような秩序問題は飽和秩序問題の条件を満たす。 ∎
飽和秩序問題は秩序問題の限界を示す。ここで混沌問題との違いを明確化するために、下記を定義する。
###### Definition 11.
飽和秩序問題を検証問題とする検定問題を、飽和混沌問題と呼ぶ。
###### Theorem 12.
飽和混沌問題は存在する。
###### 証明.
飽和秩序問題を検証問題とする検定問題を考える。飽和秩序問題の総称文字は検定問題の特定文字となるが、飽和秩序問題の総称文字を検定問題の総称文字に置き換えたとしても計算結果は変化しない。
しかし、飽和秩序問題は、9より全ての総称文字と特定文字の組み合わせを網羅していない。よって、一部の文字が一致する全ての検定情報を集めた時に、異なる文字の部分が全て特定文字となり総称文字を含まない検定情報の集合が存在する。このような部分を検定問題の総称文字に置き換えた検定情報は飽和秩序問題の検定問題には含まれない。
よって、飽和秩序問題を検証問題とした検定問題には、飽和秩序問題の検定情報に含まれない検定情報が含まれる別の問題となる。 ∎
飽和混沌問題は、飽和秩序問題よりも複雑であり、総称文字の指数規模の計算時間が必要となる。
###### Theorem 13.
飽和秩序問題の検証問題がPの時、飽和混沌問題は拡張文字の指数規模の計算時間が必要となる。
###### 証明.
飽和秩序問題には現れず、飽和混沌問題にのみ現れる総称文字について考える。この総称文字を含む検定情報は飽和秩序問題には含まれない。よって、この総称文字をそのまま計算する秩序問題は存在しない。また、この総称文字を特定文字に変更した検定問題は、飽和秩序問題の検証問題がPであることよりP完全となる。総称文字の構成する拡張文字は文字幅の指数規模の種類が存在するため、飽和混沌問題の計算時間は拡張文字の指数規模となる。
∎
ここで、飽和秩序問題の検証問題をPとすると、飽和秩序問題はP完全に、飽和混沌問題を含む検定問題はNP完全となる。また9と13より、飽和混沌問題の拡張文字の文字長に上限はなく、また計算時間も指数規模となることを考慮すると、下記を結論付けることができる。
###### Theorem 14.
$P\neq NP$
|
arxiv-papers
| 2010-08-13T02:29:55 |
2024-09-04T02:49:12.168117
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Koji Kobayashi",
"submitter": "Koji Kobayashi",
"url": "https://arxiv.org/abs/1008.2247"
}
|
1008.2299
|
# Laser Dressed Scattering of an Attosecond Electron Wave Packet
Justin Gagnon justin.gagnon@mpq.mpg.de Max-Planck-Institut für Quantenoptik,
Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Ludwig-Maximilians-
Universität München, Am Coulombwall 1, D-85748 Garching, Germany Ferenc
Krausz Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748
Garching, Germany Ludwig-Maximilians-Universität München, Am Coulombwall 1,
D-85748 Garching, Germany Vladislav S. Yakovlev Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Ludwig-
Maximilians-Universität München, Am Coulombwall 1, D-85748 Garching, Germany
###### Abstract
We theoretically investigate the scattering of an attosecond electron wave
packet launched by an attosecond pulse under the influence of an infrared
laser field. As the electron scatters inside a spatially extended system, the
dressing laser field controls its motion. We show that this interaction, which
lasts just a few hundreds of attoseconds, clearly manifests itself in the
spectral interference pattern between different quantum pathways taken by the
outgoing electron. We find that the Coulomb-Volkov approximation, a standard
expression used to describe laser-dressed photoionization, cannot properly
describe this interference pattern. We introduce a quasi-classical model,
based on electron trajectories, which quantitatively explains the laser-
dressed photoelectron spectra, notably the laser-induced changes in the
spectral interference pattern.
photoionization,scattering,laser,attosecond
††preprint: APS/123-QED
When an electron scatters from an atom in the presence of a laser field,
exotic effects can be observed which are not accessible in ordinary electron-
atom scattering Mittleman (1982). Laser-assisted electron-atom collisions were
mostly studied using monochromatic electron and laser beams Ehlotzky _et al._
(1998). In such experiments, an atom and an electron take part in a collision
at some random time in a monochromatic laser field, which acts as a
perturbation to the scattering event. Recent progress in the generation of
few-cycle laser pulses Baltus̆ka _et al._ (2003) has enabled a new class of
experiments, where an electron wave packet is _coherently_ launched, e.g. by
strong-field ionization or by an attosecond light pulse, and steered in the
field of an intense laser pulse Corkum _et al._ (1989); Itatani _et al._
(2002); Kienberger _et al._ (2004). This remarkable degree of control over
electron motion, together with the attosecond timing precision between the
light field and the electron wave packet have enabled fundamental experimental
Drescher _et al._ (2002); Niikura (2002); Itatani _et al._ (2004); Baker
_et al._ (2006) and theoretical Lucchese _et al._ (1982); Lein _et al._
(2002); Yudin _et al._ (2007); Gräfe _et al._ (2008) developments in atomic,
molecular, and optical physics.
In this paper, we theoretically study how a laser field affects the scattering
of an attosecond electron wave packet as it travels inside a spatially
extended system. Experimentally, this can be realized by ionizing a localized
electronic state of a molecule with an attosecond extreme-ultraviolet (XUV)
pulse. As it exits the molecule, the photoelectron will be scattered by other
atoms in the molecule before heading to the detector. In particular, the
scattering of a photoelectron within a molecule has recently attracted a
significant amount attention on its own: in addition to the Cohen-Fano
oscillations Cohen and Fano (1966), ionization from a localized core orbital
of CO also produces a modulation in the momentum-resolved cross sections
Zimmermann _et al._ (2008) arising from the interference between trajectories
taken by the outgoing electron, referred to as _intra-molecular scattering_.
The interference pattern produced at the detector can be interpreted as a
holographic image Krasniqi _et al._ (2010), and can be used to retrieve the
molecular structure seen by the outgoing electron. In this paper, we show that
a near-infrared (NIR) laser wave form, temporally synchronized to the
collision event, can be used to control the paths taken by the outgoing
photoelectron, which can be observed by measuring the interference in the
photoelectron spectrum.
Figure 1: The ionic potential (solid line) dips at $q=0$ and
$q=-24\,\textrm{a.u.}$ The initial state (dotted line) is localized at $q=0$.
The reflected and direct trajectories are labeled with “R” and “D”.
Since we are interested in the most general features of laser-dressed
scattering, we consider a one-dimensional model system, akin to a nanometer-
scale Fabry-Pérot etalon for the free electron wave packet. Our system is
composed of two potential wells chosen such that the electron is initially
localized in one of them (Fig. 1). The initial (bound) state
$|\psi_{0}\rangle$ is the first excited state of the double-well system
(ionization potential $W\approx 12.17\,\textrm{eV}$). The Hamiltonian of the
electron interacting with the potentials and the electromagnetic radiation,
assuming the dipole approximation, is
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{p^{2}}{2}+V_{\mathrm{N}}(q)+\Big{(}F_{\mathrm{L}}(t)+F_{\mathrm{X}}(t)\Big{)}\,q,$
(1) $\displaystyle V_{\mathrm{N}}(q)$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{1}+Z_{2}}\left(-\frac{Z_{1}}{\sqrt{q^{2}+a^{2}}}-\frac{Z_{2}}{\sqrt{(q-q_{\mathrm{r}})^{2}+a^{2}}}\right)$
with $a\approx 0.2236\,\textrm{a.u.}$, $q_{\mathrm{r}}=-24\,\textrm{a.u.}$,
$Z_{1}=2\,\textrm{a.u.}$ and $Z_{2}=5\,\textrm{a.u.}$ $p$ is the electron’s
momentum, $V_{\mathrm{N}}(q)$ is the potential due to the nuclei, with nuclear
charges $Z_{1}$ and $Z_{2}$. $F_{\mathrm{L}}(t)$ and $F_{\mathrm{X}}(t)$
represent the electric fields of the NIR laser and attosecond XUV pulses,
respectively. They are given explicitly by
$\displaystyle F_{\mathrm{X}}(t)$ $\displaystyle=$ $\displaystyle
10^{-5}\int_{-\infty}^{\infty}G_{\kappa,\theta}(\omega-W)\cos{(\omega
t)}\,\mathrm{d}\omega,$ (2) $\displaystyle F_{\mathrm{L}}(t)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{F_{0}}{\omega_{\mathrm{L}}}\cos^{4}{\left(\frac{\pi
t}{2\tau_{\mathrm{L}}}\right)\sin{(\omega_{\mathrm{L}}t)}}\right),\textrm{
}|t|\leq\tau_{\mathrm{L}},$ (3)
while $F_{\mathrm{L}}=0$ for $|t|>\tau_{\mathrm{L}}$. The XUV spectrum is
$G_{\kappa,\theta}(\omega-W)$, the Gamma distribution with mean $\kappa\theta$
and variance $\kappa\theta^{2}$. The Gamma distribution was chosen to avoid
populating Rydberg states. The parameters $\kappa$ and $\theta$ produce an XUV
spectrum peaked at an energy
$\omega_{\mathrm{X}}=80\,\textrm{eV}+W=92.2\,\textrm{eV}$, with a FWHM
bandwidth $\delta\omega_{\mathrm{X}}\approx 32.4\,\textrm{eV}$, yielding a
$55.4\,\textrm{as}$ pulse. The attosecond XUV pulse temporally overlaps with
the center of the laser pulse, at the extremum of $F_{\mathrm{L}}(t)$.
$\tau_{\mathrm{L}}$ and $\omega_{\mathrm{L}}$ are chosen to produce a laser
pulse with a full-width at half-maximum (FWHM) duration of $3\,\textrm{fs}$
and a central wavelength of $800\,\textrm{nm}$; $F_{0}$ is the laser field’s
peak amplitude. The laser electric field $F_{\mathrm{L}}(t)$, which we refer
as the _control field_ , is therefore a cosine pulse. The laser field
amplitude $F_{0}$ is the variable parameter in our analysis. It influences the
trajectory taken by the electron inside the system.
In the absence of a laser field, $F_{0}=0$, the solution of the time-dependent
Schrödinger equation (TDSE) can be formally written for positive energies, and
up to a constant phase, as
$\displaystyle\langle p|\psi(t_{\mathrm{f}})\rangle$ $\displaystyle=$
$\displaystyle\mathrm{e}^{-\frac{\mathrm{i}}{2}p^{2}t_{\mathrm{f}}}\int_{-\infty}^{t_{\mathrm{f}}}F_{\mathrm{X}}(t)d(p)e^{\mathrm{i}\big{(}\frac{p^{2}}{2}+W\big{)}t}\mathrm{d}t,$
(4) assuming $\displaystyle
F_{\mathrm{X}}(t)\langle\phi_{p_{2}}|q|\phi_{p_{1}}\rangle\approx 0,$ (5)
and $W$ is the ionization energy. The dipole matrix elements
$d(p)=\langle\phi_{p}|q|\psi_{0}\rangle$ are evaluated between the initial
(bound) state $|\psi_{0}\rangle$ and positive-energy eigenstates
$\langle\phi_{p}|$ of the Hamiltonian $H_{0}=p^{2}/2+V_{\mathrm{N}}(q)$, with
asymptotic momenta $p$. The eigenstates $|\phi_{p}\rangle$ satisfy the
Lippmann-Schwinger equation with an advanced Green’s function,
$\langle
q|\phi_{p}\rangle=e^{\mathrm{i}pq}-\frac{2\mathrm{i}}{|p|}\int_{-\infty}^{\infty}e^{-\mathrm{i}|p(q-q^{\prime})|}V_{\mathrm{N}}(q^{\prime})\phi_{p}(q^{\prime})\mathrm{d}q^{\prime},$
(6)
which we solve by computing the Born series until convergence.
Now, for the system under present scrutiny (Fig. 1), laser-dressed scattering
is clearly more pronounced on the right-going wave packet. It manifests itself
as a modulation of the photoelectron spectrum for positive momenta,
corresponding to a characteristic length of $\approx 49.3\,\textrm{a.u.}$,
which is about twice the internuclear spacing. Quantum-mechanically, this
modulation is explained by the fact that the matrix elements $d(p)$ are
spectrally modulated for positive momenta. In the presence of a NIR laser
field, there is a standard amendment to (4), which can describe laser-dressed
single-photon ionization. It is the Coulomb-Volkov approximation (CVA)
Duchateau _et al._ (2002); Kornev and Zon (2002). The CVA is used to account
for the action of the laser field on the ejected electron. We use the version
of CVA that reads
$\displaystyle\langle p|\psi_{\mathrm{CVA}}(t_{\mathrm{f}})\rangle$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-\frac{\mathrm{i}}{2}p^{2}t_{\mathrm{f}}}\int_{-\infty}^{t_{\mathrm{f}}}F_{\mathrm{X}}(t)d(p+A(t))$
(7) $\displaystyle\times$
$\displaystyle\mathrm{e}^{\mathrm{i}\left(-\frac{1}{2}\int_{t}^{t_{\mathrm{f}}}\left(p+A(t^{\prime})\right)^{2}\mathrm{d}t^{\prime}+Wt\right)}\mathrm{d}t,$
where $A(t)$ is the vector potential of the NIR field. The Coulomb-Volkov
approximation relies on a couple of intuitive arguments. First, an electron
trajectory ending with a momentum $p$, e.g. at the detector, must have been
launched with an energy $\left(p+A(t)\right)^{2}/2$ at the moment of
ionization $t$. Therefore, $\langle\phi_{p}|$ should back-propagate to
$\langle\phi_{p+A(t^{\prime})}|$ at the moment of ionization, which is why the
matrix element $d(p+A(t))$ is used in (7). Second, the electron’s evolution
under the laser field is accounted for by the Volkov phase, which is the
quantum phase acquired by a free electron in an electromagnetic field.
Figure 2: Laser-dressed photoelectron spectra numerically evaluated (a) from
the temporal Schrödinger equation and (b) from the Coulomb-Volkov
approximation. The CVA does predict a fringe pattern, but fails to account for
its change under the laser field.
The CVA approximation (7) is known to adequately describe laser-dressed
photoionization of atoms Duchateau _et al._ (2002). However, it cannot
properly describe the laser-dressed photoionization if the system is too
large. In Figure 2, a series of laser-dressed photoelectron spectra computed
by numerically solving the TDSE (panel a) are compared to those obtained by
evaluating the CVA expression (7), for different values of the control field’s
intensity (panel b). Since the attosecond XUV pulse is centered at the peak of
the laser electric field, hardly any momentum shift is expected for the
outgoing electron. Nevertheless, the TDSE predicts a noticeable shift of the
interference pattern to larger momenta as a function of the strength $F_{0}$
of the control field. This effect is not accounted for by the CVA. Now, since
the CVA is a semi-classical modification of the quasi-exact expression (4),
this would appear to preclude an intuitive classical interpretation of laser-
dressed photoelectron scattering based on the simple trajectories shown in
Figure 1.
To describe laser-dressed photoelectron scattering, we present an intuitive
theoretical model based on trajectories Smirnova _et al._ (2008); Heller
(1981). As will be clear from the subsequent analysis, our model
quantitatively accounts for the effects of the laser field on the spectral
interference. In order to gain a deeper understanding of the dynamics of the
electron as it exits the system, we take a closer look at the final positive-
energy components of the electron’s wave function. It is useful to think of
the propagated state $|\psi(t_{\mathrm{f}})\rangle$ as being composed of a sum
of two two parts corresponding to two sets of trajectories taken by the
outgoing electron: those for which the electron heads directly to the
detector, and those which make it re-scatter off the adjacent nucleus before
going to the detector. For our subsequent analysis, we further simplify this
picture into a rather classical one by considering strictly two trajectories:
a _direct_ trajectory and a _reflected_ trajectory, which will be described
below.
We can separate these two trajectories from the total positive-energy wave
packet by applying a simple unitary transformation corresponding to the back-
propagation of a free particle:
$|w(t_{\mathrm{f}})\rangle=\exp\left(\frac{\mathrm{i}}{2}p^{2}t_{\mathrm{f}}\right)|\psi(t_{\mathrm{f}})\rangle.$
(8)
Figure 3: The amplitude (solid line) and phase (dashed line) of the wave
parcel shows two separate contributions
$|w^{(0)}_{\mathrm{R}}(t_{\mathrm{f}})\rangle$ and
$|w^{(0)}_{\mathrm{D}}(t_{\mathrm{f}})\rangle$. The wave parcel is evaluated
in the absence of the control field, as indicated by the superscript “$(0)$”.
We removed the central momentum from the wave parcel to better visualize the
phases of the direct and reflected components. Once projected into real space,
the wave parcels $w^{(0)}_{\mathrm{D}}(t_{\mathrm{f}},q)$ and
$w^{(0)}_{\mathrm{R}}(t_{\mathrm{f}},q)$ are located at the _apparent_
starting points of the electron trajectories.
The projection of $|w(t_{\mathrm{f}})\rangle$ in configuration space allows us
to define a quantity which we call the _wave parcel_ ,
$\displaystyle w(t_{\mathrm{f}},q)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\langle p|\psi(t_{\mathrm{f}})\rangle
e^{\frac{\mathrm{i}}{2}p^{2}t_{\mathrm{f}}}e^{\mathrm{i}pq}\mathrm{d}p;$ (9)
the states $\langle p|$ are just free-particle eigenstates. The field-free
wave parcel $w^{(0)}(t_{\mathrm{f}},q)$ is plotted in Fig. 3. It has been
evaluated with the control field turned off ($F_{0}=0$) as indicated by the
superscript “$(0)$”. The wave parcel clearly consists of two parts. It has a
large hump, denoted by $w^{(0)}_{\mathrm{D}}(t_{\mathrm{f}},q)$ in Fig. 3,
centered about the origin, and a smaller hump
$w^{(0)}_{\mathrm{R}}(t_{\mathrm{f}},q)$ centered at
$\sim-45.6\,\textrm{a.u.}$ As will be clear from the subsequent analysis, the
large and small humps correspond, respectively, to the direct and reflected
trajectories taken by the outgoing electron.
Since the wave parcel is obtained by propagating the final positive-energy
wave function backward in time as a free particle, the position of the wave
parcel represents the _apparent_ starting point of the electron from the
perspective of an observer measuring the electron at the final time
$t_{\mathrm{f}}$. This apparent starting position is analogous to a _delay_ of
the wave parcel. Undoing the unitary transformation to the individual wave
parcel components thus gives the direct and reflected wave packets at the end
of the propagation, i.e.
$|\psi_{\mathrm{R,D}}(t_{\mathrm{f}})\rangle=\exp\left(-\frac{\mathrm{i}}{2}p^{2}t_{\mathrm{f}}\right)|w_{\mathrm{R,D}}(t_{\mathrm{f}})\rangle$.
Here we also note an important property of the wave parcel. For
$t_{\mathrm{f}}$ sufficiently large, the positive-energy part of the wave
function essentially propagates as a free particle, rendering the associated
wave parcel time-independent,
$\lim_{t_{\mathrm{f}}\rightarrow\infty}\frac{\mathrm{d}}{\mathrm{d}t}|w(t_{\mathrm{f}})\rangle=0.$
(10)
Thus, we henceforth drop the time argument of the wave parcel because we
assume the electron is measured long after its interaction with the fields and
the ionic potential, so that its wave parcel $|w\rangle$ no longer depends on
the time of measurement.
Figure 4: In panel a, the reflection probability is plotted against the
control field intensity, while panel b shows the momentum spectra of the
reflected wave parcel, along with the classically-expected final momentum of
the reflected trajectory (dashed line).
For the system under consideration, the net momentum shift of the direct
trajectory is negligible since the ionization takes place at a zero-crossing
of the control field’s vector potential. The interesting physics occurs during
the re-scattering event, experienced by the smaller hump of the wave parcel
(the reflected wave parcel). Figure 4 shows the parameters of the reflected
wave parcel as a function of the intensity of the control field. Since
photoionization takes place under a positive laser field the rescattered
electron is initially accelerated toward the adjacent potential. Since the
rescattering probability decreases with larger incident momentum, the
probability of reflection naturally decreases with the control field strength,
as displayed in Fig. 4-a. Furthermore, the spectra of the reflected wave
parcel, shown in Fig. 4-b, are progressively shifted to larger momenta for
increasing strengths of the control field.
The dependence of the reflected momentum on the laser field is well explained
by classical mechanics. The dashed line plotted in Fig. 4-b represents final
momenta computed by classically propagating an electron along the reflected
trajectory. We conduct calculations of both the reflected (R) and direct (D)
trajectories by launching them at the center of the initial potential
($Z_{1}$), at $x_{\mathrm{R,D}}(0)=0$, with initial velocities
$v_{j}(0)=\pm\sqrt{\left(v^{(0)}_{j}(t_{\mathrm{f}})\right)^{2}+2\Big{(}V_{\mathrm{N}}\big{(}x^{(0)}_{j}(t_{\mathrm{f}})\big{)}-V_{\mathrm{N}}(0)\Big{)}},$
(11)
where the subscript $j$ stands for the direct (D) or reflected (R) trajectory,
and the final positions
$x^{(0)}_{j}(t_{\mathrm{f}})=\langle\psi^{(0)}_{j}(t_{\mathrm{f}})|q|\psi^{(0)}_{j}(t_{\mathrm{f}})\rangle$
and velocities
$v^{(0)}_{j}(t_{\mathrm{f}})=\langle\psi^{(0)}_{j}(t_{\mathrm{f}})|p|\psi^{(0)}_{j}(t_{\mathrm{f}})\rangle$
are extracted from the _field-free_ reflected wave packets
$|\psi^{(0)}_{j}(t_{\mathrm{f}})\rangle$. The $\pm$ sign in (11) indicates
that direct trajectories are launched to the right ($+$) while reflected
trajectories are launched to the left ($-$). The re-scattered electron thus
initially travels to the left toward the adjacent potential ($Z_{2}$). Once
inside the scattering potential, at $x_{\mathrm{R}}=-24\,\textrm{a.u.}$, the
electron abruptly reverses its direction as if it elastically bounces off a
wall, leading it back towards the detector. In the absence of the control
field, the reflected electron crosses the initial potential $Z_{1}$ $\sim
480\,\textrm{as}$ later. When the control field is turned on, this delay
changes by $\pm 20\,\textrm{as}$ for the field intensities considered herein.
We can also explain the change in the fringe pattern as a function of the
control field strength. Just as the CVA explains laser-dressed spectra by
amending the field-free expression (4) to account for the action of the laser
field, we explain the laser-dressed spectra by adjusting the field-free wave
parcel using our classically evaluated trajectories.
To explain the laser-dressed interference pattern, we need to consider both
the direct and reflected trajectories. We launch the direct trajectories of
the classical particle at the center of the initial potential ($Z_{1}$), at
position $x=0$ with initial velocity given by (11). The classical simulations
produce the analogues of the phases $\Delta
S_{\mathrm{R},\mathrm{D}}^{(\mathrm{L})}$, momenta
$p_{\mathrm{R},\mathrm{D}}^{(\mathrm{L})}$ and positions
$q_{\mathrm{R},\mathrm{D}}^{(\mathrm{L})}$ of the direct (D) and reflected (R)
wave parcels. These three quantities set, respectively, the position, the bias
and the spacing of the interference pattern in the photoelectron spectrum, and
are deduced from the classical trajectories according to the following
relations:
$\displaystyle q_{j}^{\mathrm{(L)}}$ $\displaystyle=$ $\displaystyle
x_{j}(t_{\mathrm{f}})-v_{j}(t_{\mathrm{f}})t_{\mathrm{f}},\textrm{
}p_{j}^{\mathrm{(L)}}=v_{j}(t_{\mathrm{f}}),\textrm{ }$ (12)
$\displaystyle\Delta S_{j}^{\mathrm{(L)}}$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{\mathrm{f}}}{L\big{(}x_{j}(t),v_{j}(t),t\big{)}\mathrm{d}t}-\frac{1}{2}v_{j}^{2}(t_{\mathrm{f}})t_{\mathrm{f}},$
(13)
and $L(x_{j}(t),v_{j}(t),t)$ denotes the Lagrangian evaluated along the
reflected or direct trajectory, parameterized by $x_{j}(t)$ and $v_{j}(t)$.
Again, the index $j\in\\{\mathrm{R},\mathrm{D}\\}$ refers to the direct (D) or
reflected (R) trajectory. Since the wave parcel is obtained by back-
propagating the wave packet as a free particle, the classical parameters
$\Delta S_{j}^{\mathrm{(L)}}$, $p_{j}^{\mathrm{(L)}}$, and
$q_{j}^{\mathrm{(L)}}$ also include the effects of free-particle back-
propagation.
Using the classical quantities, we explain the laser-dressed photoelectron
spectrum by modifying the field-free direct and reflected wave parcels,
$w_{\mathrm{D}}^{(0)}(q)$ and $w_{\mathrm{R}}^{(0)}(q)$ respectively. We
obtain the laser dressed wave parcels
$w_{\mathrm{R},\mathrm{D}}^{(\mathrm{L})}(q)$ according to the prescription
$\displaystyle w_{j}^{(\mathrm{L})}(q)$ $\displaystyle=$ $\displaystyle
w_{j}^{(0)}\left(q-q_{j}^{(\mathrm{L})}+q_{j}^{(0)}\right)$
$\displaystyle\times$
$\displaystyle\mathrm{e}^{\mathrm{i}\Big{(}\big{(}p_{j}^{\mathrm{(L)}}-p_{j}^{(0)}\big{)}\big{(}q-q_{j}^{\mathrm{(L)}}\big{)}+\Delta
S_{j}^{\mathrm{(L)}}-\Delta S_{j}^{(0)}\Big{)}},$
where $q_{j}^{(0)}$ and $p_{j}^{(0)}$ are respectively the positions and
momenta of the field-free wave parcels, given by (12). As indicated by this
transformation, the field-free wave parcel is first centered at position
$q_{j}^{\mathrm{(L)}}$, evaluated from the classical trajectory. Its momentum
and phase offset are then set in position space with the classically evaluated
parameters $p_{j}^{\mathrm{(L)}}$ and $\Delta S_{j}^{\mathrm{(L)}}$,
respectively. Thus, the transformation (Laser Dressed Scattering of an
Attosecond Electron Wave Packet) makes use of purely classical information to
account for the control field. This classical information is sufficient to
explain the effect of the control field on the fringes in the photoelectron
spectra, as shown in Fig. 5. Indeed, the spectra evaluated using our classical
model represent a marked improvement to those erroneously predicted by the CVA
(cf. Fig. 2).
Figure 5: The classically-adjusted laser-dressed photoelectron spectra (b),
based on the transformation (Laser Dressed Scattering of an Attosecond
Electron Wave Packet) reproduce the correct fringe patterns predicted by the
TDSE (a). It is the difference in the back-propagated action $\Delta S$,
between the reflected and direct trajectories, that sets the position of the
fringes. The classically-adjusted spectra shown in panel b above are to be
compared to those evaluated from the CVA (Fig. 2-b).
For a given laser field strength, the fringe patterns are reproduced over a
wide range of momenta, despite the fact that only a single initial momentum
was used for each classical trajectory. The classical simulations neglect two
purely quantum-mechanical effects: the influence of the control field on the
reflection probability and the phase acquired upon reflection (i.e. the
scattering phase shift for the backward direction). Consequently, the position
and contrast of the spectral fringes predicted from our model is slightly off
at larger control field strengths. These purely quantum mechanical effects
cannot be explained by the classical model.
In order to clearly illustrate the key physics and to make the relevant
effects more discernible, we considered a rather large system. Such a system
might be a dissociating diatomic molecule, a dimer, an excimer, or a nano-
structure composed of two spatially separated entities. Our analysis also
applies to smaller systems. A smaller system would result in a broader
spectral modulation, requiring a larger XUV bandwidth to capture enough
fringes; or equivalently stated, it would require a shorter attosecond pulse
so that the wave parcel is made up of two spatially distinct portions
$w_{\mathrm{D}}(q)$ and $w_{\mathrm{R}}(q)$. Our approach applies more
generally. For instance, in the case of a delocalized initial state, the
starting points of the classical trajectories should be located near the peaks
of the initial state, with all first-order re-scattering events considered for
each trajectory.
In conclusion, we have shown that an external NIR laser field controls the re-
scattering of an electron, which can be observed by measuring the
photoelectron spectrum for different NIR field intensities. The NIR field
mainly affects three parameters of the re-scattered wave packet: it changes
its momentum, its action and its apparent starting position, the latter of
which corresponds to a _delay_ when considered in the time domain. On the
other hand, for moderate intensities the control field hardly affects the
scattering phase shift of the re-scattered electron. Moreover, we found that
the probability of re-scattering is affected by the strength of the control
field. This might provide a means to generate and control a spatially and
temporally confined electric current on a single atom by launching a free
electron wave packet with an attosecond pulse in the presence of a controlled
NIR wave form.
As evidenced in our study, the semi-classical Coulomb-Volkov approximation
cannot describe these effects, and indeed breaks down for such a spatially
extended system. In order to uphold a physically intuitive picture of laser-
dressed scattering, we presented a new model based on two classical
trajectories that quantitatively explains the influence of the NIR control
field on the photoelectron interference pattern. Our model is generalizable to
larger systems, and thus constitutes a powerful tool for interpreting this new
kind of spectroscopic measurement, where a spatially extended system is
monitored or characterized using its own outgoing electron.
###### Acknowledgements.
The authors are grateful to E. Goulielmakis, A. Kamarou, M. Korbman and N.
Karpowicz for valuable discussions. This work was supported by the DFG Cluster
of Excellence: Munich-Centre for Advanced Photonics.
## References
* Mittleman (1982) M. H. Mittleman, _Introduction to the Theory of Laser-Atom Interactions_ (Plenum Pub Corp, New York, 1982).
* Ehlotzky _et al._ (1998) F. Ehlotzky _et al._ , Phys. Rep., 297, 64 (1998).
* Baltus̆ka _et al._ (2003) A. Baltus̆ka _et al._ , Nature, 421, 611 (2003).
* Corkum _et al._ (1989) P. B. Corkum _et al._ , Phys. Rev. Lett., 62, 1259 (1989).
* Itatani _et al._ (2002) J. Itatani _et al._ , Phys. Rev. Lett., 88, 173903 (2002).
* Kienberger _et al._ (2004) R. Kienberger _et al._ , Nature, 427, 817 (2004).
* Drescher _et al._ (2002) M. Drescher _et al._ , Nature, 419, 803 (2002).
* Niikura (2002) H. Niikura, Nature, 417, 917 (2002).
* Itatani _et al._ (2004) J. Itatani _et al._ , Nature, 432, 867 (2004).
* Baker _et al._ (2006) S. Baker _et al._ , Science, 312, 424 (2006).
* Lucchese _et al._ (1982) R. R. Lucchese _et al._ , Phys. Rev. A, 25, 2572 (1982).
* Lein _et al._ (2002) M. Lein _et al._ , Phys. Rev. Lett., 88 (2002).
* Yudin _et al._ (2007) G. L. Yudin _et al._ , J. Phys. B, 40, F93 (2007).
* Gräfe _et al._ (2008) S. Gräfe _et al._ , Phys. Rev. Lett., 101 (2008).
* Cohen and Fano (1966) H. D. Cohen and U. Fano, Phys. Rev., 150, 30 (1966).
* Zimmermann _et al._ (2008) B. Zimmermann _et al._ , Nature Physics, 4, 649 (2008).
* Krasniqi _et al._ (2010) F. Krasniqi _et al._ , Phys. Rev. A, 81, 033411 (2010).
* Duchateau _et al._ (2002) G. Duchateau _et al._ , Phys. Rev. A, 66, 023412 (2002).
* Kornev and Zon (2002) A. Kornev and B. Zon, J. Phys. B, 35, 2451 (2002).
* Smirnova _et al._ (2008) O. Smirnova _et al._ , Phys. Rev. A, 77, 033407 (2008).
* Heller (1981) E. J. Heller, J. Chem. Phys., 75, 2923 (1981).
|
arxiv-papers
| 2010-08-13T11:56:38 |
2024-09-04T02:49:12.176509
|
{
"license": "Public Domain",
"authors": "Justin Gagnon, Ferenc Krausz and Vladislav S. Yakovlev",
"submitter": "Justin Gagnon",
"url": "https://arxiv.org/abs/1008.2299"
}
|
1008.2374
|
# Semicrossed products of operator algebras and their
$\mathrm{C}^{*}$-envelopes
Evgenios Kakariadis Department of Mathematics
University of Athens
15784 Athens
GREECE mavro@math.uoa.gr and Elias G. Katsoulis Department of Mathematics
University of Athens
15784 Athens
GREECE Alternate address: Department of Mathematics
East Carolina University
Greenville, NC 27858
USA katsoulise@ecu.edu
###### Abstract.
Let ${\mathcal{A}}$ be a unital operator algebra and let $\alpha$ be an
automorphism of ${\mathcal{A}}$ that extends to a $*$-automorphism of its
$\mathrm{C}^{*}$-envelope $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$. In
this paper we introduce the isometric semicrossed product
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$ and we
show that
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+})\simeq\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}$.
In contrast, the $\mathrm{C}^{*}$-envelope of the familiar contractive
semicrossed product ${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ may not
equal $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}$.
Our main tool for calculating $\mathrm{C}^{*}$-envelopes for semicrossed
products is the concept of a relative semicrossed product of an operator
algebra, which we explore in the more general context of injective
endomorphisms.
As an application, we extend the main result of [9] to tensor algebras of
$\mathrm{C}^{*}$-correspondences. We show that if
${\mathcal{T}}_{{\mathcal{X}}}^{+}$ is the tensor algebra of a
$\mathrm{C}^{*}$-correspondence $({\mathcal{X}},{\mathfrak{A}})$ and $\alpha$
a completely isometric automorphism of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
that fixes the diagonal elementwise, then the contractive semicrossed product
satisfies
$\mathrm{C}^{*}_{\text{env}}({\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}{\mathbb{Z}}^{+})\simeq{\mathcal{O}}_{{\mathcal{X}}}\times_{\alpha}{\mathbb{Z}}$,
where ${\mathcal{O}}_{{\mathcal{X}}}$ denotes the Cuntz-Pimsner algebra of
$({\mathcal{X}},{\mathfrak{A}})$.
2000 Mathematics Subject Classification. 47L55, 47L40, 46L05, 37B20
Key words and phrases: semicrossed product, crossed product
Second author was partially supported by a grant from ECU
## 1\. Introduction and preliminaries
In this paper, we offer three choices for defining the semicrossed product of
an operator algebra ${\mathcal{A}}$ by a unital, completely contractive
endomorphism $\alpha$ of ${\mathcal{A}}$ (Definitions 1.1 and 1.2.) In all
cases, the resulting algebras contain a completely isometric copy of
${\mathcal{A}}$ and a ”universal” operator that implements the covariance
relations. In the case where ${\mathcal{A}}$ is a $\mathrm{C}^{*}$-algebra and
$\alpha$ preserves adjoints, all three choices produce the same operator
algebra, Peters’ semicrossed product of a $\mathrm{C}^{*}$-algebra [25] by an
endomorphism. (Semicrossed products of $\mathrm{C}^{*}$-algebras have been
under investigation by various authors [1, 2, 4, 6, 7, 8, 14, 22, 26],
starting with the work of Arveson [2] in the late sixties.) In the general
(non-selfadjoint) case however, the semicrossed products we introduce here may
lead to non-isomorphic operator algebras. The main objective of this paper is
to clarify the relation between the three semicrossed products and calculate
their $\mathrm{C}^{*}$-envelope, whenever possible.
The present paper is a continuation of the recent work of Davidson and the
second named author in [9]. In the language of the present paper, the main
objective of [9] was to show that in the special case where ${\mathcal{A}}$ is
Popescu’s non-commutative disc algebra [27] and $\alpha$ a completely
isometric automorphism of ${\mathcal{A}}$, all three semicrossed products
coincide. One of the main results of this paper, Theorem 2.6, shows that two
of these semicrossed products, the isometric and the relative one, coincide
for any operator algebra ${\mathcal{A}}$ and any completely isometric
automorphism $\alpha$ of ${\mathcal{A}}$. To prove this, we had to abandon the
rather intricate but ad-hoc arguments of the second half of [9] and instead
adopt an abstract approach. Theorem 2.6 focuses now any further research on
semicrossed products to the study of the other two, the isometric and the
contractive semicrossed product. For these two, there are examples to show
that they do not coincide in general (Remark 2.8). Nevertheless, with Theorem
2.6 in hand, we show that they do coincide in the case of a tensor algebra of
a $\mathrm{C}^{*}$-correspondence and and a completely isometric isomorphism
of the algebra that fixes its diagonal (Corollary 2.6). This not only
generalizes the main result of [9] to a broader context but also paves the way
for additional results of this kind to come in the future.
The various semicrossed products we define in this paper are actually closed
images of the following universal semicrossed product under concrete
representations.
###### Definition 1.1.
Let $\alpha$ be a unital, completely contractive endomorphism of an operator
algebra ${\mathcal{A}}$. A contractive (isometric) covariant representation
$(\pi,K)$ of $({\mathcal{A}},\alpha)$ is a completely contractive
representation $\pi$ of ${\mathcal{A}}$ on a Hilbert space ${\mathcal{H}}$ and
a contraction (resp. isometry) $K\in{\mathcal{B}}({\mathcal{H}})$ so that
$\pi(A)K=K\pi(\alpha(A))\quad\text{for all}\,A\in{\mathcal{A}}.$
The contractive (resp. isometric) semicrossed product
${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ (resp.
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$) for the
system $({\mathcal{A}},\alpha)$ is the universal operator algebra generated by
a copy of ${\mathcal{A}}$ and a contraction (resp. isometry) ${\mathfrak{V}}$
so that $A{\mathfrak{V}}={\mathfrak{V}}\alpha(A)$, for all
$A\in{\mathcal{A}}$.
The contractive semicrossed product has, by definition, a rich representation
theory which unfortunately makes it very intractable. This was first observed
in [9] based on the famous example of Varopoulos [28] regarding three
commuting contractions that do not satisfy the usual von Neumann inequality.
Nevertheless, there are significant cases where the contractive semicrossed
product has been completely identified. These include the case where
${\mathcal{A}}$ is a $\mathrm{C}^{*}$-algebra [17, 25] and the case where
${\mathcal{A}}$ is the non-commutative disc algebra ${\mathfrak{A}}_{n}$ and
$\alpha$ is an isometric automorphism [9].
The isometric semicrossed product is the (closed) image of
${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ under the representation which
restricts to the entries where the contractions $K$ are actually isometries.
We believe that this is a more tractable object and as we shall see, in the
case where $\alpha$ is a completely isometric automorphism, i.e., it extends
to an automorphism of the $\mathrm{C}^{*}$-envelope
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$ of ${\mathcal{A}}$, we can
identify the $\mathrm{C}^{*}$-envelope of
${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ as the crossed product
$\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}$. The
main tool for establishing this result is the concept of a relative
semicrossed product.
Recall that a $\mathrm{C}^{*}$ algebra ${\mathcal{C}}$ is said to be a
$\mathrm{C}^{*}$-cover for a subalgebra ${\mathcal{A}}\subseteq{\mathcal{C}}$
provided that ${\mathcal{A}}$ generates ${\mathcal{C}}$ as a
$\mathrm{C}^{*}$-algebra, i.e., ${\mathcal{C}}=C^{*}({\mathcal{A}})$. If
${\mathcal{C}}$ is a $\mathrm{C}^{*}$-cover for ${\mathcal{A}}$, then
${\mathcal{J}}_{{\mathcal{A}}}$ will denote the Šilov ideal of ${\mathcal{A}}$
in ${\mathcal{C}}$. Therefore,
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})={\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
and the restriction of the natural projection
$q:{\mathcal{C}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ on
${\mathcal{A}}$ is a completely isometric representation of ${\mathcal{A}}$.
(Any ideal ${\mathcal{J}}\subseteq{\mathcal{C}}$, with the property that the
restriction of the natural projection
${\mathcal{C}}\rightarrow{\mathcal{C}}/{\mathcal{J}}$ on ${\mathcal{A}}$ is a
complete isometry, is called a boundary ideal and
${\mathcal{J}}_{{\mathcal{A}}}$ is the largest such ideal.)
###### Definition 1.2.
Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let $\alpha$ be an
$*$-endomorphism of ${\mathcal{C}}$ that leaves ${\mathcal{A}}$ invariant. The
subalgebra of Peters’ semicrossed product
${\mathcal{C}}\times_{\alpha}{\mathbb{Z}}^{+}$, which is generated by
${\mathcal{A}}\subseteq{\mathcal{C}}\subseteq{\mathcal{C}}\times_{\alpha}{\mathbb{Z}}^{+}$
and the universal isometry
${\mathfrak{V}}\in{\mathcal{C}}\times_{\alpha}{\mathbb{Z}}^{+}$, is denoted by
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$ and is said to be
a relative semicrossed product for the system $({\mathcal{A}},\alpha)$.
Therefore, the relative semicrossed product
${\mathcal{A}}\times_{{\mathcal{C}}\,,\alpha}{\mathbb{Z}}^{+}$ comes from the
representation of ${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ that
restricts to the entries where $\pi$ and $\alpha$ are $*$-extendable to
${\mathcal{C}}$ and the contraction $K$ satisfies the covariance relation with
these extensions. It seems plausible that non-isomorphic
$\mathrm{C}^{*}$-covers for ${\mathcal{A}}$ and varying extensions for the
endomorphism $\alpha$ could produce non-isomorphic relative semicrossed
products. It turns out that under a reasonable technical requirement, i.e.,
invariance of the Shilov ideal, all such relative semicrossed products are
completely isometrically isomorphic (Proposition 2.3). In particular, this
requirement is satisfied when $\alpha$ is a completely isometric automorphism
of ${\mathcal{A}}$; in that case all relative semicrossed products for
$({\mathcal{A}},\alpha)$ are completely isometrically isomorphic to each
other.
## 2\. The relative semicrossed product and its $\mathrm{C}^{*}$-envelope
We begin this section with some preliminary results. The first one is a
standard result that shows how to lift an injective $*$-endomorphism of a
$\mathrm{C}^{*}$-algebra to an automorphism of a possibly larger
$\mathrm{C}^{*}$-algebra.
###### Proposition 2.1.
If $\alpha$ is an injective endomorphism of a C*-algebra ${\mathfrak{A}}$,
then there is a unique triple $({\mathfrak{B}},\beta,j)$ (up to isomorphism)
where ${\mathfrak{B}}$ is a C*-algebra, $\beta$ is an automorphism of
${\mathfrak{B}}$ and $j$ is a $*$-monomorphism of ${\mathfrak{A}}$ into
${\mathfrak{B}}$ such that $\beta j=j\alpha$ and
${\mathfrak{B}}=\overline{\bigcup_{k\geq 0}\beta^{-k}j({\mathfrak{A}})}$.
To paraphrase, there is a unique minimal C*-algebra ${\mathfrak{B}}$
containing ${\mathfrak{A}}$ with an automorphism $\beta$ satisfying
$\beta|_{\mathfrak{A}}=\alpha$.
###### Proof..
Consider the inductive limit ${\mathfrak{B}}$ of the system
$\begin{CD}{\mathfrak{A}}_{1}@>{\alpha_{1}}>{}>{\mathfrak{A}}_{2}@>{\alpha_{2}}>{}>{\mathfrak{A}}_{3}@>{\alpha_{3}}>{}>{\mathfrak{A}}_{4}@>{\alpha_{4}}>{}>\cdots,\end{CD}$
where ${\mathfrak{A}}_{i}={\mathfrak{A}}$ and $\alpha_{i}=\alpha$, for all
$i\in{\mathbb{N}}$. Let $j_{i}$ be the associated $*$-monomorphism from
${\mathfrak{A}}={\mathfrak{A}}_{i}$ to ${\mathfrak{B}}$. This map is defined
as $j_{i}(A)=(0,0,\dots,0,A,\alpha(A),\alpha^{2}(A),\dots)$, with the
understanding that the infinite tuple in the definition signifies the
appropriate equivalence class. Define $j=j_{1}$.
The system
$\begin{CD}{\mathfrak{A}}_{1}@>{\alpha_{1}}>{}>{\mathfrak{A}}_{2}@>{\alpha_{2}}>{}>{\mathfrak{A}}_{3}@>{\alpha_{3}}>{}>\cdots\\\
@V{\alpha}V{}V@V{\alpha}V{}V@V{\alpha}V{}V\\\
{\mathfrak{A}}_{1}@>{\alpha_{1}}>{}>{\mathfrak{A}}_{2}@>{\alpha_{2}}>{}>{\mathfrak{A}}_{3}@>{\alpha_{3}}>{}>\cdots\end{CD}$
gives rise to an $*$-automorphism $\beta$ of ${\mathfrak{B}}$ defined as
$\beta(A_{1},A_{2},A_{3},\dots)=(\alpha(A_{1}),\alpha(A_{2}),\alpha(A_{3}),\dots),\quad
A_{i}\in{\mathfrak{A}}_{i},i\in{\mathbb{N}}.$
Clearly, $\beta j=j\alpha$. The inverse of $\beta$ on $\bigcup_{k\geq
1}j_{k}({\mathfrak{A}}_{k})$ satisfies
$\beta^{-1}(A_{1},A_{2},A_{3},\dots)=(0,\alpha(A_{1}),\alpha(A_{2}),\alpha(A_{3}),\dots),\quad
A_{i}\in{\mathfrak{A}}_{i},i\in{\mathbb{N}}.$
and so, if $A\in{\mathfrak{A}}$, then
$\beta^{-k}(A,0,0,\dots)=(0,0,\dots,0,A,\alpha(A),\alpha^{2}(A),\dots).$
Therefore, $\bigcup_{k\geq 0}\beta^{-k}j({\mathfrak{A}})$ is dense in
${\mathfrak{B}}$. ∎
We now fix some notation and use the previous result to construct a useful
embedding of ${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.
Let ${\mathcal{A}}$ be an operator algebra and let $\alpha$ be a completely
contractive endomorphism of ${\mathcal{A}}$. If $\pi$ is a completely
contractive representation of ${\mathcal{A}}$ on a Hilbert space
${\mathcal{H}}$, we define
$\widetilde{\pi}:{\mathcal{A}}\longrightarrow{\mathcal{B}}({\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}^{+}))$
so that
(1)
$\widetilde{\pi}(A)\equiv(\pi(A),\pi(\alpha(A)),\pi(\alpha^{2}(A)),\dots),\quad
A\in{\mathcal{A}}.$
Let $V_{{\mathcal{H}}}\equiv I\otimes V$, where $V$ denotes the unilateral
shift on $\ell^{2}({\mathbb{Z}}^{+})$. The pair
$(\widetilde{\pi},V_{{\mathcal{H}}})$ forms a contractive covariant
representation of $({\mathcal{A}},\alpha)$ and the associated representation
of ${\mathcal{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ is denoted as
$\widetilde{\pi}\times V_{{\mathcal{H}}}$. If $\alpha$ happens to be a
completely isometric automorphism of ${\mathcal{A}}$, we also have the
representation
$\widehat{\pi}:{\mathcal{A}}\rightarrow{\mathcal{B}}(H\otimes\ell^{2}({\mathbb{Z}}))$,
such that
$\widehat{\pi}(A)\equiv(\dots,\pi(\alpha^{-1}(A)),\pi(A),\pi(\alpha(A)),\pi(\alpha^{2}(A)),\dots),\quad
A\in{\mathcal{A}},$
the unitary $U_{{\mathcal{H}}}=I\otimes U$, where $U$ is the bilateral shift
on $\ell^{2}({\mathbb{Z}})$ and the associated representation
$\widehat{\pi}\times U_{{\mathcal{H}}}$ of
${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.
###### Proposition 2.2.
Let $\alpha$ is an injective endomorphism of a C*-algebra ${\mathfrak{A}}$ and
let $({\mathfrak{B}},\beta,j)$ be the triple of Proposition 2.1. Then
${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$ embeds completely
isometrically in ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. Furthermore,
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$ becomes a $\mathrm{C}^{*}$-cover
for ${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.
###### Proof..
Let $\pi$ be a faithful representation of ${\mathfrak{A}}$ on a Hilbert space
${\mathcal{H}}$. Since every representation of ${\mathfrak{A}}$ is a direct
sum of cyclic representations, the GNS construction implies that there exists
a representation $\pi_{\beta}$ of ${\mathfrak{B}}$ on a Hilbert space
${\mathcal{H}}_{\beta}\supseteq{\mathcal{H}}$ so that ${\mathcal{H}}$ is
reducing for $\pi_{\beta}(j({\mathfrak{A}}))$ and
$\pi_{\beta}(j(A))\mid_{{\mathcal{H}}}=\pi(A)$ for all $A\in{\mathfrak{A}}$.
By gauge invariance, $\widetilde{\pi}\times V_{{\mathcal{H}}}$ is a completely
isometric representation for ${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$;
therefore the same is true for the representation
$\widehat{\pi_{\beta}}\,j\times U_{{\mathcal{H}}_{\beta}}$. Now notice that
the representation $\widehat{\pi_{\beta}}$ is faithful on $\bigcup_{k\geq
0}\beta^{-k}j({\mathfrak{A}})$ and so, by inductivity, on all of
${\mathfrak{B}}$. By gauge invariance, the representation
$\widehat{\pi_{\beta}}\times U_{{\mathcal{H}}_{\beta}}$ is also faithful on
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. The proposition now follows by
comparing the ranges of $\widehat{\pi_{\beta}}\,j\times
U_{{\mathcal{H}}_{\beta}}$ and $\widehat{\pi_{\beta}}\times
U_{{\mathcal{H}}_{\beta}}$. ∎
As we shall see in Theorem 2.5, ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$ is
actually the $\mathrm{C}^{*}$-envelope of
${\mathfrak{A}}\times_{\alpha}{\mathbb{Z}}^{+}$.
Let ${\mathcal{A}}$ be an operator algebra and let ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$. Let $\alpha$ be a $*$-endomorphism
of ${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$ and
${\mathcal{J}}_{{\mathcal{A}}}$ and let
$\dot{\alpha}:{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
be defined as
$\dot{\alpha}(X+{\mathcal{J}}_{{\mathcal{A}}})=\alpha(X)+{\mathcal{J}}_{{\mathcal{A}}}$,
$X\in{\mathcal{C}}$. In this context, there are two relative semicrossed
products to be considered,
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$ and
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
The following proposition, which clarifies the relation between these two
semicrossed product, is an application of two significant recent results in
the theory of maximal dilations for completely contractive maps. First,
Dritschel and McCullough [11] have recently proven that every completely
contractive map $\varphi:{\mathcal{A}}\rightarrow B({\mathcal{H}})$ admits a
maximal dilation $(\Phi,{\mathcal{K}})$, i.e., a dilation
$\Phi:{\mathcal{A}}\rightarrow B({\mathcal{K}})$ so that any further dilation
of $\Phi$ has $\Phi$ as a direct summand. Furthermore, Muhly and Solel [24]
have shown that any such maximal dilation $\Phi$ extends (uniquely) to a
$*$-representation of any $\mathrm{C}^{*}$-cover of ${\mathcal{A}}$.
###### Proposition 2.3.
Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let $\alpha$ be a
$*$-endomorphism of ${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$
and ${\mathcal{J}}_{{\mathcal{A}}}$. Then the relative semicrossed products
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$ and
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$
are completely isometrically isomorphic.
###### Proof..
Let
$F=\sum_{n=0}^{k}\,{\mathfrak{V}}^{n}A_{n}\in{\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$
and
$F^{\prime}=\sum_{n=0}^{k}\,{\mathfrak{V}}^{n}(A_{n}+{\mathcal{J}}_{{\mathcal{A}}})\in{\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
We have to show that the homomorphism $F\mapsto F^{\prime}$ is a completely
isometric map.
Let $\pi$ be a faithful representation of ${\mathcal{C}}$ on a Hilbert space
${\mathcal{H}}$ and let $(\widetilde{\pi},V_{{\mathcal{H}}})$ be as in the
beginning of the section (see (1)). Consider the completely isometric map
$\varphi:{\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\longrightarrow{\mathcal{B}}({\mathcal{H}}):A+{\mathcal{J}}_{{\mathcal{A}}}\longmapsto\pi(A),\quad
A\in{\mathcal{A}}.$
According to our earlier discussion, there is a maximal dilation
$(\Phi,{\mathcal{K}})$ of $\varphi$ which extends uniquely to a representation
of ${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ such that
$P_{{\mathcal{H}}}\Phi(A+{\mathcal{J}}_{{\mathcal{A}}})|_{{\mathcal{H}}}=\varphi(A+{\mathcal{J}}_{{\mathcal{A}}})=\pi(A),$
for all $A\in A$. Since
$P_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}=P_{{\mathcal{H}}}\otimes
I$, we have that
$P_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}\widetilde{\Phi}(A+{\mathcal{J}}_{{\mathcal{A}}}))|_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}=\widetilde{\pi}(A+{\mathcal{J}}_{{\mathcal{A}}}),$
for all $A\in A$. Also,
$V_{{\mathcal{K}}}|_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}=V_{{\mathcal{H}}}$
and so
$\displaystyle\|F\|$
$\displaystyle=\|\sum_{n=0}^{k}\,V^{n}_{{\mathcal{H}}}\widetilde{\pi}(A_{n})\|$
$\displaystyle=\|P_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}\left(\sum_{n=0}^{k}\,V^{n}_{{\mathcal{K}}}\widetilde{\Phi}(A_{n}+{\mathcal{J}}_{{\mathcal{A}}})\right)|_{{\mathcal{H}}\otimes\ell^{2}({\mathbb{Z}}_{+})}\|$
$\displaystyle\leq\|\sum_{n=0}^{k}\,V^{n}_{{\mathcal{K}}}\widetilde{\Phi}(A_{n}+{\mathcal{J}}_{{\mathcal{A}}})\|\leq\|F^{\prime}\|.$
The same is also true for all the matrix norms and so the map
$F^{\prime}\mapsto F$ is well defined and completely contractive. By reversing
the roles of ${\mathcal{A}}$ and ${\mathcal{A}}/{\mathcal{J}}({\mathcal{A}})$
in the previous arguments, we can also prove that $F\mapsto F^{\prime}$ is
completely contractive, and the conclusion follows. ∎
Now we wish to identify the $C^{*}$-envelope of
${\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$. From the
previous result we know that it coincides with the $C^{*}$-envelope of
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
In the following we consider the case where
$\alpha:{\mathcal{C}}\rightarrow{\mathcal{C}}$ is injective. This is easily
seen to imply that
$\dot{\alpha}:{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
is an injective $*$-homomorphism. Indeed,
###### Lemma 2.4.
Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let $\alpha$ be an injective
$*$-endomorphism of ${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$
and ${\mathcal{J}}_{{\mathcal{A}}}$. Then
$\dot{\alpha}:{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\rightarrow{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$
is an injective $*$-homomorphism.
###### Proof..
In that case, $\alpha$ is a completely isometric map. Therefore,
$\left\|q(A)+\ker\dot{\alpha}\right\|=\left\|\dot{\alpha}(A+{\mathcal{J}}_{{\mathcal{A}}})\right\|=\left\|\alpha(A)+{\mathcal{J}}_{{\mathcal{A}}}\right\|=\left\|\alpha(A)\right\|=\left\|A\right\|,$
since ${\mathcal{J}}_{{\mathcal{A}}}$ is a boundary ideal and
$\alpha({\mathcal{A}})\subseteq{\mathcal{A}}$. The same argument holds for all
the matrix norms. Thus $\ker\dot{\alpha}$ is a boundary ideal of
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$. However,
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ is the $\mathrm{C}^{*}$-envelope
of ${\mathcal{A}}$ and so it contains no non-trivial boundary ideals for
${\mathcal{A}}$. Thus $\ker\dot{\alpha}=(0)$. ∎
The following is the main technical result of the section.
###### Theorem 2.5.
Let ${\mathcal{A}}$ be an operator algebra, ${\mathcal{C}}$ be a
$\mathrm{C}^{*}$-cover of ${\mathcal{A}}$ and let
${\mathcal{J}}_{{\mathcal{A}}}$ be the Šilov ideal of ${\mathcal{A}}$ in
${\mathcal{C}}$. Let $\alpha$ be an injective $*$-endomorphism of
${\mathcal{C}}$ that leaves invariant both ${\mathcal{A}}$ and
${\mathcal{J}}_{{\mathcal{A}}}$. Then
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+})\simeq{\mathfrak{B}}\times_{\beta}{\mathbb{Z}}\,,$
where $({\mathfrak{B}},\beta,j)$ is the unique triple of Proposition 2.1
associated with the injective $*$-endomorphism $\dot{\alpha}$ of
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$.
###### Proof..
Proposition 2.3 shows that it suffices to identify the $C^{*}$-envelope of
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
If $({\mathfrak{B}},\beta,j)$ is the unique triple of Proposition 2.1
associated with the injective $*$-endomorphism $\dot{\alpha}$ of
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$ then Proposition 2.2 shows that
${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{\dot{\alpha}}{\mathbb{Z}}^{+}$,
and therefore
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$,
embeds completely isometrically in ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$.
Moreover, ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$ is a
$\mathrm{C}^{*}$-cover for
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$.
Let ${\mathcal{J}}$ be the Šilov ideal of
${\mathcal{A}}/{\mathcal{J}}_{{\mathcal{A}}}\times_{{\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}},\dot{\alpha}}{\mathbb{Z}}^{+}$
in ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. We are to show that
${\mathcal{J}}=\\{0\\}$.
Assume to the contrary that ${\mathcal{J}}\neq\\{0\\}$. Since ${\mathcal{J}}$
is invariant by automorphisms of the $\mathrm{C}^{*}$-cover, it remains
invariant by the natural gauge action on
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. Therefore it has non-trivial
intersection with the fixed point algebra of the natural gauge action, i.e.,
${\mathcal{J}}\cap{\mathfrak{B}}\neq\\{0\\}$. However
${\mathfrak{B}}=\overline{\bigcup_{k\geq
0}\beta^{-k}j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})}$
and therefore by inductivity there exists $k\in{\mathbb{N}}$ so that
${\mathcal{J}}\cap\beta^{-k}j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})\neq\\{0\\}.$
However, $\beta$ acts by conjugating with a unitary in
${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$. Since ${\mathcal{J}}$ is an ideal
of ${\mathfrak{B}}\times_{\beta}{\mathbb{Z}}$, the above implies that
${\mathcal{J}}\cap j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})\neq\\{0\\}.$
But then $j^{-1}\left({\mathcal{J}}\cap
j({\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}})\right)$ is a non-zero boundary
ideal for ${\mathcal{A}}$ in ${\mathcal{C}}/{\mathcal{J}}_{{\mathcal{A}}}$, a
contradiction. ∎
In [9] Davidson and the second named author proved that
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}_{n}\times_{\alpha}{\mathbb{Z}}^{+})={\mathcal{O}}_{n}\times_{\alpha}{\mathbb{Z}},$
where ${\mathcal{A}}_{n}$ is Popescu’s non-commutative disc algebra [27],
$\alpha$ a is a (completely) isometric automorphism of ${\mathcal{A}}_{n}$ and
${\mathcal{O}}_{n}$ denotes the Cuntz algebra generated by $n$ isometries. A
dilation result in the first half of [9] reduces the problem of calculating
the $\mathrm{C}^{*}$-envelope of
${\mathcal{A}}_{n}\times_{\alpha}{\mathbb{Z}}^{+}$ to essentially verifying
that
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}_{n}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+})={\mathcal{O}}_{n}\times_{\alpha}{\mathbb{Z}}$.
It takes the second half of [9] and intricate use of the representation theory
for ${\mathcal{O}}_{n}$ to verify that claim. The next result establishes the
same claim for arbitrary operator algebras using only abstract arguments.
###### Theorem 2.6.
Let ${\mathcal{A}}$ be an operator algebra and $\alpha$ be an automorphism of
${\mathcal{A}}$ that extends to a $*$-automorphism of
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$. Then, any relative semicrossed
product for $({\mathcal{A}},\alpha)$ is completely isometrically isomorphic to
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$. Hence,
$\mathrm{C}^{*}_{\text{env}}({\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+})\simeq\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})\times_{\alpha}{\mathbb{Z}}.$
###### Proof..
In light of Theorem 2.5, it suffices to show that
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$ dilates
to a relative semicrossed product. This is done as follows.
Let
${\mathfrak{V}}\in{\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$
be the universal isometry acting on a Hilbert space ${\mathfrak{H}}$ and let
${\mathcal{H}}$ be the direct limit Hilbert space of the inductive system
$\begin{CD}{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>\cdots.\end{CD}$
For each $A\in{\mathcal{A}}$, the commutative diagram
$\begin{CD}{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>\cdots\\\
@V{A}V{}V@V{\alpha^{-1}(A)}V{}V@V{\alpha^{-2}(A)}V{}V\\\
{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>{\mathfrak{H}}@>{{\mathfrak{V}}}>{}>\cdots\end{CD}$
defines an operator $\pi(A)\in{\mathcal{B}}({\mathcal{H}})$. It is easily seen
that $\pi$ defines a completely isometric representation of ${\mathcal{A}}$ on
${\mathcal{H}}$. Consider now the unitary $U\in{\mathcal{B}}({\mathcal{H}})$
defined as
$U(h_{1},h_{2},h_{3},\dots)=(h_{2},h_{3},\dots),\quad
h_{i}\in{\mathfrak{H}},i\in{\mathbb{N}}.$
and notice that $\pi(\alpha(A))=U^{*}\pi(A)U$, $A\in{\mathcal{A}}$. Therefore,
the conjugation by $U$ defines a $*$-automorphism of
${\mathcal{C}}\equiv\mathrm{C}^{*}(\pi({\mathcal{A}}))$, which extends
$\alpha$ and is denoted by the same symbol as well. Therefore,
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}\simeq{\mathcal{A}}\times_{{\mathcal{C}},\alpha}{\mathbb{Z}}^{+}$
and the conclusion follows from Theorem 2.5. ∎
###### Remark 2.7.
In light of Theorem 2.6, we wonder whether one can compute the
$\mathrm{C}^{*}$-envelope of
${\mathcal{A}}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$ in the
case where $\alpha$ is an endomorphism of ${\mathcal{A}}$ that extends to an
injective $*$-endomorphism of $\mathrm{C}^{*}_{\text{env}}({\mathcal{A}})$. To
do this, one will have to prove an analogue of Theorem 2.5 in the case where
$\alpha$ may not preserve the Šilov ideal ${\mathcal{J}}_{{\mathcal{A}}}$ of
${\mathcal{A}}$ in ${\mathcal{C}}$.
###### Remark 2.8.
An observation from [9] shows that Theorem 2.6 fails for the contractive
semicrossed product, thus showing that the isometric and the contractive
semicrossed product are not completely isometrically isomorphic in general.
Indeed, the bidisk algebra $A({\mathbb{D}}^{2})$ sits inside
${\mathrm{C}}({\mathbb{T}}^{2})$, which is its C*-envelope by Ando’s theorem.
Consider the identity automorphism ${\operatorname{id}}$. Ando’s theorem also
shows that the completely contractive representations of $A({\mathbb{D}}^{2})$
are determined by an arbitrary pair $T_{1},T_{2}$ of commuting contractions. A
covariant representation of $(A({\mathbb{D}}^{2}),{\operatorname{id}})$ is
given by such a pair and a third contraction $T_{3}$ which commutes with
$T_{1}$ and $T_{2}$. If it were true that the C*-envelope of this system was
${\mathrm{C}}({\mathbb{T}}^{2})\times_{\operatorname{id}}{\mathbb{Z}}\simeq{\mathrm{C}}({\mathbb{T}}^{3})$,
then it would be true that every commuting triple of contractions satisfies
the usual von Neumann inequality. This has been disproved by Varopoulos [28].
## 3\. An application to tensor algebras
In spite of Remark 2.8, there are special cases where the contractive and
isometric semicrossed products coincide. The purpose of this section is to
verify this in the case where ${\mathcal{A}}$ is the tensor algebra of a
$\mathrm{C}^{*}$-correspondence and $\alpha$ a completely isometric
isomorphism of ${\mathcal{A}}$ that fixes its diagonal elementwise (Corollary
2.6).
The tensor algebras for $\mathrm{C}^{*}$-correspondences were introduced by
Muhly and Solel in [23]. This is a broad class of non-selfadjoint operator
algebras which includes as special cases Peters’ semicrossed products [26],
Popescu’s non-commutative disc algebras [27], the tensor algebras of graphs
(introduced in [23] and further studied in [19, 20]) and the tensor algebras
for multivariable dynamics [10], to mention but a few.
Let ${\mathfrak{A}}$ be a $\mathrm{C}^{*}$-algebra and ${\mathcal{X}}$ be a
(right) Hilbert ${\mathfrak{A}}$-module, whose inner product is denoted as
$\langle\,.\mid.\,\rangle$. Let ${\mathcal{L}}({\mathcal{X}})$ be the
adjointable operators on ${\mathcal{X}}$ and let
${\mathcal{K}}({\mathcal{X}})$ be the norm closed subalgebra of
${\mathcal{L}}({\mathcal{X}})$ generated by the operators $\theta_{\xi,\eta}$,
$\xi,\eta\in{\mathcal{X}}$, where
$\theta_{\xi,\eta}(\zeta)=\xi\langle\eta|\zeta\rangle$,
$\zeta\in{\mathcal{X}}$.
A Hilbert ${\mathfrak{A}}$-module ${\mathcal{X}}$ is said to be a
$\mathrm{C}^{*}$-correspondence over ${\mathfrak{A}}$ provided that there
exists a $*$-homomorphism
$\varphi_{{\mathcal{X}}}:{\mathfrak{A}}\rightarrow{\mathcal{L}}({\mathcal{X}})$.
We refer to $\varphi_{{\mathcal{X}}}$ as the left action of ${\mathfrak{A}}$
on ${\mathcal{X}}$. From a given $\mathrm{C}^{*}$-correspondence
${\mathcal{X}}$ over ${\mathfrak{A}}$, one can form new
$\mathrm{C}^{*}$-correspondences over ${\mathfrak{A}}$, such as the $n$-fold
ampliation or direct sum ${\mathcal{X}}^{(n)}$ ([21, page 5]) and the $n$-fold
interior tensor product ${\mathcal{X}}^{\otimes
n}\equiv{\mathcal{X}}\otimes_{\varphi_{{\mathcal{X}}}}{\mathcal{X}}\otimes_{\varphi_{{\mathcal{X}}}}\dots\otimes_{\varphi_{{\mathcal{X}}}}{\mathcal{X}}$
([21, page 39], $n\in{\mathbb{N}}$, (${\mathcal{X}}^{\otimes
0}\equiv{\mathfrak{A}}$). These operation are defined within the category of
$\mathrm{C}^{*}$-correspondences over ${\mathfrak{A}}$. (See [21] for more
details.)
A representation $(\pi,t)$ of a $\mathrm{C}^{*}$-correspondence
${\mathcal{X}}$ over ${\mathfrak{A}}$ on a $\mathrm{C}^{*}$-algebra
${\mathcal{B}}$ consists of a $*$-homomorphism
$\pi:{\mathfrak{A}}\rightarrow{\mathcal{B}}$ and a linear map
$t:{\mathcal{X}}\rightarrow{\mathcal{B}}$ so that
* (i)
$t(\xi)^{*}t(\eta)=\pi(\langle\xi|\,\eta\rangle)$, for
$\xi,\eta\in{\mathcal{X}}$,
* (ii)
$\pi(A)t(\xi)=t(\varphi_{{\mathcal{X}}}(A)\xi)$, for $A\in{\mathfrak{A}}$,
$\xi\in{\mathcal{X}}$.
For a representation $(\pi,t)$ of a $\mathrm{C}^{*}$-correspondence
${\mathcal{X}}$ there exists a $*$-homomorphism
$\psi_{t}:{\mathcal{K}}({\mathcal{X}})\rightarrow{\mathcal{B}}$ so that
$\psi_{t}(\theta_{\xi,\eta})=t(\xi)t(\eta)^{*}$, for
$\xi,\eta\in{\mathcal{X}}$. Following Katsura [18], we say that the
representation $(\pi,t)$ is covariant iff
$\psi_{t}(\varphi_{{\mathcal{X}}}(A))=\pi(A)$, for all
$A\in{\mathcal{J}}_{{\mathcal{X}}}$, where
${\mathcal{J}}_{{\mathcal{X}}}\equiv\varphi_{{\mathcal{X}}}^{-1}({\mathcal{K}}({\mathcal{X}}))\cap(\ker\varphi_{{\mathcal{X}}})^{\perp}.$
If $(\pi,t)$ is a representation of ${\mathcal{X}}$ then the
$\mathrm{C}^{*}$-algebra (resp. norm closed algebra) generated by the images
of $\pi$ and $t$ is denoted as $\mathrm{C}^{*}(\pi,t)$ (resp.
$\operatorname{Alg}((\pi,t)$). There is a universal representation
$(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$ for
${\mathcal{X}}$ and the $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$
is the Toeplitz-Cuntz-Pimsner algebra ${\mathcal{T}}_{{\mathcal{X}}}$.
Similarly, the Cuntz-Pimsner algebra ${\mathcal{O}}_{{\mathcal{X}}}$ is the
$\mathrm{C}^{*}$-algebra generated by the image of the universal covariant
representation $(\pi_{{\mathfrak{A}}},t_{{\mathcal{X}}})$ for ${\mathcal{X}}$.
A concrete presentation of both ${\mathcal{T}}_{{\mathcal{X}}}$ and
${\mathcal{O}}_{{\mathcal{X}}}$ can be given in terms of the generalized Fock
space ${\mathcal{F}}_{{\mathcal{X}}}$ which we now describe. The Fock space
${\mathcal{F}}_{{\mathcal{X}}}$ over the correspondence ${\mathcal{X}}$ is
defined to be the direct sum of the ${\mathcal{X}}^{\otimes n}$ with the
structure of a direct sum of $\mathrm{C}^{*}$-correspondences over
${\mathfrak{A}}$,
${\mathcal{F}}_{{\mathcal{X}}}={\mathfrak{A}}\oplus{\mathcal{X}}\oplus{\mathcal{X}}^{\otimes
2}\oplus\dots.$
Given $\xi\in{\mathcal{X}}$, the (left) creation operator
$t_{\infty}(\xi)\in{\mathcal{L}}({\mathcal{F}}_{{\mathcal{X}}})$ is defined by
the formula
$t_{\infty}(\xi)(A,\zeta_{1},\zeta_{2},\dots)=(0,\xi
A,\xi\otimes\zeta_{1},\xi\otimes\zeta_{2},\dots),$
where $\zeta_{n}\in{\mathcal{X}}^{\otimes n}$, $n\in{\mathbb{N}}$, and
$A\in{\mathfrak{A}}$. Also, for $A\in{\mathfrak{A}}$, we define
$\pi_{\infty}(A)\in{\mathcal{L}}({\mathcal{F}}_{{\mathcal{X}}})$ to be the
diagonal operator with $\varphi_{{\mathcal{X}}}(A)\otimes id_{n-1}$ at its
${\mathcal{X}}^{\otimes n}$-th entry. It is easy to verify that
$(\pi_{\infty},t_{\infty})$ is a representation of ${\mathcal{X}}$ which is
called the Fock representation of ${\mathcal{X}}$. Fowler and Raeburn [13]
(resp. Katsura [18]) have shown that the $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\pi_{\infty},t_{\infty})$ (resp
$\mathrm{C}^{*}(\pi_{\infty},t_{\infty})/{\mathcal{K}}({\mathcal{F}}_{{\mathcal{X}}{\mathcal{J}}_{{\mathcal{X}}}})$)
is isomorphic to ${\mathcal{T}}_{{\mathcal{X}}}$ (resp.
${\mathcal{O}}_{{\mathcal{X}}}$).
###### Definition 3.1.
The tensor algebra of a $\mathrm{C}^{*}$-correspondence ${\mathcal{X}}$ over
${\mathfrak{A}}$ is the norm-closed algebra
$\operatorname{alg}(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$
and is denoted as ${\mathcal{T}}_{{\mathcal{X}}}^{+}$.
According to [13, 18], the algebras
${\mathcal{T}}_{{\mathcal{X}}}^{+}\equiv\operatorname{alg}(\overline{\pi}_{{\mathfrak{A}}},\overline{t}_{{\mathcal{X}}})$
and $\operatorname{alg}(\pi_{\infty},t_{\infty})$ are completely isometrically
isomorphic and we will therefore identify them.
In order to prove the main results of this section, we follow the strategy of
the first half of [9]. However, the generality in which we are working with,
presents new difficulties and requires innovation. One such innovation is the
following.
###### Lemma 3.2.
Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\alpha$ be a completely isometric automorphism of the associated tensor
algebra ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ and assume that $\alpha(A)=A$, for
all $A\in{\mathfrak{A}}$. Let
$\pi:{\mathcal{T}}_{{\mathcal{X}}}^{+}\rightarrow B({\mathcal{H}})$ be a
completely contractive representation of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
and let $X\in B({\mathcal{H}})$ be a contraction satisfying,
$\pi(L)X=X\pi(\alpha(L)),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
Then there exist isometric co-extensions $\pi^{\prime}$ and
$(\pi\circ\alpha)^{\prime}$, of $\pi$ and $\pi\circ\alpha$ respectively, and
an isometric co-extension $X^{\prime}$ of $X$, all acting on some Hilbert
space ${\mathcal{H}}^{\prime}$ and satisfying
(2) $\pi^{\prime}(L)X^{\prime}=X^{\prime}(\pi\circ\alpha)^{\prime}(L),\mbox{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+},$
and
(3) $\pi^{\prime}(A)=(\pi\circ\alpha)^{\prime}(A),\mbox{ for all
}A\in{\mathfrak{A}}.$
###### Proof..
First we construct isometric co-extensions $\widehat{\pi}_{1}$ and
$\widehat{\pi}_{2}$, of $\pi$ and $\pi\circ\alpha$ respectively, and an
isometric co-extension $\widehat{X}$ of $X$, with the property that
(4) $\widehat{\pi}_{1}(A)=\widehat{\pi}_{2}(A)$
and
(5)
$\phantom{XXiXXX}\widehat{X}\widehat{\pi}_{i}(A)=\widehat{\pi}_{i}(A)\widehat{X},\quad
i=1,2,$
for all $A\in{\mathfrak{A}}$.
To do this, notice that $X$ commutes with $\pi({\mathfrak{A}})$. co-extend $X$
to its Schaeffer dilation
$S_{X}\simeq\begin{bmatrix}K&0&0&0&\dots\\\ D_{K}&0&0&0&\dots\\\
0&I&0&0&\dots\\\ 0&0&I&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}\in
B({\mathcal{H}}^{(\infty)}),$
where $D_{K}=(I-K^{*}K)^{1/2}$. Let $\pi^{(\infty)}$ be the infinite
ampliation of $\pi$ and notice that $S_{X}$ commutes with
$\pi^{(\infty)}({\mathfrak{A}})$. Subsequently, using [23, Theorem 3.3], we
obtain some isometric co-extension $\widehat{\pi}$ of $\pi^{(\infty)}$, on
some Hilbert space
${\mathcal{K}}={\mathcal{H}}^{(\infty)}\oplus{\mathcal{M}}$, and let
$\widehat{\pi}_{1}=\widehat{\pi}$,
$\widehat{\pi}_{2}=\widehat{\pi}\circ\alpha$ and $\widehat{X}=S_{X}\oplus
I_{{\mathcal{M}}}$. These $\widehat{\pi}_{1},\widehat{\pi}_{2}$ and
$\widehat{X}$ satisfy (4) and (5).
Since $\widehat{X}$ satisfies (5), the pairs
$(\widehat{t}_{i},\widehat{\pi}_{i}|_{{\mathfrak{A}}})$, $i=1,2$, where,
$\displaystyle\widehat{t}_{1}(\xi)$
$\displaystyle=\widehat{\pi}_{1}(t_{\infty}(\xi))\widehat{X},$
$\displaystyle\widehat{t}_{2}(\xi)$
$\displaystyle=\widehat{X}\widehat{\pi}_{2}(t_{\infty}(\xi)),\quad\xi\in{\mathcal{X}},$
define isometric representations of $({\mathcal{X}},{\mathfrak{A}})$ and so
there exist $*$-representations
$\rho_{i}:{\mathcal{T}}_{{\mathcal{X}}}\rightarrow B({\mathcal{K}})$ which
integrate $(\widehat{t}_{i},\widehat{\pi}_{i}|_{{\mathfrak{A}}})$, $i=1,2$.
Since,
(6)
$P_{{\mathcal{H}}}\rho_{1}(L)|_{{\mathcal{H}}}=P_{{\mathcal{H}}}\rho_{2}(L)|_{{\mathcal{H}}},\text{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+},$
the representations $\rho_{i}$ co-extend the same contractive representation
of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ (appearing in (6)). By the uniqueness
of the minimal isometric co-extension [23, Proposition 3.2], there exist
projections $Q_{i}$ commuting with
$\rho_{i}({\mathcal{T}}_{{\mathcal{X}}}^{+})$, $i=1,2$, (hence commuting with
$\widehat{\pi}_{i}({\mathfrak{A}})$, $i=1,2$) and a unitary
$W:Q_{1}({\mathcal{K}})\rightarrow Q_{2}({\mathcal{K}})$, so that
(7)
$W\rho_{1}(L)|_{Q_{1}({\mathcal{K}})}W^{*}=\rho_{2}(L)|_{Q_{2}({\mathcal{K}})},\text{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
Furthermore, for each $i=1,2$, ${\mathcal{H}}\subseteq Q_{i}({\mathcal{K}})$
and $W$ fixes ${\mathcal{H}}$, because both $\rho_{1}$ and $\rho_{2}$ co-
extend the same completely contractive representation of
${\mathcal{T}}_{{\mathcal{X}}}^{+}$, which acts on ${\mathcal{H}}$.
For $i=1,2$, let
$\widetilde{\pi}_{i}(L)=\widehat{\pi}_{i}(L)\oplus\left(\rho_{1}(L)|_{Q_{1}^{\perp}({\mathcal{K}})}\right)^{(\infty)}\oplus\left(\rho_{2}(L)|_{Q_{2}^{\perp}({\mathcal{K}})}\right)^{(\infty)},\,\,L\in{\mathcal{T}}_{{\mathcal{X}}}^{+},$
and let
$\widetilde{X}=\widehat{X}\oplus
I_{Q_{1}^{\perp}({\mathcal{K}})}^{(\infty)}\oplus
I_{Q_{2}^{\perp}({\mathcal{K}})}^{(\infty)},$
all of them acting on
${\mathcal{H}}^{\prime}={\mathcal{K}}\oplus
Q_{1}^{\perp}({\mathcal{K}})^{(\infty)}\oplus
Q_{2}^{\perp}({\mathcal{K}})^{(\infty)}.$
Because of (7), there exists a unitary $U\in B({\mathcal{H}}^{\prime})$ which
fixes ${\mathcal{H}}$, commutes with
$\widetilde{\pi}_{1}({\mathfrak{A}})=\widetilde{\pi}_{2}({\mathfrak{A}})$ and
satisfies
$U\widetilde{\pi}_{1}(L)\widetilde{X}U^{*}=\widetilde{X}\widetilde{\pi}_{2}(L),\text{
for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
Consider the isometric representations
$(t_{i},\widetilde{\pi}_{i}|_{{\mathfrak{A}}})$ of
$({\mathcal{X}},{\mathfrak{A}})$, where,
$\displaystyle t_{1}(\xi)$
$\displaystyle=\widetilde{\pi}_{1}(t_{\infty}(\xi))U,\mbox{ and,}$
$\displaystyle t_{2}(\xi)$
$\displaystyle=\widetilde{\pi}_{2}(t_{\infty}(\xi))U,\xi\in{\mathcal{X}},$
and let $\pi_{i}^{\prime}$, $i=1,2$, be the $*$-representations of
${\mathcal{T}}_{{\mathcal{X}}}$ which integrate them. Let
$X^{\prime}=U^{*}\widetilde{X}$ and notice that for any
$L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}$ we have
$\pi_{1}^{\prime}(L)X^{\prime}=\widehat{\pi}_{1}(L)UU^{*}\widetilde{X}=\widehat{\pi}_{1}(L)\widetilde{X}$
while
$X^{\prime}\pi_{2}^{\prime}(L)=U^{*}\widetilde{X}\widehat{\pi}_{2}(L)U=\widehat{\pi}_{1}(L)\widetilde{X},$
and the conclusion follows. ∎
###### Remark 3.3.
In the previous Lemma, one may take the isometric co-extension $X^{\prime}$ to
be the minimal isometric co-extension $X_{m}$ of $X$. In that case however,
the co-extensions $\pi^{\prime}$ and $(\pi\circ\alpha)^{\prime}$ can only be
considered completely contractive and not necessarily isometric.
Indeed, using Lemma 3.2, we obtain isometric co-extensions
$\pi^{\prime},(\pi\circ\alpha)^{\prime}$ and $X^{\prime}$ on some Hilbert
space ${\mathcal{H}}^{\prime}$ that satisfy (2) and (3). Let $Q$ be the
reducing projection for $X^{\prime}$ so that
$QX^{\prime}|_{Q({\mathcal{H}}^{\prime})}\simeq X_{m}$. By (3), the projection
$Q$ commutes with $\pi^{\prime}({\mathfrak{A}})$ and
$(\pi\circ\alpha)^{\prime}({\mathfrak{A}})$ and so the completely contractive
representations of $({\mathcal{X}},{\mathfrak{A}})$, determined by the
representations $\pi^{\prime}$ and $(\pi\circ\alpha)^{\prime}$ of
${\mathfrak{A}}$ and the mappings
$\displaystyle{\mathcal{X}}\ni\xi$ $\displaystyle\longmapsto
Q\pi^{\prime}(t_{\infty}(\xi))|_{Q({\mathcal{H}}^{\prime})}$
$\displaystyle{\mathcal{X}}\ni\xi$ $\displaystyle\longmapsto
Q(\pi\circ\alpha)^{\prime}(t_{\infty}(\xi))|_{Q({\mathcal{H}}^{\prime})},$
can be integrated to the desired contractive representations of
${\mathcal{T}}_{{\mathcal{X}}}^{+}$, satisfying the analogues of (2) and (3)
with $X_{m}$ instead of $X^{\prime}$.
If we take $\alpha={\operatorname{id}}$ in Lemma 3.2, then we obtain the
commutant lifting Theorem of Muhly and Solel [23], without using the ”one-
step” extension in the proof. (In [23, page 418] the authors ask for such a
proof.) Indeed
###### Corollary 3.4.
Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\pi:{\mathcal{T}}_{{\mathcal{X}}}^{+}\rightarrow B({\mathcal{H}})$ be a
completely contractive representation of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
and let $X\in B({\mathcal{H}})$ be a contraction satisfying,
$\pi(L)X=X\pi(L),\mbox{ for all }L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
If $\pi_{m}$ is the minimal isometric co-extension of $\pi$, then there exists
a contraction $X^{\prime}$ co-extending $X$ and satisfying
$\pi_{m}(L)X^{\prime}=X^{\prime}\pi_{m}(L),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
###### Proof..
Use Lemma 3.2 to obtain isometric co-extensions
$\pi^{\prime},X^{\prime\prime}$ on some Hilbert space ${\mathcal{H}}^{\prime}$
that do the job. (Note however that $\pi^{\prime}$ may be ”larger” than the
minimal isometric co-extension.) There exists now a reducing subspace
${\mathcal{K}}\subseteq{\mathcal{H}}^{\prime}$ for $\pi^{\prime}$ so that
$\pi^{\prime}|_{{\mathcal{K}}}\equiv\pi_{m}$. Letting $X^{\prime}$ be the
compression of $X^{\prime\prime}$ on ${\mathcal{K}}$, the conclusion follows.
∎
A familiar $2\times 2$ matrix trick also establishes the intertwining form of
the commutant lifting theorem for minimal isometric co-extensions.
###### Theorem 3.5.
Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\alpha$ be a completely isometric automorphism of the associated tensor
algebra ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ and assume that $\alpha(A)=A$, for
all $A\in{\mathfrak{A}}$. Let
$\pi:{\mathcal{T}}_{{\mathcal{X}}}^{+}\rightarrow B({\mathcal{H}})$ be a
completely contractive representation of ${\mathcal{T}}_{{\mathcal{X}}}^{+}$
and let $X\in B({\mathcal{H}})$ be a contraction satisfying,
$\pi(L)X=X\pi(\alpha(L)),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
Then there exist an isometric co-extension $\pi_{1}$ of $\pi$ and an isometric
co-extension $Z$ of $X$, so that
$\pi_{1}(L)Z=Z\pi_{1}(\alpha(L)),\mbox{ for all
}L\in{\mathcal{T}}_{{\mathcal{X}}}^{+}.$
###### Proof..
Notice that if $\pi_{m}$ is the minimal isometric dilation of $\pi$, then
$\pi_{m}\circ\alpha$ is the minimal isometric dilation of $\pi\circ\alpha$.
Therefore, by applying commutant lifting to the covariance relations, we
obtain a contraction $X_{1}$ on a Hilbert space ${\mathcal{H}}_{1}$,
satisfying
$\pi_{m}(L)X_{1}=X_{1}(\pi_{m}\circ\alpha)(L),\text{ for all
}L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$
Let $X_{1,m}$ be the minimal dilation of $X_{1}$, i.e.,
(8) $X_{1,m}\simeq\begin{bmatrix}X_{1}&0&0&0&\dots\\\ D_{X_{1}}&0&0&0&\dots\\\
0&I&0&0&\dots\\\ 0&0&I&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}$
where $D_{X_{1}}=(I-X_{1}^{*}X_{1})^{1/2}$. We apply now Remark 3.3 to obtain
completely contractive representations $\widehat{\pi}_{m}$ and
$\widehat{\pi_{m}\circ\alpha}$, which co-extend $\pi_{m}$ and
$\pi_{m}\circ\alpha$, coincide on ${\mathfrak{A}}$, and satisfy
(9)
$\widehat{\pi}_{m}(L)X_{1,m}=X_{1,m}\widehat{\pi_{m}\circ\alpha}(L)\quad\text{for}\quad
L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$
Assume that these dilations have the form
$\widehat{\pi}_{m}(L)=\begin{bmatrix}\pi_{m}(L)&0\\\
Y^{(L)}&[Y^{(L)}_{jk}]_{j,k\geq 1}\end{bmatrix}\text{ and \,
}\widehat{\pi_{m}\circ\alpha}(L)=\begin{bmatrix}\pi_{m}\circ\alpha(L)&0\\\
Z^{(L)}&[Z^{(L)}_{jk}]_{j,k\geq 1}\end{bmatrix}$
with regards to the decomposition of the Hilbert space that corresponds to the
matricial form of $X_{1,m}$ in (8).
Claim: $Y^{(L)}=Z^{(L)}=0$, for all $L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$.
Indeed, the claim is true in the case where $L=\pi_{\infty}(A)$,
$A\in{\mathfrak{A}}$, since the restrictions of $\widehat{\pi}_{m}$ and
$\widehat{\pi_{m}\circ\alpha}$ on ${\mathfrak{A}}$ are $*$-homomorphisms
dilating the $*$-homomorphisms $\pi_{m}$ and $\pi_{m}\circ\alpha$
respectively. Hence it suffices to prove the claim in the case where
$L=t_{\infty}(\xi)$, $\xi\in{\mathcal{X}}$. We show that $Y^{(\xi)}=0$; a
similar argument will show that $Z^{(t_{\infty}(\xi))}=0$. By the Schwarz
inequality for completely contractive maps on unital operator algebras we have
$\widehat{\pi}_{m}\left(t_{\infty}(\xi)\right)^{*}\widehat{\pi}_{m}(t_{\infty}(\xi))\leq\widehat{\pi}_{m}\big{(}\pi_{\infty}(\langle\,\xi\mid\xi\,\rangle)\big{)}.$
By taking into account the matricial form of $\widehat{\pi}_{m}$ and comparing
$(1,1)$-entries in the above inequality, we obtain
(10)
$\pi_{m}\left(t_{\infty}(\xi)\right)^{*}\pi_{m}(t_{\infty}(\xi))+(Y^{(t_{\infty}(\xi))})^{*}Y^{(t_{\infty}(\xi))}\leq\pi_{m}\big{(}\pi_{\infty}(\langle\,\xi\mid\xi\,\rangle)\big{)}.$
However the map $\pi_{m}$ is an isometric representation and so
$\pi_{m}\left(t_{\infty}(\xi)\right)^{*}\pi_{m}(t_{\infty}(\xi))=\pi_{m}\big{(}\pi_{\infty}(\langle\,\xi\mid\xi\,\rangle)\big{)}.$
and so (10) obtains
$(Y^{(t_{\infty}(\xi))})^{*}Y^{(t_{\infty}(\xi))}\leq 0,$
which proves the claim.
By comparing $(2,i)$-entries, $i=2,3,\dots$, in the covariance relation (9) we
also obtain
$Y_{2,i}^{(L)}=0,\quad\text{for all }i\geq 2.$
In addition, by comparing $(i,1)$-entries $i=3,4,\dots$, in (9) we obtain
$Y_{i,2}^{(L)}D_{X_{1}}=0,\quad\text{for all }i\geq 3,$
and so $Y_{i,2}^{(L)}=0,\text{for all }i\geq 3$. This combined with the Claim
implies that the second row and column of $\widehat{\pi}_{m}(L)$,
$L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$, are equal to zero, except perhaps
from $Y_{2,2}^{(L)}$. Therefore, the map
$\rho:{\mathcal{T}}^{+}_{{\mathcal{G}}}\longrightarrow
B(D_{X_{1}}({\mathcal{H}}_{1})),\,\,L\longmapsto Y_{2,2}^{(L)}$
is a completely contractive representation of
${\mathcal{T}}^{+}_{{\mathcal{X}}}$. By comparing $(2,1)$-entries in the
covariance relation (9), we now obtain
(11) $\rho(L)D_{X_{1}}=D_{X_{1}}\pi_{m}\circ\alpha(L),\quad\text{for}\quad
L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$
For any $L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$, we now define
$\pi_{m}^{\prime}(L)=\begin{bmatrix}\pi_{m}(L)&0&0&0&\dots\\\
0&\rho(L)&0&0&\dots\\\ 0&0&\rho(\alpha(L))&0&\dots\\\
0&0&0&\rho(\alpha^{(2)}(L))&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}$
By (11), this is a completely contractive representation $\pi_{m}^{\prime}$ on
a Hilbert space ${\mathcal{H}}_{2}$ so that
$\pi_{m}^{\prime}(L)X_{1,m}=X_{1,m}\pi_{m}^{\prime}\circ\alpha(L)\quad\text{for}\quad
L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}.$
Continuing in this fashion, we obtain a sequence
$(\pi,X),\,(\pi_{m},X_{1}),\,(\pi_{m}^{\prime},X_{1,m}),\,((\pi_{m}^{\prime})_{m},X_{2}),\,(((\pi_{m}^{\prime})_{m})^{\prime},X_{2,m})\dots$
of pairs of operators and representations acting on Hilbert spaces
${\mathcal{H}}\subseteq{\mathcal{H}}_{1}\subseteq{\mathcal{H}}_{2}\dots,$ co-
extending $\pi$ and $X$ and satisfying the covariance relations. Let
${\mathcal{H}}=\bigvee_{j}{\mathcal{H}}_{j}$, and consider these pairs as
acting on ${\mathcal{H}}$ by extending them to be zero on the complement. Let
$Z=\operatorname*{\textsc{sot}--lim}X_{j}=\operatorname*{\textsc{sot}--lim}X_{m,j}$
and define $\pi_{1}(L)$, $L\in{\mathcal{T}}^{+}_{{\mathcal{X}}}$, as a strong
limit in a similar fashion. These limits evidently exist as in both cases the
sequences consist of either isometries or isometric representations that
decompose as infinite direct sums. Multiplication is sot-continuous on the
ball, hence the covariance relations hold in the limit. ∎
Combining the Proposition above with Theorem 2.6 we obtain the main result of
the section,
###### Corollary 3.6.
Let $({\mathcal{X}},{\mathfrak{A}})$ be a $\mathrm{C}^{*}$-correspondence, let
$\alpha$ be a completely isometric automorphism of the associated tensor
algebra ${\mathcal{T}}_{{\mathcal{X}}}^{+}$ and assume that $\alpha(A)=A$, for
all $A\in{\mathfrak{A}}$. Then
${\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}{\mathbb{Z}}^{+}$ and
${\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}^{{\operatorname{is}}}{\mathbb{Z}}^{+}$
are completely isometrically isomorphic and
$\mathrm{C}^{*}_{\text{env}}({\mathcal{T}}_{{\mathcal{X}}}^{+}\times_{\alpha}{\mathbb{Z}}^{+})\simeq{\mathcal{O}}_{{\mathcal{X}}}\times_{\alpha}{\mathbb{Z}}.$
In particular, the above corollary recaptures the main result of [9] with a
different proof.
## References
* [1] M. Alaimia and J. Peters, Semicrossed products generated by two commuting automorphisms J. Math. Anal. Appl. 285 (2003), 128–140.
* [2] W. Arveson, Operator algebras and measure preserving automorphisms, Acta Math. 118, (1967), 95–109.
* [3] W. Arveson, Subalgebras of C*-algebras III, Acta Math. 181 (1998), 159–228.
* [4] W. Arveson and K. Josephson, Operator algebras and measure preserving automorphisms II, J. Functional Analysis 4, (1969), 100–134.
* [5] D. Buske and J. Peters, Semicrossed products of the disk algebra: contractive representations and maximal ideals, Pacific J. Math. 185 (1998), 97–113.
* [6] L. DeAlba and J. Peters, Classification of semicrossed products of finite-dimensional C*-algebras, Proc. Amer. Math. Soc. 95 (1985), 557–564.
* [7] K. Davidson, E. Katsoulis, Isomorphisms between topological conjugacy algebras, J. reine angew. Math. 621 (2008), 29-51.
* [8] K. Davidson, E. Katsoulis, Nonself-adjoint crossed products and dynamical systems, Contemporary Mathematics, to appear.
* [9] K. Davidson, E. Katsoulis, Dilating covariant representations of the non-commutative disc algebras, J. Funct. Anal. 259 (2010), Pages 817-831.
* [10] K. Davidson, E. Katsoulis, Operator algebras for multivariable dynamics, Mem. Amer. Math. Soc., to appear.
* [11] M. Dritschel and S. McCullough, Boundary representations for families of representations of operator algebras and spaces, J. Operator Theory 53 (2005), 159–167
* [12] S. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), 300–304.
* [13] N. Fowler, I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), 155–181.
* [14] D. Hadwin and T. Hoover, Operator algebras and the conjugacy of transformations., J. Funct. Anal. 77 (1988), 112–122.
* [15] T. Hoover, Isomorphic operator algebras and conjugate inner functions, Michigan Math. J. 39 (1992), 229–237.
* [16] T. Hoover, J. Peters and W. Wogen, Spectral properties of semicrossed products, Houston J. Math. 19 (1993), 649–660.
* [17] E. Kakariadis, Semicrossed products and reflexivity, J. Operator Theory, to appear.
* [18] T. Katsura, On $\mathrm{C}^{*}$-algebras associated with $\mathrm{C}^{*}$-correspondences, J. Funct. Anal. 217 (2004), 366–401.
* [19] E. Katsoulis and D. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann. 330, (2004), 709–728.
* [20] D. Kribs and S. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc., textbf19 (2004), 75–117.
* [21] E. C. Lance, Hilbert $C^{*}$-modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995.
* [22] M. McAsey and P. Muhly, Representations of nonselfadjoint crossed products, Proc. London Math. Soc. (3) 47 (1983), 128–144.
* [23] P.S. Muhly, B. Solel, Tensor algebras over $\mathrm{C}^{*}$-correspondences: representations, dilations and $\mathrm{C}^{*}$-envelopes J. Funct. Anal. 158 (1998), 389–457.
* [24] P. Muhly and B. Solel, An algebraic characterization of boundary representations, Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics, Birkh user Verlag, Basel 1998, pp. 189–196.
* [25] J. Peters, Semicrossed products of C*-algebras, J. Funct. Anal. 59 (1984), 498–534.
* [26] J. Peters, The ideal structure of certain nonselfadjoint operator algebras, Trans. Amer. Math. Soc. 305 (1988), 333–352.
* [27] G. Popescu, Non-commutative disc algebras and their representations Proc. Amer. Math. Soc. 124, (1996), 2137–2148.
* [28] N. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100.
|
arxiv-papers
| 2010-08-13T19:10:46 |
2024-09-04T02:49:12.182931
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Evgenios Kakariadis and Elias Katsoulis",
"submitter": "Elias Katsoulis",
"url": "https://arxiv.org/abs/1008.2374"
}
|
1008.2438
|
# Chemical Examples in Hypergroups
B. Davvaz and A. Dehghan-Nezhad
Department of Mathematics, University of Yazd,
Yazd, Iran
E-mail: davvaz@yazduni.ac.ir
###### Abstract
Hypergroups first were introduced by Marty in 1934. Up to now many researchers
have been working on this field of modern algebra and developed it. It is
purpose of this paper to provide examples of hypergroups associated with
chemistry. The examples presented are connected to construction from chain
reactions.
## 1 Introuduction
The theory of algebraic hyperstructures which is a generalization of the
concept of algebraic structures first was introduced by Marty in 1934 [4], and
had been studied in the following decades and nowadays by many mathematicians,
and many papers concerning various hyperstructures have appeared in the
literature, for example see [2,3,6,8]. The basic definitions of the object can
be found in [1,7].
### 1.1 Hypergroups and $H_{v}$-groups
An algebraic hyperstructure is a non-empty set $H$ together with a function
$\cdot:H\times H\longrightarrow{p}^{*}(H)$ called hyperoperation, where
${p^{*}(H})$ denotes the set of all non-empty subsets of $H$. If $A,B$ are
non-empty subsets of $H$ and $x\in H$, then we define
$A\cdot B=\displaystyle\bigcup_{a\in A,b\in B}a\cdot b,\ \ x\cdot
B=\\{x\\}\cdot B,\ \ {\rm and}\ \ A\cdot x=A\cdot\\{x\\}.$
The hyperoperation $(\cdot)$ is called associative in $H$ if
$(x\cdot y)\cdot z=x\cdot(y\cdot z)\ {\rm for\ all}\ x,y,z\ {\rm in}\ H,$
which means that
$\displaystyle\bigcup_{u\in x\cdot y}u\cdot z=\bigcup_{v\in y\cdot z}x\cdot
v.$
We say that a semihypergroup $(H,\cdot)$ is a hypergroup if for all $x\in H$,
we have $x\cdot H=H\cdot x=H.$ A hypergroupoid $(H,\cdot)$ is an
$H_{v}$-group, if for all $x,y,z\in H,$ the following conditions hold:
* (1)
$x\cdot(y\cdot z)\cap(x\cdot y)\cdot z\not=\emptyset$, (weak associative)
* (2)
$x\cdot H=H\cdot x=H$.
A non-empty subset $K$ of a hypergroup (respectively, $H_{v}$-group) $H$ is
called a subhypergroup (respectively, $H_{v}$-subgroup) of $H$ if $a\cdot
K=K\cdot a=K$ for all $a\in K$.
In this paper, we will give some examples of hypergroups associated with
chemistry. The examples presented are connected to construction from chian
reactions.
## 2 Prelimiaries
a) Chain reactions
An atom of group of atoms possessing an odd (unpaired) electron is called a
free radical, such as
$Cl,\ CH_{3},\ C_{2}H_{5}$
The chlorination of methane is an example of a chain reaction, a reaction that
involves a series of setps, each of which generates a reactive substance that
brings about the next step. While chain reactions may vary widely in their
details, they all have certain fundamental characteristics in common.
* 1)
$Cl_{2}\longrightarrow 2Cl^{o}$
(1) is called Chain-initiating step.
* 2)
$Cl^{o}+CH_{4}\longrightarrow HCl+CH_{3}^{o}$
* 3)
$CH_{3}^{o}+Cl_{2}\longrightarrow CH_{3}Cl+Cl^{o}$
then (2), (3), (2), (3), etc, until finatly:
(2) and (3) are called Chain-propagating steps.
* 4)
$Cl^{o}+Cl^{o}\longrightarrow Cl_{2}\ \ \ $ or
* 5)
$CH_{3}^{o}+CH_{3}^{o}\longrightarrow CH_{3}CH_{3}\ \ \ $ or
* 6)
$CH_{3}^{o}+Cl^{o}\longrightarrow CH_{3}Cl$.
(4),(5) and (6) are called Chain-terminating steps.
First in the chain of reactions is a chain-initiating step, in which energy is
absorbed and a reactive particle generated; in the present reaction it is the
cleavage of chlorine into atoms (step 1).
There are one or more chain-propagating steps, each of which consumes a
reactive particle and generates another; there they are the reaction of
chlorine atoms with methane (step 2), and of methyl radicals with chlorine
(step 3).
Finally, there are chain-terminating steps, in which reactive particles are
consumed but not generated; in the chlorination of methane these would involve
the union of two of the reactive particles, or the capture of one of them by
the walls of the reaction vessel.
b) The Halogens F, CL, Br, and I
The halogens are all typical non-metals. Although their physical forms differ-
fluorine and chlorine are gases, bromine is a liquid and iodine is a solid at
room temprature, each consists of diatomic molecules; $F_{2},Cl_{2},Br_{2}$
and $I_{2}$. The halogens all react with hydrogen to form gaseous compounds,
with the formulas $HF,HCL,HBr,$ and $HI$ all of which are very soluble in
water. The halogens all react with metals to give halides.
$:\ddot{F}$.. \- $\ddot{F}:,$.. $:\ddot{CL}$.. \- $\ddot{CL}:,$..
$:\ddot{Br}$.. \- $\ddot{Br}:,$.. $:\ddot{I}$.. \- $\ddot{I}:$..
The reader will find in [5] a deep discussion of chain reactions and halogens.
## 3 Chemical Hypergroups
In during chain reaction
$A_{2}+B_{2}\stackrel{{\scriptstyle\rm Heat\ or\
Light}}{{\longleftrightarrow}}2AB$
there exist all molecules $A_{2},B_{2},AB$ and whose fragment parts
$A^{o},B^{o}$ in experiment. Elements of this colletion can by combine with
each other. All combinational probability for the set ${\cal
H}=\\{A^{o},B^{o},A_{2},B_{2},AB\\}$ to do without energy can be displayed as
follows:
$+$ | $A^{o}$ | $B^{o}$ | $A_{2}$ | $B_{2}$ | $AB$
---|---|---|---|---|---
$A^{o}$ | $A^{o},A_{2}$ | $A^{o},B^{o},AB$ | $A^{o},A_{2}$ | $A^{o},B_{2},B^{o},AB$ | $A^{o},AB,A_{2},B^{o}$
$B^{o}$ | $A^{o},B^{o},AB$ | $B^{o},B_{2}$ | $A^{o},B^{o},AB,A_{2}$ | $B^{o},B_{2}$ | $A^{o},B^{o},AB,B_{2}$
$A_{2}$ | $A^{o},A_{2}$ | $A^{o},B^{o},AB,A_{2}$ | $A^{o},A_{2}$ | $A^{o},B^{o},A_{2},B_{2},AB$ | $A^{o},B^{o},A_{2},AB$
$B_{2}$ | $A^{o},B^{o},B_{2},AB$ | $B^{o},B_{2}$ | $A^{o},B^{o},A_{2},B_{2},AB$ | $B^{o},B_{2}$ | $A^{o},B^{o},B_{2},AB$
$AB$ | $A^{o},AB,A_{2},B^{o}$ | $A^{o},B^{o},AB,B_{2}$ | $A^{o},B^{o},A_{2},AB$ | $A^{o},B^{o},B_{2},AB$ | $A^{o},B^{o},A_{2},B_{2},AB$
Theorem. $({\cal H},+)$ is an $H_{v}$-group.
Proof. Clearly reproduction axiom and weak associativity are valid. As a
sample of how to calculate the weak associativity, we illustrate some cases:
$\left\\{\begin{array}[]{l}(AB+A_{2})+B_{2}=\\{AB,A_{2},A^{o},B^{o}\\}+B_{2}=\\{B_{2},AB,A_{2},A^{o},B^{o}\\},\\\
AB+(A_{2}+B_{2})=AB+\\{A_{2},B_{2},A^{o},B^{o},AB\\}=\\{A_{2},B_{2},AB,A^{o},B^{o}\\},\end{array}\right.$
$\ \ \
\left\\{\begin{array}[]{l}(AB+A^{o})+A^{o}=\\{AB,A^{o},A_{2},B^{o}\\}+A^{o}=\\{A_{2},A^{o},AB,B^{o}\\},\\\
AB+(A^{o}+A^{o})=AB+\\{A_{2},A^{o}\\}=\\{A_{2},AB,A^{o},B^{o}\\},\end{array}\right.$
$\left\\{\begin{array}[]{l}(A_{2}+B^{o})+B_{2}=\\{AB,A^{o},A_{2},B^{o}\\}+B_{2}=\\{B_{2},AB,B^{o},A^{o},A_{2}\\},\\\
A_{2}+(B^{o}+B_{2})=A_{2}+\\{B_{2},B^{o}\\}=\\{A_{2},A^{o},AB,B^{o},B_{2}\\}.\end{array}\right.$
Corollary. ${\cal H}_{1}=\\{A^{0},A_{2}\\}$ and ${\cal
H}_{2}=\\{B^{0},B_{2}\\}$ are only subhypergroups of $({\cal H},+)$.
If we consider $A=H$ and $B\in\\{F,CL,Br,I\\}$ (for example $B=I$), the
complete reaction table becomes:
$+$ | $H^{o}$ | $I^{o}$ | $H_{2}$ | $I_{2}$ | $HI$
---|---|---|---|---|---
$H^{o}$ | $H^{o},H_{2}$ | $H^{o},I^{o},HI$ | $H^{o},H_{2}$ | $H^{o},I_{2},I^{o},HI$ | $H^{o},HI,H_{2},I^{o}$
$I^{o}$ | $H^{o},I^{o},HI$ | $I^{o},I_{2}$ | $H^{o},I^{o},HI,H_{2}$ | $I^{o},I_{2}$ | $H^{o},I^{o},HI,I_{2}$
$H_{2}$ | $H^{o},H_{2}$ | $H^{o},I^{o},HI,I_{2}$ | $H^{o},H_{2}$ | $H^{o},I^{o},H_{2},I_{2},HI$ | $H^{o},I^{o},H_{2},HI$
$I_{2}$ | $H^{o},I^{o},I_{2},HI$ | $H^{o},I_{2}$ | $H^{o},I^{o},H_{2},I_{2},HI$ | $H^{o},I_{2}$ | $H^{o},I^{o},I_{2},HI$
$HI$ | $H^{o},HI,H_{2},I^{o}$ | $H^{o},I^{o},HI,I_{2}$ | $H^{o},I^{o},H_{2},HI$ | $H^{o},I^{o},H_{2},HI$ | $H^{o},I^{o},H_{2},I_{2},HI$
Acknowledgment
We appreciate the assistance and suggestions of Dr. A. Gorgi at the Department
of Chemistry.
## References
* [1] P. Corsini, Prolegomena of hypergroup theory, second edition, Aviani editor, (1993).
* [2] M.R. Darafsheh and B. Davvaz, $H_{v}$-ring of fractions, Italian J. Pure Appl. Math., 5 (1999) 25-34.
* [3] B. Davvaz, Weak Polygroups, Proc. 28th Annual Iranian Math. Conf, (1977), 139-145.
* [4] F. Marty, Sur une generalization de la notion de groupe, $8^{iem}$ congres Math. Scandinaves, Stocklhom, (1934), 45-49.
* [5] Morrison and Boyd, Organic Chemistry, Sixth Eddition, Prentice-Hall, Inc, 1992.
* [6] T. Vougiouklis, A new class of hyperstructures, J. Combin. Inf. system Sci, 20, (1995), 229-235.
* [7] T. Vougiouklis, Hyperstructures and their repesentations, Hadronic Press Inc, (1994).
* [8] T. Vougiouklis, Convolution on WASS hyperstructures, Discrete Math., (1997), 347-355.
|
arxiv-papers
| 2010-08-14T12:35:05 |
2024-09-04T02:49:12.190564
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bijan Davvaz and Akbar Dehghan-Nezhad",
"submitter": "Akbar Dehghan Nezhad",
"url": "https://arxiv.org/abs/1008.2438"
}
|
1008.2480
|
Any component of moduli
of polarized hyperkähler manifolds
is dense in its deformation space
Sasha Anan′in, Misha Verbitsky111Misha Verbitsky is partially supported by the
RFBR grant 10-01-93113-NCNIL-a, RFBR grant 09-01-00242-a, Science Foundation
of the SU-HSE award No. 10-09-0015 and AG Laboratory HSE, RF government grant,
ag. 11.G34.31.0023
Abstract
Let $M$ be a compact hyperkähler manifold, and $W$ the coarse moduli of
complex deformations of $M$. Every positive integer class $v$ in $H^{2}(M)$
defines a divisor $D_{v}$ in $W$ consisting of all algebraic manifolds
polarized by $v$. We prove that every connected component of this divisor is
dense in $W$.
###### Contents
1. 1 Introduction
1. 1.1 Hyperkähler manifolds and moduli spaces
2. 1.2 Lelong numbers, SYZ conjecture and Gromov’s precompactness theorem
3. 1.3 Bogomolov-Beauville-Fujiki form and the mapping class group
4. 1.4 Teichmüller space and the moduli space
5. 1.5 The polarized Teichmüller space
2. 2 Torelli theorem and polarizations
3. 3 Arithmetic subgroups in $\mathop{\text{\rm O}}(p,q)$
## 1 Introduction
### 1.1 Hyperkähler manifolds and moduli spaces
Throughout this paper, a hyperkähler manifold means a “compact complex
manifold admitting a Kähler structure and a holomorphically symplectic form.”
A hyperkähler manifold $M$ is called simple if $\pi_{1}(M)=0$ and
$H^{2,0}(M)=\mathbb{C}$. By Bogomolov’s theorem (see [Bes] and [Bo1]), any
hyperkähler manifold has a finite covering which is a product of simple
hyperkähler manifolds and compact tori. Throughout this paper, we shall
silently assume that all our hyperkähler manifolds are simple. The results
that we prove can be stated and proven for general hyperkähler manifolds, but
to do so would destroy the clarity of the exposition.
For a background story on hyperkähler manifolds, their construction, and
properties, please see [Bea] and [Bes]. The moduli spaces of hyperkähler
manifolds are discussed at great length in [V2].
The moduli space of complex structures on a given smooth oriented manifold $M$
is defined, following Kodaira and Spencer, as the quotient of the Frèchet
manifold of all integrable almost complex structures $\operatorname{Comp}$ by
the action of the group of orientation-preserving diffeomorphisms
$\operatorname{Diff}^{+}$, which is considered as a Frèchet Lie group. We
denote by $\operatorname{Comp}_{0}\subset\operatorname{Comp}$ the open set
consisting of all complex structures on $M$ admitting a compatible Kähler
metric and a compatible holomorphically symplectic structure. The quotient
$\operatorname{Mod}:=\operatorname{Comp}_{0}/\operatorname{Diff}^{+}$ is
called a coarse moduli space of hyperkähler manifolds. It is a complex
analytic space, usually non-Hausdorff.
It is well known that a generic point $I\in\operatorname{Mod}$ corresponds to
a non-algebraic complex structure on $M$. In fact, the manifold $(M,I)$ has no
divisors, because the corresponding Neron-Severi group
$H^{1,1}(M,\mathbb{Z}):=H^{1,1}(M)\break\cap H^{2}(M,\mathbb{Z})$ is zero (see
[F]). The algebraic points of $\operatorname{Mod}$ sit on a countable union of
divisors in $\operatorname{Mod}$, which is known to be dense in
$\operatorname{Mod}$ ([F], [V0]).
In this paper we prove that each of these divisors is itself dense in
$\operatorname{Mod}$. This result is known when $M$ is a K3 surface (this
follows from a statement known as “Eichler Criterion”; see 2).
### 1.2 Lelong numbers, SYZ conjecture and Gromov’s precompactness theorem
The original motivation for this work came form a research on the so-called
hyperkähler SYZ conjecture ([V3]). This conjecture, which is a version of a
(more general) abundance conjecture of Kawamata, states that a nef bundle on a
hyperkähler manifold is semiample. More specifically, one is interested in
holomorphic line bundles $L$ which are nef, and for which the Bogomolov-
Beauville-Fujiki square of $c_{1}(L)$ vanishes: $q(c_{1}(L),c_{1}(L))=0$ (for
a definition of Bogomolov-Beauville-Fujiki form, see Subsection 1.3). Such
line bundles are called parabolic. Any nef bundle admits a singular metric
with semipositive curvature (this follows from general results on weak
compactness of positive currents). If this metric is not “very singular”, $L$
is effective ([V3], [V4]). The “not very singular” above refers to the
vanishing of the so-called Lelong numbers of the curvature current; these
numbers, defined for positive closed $(p,p)$-currents, vanish for all smooth
currents, and measure the geometric “strength” of its singularities in the
general case, taking values in $\mathbb{R}^{\geqslant 0}$.
The Lelong numbers are known to be upper semicontinuous in the current
topology. This means, in particular, that any cohomology class $\eta$ which is
represented as a limit of currents with Lelong numbers bounded from below
would have positive Lelong numbers.
Suppose now that $\eta\in H^{1,1}(M,\mathbb{R})$ is a nef class on a non-
algebraic hyperkähler manifold satisfying $q(\eta,\eta)=0$ (such class is also
called parabolic). It is proven in [V4] that the Lelong sets (sets where the
Lelong numbers are bounded from below by a positive number) of $\eta$ are
coisotropic with respect to the holomorphic symplectic structure. However, all
complex subvarieties of a generic non-algebraic hyperkähler manifold are
hyperkähler ([V1]), hence they cannot be coisotropic. This means that any
parabolic nef current on a generic non-algebraic manifold has vanishing Lelong
numbers.
To apply this argument, we need to approximate a given non-algebraic manifold
with a nef current by a sequence of algebraic manifolds with a rational
parabolic current, in a controlled way. To keep this approximation controlled,
the manifolds should belong to the same algebraic family.
Generally speaking, such a sequence is hard to produce. For a K3 such
approximations are well known, and much used since the earliest works on K3 in
the 1960-es. It is known, in particular, that the variety of quartic surfaces
is dense in the moduli of all (non-algebraic) K3 surfaces.
In this paper, we generalize this theorem, proving that the moduli of
polarized hyperkähler manifolds is a dense subset in the moduli of all (non-
algebraic) deformations. More precisely, given a rational cohomology class
$\eta\in H^{2}(M,\mathbb{Q})$, satisfying $q(\eta,\eta)>0$, we show that any
given $M$ can be approximated by deformations of $M$ which satisfy $\eta\in
H^{1,1}(M_{1})$.
It is interesting that even this result seems to be quite hard to prove. Our
proof relies on rationality of $\eta$ and does not work when $\eta$ is
irrational, or $q(\eta,\eta)\leqslant 0$, though the statement is most likely
true in this case as well.
Another interesting subject connected to the present problem is the Gromov
precompactness theorem. Consider the Gromov-Hausdorff distance, which is a
metric on the set of all compact metric spaces. Gromov has shown that the set
$\mathfrak{S}$ of all Riemannian manifolds of semipositive Ricci curvature and
bounded diameter is precompact, which means that its completion with respect
to the Gromov-Hausdorff distance is compact. This result highlights the notion
of Gromov-Hausdorff limits, that is, the metric spaces appearing as limits of
some family of Riemannian manifolds with respect to this distance.
For a finite-dimensional set of Ricci-flat manifolds, the Gromov-Hausdorff
limits are especially interesting because of the Gromov precompactness
theorem.
Let now $W_{\eta}$ be the moduli of polarized hyperkähler manifdolds, with
$\eta\in H^{2}(M)$ being a polarization. The space $W_{\eta}$ is known to be
quasiprojective (see [Vi]); moreover, it is locally symmetric ([GHS1]). To
compactify $W_{\eta}$, one usually uses the so-called Baily-Borel
compactification. However, points of $W_{\eta}$ correspond to polarized
hyperkähler manifolds, which are equipped with a canonical Ricci-flat Kähler
metric in the same cohomology class as $\eta$ ([Bes]). This allows one to
define the Gromov-Hausdorff compactification of the family associated with
$W_{\eta}$. It is unknown (except for the K3, where some partial results are
obtained) how the Gromov-Hausdorff completion of $W_{\eta}$ corresponds to the
Baily-Borel one.
However, the most interesting limits occur when one varies the Kähler class of
a Ricci-flat Kähler metric on $M$. For instance, when one takes a limit of
Ricci-flat Kähler metrics with a Kähler classes $\omega_{i}$ converging to
$\eta$ with $q(\eta,\eta)=0$. The hyperkähler SYZ conjecture predicts that the
Gromov-Hausdorff limit of the corresponding Ricci-flat metrics would give
$\mathbb{C}P^{n}$, $n=\frac{1}{2}\dim_{\mathbb{C}}M$ ([KS]).
Arguing this way, one would necessarily come to study the set of all Ricci-
flat Kähler metric on $M$ and their Gromov-Hausdorff limits. However, for the
whole moduli space $W$ of deformations there is no analogue of Baily-Borel
compactification, hence the Gromov-Hausdorff compactification has no obviouos
algebraic counterpart.
In the present paper we show that $W_{\eta}$ is dense in $W$, for rational
$\eta$. Together with the known compactification results for $W_{\eta}$, this
result could lead, at least in theory, to a better understanding of the
Gromov-Hausdorff compactification of the space of all hyperkaehler metrics.
### 1.3 Bogomolov-Beauville-Fujiki form and the mapping class group
For a better understanding of the moduli space geometry, some basic facts
about topology of hyperkähler manifolds should be stated. We follow [V2].
Let $\Omega$ be a holomorphic symplectic form on a hyperkähler manifold $M$.
Bogomolov [Bo2] and Beauville [Bea] defined the following bilinear symmetric
$2$-form on $H^{2}(M)$ :
$\displaystyle\tilde{q}(\eta,\eta^{\prime}):=$
$\displaystyle\int_{M}\eta\wedge\eta^{\prime}\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}-$
(1.1)
$\displaystyle-\frac{(n-1)}{n}\frac{\Big{(}\int_{M}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\Big{)}\cdot\Big{(}\int_{M}\eta^{\prime}\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\Big{)}}{\int_{M}\Omega^{n}\wedge\overline{\Omega}^{n}},$
where $4n=\dim_{\mathbb{R}}M$.
The form $\tilde{q}$ is topological by its nature.
Theorem 1.1 [F]: Let $M$ be a simple hyperkähler manifold of real dimension
$4n$. Then there exist a bilinear, symmetric, primitive non-degenerate integer
$2$-form $q:H^{2}(M,\mathbb{Z})\otimes
H^{2}(M,\mathbb{Z}){\>\longrightarrow\>}\mathbb{Z}$ and a positive constant
$c\in\mathbb{Z}$ such that $\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$ for all
$\eta\in H^{2}(M)$. Moreover, $q$ is proportional to the form $\tilde{q}$ of
(1.1), and has signature $(3,b_{2}-3)$ (with $3$ pluses and $b_{2}-3$
minuses).
Let $\operatorname{Diff}^{+}$ denote the group of orientation-preserving
diffeomorphisms of $M$, and $\operatorname{Diff}^{0}$ its connected component,
also known as a group of isotopies. The quotient group
$\Gamma:=\operatorname{Diff}^{+}/\operatorname{Diff}^{0}$ is called the
mapping class group of $M$. In [V2] it was shown that $\Gamma$ preserves the
Bogomolov-Beauville-Fujiki form on $H^{2}(M)$ and that the corresponding
homomorphism to the orthogonal group
$\Gamma{\>\longrightarrow\>}{\mathop{\text{\rm O}}}\big{(}H^{2}(M),q\big{)}$
has finite kernel. It was also shown that the image of $\Gamma$ in
${\mathop{\text{\rm O}}}\big{(}H^{2}(M),q\big{)}$ is commensurable to the
group ${\mathop{\text{\rm O}}}\big{(}H^{2}(M,\mathbb{Z}),q\big{)}$ of
isometries of the integer lattice.
### 1.4 Teichmüller space and the moduli space
To state our main result in precise terms, we have to give a more explicit
description of the moduli space of a hyperkähler manifold. We follow [V2].
Let $M$ be a hyperkähler manifold (compact and simple, as usual), and
$\operatorname{Comp}_{0}$ the Frèchet manifold of all complex structures of
hyperkähler type on $M$. The quotient
$\operatorname{Teich}:=\operatorname{Comp}_{0}/\operatorname{Diff}^{0}$ of
$\operatorname{Comp}_{0}$ by isotopies is a finite-dimensional complex
analytic space by the same Kodaira-Spencer arguments as used to show that
$\operatorname{Mod}=\operatorname{Comp}/\operatorname{Diff}^{+}$ is complex
analytic, where $\operatorname{Comp}$ is a Frèchet manifold of all integrable
complex, oriented structures on $M$. This quotient is called a Teichmüller
space of $M$. When $M$ is a complex curve, the quotient
$\operatorname{Comp}/\operatorname{Diff}^{0}$ is the Teichmüller space of this
curve.
The mapping class group
$\Gamma=\operatorname{Diff}^{+}/\operatorname{Diff}^{0}$ acts on
$\operatorname{Teich}$ in a usual way, and its quotient is the moduli space of
$M$.
As shown in [H2], $\operatorname{Teich}$ has a finite number of connected
components. Take a connected component $\operatorname{Teich}^{I}$ containing a
given complex structure $I$, and let $\Gamma^{I}\subset\Gamma$ be the set of
elements of $\Gamma$ fixing this component. Since $\operatorname{Teich}$ has
only a finite number of connected components, $\Gamma^{I}$ has finite index in
$\Gamma$. On the other hand, as shown in [V2], the image of the group $\Gamma$
is commensurable to ${\mathop{\text{\rm
O}}}\big{(}H^{2}(M,\mathbb{Z}),q\big{)}$.
In [V2, Lemma 2.6] it was proved that any hyperkähler structure on a given
simple hyperkähler manifold is also simple. Therefore,
$H^{2,0}(M,I^{\prime})=\mathbb{C}$ for all $I^{\prime}\in\operatorname{Comp}$.
This observation is a key to the following well-known definition.
Definition 1.2: Let $(M,I)$ be a hyperkähler manifold, and
$\operatorname{Teich}$ its Teichmüller space. Consider a map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\mathbb{P}H^{2}(M,\mathbb{C})$,
sending $J$ to the line $H^{2,0}(M,J)\in\mathbb{P}H^{2}(M,\mathbb{C})$. It is
easy to see that $\operatorname{\sf Per}$ maps $\operatorname{Teich}$ into the
open subset of a quadric, defined by
$\operatorname{{\mathbb{P}}\sf
er}:=\big{\\{}l\in\mathbb{P}H^{2}(M,\mathbb{C})\ \big{|}\ q(l,l)=0,\
q(l,\overline{l})>0\big{\\}}.$
The map $\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is called the period map, and the set $\operatorname{{\mathbb{P}}\sf er}$
the period space.
The following fundamental theorem is due to F. Bogomolov [Bo2].
Theorem 1.3: Let $M$ be a simple hyperkähler manifold, and
$\operatorname{Teich}$ its Teichmüller space. Then the period map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is a local diffeomorphism (that is, an etale map). Moreover, it is
holomorphic.
Remark 1.4: Bogomolov’s theorem implies that $\operatorname{Teich}$ is smooth.
However, it is not necessarily Hausdorff (and it is non-Hausdorff even in the
simplest examples).
### 1.5 The polarized Teichmüller space
In [V4, Corollary 2.6], the following proposition was deduced from [Bou] and
[DP].
Theorem 1.5: Let $M$ be a simple hyperkähler manifold, such that all integer
$(1,1)$-classes satisfy $q(\nu,\nu)\geqslant 0$. Then its Kähler cone is one
of two connected components of the set $K:=\big{\\{}\nu\in
H^{1,1}(M,\mathbb{R})\ \big{|}\ q(\nu,\nu)>0\big{\\}}$.
Consider an integer vector $\eta\in H^{2}(M)$ which is positive, that is,
satisfies $q(\eta,\eta)>0$. Denote by $\operatorname{Teich}^{\eta}$ the set of
all $I\in\operatorname{Teich}$ such that $\eta$ is of type $(1,1)$ on $(M,I)$.
The space $\operatorname{Teich}^{\eta}$ is a closed divisor in
$\operatorname{Teich}$. Indeed, by Bogomolov’s theorem, the period map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is etale, but the image of $\operatorname{Teich}^{\eta}$ is the set of
all $l\in\operatorname{{\mathbb{P}}\sf er}$ which are orthogonal to $\eta$;
this condition defines a closed divisor $C_{\eta}$ in
$\operatorname{{\mathbb{P}}\sf er}$, hence
$\operatorname{Teich}^{\eta}=\operatorname{\sf Per}^{-1}(C_{\eta})$ is also a
closed divisor.
When $I\in\operatorname{Teich}^{\eta}$ is generic, Bogomolov’s theorem implies
that the space of rational $(1,1)$-classes $H^{1,1}(M,\mathbb{Q})$ is one-
dimensional and generated by $\eta$. This is seen from the following argument.
Locally around a given point $I$ the period map
$\operatorname{Teich}^{\eta}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is surjective on the set $\operatorname{{\mathbb{P}}\sf er}^{\eta}$ of
all $I\in\operatorname{{\mathbb{P}}\sf er}$ for which $\eta\in H^{1,1}(M,I)$.
However, the Hodge-Riemann relations give
$\operatorname{{\mathbb{P}}\sf
er}^{\eta}=\big{\\{}l\in\operatorname{{\mathbb{P}}\sf er}\ \big{|}\
q(\eta,l)=0\big{\\}}.$ (1.2)
Denote the set of such points of $\operatorname{Teich}^{\eta}$ by
$\operatorname{Teich}^{\eta}_{\text{gen}}$. It follows from 1.5 that, for any
$I\in\operatorname{Teich}^{\eta}_{\text{gen}}$, either $\eta$ or $-\eta$ is a
Kähler class on $(M,I)$.
Consider a connected component $\operatorname{Teich}^{\eta,I}$ of
$\operatorname{Teich}^{\eta}$. Changing the sign of $\eta$ if necessary, we
may assume that $\eta$ is Kähler on $(M,I)$. By Kodaira’s theorem about
stability of Kähler classes, $\eta$ is Kähler in some neighbourhood
$U\subset\operatorname{Teich}^{\eta,I}$ of $I$. Therefore, the sets
$V_{+}:=\big{\\{}I\in\operatorname{Teich}^{\eta}_{\text{gen}}\ \big{|}\
\eta\text{ is K\"{a}hler on }(M,I)\big{\\}}$
and
$V_{-}:=\big{\\{}I\in\operatorname{Teich}^{\eta}_{\text{gen}}\ \big{|}\
-\eta\text{ is K\"{a}hler on }(M,I)\big{\\}}$
are open in $\operatorname{Teich}^{\eta}_{\text{gen}}$. It is easy to see that
$\operatorname{Teich}^{\eta}_{\text{gen}}$ is a complement to a union of
countably many divisors in $\operatorname{Teich}^{\eta}$ corresponding to the
points $I^{\prime}\in\operatorname{Teich}^{\eta}$ with $\mathop{\text{\rm
rk}}\operatorname{Pic}(M,I^{\prime})>1$. Therefore, for any connected open
subset $U\subset\operatorname{Teich}^{\eta}$, the intersection
$U\cap\operatorname{Teich}^{\eta}_{\text{gen}}$ is connected. Since
$\operatorname{Teich}^{\eta}_{\text{gen}}$ is represented as a disjoint union
of open sets $V_{+}\sqcup V_{-}$, every connected component of
$\operatorname{Teich}^{\eta}_{\text{gen}}$ and of
$\operatorname{Teich}^{\eta}$ is contained in $V_{+}$ or in $V_{-}$. We
obtained the following corollary.
Corollary 1.6: Let $\eta\in H^{2}(M)$ be a positive integer vector,
$\operatorname{Teich}^{\eta}$ the corresponding divisor in the Teichmüller
space, and $\operatorname{Teich}^{\eta,I}$ a connected component of
$\operatorname{Teich}^{\eta}$ containing a complex structure $I$. Assume that
$\eta$ is Kähler on $(M,I)$. Then $\eta$ is Kähler for all
$I^{\prime}\in\operatorname{Teich}^{\eta,I}$ which satisfy $\mathop{\text{\rm
rk}}H^{1,1}(M,\mathbb{Q})=1$.
We call the set $\operatorname{Teich}^{\eta}_{\text{pol}}$ of all
$I\in\operatorname{Teich}^{\eta}$ for which $\eta$ is Kähler the polarized
Teichmüller space, and $\eta$ its polarization. From the above arguments it is
clear that the polarized Teichmüller space is open and dense in
$\operatorname{Teich}^{\eta}$.
The quotient ${\cal M}_{\eta}$ of $\operatorname{Teich}^{\eta}_{\text{pol}}$
by the subgroup of a mapping class group fixing $\eta$ is called the moduli of
polarized hyperkähler manifolds. It is known (due to the general theory which
goes back to Viehweg and Grothendieck that ${\cal M}_{\eta}$ is Hausdorff and
quasiprojective (see e.g. [Vi] and [GHS1]).
We conclude that there are countably many quasiprojective divisors ${\cal
M}_{\eta}$ immersed in the moduli space $\operatorname{Mod}$ of hyperkähler
manifolds. Moreover, every algebraic complex structure belongs to one of these
divisors. However, these divisors need not to be closed. Indeed, as we prove
in this paper, each of the ${\cal M}_{\eta}$ is dense in $\operatorname{Mod}$.
The main result of the present paper is the following theorem.
Theorem 1.7: Let $M$ be a compact, simple hyperkähler manifold,
$\operatorname{Teich}^{I}$ a connected component of its Teichmüller space, and
$\operatorname{Teich}^{I}\stackrel{{\scriptstyle\Psi}}{{{\>\longrightarrow\>}}}\operatorname{Teich}^{I}/\Gamma^{I}=\operatorname{Mod}$
its projection to the moduli of complex structures. Consider a positive vector
$\eta\in H^{2}(M,\mathbb{Z})$, and let $\operatorname{Teich}^{I,\eta}$ be the
corresponding connected component of the polarized Teichmüller space. Assume
that $b_{2}(M)>3$. Then the image $\Psi(\operatorname{Teich}^{I,\eta})$ is
dense in $\operatorname{Mod}$.
We deduce 1.5 from 3 in Section 2, and prove 3 in Section 3.
Remark 1.8: We assumed positivity of $\eta$ in the statement of 1.5, but this
assumption is completely unnecessary. In fact, for $\eta$ non-positive, the
proof of 1.5 becomes easier (3).
## 2 Torelli theorem and polarizations
In this Section, we reduce 1.5 to a statement about lattices and arithmetic
groups, proven in Section 3.
Let $M$ be a topological space, not necessarily Hausdorff. We say that points
$x,y\in M$ are inseparable (denoted $x\sim y$) if for any open subsets $U\ni
x,V\ni y$, one has $U\cap V\neq\varnothing$.
Theorem 2.1 [V2, Theorem 1.14, Theorem 1.16]: Let $\operatorname{Teich}$ be a
Teichmüller space of a hyperkähler manifold, and $\sim$ the inseparability
relation defined above. Then $\sim$ is an equivalence relation, and the
quotient $\operatorname{Teich}_{b}:={\operatorname{Teich}}/{\sim}$ is a
smooth, Hausdorff, complex analytic manifold. Moreover, the period map
$\operatorname{\sf
Per}:\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ induces a complex analytic diffeomorphism
$\operatorname{Teich}_{b}^{I}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ for each connected component $\operatorname{Teich}_{b}^{I}$ of
$\operatorname{Teich}_{b}$.
Remark 2.2: As shown by Huybrechts [H1], inseparable points on a Teichmüller
space correspond to bimeromorphically equivalent hyperkählermanifolds. The
Hausdorff quotient $\operatorname{Teich}_{b}={\operatorname{Teich}}/{\sim}$ is
called the birational Teichmüller space of $M$.
By construction, the action of the mapping class group $\Gamma$ on
$\operatorname{Teich}_{b}$ is compatible with the natural action of
${\mathop{\text{\rm O}}}\big{(}H^{2}(M,\mathbb{Z}),q\big{)}$ on
$\operatorname{{\mathbb{P}}\sf er}$. Define the birational moduli space as
$\operatorname{Mod}_{b}:=\operatorname{Teich}_{b}/\Gamma$. The space
$\operatorname{Mod}_{b}$ is obtained by gluing together some (not all)
inseparable points in $\operatorname{Mod}$. By 2,
$\operatorname{Mod}_{b}=\operatorname{{\mathbb{P}}\sf er}/\Gamma^{I}$, where
$\Gamma^{I}$ is a subgroup of $\Gamma$ fixing a connected component
$\operatorname{Teich}^{I}$ of the Teichmüller space. As follows from [V2,
Theorem 3.5] (see also Subsection 1.3), the image of $\Gamma^{I}$ in
$\operatorname{Aut}(\operatorname{{\mathbb{P}}\sf er})$ is a finite index
subgroup in ${\mathop{\text{\rm O}}}\big{(}H^{2}(M,\mathbb{Z}),q\big{)}$.
It is well known that the homogeneous space
$\operatorname{{\mathbb{P}}\sf er}=\big{\\{}l\in\mathbb{P}H^{2}(M,\mathbb{C})\
\big{|}\ q(l,l)=0,\ q(l,\overline{l})>0\big{\\}}$
is naturally identified with the Grassmanian
${\mathop{\text{\rm
Gr}}}^{++}\big{(}H^{2}(M,\mathbb{R})\big{)}\cong\mathop{\text{\rm
SO}}(3,b_{2}-3)/\mathop{\text{\rm SO}}(2)\times\mathop{\text{\rm
SO}}(1,b_{2}-3)$
of oriented positive 2-dimensional planes in $H^{2}(M,\mathbb{R})$. This
identification is performed as follows: to each line
$l\in\mathbb{P}H^{2}(M,\mathbb{C})$ one associates the plane spanned by
$\operatorname{Re}(l),\operatorname{Im}(l)$. Under this identification, the
image of the polarized Teichmüller space $\operatorname{Teich}^{\eta}$ is the
space of all $2$-dimensional planes $P\in\mathop{\text{\rm
Gr}}^{++}\big{(}H^{2}(M,\mathbb{R})\big{)}$ orthogonal to $\eta$ (see (1.2)).
Then 1.5 is implied by the following statement.
Theorem 2.3: Let $M$ be a simple, compact hyperkähler manifold,
$V:=H^{2}(M,\mathbb{R})$ its second cohomology, $L:=H^{2}(M,\mathbb{Z})$, and
$q$ the Bogomolov-Beauville-Fujiki form on $V$. Given a positive integer
vector $\eta\in L$, denote by $\mathop{\text{\rm
Gr}}^{++}(\eta^{\bot})\subset\mathop{\text{\rm Gr}}^{++}(V)$ the space of all
planes orthogonal to $\eta$. Consider a finite index subgroup
$G\subset\mathop{\text{\rm SO}}\big{(}H^{2}(M,\mathbb{Z}),q\big{)}$ acting on
$\mathop{\text{\rm Gr}}^{++}(V)$ in a natural way. Then
$G\cdot\mathop{\text{\rm Gr}}^{++}(\eta^{\bot})$ is dense in
$\mathop{\text{\rm Gr}}^{++}(V)=\operatorname{{\mathbb{P}}\sf er}$.
2 is implied by a more general 3 proven in the next section using the
framework laid down in [AGr].
Remark 2.4: When $M$ is a K3 surface, the Bogomolov-Beauville-Fujiki form is
unimodular, and the mapping class group is generated by appropriate
reflections. From a statement known as “Eichler’s criterion” (see [GHS2,
Proposition 3.3(i)]), the mapping class group acts transitively on the set of
integer vectors of a given length in $H^{2}(M)$. 2 follows from this
observation easily. When the Eichler’s criterion cannot be applied, its proof
is more complicated.
## 3 Arithmetic subgroups in $\mathop{\text{\rm O}}(p,q)$
Let $V$ be a finite-dimensional $\mathbb{R}$-vector space equipped with a non-
degenerate symmetric form $\langle\cdot,\cdot\rangle$ and $W$ an
$\mathbb{R}$-vector subspace in $V$. Denote by $\mathop{\text{\rm
Gr}}_{++}(W)$ (respectively, by $\mathop{\text{\rm Gr}}_{+-}(W)$) the part of
the Grassmannian $\mathop{\text{\rm Gr}}_{\mathbb{R}}(2,V)$ of $2$-dimensional
$\mathbb{R}$-subspaces in $V$ formed by the subspaces of signature $++$
(respectively, $+-$) in $W$.
Definition 3.1: We shall call a discrete, additive subgroup $L\subset V$ a
lattice if $V=\mathbb{R}\otimes_{\mathbb{Z}}L$ and $\langle
l_{1},l_{2}\rangle\in\mathbb{Q}$ for all $l_{1},l_{2}\in L$. Denote by
$\mathop{\text{\rm O}}(V)$ and $\mathop{\text{\rm O}}(L)$ the corresponding
orthogonal groups:
$\displaystyle\mathop{\text{\rm O}}(V):=$
$\displaystyle\Big{\\{}g\in\mathop{\text{\rm GL}}(V)\ \Big{|}\
\big{\langle}g(v_{1}),g(v_{2})\big{\rangle}=\langle v_{1},v_{2}\rangle\text{
for all }v_{1},v_{2}\in V\Big{\\}},$ $\displaystyle\mathop{\text{\rm O}}(L):=$
$\displaystyle\big{\\{}g\in\mathop{\text{\rm O}}(V)\ \big{|}\
g(L)=L\big{\\}}.$
Clearly, $\mathop{\text{\rm O}}(V)$ acts on $\mathop{\text{\rm Gr}}_{++}(V)$.
For $S\subset V$, we denote
$S^{\perp}:=\big{\\{}v\in V\ \big{|}\ \langle v,S\rangle=0\big{\\}}.$
The purpose of the present section is to prove
Proposition 3.2: Let $V$ be an $\mathbb{R}$-vector space equipped with a non-
degenerate symmetric form of signature $(s_{+},s_{-})$ with $s_{+}\geq 3$ and
$s_{-}\geq 1$. Consider a lattice $L\subset V$. Let $\Gamma$ be a subgroup of
finite index in $\mathop{\text{\rm O}}(L)$, and $l\in L$ a positive vector,
i.e., one which satisfies $\langle l,l\rangle>0$. Then
$\Gamma\cdot\mathop{\text{\rm Gr}}_{++}(l^{\perp})$ is dense in
$\mathop{\text{\rm Gr}}_{++}(V)$.
The proof of 3 takes the rest of this Section.
Proof of 3: Step 1: We reduce 3 to a case of a space $V$ of signature $(3,1)$.
A subspace $W\subset V$ is called rational if $\mathop{\text{\rm rk}}(W\cap
L)=\dim_{\mathbb{R}}W$ or, equivalently, if $W=\mathbb{R}W_{0}$ with a
$\mathbb{Q}$-subspace $W_{0}\subset\mathbb{Q}L$. Since the rational subspaces
are dense in $\mathop{\text{\rm Gr}}_{++}(V)$, it suffices to show that an
arbitrary rational $2$-plane $C\in\mathop{\text{\rm Gr}}_{++}(V)$ belongs to
the closure of $\Gamma\cdot\mathop{\text{\rm Gr}}_{++}(l^{\perp})$. We have
$C=\mathbb{R}C_{0}$ for some $\mathbb{Q}$-subspace $C_{0}\subset\mathbb{Q}L$.
Obviously, $\mathbb{Q}L$ has signature $(s_{+},s_{-})$. Applying to
$\mathbb{Q}$-subspaces in $\mathbb{Q}L$ the standard orthogonalization
arguments, we can find a $\mathbb{Q}$-subspace $U_{0}\subset\mathbb{Q}L$ of
signature $+++-$ that contains both $l$ and $C_{0}$. Indeed, we have
$l=c_{0}+c_{1}$, where $c_{0}\in C_{0}$ and $c_{1}\in\mathbb{Q}L\cap
C_{0}^{\perp}$ with $\mathbb{Q}L\cap C_{0}^{\perp}$ of signature
$(s_{+}-2,s_{-})$. We can always pick a $2$-dimensional $\mathbb{Q}$-subspace
$C_{1}\subset\mathbb{Q}L\cap C_{0}^{\perp}$ of signature $+-$ that includes
$c_{1}$ and put $U_{0}:=C_{0}\oplus C_{1}$. So, $U:=\mathbb{R}U_{0}$ is
rational of signature $+++-$ and $L_{0}:=U\cap L$ is a lattice in $U$. To
prove that the $2$-plane $C$ belongs to the closure of
$\Gamma\cdot\mathop{\text{\rm Gr}}_{++}(l^{\perp})$, it would suffice to show
that the set $(\Gamma\cap\Gamma^{\prime})\cdot\mathop{\text{\rm
Gr}}_{++}(l^{\perp}\cap U)$ is dense in the corresponding $++$-Grassmannian
$\mathop{\text{\rm Gr}}_{++}(U)$, where $\Gamma^{\prime}:=\mathop{\text{\rm
O}}(L_{0})$.
Step 2: We prove that the orthogonal groups $\mathop{\text{\rm O}}(L)$ and
$\mathop{\text{\rm O}}(L^{\prime})$ are commensurable, i.e., the subgroup
$\mathop{\text{\rm O}}(L)\cap\mathop{\text{\rm O}}(L^{\prime})$ has finite
index in $\mathop{\text{\rm O}}(L)$ and in $\mathop{\text{\rm
O}}(L^{\prime})$, if lattices $L,L^{\prime}\subset V$ are commensurable.
Taking $L\cap L^{\prime}$ for $L^{\prime}$, we can assume that $mL\subset
L^{\prime}\subset L$ for some $0\neq m\in\mathbb{Z}$. Put
$\overline{L^{\prime}}:=L^{\prime}/mL\subset\overline{L}:=L/mL$ and note that
$\mathop{\text{\rm O}}(L)$ acts on $\overline{L}$ because $\mathop{\text{\rm
O}}(L)=\mathop{\text{\rm O}}(mL)$. We can see that the group
$\mathop{\text{\rm O}}(L)\cap\mathop{\text{\rm
O}}(L^{\prime})=\big{\\{}g\in\mathop{\text{\rm O}}(L)\ \big{|}\
g(L^{\prime})=L^{\prime}\big{\\}}$ coincides with the stabilizer
$\operatorname{St}_{\mathop{\text{\rm O}}(L)}\overline{L^{\prime}}$. Hence,
$\mathop{\text{\rm O}}(L)\cap\mathop{\text{\rm O}}(L^{\prime})$ has finite
index in $\mathop{\text{\rm O}}(L)$. Since $m(\frac{1}{m}L^{\prime})\subset
L\subset\frac{1}{m}L^{\prime}$ and $\mathop{\text{\rm
O}}(\frac{1}{m}L^{\prime})=\mathop{\text{\rm O}}(L^{\prime})$, we infer as
well that $\mathop{\text{\rm O}}(L)\cap\mathop{\text{\rm O}}(L^{\prime})$ has
finite index in $\mathop{\text{\rm O}}(L^{\prime})$.
Step 3: Let $W\subset V$ be a rational non-degenerate subspace. Then we have
an orthogonal decomposition $V=W\oplus W^{\perp}$ and $W^{\perp}$ is rational.
Define $L_{0}:=W\cap L$, $L_{1}:=W^{\perp}\cap L$, and
$L^{\prime}:=L_{0}+L_{1}$. It is immediate that $L^{\prime}$ is a lattice in
$V$ such that $\mathbb{Q}L^{\prime}=\mathbb{Q}L$. By Step 2, the orthogonal
groups $\mathop{\text{\rm O}}(L)$ and $\mathop{\text{\rm O}}(L^{\prime})$ are
commensurable. Since $\mathop{\text{\rm O}}(L_{0})\times\mathop{\text{\rm
O}}(L_{1})\subset\mathop{\text{\rm O}}(L^{\prime})$, there exists a subgroup
$\Gamma_{0}$ of finite index in $\mathop{\text{\rm O}}(L_{0})$ such that
$\Gamma_{0}\subset\Gamma$.
Step 4: We reduce 3 to 3 below.
Applying Steps 1 and 3, we can assume that $(s_{+},s_{-})=(3,1)$. Indeed, by
Step 1, we need only to show that
$(\Gamma\cap\Gamma^{\prime})\cdot\mathop{\text{\rm Gr}}_{++}(l^{\perp}\cap U)$
is dense in $\mathop{\text{\rm Gr}}_{++}(U)$, where
$\Gamma^{\prime}:=\mathop{\text{\rm O}}(L_{0})$, $L_{0}:=U\cap L$, and
$U\subset V$ is a rational subspace of signature $+++-$. Taking $W:=U$ in Step
3, we find a subgroup $\Gamma_{0}$ of finite index in $\Gamma^{\prime}$ such
that $\Gamma_{0}\subset\Gamma$.
Now using the homeomorphism $\mathop{\text{\rm
Gr}}_{++}(V)\to\mathop{\text{\rm Gr}}_{+-}(V)$, $G\mapsto G^{\perp}$, i.e.,
taking instead of subspaces of signature $++$, their orthogonal complements
(of signature $+-$), we reformulate 3 as follows:
Every rational $G_{0}\in\mathop{\text{\rm Gr}}_{+-}(V)$ belongs to
the closure of $\Gamma\cdot\big{\\{}G\in\mathop{\text{\rm Gr}}_{+-}(V)\
\big{|}\ G\ni l\big{\\}}$.
The subspace $W$ spanned by $l,G_{0}$ is rational of signature $++-$. Again
using Step 3, we reduce 3 to
Lemma 3.3: Let $V$ be an $\mathbb{R}$-vector space equipped with a symmetric
form of signature $++-$, $\Gamma$ a subgroup of finite index in
$\mathop{\text{\rm O}}(L)$, where $L$ is a lattice in $V$, and $l\in V$ a
positive vector. Then $\Gamma\cdot\big{\\{}G\in\mathop{\text{\rm Gr}}_{+-}(V)\
\big{|}\ G\ni l\big{\\}}$ is dense in $\mathop{\text{\rm Gr}}_{+-}(V)$.
Till the end of this Section we fix $\Gamma$ as in 3.
In fact, we deal now with a hyperbolic plane
$\overline{\mathbb{H}}_{\mathbb{R}}^{2}=\mathbb{H}_{\mathbb{R}}^{2}\sqcup\partial\mathbb{H}_{\mathbb{R}}^{2}$.
Let us state in 3, 3, and 3 a few simple and well-known facts concerning the
hyperbolic plane (see e.g. [AGr]).
Claim 3.4: The plane $\overline{\mathbb{H}}_{\mathbb{R}}^{2}$ can be
identified with the set of all nonpositive points in the real projective plane
$\mathbb{P}_{\mathbb{R}}V$, where the isotropic ones form the absolute
$\partial\mathbb{H}_{\mathbb{R}}^{2}$. In the affine chart related to
orthonormal coordinates on $V$, the plane
$\overline{\mathbb{H}}_{\mathbb{R}}^{2}$ is nothing but a closed unitary disc.
In this way, we obtain the Beltrami-Klein model of a hyperbolic plane, where
geodesics are chords of the disc. In other words, we can describe a geodesic
in $\overline{\mathbb{H}}_{\mathbb{R}}^{2}$ as the projectivization
$\mathbb{P}_{\mathbb{R}}G\cap\overline{\mathbb{H}}_{\mathbb{R}}^{2}$ of a
subspace $G\in\mathop{\text{\rm Gr}}_{+-}(V)$. We keep denoting this geodesic
by $G$. Of course, every geodesic $G$ can be described via its vertices
$v,v^{\prime}\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ as $G=[v,v^{\prime}]$. In
terms of $V$, this means that the $\mathbb{R}$-vector subspace $G$ is spanned
by $v,v^{\prime}$.
Claim 3.5: Let $G^{\prime}\subset\overline{\mathbb{H}}_{\mathbb{R}}^{2}$ be a
geodesic not passing through a point
$v\in\partial\mathbb{H}_{\mathbb{R}}^{2}$, i.e., $v\notin G^{\prime}$. Then,
reflecting $v$ in $G^{\prime}$, we obtain a point
$v^{\prime}\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ such that the geodesics
$G^{\prime}$ and $[v,v^{\prime}]$ are orthogonal.
Claim 3.6: The group $\mathop{\text{\rm O}}(V)$ acts naturally on
$\overline{\mathbb{H}}_{\mathbb{R}}^{2}$. On $\mathbb{H}_{\mathbb{R}}^{2}$,
the group $\mathop{\text{\rm O}}(V)$ acts by isometries.
We can now reduce 3 to the following statement about the hyperbolic plane:
Lemma 3.7: Let $G^{\prime}$ be a geodesic on the hyperbolic plane
$\mathbb{H}_{\mathbb{R}}^{2}$, and $\Gamma\cdot G^{\prime}$ the set of all
geodesics obtained from $G^{\prime}$ by the action of $\Gamma$. Then the set
of all geodesics orthogonal to some $G^{\prime\prime}\in\Gamma\cdot
G^{\prime}$ is dense in the set of all geodesics in
$\mathbb{H}_{\mathbb{R}}^{2}$.
Reduction of 3 to 3. Let $G^{\prime}$ be the orthogonal complement of
$l\in\mathbb{H}_{\mathbb{R}}^{2}\subset{\mathbb{P}}V$, considered as a
geodesic in $\mathbb{H}_{\mathbb{R}}^{2}$. It is easy to see that the
inclusion $G\ni l$ is equivalent to the fact that the geodesics $G$ and
$G^{\prime}:=l^{\perp}$ are orthogonal (see, for instance, the duality
described in the introductory [AGr, Section 1] shortly after Example 1.7). For
this choice of $G^{\prime}$, 3 is clearly equivalent to 3.
We reduce 3 further, obtaining a simpler statement about the hyperbolic plane:
Lemma 3.8: Let $v,v^{\prime}\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ be
distinct points on the absolute and $G^{\prime}$ a geodesic. For every
$\gamma\in\Gamma$, denote by $R_{\gamma}$ the reflection in the geodesic
$\gamma(G^{\prime})$. Then $v^{\prime}$ belongs to the closure of the set
$\big{\\{}R_{\gamma}(v)\ \big{|}\ \gamma\in\Gamma,\
v\notin\gamma(G^{\prime})\big{\\}}$ formed by the reflections of $v$ in those
geodesics $\gamma(G^{\prime})$ that do not pass through $v$.
Reduction of 3 to 3: By 3, the geodesic $[v,R_{\gamma}(v)]$ is orthogonal to
$\gamma(G^{\prime})$. To prove 3, it suffices to show that the set of such
geodesics is dense in the set of all geodesics of the form $[v,v^{\prime}]$,
where $v$ is fixed. 3 says that we are able to approximate $v^{\prime}$ by
$R_{\gamma}(v)$ for an appropriate $\gamma\in\Gamma$. Hence, we can
approximate the geodesic $[v,v^{\prime}]$ by geodesics orthogonal to some
$G^{\prime\prime}\in\Gamma\cdot G^{\prime}$.
We deduce 3 from two easy lemmas below, 3 and 3. First, we need a few more
simple and well-known facts concerning the hyperbolic plane:
$\bullet$ The nontrivial orientation-preserving isometries of
$\mathbb{H}_{\mathbb{R}}^{2}$ are classified with respect to the location of
their fixed points: an elliptic one has a (unique) fixed point in
$\mathbb{H}_{\mathbb{R}}^{2}$; a hyperbolic one has exactly two fixed points
on the absolute; and a parabolic one has exactly one fixed point on the
absolute.
$\bullet$ Let $p\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ be the fixed point of
a parabolic isometry $\gamma$ and let
$v\in\partial\mathbb{H}_{\mathbb{R}}^{2}$. Then $\gamma^{n}(v)\to p$ as
$n\to\infty$.
$\bullet$ The fixed points in $\partial\mathbb{H}_{\mathbb{R}}^{2}$ of a
hyperbolic isometry $\gamma$ are the repeller $p_{1}$ and the attractor
$p_{2}$. This means that, for every $v\in\partial\mathbb{H}_{\mathbb{R}}^{2}$
such that $v\neq p_{1}$, we have $\gamma^{n}(v)\to p_{2}$ as $n\to\infty$.
When taking $\gamma^{-1}$ in place of $\gamma$, the repeller becomes the
attractor and vice versa.
We arrive at the following remark needed in the proof of 3.
Remark 3.9: Let $\gamma$ be a hyperbolic or parabolic isometry,
$p\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ a fixed point of $\gamma$, and
$u,u^{\prime}\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ points not fixed by
$\gamma$. Then, for $n\to\infty$ or for $n\to-\infty$, both limits
$\lim\gamma^{n}(u)$ and $\lim\gamma^{n}(u^{\prime})$ exist and are equal to
$p$.
Lemma 3.10: The set $F:=\big{\\{}p\in\partial\mathbb{H}_{\mathbb{R}}^{2}\
\big{|}\ \gamma(p)=p\text{ \rm for some }1\neq\gamma\in\Gamma\big{\\}}$ of
points on the absolute fixed by some nontrivial $\gamma\in\Gamma$ is dense in
$\partial\mathbb{H}_{\mathbb{R}}^{2}$.
Proof: Suppose that there exists an open arc
$A\subset\partial\mathbb{H}_{\mathbb{R}}^{2}$ such that $A\cap F=\varnothing$.
By the Zorn lemma, we can take maximal $A$ with this property. By
construction, $\gamma(A)$ also enjoys the property of the maximality for every
$\gamma\in\Gamma$. Every point on the boundary $\partial A$ belongs to the
closure of $F$. Let $\gamma\in\Gamma$. Then $A\cap\gamma(A)=\varnothing$ or
$A=\gamma(A)$ because otherwise $\gamma(A)$ contains an open neighbourhood of
one end of $A$, which intersects $F$.
Due to B. A. Venkov (see [VGSh, Example 7.5, p. 33]), $\mathop{\text{\rm
O}}(L)$ is known to act discretely on $\mathbb{H}_{\mathbb{R}}^{2}$, is
finitely generated, and is of finite coarea. Note that Selberg’s Theorem
[VGSh, Theorem 3.2, p. 18] claims that every finitely generated matrix group
over a field of characteristic $0$ has a subgroup of finite index without
torsion. Therefore, we can at the very beginning pass to a torsion-free
subgroup of finite index in $\Gamma$ thus assuming that all isometries in
$\Gamma$ are orientation-preserving and that there are no elliptic isometries
in $\Gamma$.
Let $\partial A=\\{p,p^{\prime}\\}$. Since $\Gamma$ has no elliptic isometries
and all isometries in $\Gamma$ are orientation-preserving, the stabilizer
$\Gamma^{\prime\prime}:=\operatorname{St}_{\Gamma}A$ of $A$ in $\Gamma$ is a
discrete
group of orientation-preserving isometries of the geodesic $[p,p^{\prime}]$.
Hence, $\Gamma^{\prime\prime}$ is cyclic, generated by some $\gamma_{0}\neq
1$. Let $G$ be a geodesic perpendicular to $[p,p^{\prime}]$. Consider the open
region $D\subset\mathbb{H}_{\mathbb{R}}^{2}$ limited by $A\cup
G\cup[p,p^{\prime}]\cup\gamma_{0}(G)$. It is easy to see that
$D\cap\gamma(D)=\varnothing$ for every $1\neq\gamma\in\Gamma^{\prime\prime}$.
For any $\gamma\in\Gamma\setminus\Gamma^{\prime\prime}$, we have
$A\cap\gamma(A)=\varnothing$, which again implies
$D\cap\gamma(D)=\varnothing$. Therefore, $D$ is a part of a fundamental domain
for $\Gamma$. Since the area of $D$ is infinite, we arrive at a contradiction.
Lemma 3.11: Let $u,u^{\prime}\in\partial\mathbb{H}_{\mathbb{R}}^{2}$ be
distinct points. Then there exists a hyperbolic or parabolic
$\gamma_{0}\in\Gamma$ such that $\gamma_{0}(u)\neq u$ and
$\gamma_{0}(u^{\prime})\neq u^{\prime}$.
Proof: As in the proof of 3, we assume $\Gamma$ torsion-free. Suppose that
$\gamma_{0}(u)=u$ or $\gamma_{0}(u^{\prime})=u^{\prime}$ for every
$\gamma_{0}\in\Gamma$. If $\gamma,\gamma^{\prime}\in\Gamma$ fix respectively
$u,u^{\prime}$ and do not fix respectively $u^{\prime},u$, then
$\gamma\gamma^{\prime}$ does not fix both $u$ and $u^{\prime}$. Therefore, we
can assume that $\gamma(u)=u$ for all $\gamma\in\Gamma$. It is well known
(consider the upper half-plane model with $u=0$) that the group of all
orientation-preserving isometries of $\mathbb{H}_{\mathbb{R}}^{2}$ is
isomorphic to $\mathop{\text{\rm PSL}}_{2}(\mathbb{R})$ and that
$S:=\operatorname{St}_{\mathop{\text{\rm
PSL}}_{2}(\mathbb{R})}u\simeq\left\\{\left[\begin{smallmatrix}\alpha&0\\\
a&\alpha^{-1}\end{smallmatrix}\right]\Big{|}\ a,\alpha\in\mathbb{R},\
\alpha>0\right\\}$. Since $S$ is Zariski closed, the inclusion $\Gamma\subset
S$ would contradict the Borel density theorem [VGSh, Theorem 8.2, p. 37] which
implies that $\Gamma$ should be Zariski dense in $\mathop{\text{\rm
PSL}}_{2}(\mathbb{R})$.
Proof of 3: For suitable distinct points
$u,u^{\prime}\in\partial\mathbb{H}_{\mathbb{R}}^{2}$, the geodesic
$G^{\prime}$ in 3 has the form $G^{\prime}=[u,u^{\prime}]$.
Let $A$ be a small connected open neighbourhood of $v^{\prime}$ in
$\partial\mathbb{H}_{\mathbb{R}}^{2}$. In other words,
$A\subset\partial\mathbb{H}_{\mathbb{R}}^{2}$ is an open arc containing
$v^{\prime}$ and not containing $v$. By 3, for a suitable $p\in A\cap F$ and
for some $1\neq\gamma\in\Gamma$, we have $\gamma(p)=p$.
We consider two cases. The first case: $u,u^{\prime}$ are not fixed by
$\gamma$. Then, taking into account that $\gamma$ is hyperbolic or parabolic,
we conclude by 3 that $\gamma^{n}(u)\to p$ and $\gamma^{n}(u^{\prime})\to p$
for $n\to\infty$ or for $n\to-\infty$. Hence,
$\gamma^{n}(u),\gamma^{n}(u^{\prime})\in A$ for some $n\in\mathbb{Z}$.
Therefore, $R_{\gamma^{n}}(v)\in A$.
The second case: one of $u,u^{\prime}$ is fixed by $\gamma$. By 3, there
exists $\gamma_{0}\in\Gamma$ such that the points
$\gamma_{0}(u),\gamma_{0}(u^{\prime})$ are not fixed by $\gamma$. Now, by 3,
we have $\gamma^{n}\gamma_{0}(u)\to p$ and
$\gamma^{n}\gamma_{0}(u^{\prime})\to p$ for $n\to\infty$ or for $n\to-\infty$.
This implies $R_{\gamma^{n}\gamma_{0}}(v)\in A$ for some $n\in\mathbb{Z}$.
For an arbitrarily small open arc $A$ containing $v^{\prime}$, we found, in
either case, some $\gamma^{\prime}\in\Gamma$ such that
$R_{\gamma^{\prime}}(v)\in A$ and $v\notin\gamma^{\prime}(G^{\prime})$. This
implies 3.
Remark 3.12: We stated 3 in assumption that $(l,l)>0$ (this assumption was
geometrically motivated). But, in fact, this assumption is completely
unnecessary. Moreover, as the following result implies, the proof of 3 becomes
much easier when $(l,l)\leqslant 0$.
Proposition 3.13: The condition $\langle l,l\rangle>0$ in 3 is unnecessary.
Proof: To see this, we repeat the proof of 3 literally until 3. To obtain 3,
we need to check a version of 3 when the vector $l$ is not assumed to be
positive.
We reduce the case of $\langle l,l\rangle<0$ to the case of $\langle
l,l\rangle=0$. Let $l_{0}$ be a limit point of the orbit $\Gamma\cdot l$.
Since $\Gamma$ is a discrete subgroup in $\mathop{\text{\rm
PSL}}_{2}(\mathbb{R})$, this limit lies on the absolute, and we have $\langle
l_{0},l_{0}\rangle=0$. It suffices to show that any geodesic $G$ passing
through $l_{0}$ lies in the closure of the set of all geodesics $G^{\prime}$
that pass through $\gamma(l)$ for some $\gamma\in\Gamma$. For a given point
$\gamma(l)$, we denote by $G^{\prime}_{\gamma}$ the Euclidean parallel to $G$
passing through $\gamma(l)$. For this choice of $G^{\prime}_{\gamma}$, the
limit $\gamma(l)\to l_{0}$ implies the limit $G^{\prime}_{\gamma}\to G$.
It remains now to prove 3 when $\langle l,l\rangle=0$. Since $F$ is dense in
$\partial\mathbb{H}_{\mathbb{R}}^{2}$ by 3, the subset $\Gamma\cdot l$ is also
dense in $\partial\mathbb{H}_{\mathbb{R}}^{2}$. So, fixing one end of an
arbitrary geodesic $G\in\mathop{\text{\rm Gr}}_{+-}(V)$, we can approximate
the other one by a point in $\Gamma\cdot l$.
Acknowledgements: We are grateful to Eyal Markman and Vyacheslav Nikulin for
an email correspondence.
## References
* [AGr] Anan′in, S., Grossi, C. H., Coordinate-free classic geometries, Mosc. Math. J. 11 (2011), 633–655, arXiv:math/0702714
* [Bea] Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.
* [Bes] Besse, A. L., Einstein Manifolds, Springer-Verlag, Berlin, Heidelberg, New York 1987, 516 pp.
* [Bo1] Bogomolov, F. A., On the decomposition of Kähler manifolds with trivial canonical class, Matt. USSR-Sb. 22 (1974), no. 4, 580–583.
* [Bo2] Bogomolov, F. A., Hamiltonian Kähler manifolds, Sov. Math. Dokl. 19 (1978), 1462–1465.
* [Bou] Boucksom, S., Higher dimensional Zariski decompositions, Ann. Sci. Ecole Norm. Sup. (4)37 (2004), no. 1, 45–76, arXiv:math/0204336
* [DP] Demailly, J.-P., Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. 159 (2004), 1247–1274, arXiv:math/0105176
* [F] Fujiki, A., On the de Rham Cohomology Group of a Compact Kähler Symplectic Manifold, Adv. Stud. Pure Math. 10 (1987), 105–165.
* [GHS1] Gritsenko, V., Hulek, K., Sankaran, G. K., Moduli spaces of irreducible symplectic manifolds, Compositio Mathematica 146 (2010), no. 2, 404–434, arXiv:0802.2078
* [GHS2] Gritsenko, V., Hulek, K., Sankaran, G. K., Abelianisation of orthogonal groups and the fundamental group of modulae varieties, arXiv:0810.1614, 21 pp.
* [H1] Huybrechts, D., Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), no. 1, 63–113, arXiv:alg-geom/9705025
* [H2] Huybrechts, D., Finiteness results for hyperkähler manifolds, J. Reine Angew. Math. 558 (2003), 15–22, arXiv:math/0109024
* [KS] Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrations, arXiv:math/0011041, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 203-263.
* [M] Markman, E., Integral constraints on the monodromy group of the hyperkähler resolution of a symmetric product of a K3 surface, International Journal of Mathematics 21 (2010), no. 2, 169–223, arXiv:math/0601304
* [T] Todorov, A., Ray Singer Analytic Torsion of CY Manifolds II, arXiv: math/0004046, 18 pp.
* [V0] Verbitsky, M., Algebraic structures on hyperkähler manifolds, Math. Res. Lett., 3 (1996), 763–767.
* [V1] Verbitsky, M., Trianalytic subvarieties of hyperkaehler manifolds, GAFA vol. 5 no. 1 (1995) pp. 92-104.
* [V2] Verbitsky, M., A global Torelli theorem for hyperkähler manifolds, arXiv: 0908.4121, 47 pp.
* [V3] Verbitsky, M., Hyperkahler SYZ conjecture and semipositive line bundles, arXiv:0811.0639, GAFA 19, No. 5, 1481-1493 (2010)
* [V4] Verbitsky, M., Parabolic nef currents on hyperkaehler manifolds, arXiv:0907.4217, 19 pages.
* [VGSh] Vinberg, E. B., Gorbatsevich, V. V., Shvartsman, O. V., Discrete Subgroups of Lie Groups, Encyclopaedia of Mathematical Sciences 21, Springer-Verlag, 2000, 224 pp.
* [Vi] Viehweg, E., Quasi-projective Moduli for Polarized Manifolds, Springer-Verlag, Berlin, Heidelberg, New York, 1995, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 30, also available at http://www.uni-due.de/$\widetilde{\phantom{a}}$mat903/books.html
Misha Verbitsky
Laboratory of Algebraic Geometry, NRU-HSE,
7 Vavilova Str. Moscow, Russia, 117312
verbit@maths.gla.ac.uk, verbit@mccme.ru
Sasha Anan′in
IMECC – UNICAMP, Departamento de Matemática,
Caixa Postal 6065 13083-970 Campinas-SP, Brazil
ananin_sasha@yahoo.com
|
arxiv-papers
| 2010-08-14T21:29:05 |
2024-09-04T02:49:12.195714
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sasha Anan'in, Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1008.2480"
}
|
1008.2505
|
# Introduction to co-split Lie algebras
Limeng Xia
Faculty of Science, Jiangsu University
Zhenjiang 212013, Jiangsu Province, P.R. China
Naihong Hu111Corresponding author: nhhu@math.ecnu.edu.cn.
Department of Mathematics, East China Normal University
Minhang Campus, Dongchuan Road 500, Shanghai 200241, P.R. China
###### Abstract
In this work, we introduce a new concept which is obtained by defining a new
compatibility condition between Lie algebras and Lie coalgebras. With this
terminology, we describe the interrelation between the Killing form and the
adjoint representation in a new perspective.
## 1 Introduction
During the past decade, there have appeared a number of papers on the study of
Lie bialgebras (see [EK], [ES] and references therein, etc). It is well-known
that a Lie bialgebra is a vector space endowed simultaneously with a Lie
algebra structure and a Lie coalgebra structure, together with a certain
compatibility condition, which was suggested by a study of Hamiltonian
mechanics and Poisson Lie groups ([ES]).
In the present work, we consider a new [Lie algebra]-[Lie coalgebra]
structure, say, a co-split Liealgebra. Using this concept, we can easily study
the Lie algebra structure on the dual space of a semi-simple Lie algebra from
another point of view.
This paper is arranged as follows: At first we recall some concepts and study
the relations between Lie algebras and Lie coalgebras. Then we give the
definition of a co-split Lie algebra. In section 4, we prove that
$sl_{n+1}(\mathbf{C})$ is a co-split Lie algebra. Then we discuss the
interrelation of the Killing form and the adjoint representation of
$sl_{n+1}(\mathbf{C})$. Finally, the results are proved to hold for all finite
dimensional complex semi-simple Lie algebras.
## 2 Basics
In this section, we mainly recall the definitions of Lie algebras, Lie
coalgebras and Lie bialgebras, and also their relationship. For more
information, one can see [EK], [ES] and references therein.
A Lie algebra is a pair $(L,[,])$, where $L$ is a linear space and
$[,]:L\times L\longrightarrow L$ is a bilinear map (in fact, it is a linear
map from $L\otimes L$ to $L$) satisfying:
(L1) [a,b]+[b,a]=0, (L2) [a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0.
For any spaces $U,V,W$, define maps
$\displaystyle\tau:$ $\displaystyle U\otimes V\longrightarrow V\otimes U$
$\displaystyle u\otimes v\longmapsto v\otimes u,$ $\displaystyle\xi:$
$\displaystyle U\otimes V\otimes W\longrightarrow V\otimes W\otimes U$
$\displaystyle u\otimes v\otimes w\longmapsto v\otimes w\otimes u.$
A Lie coalgebra is a pair $(L,\delta)$, where $L$ is a linear space and
$\delta:L\longrightarrow L\otimes L$ is a linear map satisfying:
(Lc1) | $(1+\tau)\circ\delta=0$,
---|---
(Lc2) | $(1+\xi+\xi^{2})\circ(1\otimes\delta)\circ\delta=0$.
A Lie bialgebra is a triple $(L,[,],\delta)$ such that
(Lb1) | $(L,[,])$ is a Lie algebra,
---|---
(Lb2) | $(L,\delta)$ is a Lie coalgebra,
(Lb3) | For any $x,y\in L$, $\delta([x,y])=x\cdot\delta(y)-y\cdot\delta(x)$.
The compatibility condition (Lb3) shows that $\delta$ is a derivation map.
In the following lemmas, $c$ is an arbitrary constant.
###### Lemma 2.1
For any finite dimensional Lie algebra $(L,[,])$, the dual space $L^{*}$ has a
Lie coalgebra structure defined by
$\delta_{L^{*}}(f^{*})=\sum_{(f)}f_{1}\otimes f_{2}:x\otimes y\longmapsto
f_{1}(x)f_{2}(y)=cf^{*}([x,y]).$
###### Lemma 2.2
For any finite dimensional Lie coalgebra $(L,\delta)$, the dual space $L^{*}$
has a Lie algebra structure defined by
$[f^{*},g^{*}]:x\longmapsto cf^{*}(x_{1})g^{*}(x_{2}),$
where $\delta(x)=\sum_{(x)}x_{1}\otimes x_{2}.$
These two lemmas are natural conclusions and easy to be verified.
## 3 What is a co-split Lie
###### Definition 3.1
Suppose that $(L,[\cdot,\cdot])$ is a Lie algebra and $(L,\delta)$ is a Lie
coalgebra. A triple $(L,[\cdot,\cdot],\delta)$ is called a co-split Lieif the
compatibility condition
$[\cdot,\cdot]\circ\delta=\hbox{\rm id}_{L}$
holds, and $\delta$ is called a co-splitting of $L$.
If in the compatibility condition, $\hbox{\rm id}_{L}$ is replaced by a non-
degenerate diagonal matrix, then $(L,[,],\delta)$ is called a weak co-split
Liealgebra and $\delta$ is called a weak co-splitting .
###### Remark 3.1
Obviously, a co-split Lie$L$ should satisfies $[L,L]=L$.
###### Remark 3.2
If $(L,[,],\delta)$ is a finite dimensional (weak) co-split Liealgebra, so is
$(L^{*},\delta^{*},[,]^{*})$, where
$\displaystyle\delta^{*}(f\otimes g)(x)$ $\displaystyle=$
$\displaystyle(f\otimes g)\delta(x),$ $\displaystyle{[,]}^{*}(f)(x\otimes y)$
$\displaystyle=$ $\displaystyle f({[x,y]}),$
for all $x,y\in L$ and $f,g\in L^{*}$. This follows from the fact that
$V\longrightarrow V^{*}$ is a contravariant functor.
## 4 Co-split Lie algebras of type $A$
Suppose that $L$ is a complex simple Lie algebra of type $A_{n}$, then it can
be realized as the special linear Lie algebra $sl_{n+1}(\mathbf{C})$ with
basis
$\\{E_{i,j},E_{j,i},E_{i,i}-E_{j,j}\mid 1\leq i<j\leq n+1\\}.$
The Lie bracket is the commutator
$[E_{i,j},E_{k,l}]=\delta_{j,k}E_{i,l}-\delta_{l,i}E_{k,j}.$
Define a linear map $\delta:sl_{n+1}(\mathbf{C})\longrightarrow
sl_{n+1}(\mathbf{C})\otimes sl_{n+1}(\mathbf{C})$ as
$\displaystyle\delta(E_{i,j})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k=1}^{n+1}(E_{i,k}\otimes
E_{k,j}-E_{k,j}\otimes E_{i,k}).$
###### Proposition 4.1
$\delta$ is well-defined.
Proof. Assume that $i\not=j$, then
$\displaystyle\delta(E_{i,j})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k=1}^{n+1}(E_{i,k}\otimes
E_{k,j}-E_{k,j}\otimes E_{i,k})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k\not=i,j}(E_{i,k}\otimes
E_{k,j}-E_{k,j}\otimes E_{i,k})$
$\displaystyle+\frac{1}{2n+2}[(E_{i,i}-E_{j,j})\otimes
E_{i,j}-E_{i,j}\otimes(E_{i,i}-E_{j,j})].$
$\displaystyle\delta(E_{i,i}-E_{j,j})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k=1}^{n+1}(E_{i,k}\otimes
E_{k,i}-E_{j,k}\otimes E_{k,j})$
$\displaystyle-\frac{1}{2n+2}\sum_{k=1}^{n+1}(E_{k,i}\otimes
E_{i,k}-E_{k,j}\otimes E_{j,k})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k\not=i}(E_{i,k}\otimes
E_{k,i}-E_{k,i}\otimes E_{i,k})$
$\displaystyle-\frac{1}{2n+2}\sum_{k\not=j}(E_{j,k}\otimes
E_{k,j}-E_{k,j}\otimes E_{j,k}).$
Hence $\delta$ is well-defined. $\square$
###### Theorem 4.1
$(sl_{n+1}(\mathbf{C}),\delta)$ is a Lie coalgebra.
Proof. At first, it is clear that $(1+\tau)\circ\delta=0$. By a direct
calculation, we have
$\displaystyle((1\otimes\delta)\circ\delta)(E_{i,j})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}(1\otimes\delta)\left(\sum_{k=1}^{n+1}(E_{i,k}\otimes
E_{k,j}-E_{k,j}\otimes E_{i,k})\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4(n+1)^{2}}\sum_{1\leq k,l\leq n+1}(E_{i,k}\otimes
E_{k,l}\otimes E_{l,j}-E_{i,k}\otimes E_{l,j}\otimes E_{k,l})$
$\displaystyle-\frac{1}{4(n+1)^{2}}\sum_{1\leq k,l\leq n+1}(E_{k,j}\otimes
E_{i,l}\otimes E_{l,k}-E_{k,j}\otimes E_{l,k}\otimes E_{i,l})$
$\displaystyle=$ $\displaystyle\frac{1}{4(n+1)^{2}}\sum_{1\leq k,l\leq
n+1}(E_{i,k}\otimes E_{k,l}\otimes E_{l,j}-E_{l,j}\otimes E_{i,k}\otimes
E_{k,l})$ $\displaystyle-\frac{1}{4(n+1)^{2}}\sum_{1\leq k,l\leq
n+1}(E_{i,l}\otimes E_{k,j}\otimes E_{l,k}-E_{k,j}\otimes E_{l,k}\otimes
E_{i,l}).$
Hence,
$(1+\xi+\xi^{2})\circ(1\otimes\delta)\circ\delta=0,$
that is, $\delta$ satisfies the anti-symmetriy property and the Jacobi
identity. Then $(sl_{n+1}(\mathbf{C}),\delta)$ is a Lie coalgebra. $\square$
###### Theorem 4.2
$(sl_{n+1}(\mathbf{C}),[\cdot,\cdot],\delta)$ is a co-split Liealgebra.
Proof. For $i\not=j$, it is easy to check that
$\displaystyle([\cdot,\cdot]\circ\delta)(E_{i,j})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k=1}^{n+1}([E_{i,k},E_{k,j}]-[E_{k,j},E_{i,k}])$
$\displaystyle=$ $\displaystyle E_{i,j},$
$\displaystyle([\cdot,\cdot]\circ\delta)(E_{i,i}-E_{j,j})$ $\displaystyle=$
$\displaystyle\frac{1}{2n+2}\sum_{k=1}^{n+1}([E_{i,k},E_{k,i}]-[E_{k,i},E_{i,k}])$
$\displaystyle-\frac{1}{2n+2}\sum_{k=1}^{n+1}([E_{j,k},E_{k,j}]-[E_{k,j},E_{j,k}])$
$\displaystyle=$ $\displaystyle E_{i,i}-E_{j,j},$
that is, $[\cdot,\cdot]\circ\delta=\hbox{\rm id}$, also by Theorem 4.1. So,
the theorem holds. $\square$
## 5 Dual Lie algebras, Killing form and adjoint representation
In this section, we discuss the interrelation of the Killing form and the
adjoint representation for the Lie algebra of type $A$ within our new
terminology.
###### Theorem 5.1
$((sl_{n})^{*},-2n\delta^{*})$ is a Lie algebra isomorphic to $sl_{n}$, the
isomorphism is given by
$B:f_{i,j}\longmapsto E_{j,i},$
where $\\{f_{i,j}\mid 1\leq i,j\leq n\\}$ forms a basis of
$(gl_{n})^{*}\supset(sl_{n})^{*}$, and
$f_{i,j}(E_{k,l})=\delta_{i,k}\delta_{j,l}.$
Proof. By definition, we have
$\displaystyle-2n\delta^{*}(f_{i,j}\otimes f_{k,l})(E_{s,t})$ $\displaystyle=$
$\displaystyle-\sum_{r=1}^{n}(f_{i,j}(E_{s,r})f_{k,l}(E_{r,t})-f_{i,j}(E_{r,t})f_{k,l}(E_{s,r}))$
$\displaystyle=$
$\displaystyle-\delta_{j,k}(f_{i,j}(E_{s,j})f_{j,l}(E_{j,t})+\delta_{i,l}f_{i,j}(E_{i,t})f_{k,i}(E_{s,i})$
$\displaystyle=$
$\displaystyle-(\delta_{j,k}f_{i,l}-\delta_{i,l}f_{k,j})(E_{s,t}),$
then $(sl_{n})^{*}$ is a Lie algebra under bracket $-2n\delta^{*}$, and $B$ is
an isomorphism. $\square$
Define a bilinear form $(,)_{B}:sl_{n}\times sl_{n}\longrightarrow\mathbf{C}$
as $(x,y)_{B}=B^{-1}(x)(y)$.
###### Theorem 5.2
$(,)_{B}$ is just a non-zero scalar of the Killing form.
Proof. This result is direct. $\square$Now we can consider the following maps:
$\textstyle{sl_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2n\delta\quad}$$\textstyle{sl_{n}\otimes
sl_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hbox{\rm
id}_{sl_{n}}\otimes
B^{-1}\quad}$$\textstyle{sl_{n}\otimes(sl_{n})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\hbox{\bf
End}(sl_{n})}$
where $\eta$ is an isomorphism, $\eta(x\otimes f)(y)=f(y)x$.
###### Theorem 5.3
For the adjoint representation
$\hbox{\rm ad}\ :sl_{n}\longrightarrow\hbox{\bf End}(sl_{n}),$
we have
$\hbox{\rm ad}\ =2n\eta\circ(\hbox{\rm id}_{sl_{n}}\otimes
B^{-1})\circ\delta.$
Proof. For any $E_{i,j},E_{k,l}$, we have
$\displaystyle 2n\eta\circ(\hbox{\rm id}_{sl_{n}}\otimes
B^{-1})\circ\delta(E_{i,j})(E_{k,l})$ $\displaystyle=$
$\displaystyle\sum_{s=1}^{n}(f_{j,s}(E_{k,l})E_{i,s}-f_{s,i}(E_{k,l})E_{s,j})$
$\displaystyle=$ $\displaystyle\delta_{j,k}E_{i,l}-\delta_{i,l}E_{k,j}$
$\displaystyle=$ $\displaystyle\hbox{\rm ad}\ (E_{i,j})(E_{k,l}).$
$\square$
###### Remark 5.1
For convenience, many computations are made in $gl_{n}$ or $(gl_{n})^{*}$, but
the results always hold in $sl_{n}$ or $(sl_{n})^{*}$.
## 6 Co-splitting Theorem
In this section, we prove the following theorem:
###### Theorem 6.1
Any finite dimensional complex simple Lie algebra has a co-split Liestructure.
Proof. For a simple Lie algebra $L$ of type $X_{l}$ rather than of type $A$,
our proof is divided into following steps.
Step 1:
Suppose that $V$ is a non-trivial irreducible $X_{l}$-module of dimension $n$.
Then there is an injection
$\rho:L\longrightarrow sl_{n}\subset\hbox{\bf End}(V),$
and it is easy to check that the bilinear form $(,)_{B}$ of $sl_{n}$ is still
non-degenerate over $\rho(L)$.
Step 2:
Let $M$ be the orthogonal complement of $\rho(L)$ with respect to $(,)_{B}$,
that is,
$M=\\{m\in sl_{n}\mid(m,\rho(L))_{B}=0\\}.$
Then $M$ is a $\rho(L)$-submodule and $sl_{n}=\rho(L)\bigoplus M$.
Step 3:
For any element $a=x+v\in sl_{n}\otimes sl_{n}$, if we have
$x\in\rho(L)\otimes\rho(L)$ and $v\in\rho(L)\otimes M+M\otimes\rho(L)+M\otimes
M$, the projective map $\hbox{\bf Proj}^{sl_{n}\otimes
sl_{n}}_{\rho(L)\otimes\rho(L)}$ is defined to map $a$ to $x$. Now we write
$\delta_{res}=:\hbox{\bf Proj}^{sl_{n}\otimes
sl_{n}}_{\rho(L)\otimes\rho(L)}\circ\delta|_{\rho(L)}$, where $\delta$ is
given in Section 4., then we have
###### Lemma 6.1
$(\rho(L),\delta_{res})$ is a Lie coalgebra.
Proof. At first, it is easy to show that $\delta$ is an injective map of
$sl_{n}$-module, hence of $\rho(L)$-modules.
By Theorem 5.3, $\delta$ is equivalent to the adjoint representation, so it is
easy to know that $\delta(\rho(L))\subset\rho(L)\otimes\rho(L)+M\otimes
M\cong\rho(L)\otimes\rho(L)^{*}+M\otimes M^{*}$. Now the skew-symmetry of
$\delta_{res}$ is clear.
Furthermore, we have
$(1\otimes\delta_{res})\circ\delta_{res}=\hbox{\bf Proj}^{sl_{n}\otimes
sl_{n}\otimes
sl_{n}}_{\rho(L)\otimes\rho(L)\otimes\rho(L)}\circ[(1\otimes\delta)\circ\delta]|_{\rho(L)},$
it is obvious by the contained relation
$(1\otimes\delta)\circ\delta(\rho(L))\subset\rho(L)\otimes\rho(L)\otimes\rho(L)+\rho(L)\otimes
M\otimes M+M\otimes\rho(L)\otimes M+M\otimes M\otimes\rho(L),$
thus we have proved this lemma. $\square$
Step 4:
###### Lemma 6.2
$[,]\circ\delta_{res}=\hbox{\it a non-zero scalar of}\quad\hbox{\rm
id}_{\rho(L)}.$
Proof. Suppose that $\Delta^{+}$ is the positive root system of $X_{l}$ and
$\gamma$ is the highest root. It is easy to find a basis of $\rho(L)$
$\\{X_{\pm\alpha},h_{i}\mid i=1,\cdots,l;\alpha\in\Delta^{+}\\}$
such that $(h_{i},h_{j})_{B}=\delta_{i,j}$ and $\alpha(h_{i})\in\mathbf{R}$.
Since $\gamma$ is the highest root, then for any $\alpha\in\Delta^{+}$,
$[E_{\gamma},E_{\alpha}]=0$. By the property of $\delta$ (Theorem 5.3) and
definition of $\delta_{res}$, we have
$2n\delta_{res}(X_{\gamma})=\sum_{i=1}^{l}[X_{\gamma},h_{i}]\otimes
h_{i}+\sum_{\alpha\in\Delta^{+}}[X_{\gamma},X_{-\alpha}]\otimes\frac{X_{\alpha}}{(X_{\alpha},X_{-\alpha})_{B}},$
hence
$\displaystyle 2n{[,]}\circ\delta_{res}(X_{\gamma})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{l}[[X_{\gamma},h_{i}],h_{i}]+\sum_{\alpha\in\Delta^{+}}\frac{[[X_{\gamma},X_{-\alpha}],X_{\alpha}]}{(X_{\alpha},X_{-\alpha})_{B}}$
$\displaystyle=$
$\displaystyle\left[\sum_{i=1}^{l}\gamma(h_{i})^{2}+\sum_{\alpha\in\Delta^{+}}\gamma(h_{\alpha})\right]X_{\gamma},$
where
$h_{\alpha}=[X_{\alpha},X_{-\alpha}]/(X_{\alpha},X_{-\alpha})_{B}=\frac{2\alpha}{(\alpha,\alpha)},$
the second assertion holds by $\rho(L)\cong\rho(L)^{*}$.
Clearly,
$\sum_{i=1}^{l}\gamma(h_{i})^{2}+\sum_{\alpha\in\Delta^{+}}\gamma(h_{\alpha})>0,$
then ${[,]}\circ\delta_{res}(X_{\gamma})\not=0$.
Secondly, $\delta_{res}(X_{\gamma})$ is a highest weight vector of $L$-module
$\rho(L)\otimes\rho(L)\cong L\otimes L$, thus the equation in this lemma
holds. $\square$
Up to now, we have completed the proof of Theorem 6.1.
We also obtain the following result.
###### Theorem 6.2
For any type $X_{l}$, we have
$2n\eta\circ(\hbox{\rm id}_{sl_{n}}\otimes
B^{-1})\circ(\delta_{res})=\hbox{\rm ad}\ _{\rho(L)}|_{\rho(L)},$
where $\hbox{\rm ad}\ _{\rho(L)}|_{\rho(L)}(\rho(x)+m)=\hbox{\rm ad}\
(\rho(x))|_{\rho(L)}$ for any $x\in L,m\in M$.
Proof. For any $x,y\in L$, by the obvious fact $[\rho(x),\rho(y)]\in\rho(L)$,
we have
$\displaystyle\left[2n\eta\circ(\hbox{\rm id}_{sl_{n}}\otimes
B^{-1})\circ\delta\right](\rho(x))(\rho(y))$ $\displaystyle=$
$\displaystyle\left[2n\eta\circ(\hbox{\rm id}_{sl_{n}}\otimes
B^{-1})\circ(\delta_{res})\right](\rho(x))(\rho(y))$ $\displaystyle=$
$\displaystyle\hbox{\rm ad}\ (\rho(x))(\rho(y))$ $\displaystyle=$
$\displaystyle\hbox{\rm ad}\ _{\rho(L)}(\rho(x))|_{\rho(L)}(\rho(y)),$
and by the definition of $\delta_{res}$,
$[2n\eta\circ(\hbox{\rm id}_{sl_{n}}\otimes
B^{-1})\circ(\delta_{res})](\rho(x))(M)=0.$
So, the claim is true. $\square$
###### Remark 6.1
This work shows that for any finite dimensional semi-simple Lie algebra $L$
over the complex field $\mathbf{C}$ (or, equivalently, over any algebraically
closed field with characteristic zero), there exists some important relation
between its Killing form and adjoint action. Hence our new algebraic structure
is proved to be very useful. However, much more problems about it need to be
solved.
Acknowledgements
The first author is supported by the Science Foundation of Jiangsu University.
He is grateful to Prof. R. Farnsteiner for his valuable suggestion and
discussion.
The second author is supported in part by the NNSF of China (Grants: 10971065,
10728102), the PCSIRT and the RFDP from the MOE of China, the National &
Shanghai Leading Academic Discipline Projects (Project Number: B407).
## References
* [EK] P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, I, II, Selecta Math. (N.S.), 2 (1996), no. 1, 1-41; 4 (1998), no. 2, 213-231.
* [ES] P. Etingof and O. Schiffmann, Lectures on Quantum Groups, Internat. Press, Boston, MA, 1998.
|
arxiv-papers
| 2010-08-15T08:49:51 |
2024-09-04T02:49:12.202873
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Limeng Xia and Naihong Hu",
"submitter": "Naihong Hu",
"url": "https://arxiv.org/abs/1008.2505"
}
|
1008.2509
|
# Reionization and feedback in overdense regions at high redshift
Girish Kulkarni and T. Roy Choudhury
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019,
India E-mail: girish@hri.res.inE-mail: tirth@hri.res.in
###### Abstract
Observations of galaxy luminosity function at high redshifts typically focus
on fields of view of limited sizes preferentially containing bright sources.
These regions possibly are overdense and hence biased with respect to the
globally averaged regions. Using a semi-analytic model based on Choudhury &
Ferrara (2006) which is calibrated to match a wide range of observations, we
study the reionization and thermal history of the universe in overdense
regions. The main results of our calculation are: (i) Reionization and thermal
histories in the biased regions are markedly different from the average ones
because of enhanced number of sources and higher radiative feedback. (ii) The
galaxy luminosity function for biased regions is markedly different from those
corresponding to average ones. In particular, the effect of radiative feedback
arising from cosmic reionization is visible at much brighter luminosities.
(iii) Because of the enhanced radiative feedback within overdense locations,
the luminosity function in such regions is more sensitive to reionization
history than in average regions. The effect of feedback is visible for
absolute AB magnitude $M_{AB}\gtrsim-17$ at $z=8$, almost within the reach of
present day observations and surely to be probed by the James Webb Space
Telescope (JWST). This could possibly serve as an additional probe of
radiative feedback and hence reionization at high redshifts.
###### keywords:
intergalactic medium cosmology: theory large-scale structure of Universe.
††pubyear: 2010
## 1 Introduction
Deep surveys have now discovered galaxies at redshifts close to the end of
reionization (Bouwens & Illingworth, 2006; Iye et al., 2006; Bouwens et al.,
2007; Henry et al., 2007; Stark et al., 2007; Bouwens et al., 2008; Bradley et
al., 2008; Henry et al., 2008; Ota et al., 2008; Richard et al., 2008; Bunker
et al., 2009; Bouwens et al., 2009; Henry et al., 2009; McLure et al., 2009;
Oesch et al., 2009; Ouchi et al., 2009; Bouwens et al., 2009; Ouchi et al.,
2009; Sobral et al., 2009; Zheng et al., 2009; Oesch et al., 2010; Castellano
et al., 2010; Bouwens et al., 2010; Hickey et al., 2010; McLure et al., 2010).
Luminosity function of these galaxies, and its evolution, can answer important
questions about reionization. Indeed, much work has been done on constructing
self-consistent models of structure formation and the evolution of ionization
and thermal state of the IGM that explain these observations (Choudhury &
Ferrara, 2005; Haiman & Cen, 2005; Wyithe & Loeb, 2005; Choudhury & Ferrara,
2006; Dijkstra et al., 2007; Samui et al., 2007; Iliev et al., 2008; Samui et
al., 2009). Studies of the Gunn-Peterson trough (Gunn & Peterson, 1965) at
$z\geq 6$ have established that the mean neutral hydrogen fraction is higher
than $10^{-4}$ (e. g. Fan et al. 2006) and it is most likely that the IGM is
still highly ionized at these redshifts (Gallerani et al., 2008a, b).
Furthermore, CMB observations indicate the electron scattering optical depth
to the last scattering surface to be $\tau_{e}=0.088\pm 0.015$ based on the
WMAP seven year data. A combination of high redshift luminosity function data
with the data from these absorption systems and CMB observations favour an
extended epoch of reionization that begins at $z\approx 20$ and ends at
$z\approx 6$ (Choudhury & Ferrara, 2006).
Nonetheless, interpreting high redshift luminosity functions is not
straightforward and detailed modelling is required. For instance, local HII
regions around these galaxies can affect luminosity function evolution (Cen et
al., 2005) and clustering of galaxies can enhance this effect (Cen, 2005).
Another complication is because of the fact that these surveys can detect only
the brightest galaxies at these high redshifts ($z\gtrsim 6$). Such galaxies
can form only in highly overdense regions and therefore the surveyed volume is
far from average. An important question in that case is whether reionization
proceeds differently in such regions (Wyithe & Loeb, 2007).
Galaxy formation is enhanced in overdense regions because of a positive bias
in abundance of dark matter haloes. The enhancement in the number of galaxies
is proportional to the mass overdensity in the region, with the constant of
proportionality (‘bias’) related to halo masses and collapse redshifts (Cooray
& Sheth, 2002). This increases the number density of sources of ionising
radiation and aids reionization of the intergalactic medium (IGM) in overdense
regions. However, an increase in the IGM density also adds to radiative
recombination. Furthermore, reionization is accompanied by radiative feedback
(Thoul & Weinberg, 1996). Radiative feedback heats the IGM and suppresses
formation of low mass galaxies. This increase in radiative recombinations and
feedback works against the process of reionization and the two effects need
not cancel out. Relative significance of these negative and positive
contributions will determine how differently reionization evolves in overdense
regions.
|
---|---
|
Figure 1: Behaviour of various quantities in our fiducial model, in the
average and overdense regions are shown by dashed and solid lines
respectively. The top left panel shows the photoionisation rate, with data
points taken from (Bolton & Haehnelt, 2007). The top right panel shows the
mass-averaged temperature for ionised regions, which essentially determine the
radiative feedback. The bottom left panel is for the volume filling factor of
ionised regions. The bottom right panel shows the cosmic star formation rate.
Note that the overdense region that we consider here collapses at $z=6.8$. The
vertical dotted line in the top left panel highlights this. We cannot evolve
our reionization model for smaller redshifts.
Recently, Kim et al. (2009) studied a sample of $i_{775}$-dropout candidates
identified in five Hubble Advanced Camera for Surveys (ACS) fields centred on
Sloan Digital Sky Survey (SDSS) QSOs at redshifts $z\approx 6$. They compared
results with those from equally deep Great Observatory Origins Deep Survey
(GOODS) observations of the same fields in order to find an enhancement or
suppression in source counts in ACS fields. An enhancement would imply that
bias wins over negative feedback in these overdense regions. They found the
ACS populations to be overdense in two fields, underdense in two field, and
equally dense as the GOODS populations in one field. Somewhat surprisingly,
they did not find a clear correlation between density of $i_{775}$ dropouts
and the region’s overdensity.
In this paper, we use semi-analytic models to study reionization within
overdense regions. The main aim of this work is to quantify the effects of
enhancement in the number of sources and radiative feedback within such
regions and explore the possibility whether the galaxy luminosity function in
overdense regions can be used as potential probe of feedback and reionization
history. It is known that if ionization feedback is the main contributor to
the suppression of star formation in low mass haloes then one can distinguish
between early and late reionization histories by constraining the epoch at
which feedback-related low-luminosity flattening occurs in the galactic
luminosity function. The effect of reionization feedback on the high redshift
galaxy luminosity function was first demonstrated using semi-analytic models
by Samui et al. (2007). We apply their method to study the luminosity function
in overdense regions.
We describe our model for reionization and give details about various
parameters and their calibration in §2. We explain our method of identifying
biased regions and outline modification to our reionization model in such
regions in §3. Details of our luminosity function calculation appear in §4. We
present and discuss our results in §5 and §6. Throughout the paper, we use the
best-fit cosmological parameters from the 7-year WMAP data (Larson et al.,
2010), i.e., a flat universe with $\Omega_{m}=0.26$, $\Omega_{\Lambda}=0.73$,
$\Omega_{K}=0.044$ and $\Omega_{b}=0.045$, and $h=0.713$. The parameters
defining the linear dark matter power spectrum are $\sigma_{8}=0.80$,
$n_{s}=0.96$, ${\rm d}n_{s}/{\rm d}\ln k=-0.034$.
## 2 Description of the Semi-analytic model
In this section, we first summarise the basic features of the semi-analytic
model used for studying the globally averaged reionization history. We then
describe in detail the modifications made to this model in order to study
reionization in biased regions.
### 2.1 Globally averaged reionization
Our model for reionization and thermal history of the average IGM is
essentially that developed in Choudhury & Ferrara 2005 (CF05). The main
features of this model are as follows.
The model accounts for IGM inhomogeneities by adopting a lognormal
distribution with the evolution of volume filling factor of ionized hydrogen
(Hii) regions $Q_{\rm HII}(z)$ being calculated according to the method
outlined in Miralda-Escudé et al. (2000); reionization is said to be complete
once all the low-density regions (say, with overdensities $\Delta<\Delta_{\rm
crit}\sim 60$) are ionised. We follow the ionization and thermal histories of
neutral and HII regions simultaneously and self-consistently, treating the IGM
as a multi-phase medium. In this work, we do not consider the reionization of
singly ionised helium as it occurs much later ($z\sim 3$) than redshifts of
our interest.
The number of ionising photons depends on the assumptions made regarding the
sources. In this work, we have assumed that reionization of hydrogen is driven
by stellar sources. The rate of ionising photons injected into the IGM per
unit time per unit volume at redshift $z$ is denoted by
$\dot{n}_{\mathrm{ph}}(z)$ and is essentially determined by the star formation
rate (SFR) density $\dot{\rho}_{*}(z)$. The first step in this calculation is
to evaluate the comoving number density $N(M,z,z_{c})dMdz_{c}$ at redshift $z$
of collapsed halos having mass in the range $M$ and $M+dM$ and redshift of
collapse in the range $z_{c}$ and $z_{c}+dz_{c}$ (Sasaki, 1994):
$\begin{split}N(M,z,z_{c})dMdz_{c}=N(&M,z_{c})\nu^{2}(M,z_{c})\frac{\dot{D}(z_{c})}{D(z_{c})}\\\
&\times p_{\mathrm{surv}}(z,z_{c})\frac{dt}{dz_{c}}dz_{c}dM,\end{split}$ (1)
where $N(M,z_{c})dM$ is the comoving number density of collapsed halos with
mass between $M$ and $M+dM$, also known as the Press-Schechter (PS) mass
function (Press & Schechter, 1974), and $p_{\mathrm{surv}}(z,z_{c})$ is the
probability of a halo collapsed at redshift $z_{c}$ surviving without merger
till redshift $z$. This survival probability is simply given by
$p_{\mathrm{surv}}(z,z_{c})=\frac{D(z_{c})}{D(z)},$ (2)
where $D(z)$ is growth function of matter perturbations. Furthermore,
$\nu(M,z_{c})$ is given by $\delta_{c}/[D(z_{c})\sigma(M)]$, where $\sigma(M)$
is the rms value of density fluctuations at the comoving scale corresponding
to mass $M$ and $\delta_{c}$ is the critical overdensity for collapse of the
halo. Next, we assume that the SFR of a halo of mass $M$ that has collapsed at
an earlier redshift $z_{c}$ peaks around a dynamical time-scale of the halo
and has the form
$\begin{split}\dot{M}_{*}(M,z,z_{c})=f_{*}\left(\frac{\Omega_{b}}{\Omega_{m}}M\right)&\frac{t(z)-t(z_{c})}{t^{2}_{\mathrm{dyn}}(z_{c})}\\\
&\times\exp\left[-\frac{t(z)-t(z_{c})}{t_{\mathrm{dyn}}(z_{c})}\right].\end{split}$
(3)
where $f_{*}$ denotes the fraction of the total baryonic mass of the halo that
gets converted into stars. The global SFR density at redshift $z$ is then
$\dot{\rho}_{*}(z)=\int_{z}^{\infty}\\!\\!dz_{c}\int_{M_{\mathrm{min}}(z_{c})}^{\infty}\\!\\!dM\dot{M}_{*}(M,z,z_{c})N(M,z,z_{c}),$
(4)
where the lower limit of the mass integral, $M_{\mathrm{min}}(z_{c})$,
prohibits low-mass halos from forming stars; its value is decided by different
feedback processes. In this work, we exclusively consider radiative feedback.
For neutral regions, we assume that this quantity is determined by atomic
cooling of gas within haloes (we neglect cooling via molecular hydrogen).
Within ionised regions, photo-heating of the gas can result in a further
suppression of star formation in low-mass haloes. We compute such (radiative)
feedback self-consistently from the evolution of the thermal properties of the
IGM, as discussed in Section 2.3.
We can then write the rate of emission of ionising photons per unit time per
unit volume per unit frequency range, $\dot{n}_{\nu}(z)$, as
$\dot{n}_{\nu}(z)=N_{\gamma}(\nu)f_{\mathrm{esc}}\dot{\rho}_{*}(z),$ (5)
where $N_{\gamma}(\nu)$ is the total number of ionising photons emitted per
unit frequency range per unit stellar mass and $f_{\mathrm{esc}}$ is the
escape fraction of photons from the halo. The quantity $N_{\gamma}(\nu)$ can
be calculated using population synthesis, given the initial mass function and
spectra of stars of different masses (Samui et al., 2007). In this paper we
have used the population synthesis code Starburst99 (Leitherer et al., 1999;
Vázquez & Leitherer, 2005) to calculate $N_{\gamma}(\nu)$ by evolving a
stellar population of total mass $10^{6}$ M⊙ with a $0.1-100.0$ M⊙ Salpeter
IMF and metallicity $0.001$ ($0.05$ times the solar metallicity,
$Z_{\odot}=0.02$). The total rate of emission of ionising photons per unit
time per unit volume is obtained simply integrating by Equation (5) over
suitable frequency range.
Given the above model, we obtain best-fit parameters by comparing with the
redshift evolution of photoionisation rate obtained from the Ly$\alpha$ forest
(Bolton & Haehnelt, 2007) and the electron scattering optical depth (Larson et
al., 2010). We should mention here that any model containing only a single
population of atomic-cooled stellar sources with non-evolving $f_{*}f_{\rm
esc}$ cannot match both the Ly$\alpha$ forest and WMAP constraints (Choudhury
et al., 2008; Bolton & Haehnelt, 2007). In this work, we choose the model
which satisfies the Ly$\alpha$ constraints but underpredicts $\tau_{e}$. In
order to match both the constraints, one has to invoke either molecular
cooling in minihaloes and/or metal-free stars and/or other unknown sources of
reionization. This model is described by the parameter values $f_{*}=0.2$ and
$f_{\mathrm{esc}}=0.135$, and gives $\tau_{e}$ of 0.072. Figure 1 shows
evolution of the filling factor of ionised regions, global star formation rate
density, mass-weighted average temperature in ionised regions and average
hydrogen photoionisation rate in this model (dashed curves in all the panels).
The filling factor of ionized regions is seen to rise monotonically from
$z\approx 15$ and takes values close to unity at redshifts $z\approx 6$.
Temperature of ionized regions also rises rapidly during reionization and
flattens out to a few times $10^{4}$ K at redshift $z\lesssim 4$ (not shown
here). Lastly, the photoionisation rate also increases during reionization as
the star formation rate builds up. However, the photoionisation rate increases
rapidly with a sudden jump at $z\approx 6$ when the ionized regions overlap
(filling factor becomes close to unity). This is because a given region in
space starts receiving ionizing photos from multiple sources and as a result,
the ionizing flux suddenly increases. This is our fiducial model, which
satisfies observational constraints from Ly$\alpha$ forest, observations of
star formation rate history, number density of Lyman-limit systems at high
redshift and of the IGM temperature. In this model, reionization starts at
$z\approx 15$ and is 90% complete by $z\approx 7$. Evolution of
$x_{\mathrm{HII}}$ is consistent with constraints from Ly$\alpha$ emitters and
the GP optical depths.
---
Figure 2: Luminosity function from our fiducial model at $z=6$, $7$ and $8$.
This is the average case. Data points are from Bouwens & Illingworth (2006)
($z=6$) and Bouwens et al. (2010) ($z=7$, $8$).
Having set up the reionization model, we then calculate the predicted
luminosity function of galaxies in this model. Luminosity functions of objects
are usually preferred for comparing theory with observations because of its
directly observable nature. In this work, we closely follow the approach
presented by Samui et al. (2007) to calculate the luminosity function. We
obtain luminosity per unit mass, $l_{1500}(t)$, at 1500 Å as a function of
time from population synthesis for an instantaneous burst. In our model, star
formation does not happen in a burst, but is a continuous process spread out
over a dynamical time-scale. Therefore, in order to determine the luminosity
of a halo, $L_{1500}(t)$ with this kind of star formation, we convolve
$l_{1500}(t)$ with the halo’s star formation rate using
$L_{1500}(M,T)=\int_{T}^{0}d\tau\dot{M}_{\mathrm{*}}(M,T-\tau,z_{c})l_{1500}(\tau),$
(6)
where $T$ is the age of the halo, which has mass $M$ and which collapsed at
redshift $z_{c}$. This luminosity can be converted to absolute AB magnitude
using
$M_{AB}=-2.5\log_{10}(L_{\nu 0})+51.60,$ (7)
where the luminosity is in units of erg s-1 Hz-1 (Oke & Gunn, 1983). One can
compute the luminosity evolution for any halo that collapses at redshift
$z_{c}$ and undergoes star formation according to Equation (3). The luminosity
function at redshift $z$, $\Phi(M_{AB},z)$, is now given by
$\begin{split}\Phi(&M_{AB},z)dM_{AB}\\\
&=\int_{z}^{\infty}dz_{c}N(M,z,z_{c})\frac{dM}{dL_{1500}}\frac{dL_{1500}}{dM_{AB}}dM_{AB},\end{split}$
(8)
where $N(M,z,z_{c})$ is the number density at redshift $z$ of halos of mass
$M$ collapsed at redshift $z_{c}$. We will use Equation (8) to study effect of
overdensity on the luminosity function and to compare the luminosity function
in our model with observations in the next section.
Figure 2 shows the globally averaged luminosity function calculated using our
model for different redshifts in comparison with observations presented by
Bouwens & Illingworth (2006). We find that our model reproduces the observed
luminosity functions at high redshifts reasonably well. In particular, the
match at $z=6$ is remarkably good while the model predicts less number of
galaxies than what is observed at $z=7$ and 8. This could indicate that the
star-forming efficiency $f_{*}$ increases with $z$, and/or the time-scale of
star formation is lower than $t_{\rm dyn}$ at higher redshifts. The match of
the model with the data can be improved by tuning these parameters suitably,
however we prefer not to introduce additional freedom in constraining the
parameters; rather our focus is to estimate the effect of reionization and
feedback on the luminosity function.
In our calculation of luminosities, we do not make any correction for dust.
This is partly because of indications from observed very blue UV-continuum
slopes (Bouwens et al., 2010; Oesch et al., 2010; Finkelstein et al., 2010;
Bunker et al., 2010) that dust extinction in $z\gtrsim 7$ is small. As
discussed in the next section, we exclusively work with luminosity functions
at these redshifts. Also, the effect of dust is degenerate with $f_{*}$ to
some extent. Therefore, the exclusion of dust extinction does not affect the
general results of our calculation.
Figure 3: Effect of overdensity on the luminosity function via feedback at
z=8.0. This is for $\delta=8.8$ and $R_{L}=1.482$ Mpc.
### 2.2 Biased regions
We have mentioned that galaxy luminosity functions provide valuable
information regarding reionization. However, observations are carried out over
relatively small fields of view. The bright sources in these fields are
typically hosted by high mass haloes. Hence, it is most likely that these
fields are biased tracers of luminosity function and reionization. In this
section, we extend our model to study reionization within such biased regions
and quantify the departure of various quantities from their globally averaged
trends. Reionization in biased regions has been discussed in the literature.
Wyithe & Loeb (2007) studied the correlation between high redshift galaxy
distribution and the neutral Hydrogen 21 cm emission by considering
reionization in the vicinity of these galaxies. Wyithe et al. (2008)
considered the ionization background near high redshift quasars. Geil & Wyithe
(2009) studied effect of reionization around high redshift quasars on the
power spectrum of 21 cm emission (see also Pritchard & Furlanetto 2007). The
general conclusion of these studies is that overdense regions are ionised
earlier. In this work, we will consider the behaviour of luminosity functions
in such regions.
Overdense regions are characterised by their comoving Lagrangian size $R$, and
their linearly extrapolated overdensity $\delta$. At a given redshift, we can
take a scale $R$ corresponding to an observed field of view (e.g., WFC3/IR
field in HST) and then determine $\delta$ by identifying the presence of a
massive object, for example, a quasar or a bright galaxy. We follow a
prescription discussed by Muñoz & Loeb (2008).
Note that if a galaxy with luminosity $L_{1500}$ is observed at redshift
$z_{g}$ then we can assign a certain mass to the dark matter halo containing
the galaxy, say $M$. The halo mass $M$ has to be obtained from galaxy
luminosity $L_{1500}$ by using some prescription or by fitting the galaxy’s
spectral energy distribution (SED; Vale & Ostriker 2004). In this work,
however, since uncertainty in the value of overdensity $\delta$ is expected to
be larger than any uncertainty in the mass of the galaxy’s host halo, we
choose to calculate this mass in an alternate, simpler manner. Note that we
can invert equations (6) and (3) to obtain $M$ given an observed value of
$L_{1500}$ if we have an estimate of the redshift at which the galaxy formed.
In fact, we obtain the halo mass $M$ such that the halo luminosity after a
dynamical time from halo formation time, calculated according to our model, is
equal to $L_{1500}$. In other words, we break the degeneracy between halo mass
and formation redshift by assuming that the galaxy’s age is equal to the
dynamical time of the halo.
From this analysis, we conclude that a collapsed object with mass $M$ exists
at redshift $z_{g}$. Now suppose our field of observation corresponds to some
linear scale $R_{o}$ at this redshift. Then the linearly extrapolated
overdensity at this scale can be obtained using the excursion set prescription
(Muñoz & Loeb, 2008). Recall that the probability distribution of the
extrapolated Gaussian density field smoothed over scale $R$ is also Gaussian
$Q(\delta_{R},\sigma^{2}(R))d\delta_{R}=\frac{1}{\sqrt{2\pi\sigma^{2}(R)}}\exp\left[-\frac{\delta_{R}^{2}}{2\sigma^{2}(R)}\right]d\delta_{R}.$
(9)
The conditional probability distribution of overdensity $\delta_{1}$ on a
scale specified by the variance $\sigma^{2}_{1}$, given a value of overdensity
$\delta_{2}$ on a larger scale specified by the variance
$\sigma^{2}_{2}<\sigma^{2}_{1}$ is given by
$Q(\delta_{1},\sigma^{2}_{1}|\delta_{2},\sigma^{2}_{2})=Q(\delta_{1}-\delta_{2},\sigma^{2}_{1}-\sigma^{2}_{2}).$
(10)
Conversely, when the value of overdensity $\delta_{1}$ on a smaller scale
specified by variance $\sigma^{2}_{1}$ is given, the conditional probability
distribution of $\delta_{2}$ can be obtained using Bayes theorem as
$Q(\delta_{2},\sigma^{2}_{2}|\delta_{1},\sigma^{2}_{1})\propto
Q(\delta_{1},\sigma^{2}_{1}|\delta_{2},\sigma^{2}_{2})Q(\delta_{2},\sigma^{2}_{2})d\delta_{2}.$
(11)
If we now set the smaller scale to be that of the observed collapsed halo, and
the overdensity at that scale to be the critical overdensity for spherical
collapse, Equation (11) will give the resulting overdensity at any larger
scale due to the presence of this massive galaxy. In other words, we set
$\delta_{1}=\delta_{c}(z_{g})$ and $\sigma^{2}_{1}=\sigma^{2}(M)$ in Equation
(11).
The larger scale corresponds to the field of observation. In order to
calculate that, we first note that the excursion set principle functions
entirely in Lagrangian coordinates. As a region evolves towards eventual
collapse its Lagrangian size stays unchanged while its Eulerian size changes.
For a spherical region the Eulerian evolution will follow the solution of the
spherical collapse model. However, since the Eulerian and Lagrangian sizes of
the region coincide at the initial instant, the spherical collapse solution is
also a relationship between these two sizes. Thus we have
$R_{E}=\frac{3}{10}\frac{1-\cos\theta}{\delta_{L}}\frac{D(z=0)}{D(z)}R_{L},$
(12)
where $\theta$ is a parameter given by
$\frac{1}{1+z}=\frac{3\times
6^{2/3}}{20}\frac{(\theta-\sin\theta)^{2/3}}{\delta_{L}}.$ (13)
In our case, the Eulerian size of the region of interest is just the comoving
distance corresponding to the angular field of view, which is just the angular
diameter distance at the relevant redshift multiplied by the angular field of
view. The WFC3/IR field is $136^{{}^{\prime\prime}}\times
123^{{}^{\prime\prime}}$. For the best fit $\Lambda$CDM cosmology the diagonal
size of this field corresponds to a comoving Eulerian distance $R_{E}=1.365$
Mpc at $z=8$. In a WFC3/IR field centred on the object UDFy-42886345 at
redshift $8.0$ and apparent magnitude $H_{160,AB}=28.0$ we obtain a halo mass
$M=2.52\times 10^{11}$ M⊙ and luminosity $L_{1500}=2.21\times 10^{29}$ erg s-1
Hz-1.
Notice, however, that since $\delta_{L}$ is unknown, Equation (12) implies
that the relation between the Eulerian size $R_{E}$ and Lagrangian size
$R_{L}$ is not one-to-one. Thus, for the probability distribution of linearly
extrapolated overdensity $\delta$ given the halo mass $M$, we can only write
$\frac{dP(\delta|M)}{d\delta}\propto
Q[\delta,R_{L}(\delta,R_{E})|\delta_{c}(z),R(M)],$ (14)
where the constant of proportionality is calculated by using the normalization
condition $\int[dP(\delta|M)/d\delta]d\delta=1$.
In our calculations, we work with the value of $\delta$ for which
$dP(\delta|M)/d\delta$ is maximum. For the WFC3/IR field at $z=8$, this turns
out to be $\delta=8.86$ (linearly extrapolated to $z=0$), which results in a
Lagrangian size $R_{L}=1.482$ Mpc for the region of interest. Notice that
since $\delta>\delta_{c}$ the region must have collapsed at some redshift
$z\lesssim 6.5$.
In order to incorporate this overdensity into our reionization model, note
first that the number density of collapsed objects in such overdense regions
is enhanced with respect to that in a region with average density. This
enhancement can be calculated using the excursion set formalism (Bond et al.,
1991). It is then shown in the Appendix that the comoving number density
$N(M,z,z_{c})dMdz_{c}$ at redshift $z$ of collapsed halos having mass in the
range $M$ and $M+dM$ and redshift of collapse in the range $z_{c}$ and
$z_{c}+dz_{c}$ is given in this case by
$\begin{split}N(M,z,z_{c})dMdz_{c}=N(&M,z_{c})\left(\frac{\nu^{2}\delta_{c}}{\delta_{c}/D(z_{c})-\delta}\right)\frac{\dot{D}(z_{c})}{D^{2}(z_{c})}\\\
&\times p_{\mathrm{surv}}(z,z_{c})\frac{dt}{dz_{c}}dz_{c}dM,\end{split}$ (15)
where $N(M,z_{c})$ is the PS mass function. The scale $R$ enters via the
definition of $\nu(M,z_{c})$, which is now given by
$\nu(M,z_{c})=\frac{\delta_{c}/D(z_{c})-\delta}{\sqrt{\sigma^{2}(M)-\sigma^{2}_{R}}}.$
(16)
The survival probability $p_{\mathrm{surv}}(z,z_{c})$ is given by
$p_{\mathrm{surv}}(z,z_{c})=\frac{\delta_{c}/D(z)-\delta}{\delta_{c}/D(z_{c})-\delta}.$
(17)
Another change when our reionization model is applied to overdense regions is
that we now normalize the probability distribution of inhomogeneities in the
IGM such that the average density in the region is $\Delta=\delta+1$.
---
Figure 4: Effect of reionization history on luminosity function at z=8.0.
Solid, dashed and dot-dashed lines have $\tau=$ 0.073, 0.058 and 0.088
respectively. Left panel shows the average case. Right panel shows the
overdense case.
### 2.3 Radiative feedback
As we argue in the next section, the luminosity function of galaxies in an
overdense region could carry an enhanced signature of feedback. We therefore
highlight our feedback model in this subsection.
Radiation from stars in the first galaxies is expected to ionize and heat the
surrounding medium. This increases the mass scale above which baryons can
collapse in haloes. Also, as a result, the minimum mass of haloes that are
able to cool is much higher in ionized regions than in the neutral ones. In
our calculations, feedback appears through the quantity $M_{\mathrm{min}}(z)$
in Equation 4. The temperature evolution of both regions is calculated self-
consistently. In the ionized regions, we fix the cut-off mass to that
corresponding to a virial temperature of $10^{4}$ K or the local Jeans mass,
whichever is higher. In the neutral regions, since the Jeans mass is always
low, the cut-off mass always corresponds to the virial temperature of $10^{4}$
K. The minimum mass corresponds to the circular velocity of
$v_{c}^{2}=\frac{2k_{\mathrm{boltz}}T}{\mu m_{p}},$ (18)
where $\mu$ is the mean molecular weight. For a temperature of $\approx
10^{4}$ K, the minimum circular velocity is $\approx 25$ km s-1. Note that
this value is comparable to values obtained in simulations (Gnedin, 2000) but
is somewhat higher than that taken in the semi-analytic prescription of Samui
et al. (2007).
We find that $M_{\mathrm{min}}(z)$ increases with time taking values of
$\approx 10^{7}$ M⊙ at $z\approx 10$ and $\approx 10^{8}$ M⊙ at $z\approx 7$.
In overdense regions the minimum mass is enhanced to about $10^{10}$ M⊙.
Figure 5 shows the evolution of the minimum mass.
## 3 Results
The results for reionization and thermal histories within overdense regions
are presented in this section.
Figure 5: Evolution of the minimum mass $M_{\mathrm{min}}(z)$ of haloes that
can host galaxies in ionized regions in the average case (dashed line) and the
overdense case (solid line). The dot-dashed line shows the minimum mass in
neutral regions, which is same for the average and overdense cases.
### 3.1 Effect of overdensity on reionization history
We first consider the effect of overdensity on reionization history for our
fiducial model. As is well known, reionization proceeds differently in
overdense regions. The solid lines in Figure 1 show the evolution of the
photoionisation rate, temperature in ionised regions, star formation rate
density and the volume filling factor of ionised regions in an overdense
region with size $R_{L}=1.482$ Mpc and linearly extrapolated overdensity
$\delta=8.86$. This corresponds to the HUDF WFC3/IR field centred at the
brightest source in Bouwens et al. (2010). (See Section 2.2.) Clearly while
the average region is completely ionised at $z\approx 6$, the biased region is
ionised much earlier, at $z\approx 7.5$. This result agrees with Wyithe & Loeb
(2007), although note that unlike that work, here we calculate the clumping
factor from a physical model for inhomogeneities. The reason for early
reionization in overdense regions is the enhanced number of sources, which is
clear from the plots of photoionisation rate and the star-formation rate, both
of which are $\sim 5$ times higher than the corresponding globally averaged
values. However, these overdense regions have more recombinations, which
results in enhanced temperatures as is clear from the top right panel. This
results in enhanced negative radiative feedback which will suppress star
formation in low mass galaxies and hence affect the shape of the luminosity
function. In fact, for the average case, haloes in ionised regions with masses
below $10^{8}$ M⊙ cannot form stars, whereas this cutoff mass rises to close
to $10^{10}$ M⊙ in the overdense case. Clearly feedback is enhanced in
overdense regions.
### 3.2 Effect of overdensity on luminosity function
We now discuss the effect of overdensity on luminosity function. Clearly,
overdense regions have enhanced number of sources, hence it is natural that
the amplitude of the luminosity function for such regions should be higher
than the globally averaged values. However, the overdense regions have
enhanced radiative feedback too, and hence we expect a decrease in the number
of sources, particularly towards the fainter end.
Figure 3 shows the effect of overdensity on the luminosity function at $z=8$
for our fiducial model. The ionised volume filling factor within the overdense
region is $Q\approx 1.0$ for the overdense region under consideration at this
redshift. The average region luminosity function (dashed line) is clearly very
different from the luminosity function in the overdense region (solid line) at
that redshift. Firstly, we can clearly see an enhancement in the source counts
for brighter galaxies, which is as expected. In addition, there is a clear
sign of a flattening for magnitudes $M_{\rm AB}\gtrsim-17$, which is a
signature of radiative feedback. In comparison, the effect of feedback for
average regions occurs at much fainter magnitudes $M_{\rm AB}\sim-12$. Note
that there is no complete suppression of star formation for halo masses lower
than the feedback threshold, rather the luminosity function for magnitudes
below the knee continues to grow in the flattened region. This is simply due
to the continued star formation in haloes with mass less than the cutoff mass
at $z=8.0$, but which collapsed at higher redshifts when the feedback
threshold mass was lower. Thus, for instance, if star formation is allowed to
happen in a halo for only for a fraction of the dynamical time [see equation
(3)], the luminosity function will rise less steeply at the fainter end. For
small enough star formation time scale, the luminosity function will show an
abrupt cutoff. Of course, an abrupt cutoff is always seen at low enough
luminosities, which are not shown in the figure here.
It is important to understand here that the data points in Figure 2 do not
represent luminosity function of the overdense region. Instead, those data
points represent the globally averaged luminosity function derived using a
maximum likelihood procedure from the observed luminosity distribution of
sources. In this procedure, a likelihood function is defined, which describes
the step-wise shape of the luminosity function that is most likely given the
observed luminosity distribution in the search fields. Details of this
procedure are described, for example, in Section 5.1 of Bouwens et al. (2010)
and references therein.
### 3.3 Luminosity function as a probe of reionization
Given the fact that the effect of radiative feedback shows up at brighter
magnitudes for overdense regions, it is possible to use this feature for
studying feedback using near-future observations. For this purpose, we
consider two additional models (other than the fiducial one) of reionization.
These models have parameter values ($f_{*}$, $f_{\rm esc}$) = ($0.06$, $0.3$)
and ($0.2,0.07$) and we obtain $\tau_{e}=0.088$ and $0.058$ respectively for
these models. We fix $f_{*}$ and only change the value of $f_{\rm esc}$ to
ensure that any effect on the luminosity function is purely due to feedback.
These two models predict photoionisation rates greater and lesser respectively
than what are presented by Bolton & Haehnelt (2007).
The right panel of Figure 4 shows the luminosity function at $z=8$ within the
overdense region for three different reionization histories, which can be
compared with the corresponding luminosity function in average region (shown
in the left panel). In both cases a distinct “knee” is seen in the luminosity
function as a signature of feedback. The luminosity function flattens at this
luminosity, and is suppressed to very low values at much lower luminosities.
As described in the previous section, this signature of feedback appears at
brighter magnitudes for the overdense region. This is expected, because the
cutoff mass depends directly on the temperature, which is enhanced in the
overdense region. We also note that in the case of the first model the
flattening occurs for $M_{\rm AB}\gtrsim-19$ whereas for the second model at a
fainter luminosity of $M_{\rm AB}\simeq-16$. This is due to the fact that the
photoionisation feedback is enhanced in the first model due to enhanced flux.
The evolution of the filling factor affects this result through the average
temperature which sets the cutoff mass. Thus, early and late reionization
models are distinguished by the difference in the nature of flattening in both
cases. This also affects the evolution of the luminosity function.
We find that the reionization history has a strong effect on the luminosity
function at the faint end. It is known that the bright end of the luminosity
function is affected primarily by the star formation mode of a halo, and the
overall bias, whereas its faint end is affected by the reionization history.
However, we also find, from Figure 4, that the effect of reionization history
is much stronger in the case of overdense regions. This is because of the
enhanced photoionisation feedback, which is more sensitive to changes in
reionization history. This order of magnitude change in the overdense region
luminosity function should be visible to the James Webb Space Telescope, which
can observe up to $m_{\rm AB}\approx 31.5$ ($M_{\rm AB}\approx-16.0$ at
redshifts of interest; Windhorst et al. 2006).
## 4 Discussion and Summary
We have used a semi-analytic model, based on Choudhury & Ferrara (2005, 2006)
to study reionization and thermal history of an overdense region. Studying
such regions is important because observations of galaxy luminosity function
at high redshifts typically focus fields of view of limited sizes
preferentially containing bright sources; these regions possibly are overdense
and hence biased with respect to the globally averaged regions. In particular,
we study the effect of radiative feedback arising from reionization on the
shape of galaxy luminosity function.
In summary, we find that
1. 1.
Reionization proceeds differently in overdense regions. Overdense regions are
ionised earlier because of enhanced number of sources and star formation. In
addition, these regions have higher temperatures because of enhanced
recombinations and hence effect of radiative feedback is enhanced too.
2. 2.
In particular, the shape of the galaxy luminosity function for biased regions
is very different from that for average regions. There is a significant
enhancement in the number of high-mass galaxies because of bias, while there
is a reduction in low-mass galaxies resulting from enhanced radiative
feedback.
3. 3.
Luminosity function in overdense regions is more sensitive to reionization
history compared to average regions. The effect of radiative feedback shows up
at absolute AB magnitudes $M_{\rm AB}\gtrsim-17$ in these regions, while it
occurs at much fainter magnitudes $M_{\rm AB}\sim-12$ for average regions.
This order of magnitude change in the overdense region luminosity function
should be visible to the James Webb Space Telescope in future.
Finally, we critically examine some of the simplifying assumptions made in
this work and how they are likely to affect our conclusions. Firstly, we have
seen that the presence of a high mass galaxy within a region if size $R$ does
not uniquely specify the value of the overdensity $\delta$. Rather we obtain a
probability density (which is Gaussian in shape) and work with the value where
this probability is maximum. In reality, however, the actual value of $\delta$
could be different and this may possibly affect the predicted luminosity
function. Note that the luminosity function at the brighter end is almost
independent of the details of reionization history, and this, in principle,
can be used for constraining the value of $\delta$. The effect of feedback can
then be studied using the faint end of the luminosity function.
The radiative feedback prescription used in this paper is based on a Jeans
mass calculation (Choudhury & Ferrara, 2005). However, alternate prescriptions
for feedback exist in literature, e.g., Gnedin (2000) and hence the shape of
the luminosity function at faint ends as predicted by our model may not be
robust. Interestingly, the presence of a “knee” in the luminosity function can
be used to estimate the value of the halo mass below which star formation can
be suppressed (which in turn can indicate the temperature) while the shape of
the function below this knee should indicate the nature of feedback. This
study can also be complemented with proposed for studying feedback using other
observations, e.g., 21 cm observation (Schneider et al., 2008) and CMBR
(Burigana et al., 2008).
Finally, we have neglected the presence of other sources of reionization,
e.g., metal-free stars, minihaloes, and so on. It is expected that these
sources would be too faint to affect the luminosity function in the ranges we
are considering. However, these sources may affect the thermal history of the
medium, e.g, the metal-free stars would produce higher temperatures because of
harder spectra. In such cases, it is most likely that feedback would occur at
magnitude brighter than what we have indicated and hence would possibly be
easier to detect.
## Acknowledgements
GK acknowledges useful discussion with Prof. Jasjeet S. Bagla. Computational
work for this study was carried out at the cluster computing facility in the
Harish-Chandra Research Institute (http://cluster.hri.res.in/index.html). We
would also like to thank the referee for suggestions that improved this
paper’s quality.
## References
* Bolton & Haehnelt (2007) Bolton J. S., Haehnelt M. G., 2007, MNRAS, 382, 325
* Bond et al. (1991) Bond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440
* Bouwens & Illingworth (2006) Bouwens R., Illingworth G., 2006, New Astronomy Review, 50, 152
* Bouwens et al. (2009) Bouwens R. J., Illingworth G. D., Franx M., Chary R., Meurer G. R., Conselice C. J., Ford H., Giavalisco M., van Dokkum P., 2009, ApJ, 705, 936
* Bouwens et al. (2007) Bouwens R. J., Illingworth G. D., Franx M., Ford H., 2007, ApJ, 670, 928
* Bouwens et al. (2008) Bouwens R. J., Illingworth G. D., Franx M., Ford H., 2008, ApJ, 686, 230
* Bouwens et al. (2010) Bouwens R. J., Illingworth G. D., Gonzalez V., Labbe I., Franx M., Conselice C. J., Blakeslee J., van Dokkum P., Ford H., Holden B., Marchesini D., Magee D., Zheng W., 2010, ArXiv e-prints
* Bouwens et al. (2009) Bouwens R. J., Illingworth G. D., Labbe I., Oesch P. A., Carollo M., Trenti M., van Dokkum P. G., Franx M., Stiavelli M., Gonzalez V., Magee D., 2009, ArXiv e-prints
* Bouwens et al. (2010) Bouwens R. J., Illingworth G. D., Oesch P. A., Labbe I., Trenti M., van Dokkum P., Franx M., Stiavelli M., Carollo C. M., Magee D., Gonzalez V., 2010, ArXiv e-prints
* Bouwens et al. (2010) Bouwens R. J., Illingworth G. D., Oesch P. A., Stiavelli M., van Dokkum P., Trenti M., Magee D., Labbé I., Franx M., Carollo C. M., Gonzalez V., 2010, ApJ, 709, L133
* Bouwens et al. (2010) Bouwens R. J., Illingworth G. D., Oesch P. A., Trenti M., Stiavelli M., Carollo C. M., Franx M., van Dokkum P. G., Labbé I., Magee D., 2010, ApJ, 708, L69
* Bradley et al. (2008) Bradley L. D., Bouwens R. J., Ford H. C., Illingworth G. D., Jee M. J., Benítez N., Broadhurst T. J., Franx M., Frye B. L., Infante L., Motta V., Rosati P., White R. L., Zheng W., 2008, ApJ, 678, 647
* Bunker et al. (2009) Bunker A., Wilkins S., Ellis R., Stark D., Lorenzoni S., Chiu K., Lacy M., Jarvis M., Hickey S., 2009, ArXiv e-prints
* Bunker et al. (2010) Bunker A. J., Wilkins S., Ellis R. S., Stark D. P., Lorenzoni S., Chiu K., Lacy M., Jarvis M. J., Hickey S., 2010, MNRAS, pp 1378–+
* Burigana et al. (2008) Burigana C., Popa L. A., Salvaterra R., Schneider R., Choudhury T. R., Ferrara A., 2008, MNRAS, 385, 404
* Castellano et al. (2010) Castellano M., Fontana A., Boutsia K., Grazian A., Pentericci L., Bouwens R., Dickinson M., Giavalisco M., Santini P., Cristiani S., Fiore F., Gallozzi S., Giallongo E., Maiolino R., Mannucci F., and others 2010, A&A, 511, A20+
* Cen (2005) Cen R., 2005, ArXiv Astrophysics e-prints
* Cen et al. (2005) Cen R., Haiman Z., Mesinger A., 2005, ApJ, 621, 89
* Choudhury & Ferrara (2005) Choudhury T. R., Ferrara A., 2005, MNRAS, 361, 577
* Choudhury & Ferrara (2006) Choudhury T. R., Ferrara A., 2006, MNRAS, 371, L55
* Choudhury et al. (2008) Choudhury T. R., Ferrara A., Gallerani S., 2008, MNRAS, 385, L58
* Cooray & Sheth (2002) Cooray A., Sheth R., 2002, Phys. Rep., 372, 1
* Dijkstra et al. (2007) Dijkstra M., Wyithe J. S. B., Haiman Z., 2007, MNRAS, 379, 253
* Fan et al. (2006) Fan X., Strauss M. A., Becker R. H., White R. L., Gunn J. E., Knapp G. R., Richards G. T., Schneider D. P., Brinkmann J., Fukugita M., 2006, AJ, 132, 117
* Finkelstein et al. (2010) Finkelstein S. L., Papovich C., Giavalisco M., Reddy N. A., Ferguson H. C., Koekemoer A. M., Dickinson M., 2010, ApJ, 719, 1250
* Gallerani et al. (2008a) Gallerani S., Ferrara A., Fan X., Choudhury T. R., 2008a, MNRAS, 386, 359
* Gallerani et al. (2008b) Gallerani S., Salvaterra R., Ferrara A., Choudhury T. R., 2008b, MNRAS, 388, L84
* Geil & Wyithe (2009) Geil P. M., Wyithe J. S. B., 2009, MNRAS, 399, 1877
* Gnedin (2000) Gnedin N. Y., 2000, ApJ, 542, 535
* Gunn & Peterson (1965) Gunn J. E., Peterson B. A., 1965, ApJ, 142, 1633
* Haiman & Cen (2005) Haiman Z., Cen R., 2005, ApJ, 623, 627
* Henry et al. (2008) Henry A. L., Malkan M. A., Colbert J. W., Siana B., Teplitz H. I., McCarthy P., 2008, ApJ, 680, L97
* Henry et al. (2007) Henry A. L., Malkan M. A., Colbert J. W., Siana B., Teplitz H. I., McCarthy P., Yan L., 2007, ApJ, 656, L1
* Henry et al. (2009) Henry A. L., Siana B., Malkan M. A., Ashby M. L. N., Bridge C. R., Chary R., Colbert J. W., Giavalisco M., Teplitz H. I., McCarthy P. J., 2009, ApJ, 697, 1128
* Hickey et al. (2010) Hickey S., Bunker A., Jarvis M. J., Chiu K., Bonfield D., 2010, MNRAS, 404, 212
* Iliev et al. (2008) Iliev I. T., Shapiro P. R., McDonald P., Mellema G., Pen U., 2008, MNRAS, 391, 63
* Iye et al. (2006) Iye M., Ota K., Kashikawa N., Furusawa H., Hashimoto T., Hattori T., Matsuda Y., Morokuma T., Ouchi M., Shimasaku K., 2006, Nature, 443, 186
* Kim et al. (2009) Kim S., Stiavelli M., Trenti M., Pavlovsky C. M., Djorgovski S. G., Scarlata C., Stern D., Mahabal A., Thompson D., Dickinson M., Panagia N., Meylan G., 2009, ApJ, 695, 809
* Larson et al. (2010) Larson D., Dunkley J., Hinshaw G., Komatsu E., Nolta M. R., Bennett C. L., Gold B., Halpern M., Hill R. S., Jarosik N., Kogut A., Limon M., Meyer S. S., Odegard N., Page L., 2010, ArXiv e-prints
* Leitherer et al. (1999) Leitherer C., Schaerer D., Goldader J. D., González Delgado R. M., Robert C., Kune D. F., de Mello D. F., Devost D., Heckman T. M., 1999, ApJS, 123, 3
* McLure et al. (2009) McLure R. J., Cirasuolo M., Dunlop J. S., Foucaud S., Almaini O., 2009, MNRAS, 395, 2196
* McLure et al. (2010) McLure R. J., Dunlop J. S., Cirasuolo M., Koekemoer A. M., Sabbi E., Stark D. P., Targett T. A., Ellis R. S., 2010, MNRAS, 403, 960
* Miralda-Escudé et al. (2000) Miralda-Escudé J., Haehnelt M., Rees M. J., 2000, ApJ, 530, 1
* Muñoz & Loeb (2008) Muñoz J. A., Loeb A., 2008, MNRAS, 385, 2175
* Oesch et al. (2010) Oesch P. A., Bouwens R. J., Illingworth G. D., Carollo C. M., Franx M., Labbé I., Magee D., Stiavelli M., Trenti M., van Dokkum P. G., 2010, ApJ, 709, L16
* Oesch et al. (2009) Oesch P. A., Carollo C. M., Stiavelli M., Trenti M., Bergeron L. E., Koekemoer A. M., Lucas R. A., Pavlovsky C. M., Beckwith S. V. W., Dahlen T., Ferguson H. C., Gardner J. P., Lilly S. J., Mobasher B., Panagia N., 2009, ApJ, 690, 1350
* Oke & Gunn (1983) Oke J. B., Gunn J. E., 1983, ApJ, 266, 713
* Ota et al. (2008) Ota K., Iye M., Kashikawa N., Shimasaku K., Kobayashi M., Totani T., Nagashima M., Morokuma T., Furusawa H., Hattori T., Matsuda Y., Hashimoto T., Ouchi M., 2008, ApJ, 677, 12
* Ouchi et al. (2009) Ouchi M., Mobasher B., Shimasaku K., Ferguson H. C., Fall S. M., Ono Y., Kashikawa N., Morokuma T., Nakajima K., Okamura S., Dickinson M., Giavalisco M., Ohta K., 2009, ApJ, 706, 1136
* Ouchi et al. (2009) Ouchi M., Ono Y., Egami E., Saito T., Oguri M., McCarthy P. J., Farrah D., Kashikawa N., Momcheva I., Shimasaku K., Nakanishi K., Furusawa H., Akiyama M., Dunlop J. S., and others 2009, ApJ, 696, 1164
* Press & Schechter (1974) Press W. H., Schechter P., 1974, ApJ, 187, 425
* Pritchard & Furlanetto (2007) Pritchard J. R., Furlanetto S. R., 2007, MNRAS, 376, 1680
* Richard et al. (2008) Richard J., Stark D. P., Ellis R. S., George M. R., Egami E., Kneib J., Smith G. P., 2008, ApJ, 685, 705
* Samui et al. (2007) Samui S., Srianand R., Subramanian K., 2007, MNRAS, 377, 285
* Samui et al. (2009) Samui S., Srianand R., Subramanian K., 2009, MNRAS, 398, 2061
* Sasaki (1994) Sasaki S., 1994, PASJ, 46, 427
* Schneider et al. (2008) Schneider R., Salvaterra R., Choudhury T. R., Ferrara A., Burigana C., Popa L. A., 2008, MNRAS, 384, 1525
* Sobral et al. (2009) Sobral D., Best P. N., Geach J. E., Smail I., Kurk J., Cirasuolo M., Casali M., Ivison R. J., Coppin K., Dalton G. B., 2009, MNRAS, 398, L68
* Stark et al. (2007) Stark D. P., Ellis R. S., Richard J., Kneib J., Smith G. P., Santos M. R., 2007, ApJ, 663, 10
* Thoul & Weinberg (1996) Thoul A. A., Weinberg D. H., 1996, ApJ, 465, 608
* Vale & Ostriker (2004) Vale A., Ostriker J. P., 2004, MNRAS, 353, 189
* Vázquez & Leitherer (2005) Vázquez G. A., Leitherer C., 2005, ApJ, 621, 695
* Windhorst et al. (2006) Windhorst R. A., Jansen R. A., Cohen S. H., Mechtley M., Yan H., Conselice C., 2006, in Bulletin of the American Astronomical Society Vol. 38 of Bulletin of the American Astronomical Society, How can the James Webb Space Telescope measure First Light, Reionization, and Galaxy Assembly?. pp 1188–+
* Wyithe et al. (2008) Wyithe J. S. B., Bolton J. S., Haehnelt M. G., 2008, MNRAS, 383, 691
* Wyithe & Loeb (2005) Wyithe J. S. B., Loeb A., 2005, ApJ, 625, 1
* Wyithe & Loeb (2007) Wyithe J. S. B., Loeb A., 2007, MNRAS, 375, 1034
* Zheng et al. (2009) Zheng W., Bradley L. D., Bouwens R. J., Ford H. C., Illingworth G. D., Benítez N., Broadhurst T., Frye B., Infante L., Jee M. J., Motta V., Shu X. W., Zitrin A., 2009, ApJ, 697, 1907
## Appendix: Formation rate and survival probability of haloes in overdense
regions
As expressed in Equation (5), the number density of ionizing photons produced
per unit time is related to the SFR density, which in turn depends on the SFR
in each halo, given by Equation (4), and the number density of haloes of a
certain age, given by Equation (1) for average regions, and by Equation (15)
for overdense regions. We derive Equation (15) in this appendix.
We denote the number density at redshift $z$ of haloes formed between
redshifts $z_{c}$ and $z_{c}+dz_{c}$, with mass between $M$ and $M+dM$, by
$N(M,z,z_{c})dMdz_{c}$. This quantity is related to (1) the formation rate at
redshift $z_{c}$ of haloes with mass between $M$ and $M+dM$, denoted by
$\dot{N}_{\mathrm{form}}(M,z_{c})dM$, and (2) the probability of their
survival at redshift $z$, denoted by $p_{\mathrm{surv}}(z,z_{c})$. We
calculate these two quantities using a technique given by Sasaki (1994),
applied to an overdense region with overdensity $\delta$ and size $R$.
Recall that in extended Press-Schechter theory (Bond et al., 1991), the mass
function of dark matter haloes is defined as the comoving number density of
haloes with mass between $M$ and $M+dM$. At redshift $z$, this quantity is
given by
$N(M,z)dM=\sqrt{\frac{2}{\pi}}\frac{\bar{\rho}_{m}}{M}\exp\left(\frac{-\nu^{2}}{2}\right)\frac{d\nu}{dM}dM,$
(19)
where $\bar{\rho}_{m}$ is the average matter density, and, as before,
$\nu(M,z)\equiv\delta_{c}/[D(z)\sigma(M)]$. The critical overdensity of
collapse of a halo is denoted by $\delta_{c}$, $D(z)$ is the growth function
of density perturbations, and $\sigma(M)$ is the rms value of density
perturbations at the comoving scale corresponding to mass $M$. In a region
with overdensity $\delta$ and linear size $R$, the mass function is enhanced.
This enhancement can be calculated using the excursion set formalism (Bond et
al., 1991). The resulting mass function is again given by Equation (19),
except that now the quantity $\nu(M,z)$ is defined as
$\nu(M,z)\equiv\frac{\delta_{c}/D(z)-\delta}{\sqrt{\sigma^{2}(M)-\sigma^{2}_{R}}},$
(20)
where $\sigma_{R}$ is the rms value of density perturbations at comoving scale
$R$. Closely following Sasaki (1994), we can write
$\dot{N}(M,z)=\dot{N}_{\mathrm{form}}(M,z)-\dot{N}_{\mathrm{dest}}(M,z),$ (21)
where $\dot{N}_{\mathrm{dest}}(M,z)dM$ is the destruction rate at redshift $z$
of haloes of mass between $M$ and $dM$. (The halo formation rate is defined as
the number density of haloes formed per unit time from mergers of lower mass
haloes. Similarly the halo destruction rate is defined as the number density
of haloes destroyed per unit time due to mergers with other haloes.) Here, an
overdot denotes the time derivative. We can write the destruction rate as
$\displaystyle\dot{N}_{\mathrm{dest}}(M,z)$ $\displaystyle=$
$\displaystyle\int_{M}^{\infty}N(M,z)\tilde{Q}(M,M^{\prime},z)dM^{\prime},$
(22) $\displaystyle\equiv$ $\displaystyle\phi(M,z)N(M,z),$ (23)
and the formation rate as
$\dot{N}_{\mathrm{form}}(M,z)=\int_{M_{\mathrm{min}}}^{M}N(M^{\prime},z)Q(M^{\prime},M,z)dM^{\prime},\\\
$ (24)
where $\tilde{Q}(M,M^{\prime},z)$ is the probability that a halo of mass $M$
merges with another halo to result in a halo of mass $M^{\prime}$ per unit
time, and $Q(M^{\prime},M,z)$ that an halo of mass $M$ forming at redshift $z$
has a progenitor of mass $M^{\prime}$. The threshold mass $M_{\mathrm{min}}$
is introduced at this stage to avoid divergence. This gives
$\dot{N}_{\mathrm{form}}(M,z)=\dot{N}(M,z)+\phi(M,z)N(M,z).$ (25)
We now assume that $\phi$ has no characteristic mass scale so that
$\phi(M,z)=M^{\alpha}\tilde{\phi}(z)$. This gives
$\tilde{\phi}(z)=\frac{-\dot{N}(M,z)+\dot{N}_{\mathrm{form}}(M,z)}{N(M,z)M^{\alpha}}.$
(26)
But since the left hand side of Equation (26) is a function of time alone
(through the redshift), the right hand side of this equation also has to be
independent of mass. In particular, we can then set $M=M_{\mathrm{min}}$ in
this equation, giving us
$\tilde{\phi}(z)=\frac{-\dot{N}(M_{\mathrm{min}},z)}{N(M_{\mathrm{min}},z)M_{\mathrm{min}}^{\alpha}}.$
(27)
Now, in the case of the overdense region that we are considering here, we have
$\dot{N}(M,z)=N(M,z)\frac{\dot{D}(z)}{D^{2}(z)}\frac{\delta_{c}}{\delta_{c}/D(z)-\delta}[\nu^{2}(m,z)-1],$
(28)
which gives
$\tilde{\phi}=\frac{\dot{D}}{D^{2}}\frac{\delta_{c}}{\delta_{c}/D(z)-\delta}[\nu^{2}(M_{\mathrm{min}},z)-1]M_{\mathrm{min}}^{-\alpha}.$
(29)
Since our choice of threshold mass $M_{\mathrm{min}}$ is arbitrary, we now
need to take the limit $M_{\mathrm{min}}\rightarrow 0$. However, since
$\nu\rightarrow 0$ in this limit, $\tilde{\phi}$ becomes indeterminate, except
when $\alpha=0$. This implies that we must set $\alpha=0$ for consistency.
This gives $\phi(M,z)=\tilde{\phi}(z)$. Substituting the resultant expression
in Equation (25), we get
$\dot{N}_{\mathrm{form}}(M,z)=N(M,z)\frac{\dot{D}}{D^{2}}\frac{\delta_{c}}{\delta_{c}/D(z)-\delta}\nu^{2}(M,z).$
(30)
This the required formation rate of haloes in an overdense region.
Furthermore, from our definitions of probabilities in Equations (23) and (24),
we can write the probability that a halo that has formed at redshift $z_{c}$
continues to exist at redshift $z$ as
$p_{\mathrm{surv}}(z,z_{c})=\exp\left[-\int_{t(z_{c})}^{t(z)}\phi(t^{\prime})dt^{\prime}\right],$
(31)
which in our case results in
$p_{\mathrm{surv}}(z,z_{c})=\frac{\delta_{c}/D(z)-\delta}{\delta_{c}/D(z_{c})-\delta}.$
(32)
From Equations (30) and (32), we can now write the the comoving number density
$N(M,z,z_{c})dMdz_{c}$ at redshift $z$ of collapsed halos having mass in the
range $M$ and $M+dM$ and redshift of collapse in the range $z_{c}$ and
$z_{c}+dz_{c}$ as
$\begin{split}N(M,z,z_{c})dMdz_{c}=N(&M,z_{c})\left(\frac{\nu^{2}\delta_{c}}{\delta_{c}/D(z_{c})-\delta}\right)\frac{\dot{D}(z_{c})}{D^{2}(z_{c})}\\\
&\times p_{\mathrm{surv}}(z,z_{c})\frac{dt}{dz_{c}}dz_{c}dM,\end{split}$ (33)
This is our Equation (15).
It is worth pointing out that Equations (30) and (32) reduce to the average
forms for halo formation rate and survival probability in the limit
$\delta\rightarrow 0$ and $R\rightarrow\infty$.
|
arxiv-papers
| 2010-08-15T10:06:14 |
2024-09-04T02:49:12.207530
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Girish Kulkarni and T. Roy Choudhury (Harish-Chandra Research\n Institute)",
"submitter": "Girish Kulkarni",
"url": "https://arxiv.org/abs/1008.2509"
}
|
1008.2704
|
# Lepto-Hadronic Origin of $\gamma$-rays from the G54.1+0.3 Pulsar Wind Nebula
Hui Li1, Yang Chen1,2 and Li Zhang3,
1Department of Astronomy, Nanjing University, Nanjing 210093, P. R. China
2Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,
Ministry of Education, China
3Department of Physics, Yunnan University, Kunming,P. R. China
E-mail: ygchen@nju.edu.cn
(Accepted . Received ; in original form)
###### Abstract
G54.1+0.3 is a Crab-like pulsar wind nebula (PWN) with the highest
$\gamma$-ray to X-ray luminosity ratio among all the nebulae driven by young
rotation-powered pulsars. We model the spectral evolution of the PWN and find
it difficult to match the observed multi-band data with leptons alone using
reasonable model parameters. In lepton-hadron hybrid model instead, TeV
photons come mainly from $\pi^{0}$ decay in proton-proton interaction and the
observed photon spectrum can be well reproduced. The newly discovered infrared
loop and molecular cloud in or closely around the PWN can work as the target
for the bombardment of the PWN protons.
###### keywords:
gamma rays: theory – ISM: individual (G54.1+0.3) – radiation mechanisms: non-
thermal
††pagerange: Lepto-Hadronic Origin of $\gamma$-rays from the G54.1+0.3 Pulsar
Wind Nebula–References††pubyear: 2009
## 1 INTRODUCTION
Pulsar wind nebulae (PWNe) are thought to be an efficient accelerator for
cosmic rays with energy above the “knee”. Pulsar, located in the center of
PWN, loses its energy by driving ultra-relativistic wind of electrons,
positrons, and ions. However, it is hard to know the fraction of energy
division of different particle components. The extended $\gamma$-ray emission
from PWN provides an exciting opportunity for studying the acceleration and
radiation mechanism of particles in ultra-relativistic shocks. It has been
long debated whether the very high energy (VHE) emission from PWNe as well as
from supernova remnants (SNRs) is leptonic or hadronic origin. Theoretically,
it has been suggested that some fraction of the pulsar’s spin-down energy can
be converted into nuclei (Cheng et al. 1990; Arons & Tavani 1994), which
indicates that TeV emission from PWNe may contain contribution from both
leptons and hadrons. Indeed, nucleonic models have been used to reproduce the
$\gamma$-rays from Crab and Vela X, respectively (Atoyan et al. 1996; Horns et
al. 2006). Recently, the discovery of TeV emission from G54.1+0.3 by VERITAS
(Acciari et al. 2010) presents a brand new case for highlighting the relative
significance of hadrons in PWNe.
G54.1+0.3 is a Crab-like (Lu et al. 2002) SNR with properties very similar to
the Crab Nebula in both morphology and photon spectral indices. The central
pulsar, PSR J1930+1852, has a period of $P=137$ ms and a period derivative of
$\dot{P}=7.5\times 10^{-13}\rm s~{}\rm s^{-1}$, corresponding to a current
spin-down luminosity of $L_{sd}=1.2\times 10^{37}\rm erg~{}\rm s^{-1}$ and a
characteristic age $\tau_{c}\approx 2900\rm yr$ (Camilo et al. 2002). A faint
X-ray shell was most recently detected surrounding the PWN up to $\sim
6^{\prime}$ from the pulsar (Bocchino, Bandiera, & Gelfand 2009). The SNR has
been suggested to be at a distance of $6.2~{}\rm kpc$ by the HI absorption and
morphological association with a molecular cloud (Leahy et al. 2008).
Recent AKARI observation discovered an infrared (IR) loop, which is explained
to be a star-formation loop around the G54.1+0.3 PWN (Koo et al. 2008) and is
alternatively explained to be the freshly-formed dust in the supernova ejecta
(Temim et al. 2009). Using VLA radio polarization and Spitzer mid-IR
observations, Lang et al. (2009) found a molecular cloud located at the
southern edge of the PWN and suggested an interaction between the PWN and the
cloud. In $\gamma$-rays, VERITAS observed the VHE TeV emission from G54.1+0.3
and found that the efficiency of converting the spin-down energy to
$\gamma$-ray emission is high and the ratio of $\gamma$-ray to X-ray
luminosity is as large as 0.7. This ratio, two orders of magnitudes higher
than that of the Crab, is the highest among all the nebulae supposedly driven
by young rotation-powered pulsars (Acciari et al. 2010). This may imply that
the VHE TeV emission has extra components in addition to the contribution from
commonly-acknowledged energetic leptons scattering background photons. The
newly discovered IR loop and/or molecular cloud around the G54.1+0.3 PWN may
act as an appropriate target for the energetic protons to account for high-
efficiency $\gamma$-ray production from this unusual source (Bartko & Bednarek
2008).
In this letter, we show that the TeV emission from G54.1+0.3 cannot be
accounted for by leptons alone, but can be naturally explained by introduction
of a hadronic component.
## 2 MODEL AND RESULTS
### 2.1 The Pure-Lepton Case
We first try to reproduce the wide-range radiation spectrum of G54.1+0.3 from
radio to TeV using a pure lepton component. For calculating the spectral
evolution of the PWN, we specify the evolution of the time-dependent injection
spectrum and that of the magnetic field in the following.
Let us consider the relativistic wind of leptons produced within the light
cylinder of the pulsar where the spin-down power L(t) is injected into PWN. A
termination shock is formed in the outflowing relativistic wind, where the ram
pressure is balanced by the pressure of surrounding medium, and accelerates
particles to high energies. The leptons produced inside the light cylinder of
the pulsar account for the radio emission, while the wind leptons accelerated
by the shock have a Fermi-type energy spectrum and contribute to the X-ray
emission.
As usual, we assume that the injection spectrum of the relativistic particles
$Q_{\rm inj}(\gamma,t)$ obeys a broken power-law
$Q_{\rm inj}(\gamma,t)=\left\\{\begin{array}[]{ll}Q_{\rm
0}(t)(\gamma/\gamma_{\rm{b}})^{-p_{\rm 1}}&\mbox{ for $\gamma_{\rm
min}\leq\gamma\leq\gamma_{\rm b}$ ,}\\\ Q_{\rm
0}(t)(\gamma/\gamma_{\rm{b}})^{-p_{\rm 2}}&\mbox{ for $\gamma_{\rm
b}\leq\gamma\leq\gamma_{\rm max}$ ,}\end{array}\right.$ (1)
where $Q_{0}$ is normalization coefficient, $\gamma$ is the Lorentz factor of
the relativistic electrons and positrons, and the minimum ($\gamma_{\rm
min}$), maximum ($\gamma_{\rm max}$), and break ($\gamma_{\rm b}$) Lorentz
factors together with the energy indices ($p_{1}$ and $p_{2}$) are assumed
time-independent. Parameter $\gamma_{\rm max}$ is obtained so as to confine
the accelerated electrons within the PWN (i.e., the electrons’s Larmor radius
must be less than the radius of the PWN) (Venter & de Jager 2006)
$\gamma_{\rm max}\approx\frac{e}{2m_{\rm e}c^{2}}\sqrt{\frac{\sigma
L(t)}{(1+\sigma)c}},$ (2)
where magnetization parameter $\sigma$ is the ratio of the electromagnetic
energy flux to the lepton energy flux at the wind shock of the PWN. Parameter
$\gamma_{\rm min}=100$ is assumed so as to reproduce the flux of the observed
minimum frequency at radio wavelengths. Bucciantini et al. (2010) found that
$\gamma_{\rm b}$ is at a similar value in a narrow range of $10^{5}$–$10^{6}$
for several PWNe of a variety of ages, which is closely related to the working
of pulsar magnetospheres, pair multiplicity, and the particle acceleration
mechanisms. Therefore, here we adopt $\gamma_{\rm b}=5\times 10^{5}$ without
loss of generality.
The injection spectrum can be related to the spin-down power $L(t)$ of the
pulsar at given time $t$ by assuming that a fraction ($\eta_{e}$) of the spin-
down power is converted into lepton luminosity: $\eta_{e}L(t)=\int
Q(\gamma,t)\gamma m_{e}c^{2}d\gamma$. For a spin-down pulsar,
$L(t)=L_{0}[1+(t/\tau_{0})]^{-(n+1)/(n-1)}$, where $L_{0}$ is the initial
spin-down power, $\tau_{0}$ the characteristic timescale, and $n$ the breaking
index (here we adopt $n=3$ for simplicity). Thus the normalization parameter
$Q_{\rm 0}(t)$ can be derived as (Tanaka & Takahara 2010)
$\displaystyle Q_{\rm 0}(t)$ $\displaystyle=\frac{L_{\rm 0}\eta_{e}}{m_{\rm
e}c^{2}}\left(1+\frac{t}{\tau_{\rm 0}}\right)^{-2}\times$ (3)
$\displaystyle\left[\frac{\gamma_{\rm b}^{2}(p_{\rm 1}-p_{\rm 2})}{(2-p_{\rm
1})(2-p_{\rm 2})}+\frac{\gamma_{\rm b}^{p_{\rm 2}}\gamma_{\rm max}^{2-p_{\rm
2}}}{2-p_{\rm 2}}-\frac{\gamma_{\rm b}^{p_{\rm 1}}\gamma_{\rm min}^{2-p_{\rm
1}}}{2-p_{\rm 1}}\right]^{-1}.$
On the assumption of magnetic-field energy conservation (see Tanaka & Takahara
2010 for the comparison of various approximations of magnetic field
evolution),
$\frac{4\pi}{3}R_{\rm{PWN}}^{3}(t)\cdot\frac{B^{2}(t)}{8\pi}=\int_{0}^{t}\eta_{B}L(t^{\prime})dt^{\prime},$
(4)
the time-varying field strength of the nebula is given by
$B(t)=\left[\frac{6\eta_{B}L_{0}\tau_{0}t}{R_{\rm
PWN}^{3}(t+\tau_{0})}\right]^{1/2}$ (5)
where $\eta_{B}$ is the fraction of spin-down energy converted to the magnetic
energy and $R_{\rm PWN}$ the average radius of the PWN. (In parenthesis, the
magnetization parameter is thus essentially $\sigma\sim\eta_{\rm B}/\eta_{\rm
e}$.) Because the young G54.1+0.3 PWN ($\sim 2900$yr) may be in an evolution
stage before the reverse shock passage (typically at $1\times 10^{4}$yr, e.g.,
Reynolds & Chevalier; Gelfand 2009), we also assume that the PWN is freely
expanding at velocity $v_{\rm PWN}$ and thus have $v_{\rm PWN}\sim 550(R_{\rm
PWN}/1.8{\rm pc})(t/2900{\rm yr})^{-1}$ km s-1.
The volume-integrated particle number as a function of energy is described by
the continuity equation in the energy space:
$\frac{\partial}{\partial
t}N(\gamma,t)+\frac{\partial}{\partial\gamma}\left[\dot{\gamma}(\gamma,t)N(\gamma,t)\right]=Q_{\mathrm{inj}}(\gamma,t)-\frac{N(\gamma,t)}{\tau_{esc}(t)}$
(6)
where $\dot{\gamma}(\gamma,t)$ is the cooling rates of the relativistic
leptons including the synchrotron radiation, the inverse Compton scattering
off the cosmic microwave background (CMB) and ambient IR radiation, and the
adiabatic expansion, i.e.,
$\dot{\gamma}(\gamma,t)=\dot{\gamma}_{\mathrm{syn}}(\gamma,t)+\dot{\gamma}_{\mathrm{IC}}(\gamma)+\dot{\gamma}_{\mathrm{ad}}(\gamma,t),$
(7)
and $\tau_{esc}$ is the escape timescale and can be estimated as in Bohm
diffusion (e.g., Zhang et al. 2008),
$\tau_{esc}\approx 9\times 10^{5}\bigg{[}\frac{B(t)}{80\rm\mu
G}\bigg{]}\bigg{(}\frac{E_{e}}{10\rm TeV}\bigg{)}^{-1}\bigg{[}\frac{R_{\rm
PWN}(t)}{1.8\rm pc}\bigg{]}^{2}\rm yr,$ (8)
where the current magnetic field strength $80\mu{\rm G}$ (see §3.1) is used.
The adiabatic loss $\dot{\gamma}_{\rm ad}=-\gamma/t$ is the dominant cooling
process for the low energy particles and insignificant for the high energy
ones.
The time-dependent lepton distribution is numerically solved from the
continuity equation (6). Then multi-wavelength non-thermal emission can be
calculated for the process of synchrotron radiation and inverse Compton
scattering, with photon spectra plotted in Figures 1 and 2 (as described
below).
Here the $\gamma$-rays are considered to purely come from leptons scattering
the soft radiation field (CMB, IR and optical photons in the Galactic plane,
and the IR-optical-UV emission of possible young stellar objects (YSOs) in the
IR loop). The IR background at the Galactic disc is characterized by
temperature $25$K and energy density two times larger than the CMB, while the
optical background by temperatures between 5000 and $10^{4}$ K and energy
densities equal to the CMB. The incident IR photons from the SNR are defined
by a $\sim 90$ K blackbody radiation with the energy density $\sim 5.3\times
10^{-12}\rm erg~{}cm^{-3}$ based on the Spitzer IRAC fluxes at $24\mu$m and
$70\mu$m from Temim et al. (2009), a factor of roughly 5 larger than the IR
energy density in the Crab Nebula and 13 larger than the energy density in the
CMB. In the calculation we also take into account the possible IR-optical-UV
starlight from 11 possible YSOs, which has an energy density $\sim 4.4\times
10^{-11}\rm erg~{}cm^{-3}$ with a blackbody temperature $T\sim 35000K$ (Koo et
al. 2008). The IC flux is dominated by scattering with the IR photons from the
SNR, while the IC scattering with other components are insignificant by
comparison. Note that the power of synchrotron self-Compton emission to
synchrotron emission $P_{\rm SSC}/P_{\rm syn}=U_{\rm syn}/U_{B}<10^{-2}$ (here
eq.(27) in Tanaka & Takahara 2010 is used), the contribution of $\gamma$-ray
emission for G54.1+0.3 PWN from IC scattering off the synchrotron radiation is
negligible.
Figure 1: Comparison of the predicted spectra in the pure-lepton Models A
(solid line) and B (dashed line) with the observed data for G54.1+0.3 in radio
(Natasha et al. 2008; Lang et al. 2009), X-rays (Lang et al. 2009) and
$\gamma$-rays (Acciari et al. 2010). The model parameters are described in the
text of §2.1. The red dashed line shows the 1 year, 5$\sigma$ sensitivity for
the Fermi LAT (Fermi LAT 2007).
For the physical parameters of the PWN, we set $L_{0}\approx 1.4\times
10^{39}\rm ergs~{}s^{-1}$, $\gamma_{b}=5\times 10^{5}$, and $p_{1}=1.2$
according to previous studies (Camilo et al. 2002; Lang et al. 2009;
Bucciantini et al. 2010) and leave other three parameters, $\eta_{e}$,
$\eta_{B}$, and $p_{2}$, adjustable. For comparison, we develop three sets of
parameters for leptonic model. In Model A, we reproduce the observed results
of radio to X-ray emission (which are synchrotron) and get $\eta_{e}=6\%$,
$\eta_{B}=8\%$, and $p_{2}=2.4$. As can be seen in Figure 1 (the solid line),
the resulting TeV emission from leptons is lower than the observed flux by
more than an order of magnitude. In Model B, we change parameters to reproduce
the observed TeV emission by IC scattering soft photon fields described above.
The adopted parameters are $\eta_{e}=92\%$, $\eta_{B}=8\%$, and $p_{2}=2.1$.
The resulting synchrotron radio and X-ray emission (the dashed line in Figure
1) excess the observation data by more than an order of magnitude. The current
magnetic field strength for Model A and B, $80\mu\rm G$ (derived from
observation, see §3.1), has been used in Eq.(5). In order to match both the
synchrotron and IC emission to the observed data, we explore the parameter
space and obtain the third model (Model C) (Figure 2) with $\eta_{\rm
e}=99.8\%$, $\eta_{\rm B}=0.15\%$, and $p_{2}=2.8$. However, this corresponds
to a weak magnetic field $\sim 10\mu\rm G$. If we adopt an age of $2000\,{\rm
yr}$ for this PWN as obtained by Bocchino et al.(2009) in their dynamic
evolution model, other than $2900\,{\rm yr}$, then lower field strength would
be needed in Model C. Such low values of the field strength are inconsistent
with that derived from observation, as will be discussed in §3.1. Therefore,
it is hard for a pure-lepton model to reproduce the radio, X-ray, and TeV data
simultaneously, and thus the leptons alone cannot account for the $\gamma$-ray
emission.
Figure 2: The same as Figure 1, but for pure-lepton Model C (solid line). The
parameters are described in the text of §2.1. The IC flux (the solid line on
the right side) is dominated by scattering with the IR photons from the SNR,
while the IC scattering with the IR photons from Galactic diffusion (dashed),
the CMB (dotted), and the starlight of the possible YSOs (dashed-dotted) are
also shown.
### 2.2 The Lepton-Hadron Hybrid Case
We now consider the contribution to the TeV emission from a hadronic component
besides the leptonic contribution. In this model, both leptons and ions
extracted from the charged polar cap region are accelerated in the rotating
magnetospheres of neutron stars and PWN termination shocks (e.g., Zhang et al.
2009). For simplicity, we assume the protons gain energy from central pulsar
and are represented by a power-law spectrum which is common for the
acceleration process. Then the total energy of protons of the PWN is
$W_{p}=\int
A_{p}E_{p}^{-\alpha_{p}}E_{p}dE_{p}=\int_{0}^{t}\eta_{p}L(t^{\prime})dt^{\prime}$,
where $\eta_{p}$ is the energy fraction converted to protons, $A_{\rm p}$ the
normalization coefficient and $\alpha_{p}$ the spectral index of accelerated
protons. So the energy released from the pulsar consists of the kinetic energy
of particles ($\eta_{e}$ and $\eta_{p}$) and the magnetic energy ($\eta_{B}$).
For the energy, $E_{p}$, of the accelerated protons, the rest energy of
protons ($9.4\times 10^{8}$ eV) is adopted as minimum and the energy at the
“knee” ($3\times 10^{15}$ eV) as the maximum. Note that the energy converted
into leptons and magnetic field in Model A is only a small fraction
($\eta_{e}=5\%$ and $\eta_{B}=8\%$, respectively) of the total spin-down
energy of central pulsar. In fact, in the study of the Vela X PWN, Horns et
al. (2006) have questioned where the remaining energy injected from pulsar is
and suggested a hadronic origin of TeV emission. Hence, we assume
$\eta_{p}=87\%$ in the lepton-hadron hybrid case (denoted as Model D).
The Bohm diffusion timescale of the PWN particles determined from Eq. (8)
($\sim 10^{4}\,{\rm yr}$) is much longer than the PWN age. Therefore, the
protons are considered to be well confined in the PWN and the escape losses of
protons are negligible. The cooling time of p-p interaction is (e.g.,
Aharonian 2004) $t_{\rm pp}\approx 1.8\times 10^{6}(n_{\rm b}/30\,{\rm
cm}^{-3})^{-1}\,{\rm yr}$, much longer than the age of G54.1+0.3, where
$n_{\rm b}$ is the average density of target baryons in the PWN (see below).
Hence the collision losses of the PWN protons are negligible as well. Also
because $t_{\rm pp}$ is almost energy-independent in the energy region above
$1\rm GeV$, the total spectrum of protons remains unchanged (Aharonian 2004).
The contribution from the secondary leptons that are created by protons
interaction to the overall spectrum is negligible too, as compared with the
dominant contribution of the primary leptons (Horns et al. 2006; Zhang et al.
2009).
In addition to the contribution from the leptons as given in Model A, we
calculate that from p-p interaction so as to match the observed TeV flux. For
the $\pi^{0}$ decay ensuing from p-p collision, the analytic emissivity
developed by Kelner et al. (2006) is used. It is difficult to determine the
detail process of energetic protons captured by the baryonic targets, since
this process depends on geometry of the PWN and the targets and anisotropy of
the magnetic field and diffusion coefficient. Thus, we assume that a small
fraction ($\xi$) of all hadrons is captured by baryonic targets (as suggested
by Bartko & Bednarek 2008). The wide-range spectrum of the PWN can now be well
reproduced with $\xi\sim 8\times 10^{-3}(n_{\rm b}/30\,{\rm cm}^{-3})^{-1}$
and the results are shown in Figure 3. Here a target baryon density $\sim
30\,{\rm cm}^{-3}$ has been adopted from the estimate of the IR clump density
(Temin et al. 2010); this number can also be typical of the density of the
molecular materials, which Koo et al. (2009) and Lang et al. (2010) reported
to detect. Apparently, even such a low capture efficiency is sufficient for
hadrons to produce the observed flux of TeV emission.
Figure 3: The same as Fig.1, but for lepton-hadron hybrid Model D. The
parameters are described in the test of §2.2. The solid line on the right side
is dominated by $\pi^{0}$ decay ensuing from p-p interaction. The inverse
Compton scattering with IR photos from SNR (dashed-dotted line), IR photos
from Galactic diffusion (dashed), starlight of the possible YSOs (dashed-
dotted-dotted) and the CMB (dotted) are also shown.
## 3 Discussion
In the pure-lepton case, Model C seems to marginally match the wide-range
spectrum of the G54.1+0.3 PWN; by comparison, however, the lepton-hadron
hybrid case (Model D) can reproduce the spectrum better and more physical in
the following aspects.
### 3.1 Magnetic field
In §2.1, the field strength obtained in Model C (the lepton case) is
$10\mu{\rm G}$ or even lower. Such values of field strength are actually
weaker than that derived from observation. Based on radio luminosity, Lang et
al. (2009) derived an equipartition field of $38\mu{\rm G}$. However, they
suggested stronger field in the light of the strong polarization which is
organized on large scales of the nebula and implies the PWN is filled with
magnetically-dominated plasma. They also found an alternative field strength
of 80–$200\mu{\rm G}$ by using the lifetime of the X-ray emitting particles.
In Model D (the lepton-hadron case), however, we use $80\mu{\rm G}$ which can
typify the field strength estimated by Lang et al.
### 3.2 TeV index
In Model C, the calculated TeV slope ($\sim 2.6$–3) of the IC spectrum cannot
well match the VERITAS data point (with photon index 2.4, Acciari et al.
2010). Matching the TeV slope would entail a lepton ensemble with a
unreasonable large energy index 3.8. Even if the energy losses in high energy
leptons are considered, we, using the time-dependent model, find the energy
index of accelerated leptons by relativistic shock is 2.8, still considerably
higher than the universal power-law index $2.2$–2.3 for Fermi-type
acceleration by the shock of large Lorentz factor using different approaches
(e.g., Horns et al. 2007). As a contrast, the observed slope is easily
reproduced by protons p-p interaction with a mild proton index
$\alpha_{p}=2.4$. This proton index is fortuitously similar to the lepton
index that is used to reproduce the synchrotron X-rays in Model A.
### 3.3 Baryonic targets
The IR loop closely around the G54.1+0.3 PWN discovered by AKARI was suggested
to be star-forming region (Koo et al. 2008), while it was also argued to be
the freshly formed supernova dust heated by early-type stars belonging to a
cluster in which the supernova exploded (Temim et al.2009). It was also
reported that a molecular cloud is found to be located at the southern edge of
the PWN by the VLA radio and Spitzer mid-IR observations and thus an
interaction between the PWN with the cloud was suggested (Lang et al. 2009).
These components within or surrounding the PWN, whatever they are, may readily
be a baryonic target for the bombardment of the PWN protons, and therefore it
is very reasonable to expect the $\gamma$-ray contribution from the hadron
interaction. This is the very case that we address in Model D. This scenario
seems to naturally explain the exceptionally high $\gamma$-ray to X-ray
luminosity ratio of G54.1+0.3 among all the rotation-powered PWNe.
The Fermi observation at GeV band will be important to discriminate between
the leptonic model and the hadronic model. In the pure-lepton model (cases A,
B, and C; see Figures 1 and 2), the theoretical GeV $\gamma$-ray flux of the
PWN is basically below the 1 year, $5\sigma$ sensitivity of the Fermi LAT,
while the lepton-hadron hybrid model (case D; see Figure 3) predicts a GeV
flux above the sensitivity.
## 4 CONCLUSION
We have calculated the multi-band non-thermal emission from the G54.1+0.3 PWN
in both the pure-lepton case and the lepton-hadron hybrid case. In the lepton
case, we find that the leptons that are responsible for the radio and X-ray
synchrotron cannot alone account for the TeV $\gamma$-ray emission by IC
scattering. An addition of hadron contribution by p-p interaction can well
reproduce the observation spectrum. The lepton-hadron hybrid scenario is
strongly supported by the most recently discovered IR loop and molecular cloud
in or closely around the PWN. This scenario can also shed light on the study
of the PWNe with high $\gamma$-ray to X-ray luminosity ratios.
## Acknowledgments
We thank Q. Daniel Wang, Rino Bandiera, and the anonymous referee for helpful
comments on the manuscripts. Y.C. acknowledges support from NSFC grant
10725312. L.Z. acknowledges support from NSFC grants 10778702 and 10803005 and
Yunnan Province under grant 2009 OC. The authors also acknowledge support from
the 973 Program grant 2009CB824800.
## References
* Acciari et al. (2010) Acciari, V. A., et al. 2010, arXiv:1005.0032
* Aharonian (2004) Aharonian, F. A. 2004, Very high energy cosmic gamma radiation : a crucial window on the extreme Universe, by F.A. Aharonian. River Edge, NJ: World Scientific Publishing, 2004,
* Arons & Tavani (1994) Arons, J., & Tavani, M. 1994, ApJS, 90, 797
* Atoyan & Aharonian (1996) Atoyan, A. M., & Aharonian, F. A. 1996, MNRAS, 278, 525
* Bartko & Bednarek (2008) Bartko, H., & Bednarek, W. 2008, MNRAS, 385, 1105
* Bocchino et al. (2010) Bocchino, F., Bandiera, R., & Gelfand, J. 2010, arXiv:1004.3515
* Bucciantini et al. (2010) Bucciantini, N., Arons, J., & Amato, E. 2010, arXiv:1005.1831
* Camilo et al. (2002) Camilo, F., Lorimer, D. R., Bhat, N. D. R., Gotthelf, E. V., Halpern, J. P., Wang, Q. D., Lu, F. J., & Mirabal, N. 2002, ApJL, 574, L71
* Cheng et al. (1990) Cheng, K. S., Cheung, T., Lau, M. M., Yu, K. N., & Kwok, P. W. 1990, Journal of Physics G Nuclear Physics, 16, 1115
* Fermi. (2007) Fermi LAT Performance 2007, http://www-glast.slac. stanford.edu/software/IS/glast_lat_performance.htm
* Gelfand et al. (2009) Gelfand, J. D., Slane, P. O., & Zhang, W. 2009, ApJ, 703, 2051
* Horns et al. (2006) Horns, D., Aharonian, F., Santangelo, A., Hoffmann, A. I. D., & Masterson, C. 2006, A&Ap, 451, L51
* Horns et al. (2007) Horns, D., Aharonian, F., Hoffmann, A. I. D., & Santangelo, A. 2007, Ap&SS, 309, 189
* Hurley-Walker et al. (2009) Hurley-Walker, N., et al. 2009, MNRAS, 396, 365
* Kelner et al. (2006) Kelner, S. R., Aharonian, F. A., & Bugayov, V. V. 2006, PRvd, 74, 034018
* Koo et al. (2008) Koo, B.-C., et al. 2008, ApJL, 673, L147
* Lang et al. (2010) Lang, C. C., Wang, Q. D., Lu, F., & Clubb, K. I. 2010, ApJ, 709, 1125
* Leahy et al. (2008) Leahy, D. A., Tian, W., & Wang, Q. D. 2008, AJ, 136, 1477
* Lu et al. (2002) Lu, F. J., Wang, Q. D., Aschenbach, B., Durouchoux, P., & Song, L. M. 2002, ApJL, 568, L49
* Reynolds & Chevalier (1984) Reynolds, S. P., & Chevalier, R. A. 1984, ApJ, 278, 630
* Tanaka & Takahara (2010) Tanaka, S. J., & Takahara, F. 2010, ApJ, 715, 1248
* Temim et al. (2010) Temim, T., Slane, P., Reynolds, S. P., Raymond, J. C., & Borkowski, K. J. 2010, ApJ, 710, 309
* Venter et al. (2006) Venter, C., & de Jager, O. C. 2006, in Proc. 363rd WE–Heraeus Seminar on Neutron stars and pulsars, ed. W. Becker & H.-H. Huang (MPE Rep. 291) (Garching: MPI extraterr. Phys.), 40
* Zhang et al. (2008) Zhang, L., Chen, S. B., & Fang, J. 2008, ApJ, 676, 1210
* Zhang & Yang (2009) Zhang, L., & Yang, X. C. 2009, ApJL, 699, L153
|
arxiv-papers
| 2010-08-16T16:16:59 |
2024-09-04T02:49:12.217764
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui Li (NJU), Yang Chen (NJU) and Li Zhang (YNU)",
"submitter": "Yang Chen",
"url": "https://arxiv.org/abs/1008.2704"
}
|
1008.2829
|
# Pseudoscalar mixing in $J/\psi$ and $\psi(2S)$ decay
Dai-Hui Wei, Yong-Xu Yang
College of Physics and Technology, Guangxi Normal University, Guilin 541004,
China
Based on the branching fractions of $J/\psi(\psi(2S))\rightarrow VP$ from
different collaborations, the pseudoscalar mixing is extensively discussed
with a well established phenomenological model. The mixing angle is determined
to be $-14^{\circ}$ by fitting to the new world average if only quark content
is considered. After taking into account the gluonic content in $\eta$ and
$\eta^{\prime}$ simultaneously, the investigation shows that $\eta$ favors
only consisting of light quarks, while the gluonic content of $\eta^{\prime}$
is $Z_{\eta^{\prime}}^{2}=0.30\pm 0.24$.
PACS numbers: 13.25.Gv,14.40.Ag
## 1 Introduction
As the ground pseudoscalar nonet, $\pi$, $K$, $\eta$ and $\eta^{\prime}$, in
the constituent quark model, their masses and widths are determined with high
precision and the main decay modes are also observed[1] in addition to the
forbidden and rare decays. However there is one issue, pseudoscalar mixing,
remains not completely settled, which has been discussed for many times with
different transitions. The linear Gell-Mann-Okubo(GMO) mass relation[2] gives
a mixing angle, $\theta_{P}=-11^{\circ}$, which is hardly consistent with the
value, $\theta_{P}=-24.6^{\circ}$, obtained from the quadratic GMO mass
formula by replacing the meson masses by their squares. The full set of
$J/\psi$ decays into a vector and a pseudoscalar was measured by MarkIII, and
the phenomenological analysis of mixing angle is determined to be
$\theta_{P}=(-19.2\pm 1.4)^{\circ}$[3], which was confirmed by DM2[4]. Both of
them got the conclusion that $\eta$ and $\eta^{\prime}$ consist of light
quarks, with no contribution from gluonium or radial excitation states. After
that an important work was performed by Bramon and Scadron[5, 6], taking into
account $\omega$-$\phi$ mixing in the analysis for $J/\psi\rightarrow VP$, a
weighted $\theta_{P}$ is calculated to be $(-15.5\pm 1.3)^{\circ}$ based on
many different transitions. For a nice review based on the discussions before
2000, see[7] in which the reasonable range of $\eta$-$\eta^{\prime}$ mixing
angle is believed to be $-20^{\circ}\sim-10^{\circ}$.
Recently the new experimental data on $J/\psi\rightarrow VP$ and
$\psi(2S)\rightarrow VP$ were reported by BES[8, 9, 10, 11, 12, 13, 14],
BABAR[15, 16, 17, 18] and CLEO[19]. It is worth pointing out that some of the
new measurements are not well consistent with the previous works. Take
$J/\psi\rightarrow\rho\pi$ for example, the branching fractions measured by
BES is $(2.10\pm 0.12)\%$[8], subsequently confirmed by BABAR[15], which is
larger than the world average $(1.28\pm 0.10)\%$[20], about 64%. This
significant change stimulates new interest in this issue[21, 22, 23, 24]. The
analysis in Ref.[23] indicates it is difficult to get reasonable results with
the updated branching fractions of $J/\psi\rightarrow\rho\pi$, however, the
results in Ref.[24] performed with the same data and phenomenological model
seems reasonable. This discrepancy motivated us to reanalyze the full set of
$J/\psi\rightarrow VP$ data. Actually it is difficult for us to compare the
results obtained with different sets of parameters in one time, in this paper
we would like to discuss this issue for different cases, e.g. fix SU(3)
breaking term $x$ to 0.64, 0.82 or 1.
## 2 Notation
The physical eigenstates $\eta$, $\eta^{\prime}$ are the mixture of octet,
singlet and gluonium. And they are defined as,
$\displaystyle|\eta>=X_{\eta}|N>+Y_{\eta}|S>+Z_{\eta}|G>,$ (1)
$\displaystyle|\eta^{\prime}>=X_{\eta}^{\prime}|N>+Y_{\eta}^{\prime}|S>+Z_{\eta}^{\prime}|G>.$
where, $N=\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$, $S=s\bar{s}$ and $G$ for
gluonium; $X_{i}$, $Y_{i}$ and $Z_{i}$ denote the magnitude of non-strange,
strange contents and gluonium in $\eta$ and $\eta^{\prime}$.
The above form can be written in terms of the three Euler angles, with
$\displaystyle X_{\eta}=\cos\phi_{p}\cos\phi_{G1},$ (2) $\displaystyle
Y_{\eta}=-\sin\phi_{p}\cos\phi_{G1},$ $\displaystyle Z_{\eta}=-\sin\phi_{G1},$
$\displaystyle
X_{\eta^{\prime}}=\cos\phi_{P}\cos\phi_{G2}-\sin\phi_{P}\sin\phi_{G2}\sin\phi_{G1},$
$\displaystyle
Y_{\eta^{\prime}}=\sin\phi_{P}\cos\phi_{G2}+\cos\phi_{P}\sin\phi_{G2}\sin\phi_{G1},$
$\displaystyle Z_{\eta^{\prime}}=-\sin\phi_{G2}\cos\phi_{G1}.$
If we only consider the simplest case and neglect possible mixing of the
$\eta$ and $\eta^{\prime}$ with other pseudoscalar states,
$\eta$-$\eta^{\prime}$ mixing is characterized by a single mixing angle
$\theta_{P}$.
$\displaystyle|\eta>=\cos\theta_{P}|\eta_{8}>-\sin\theta_{P}|\eta_{0}>,$ (3)
$\displaystyle|\eta^{\prime}>=\sin\theta_{P}|\eta_{8}>+\cos\theta_{P}|\eta_{0}>.$
where $\eta$ and $\eta^{\prime}$ are the orthogonal mixture of the respective
singlet and octet iso-spin zero states. $\eta_{0}$ and $\eta_{8}$ are SU(3)
quark basis states which are denoted as
$\eta_{0}=\frac{1}{\sqrt{3}}|u\bar{u}+d\bar{d}+s\bar{s}>$ and
$\eta_{8}=\frac{1}{\sqrt{6}}|u\bar{u}+d\bar{d}-2s\bar{s}>$ respectively.
In terms of quark basis, the $\eta$ and $\eta^{\prime}$ include non-strange
and strange contents. In the flavor SU(3) quark model, they are defined
through quark-antiquark($q\bar{q}$) basis states as,
$\displaystyle
X_{\eta}=Y_{\eta^{\prime}}=\sqrt{\frac{1}{3}}\cos\theta_{P}-\sqrt{\frac{2}{3}}\sin\theta_{p}=\cos\phi_{P},$
(4) $\displaystyle
X_{\eta^{\prime}}=-Y_{\eta}=\sqrt{\frac{1}{3}}\sin\theta_{P}+\sqrt{\frac{2}{3}}\cos\theta_{P}=\sin\phi_{P}.$
where, $\theta_{P}=\phi_{P}-54.7^{\circ}$.
## 3 Phenomenological model
$J/\psi$ and $\psi(2S)$ have the similar decay mechanism and are suppressed by
Okubo-Zweig-Iizuka (OZI) rule. Both of them decay into a vector and
pseudoscalar meson via three gluon annihilation and electromagnetic decays.
Therefore, in this paper, the phenomenological model for $J/\psi\rightarrow
VP$ in Ref.[25] is simply applied in $\psi(2S)$ decays to discuss the
$\eta$-$\eta^{\prime}$ mixing and other physics.
A first-order parameterization of the amplitudes appears in Ref.[25] and is
described there in detail. The amplitude, which has contributions from both
the three gluon annihilation and electromagnetic processes can be expressed in
terms of an SU(3) symmetric single-OZI(SOZI) amplitude $g$, an electromagnetic
amplitude $e$ (the coupling strength $e$ has a relative phase $\theta_{e}$ to
the strength $g$ because these are produced from different origins) and the
nonet-symmetry-breaking double-OZI(DOZI) amplitude $r$, relative to $g$. SU(3)
violation has been accounted for by a pure octet SU(3) breaking term. The
SU(3) breaking term in strong interaction and electromagnetic process are
expressed by (1-$s$) and $x$, respectively. A factor (1-$s$) for every strange
quark contributing to $g$ and a factor for $x$ for a strange quark
contributing to $e$. The factor $s_{v}$($s_{p}$) is for the strange
vector(pseudoscalar) contributing to $r$.
In spite of these simplified assumptions this phenological model contains a
rather large number of parameters ($g$, $e$, $r$, $s$, $s_{p}$, $s_{v}$, $x$,
$\theta_{P}$ and $\theta_{e}$). This $x$ can be well determined via
$V\rightarrow P\gamma$ and $P\rightarrow V\gamma$ data, we reanalyzed it using
the phenomenological model in Ref.[26] and the branching fractions of
$V\rightarrow P\gamma$ and $P\rightarrow V\gamma$ in Ref.[1], $x$ is
determined to be $0.82\pm 0.05$ and $\theta_{V}=(3.2\pm 0.9)^{\circ}$ and
$\theta_{P}=(-12.9\pm 0.5)^{\circ}$ which are in good agreement in those in
Ref.[21]. To further simplify it again, $s_{p}$ is ignored in this paper and
$s_{v}$ is discussed below with two assumptions($s_{v}=0$ and $s_{v}=s$).
Table 1: Branching fractions of $J/\psi\rightarrow VP$($\times 10^{-4}$) Decay Modes | MarkIII | DM2 | BES | BABAR | PDG2010
---|---|---|---|---|---
$\rho\pi$ | $142\pm 1\pm 9$ | $132\pm 20$ | $210\pm 12\pm 20.1$ | $218\pm 19$ | $169\pm 15$
$\rho\eta$ | $1.93\pm 0.13\pm 0.29$ | $1.94\pm 0.17\pm 0.29$ | | | $1.93\pm 0.23$
$\rho\eta^{\prime}$ | $1.14\pm 0.14\pm 0.16$ | $0.83\pm 0.30\pm 0.12$ | | | $1.05\pm 0.18$
$\phi\pi^{0}$ | $<0.068$ | | $<0.064$ | | $<0.064$
$\phi\eta$ | $6.61\pm 0.45\pm 0.78$ | $6.4\pm 0.4\pm 1.1$ | $8.98\pm 0.24\pm 0.89$ | $14\pm 6\pm 1$ | $7.5\pm 0.8$
$\phi\eta^{\prime}$ | $3.08\pm 0.34\pm 0.36$ | $4.1\pm 0.3\pm 0.8$ | $5.46\pm 0.31\pm 0.56$ | | $4.0\pm 0.7$
$\omega\pi^{0}$ | $4.82\pm 0.19\pm 0.64$ | $3.6\pm 0.28\pm 0.54$ | $5.38\pm 0.12\pm 0.65$ | | $4.5\pm 0.5$
$\omega\eta$ | $17.1\pm 0.8\pm 2.0$ | $14.3\pm 1.0\pm 2.1$ | $23.52\pm 2.73$ | $14.4\pm 4.0\pm 1.4$ | $17.4\pm 2.0$
$\omega\eta^{\prime}$ | $1.66\pm 0.17\pm 0.19$ | $1.8^{+1.0}_{-0.8}\pm 0.3$ | $2.26\pm 0.43$ | | $1.82\pm 0.21$
$K^{*-}K^{+}+c.c.$ | $52.6\pm 1.3\pm 5.3$ | $45.7\pm 1.7\pm 7.0$ | | $52\pm 4\pm 1$ | $51.2\pm 3.0$
$K^{*0}\bar{K^{o}}+c.c.$ | $43.3\pm 1.2\pm 4.5$ | $39.6\pm 1.5\pm 6.0$ | | $48\pm 5\pm 1$ | $43.9\pm 3.1$
## 4 Results
The experimental data sets shown in Table 1 are analyzed with the least
squares method to determine the coupling strengths and mixing angle. To
clarify the results obtained from different data set, we divided it into
several subsections to investigate the pseudoscalar mixing.
### 4.1 Analysis of $J/\psi\rightarrow VP$ from MarkIII and DM2
We start to perform the fit to experimental data with the simplest case, the
$\omega$-$\phi$ mixing and gluon content are ignored. Actually the treatment
on SU(3)-breaking parameter $x$ and the second order corrections $s_{v}$ in
Ref.[3] and Ref.[4] are different. The $x$ is set to 1 and the correction
terms $s_{v}=0$, is ignored in MarkIII analysis, while $x$ is fixed to 0.64
and the correction terms $s_{v}=s$ is included in DM2 analysis. To clearly
compare the difference between them, all the possible combinations are
considered to perform the fit.
A fit to the data without considering SU(3) breaking as well as in MarkIII
analysis yields $\theta_{P}=(-13.95\pm 2.39)^{\circ}$ with
$\chi^{2}/d.o.f=9.0/4$, which is obviously inconsistent with the value
$(-19.2\pm 1.4)^{\circ}$[3]. After tuning the parameter, we also get a
reasonable results which are the same as those in Ref.[3],
$g=1.10\pm 0.03$, $s=0.12\pm 0.03$, $e=0.122\pm 0.005$, $\theta_{e}=1.25\pm
0.12$, $\theta_{p}=(-19.34\pm 1.40)^{\circ}$, $r=-0.15\pm 0.09$.
But the goodness of fit, $\chi^{2}/d.o.f=10.1/4$, seems slightly worse.
Compared with the results listed in the first column of Table 2, $s$ and $r$
also change significantly. The results of the fits performed with $x=0.64$ and
$x=0.82$ are also given in Table 2. Apart from the mixing angle, the values of
other parameters are also consistent with the previous fit.
If $s_{v}$ is replaced with $s$ and $x$ is fixed to 0.64, the fit gives
$\theta_{P}=(-18.59\pm 1.40)^{\circ}$ with $\chi^{2}/d.o.f=9.0/4$ which is in
good agreement with DM2’s result $\theta_{P}=(-19.1\pm 1.4)^{\circ}$.
Meanwhile we also checked the fits with $x=1$ and $x=0.82$ and the results are
listed in Table 2. Compared with the results without considering the
contribution of $s_{v}$, the results change significantly, in particular for
$s$, $r$ and $\theta_{p}$. This is reasonable because the two phenomenological
models are slightly different. The fit to DM2 data is also performed to check
the discrepancy discussed above. In the DM2’s analysis, the common error of
the branching fractions is removed, so the fitting error here is larger than
those in Ref.[4]. Here it is clear that the reasonable results can also be
obtained $\theta_{P}=(-14.84\pm 4.35)^{\circ}$, with $\chi^{2}/d.o.f=1.9/4$ in
the case of $s_{v}=0$ and $x=1$.
Based on the above results, we can get the conclusion that $s_{v}$ plays an
important role in the fit to extract the mixing angle. The mixing angle in DM2
analysis is consistent with that in MarkIII because the latter is not from the
best fit.
### 4.2 Analysis of $J/\psi\rightarrow VP$ from BES, BABAR and PDG2010
Until now the pseudoscalar mixing is investigated with the well established
models and the data measured about 20 years ago. The new measurements reported
by BES, BABAR and the new world average of 2010 are listed in Table.1. Each
branching fraction is regarded as one constraint in the fit to BES and BABAR
data. The amplitude of $J/\psi\rightarrow\rho\eta$ and
$J/\psi\rightarrow\rho\eta^{\prime}$ is removed from the fit because no new
measurements are available. The results of the fits with $s_{v}=0$ yields the
mixing angle $\theta_{p}\sim-17^{\circ}$, which is still consistent with the
above results within one standard deviation. This value is also in agreement
with the previous work in Ref.[21, 22, 24]. The fit with $s_{v}=s$ is
performed, but the quality of fit is very poor.
The further check is performed using the world average of 2010[1], and the
results are shown in Table 3. As we expected, the results are fine for the fit
with $s_{v}=0$ and the mixing angle $\theta_{p}$ favors $\sim 14^{\circ}$. The
goodness of the fit with $s_{v}=s$ is still worse because of the weight of new
measurements in the world average.
### 4.3 Analysis of $\psi(2S)\rightarrow VP$
We now turn to examine the full set of $\psi(2S)\rightarrow VP$ to get the
pseudoscalar mixing using the same phenomenological model. At present the
measurements of $\psi(2S)\rightarrow VP$ mainly come from BES and CLEO’s
reports which are shown in Table 4. We have omitted the known upper limit for
the $\psi(2S)\rightarrow\phi\pi$ and $\psi(2S)\rightarrow\omega\eta$ branching
fractions in our analysis because they are the upper limits at 90% confidence
level rather than branching fractions. As previously stated, we just consider
the mixing angle between $\eta$ and $\eta^{\prime}$ and assume the mixing of
$\omega$ and $\phi$ is ideal. The results listed in Table 5 indicate that both
of the above two slightly different models are reasonable and the $\theta_{p}$
favors $\sim-12^{\circ}$ with large uncertainty. Without considering the
branching fraction of $\psi(2S)\rightarrow\rho\pi$, the fit was also performed
in Ref.[24], the mixing angle is calculated to be $-10^{+7}_{-8}$ which is in
agreement with our result. But the branching fraction of $\psi(2S)\rightarrow
K^{*+}K^{-}$, $(8.5\pm 4.0)\times 10^{-5}$, applied in the analysis is not
correct. Therefore the values of parameters listed in Table 5 are inconsistent
with those in Ref.[24].
Table 2: Results of fit to MarkIII data Parameter | $s_{v}=0$,x=1 | $s_{v}=0$,x=0.64 | $s_{v}=0$,x=0.82 | $s_{v}=s$,x=1 | $s_{v}=s$,x=0.64 | $s_{v}=s$,x=0.82
---|---|---|---|---|---|---
g | $1.30\pm 0.04$ | $1.31\pm 0.04$ | $1.30\pm 0.04$ | $1.12\pm 0.04$ | $1.11\pm 0.04$ | $1.11\pm 0.04$
s | $0.27\pm 0.02$ | $0.28\pm 0.02$ | $0.27\pm 0.02$ | $0.13\pm 0.03$ | $0.13\pm 0.02$ | $0.13\pm 0.03$
e | $0.124\pm 0.005$ | $0.123\pm 0.05$ | $0.124\pm 0.05$ | $0.123\pm 0.005$ | $0.123\pm 0.005$ | $0.123\pm 0.005$
$\theta_{e}$ | $1.21\pm 0.12$ | $1.29\pm 0.12$ | $1.27\pm 0.12$ | $1.27\pm 0.12$ | $1.30\pm 0.12$ | $1.29\pm 0.12$
r | $-0.37\pm 0.01$ | $-0.37\pm 0.01$ | $-0.37\pm 0.01$ | $-0.16\pm 0.01$ | $-0.15\pm 0.01$ | $-0.15\pm 0.01$
$\theta_{P}$ | $-13.95\pm 2.39$ | $-13.17\pm 2.40$ | $-13.49\pm 2.38$ | $-18.29\pm 1.43$ | $-18.59\pm 1.40$ | $-18.47\pm 1.41$
$\chi^{2}/d.o.f$ | 9.0/4 | 7.9/4 | 8.3/4 | 8.1/4 | 9.0/4 | 8.6/4
Table 3: Results of fit PDG2010 data Parameter | $s_{v}=0$,x=1 | $s_{v}=0$,x=0.64 | $s_{v}=0$,x=0.82 | $s_{v}=s$,x=1 | $s_{v}=s$,x=0.64 | $s_{v}=s$,x=0.82
---|---|---|---|---|---|---
g | $1.35\pm 0.04$ | $1.36\pm 0.04$ | $1.36\pm 0.04$ | $1.15\pm 0.04$ | $1.14\pm 0.04$ | $1.14\pm 0.04$
s | $0.30\pm 0.02$ | $0.30\pm 0.03$ | $0.30\pm 0.02$ | $0.15\pm 0.03$ | $0.14\pm 0.03$ | $0.14\pm 0.03$
e | $0.120\pm 0.005$ | $0.121\pm 0.04$ | $0.121\pm 0.04$ | $0.119\pm 0.005$ | $0.119\pm 0.005$ | $0.119\pm 0.005$
$\theta_{e}$ | $1.31\pm 0.12$ | $1.36\pm 0.12$ | $1.34\pm 0.12$ | $1.35\pm 0.12$ | $1.38\pm 0.12$ | $1.36\pm 0.12$
r | $-0.37\pm 0.01$ | $-0.37\pm 0.01$ | $-0.37\pm 0.01$ | $-0.16\pm 0.01$ | $-0.15\pm 0.01$ | $-0.15\pm 0.01$
$\theta_{P}$ | $-14.27\pm 2.44$ | $-13.90\pm 2.35$ | $-14.04\pm 2.37$ | $-17.66\pm 1.81$ | $-17.96\pm 1.77$ | $-17.84\pm 1.78$
$\chi^{2}/d.o.f$ | 3.1/4 | 3.5/4 | 3.3/4 | 16.5/4 | 18.1/4 | 17.4/4
Table 4: Branching fractions of $\psi(2S)\rightarrow VP$($\times 10^{-5}$) Decay modes | BES | CLEO | PDG2010
---|---|---|---
$\rho\pi$ | $5.1\pm 0.7\pm 1.1$ | $2.4\pm 0.8\pm 0.2$ | $3.2\pm 1.2$
$\rho\eta$ | $1.78^{+0.67}_{-0.62}\pm 0.17$ | $3.0^{+1.1}_{-0.9}\pm 0.2$ | $2.2\pm 0.6$
$\rho\eta^{\prime}$ | $1.87^{+1.64}_{-1.11}\pm 0.33$ | | $1.9^{+1.7}_{-1.2}$
$\phi\pi^{0}$ | $<0.4$ | $<0.7$ | $<0.4$
$\phi\eta$ | $3.3\pm 1.1\pm 0.5$ | $2.0^{+1.5}_{-1.1}\pm 0.4$ | $2.8^{+1.0}_{-0.8}$
$\phi\eta^{\prime}$ | $3.1\pm 1.4\pm 0.7$ | | $3.1\pm 1.6$
$\omega\pi^{0}$ | $1.87^{+0.68}_{-0.62}\pm 0.28$ | $2.5^{+1.2}_{-1.0}\pm 0.2$ | $2.1\pm 0.6$
$\omega\eta$ | $<3.1$ | $<1.1$ | $<1.1$
$\omega\eta^{\prime}$ | $3.2^{+2.4}_{-2.0}\pm 0.7$ | | $3.2^{+2.5}_{-2.1}$
$K^{*-}K^{+}+c.c.$ | $2.9^{+1.3}_{-1.7}\pm 0.4$ | $1.3^{+1.0}_{-0.7}\pm 0.3$ | $1.7^{+0.8}_{-0.7}$
$K^{*0}\bar{K^{o}}+c.c.$ | $13.3^{+2.4}_{-2.8}\pm 1.7$ | $9.2^{+2.7}_{-2.2}\pm 0.9$ | $10.9\pm 2.0$
Table 5: Results of fit to PDG2010 data of $\psi(2S)\rightarrow VP$ Parameter | $s_{v}=0$,x=1 | $s_{v}=0$,x=0.64 | $s_{v}=0$,x=0.82 | $s_{v}=s$,x=1 | $s_{v}=s$,x=0.64 | $s_{v}=s$,x=0.82
---|---|---|---|---|---|---
g | $0.64\pm 0.11$ | $0.65\pm 0.10$ | $0.65\pm 0.10$ | $0.64\pm 0.11$ | $0.65\pm 0.04$ | $0.65\pm 0.04$
s | $0.003\pm 0.18$ | $-0.01\pm 0.19$ | $-0.05\pm 0.18$ | $0.02\pm 0.18$ | $-0.10\pm 0.19$ | $-0.10\pm 0.20$
e | $0.23\pm 0.02$ | $0.23\pm 0.02$ | $0.23\pm 0.02$ | $0.23\pm 0.02$ | $0.23\pm 0.02$ | $0.23\pm 0.02$
$\theta_{e}$ | $2.73\pm 0.62$ | $2.81\pm 0.60$ | $2.79\pm 0.63$ | $2.75\pm 0.64$ | $2.83\pm 0.64$ | $2.83\pm 0.62$
r | $0.18\pm 0.28$ | $0.17\pm 0.27$ | $0.16\pm 0.28$ | $0.14\pm 0.28$ | $0.14\pm 0.29$ | $0.14\pm 0.31$
$\theta_{P}$ | $-12.07\pm 10.42$ | $-11.94\pm 10.48$ | $-11.99\pm 10.46$ | $-11.80\pm 10.63$ | $-12.19\pm 11.59$ | $-12.19\pm 12.18$
$\chi^{2}/d.o.f$ | 4.4/3 | 4.5/3 | 4.4/3 | 4.4/3 | 4.5/3 | 4.5/3
### 4.4 $\omega$-$\phi$ mixing
In the above analysis, the $\omega$-$\phi$ mixing is ignored to simplify the
model. This fit in the case of $s_{v}=0$ and $x=0.82$ is an attempt to account
for the $\omega$-$\phi$ mixing. If $\omega$-$\phi$ mixing angle is left as a
free parameter, the fit to the world average of 2010 leads to a minimum
$\chi^{2}=3.3$ for three degrees of freedom,
$g=1.36\pm 0.04$, $s=0.30\pm 0.03$, $e=0.121\pm 0.005$, $\theta_{e}=1.33\pm
0.12$, $\theta_{p}=(-14.06\pm 2.37)^{\circ}$, $r=-0.37\pm 0.02$,
$\theta_{V}=(0.09\pm 4.13)^{\circ}$.
If we assumed $s_{v}=s$, then the fit with $\chi^{2}/d.o.f$ of 17.4/3 gives,
$g=1.14\pm 0.05$, $s=0.14\pm 0.04$, $e=0.119\pm 0.005$, $\theta_{e}=1.36\pm
0.12$, $\theta_{p}=(-17.78\pm 2.70)^{\circ}$, $r=-0.15\pm 0.01$,
$\theta_{V}=(0.11\pm 3.78)^{\circ}$.
$\theta_{V}$ is very close to zero and the uncertainty is very large compared
with other parameters. This means that there is not a significant constraint
on it. Among $J/\psi\rightarrow VP$ decays, the amplitude of
$J/\psi\rightarrow\phi\pi^{0}$ is directly related to the $\omega$-$\phi$
mixing, but it is still not observed yet. No observation of
$J/\psi\rightarrow\phi\pi^{0}$ shows that the contribution of $\omega$-$\phi$
is small. On the other hand, the values of other parameters are almost the
same as those listed in Table 3 without considering $\omega$-$\phi$ mixing.
Therefore it is reasonable that $\omega$-$\phi$ mixing is assumed to be ideal
and could be ignored in the above analysis. Further check is done by fixing
the $\omega-\phi$ mixing angle to $3.2^{\circ}$ obtained from
$V\rightarrow\gamma P$ and $P\rightarrow\gamma V$ process. The fit with
$\chi^{2}/d.o.f=3.3/4$ gives,
$g=1.36\pm 0.04$, $s=0.30\pm 0.02$, $e=0.121\pm 0.004$, $\theta_{e}=1.33\pm
0.13$, $\theta_{p}=(-14.05\pm 2.36)^{\circ}$, $r=-0.37\pm 0.01$, these values
are also in good agreement with those in the hypothesis of the ideal
$\omega$-$\phi$ mixing.
### 4.5 Gluon content in $\eta$ and $\eta^{\prime}$
At present $\eta$ is believed to be well-understood as an SU(3) flavor octet
with a small quarkonium singlet admixture, and not much room for a significant
gluonium admixture[21, 24]. Therefore the analyses[24] are usually performed
to determine the gluonic content in $\eta^{\prime}$ with the assumption of no
gluonic content in $\eta$. After taking into account the gluonic content in
$\eta$ and $\eta^{\prime}$ simultaneously, we present the fit with the above
two slightly different models. In the first case, $s_{v}$ is assumed to be
zero and the fit to the world average in 2010 yields,
$g=1.32\pm 0.06$, $s=0.27\pm 0.04$, $e=0.126\pm 0.007$, $\theta_{e}=1.34\pm
0.12$, $\theta_{p}=(-10.21\pm 4.48)^{\circ}$, $r=-0.45\pm 0.08$,
$\phi_{g1}=0.04\pm 0.05$, $\phi_{g2}=0.53\pm 0.24$, $r^{\prime}=-0.77\pm
0.46$,
with $\chi^{2}/d.o.f=1.56/1$.
The second fit is performed under the hypothesis of $s_{v}=s$, and the results
with $\chi^{2}/d.o.f=3.5/1$ are listed as follows,
$g=1.28\pm 0.06$, $s=0.24\pm 0.03$, $e=0.128\pm 0.007$, $\theta_{e}=1.35\pm
0.11$, $\theta_{p}=(-9.17\pm 4.67)^{\circ}$, $r=-0.67\pm 0.08$,
$\phi_{g1}=0.11\pm 0.04$, $\phi_{g2}=0.50\pm 0.22$, $r^{\prime}=-0.85\pm
0.56$.
The goodness of the second fit is still worse than the first fit. Based on the
results of the first fit, the magnitudes of gluon components in $\eta$ and
$\eta^{\prime}$ are calculated to be $Z^{2}_{\eta}=0.002\pm 0.002$ and
$Z^{2}_{\eta^{\prime}}=0.30\pm 0.24$, respectively. The small gluonic
contribution in $\eta$ shows there is not much room for gluonium admixture,
which is consistent with the results presented in Ref.[22]. It seems that 30%
of $\eta^{\prime}$ component could be attributed to gluonium, but further
investigation with more precisely data needed to be done due to the large
uncertainty.
## 5 Summary and outlook
A wide set of data on $J/\psi\rightarrow VP$ and $\psi(2S)\rightarrow VP$
decays are analyzed in terms of a rather general phenomenological model in an
attempt to determine the magnitudes of components in $\eta$ and
$\eta^{\prime}$. The data include the branching fractions of
$J/\psi\rightarrow VP$ which were measured nearly 20 years ago and the recent
measurements by BES and BABAR. The measurements of MarkIII and DM2 are
reanalyzed, we found that the results obtained from the two different
phenomenological models are inconsistent. The fit to the new measurements by
BES and BABAR indicates that the assumption of $s=s_{v}$ is not a good
approximation in accordance with the goodness of fit. And the mixing angle is
determined to be $-14^{\circ}$, which is in good agreement with previous
works.
The content of $\eta$ and $\eta^{\prime}$ is also examined in this paper.
After considering the gluonium content in the model, the fit to data of the
world average in 2010 yields $Z^{2}_{\eta}=0.002\pm 0.002$ and
$Z^{2}_{\eta^{\prime}}=0.30\pm 0.24$, which are the contribution of gluonium
content in $\eta$ and $\eta^{\prime}$ respectively. Although the possibility
of gluonic content can not be excluded, it is a reasonable description for
$\eta$ in terms of pure $q\bar{q}$ meson, and no much room for a significant
gluonium admixture. The magnitude of gluonium contamination in $\eta^{\prime}$
shows that $\eta^{\prime}$ has room for gluonium admixture, but the large
uncertainty prevents us from definitely saying that gluonium content is
present or not.
As we previously stated that the latest results from BES and BABAR are not
consistent with those previous works. The branching fractions shown in Table 1
still have large error, including statistical and systematic errors. The main
reason is that $J/\psi$ and $\psi(2S)$ samples are not enough and the
performance of detector need to be improved. A modern detector, BESIII[27],
has been built to meet the above requirements. Up to now, about $2.3\times
10^{8}$ $J/\psi$ and $1.2\times 10^{8}$ $\psi(2S)$ events have been
accumulated at BESIII, which provide a unique chance to study the
$\eta$-$\eta^{\prime}$ mixing and further improve these measurements with much
higher sensitivities.
## References
* [1] Nakamura K et al (Particle Data Group). J. phys. G, 2010, 37: 075021
* [2] Donoghue J F, Golowich E and Holstein B R. Dynamics of the Standard Model. New York: Cambridge Univ. Press,1992.
* [3] Coffman D et al (MarkIII collaboration). Phys. Rev. D, 1988, 38: 2695-2705
* [4] Jousset J et al (DM2 collaboration). Phys. Rev. D, 1990, 41: 1389-1400
* [5] Bramon A and Scadron D. Phys. Lett. B, 1990, 234: 346-348
* [6] Bramon A et al. Phys. Lett. B, 1997, 403: 339-343; Bramon A et al. Euro. Phys. J. C, 1999, 7: 271-278
* [7] Feldmann T. Int. J. Mod. Phys. A, 2000, 15: 159-207
* [8] Bai J Z et al (BES collaboration). Phys. Rev. D, 2004, 70: 012005
* [9] Ablikim M et al (BES collaboration). Phys. Rev. D, 2005, 71: 032003
* [10] Ablikim M et al (BES collaboration). Phys. Rev. D, 2006, 73: 052007
* [11] Ablikim M et al (BES collaboration). Phys. Rev. D, 2004, 70: 112003
* [12] Ablikim M et al (BES collaboration). Phys. Rev. D, 2004, 70: 112007
* [13] Ablikim M et al (BES collaboration). Phys. Lett. B, 2005, 614: 37-43
* [14] Ablikim M et al (BES collaboration). Phys. Lett. B, 2005, 619: 247-254
* [15] Aubert B et al (BABAR collaboration). Phys. Rev. D, 2004, 70: 072004
* [16] Aubert B et al (BABAR collaboration). Phys. Rev. D, 2006, 73: 052003
* [17] Aubert B et al (BABAR collaboration). Phys. Rev. D, 2007, 76: 092005
* [18] Aubert B et al (BABAR collaboration). Phys. Rev. D, 2008, 77: 092002
* [19] Adam N E et al (CLEO collaboration). Phys. Rev. Lett., 2005, 94: 012005
* [20] Barnett R M et al (Particle Data Group). Phys. Rev. D, 1996, 54: 1-708
* [21] Escribano R. Acta Phys. Polon. Supp., 2009, 2: 71-79
* [22] Zhao Q, Li G and Chang C H. Phys. Lett. B, 2007, 645: 173-179; Li G, Zhao Q and Chang C H. J. Phys. G, 2008, 35: 055002
* [23] Escribano R. AIP. Conf. Proc., 2008, 1030: 368-373
* [24] Thomas C E. JHEP, 2007, 0710: 026; Escribano R. Nucl. Phys. Proc. Suppl., 2008, 181-182: 226-230
* [25] Haber H E and Perrier J. Phys. Rev. D, 1985, 32: 2961-2970; Seiden A et al. Phys. Rev. D, 1988, 38: 824-836
* [26] Escribano R and Nadal J. JHEP, 2007, 0705: 006
* [27] Asner D M et al. Int. J. Mod. Phys. A Supp., 2009, 24: 1-794
|
arxiv-papers
| 2010-08-17T06:25:15 |
2024-09-04T02:49:12.224969
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Daihui and Yang Yongxu",
"submitter": "Wei Daihui",
"url": "https://arxiv.org/abs/1008.2829"
}
|
1008.2866
|
# Whether the vacuum manifold in the Minkowskian non-Abelian model quantized
by Dirac can be described with the aid of the superselection rules?
L. D. Lantsman.
18109, Rostock, Germany; Mecklenburger Allee, 7
llantsman@freenet.de
Tel. (049)-0381-7990724.
###### Abstract
We intend to show that the vacuum manifold inherent in the Minkowskian non-
Abelian model involving Higgs and Yang-Mills BPS vacuum modes and herewith
quantized by Dirac can be described with the help of the superselection rules
if and only if the “discrete” geometry for this vacuum manifold is assumed (it
is just a necessary thing in order justify the Dirac fundamental quantization
scheme applied to the mentioned model) and only in the infinitely narrow
spatial region of the cylindrical shape where topologically nontrivial
vortices are located inside this discrete vacuum manifold.
PACS: 12.38.Aw, 14.80.Bn, 14.80.Hv.
Keywords: Non-Abelian Theory, BPS Monopole, Minkowski Space, Topological
Defects, Phase transitions.
In the recent paper [1] it was argued that the so-called Dirac fundamental
quantization [2] of the Minkowskian non-Abelian model involving Higgs and
Yang-Mills (YM) vacuum BPS modes, coming to the Gauss-shell reduction of the
mentioned model in terms of topological Dirac variables, gauge invariant and
transverse functionals of YM fields 111As important “milestones” in
development of this model, it is worth to mention the papers [3, 4, 5, 6, 7,
8]. For the history of the question see also the survey [9]., is compatible
with assuming the “discrete” geometry for the appropriate vacuum manifold:
$R_{\rm YM}={\bf Z}\otimes G_{0}/U_{0}.$ (1)
Such representation for the vacuum manifold $R_{\rm YM}$ is the direct
consection of the “discrete” representations
$SU(2)\simeq G_{0}\otimes{\bf Z};\quad U(1)\simeq U_{0}\otimes{\bf Z}$ (2)
for the initial, $SU(2)$, and residual, $U(1)$, gauge symmetries groups
(respectively) in the Minkowskian non-Abelian Higgs model (we shall refer to
this model as to the YMH model henceforth in the present study).
From the topological viewpoint, the discrete representation (2) for the gauge
groups $G$ and $H$ extracts ”small” (topologically trivial) and ”large”
(corresponding to topological numbers $n\neq 0$) gauge transformations in the
complete set of appropriate gauge transformations (the idea of such
subdividing for gauge transformations was suggested in Ref. [10]).
According to the terminology [10], the complete groups $G_{0}$ and $H_{0}$
just contain ”small” gauge transformations, that implies
$\pi_{n}G_{0}=\pi_{n}H_{0}=0$ (3)
for loops in the group spaces $G_{0}$ and $H_{0}$ in all the dimensions $n\geq
1$.
Simultaneously, in definition,
$\pi_{0}G_{0}=\pi_{0}H_{0}=0,$ (4)
i.e. $G_{0}$ and $H_{0}$ are maximal connected components (in the terminology
[11]) in their gauge groups (respectively, $G$ and $H$).
Later Eq. implies [11] that
$\pi_{0}[G_{0}\otimes{\bf Z}]=\pi_{0}[G_{0}\otimes{\bf Z}]=\pi_{0}({\bf
Z})={\bf Z}.$ (5)
It becomes obvious from Eq. (1) that the ”small” coset $G_{0}/U_{0}$ is one-
connected:
$\pi_{1}(G_{0}/U_{0})=0.$
Really, the coset $G_{0}/U_{0}$ is treated as the space of $U_{0}$-orbits on
$G_{0}$; the latter space is one-connected.
One can see also the topological equivalence between $G_{0}/U_{0}$ and the
subset of one-dimensional ways on $R_{\rm YM}$ which can be contracted into a
point.
The vacuum manifold $R_{\rm YM}$ is transparently multi-connected (i.e.
discrete):
$\pi_{0}(R_{YM})={\bf Z}.$ (6)
This implies [11] that domain walls exist between different topological
sectors in the Minkowskian Higgs model with vacuum BPS monopoles quantized by
Dirac.
The origin of said domain walls is in the ”discrete” factorisation (2) of the
residual gauge symmetry group $U(1)$.
As it is well known (see e.g. §7.2 in [12] or the paper [13]), the width of a
domain (or Bloch, in the terminology [13]) wall is roughly proportional to the
inverse of the lowest mass among all the physical particles presented in the
(gauge) model considered.
In Minkowskian Higgs models (without quarks) the typical such scale is the
(effective) Higgs mass $m/\sqrt{\lambda}$. In particular, in the Minkowskian
YMH model [3, 4, 5, 6, 7, 8] with vacuum BPS monopoles quantized by Dirac,
$m/\sqrt{\lambda}$ is the only mass scale different from zero (in the “world
with quarks” this remains almost correctly at assuming [1] $m_{0}\ll
m/\sqrt{\lambda}$ for any “bare” flavour mass $m_{0}$).
Together with the “effective Higgs mass” $m/\sqrt{\lambda}$, it is possible to
write down the value roughly its inverse, i.e. having the length dimension. It
is the (typical) size $\epsilon$ of BPS monopoles.
It can be given as [4, 7, 8]
$\frac{1}{\epsilon}=\frac{gm}{\sqrt{\lambda}}\sim\frac{g^{2}<B^{2}>V}{4\pi},$
(7)
with $g$ being the YM coupling constant. Thus $\epsilon$ is inversely
proportional to the spatial volume $V\sim r^{3}$ occupied by the appropriate
YMH field configuration.
The said allows to assert that $\epsilon$ disappears in the infinite spatial
volume limit $V\to\infty$, while it is maximal at the origin of coordinates
(herewith it can be set $\epsilon(0)\to\infty$). This means, due to the above
reasoning [13], that walls between topological domains inside $R_{\rm YM}$
become infinitely wide, $O(\epsilon(0))\to\infty$, at the origin of
coordinates.
The fact $\epsilon(\infty)\to 0$ is also meaningful. This implies actual
merging topological domains inside the vacuum manifold $R_{\rm YM}$, (1), at
the spatial infinity. This promotes the infrared topological confinement
(destructive interference) of Gribov ”large” multipliers $v^{(n)}({\bf x})$ in
gluonic and quark Green functions in all the orders of the perturbation
theory. The latter fact was demonstrated utilizing the strict mathematical
language in Ref. [14] (partially these arguments [14] were reproduced in Ref.
[9]).
The nontrivial isomorphism [11]
$\pi_{1}({R}_{\rm YM})=\pi_{0}(H)\neq 0$ (8)
correct [1] for the vacuum manifold $R_{YM}$, (1) 222It is the particular case
of the general relation [11] $\pi_{i}(K)=\pi_{i}(L_{1})+\dots+\pi_{i}(L_{r})$
for a group $K$ which is the product of the groups $L_{1}\dots L_{r}$ at a
fixed $i$ (it is correctly for the Lie groups of the series $SU$, $U$ and
$SO$, with which modern theoretical physics deals). , implies the presence of
thread topological defects inside this manifold.
As it was argued in the paper [1] (with the aid of the arguments [11]), this
kind of topological defects in the Minkowskian YMH model [3, 4, 5, 6, 7, 8]
with vacuum BPS monopoles quantized by Dirac can be represented by specific
solutions in its YM and Higgs sectors: so-called (topologically nontrivial)
threads.
In particular, in the Higgs sector of the Minkowskian YMH theory [3, 4, 5, 6,
7, 8] there are [11] z-invariant (vacuum) Higgs solutions in a (small)
neighbourhood of the origin of coordinates ($\rho\to 0$):
$\Phi^{(n)}(\rho,\theta,z)=\exp(M\theta)~{}\phi(\rho)\quad(n\in{\bf
Z}),\quad\nabla_{\mu}\phi(\rho)\leq{\rm
const}~{}\rho^{-1-\delta};\quad\delta>0;\quad n\in{\bf Z};$ (9)
$\rho=\sqrt{x^{2}+y^{2}}$ is the distance from the axis $z$.
One claims for Higgs thread solutions $\Phi^{(n)}(\rho,\theta,z)$ to join
contineously and smoothly the vacuum Higgs BPS monopoles, belonging to the
same topology $n$ and disappearing [5] at the origin of coordinates. Herewith,
speaking ”in a smooth wise”, we imply that the covariant derivative $D\Phi$ of
any vacuum Higgs field $\Phi_{a}^{(n)}$ merges with the covariant derivative
of such a vacuum Higgs BPS monopole solution.
The requirement for vacuum Higgs fields $\Phi_{a}^{(n)}$ to be smooth is quite
natural if the goal is pursued, as it is done in the Minkowskian YMH model [3,
4, 5, 6, 7, 8] with vacuum BPS monopoles quantized by Dirac, to justify
various rotary effects inherent in this model.
In particular, vacuum ”electric” monopoles [4] 333They involve, firstly, the
topological varible $N(t)$ (with its time derivative $\dot{N}(t)$) introduced
[6] via the vacuum Chern-Simons functional
$\displaystyle\nu[A_{0},\Phi^{(0)}]$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\int\limits_{t_{\rm in}}^{t_{\rm
out}}dt\int
d^{3}xF^{a}_{\mu\nu}\widetilde{F}^{a\mu\nu}=\frac{\alpha_{s}}{2\pi}\int
d^{3}xF^{a}_{i0}B_{i}^{a}(\Phi^{(0)})[N(t_{\rm out})-N(t_{\rm in})]$
$\displaystyle=N(t_{\rm out})-N(t_{\rm in})=\int\limits_{t_{\rm in}}^{t_{\rm
out}}dt\dot{N}(t);\quad t_{\rm in}\to-\infty,~{}~{}t_{\rm out}\to\infty;$ and
secondly, the real, i.e. physical, topological momentum
$P_{N}={\dot{N}}I=2\pi k+\theta;\quad\theta\in[-\pi,\pi].$
$F^{a}_{i0}\equiv
E_{i}^{a}=\dot{N}(t)~{}(D_{i}(\Phi_{k}^{(0)})~{}\Phi_{(0)})^{a}=P_{N}\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a}(\Phi_{(0)})=(2\pi
k+\theta)\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a}(\Phi_{(0)});\quad
k\in{\bf Z};$ (10) $\alpha_{s}=\frac{g^{2}}{4\pi(\hbar c)^{2}};$
prove to be directly proportional to $D_{i}(\Phi_{k}^{(0)})~{}\Phi_{(0)}$.
These vacuum ”electric” monopoles, in turn, enter explicitly the action
functional
$W_{N}=\int d^{4}x\frac{1}{2}(F_{0i}^{c})^{2}=\int
dt\frac{{\dot{N}}^{2}I}{2},$ (11)
implicating the “rotary momentum” [4]
$I=\int_{V}d^{3}x(D_{i}^{ac}(\Phi_{a}^{(0)})\Phi_{(0)c})^{2}=\frac{4\pi^{2}\epsilon(\infty)}{\alpha_{s}}=\frac{4\pi^{2}}{\alpha_{s}^{2}}\frac{1}{V<B^{2}>}$
(12)
and describing, in the Dirac fundamental quantization scheme [2], collective
solid rotations inside the Minkowskian BPS monopole vacuum.
Such (smooth) sawing together appropriate vacuum Higgs modes $\Phi^{(n)}$
(which are [11] specific thread rectilinear vortices) and BPS monopoles serves
to remove the seeming contradiction between the manifest superfluid properties
of the Minkowskian BPS monopole vacuum (suffered the Dirac fundamental
quantization [2]), setting by the Bogomolny’i [7, 8, 11],
${\bf B}=\pm D\Phi,$ (13)
and Gribov ambiguity [6, 7, 8],
$[D^{2}_{i}(\Phi_{a}^{(0)})]^{ab}\Phi_{(0)b}=0,$ (14)
equations.
One can assert (following [3]), and this can be seen from (10), containing the
vacuum “magnetic” field $\bf B$ given by the Bogomolny’i equation (13), that,
due to the Bianchi identity,
$D~{}B\sim D~{}E=0$ (15)
for vacuum ”magnetic” and ”electric” tensions: $\bf B$ and $\bf E$,
respectively, in the quested YMH model [3, 4, 5, 6, 7, 8], these tensions are,
indeed, ”transverse” vectors colinear each other. This just implies the
potential nature of the ”electric” tension $\bf E$, that can be perceived as
the above contradiction, on the face of it.
Going out from this contradiction seems to be just in locating (topologically
nontrivial) threads in the infinitely narrow cylinder of the effective
diameter $\epsilon(\infty)$ around the axis $z$ and in joining (in a smooth
wise) vacuum Higgs fields $\Phi_{a}^{(n)}$ and Higgs BPS monopole solutions
(as it was explained in Ref. [1]).
In this case collective solid rotations (vortices) inside the Minkowskian BPS
monopole vacuum, occurring actually in that spatial region around the axis $z$
and described correctly by the action functional (11), become quite
”legitimate”, and simultaneously, the Gauss law constraint [6]
$[D^{2}_{i}(\Phi^{(0)})]^{ac}A_{0c}=0,$ (16)
just permitting, in the Minkowskian YMH model [3, 4, 5, 6, 7, 8] with vacuum
BPS monopoles quantized by Dirac, the family of zero mode solutions [3, 6]
$A_{0}^{c}(t,{\bf x})={\dot{N}}(t)\Phi_{(0)}^{c}({\bf x})\equiv Z^{c},$ (17)
generating “electric monopoles” (10), is satisfied outward this region with
these smooth vacuum ”electric” monopole solutions. In turn, one can refer [1]
the “electric monopoles” (10) to thread solutions since vacuum Higgs fields
$\Phi_{a}^{(n)}$ are such.
On the other hand, in the region of thread topological defects inside the
discrete vacuum manifold $R_{\rm YM}$, Eq. (15) is violated since the vacuum
”magnetic” field $\bf B$ suffers a break in this region. Really, according to
the arguments [15], the vacuum ”magnetic” field $\bf B$ set via the
Bogomol’nyi equation (13) over YM and Higgs BPS monopole solutions diverges as
$r^{-2}$ at the origin of coordinates.
Simultaneously, following [11], thread “counterparts” of YM BPS monopole
solutions $\Phi_{i}^{a{\rm BPS}}$ [7, 8] can be constructed:
$A_{\theta}(\rho,\theta,z)=\exp(iM\theta)A_{\theta}(\rho)\exp(-iM\theta),$
(18)
with $M$ being the generator of the group $G_{1}$ of rigid rotations
compensating changes in the vacuum YMH “thread” configuration
$(\Phi^{a},A_{\mu}^{a})$ (with $\Phi^{a}$ given in (9)) at rotations around
the axis $z$ of the chosen (rest) reference frame.
In (18),
$A_{\theta}(\rho)=M+\beta(\rho),$
where the function $\beta(\rho)$ approaches zero as $\rho\to\infty$.
The elements of $G_{1}$ can be set as [11]
$g_{\theta}=\exp(iM\theta).$ (19)
YM fields $A_{\theta}$ are manifestly invariant with respect to shifts along
the axis $z$.
Rectilinear threads $A_{\theta}$ don’t coincide with vacuum YM BPS monopole
solutions $\Phi_{i}^{a{\rm BPS}}$ [7, 8], and, on the contrary, there are gaps
between directions of ”magnetic” tensions vectors: ${\bf B}_{1}$,
$|{\bf B}_{1}|\sim\partial_{\rho}A_{\theta}(\rho,\theta,z),$ (20)
and $\bf B$, given by the Bogomol’nyi equation (13) (and diverging as $r^{-2}$
at the origin of coordinates).
These gaps testify in favour of the first-order phase transition [1] occurring
in the Minkowskian YMH model [3, 4, 5, 6, 7, 8] with vacuum BPS monopoles
quantized by Dirac.
The important point of our above reasoning is that the vacuum expectation
value of the Higgs field squared, $\sim<\Phi^{a}\Phi_{a}>$, cannot be treated
as an order parameter in the Minkowskian YMH model [3, 4, 5, 6, 7, 8] with
vacuum BPS monopoles quantized by Dirac. Otherwise, a flip should exist in the
plot of a Higgs field $\Phi^{a}(r)$ at the origin of coordinates, $r\to 0$, as
a sign of the first-order phase transition occuring in the Minkowskian YMH
model [4, 5, 6, 7, 8]. But then it will be impossible to “join” continiously
and smoothly Higgs solutions $\Phi^{a}(r)$ with “zero mode” solutions $Z^{a}$
[3], (17), involving Higgs BPS monopole modes. And this should contradict to
the Dirac fundamental quantization of the model [3, 4, 5, 6, 7, 8].
Vice verse, the vacuum expectation value of the ”magnetic” tension, $<B^{2}>$,
can serve as an order parameter in the quested Minkowskian YMH model [3, 4, 5,
6, 7, 8], with the first-order phase transition taking place, due to the
obvious gap between directions of the ”magnetic” tensions vectors ${\bf
B}_{1}$ and ${\bf B}$ (such assumption was made already in Refs. [7, 8], and
then it was confitmed in the paper [1]).
This distinguish the Minkowskian YMH model [3, 4, 5, 6, 7, 8] with vacuum BPS
monopoles quantized by Dirac from another YM models (for instance, the ’t
Hooft-Polyakov model [16, 17]) implying the continuous $\sim S^{2}$ vacuum
geometry, where just the vacuum expectation value of the Higgs field squared,
$<\Phi^{a}\Phi_{a}>$, serves as an order parameter). This is associated with
the second-order phase transition taking place in such non-Abelian models
(this was grounded, for example, in Ref. [18] with the help of the arguments
[12]).
The first-order phase transition taking place in the Minkowskian YMH model [3,
4, 5, 6, 7, 8] with vacuum BPS monopole solutions quantized by Dirac comes [1]
to the coexistence (in the absolute temperature limit $T\to 0$) of two
thermodynamic phases inside the vacuum of that model. These two thermodynamic
phases are the phase of collective solid rotations, set by the action
functional (11) (involving [topologically nontrivial] thread configurations
$(\Phi^{a},A_{\mu}^{a})$ and generating “electric monopoles” $E_{i}^{a}$ [4],
(10)) and the phase of superfluid potential motions set by the Bogomol’nyi
equation (13) [7, 8, 11] and the Gribov ambiguity equation (14).
The just described thermodynamic phases inside the Minkowskian YMH physical
vacuum [3, 4, 5, 6, 7, 8] can be characterized by two different scales for the
“effective” Higgs mass $m/\sqrt{\lambda}$. For instance, collective solid
rotations inside that vacuum correspond, as it is easy to see, to the zero
mass scale $m/\sqrt{\lambda}\to 0$, while superfluid potential motions
correspond to a nonzero mass scale $m/\sqrt{\lambda}\neq 0$.
At $T\to 0$ the both thermodynamic phases inside the Minkowskian physical
vacuum [3, 4, 5, 6, 7, 8] as if freeze [1], that gives a stable look to the
studied model [3, 4, 5, 6, 7, 8]. Nevertheless, it remains an important
question, in the framework of the first-order phase transition occurring
therein, which of the enumerated thermodynamic phases “belongs” to the “true”
and which to the “false” (metastable) vacuum?
In the present study we attempt to ground that, for all that, collective solid
rotations inside the Minkowskian physical vacuum [3, 4, 5, 6, 7, 8] relate to
the “true vacuum”, while superfluid potential motions therein relate to the
“false” vacuum.
The key point in this grounding will be once again the “discrete” vacuum
geometry (1) [1] us assumed for the appropriate vacuum manifold $R_{\rm YM}$.
In the coordinate region
$r=\sqrt{x^{2}+y^{2}}\to 0;\quad{\rm arbitrary}~{}~{}z$ (21)
of the Minkowski space (i.e. [infinitely] near the axis $z$ of the chosen rest
reference frame), the vacuum manifold $R_{\rm YM}$, (1), consists of
topological domains separated by infinitely thick walls of the typical
thickness $\epsilon(0)\to\infty$.
In this case the assumption is quite permissible that topological sectors
inside the vacuum manifold $R_{\rm YM}$ in the pointed spatial region can be
identified with the superselection sectors [coherent spaces] (see e.g. §6.2 in
[19]).
Indeed, to accomplish such an identification, some conditions would be
observed. Note, first of all, that the term “coherent spaces” implies [19]
constructing physical Hilbert spaces ${\cal H}_{n}$ ($n\in{\bf Z}$), which
are, from the physical viewpoint, quantum analogues of topological sectors
inside $R_{\rm YM}$. In turn, in definition, coherent Hilbert spaces ${\cal
H}_{n}$ would consist of vectors describing pure quantum states and forming
irreducible representations of these ${\cal H}_{n}$. Only thereafter, the
vacuum manifold $R_{\rm YM}$ can be represented (in the meanwhile,
theoretically!) as [19]
$R_{\rm YM}\simeq\oplus{{}_{n}}{\cal H}_{n},$ (22)
where all the ${\cal H}_{n}$ are mutually orthogonal.
Latter Eq. reflects also [19] identifying the gauge and topological charges.
It is quite justified in the Minkowskian YMH model [3, 4, 5, 6, 7, 8]
quantized by Dirac due to the nature of topological Dirac variables
$\hat{A}^{D}$ [4, 5],
$\displaystyle\hat{A}_{k}^{D}=v^{(n)}({\bf
x})T\exp\left\\{\int\limits_{t_{0}}^{t}d{\bar{t}}\hat{A}_{0}(\bar{t},{\bf
x})\right\\}\left({\hat{A}}_{k}^{(0)}+\partial_{k}\right)\left[v^{(n)}({\bf
x})T\exp\left\\{\int\limits_{t_{0}}^{t}d{\bar{t}}\hat{A}_{0}(\bar{t},{\bf
x})\right\\}\right]^{-1};\quad D^{k}\hat{A}_{k}^{D}=0;$ (23) $k=1,2,3;$
involving (“small”, “large”) gauge matrices $v^{(n)}({\bf x})$ [10].
The key point of the present reasoning is that each ${\cal H}_{n}$ consist of
vectors describing pure quantum states. But as far as it is correctly for the
vacuum manifold $R_{\rm YM}$? Obviously, in the light identifying the gauge
and topological charges in the Minkowskian YMH model [3, 4, 5, 6, 7, 8]
quantized by Dirac, each coherent physical Hilbert space ${\cal H}_{n}$ would
imply fixing a definite topology $n$ inside $R_{\rm YM}$. Then one can speak
about the pure quantum states sweeping ${\cal H}_{n}$. These pure quantum
states can be transformed each into another by means of “small” gauge matrices
$v^{(0)}({\bf x})$; on the other hand, there is a one-to-one correspondence
between this ${\cal H}_{n}$ and the set of “large” gauge matrices
$v^{(n)}({\bf x})$.
In the theoretical-group language, a one-to-one correspondence can be traced
between a Hilbert space ${\cal H}_{n}$ and the appropriate “small” orbit of
$U(1)\subset SU(2)$. The said allows, following Ref. [20], to represent a
(physical) coherent Hilbert space ${\cal H}_{n}$ as $V\otimes V_{u}$ ($u\in
U(1)$), with $V$ being the Hilbert space in the usual “classical” sence, while
$V_{u}$ being the (finite-dimensional) vector space topologically equivalent
to the nth topological sector inside $U(1)\simeq S^{1}$ group space.
There are, however, definite remarks and questions, whether and to which
extend it is posible to do this fixing a definite topology inside the vacuum
manifold $R_{\rm YM}$?
As it was discussed in Ref. [1] repeating the arguments [11], YM fields with
equal magnetic charges ${\bf m}\neq 0$ can annihilate mutually at crossing
topologically nontrivial threads which are always present inside the discrete
manifold $R_{\rm YM}$. Furthermore, topological deffects (hedgehogs and
threads in the discussed YMH model [3, 4, 5, 6, 7, 8]) can merge and
annihilate quite spontaneously, beyond the above colliding processes (see e.g.
§$\Phi$1 in [11]).
All this, on the face of it, impedes fixing a definite topology inside $R_{\rm
YM}$ (as a result, quantum states become mixed). But the reasonable way out
from this problem seems to be the following. One consider all the processes
with merging and annihilating topological defects as those violating
thermodynamic equilibrium inside $R_{YM}^{{}^{\prime}}$. In this case it is
possible to fix a definite topology $n$ inside the discrete vacuum manifold
$R_{\rm YM}$ and to construct the appropriate coherent physical Hilbert spaces
${\cal H}_{n}$ if the time $\tau$ during which merging and annihilating
topological defects proceeds is large enough (see e.g. §110 in [21]). Then
(quantum) fluctuations of physical parameters referring to $R_{\rm YM}$ will
be small and these parameters will refer to a thermodynamic equilibrium. Only
at these assumptions one can assert that the vacuum manifold $R_{\rm YM}$ is
in a pure quantum state (corresponding to the direct sum $\oplus{{}_{n}}{\cal
H}_{n}$). As it was demonstrated in [21], the above claim $\tau\to\infty$ is
equivalent to the Gaussian distribution of physical parameters characterizing
$R_{\rm YM}$.
On the other hand, the knowledge about the free energy $F$ of the vacuum
manifold $R_{\rm YM}$ is very important to decide whether physical parameters
characterizing $R_{\rm YM}$ are distributed Gaussian (that is equivalent to
finding this manifold in a pure quantum state).
The maximum entropy point of a model can be normalized to be [21] $S_{\rm
max}=S|_{x=\bar{x}=0}$ (in our case $x$ is a physical parameter characterizing
$R_{YM}^{{}^{\prime}}$ while $\bar{x}$ is its [Gibbs] average). Whence
$\frac{\partial S}{\partial x}|_{x=0}=0;\quad\frac{\partial^{2}S}{\partial
x^{2}}|_{x=0}<0.$ (24)
Then in a neighborhood of $x=0$, the entropy $S=(E-F)/T$ inherent in the
vacuum manifold $R_{\rm YM}$ can be expand in the series [21]
$S(x)\sim S(0)-\frac{\beta}{2}x^{2};\quad\beta={\rm const}>0;$ (25)
by the powers of $x$.
In this case the probability $w(x)$ for $x$ to be in the interval $[x,x+dx]$
which is directly proportional to $e^{S(x)}$:
$w(x)={\rm const}\cdot e^{S(x)},$ (26)
just results the Gaussian distribution for $x$:
$w(x)dx=Ae^{\frac{-\beta}{2}x^{2}};\quad A=\sqrt{\beta/2\pi}.$ (27)
We see thus the importance knowing the complete Hamiltonian describing $R_{\rm
YM}$, (1). In particular, it is worth to study the item in this Hamiltonian
responsible for colliding vacuum BPS monopole modes with (topologically
nontrivial) threads (i.e. YM fields $A_{\theta}$ [1, 11], (18)). It is optimal
herewith the situation when $\beta$ is small. Then the entropy $S$ go to its
maximum (that corresponds [21] to the minimum of the free energy $F$).
Thus for a system of (physical) fields it is energetically advantageous that
corrections to the free energy $F$ conditioned by merging and annihilating
topological defects are small and “belong” to the perturbation theory.
In the framework of the Minkowskian YMH model [3, 4, 5, 6, 7, 8] quantized by
Dirac, for vacuum BPS monopole modes colliding [1, 11] with (topologically
nontrivial) threads, it is important, in the light of the said above, to
understand whether it is described by a perturbation theory in the YM
effective coupling constant $\alpha_{s}$ (that corresponds to small values of
the appropriate $\beta$) or not.
If it is so, the arising radiative corrections result a shift of the “true”
vacuum. This implies, in turn, a “blurring” of the first-order phase
transition picture taking place [1] in the Minkowskian YMH model [3, 4, 5, 6,
7, 8] quantized by Dirac.
On the other hand, setting $\bar{x}=(\bar{x})^{2}=0$ refers rather to the
symmetric ($SU(2)$) phase of the quested model. But our interest in the
Minkowskian YMH model [3, 4, 5, 6, 7, 8] is its less symmetrical ($U(1)$)
phase, in which various vacuum superfluid and rotary effects are revealed (in
the framework of the first-order phase transition picture).
For example, $x\neq 0$ (then $(\bar{x})^{2}\neq 0$) can be ordering parameter
characterizing the Minkowskian YMH model [3, 4, 5, 6, 7, 8] quantized by Dirac
(it is [7, 8] the $\pm\sqrt{<B^{2}>}$ for the “magnetic” field squared ${\bf
B}^{2}$).
In this case $x$ has the nonzero dispersion
$Dx=<(x-\bar{x})>^{2}=M(x^{2})-(Mx)^{2}\neq 0$ (28)
($Mx$ is the mathematical, i.e. vacuum in the physical context, expectation
value of $x$). Thinking that $M(x)=0$ (this is an ordinary assumption in QFT),
one has $Dx=M(x^{2})\equiv<x^{2}>$.
On the other hand [21], now (at the assumption $M(x)\equiv<x>=0$)
$<(x-\bar{x})>^{2}=<x^{2}>=\int\limits_{-\infty}^{\infty}x^{2}w(x)dx=\beta^{-1}.$
(29)
Just this shows that the maximum of the entropy, corresponding to the limit
$\beta\to 0$, can be achieved in the Minkowskian YMH model [3, 4, 5, 6, 7, 8]
quantized by Dirac if the minimum of the ordering parameter $\sqrt{<B^{2}>}$
is absolute, i.e. maximally possible deep. In other words, the maximal entropy
(in the $T\to 0$ limit) is reached, obviously, over the “true” vacuum, for
which $<B^{2}>=<B_{1}^{2}>\neq 0$ (this “true” vacuum is induced by YM threads
$A_{\theta}$ [1, 11]. It corresponds to collective solid rotations of the
physical vacuum [3, 4, 5, 6, 7, 8]. Simultaneously, superfluid potential
motions inside this vacuum (set by the Bogomolny’i and Gribov ambiguity
equations) refer to the “metastable” thermodynamic phase (i.e. to the “false”
vacuum).
And moreover, it is obviously now that the above describeed gap between the
directions $\bf B$ and ${\bf B}_{1}$, referring, respectively, to the
“superfluid” and “rotary” thermodynamic phases induces the gap in the plot of
the entropy $S$. It is just the sign of the first-order phase transition
occurring in the Minkowskian YMH model [3, 4, 5, 6, 7, 8] quantized by Dirac.
The case when $x\neq 0$ is another parameter having a relation to the vacuum
manifold $R_{\rm YM}$ is not less interesting. The one of such important
parameters is $(m/\sqrt{\lambda})^{-1}$ for the effective Higgs mass [7, 8].
It is obvious now that the $\beta\to 0$ limit (at which the entropy $S$ of the
vacuum manifold $R_{\rm YM}$ is maximum according to (24)) corresponds to the
limit $(m/\sqrt{\lambda})^{-1}\to\infty$ for this parameter.
“Geometrically”, this occurs in the spatial region (21) along the axis $z$
intimately near this axis. It is just [1] the region locating (topologically
nontrivial) threads inside the vacuum manifold $R_{\rm YM}$. And moreover,
course our above discussion (repeating the arguments [1, 13]) we have
elucidatet that the value $(m/\sqrt{\lambda})^{-1}$ can be interpreted (to
within the multiplier $g^{-1}$ [7, 8]444 $\lambda\to 0,~{}~{}~{}~{}~{}~{}m\to
0:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{\epsilon}\equiv\frac{gm}{\sqrt{\lambda}}\not=0.$
) as the (effective) thickness $\epsilon(V)\propto r^{-3}$ of domain walls
inside the vacuum manifold $R_{\rm YM}$. It just approaches infinity at the
origin of coordinates [1].
Whence an interesting conclusion can be drawn that the maximum entropy in the
Minkowskian YMH model [3, 4, 5, 6, 7, 8] quantized by Dirac is achieved in the
spatial region intimately near the axis $z$ (of the chosen rest reference
frame), the region locating thread topological defects inside the vacuum
manifold $R_{\rm YM}$. On the other hand, this allows to apply the
superselection rules [19] to this manifold in order to construct the Hilbert
space $\oplus{{}_{n}}{\cal H}_{n}$: in a definite sense, the latter one is a
quantum analogue of $R_{\rm YM}$.
As to the “effective” Higgs mass $(m/\sqrt{\lambda})\sim\epsilon(V)^{-1}$ (it
is the only mass scale in the Minkowskian YMH model [3, 4, 5, 6, 7, 8]
quantized by Dirac beyond incorporating quarks in this model), it approaches
zero as $r\to 0$ ($\epsilon(0)\to\infty$) in the maximum entropy point $S(0)$
of the plot $S(r)$), i.e. there where the superselection rules [19] are valid.
Thus the model [3, 4, 5, 6, 7, 8] becomes massless (involves the zero mass
gap) in the $r\to 0$ limit.
As it was discussed in Ref. [1], in the $r\to\infty$ limit, the “geometrical”
picture of the vacuum manifold $R_{\rm YM}$ changes in a radical wise. Domain
walls become (infinitely) thin, and this promotes merging topological domains
inside $R_{\rm YM}$ in this spatial region. In this case merging
(annihilation) topological defects cannot be considered as perturbation
processes because of unsuppressed tunelling through such (infinitely) thin
domain walls 555The said resembles the visual picture when liquid helium II,
possessing superfluidity, flows in parallel capillaries with porous walls. .
Then also our above arguments of the “maximum entropy” [21] lose their
validity.
Namely against this background of tunelling effects between topological
domains inside $R_{\rm YM}$ at distances $r\gg 0$ superfluid potential motions
proceed in the Minkowskian physical vacuum [3, 4, 5, 6, 7, 8] involving BPS
monopole solutions and quantized by Dirac.
Thus to achieve the correct superselection description of the vacuum manifold
$R_{\rm YM}$, any coherent Hilbert space ${\cal H}_{n}$ would be restricted in
the Minkowskian coordinate space. The scalar product in such Hilbert spaces
looks as following:
$2\pi\int\limits_{0}^{r_{1}}f(r)g(r)r^{2}dr;\quad f(r),~{}g(r)\in{\cal
H}_{n};\quad r=\sqrt{x^{2}+y^{2}}$ (30)
(in cylindrical coordinates introduced in the Minkowskian space). The upper
limit $r_{1}$ in the above Lebesgue integral can be evaluated as $r_{1}\to
0\sim O(\epsilon(\infty))$, i.e. it is infinitely small.
It is easy to see (this, perhaps, will be done in the one of future studies
the author plans) that “thinning” domain walls inside $R_{\rm YM}$ at
distances $r\to\infty$ (with accompanying tunelling effects between
topological domains inside this vacuum manifold) promotes the infrared
topological confinement in the spirit [14] i.e. surviving only “small” Gribov
multipliers $v^{(n)}({\bf x})$ in quark and gluonic Green functions in all the
orders of the perturbation theory. And it is the one of important gains of
that “thinning”, especially because such infrared topological confinement
implies [6] the confinement of gluons and quarks in the sense as it is
realized ordinary in theoretical physic.
As it was noted in Ref. [1] (and repeated again in the present study), the
effective Higgs mass $m/\sqrt{\lambda}$ varies (in the Bogomolny’i limit $m\to
0$, $\lambda\to 0$ [7, 8, 11, 15]) in the interval from zero in the spatial
region (21) locating (topologically nontrivial) thread configurations to a
nonzero value in the infrared limit $r\to\infty$.
The limit $m/\sqrt{\lambda}\to 0$ can be treated as an ultraviolet one,
$p^{2}=(m/\sqrt{\lambda})^{2}=0.$
It corresponds to the ’cut-off’ parameter [4] $\epsilon(\infty)\to 0$ 666This
’cut-off’ parameter enters [4, 7, 8] as the lower integration limit the
expressions for the vacuum “magnetic” and “electric” energies. For instance,
the expression
$\frac{1}{2}\int\limits_{\epsilon(\infty)}^{\infty}d^{3}x[B_{i}^{a}(\Phi_{k})]^{2}\equiv\frac{1}{2}V<B^{2}>=\frac{1}{2\alpha_{s}}\int\limits_{\epsilon(\infty)}^{\infty}\frac{dr}{r^{2}}\sim\frac{1}{2}\frac{1}{\alpha_{s}\epsilon(\infty)}=2\pi\frac{gm}{g^{2}\sqrt{\lambda}}=\frac{2\pi}{g^{2}\epsilon(\infty)}$
for the vacuum “magnetic” energy. The similar computations take place also for
the “electric” vacuum energy item (11), proving to be directly proportional to
$\epsilon(\infty)$. .
In the opposite spatial region $r\to\infty$ of the infrared topological [14]
and “physical” [6] confinement, the reasonable question arises about correct
estimating the nonzero value of the effective Higgs mass $m/\sqrt{\lambda}$.
With a large probability, it is in a correlation with the typical hadronic
radius [22] $r_{h}\sim 1~{}{\rm fm}$.
Thus one can consider the diapason $[0,m(r_{h})]$ in which the effective Higgs
mass $m/\sqrt{\lambda}$ varies (where $m(r_{h})$ can be treated as an infrared
cut-off 777Indeed, infrared QCD effects refer to the interval of distances
$[r_{h},\infty[$, but any gluonic string confining a quark-antiquark pair near
each other cannot stretch to infinite distances; it will tear to a few strings
with typical lengths $\sim$ 1 fm [22]. Therefore, there are no any sence to
consider $(m/\sqrt{\lambda})|_{\infty})$, which formally approaches infinity
according to Eq. (7). ). Herewith the point $m/\sqrt{\lambda}=0$, that is
scale (renorm-group) invariant, is treated as the ultraviolet fixed point
[22]. There is, obviously, a continuous (and analytical) renorm-group
transformation connecting this zero value and $m(r_{h})$. This allows to
interpret the effective Higgs mass $m/\sqrt{\lambda}$ as a Wegner variable
[23, 24] (this circumstance was noted already in the paper [9]).
The said gives a hope, in spite the first-order phase transition occurring in
the Minkowskian YMH BPS monopole model [3, 4, 5, 6, 7, 8] quantized by Dirac,
that weak, $m/\sqrt{\lambda}\to 0$ 888One can think that it is a function of
the spatial cylindrical region (21) locating thread topological defects inside
the vacuum manifold $R_{\rm YM}$., and strong, $m/\sqrt{\lambda}\to m(r_{h})$,
coupling regions can be connecteed by an analytical line (referred to as the
critical line in the paper [23]) 999 As it was analyzed in [1], annihilating
processes for magnetic charges ${\bf m}\neq 0$ (i.e. appropriate YM BPS
monopole modes and excitations over the BPS monopole vacuum) colliding with
(topologically nontrivial) threads $A_{\theta}$ can lead (in a definite time
space) to the situation when all such magnetic charges annihilate while Higgs
vacuum modes possess arbitrary electric charges (according to the Dirac
quantization [25] of the both types of charges). In the terminology [26], one
can refer to this as to the Higgs phase (with additional screening “Higgs”
electric charges by BPS ansatzes [7, 8, 11, 15], playing the role of electric
formfactors [1]). As it is well known [26], the Higgs phase is treated as that
dual to the confinement phase, when Higgs vacuum modes are ”magnetic objects”
while quark and gluons are “electric objects”. For the “ordinary” Higgs non-
Abelian gauge theory the Fradkin-Shenker (Osterwalder-Seiler) theorem takes
place [27]. It turns out that there are no transition separating the Higgs and
confinement phases in such theory. But the proof of the Fradkin-Shenker
(Osterwalder-Seiler) theorem losses its validity in the BPS limit [7, 8, 11,
15] $\lambda\to 0$, when the Higgs potential decouples from the complete QCD
action functional. Additionaly, the Fradkin-Shenker (Osterwalder-Seiler)
theorem [27] is valid only in the non-Abelian gauge theory where the Higgs
vacuum expectation value $<\Phi>^{2}$ serves as an order parameter. This
creates definite difficulties since the Higgs and confinement phases can be
now separated each from other. In particular, it can be correctly for the
Minkowskian YMH BPS monopole model [3, 4, 5, 6, 7, 8] quantized by Dirac. Then
such “separation” will be in an agreement with the first-order phase
transition occurring therein but in a definite contradiction with the
treatment of the “effective” Higgs mass $m/\sqrt{\lambda}$ as a Wegner
variable. Also $<\Phi>^{2}$ ceases to be the order parameter in the mentioned
model; instead, the value $<B>^{2}$ for the vacuum “magnetic” field $\bf B$
acquires the sense of such a parameter. The way out from this uncertain
situation is, on the author particular opinion, is in reexamining the Fradkin-
Shenker (Osterwalder-Seiler) theorem in the BPS limit. .
In the recent paper [1] and in the present study the ways solving the mass gap
problem in the Minkowskian YMH BPS monopole model [3, 4, 5, 6, 7, 8] quantized
by Dirac and involving the discrete vacuum geometry (1) (calling to justify
the Dirac fundamental quantization scheme [2] applied to this model) are
outlined. Of course, lot of difficulties still remain in this aspect need
further study. For examle, the relation between the first-order phase
transition taking in the Minkowskian YMH model [3, 4, 5, 6, 7, 8] quantized by
Dirac and the existence therein the critical line [23] connecting weak and
strong coupling regions: more exactly, wheter these both things are compatible
each with other or not 101010Nevertheless, the author of the present study
does not aspires to solving the mass gap problem in the massless YM theory.
That problem was formulated as following [28]. Experiment and computer
simulations about the “pure”YM theory without other (quantum) fields suggest
the existence of a ”mass gap” in the solution to the quantum versions of the
YM equations. But no proof of this property is known. In the strict
mathematical language, the mass gap problem can be expressed in the following
way [28]. Since the Hamiltonian $H$ of a QFT is the element of the Lie algebra
of the Poincare group and the appropriate vacuum vector $\Omega$ is Poincare
invariant (see e.g. [19]), it is an eigenstate with zero energy, $H\Omega=0$.
The positive energy axiom (in absence of external negative potentials) asserts
that in any QFT, the spectrum of $H$ is supported in the region $[0,\infty)$.
In this terminology, a QFT has a mass gap if $H$ has no spectrum in the
interval $[0,\Delta)$ for a $\Delta>0$ The supremum of such $\Delta$ is called
the mass $m$. Then the YM mass gap problem can be formulated mathematically
[28] as proving that for any compact simple gauge group $G$, the quantum YM
theory on ${\bf R}^{4}$ exists and has a mass gap $\Delta>0$. An important
consequence of the existence of a mass gap is that for any positive constant
$C<\Delta$ and for any local quantum field operator ${\cal O}(x)$ such that
$\left\langle\Omega,{\cal O}\Omega\right\rangle=0$, one has
$|\left\langle\Omega,{\cal O}(x){\cal
O}(y)\Omega\right\rangle|\leq\exp(-C|x-y|)$ if $|x-y|$ is sufficiently large
(depending on $C$ and ${\cal O}$). As we see, in the quested Minkowskian YMH
model [3, 4, 5, 6, 7, 8] quantized by Dirac, firstly, Higgs vacuum BPS
monopole modes are present, and secondly, $\Delta\geq 0$ in the therminology
[28]. .
## References
* [1] L. D. Lantsman, ”Discrete” Vacuum Geometry as a Tool for Dirac Fundamental Quantization of Minkowskian Higgs Model, [arXiv:hep-th/0701097].
* [2] P. A. M. Dirac, Proc. Roy. Soc. A 114 (1927) 243; Can. J. Phys. 33 (1955) 650.
* [3] V. N. Pervushin, Teor. Mat. Fiz. 45, 395 (1980) [Theor. Math. Phys. 45 (1981) 1100].
* [4] D. Blaschke, V. N. Pervushin, G. R$\rm\ddot{o}$pke, Topological Gauge invariantVariables in QCD, MPG-VT-UR 191/99, [arXiv:hep-th/9909193].
* [5] D. Blaschke, V. N. Pervushin, G. R$\rm\ddot{o}$pke, Topological Invariant Variables in QCD, in Proceeding of the Int. Seminar Physical variables in Gauge Theories, Dubna, September 21-25, 1999, edited by A. M. Khvedelidze, M. Lavelle, D. McMullan and V. Pervushin (E2-2000-172, Dubna, 2000), p. 49, [arXiv:hep-th/0006249].
* [6] V. N. Pervushin, Dirac Variables in Gauge Theories, Lecture Notes in DAAD Summerschool on Dense Matter in Particle and Astrophysics, JINR, Dubna, Russia, August 20- 31, 2001; Phys. Part. Nucl. 34, 348 (2003); Fiz. Elem. Chast. Atom. Yadra 34, 679 (2003); [hep-th/0109218].
* [7] L. D. Lantsman, V. N. Pervushin, The Higgs Field as The Cheshire Cat and his Yang-Mills ”Smiles”, Proc. of 6th International Baldin Seminar on High Energy Physics Problems (ISHEPP), Dubna, Russia, 10-15 June 2002; [arXiv:hep-th/0205252];
L. D. Lantsman, Minkowskian Yang-Mills Vacuum, [arXiv:math-ph/0411080].
* [8] L. D. Lantsman, V. N. Pervushin, Yad. Fiz. 66, 1416 (2003); Physics of Atomic Nuclei 66, 1384 (2003); [arXiv:hep-th/0407195].
* [9] L. D. Lantsman, Fizika B 18 (Zagreb), 99 (2009); [arXiv:hep-th/0604004].
* [10] L. D. Faddeev, Proc. of 4th Int. Symp. on Nonlocal Quantum Field Theory, Dubna, USSR, 1976, JINR D1-9768, p. 267.
R. Jackiw, Rev. Mod. Phys. 49 (1977) 681.
* [11] A. S. Schwarz, Kvantovaja Teorija Polja i Topologija, 1st edition (Nauka, Moscow, 1989) [A. S. Schwartz, Quantum Field Theory and Topology (Springer, 1993)].
* [12] A. D. Linde, Elementary Particle Physics and Inflationary Cosmology, 1st edition (Nauka, Moscow, 1990), [arXiv: hep-th/0503203].
* [13] G. ’t Hooft, Nucl. Phys. B 138 (1978) 1.
* [14] P. I. Azimov, V. N. Pervushin, Teor. Mat. Fiz. 67 (1986) 349 [Theor. Math. Phys. 67 (1987) 546].
* [15] M. K. Prasad, C. M. Sommerfeld, Phys. Rev. Lett. 35 (1975) 760;
E. B. Bogomol’nyi, Yad. Fiz. 24 (1976) 449.
* [16] G. ’t Hooft, Nucl. Phys. B 79 (1974) 276.
* [17] A. M. Polyakov, Pisma JETP 20 (1974) 247 [Sov. Phys. JETP Lett. 20 (1974) 194]; Sov. Phys. JETP Lett. 41 (1975) 988.
* [18] L. D. Lantsman, Superfluid Properties of BPS Monopoles, [arXiv:hep-th/0605074].
* [19] N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, Obshie Prinzipi Kvantovoj Teorii Polja, 1st edn. (Nauka, Moscow 1987).
* [20] E. Witten, Nuovo Cim. A 51, 325 (1979).
* [21] L. D. Landau, E. M. Lifschitz, Lehrbuch der Theoretischen Physik (Statistishe Physik, Band 5, teil 1), in German, edited by R. Lenk and P. Ziesche (Akademie-Verlag, Berlin 1979/1987).
* [22] T. P. Cheng, L.- F. Li, Gauge Theory of Elementary Particle Physics, 3rd edn. (Oxford University Press 1988).
* [23] L. P. Kadanoff, Rev. Mod. Phys. 49, 267 (1977).
* [24] F. Wegner, Phys. Rev. B 5, 4529 (1972); Lecture Notes in Physics 37, 171 (1973).
* [25] P. A. M. Dirac, Proc. Roy. Soc. A 133 (1931) 69.
* [26] F. Bruckmann, G. ’t Hooft, Phys. Rep. 142 (1986) 357; [arXiv:hep-th/0010225].
* [27] E. Fradkin, S. Shenker, Phys. Rev. D19 (1979) 3682;
K. Osterwalder, E. Seiler, Ann. Phys. 110 (1978) 440.
* [28] A. Jaffe, E. Witten, Quantum Yang-Mills Theory, 2000.
|
arxiv-papers
| 2010-08-17T11:13:35 |
2024-09-04T02:49:12.230925
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Leonid Lantsman",
"submitter": "Leonid Lantsman",
"url": "https://arxiv.org/abs/1008.2866"
}
|
1008.2993
|
# Determination of fundamental properties of an M31 globular cluster from
main-sequence photometry
Jun Ma11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China; majun@vega.bac.pku.edu.cn
22affiliation: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, China , Zhenyu
Wu11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China; majun@vega.bac.pku.edu.cn , Song
Wang,11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China; majun@vega.bac.pku.edu.cn
33affiliation: Graduate University, Chinese Academy of Sciences, Beijing,
100039, P. R. China Zhou Fan11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn , Xu Zhou11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn , Jianghua Wu11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn , Zhaoji Jiang11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn and Jiansheng Chen11affiliation: National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, P.
R. China; majun@vega.bac.pku.edu.cn
###### Abstract
M31 globular cluster B379 is the first extragalactic cluster, the age of which
was determined by main-sequence photometry. In the main-sequence photometric
method, the age of a cluster is obtained by fitting its color-magnitude
diagram (CMD) with stellar evolutionary models. However, different stellar
evolutionary models use different parameters of stellar evolution, such as
range of stellar masses, different opacities and equations of state, and
different recipes, and so on. So, it is interesting to check whether different
stellar evolutionary models can give consistent results for the same cluster.
Brown et al. (2004a) constrained the age of B379 by comparing its CMD with
isochrones of the 2006 VandenBerg models. Using SSP models of Bruzual &
Charlot (2003) (BC03) and its multi-photometry, Ma et al. (2007) independently
determined the age of B379, which is in good agreement with the determination
of Brown et al. (2004a). The BC03 models are calculated based on the Padova
evolutionary tracks. It is necessary to check whether the age of B379 which,
being determined based on the Padova evolutionary tracks, is in agreement with
the determination of Brown et al. (2004a). So, in this paper, we re-determine
its age using isochrones of the Padova stellar evolutionary models. In
addition, the metal abundance, the distance modulus, and the reddening value
for B379 are also determined in this paper. The results obtained in this paper
are consistent with the previous determinations, which including the age
obtained by Brown et al. (2004a). So, this paper confirms the consistence of
the age scale of B379 between the Padova isochrones and the 2006 VandenBerg
isochrones, i.e. the results’ comparison between Brown et al. (2004a) and Ma
et al. (2007) is meaningful. The results obtained in this paper are: the
metallicity $\rm{[M/H]}=\log(Z/Z_{\odot})=-0.325$, the age $\tau=11.0\pm 1.5$
Gyr, the reddening value $E(B-V)=0.08$, and the distance modulus
$(m-M)_{0}=24.44\pm 0.10$.
###### Subject headings:
galaxies: individual (M31) – galaxies: globular clusters – galaxies: stellar
content
††slugcomment: PASP, in press
## 1\. Introduction
Globular clusters (GCs), relics of some of the earliest phases of star and
galaxy formation, can be analyzed to understand how soon after the Big Bang
did the various stellar systems form. The most direct method for determining
the age of a star cluster is main-sequence photometry, in which the isochrone
that minimizes the discrepancies between the observed and calculated sequences
can exactly present the estimated cluster age. However, this method has only
been applied to the Galactic GCs and GCs in the satellites of the Milky Way
(e.g., Rich et al., 2001) before Brown et al. (2004a) used this method to
constrain the age of an M31 GC B379111In Brown et al. (2004a), SKHB 312 is
used. In this paper, B379 is used following the designations from the Revised
Bologna Catalog (RBC) of M31 GCs and candidates (Galleti et al. 2004, 2006,
2007), which is the main catalog used in studies of M31 GCs. based on the CMD
reaching more than 1.5 mag below the main-sequence turn-off. The CMD of B379
was constructed from the extremely deep images with the Advanced Camera for
Surveys (ACS) on the Hubble Space Telescope (HST).
In general, ages of extragalactic star clusters are obtained by comparing
integrated photometry with models of simple stellar populations (SSPs). For
examples, Ma et al. (2001, 2002a, 2002b, 2002c) and Jiang et al. (2003)
estimated ages for star clusters in M33 and M31 by comparing the SSP models of
BC96 (Bruzual & Charlot 1996, unpublished) with their integrated photometric
measurements in the Beijing-Arizona-Taiwan-Connecticut (BATC) photometric
system; de Grijs et al. (2003a) determined ages and masses of star clusters in
the fossil starburst region B of M82 by comparing their observed cluster
spectral energy distributions (SEDs) with the model predictions for an
instantaneous burst of star formation (see also de Grijs et al., 2003b, c).
Bik et al. (2003) and Bastian et al. (2005) derived ages, initial masses and
extinctions of M51 star cluster candidates by fitting Starburst99 SSP models
(Leitherer et al., 1999) to their observed SEDs in six broad-band and two
narrow-band filters from the Wide Field Planetary Camera-2 (WFPC2) onboard the
HST. Ma et al. (2006a) estimate ages and metallicities for 33 M31 GCs by
comparing between BC03 models (Bruzual & Charlot, 2003) and their BATC multi-
band photometric data. Ma et al. (2006b) derived the age and reddening value
of the M31 GC 037-B327 based on photometric and BC03 measurements in a large
number of broad- and intermediate-band from the optical to the near-infrared.
Fan et al. (2006) determined new ages for 91 M31 GCs from Jiang et al. (2003)
based on improved photometric data and BC03 models. In particular, Ma et al.
(2007) derived the age of B379222In Ma et al. (2007), S312 is used. by
comparing its photometric data with BC03 models. The age obtained by Ma et al.
(2007) is $9.5_{-0.99}^{+1.15}$ Gyr, which is consistent with the
determination of $10_{-1}^{+2.5}$ Gyr by Brown et al. (2004a) using the main-
sequence photometry.
The nearest large GC system outside the Milky Way is that of the Andromeda
Galaxy (= M31), which is located at a distance of 770 kpc (Freedman & Madore,
1990). The first M31 GC resolved into stars was studied from the ground by
Heasley et al. (1988), who only resolved the red giant branch of G1.
Subsequently, some authors such as Ajhar et al. (1996), Fusi Pecci et al.
(1996), Rich et al. (1996), Holland et al. (1997), Jablonka et al. (2000),
Williams & Hodge (2001a), and Williams & Hodge (2001b) have used images from
the HST/WFPC2 to construct the CMDs of M31 star clusters in order to determine
their metallicities, reddening values, and ages. However, these CMDs are not
deep enough to show conspicuous main-sequence turn-offs.
Since the luminosity of the horizontal branch (HB) in stellar populations
older than about 8 Gyr is expected to be independent of age and only mildly
dependent on metallicity, it is widely used a distance indicator (see Gallart
et al., 2005, and references therein). In addition, HB stars are fundamental
standard candles for Population II systems, and consequently are important
tools for determining ages of GCs from the main-sequence turn-off luminosity
(Rich et al., 1996).
Two of the first studies about the HB for M31 GCs were that of Rich et al.
(1996) and Fusi Pecci et al. (1996). They used the observed data from the Wide
Field Planetary Camera-2 (WFPC2) and Faint Object Camera (FOC) onboard the HST
to make the first tentative detection of HB in G1, B006, B045, B225, B343,
B358, B405, and B468, deriving the apparent magnitude of the HB for these GCs
to be in the range $25.29<V<25.66$. In addition, Fusi Pecci et al. (1996)
firstly presented a direct calibration for the mean absolute magnitude of the
HB at the instability strip with varying metallicity for M31 GCs.
B379 was firstly detected by Sharov (1973) (No.19), and confirmed by Sargent
et al. (1977) ($\rm No.312=SKHB~{}312$, which was used by Brown et al.
(2004a); or $=\rm S312$, which was used by Ma et al. (2007).) and Battistini
et al. (1987) ($\rm No.379=B379$, which is used in the RBC (Galleti et al.
2004, 2006, 2007) and this paper.). B379 is located in the halo of M31, at a
projected distance of about $59\arcmin$ ($\rm=13~{}kpc$) from the galaxy’s
nucleus. It is a fact that B379 is a common halo GC, however, it is among the
first extragalactic GCs whose age was accurately estimated by main-sequence
photometry (Brown et al., 2004a) based on its CMD from the extremely deep
images observed with the HST/ACS.
In this paper, we re-determine the age, metallicity, reddening value and
distance modulus for B379 by comparing its CMD constructed by Brown et al.
(2004a) with isochrones of the Padova stellar evolutionary models. The paper
is organized as follows. In §3, we describe the results of photometric data
based on the HST/ACS observations for B379. In §4, we constrain the age,
metallicity, reddening value, and distance modulus for B379. At last, we will
give a summery in §5.
## 2\. Recent works of B379
Brown et al. (2004a) used the main-sequence photometry to determine the age of
B379 based on the CMD constructed using the extremely deep images from the
HST/ACS observations. This CMD reached more than 1.5 mag below the main-
sequence turn-off, and firstly allowed a direct age estimate from the turn-off
for an extragalactic cluster. By comparison to isochrones of VandenBerg et al.
(2006), Brown et al. (2004a) derived the age of B379 to be $10_{-1}^{+2.5}$
Gyr. Ma et al. (2007) determined the age of B379 by comparing its multi-color
photometric data which including the near-ultraviolet (NUV) from the Nearby
Galaxies Survey (NGS) of the Galaxy Evolution Explorer (GALEX) (Rey et al.,
2005, 2007), broad-band $UBVR$ (Battistini et al., 1987; Reed, Harris &
Harris, 1994), 9 BATC intermediate-band filters and Two Micron All Sky Survey
(2MASS) $JHK_{s}$, with SSP models of BC03. These photometric data constitute
the SEDs of B379 covering $2267-20000$Å. The age of B379 determined by Ma et
al. (2007) is $9.5_{-0.99}^{+1.15}$ Gyr, which is consistent with the
determination of $10_{-1}^{+2.5}$ Gyr by Brown et al. (2004a). However, BC03
models are based on the Padova evolutionary tracks. So, it is necessary to
compare the age scale between the Padova evolutionary tracks and the Victoria-
Regina isochrones used in Brown et al. (2004a), and only if these two
evolutionary tracks have the consistent age scale for B379, the results’
comparison between Brown et al. (2004a) and Ma et al. (2007) is meaningful. As
an example, Ma et al. (2007) drew the isochrones with 10 Gyr and the solar
metallicity, and found that the matching is very good in the main-sequence
(MS) and the subgiant branch (SGB) (see Ma et al., 2007, for details).
However, we should check whether the age of B379 can be estimated to be $\sim
10$ Gyr based on the Padova evolutionary tracks. This is one of the key
contributions of the present paper.
## 3\. Database
The observed data of B379 in this study are from Brown et al. (2004a), who
constructed the CMD of B379 using the images observed with the ACS
observations in the F606W and the F814W filters. Using the ACS Wide-Field
Camera (WFC), Brown et al. (2003) obtained deep optical images of a field,
$51^{\prime}$ from the nucleus on the southeast minor axis of the M31 halo
including B379, which are 39.1 hr in the F606W filter and 45.4 hr in the F814W
filter. Brown et al. (2004a) presented the CMD of B379 based on these ACS
observations. The resulting CMD reached $m_{V}\approx 30.5$ mag, which is the
first CMD of extragalactic clusters reaching more than 1.5 mag below the main-
sequence turn-off. These observations firstly allow a direct age estimate from
the turn-off for an extragalactic cluster. By comparison to isochrones of
VandenBerg et al. (2006), Brown et al. (2004a) derived the age of B379 to be
$10_{-1}^{+2.5}$ Gyr. In Brown et al. (2004a), the CMD of B379 was constructed
from stars within an annulus chosen to maximize the signal-to-noise and
minimize field contamination. Because B379 was near the field edge and the
observations were dithered, the exposure time was not uniform across the
annulus. So, Brown et al. (2004a) discarded the fraction of annulus ($<0.5\%$)
that had half of the total exposure time but kept the fraction ($<14\%$) that
was exposed for $75\%$. In addition, Brown et al. (2004a) used extensive
artificial star tests to determine the photometric scatter and completeness as
a function of color, luminosity and field position. At last, 1720 stars within
the annulus spanning $100-300$ pixels were retained to produce a much cleaner
CMD (see Brown et al., 2004a, for details). In this paper, we also take these
1720 stars as the member stars of B379 as Brown et al. (2004a) did (The data
were kindly provided by Dr. Brown).
## 4\. The age, metallicity, reddening value and distance modulus of B379
### 4.1. Isochrones of stellar evolutionary models
More than 50 years ago, Sandage (1953) presented the CMD for the Galactic GC
M3 and applied an evolutionary theory to M3 CMD to give a time interval of
$5\times 10^{9}$ years since the formation of the main-sequence. From then on,
main-sequence photometry is thought the most direct method for determining
ages of star clusters, because the turn-off of the CMD is mostly affected by
age (see Puzia et al., 2002b, and references therein). Stellar evolutionary
models from the Padova group (Bertelli et al., 1994; Girardi et al., 2000,
2002, and references therein) and the Victoria-Regina (VandenBerg et al.,
2000, 2006, and references therein) are widely used. In the Padova stellar
evolutionary models, Girardi et al. (2002) provided tables of theoretical
isochrones in such photometric systems as ABmag, STmag, VEGAmag, and a
standard star system, and derived tables of bolometric corrections for
Johnson-Cousins-Glass, HST/WFPC2, HST/NICMOS, Washington, and ESO Imaging
Survey systems. The complete data-base (Girardi et al., 2002) covers a very
large range of stellar masses (typically from 0.6 to $120~{}M_{\odot}$). As a
supplement, Girardi et al. (2008) presented several theoretical isochrones
including HST/ACS WFC. These models (Girardi et al., 2002, 2008) are computed
with updated opacities and equations of state, and moderate amount of
convective overshoot. However, the isochrones are presented for only 6 initial
chemical compositions: ${\rm[Fe/H]}=-2.2490$, $-1.6464$, $-0.6392$, $-0.3300$,
$+0.0932$ (solar metallicity), and $+0.5595$, which are evidently not dense
enough. It is fortunate that Marigo et al. (2008) provide tables for any
intermediate value of age and metallicity via an interactive web interface
(http://stev.oapd.inaf.it/cmd). We will discuss this web in detail in §4.2.
The novel feature of the Victoria-Regina models (VandenBerg et al., 2000,
2006, and references therein) is that they provide a wide range of
metallicities, i.e. VandenBerg et al. (2006) presented seventy-two grids of
stellar evolutionary tracks for 32 [Fe/H] values from $-2.31$ to $0.49$, which
are dense enough for studying properties of stellar populations with different
metallicities. In addition, in these models, convective core overshooting has
been treated using a parameterized form of the Roxbergh criterion (Roxburgh,
1978, 1989), in which the free parameter, $F_{\rm over}$ ($F_{\rm over}$ must
be calibrated using observations.), is assumed to be a function of both mass
and metal abundance.
### 4.2. Isochrone fitting
To determine the main characteristics (age and metallicity) of the population
in B379, we fit isochrones to the cluster CMD. We used the Padova theoretical
isochrones in the HST/ACS WFC STmag system (Marigo et al., 2008). Via an
interactive web interface at http://stev.oapd.inaf.it/cmd, we can construct a
grid of isochrones for different values of age and metallicity, photometric
system, and dust properties. We use the default models that involve scaled
solar abundance ratios (i.e., $[\rm{\alpha/Fe]}=0.0$). In performing, the
Salpeter initial mass function (IMF) (Salpeter, 1955) is adopted to match the
selection of Ma et al. (2007), who used the high-resolution SSP models of BC03
computed using the Salpeter (1955) IMF; and circumstellar dust is not
included. As we pointed out previously, that by comparison to isochrones of
VandenBerg et al. (2006), Brown et al. (2004a) derived the age of B379 to be
$10_{-1}^{+2.5}$ Gyr. In addition, the metallicity of B379 is available:
Huchra et al. (1991) derived $\rm[Fe/H]=-0.7\pm 0.35$ using the strengths of
six absorption features in the cluster integrated spectra; Holland et al.
(1997) used the HST/WFPC2 photometry to construct the deep CMD for B379, and
the shape of the red giant branch (RGB) gave an iron abundance of
$\rm[Fe/H]=-0.53\pm 0.03$. These two metallicities obtained from different
methods are consistent. Based on the age and metallicity of B379 obtained by
the previous authors (Brown et al., 2004a; Huchra et al., 1991; Holland et
al., 1997), we used the interactive web (http://stev.oapd.inaf.it/cmd) to
construct a fine grid of isochrones about ages and metallicities, sampling an
age range $8.0\leq\tau\leq 13.5$ Gyr at intervals of 0.5 Gyr, and a metal
abundance range $0.00250\leq Z\leq 0.00950$ at intervals of 0.00025 dex. The
total metallicity $\rm{[M/H]}=\log(Z/Z_{\odot})$ where $Z_{\odot}\approx
0.019$, so this abundance range corresponds to $-0.88\leq\rm{[M/H]}\leq-0.30$.
We followed the method of Mackey & Broby Nielsen (2007) of finding the best
fitting isochrone, i.e. we did this by locating by eye three fiducial points
on the CMD of the cluster: the magnitude and color of the turn-off, the
magnitude of the tight clump of red HB stars, and the color of the RGB at a
level 3.0 mag brighter than the level of the turn-off. This latter point was
selected simply as a point lying on the lower RGB at a level intermediate
between that of the red end of the SGB and that of the tight clump of red HB.
We then calculated the difference in magnitude between the level of the turn-
off and the level of the tight clump of red HB ($\Delta m_{\rm F814W}$), and
the difference in color between the turn-off and the RGB fiducial point
($\Delta c_{m_{\rm F606W}-m_{\rm F814W}}$). As Mackey & Broby Nielsen (2007)
pointed out that, the difference in magnitude between the level of the turn-
off and the level of the tight clump of red HB is strongly sensitive to
cluster age (and weakly to cluster metallicity), while the difference in color
between the turn-off and the RGB fiducial point is sensitive to both cluster
age and metallicity. We determined $\Delta m_{\rm F814W}=3.77\pm 0.1$ and
$\Delta c_{m_{\rm F606W}-m_{\rm F814W}}=0.28\pm 0.01$.
Second, we calculated the same intervals for all isochrones on the grid, and
selected only those with values lying within certain tolerances of the cluster
measurements. In this paper, we adopted $\pm 0.2$ mag for $\Delta m_{\rm
F814W}$ and $\pm 0.02$ for $\Delta c_{m_{\rm F606W}-m_{\rm F814W}}$. We fit
the selected isochrones to the CMD by eye. At the same time, we calculated the
offsets in magnitude and color required to align the turn-off of the isochrone
with that of the CMD, and the offsets in magnitude required to align the tight
clump of red HB of the isochrone with that of the CMD, and the offsets in
color required to align the RGB fiducial point of the isochrone with that of
the CMD. We then averaged the offsets in magnitude and in color and applied to
overplot the isochrone on the CMD, and identified the best fitting isochrone
by eye. The resulting offsets $\delta m_{\rm F814W}$ and $\delta c_{m_{\rm
F606W}-m_{\rm F814W}}$ provide estimates for the distance modulus to B379
($(m-M)_{0}$) and the reddening value ($E(B-V)$): $\delta m_{\rm
F814W}=(m-M)_{0}+A_{\rm F814W}$, and $\delta c_{m_{\rm F606W}-m_{\rm
F814W}}=A_{\rm F606W}-A_{\rm F814W}$. The reddening law from Cardelli et al.
(1989) is employed in this paper. The effective wavelengths of the ACS F606W
and F814W filters are $\lambda_{\rm eff}=5918$ and 8060 Å (Sirianni et al.,
2005), so that from Cardelli et al. (1989), $A_{\rm{F606W}}\simeq 2.8\times
E(B-V)$ and $A_{\rm{F814W}}\simeq 1.8\times E(B-V)$ (see Barmby et al., 2007,
for details). The reddening value and distance modulus for B379 obtained in
this paper are: $E(B-V)=0.08$ and $(m-M)_{0}=24.44\pm 0.10$, where the
uncertainty is the standard error of the mean.
The best-fitting Padova isochrone can be seen in Figure 1: with the metal
abundance $0.009$ in $Z$ (or $-0.325$ in $\rm[M/H]$) and 11.0 Gyr in age. The
age of B379 obtained in this paper is $11.0\pm 1.5$ Gyr, where the uncertainty
is the standard error of the mean.
Figure 1.— Best-fitting Padova isochrone overplotted on the cluster CMD. The
isochrone has the metal abundance $\rm[M/H]=-0.325$ and age 11.0 Gyr. The
isocrone has been shifted by $E(B-V)=0.08$ and $(m-M)_{0}=24.44$.
The primary purpose of this paper is to obtain the age of B379 by comparing
its CMD with isochrones of the Padova stellar evolutionary models, and to
check whether the age of B379 obtained in this paper, is in agreement with the
determination of Brown et al. (2004a). From high-resolution stellar
spectroscopy it is presented that GCs in both the halo and the bulge of our
Galaxy are $\rm{\alpha/Fe}$ enhanced with the typical values
$\rm{[\alpha/Fe]\approx 0.3\pm 0.1}$ dex (see Thomas et al., 2003, and
references therein). For M31 GCs, the estimates of $\rm{[\alpha/Fe]}$ ratios
by Beasley et al. (2005) and Puzia et al. (2005) showed it may on average be
$\sim 0.1-0.2$ dex lower than in the Milky Way (see also Colucci et al.,
2009). The Padova stellar evolutionary models do not provide isochrones with
$\rm{[\alpha/Fe]>0.0}$, however, the luminosities of turn-off, SGB, and of the
tip of the RGB are nearly unchanged by varying $\alpha$ enhancement except in
the intermediate-age regime, where $\alpha$-enhanced isochrones are slightly
fainter than scaled solar ones (see Gallart et al., 2005, and references
therein). It is generally known that the turn-off, SGB and lower RGB are the
most age-sensitive features of the CMD, so, the age of B379 obtained based on
the isochrones with $\rm{[\alpha/Fe]=0.0}$ will not change when using the
isochrones with $\rm{[\alpha/Fe]>0.0}$.
### 4.3. Comparison with the previously published results
The age of B379 ($11.0\pm 1.5$ Gyr) obtained in this paper is consistent with
the determination ($10_{-1}^{+2.5}$ Gyr) of Brown et al. (2004a). Brown et al.
(2004a) determined the age of B379 by comparing the observed CMD with
isochrones of VandenBerg et al. (2006). The result of this paper confirmed the
conclusion of Brown et al. (2004a) that B379 is 2–3 Gyr younger than the
oldest Galactic GCs. The metallicity of B379 obtained in this paper is
$\rm[M/H]=-0.325$. Taking into account an enhancement of the $\alpha$-capture
elements by $[\alpha/\rm Fe]=0.3$ (Brown et al., 2004a), and using the
relation between [M/H], [Fe/H], and $[\alpha/\rm Fe]$ from Salaris et al.
(1993), we derived $\rm[Fe/H]=-0.54$, which is in good agreement with the
determination of $\rm[Fe/H]=-0.53$ of Holland et al. (1997) based on the shape
of the RGB of the deep CMD observed by the HST/WFPC2.
B379 is located in the M31 halo, so the extinction is mainly from the
foreground Galactic reddening in the direction of M31, which was discussed by
many authors (e.g., van den Bergh, 1969; McClure & Racine, 1969; Frogel et
al., 1980; Fusi Pecci et al., 2005), and nearly similar values were determined
such as $E(B-V)=0.08$ by van den Bergh (1969), 0.11 by McClure & Racine (1969)
and Hodge (1992), 0.08 by Frogel et al. (1980). In addition, Barmby et al.
(2000) determined the reddening for each individual cluster using correlations
between optical and infrared colors and metallicity, and by defining various
“reddening-free” parameters using their large database of multi-color
photometry. Finally, Barmby et al. (2000) determined reddenings for 314
clusters, 221 of which are reliable (see Barmby et al., 2000, for details).
For B379, Barmby et al. (2000, also P. Barmby, priv. comm.) obtained its
reddening value to be $E(B-V)=0.10\pm 0.05$. It is evident that the reddening
value of $E(B-V)=0.08$ obtained in this paper is consistent with these
determinations.
Given the importance of M31 as an anchor for the extragalactic distance scale,
many studies have presented distance determinations to M31 using different
methods. Pritchet & van den Bergh (1987), Holland (1998) and Vilardell et al.
(2006) have given a detailed review. Although the stellar populations located
in different positions in M31 have different distance moduli, the dispersion
can be neglected since the distance of M31 is large enough. For example, Rich
et al. (2005) pointed out that, the clusters in M31 dispersed over a 20 kpc
radius would have up to 0.06 mag random distance uncertainty. So, the distance
modulus to B379 obtained in this paper should be consistent with the distance
of M31 previously determined within 0.06 mag random distance uncertainty. Now
we compared our determination with the most recent and/or important
measurements. Freedman & Madore (1990) derived the mean distance modulus to
M31 to be $(m-M)_{0}=24.44\pm 0.13$ based on the Cepheids in Baade’s fields I,
III, and IV (Baade & Swope, 1963, 1965) observed using the Canada-France-
Hawaii Telescope (CFHT). Holland (1998) determined the distance moduli to 14
M31 GCs by fitting theoretical isochrones to the observed RGBs including B379.
The distance modulus to B379 obtained by Holland (1998) is $(m-M)_{0}=24.45\pm
0.07$. Stanek & Garnavich (1998) estimated the distance modulus to M31 as
$(m-M)_{0}=24.471\pm 0.035$ by comparing the red clump stars with parallaxes
known to better than 10% in the Hipparcos catalog with the red clump stars in
three fields in M31 observed with the HST. A determination of Freedman et al.
(2001) based on Cepheid P–L relation suggests the distance modulus of
$(m-M)_{0}=24.38\pm 0.05$ to M31 when they performed the results of the HST
Distance Scale Key Project to measure the Hubble constant. Durrell et al.
(2001) determined the distance modulus of $(m-M)_{0}=24.47\pm 0.12$ to M31
from the luminosity of the RGB tip of over 2000 RGB halo stars in a halo field
located about 20 kpc from the M31 nucleus along the southeast minor axis.
Joshi et al. (2003) have obtained $R-$ and $I-$band observations of a
$13^{\prime}\times 13^{\prime}$ region in the disk of M31 and derived the
Cepheid period–luminosity distance modulus to be $(m-M)_{0}=24.49\pm 0.11$.
Brown et al. (2004b) determined the distance modulus of $(m-M)_{0}=24.5\pm
0.1$ to M31 based on brightness of 55 RR Lyrae stars detected on the HST/ACS
images of $\sim$ 84 hr (250 exposures over 41 days). McConnachie et al. (2005)
derived the distance modulus to M31 to be $(m-M)_{0}=24.47\pm 0.07$ based on
the method of the tip of the RGB observed using the Isaac Newton Telescope
Wide Field Camera (INT WFC). Ribas et al. (2005) derived the distance modulus
of M31 as $(m-M)_{0}=24.44\pm 0.12$ from an eclipsing binary. Very recently,
Sarajedini et al. (2009) presented the HST observations taken with the ACS WFC
of two fields near M32 located $4-6$ kpc from the center of M31, and
identified 752 RR variables with excellent photometric and temporal
completeness. Based on this large sample of M31 RR Lyrae variables, and using
a relation between RR Lyrae luminosity and metallicity along with a reddening
value of $E(B-V)=0.08\pm 0.03$, they derived the distance modulus of
$(m-M)_{0}=24.46\pm 0.11$ to M31. In order to see clearly, we list these
determinations of M31 distance moduli in Table 1. It is evident that our
determination is in good agreement with the previous determinations.
## 5\. Summary
In this paper, we re-determined the age of the M31 GC B379 by fitting its deep
photometry extending below the main-sequence turn-off to the isochrones of the
Padova group (Girardi et al., 2002, 2008; Marigo et al., 2008). The age
obtained in this paper is consistent with the determination of Brown et al.
(2004a), and confirms the conclusion of Brown et al. (2004a) that, B379 is 2–3
Gyr younger than the oldest Galactic GC. This paper also confirms the
consistence of the age scale of B379 between the Padova group isochrones used
in this paper and the 2006 VandenBerg isochrones used by Brown et al. (2004a).
So, the results’ comparison between Brown et al. (2004a) and Ma et al. (2007)
is meaningful. In addition, the metal abundance, reddening value, and distance
modulus obtained in this paper are consistent with the previous
determinations.
We are indebted to the referee for thoughtful comments and insightful
suggestions that improved this paper significantly. We are thankful to Dr.
Brown for providing the HST/ACS WFC data for B379. This work was supported by
the Chinese National Natural Science Foundation grands No. 10873016, 10633020,
10603006, and 10803007, and by National Basic Research Program of China (973
Program), No. 2007CB815403.
## References
* Ajhar et al. (1996) Ajhar, E. A., Grillmair, C. J., Lauer, T. R., Baum, W. A., Faber, S. M., Holtzman, J. A., Lynds, C. R., & O’Neil, E. J. 1996, AJ, 111, 1110
* Baade & Swope (1963) Baade, W., & Swope, H. H. 1963, AJ, 68, 435
* Baade & Swope (1965) Baade, W., & Swope, H. H. 1965, AJ, 70, 212
* Barmby et al. (2000) Barmby P., Huchra J., Brodie J., Forbes D., Schroder L., & Grillmair C. 2000, AJ, 119, 727
* Barmby et al. (2007) Barmby, P., McLaughlin, D. E., Harris, W. E., Harris, G. L. H., & Forbes, D. A. 2007, AJ, 133, 2764
* Bastian et al. (2005) Bastian, N., Gieles, M., Lamers, H.J.G.L.M., Scheepmaker, R., & de Grijs, R. 2005, A&A, 431, 905
* Battistini et al. (1987) Battistini, P., Bonoli, F., Braccesi, A., Federici, L., Fusi Pecci, F., Marano, B., & Börngen, F. 1987, A&AS, 67, 447
* Beasley et al. (2005) Beasley, M., et al. 2005, AJ, 129, 1412
* Bertelli et al. (1994) Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, A&AS, 106, 275
* Bik et al. (2003) Bik, A., Lamers, H. J. G. L. M., Bastian, N., Panagia, N., & Romaniello, M. 2003, A&A, 397, 473
* Brown et al. (2003) Brown, T. M., et al. 2003, ApJ, 592, L17
* Brown et al. (2004a) Brown, T. M., et al. 2004a, ApJ, 613, L125
* Brown et al. (2004b) Brown, T. M., Ferguson, H. C., Smith, E., Kimble, R. A., Sweigart, A. V., Renzini, A., & Rich, R. M. 2004b, AJ, 127, 2738
* Bruzual & Charlot (1996) Bruzual, A. G., & Charlot, S. 1996, unpublished
* Bruzual & Charlot (2003) Bruzual, A. G., & Charlot, S. 2003, MNRAS, 344, 1000
* Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* Colucci et al. (2009) Colucci, J. E., Bernstein, R. A., Cameron, S., McWilliam, A., & Cohen, J. G. 2009, ApJ, 704, 385
* Durrell et al. (2001) Durrell, P. R., Harris, W. E., & Pritchet, C. J. 2001, AJ, 121, 2557
* de Grijs et al. (2003a) de Grijs, R., Bastian, N., & Lamers, H. J. G. L. M. 2003a, MNRAS, 340, 197
* de Grijs et al. (2003b) de Grijs, R., Fritze-v. Alvensleben, U., Anders, P., Gallagher, J. S., Bastian, N., Taylor, V. A., & Windhorst, R. A. 2003b, MNRAS, 342, 259
* de Grijs et al. (2003c) de Grijs, R., Anders, P., Lynds, R., Bastian, N., Lamers, H. J. G. L. M., & O’Neill, E. J., Jr. 2003c, MNRAS, 343, 1285
* Fan et al. (2006) Fan, Z., Ma, J., de Grijs, R., Yang Y., & Zhou X. 2006, MNRAS, 371, 1648
* Freedman & Madore (1990) Freedman, W. L., & Madore, B. F. 1990, ApJ, 365, 186
* Freedman et al. (2001) Freedman, W. L., et al. 2001, ApJ, 553, 47
* Frogel et al. (1980) Frogel, J. A., Persson, S. E., & Cohen, J. G. 1980, ApJ, 240, 785
* Fusi Pecci et al. (1996) Fusi Pecci, F., et al. 1996, AJ, 112, 1461
* Fusi Pecci et al. (2005) Fusi Pecci, F., Bellazini, M., De Simone, E., & Federici, L. 2005, AJ, 130, 544
* Gallart et al. (2005) Gallart, C., Zoccali, M., & Aparicio, A. 2005, ARA&A, 43, 387
* Galleti et al. (2004) Galleti, S., Federici, L., Bellazzini, M., Fusi Pecci, F., & Macrina, S. 2004, A&A, 426, 917
* Galleti et al. (2006) Galleti, S., Federici, L., Bellazzini, M., Buzzoni, A., & Fusi Pecci, F. 2006, A&A, 456, 985
* Galleti et al. (2007) Galleti, S., Bellazzini, M., Federici, L., Buzzoni, A., & Fusi Pecci, F. 2007, A&A, 471, 127
* Girardi et al. (2000) Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, A&AS, 141, 371
* Girardi et al. (2002) Girardi, L., et al. 2002, A&A, 391, 195
* Girardi et al. (2008) Girardi, L., et al. 2008, PASP, 120, 583
* Heasley et al. (1988) Heasley, J. N., Friel, E. D., Christian, C. A., & Janes, K. A. 1988, AJ, 96, 1312
* Holland et al. (1997) Holland, S., Fahlman, G. G., & Richer, H. B. 1997, AJ, 114, 1488
* Holland (1998) Holland, S. 1998, AJ, 115, 1916
* Hodge (1992) Hodge, P. W. 1992, The Andromeda Galaxy (Dordrecht: Kluwer)
* Huchra et al. (1991) Huchra, J. P., Brodie, J. P., & Kent, S. M. 1991, ApJ, 370, 495
* Jablonka et al. (2000) Jablonka, P., Courbin, F., Meylan, G., Sarajedini, A., Bridges, T. J., & Magain, P. 2000, A&A, 359, 131
* Jiang et al. (2003) Jiang, L. H., Ma, J., Zhou, X., Chen, J. S., Wu, H., & Jiang, Z. J. 2003, AJ, 125, 727
* Joshi et al. (2003) Joshi, Y. C., Pandey, A. K., Narasimha, D., Sagar, R., & Giraud-Héraud, Y. 2003, A&A, 402, 113
* Leitherer et al. (1999) Leitherer, C., et al. 1999, ApJS, 123, 3
* Lupton (1989) Lupton, R. H. 1989, AJ, 97, 1350
* Ma et al. (2001) Ma, J., Zhou, X., Kong, X., Wu, H., Chen, J., Jiang, Z., Zhu, J., & Xue S. 2001, AJ, 122, 1796
* Ma et al. (2002a) Ma, J., Zhou, X., Chen, J., Wu, H., Jiang, Z., Xue, S., & Zhu, J. 2002a, A&A, 385, 404
* Ma et al. (2002b) Ma, J., Zhou, X., Chen, J., Wu, H., Jiang, Z., Xue, S., & Zhu, J. 2002b, AJ, 123, 3141
* Ma et al. (2002c) Ma, J., Zhou, X., Chen, J., Wu, H., Kong, X., Jiang, Z., Zhu, J., & Xue, S. 2002c, AcA, 52, 453
* Ma et al. (2006a) Ma J., et al. 2006a, A&A, 449, 143
* Ma et al. (2006b) Ma, J., de Grijs, R., Yang, Y., Zhou, X., Chen, J., Jiang, Z., Wu, Z., & Wu, J. 2006b, MNRAS, 368, 1443
* Ma et al. (2007) Ma, J., et al. 2007, ApJ, 659, 359
* Mackey & Broby Nielsen (2007) Mackey, A. D., & Broby Nielsen, P. 2007, MNRAS, 379, 151
* Marigo et al. (2008) Marigo, P., Girardi, L., Bressan, A., Groenewegen, M. A. T., Silva,L., & Granato, G. L. 2008, A&A, 482, 883
* McClure & Racine (1969) McClure, R. D., & Racine, R. 1969, AJ, 74, 1000
* McConnachie et al. (2005) McConnachie, A. W., Irwin, M. J., Ferguson, A. M. N., Ibata, R. A., Lewis, G. F., & Tanvir, N. 2005, MNRAS, 356, 979
* Pritchet & van den Bergh (1987) Pritchet, C. J., & van den Bergh, S. 1987, ApJ, 316, 517
* Puzia et al. (2005) Puzia, T. H., Perrett, K. M., & Bridges, T. J. 2005, A&A, 434, 909
* Puzia et al. (2002b) Puzia, T. H., Zepf, S. E., Kissler-Patig, M., Hilker, M., Minniti, D., & Goudfrooij, P. 2002, A&A, 391, 453
* Reed, Harris & Harris (1994) Reed, L. G., Harris, G. L. H., & Harris, W. E. 1994, AJ, 107, 555
* Rey et al. (2005) Rey, S.-C., et al. 2005, ApJ, 619, L119
* Rey et al. (2007) Rey, S.-C., et al. 2007, ApJS, 173, 643
* Ribas et al. (2005) Ribas, I., Jordi, C., Vilardell, F., Fitzpatrick, E. L., Hilditch, R. W., & Guinan, E. F. 2005, ApJ, 635, L37
* Rich et al. (1996) Rich, R. M., Mighell, K. J., Freedman, W. L., & Neill, J. D. 1996, AJ, 111, 768
* Rich et al. (2001) Rich, R. M., Shara, M. M., & Zurek, D. 2001, AJ, 122, 842
* Rich et al. (2005) Rich, R. M., Corsi, C. E., Cacciari, C., Federici, L., Fusi Pecci, F., Djorgovski, S. G., & Freedman, W. L. 2005, AJ, 129, 2670
* Roxburgh (1978) Roxburgh, I. W. 1978, A&A, 65, 281
* Roxburgh (1989) Roxburgh, I. W. 1989, A&A, 211, 361
* Salaris et al. (1993) Salaris, M., Chieffi, A., & Straniero, O. 1993, ApJ, 414, 580
* Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161
* Sandage (1953) Sandage, A. R. 1953, AJ, 58, 61
* Sarajedini et al. (2009) Sarajedini, A., et al. 2009, AJ, 138, 184
* Sargent et al. (1977) Sargent, W. L. W., Kowal, C. T., Hartwick, F. D. A., & van den Bergh, S. 1977, AJ, 82, 947
* Sharov (1973) Sharov, A. S. 1973, SvAL, 17, 174
* Sirianni et al. (2005) Sirianni, M., et al. 2005, PASP, 117, 1049
* Stanek & Garnavich (1998) Stanek, K. Z., & Garnavich, P. M. 1998, ApJ, 503, L131
* Thomas et al. (2003) Thomas, D., Maraston, C., & Bender, R. 2003, MNRAS, 339, 897
* van den Bergh (1969) van den Bergh, S. 1969, ApJS, 19, 145
* VandenBerg et al. (2000) VandenBerg, D. A., Swenson, F. J., Rogers, F. J., Iglesias, C. A., & Alexander, D. R. 2000, ApJ, 532, 430
* VandenBerg et al. (2006) VandenBerg, D. A., Bergbusch, P. A., & Dowler, P. D. 2006, ApJS, 162, 375
* Vilardell et al. (2006) Vilardell, F., Ribas, I., & Jordi, C. 2006, A&A, 459, 321
* Williams & Hodge (2001a) Williams, B. F., & Hodge, P. W. 2001a, ApJ, 548, 190
* Williams & Hodge (2001b) Williams, B. F., & Hodge, P. W. 2001b, ApJ, 559, 851
Table 1Distance determinations to M31 as presented in the references for
comparison.
Method | $(m-M)_{0}$ | Reference
---|---|---
| [mag] |
Cepheids | $24.44\pm 0.13$ | [1]
Red Giant Branch | $24.45\pm 0.07$ | [2]
Red Clump | $24.471\pm 0.035$ | [3]
Cepheids | $24.38\pm 0.05$ | [4]
Red Giant Branch | $24.47\pm 0.12$ | [5]
Cepheids | $24.49\pm 0.11$ | [6]
RR Lyrae | $24.5\pm 0.1$ | [7]
Tip of the RGB | $24.47\pm 0.07$ | [8]
Eclipsing binary | $24.44\pm 0.12$ | [9]
RR Lyrae | $24.46\pm 0.11$ | [10]
CMD | $24.44\pm 0.10$ | [11]
[1]: Freedman & Madore (1990); [2]: Holland (1998); [3]: Stanek & Garnavich
(1998); [4]: Freedman et al. (2001); [5]: Durrell et al. (2001); [6]: Joshi et
al. (2003); [7]: Brown et al. (2004b); [8]: McConnachie et al. (2005); [9]:
Ribas et al. (2005); [10]: Ribas et al. (2005); [10]: Sarajedini et al.
(2009); [11]: this paper.
|
arxiv-papers
| 2010-08-18T01:28:26 |
2024-09-04T02:49:12.239935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Ma (1,2), Zhenyu Wu (1), Song Wang (1,3), Zhou Fan (1), Xu Zhou\n (1), Jianghua Wu (1), Zhaoji Jiang (1), Jiansheng Chen (1) ((1) National\n Astronomical Observatories, Chinese Academy of Sciences, (2) Key Laboratory\n of Optical Astronomy, National Astronomical Observatories, Chinese Academy of\n Sciences, Beijing, 100012, China)",
"submitter": "Jun Ma",
"url": "https://arxiv.org/abs/1008.2993"
}
|
1008.2999
|
2009 Vol. 9 No. XX, 000–000
11institutetext: Physics and Information Engineering Institute, Shanxi Normal
University, Linfen 041004, China; yuanjz@sxnu.edu.cn
# First photometric study of the eclipsing binary PS Persei
Jinzhao Yuan
###### Abstract
The CCD photometric observations of the eclipsing binary PS Persei (PS Per)
were obtained on two consecutive days in 2009. The 2003 version Wilson-
Devinney code was used to analyze the first complete light curves in $V$ and
$R$ bands. It is found that PS Per is a short-period Algol-type binary with
the less massive component accurately filling its inner critical Roche lobe.
The mass ratio of $q=0.518$ and the orbital inclination of $i=89.^{\circ}86$
are obtained. On the other hand, based on all available times of primary light
minimum including two new ones, the orbital period has been improved.
###### keywords:
stars: binaries: close — stars: binaries: eclipsing — stars: individual: PS
Per
## 1 Introduction
PS Per ($\alpha_{2000.0}=02^{h}39^{m}33.^{s}3$ and
$\delta_{2000.0}=+45^{\circ}38^{{}^{\prime}}05.^{{}^{\prime\prime}}5$) was
designated by Kukarkin et al. (1968). But photographic and visual times of
light minimum have been obtained since 1926. Later, Photoelectric and CCD
times of light minimum were published by Šafář & Zejda (2000a), Šafář & Zejda
(2000b), Zejda (2002), Agerer & Hübscher (2003), Zejda (2004), Diethelm
(2005), Hübscher et al. (2006), Zejda et al. (2006), Brát et al. (2009), and
Diethelm (2010). But, no complete light curve of the binary system have been
made so far for photometric analysis.
In this paper, the first complete light curves in $V$ and $R$ bands were
presented. And the absolute physical parameters as well as orbital period were
determined.
## 2 Observations
New CCD photometric observations of PS Per in $V$ and $R$ bands were carried
out on 2009 November 13 and 14 using the 85-cm telescope at the Xinglong
Station of National Astronomical Observatory of China (NAOC), equipped with a
primary-focus multicolor CCD photometer. The telescope provides a field of
view of about $16.^{{}^{\prime}}5\times 16.^{{}^{\prime}}5$ at a scale of
$0.^{{}^{\prime\prime}}96$ per pixel and a limit magnitude of about 17 mag in
$V$ band.
The typical exposure times in $V$ and $R$ bands were 90s and 60s respectively.
The coordinates of the variable, comparison, and check stars are listed in
Table 1. The data reduction was performed by using the aperture photometry
package IRAF111IRAF is developed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under contract to the National Science
Foundation. (bias subtraction, flat-field division). Extinction corrections
were ignored as the comparison star is very close to the variable. In total,
445 CCD images in the $V$ band and 446 images in the $R$ band were obtained.
Several new times of light minimum (see Table 2) are derived from the new
observation by using a parabolic fitting method.
The first complete light curves in $V$ and $R$ bands are obtained, and
displayed in the top panel of Figure 1. The new orbital period revised in the
next section was used to calculate the phase.
Table 1: Coordinates of PS Per and its Comparison and Check Stars.
Stars | $\alpha_{2000}$ | $\delta_{2000}$
---|---|---
PS Per | $02^{h}39^{m}33.3^{s}$ | $45^{\circ}38^{\prime}05.5"$
Comparison | $02^{h}39^{m}24.1^{s}$ | $45^{\circ}42^{\prime}22.1"$
Check | $02^{h}39^{m}22.2^{s}$ | $45^{\circ}43^{\prime}34.0"$
Table 2: New CCD Times of Light Minimum for PS Per.
No. | J.D. (Hel.) (days) | Error (days) | Min. | Filter
---|---|---|---|---
1 | 2455149.2742 | $\pm 0.0004$ | I | $V$
| 2455149.2752 | $\pm 0.0005$ | I | $R$
2 | 2455149.9768 | $\pm 0.0003$ | I | $V$
| 2455149.9768 | $\pm 0.0004$ | I | $R$
3 | 2455150.3267 | $\pm 0.0004$ | II | $V$
| 2455150.3265 | $\pm 0.0004$ | II | $R$
Figure 1: Top panel: the light curves of PS Per in the $V$ and $R$ bands
obtained on 2009 November 13 and 14. The points in $R$ band has been shifted
down by 1.0 mag. Bottom panel: the differential light curves of the comparison
star relative to the check star. The points in $R$ band has been shifted up by
0.2 mag.
## 3 Orbital period study
All available times of primary light minimum seen in literature were collected
and listed in Table 3, which also includes the data in the database of
eclipsing binaries of Kreiner (2004). For my two-band light minima, a mean
time of light minimum is given. The $O-C$ values of all minimum times were
computed with the ephemeris given by Kreiner et al. (2001):
$\mathrm{Min}~{}I=2424527.2165+0^{d}.70217968\times{E},$ (1)
and listed in the fifth column of Table 3. The corresponding $O-C$ diagram,
Figure 2, shows that the data are distributed around a straight line. So, a
linear ephemeris was used to fit the $O-C$ values. The photographic and visual
data show large deviation from the straight line for their low quality. In the
fitting process, a weight of 10 is used for photoelectric and CCD data, and 1
for photographic and visual data. The CCD times, 2451841.3121 and
2452213.4662, have the weight of 1 for their large errors. A least-squares fit
to the data gave the following ephemeris:
$\mathrm{Min}~{}I=2424527.2163+0^{d}.70217977\times{E},$ (2)
The new ephemeris is plotted in Figure 2 with a solid line. The residuals with
respect to Equation (2) are listed in the sixth column of Table 3.
Table 3: Times of primary light Minimum for PS Per.
JD (Hel.) | Method | Error | E | $O-C$ | Residuals | Ref.
---|---|---|---|---|---|---
2424527.250 | p | | 0 | 0.03350 | 0.03374 | MVS 2
2428834.407 | p | | 6134 | 0.02034 | 0.02000 | MVS 2
2430972.531 | p | | 9179 | 0.00722 | 0.00660 | MVS 2
2430991.464 | p | | 9206 | -0.01863 | -0.01925 | MVS 2
2435718.545 | p | | 15938 | -0.01124 | -0.01249 | MVS 2
2435725.534 | p | | 15948 | -0.04404 | -0.04529 | MVS 2
2436114.558 | p | | 16502 | -0.02758 | -0.02888 | MVS 2
2436596.315 | p | | 17188 | 0.03416 | 0.03278 | MVS 2
2436603.330 | p | | 17198 | 0.02736 | 0.02598 | MVS 2
2436850.460 | p | | 17550 | -0.00988 | -0.01128 | MVS 2
2436852.580 | p | | 17553 | 0.00358 | 0.00217 | MVS 2
2436876.438 | p | | 17587 | -0.01253 | -0.01393 | MVS 2
2436895.406 | p | | 17614 | -0.00338 | -0.00479 | MVS 2
2437588.460 | p | | 18601 | -0.00073 | -0.00223 | MVS 2
2437939.528 | p | | 19101 | -0.02257 | -0.02412 | MVS 2
2437944.477 | p | | 19108 | 0.01117 | 0.00961 | MVS 2
2437946.551 | p | | 19111 | -0.02136 | -0.02291 | MVS 2
2437970.442 | p | | 19145 | -0.00447 | -0.00602 | MVS 2
2438321.547 | p | | 19645 | 0.01069 | 0.00908 | MVS 2
2438385.453 | p | | 19736 | 0.01834 | 0.01672 | MVS 2
2447028.565 | v | | 32045 | 0.00065 | -0.00211 | BBSAG 85
2447118.446 | v | | 32173 | 0.00266 | -0.00011 | BBSAG 86
2447170.406 | v | | 32247 | 0.00136 | -0.00142 | BBSAG 87
2447384.566 | v | | 32552 | -0.00344 | -0.00625 | BBSAG 89
2447491.294 | v | | 32704 | -0.00675 | -0.00957 | BBSAG 90
2447566.432 | v | | 32811 | -0.00198 | -0.00481 | BBSAG 91
2447894.348 | v | | 33278 | -0.00389 | -0.00677 | BBSAG 94
2448136.597 | v | | 33623 | -0.00688 | -0.00979 | BBSAG 96
2448283.362 | v | .004 | 33832 | 0.00257 | -0.00036 | BBSAG 97
2448509.458 | v | .005 | 34154 | -0.00329 | -0.00625 | BBSAG 99
2448867.573 | v | .003 | 34664 | 0.00007 | -0.00294 | BBSAG 102
2449202.511 | v | .004 | 35141 | -0.00163 | -0.00468 | BBSAG 104
2449546.575 | v | .006 | 35631 | -0.00568 | -0.00878 | BBSAG 107
2449653.312 | v | .003 | 35783 | 0.00001 | -0.00310 | BBSAG 108
2449945.401 | v | .005 | 36199 | -0.01774 | -0.02089 | BBSAG 110
2450713.6127 | cc | .0014 | 37293 | 0.00939 | 0.00612 | Šafář & Zejda (2000a)
2450721.3309 | cc | .0008 | 37304 | 0.00362 | 0.00035 | BBSAG 116
2450839.2981 | cc | .0021 | 37472 | 0.00463 | 0.00135 | Šafář & Zejda (2000b)
2450841.4035 | cc | .0021 | 37475 | 0.00349 | 0.00021 | Šafář & Zejda (2000b)
2451077.3366 | cc | .0006 | 37811 | 0.00422 | 0.00091 | BBSAG 119
2451088.572 | v | .003 | 37827 | 0.00474 | 0.00142 | BBSAG 119
2451515.501 | v | .008 | 38435 | 0.00850 | 0.00513 | BBSAG 121
2451810.419 | v | .005 | 38855 | 0.01103 | 0.00762 | BBSAG 123
2451841.3121 | cc | .0058 | 38899 | 0.00823 | 0.00481 | Zejda (2002)
2451876.4201 | cc | .0017 | 38949 | 0.00724 | 0.00382 | Zejda (2002)
2451878.533 | v | .008 | 38952 | 0.01360 | 0.01018 | BBSAG 124
2451899.5900 | cc | .0003 | 38982 | 0.00521 | 0.00179 | Agerer & Hübscher (2002)
2452190.295 | v | .003 | 39396 | 0.00783 | 0.00437 | BBSAG 126
2452204.3365 | cc | .0010 | 39416 | 0.00573 | 0.00227 | BBSAG 126
2452213.4637 | cc | .0005 | 39429 | 0.00460 | 0.00113 | BBSAG 127
2452213.4662 | cc | .0070 | 39429 | 0.00710 | 0.00363 | Zejda (2004)
2452260.516 | v | .004 | 39496 | 0.01086 | 0.00739 | BBSAG 127
2452524.525 | v | .007 | 39872 | 0.00030 | -0.00320 | Diethelm (2003)
2452531.5507 | pe | .0002 | 39882 | 0.00420 | 0.00069 | Agerer & Hübscher (2003)
2452885.446 | v | .002 | 40386 | 0.00094 | -0.00261 | Diethelm (2004)
2453302.5428 | cc | .0010 | 40980 | 0.00301 | -0.00059 | Diethelm (2005)
2453656.4422 | cc | .0003 | 41484 | 0.00385 | 0.00019 | Zejda et al. (2006)
2453705.5937 | pe | .0008 | 41554 | 0.00278 | -0.00088 | Hübscher et al. (2006)
2453988.5713 | cc | .0001 | 41957 | 0.00197 | -0.00172 | Brát et al. (2009)
2454019.4675 | cc | .0001 | 42001 | 0.00226 | -0.00144 | Brát et al. (2009)
2455114.8667 | cc | .0006 | 43561 | 0.00116 | -0.00268 | Diethelm (2010)
2455149.2747 | cc | .0003 | 43610 | 0.00236 | -0.00149 | this paper
2455149.9768 | cc | .0003 | 43611 | 0.00228 | -0.00157 | this paper
Figure 2: $O-C$ diagrams. The solid line is calculated with the new ephemeris
in Equation (2).
## 4 Photometric solutions with the W-D method
The light curves were analyzed using the 2003 version of the Wilson-Devinney
code (Wilson & Devinney 1971; Wilson 1979, 1990). Since the spectral type of
PS Per is F5, an effective temperature of $T_{1}=6750$K is assumed for the
primary component. Assuming the photospheric surface of the binary star is
convective, gravity-darkening coefficients ($g_{1}=g_{2}=0.320$) and
bolometric albedo ($A_{1}=A_{2}=0.5$) were used. According to the tables of
van Hamme (1993), the limb-darkening coefficients 0.506 for $V$ band
($x_{1V}=0.506$) and 0.414 for $R$ band ($x_{1R}=0.414$) were adopted.
Since no mass ratio has been published in literature, a $q$-search method was
used to determine the mass ratio. Solutions were carried out for a series of
values of the mass ratio $q=M_{2}/M_{1}$ ($q=0.3$, 0.4, 0.5, 0.6, 0.7, 0.8,
0.9, 1.0). Considering the light curves of EB type, mode 2 (detached mode) is
assumed. The behavior of the sum of the residuals squared, $\Sigma$, as a
function of mass ratio $q$ is plotted in Figure 3, showing that $\Sigma$
reaches the minimum value near $q=0.5$. Therefore, the mass ratio was taken as
an adjustable parameter and given the initial value of $q=0.5$. After some
differential corrections, The solution converged to mode 5 (semi-detached) and
gave the final mass ratio of $q=0.518$. The derived physical parameters are
listed in Table 4. The theoretical light curves computed with the parameters
are plotted in Figure 4 as a solid line.
Figure 3: Relation between $\Sigma$ (the sum of the residuals squared) and $q$ for PS Per. Table 4: Photometric Solutions for PS Per. Parameters | Photometric elements | Errors
---|---|---
$g_{1}=g_{2}$ | 0.32 | assumed
$A_{1}=A_{2}$ | 0.5 | assumed
$x_{1bol}$ | 0.480 | assumed
$x_{2bol}$ | 0.536 | assumed
$x_{1V}$ | 0.506 | assumed
$x_{1R}$ | 0.414 | assumed
$x_{2V}$ | 0.726 | assumed
$x_{2R}$ | 0.600 | assumed
$T_{1}$ | 6750K | assumed
q ($M_{2}/M_{1}$ ) | 0.518 | 0.003
$\Omega_{in}=\Omega_{2}$ | 2.8944 | –
$\Omega_{out}$ | 2.5907 | –
$T_{2}$ | 4822K | 7K
$i$ | $89.^{\circ}86$ | $0.^{\circ}35$
$\Omega_{2}$ | 3.590 | 0.008
$r_{1}(pole)$ | 0.3229 | 0.0008
$r_{1}(side)$ | 0.3317 | 0.0009
$r_{1}(back)$ | 0.3402 | 0.0010
$r_{2}(pole)$ | 0.3025 | 0.0004
$r_{2}(side)$ | 0.3159 | 0.0005
$r_{2}(back)$ | 0.3483 | 0.0005
$\Sigma{(O-C)^{2}}$ | 0.0022 |
Figure 4: Same as the top panel of Figure 1. But the solid curves represent
the theoretical light curves computed with the parameters in Table 4.
## 5 Discussion and Conclusions
In this paper, my photometric solution reveals that PS Per is a semi-detached
system. The Roche-geometry configuration that the less massive and cool
secondary component fills its inner Roche lobe permits a dynamical mass
transfer from the secondary to the more massive primary star, suggesting a
continuous period increase just as in AI Cru (Zhao et al. 2010) and DD Mon
(Qian et al. 2009). The orbital period of PS Per, however, does not show
continuous increase in this paper. This may be due to the low quality of
photographic and visual times, and the short span of photoelectric and CCD
times. In order to confirm the mass transfer from the secondary to the primary
star, long-term orbital timing data are required.
According to the derived physical parameters listed in Table 4 and the
Harmanec’s (1988) relation for masses and radii as functions of spectral type,
the following orbital parameters can be derived: $M_{1}=1.31~{}M_{\odot}$,
$R_{1}=1.39~{}R_{\odot}$, $L_{1}=3.62~{}L_{\odot}$, $M_{2}=0.68~{}M_{\odot}$,
$R_{2}=1.35~{}R_{\odot}$, $L_{2}=0.89~{}L_{\odot}$, and $a=4.19~{}R_{\odot}$.
In order to further verify the parameters, Spectroscopic observations of the
radial velocity curves of both components are needed.
As showed in Figure 1, the second maxima of the light curves are a little
higher than the primary maxima. The weak O’Connell effect maybe arises from a
hot spot on the primary component as a result of the impact of the gaseous
stream from the cooler, less massive secondary component. Such hot spot is
often seen in other semi-detached binary systems, such as CL Aur (Lee et al.
2010) and KQ Gem (Zhang 2010). Considering the late spectral type and fast
rotation of the secondary star, The asymmetry of the light curves can be also
attributed to a cool spot on the secondary star caused by magnetic activity.
It is a reasonable trial that the magnetic activity makes the light curves
show more variability than the impact of the gaseous stream does. So, in order
to tell the two mechanisms of magnetic activity and impact of the gaseous
stream, the investigation on long-term behaviour of the light curves is also
needed.
###### Acknowledgements.
I thank an anonymous referee for some useful suggestions. This work is
supported by Natural Science Foundation of Shanxi Normal University (No.
ZR09002).
## References
* Agerer & Hubscher (2002) Agerer, F., & Hübscher, J. 2002, , No. 5296
* Agerer & Hubscher (2003) Agerer, F., & Hübscher, J. 2003, , No. 5484
* Bratet al. (2009) Brát, L., Trnka, J., & Lehký, M. 2009, OEJV, No. 107
* Diethelm (2003) Diethelm, R. 2003, , No. 5438
* Diethelm (2004) Diethelm, R. 2004, , No. 5543
* Diethelm (2005) Diethelm, R. 2005, , No. 5653
* Diethelm (2005) Hübscher, J., Paschke, A., & Walter, F 2006, , No. 5731
* Diethelm (2010) Diethelm, R. 2010, , No. 5920
* Harmanec (1988) Harmanec, P. 1988, Bull. Astron. Inst. Czechoslovakia, 39, 329
* Kukarkinet al. (1968) Kukarkin, B. V., Efremov, Yu. N., Frolov, M. S., et al. 1968, , No. 311
* Kreineret al. (2001) Kreiner, J. M., Kim, C.-H., & Nha, I.-S. 2001, An Atlas of $O-C$ Diagrams of Eclipsing Binary Stars (Kraków: Wydawnictwo Naukowe Akad. Pedagogicznej)
* Kreiner (2001) Kreiner, J. M. 2004, Acta Astron., 54, 207
* Leeet al. (2010) Lee, J. W., Kim, C.-H., Kim, D. H., Kim, S.-L., Lee, C.-U., & Koch, R. H. 2010, AJ, 139, 2669
* Qianet al. (2009) Qian, S.-B., Zhu, L.-Y., Boonrucksar, S., Xiang, F.-Y., & He, J.-J. 2009, PASJ, 61, 333
* Safar & Zejda (2000) Šafář, J., & Zejda M. 2000a, , No. 4887
* Safar & Zejda (2000) Šafář, J., & Zejda M. 2000b, , No. 4888
* Hamme (1993) van Hamme, W. 1993, AJ, 106, 2096
* Wilson & Devinney (1971) Wilson, R. E., & Devinney E. J. 1971, ApJ, 166, 605
* Wilson (1979) Wilson, R. E. 1979, ApJ, 234, 1054
* Wilson (1990) Wilson, R. E. 1990, ApJ, 356, 613
* Zejda (2002) Zejda, M. 2002, , No. 5287
* Zejda (2004) Zejda, M. 2004, , No. 5583
* Zejdaet al. (2006) Zejda, M., Mikulášek, Z., & Wolf, M. 2006, , No. 5741
* Zhang (2010) Zhang, L.-Y. 2010, PASP, 122, 309
* Zhaoet al. (2010) Zhao, E.-G., Qian, S.-B., Fernández Lajús, E., von Essen, C., & Zhu, L.-Y. 2010, RAA(Research Astron. Astrophys.), 10, 438
|
arxiv-papers
| 2010-08-18T02:33:10 |
2024-09-04T02:49:12.245724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinzhao Yuan",
"submitter": "Yuan Jinzhao Mr",
"url": "https://arxiv.org/abs/1008.2999"
}
|
1008.3203
|
# A Simple Pendulum Determination of the Gravitational Constant
Harold V. Parks hvparks@sandia.gov JILA, University of Colorado and National
Institute of Standards and Technology, Boulder, CO 80309, USA Sandia National
Laboratories, Albuquerque, NM 87185, USA James E. Faller JILA, University of
Colorado and National Institute of Standards and Technology, Boulder, CO
80309, USA
###### Abstract
We determined the Newtonian Constant of Gravitation $G$ by interferometrically
measuring the change in spacing between two free-hanging pendulum masses
caused by the gravitational field from large tungsten source masses. We find a
value for $G$ of $(6.672\>34\pm 0.000\>14)\times
10^{-11}\>\mathrm{m}^{3}\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$. This value is in
good agreement with the 1986 Committee on Data for Science and Technology
(CODATA) value of $(6.672\>59\pm 0.000\>85)\times
10^{-11}\>\mathrm{m}^{3}\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$ [Rev. Mod. Phys.
59, 1121 (1987)] but differs from some more recent determinations as well as
the latest CODATA recommendation of $(6.674\>28\pm 0.000\>67)\times
10^{-11}\>\mathrm{m}^{3}\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$ [Rev. Mod. Phys.
80, 633 (2008)].
###### pacs:
04.80-y, 06.20.Jr
Measurements of the gravitational constant $G$ have a very long history, that
dates back to the birth of modern experimental science. This precision
measurement requires that the weak gravitational pull of a well-characterized
source mass be measured to a high accuracy. It is a supreme test of an
experimental physicist to cleanly pull this signal out of the inevitable sea
of perturbing influences.
Traditionally, $G$ is measured with a torsion balance. In 1798, Cavendish and
Michell reported numbers from a torsion balance that could be used to
calculate $G$ to within about 1% of its true value Cavendish (1798). It took
nearly 200 years to improve on this accuracy by two orders of magnitude; in
1982 Luther and Towler reported a value of $G$ with an uncertainty of slightly
less than 1 part in $10^{4}$ from a torsion balance experiment Luther and
Towler (1982). This measurement became the principal basis of the accepted
value of $G$ (Committee on Data for Science and Technology, CODATA 1986 Cohen
and Taylor (1987)) for over a decade.
However in 1995, Kuroda pointed out that anelasticity in a torsion fiber (a
frequency dependence of the restoring force due to material properties of the
fiber) had the potential to cause a significant error at the level of
uncertainty quoted by Luther and Towler Kuroda (1995). A number of new
determinations of $G$ followed. Many of these used a torsion balance in a mode
that minimized the effects of the fiber anelasicity Gundlach and Merkowitz
(2000); Armstrong and Fitzgerald (2003); Quinn _et al._ (2001); Luo _et al._
(2009); Bagley and Luther (1997); Karagioz and Izmailov (1996), while several
used other methods such as replacing the torsion balance with a simple
pendulum Kleinevoß _et al._ (1999); *wup02 or a beam balance Schlamminger
_et al._ (2006). The lowest reported uncertainties from this new slate of
measurements approach 1 part in $10^{5}$ Gundlach and Merkowitz (2000); Luo
_et al._ (2009); Schlamminger _et al._ (2006). The CODATA recommended $G$
value has now shifted by 2.5 parts in $10^{4}$ from the Luther and Towler
number, though the CODATA uncertainty remains at 1 part in $10^{4}$ because of
some conflicting results Mohr _et al._ (2008).
Our determination uses a simple pendulum method similar to that of Kleinevoß
et al. Kleinevoß _et al._ (1999); *wup02. By using a laser rather than a
microwave interferometer and by better controlling the mass geometries, we
have achieved a standard ($1\sigma$) uncertainty of 2.1 parts in $10^{5}$ for
our value of $G$, which is an order of magnitude lower than the Kleinevoß et
al. result. This uncertainty is within a factor of $\sqrt{2}$ of the lowest
uncertainty $G$ value reported to date Gundlach and Merkowitz (2000), but
differs from this number by over $10\sigma$! (We are $2.9\sigma$ below the
current CODATA value Mohr _et al._ (2008) because of its larger uncertainty.)
We base our value on data taken in 2004, and in the interim, we have been
unable to find a likely source for this discrepancy. So, having checked and
rechecked our work, we must finally report our value as we have found it. It
lies within the $1\sigma$ uncertainty band of the original Luther and Towler
number.
$\begin{array}[]{c}\includegraphics[width=231.26378pt]{JILA_G_schematic.eps}\\\
\includegraphics[width=231.26378pt]{JILA_G_signal.eps}\end{array}$
Figure 1: A schematic of the apparatus is shown on top. A Fabry-Perot
interferometer measures the spacing between the two pendulum bobs with respect
to a suspension-point-located reference cavity. The bobs are made of oxygen-
free copper and have a mass of 780 g. The pendulum length is 72 cm, and the
spacing between the bob centers is 34 cm. When the four 120 kg tungsten source
masses (which are floated on air bearings) are moved from one position to
another, the horizontal gravitational force on each pendulum bob changes by
480 nN, giving rise to a change in pendulum bob separation. Not pictured is
the vacuum chamber that encloses the pendulums but not the source masses.
Magnets (not shown) outside of the vacuum system and below the pendulum bobs
damp the swinging motion of the pendulums so that the static deflection due to
the gravitational pull of the source masses can be measured. The gravitational
signal is plotted on the bottom as the source masses are moved between the
inner and outer positions several times (with the source masses pausing at
each position for 80 s). The 125 MHz change in the beat frequency between the
laser locked to the pendulum cavity and the laser locked to the reference
cavity corresponds to a 90 nm change in the pendulum bobs’ separation.
A schematic of our apparatus is shown in Fig. 1. We find $G$ by balancing the
gravitational pull of tungsten source masses against the restoring force of a
simple pendulum:
$G\int\frac{\hat{\mathbf{z}}\cdot(\mathbf{x}-\mathbf{x}^{\prime})\rho_{s}(\mathbf{x})\rho_{t}(\mathbf{x}^{\prime})}{|\mathbf{x}-\mathbf{x}^{\prime}|^{3}}\,d^{3}x\,d^{3}x^{\prime}=-kz,$
(1)
where the source mass distribution $\rho_{s}$ corrected for displaced air and
the test (pendulum bob) mass distribution $\rho_{t}$ are known. The pendulum
spring constant, up to some small corrections, is given by $k=m\omega^{2}$
with $m$ the bob mass and $\omega$ the angular frequency of the pendulum when
it is set into free oscillation in a separate experiment. The four-wire
pendulum design causes the bob to translate with very little rotation and
constrains the bob to move only along the $\hat{\mathbf{z}}$ axis. Since most
(99.87%) of the pendulum restoring force is from earth’s gravity rather than
from the material properties of a fiber, the pendulums behave very much like
perfect springs for small displacements. These springs are stiff compared to a
torsion fiber, but this stiffness is offset by the ability of the laser
interferometer to measure very accurately the change in the distance between
the two pendulum bobs that occurs when the source masses are moved from one
position to another.
The most difficult aspect of any precision measurement experiment is
understanding and controlling the major sources of uncertainty. Though
conceptually the experiment is very simple, Nature’s cunning is in the
details. We sketch out the uncertainty sources here but a longer follow-up
paper is planned to more fully describe the experimental details. The
uncertainties are summarized in Table 1 and are dominated by components
related to the mass distributions.
Table 1: The major components of uncertainty are listed here expressed in terms of each contribution to $\delta G/G$ in parts in $10^{5}$ at the $1\sigma$ level. The uncertainties in this table, along with all other uncertainties in this paper, are expressed as standard ($1\sigma$) uncertainties. Uncertainty Component | $\delta G/G(\times 10^{-5})$
---|---
Six critical dimensions | 1.4
All other dimensions | 0.8
Source mass density inhomogeneities | 0.8
Pendulum spring constants | 0.7
Total mass measurement | 0.6
Interferometer | 0.6
Tilt due to source mass motion | 0.1
Day-to-day scatter | 0.4
Combined uncertainty | 2.1
The source masses are arranged so that, in both measuring positions, the
pendulum bobs are at a saddle point in the gravitational field from the source
masses. This makes the gravitational signal quite insensitive to the position
of the pendulum bobs relative to the source masses, though the signal does
depend critically on the distance - perpendicular to the interferometer axis -
between the two opposite pairs of source mass cylinders as well as the along-
axis distance between the two adjacent source masses when they are in the
inner position. This geometry reduces the hardest part of defining the three-
dimensional mass distribution to just six one-dimensional measurements. We
constructed a large caliper with a movable stand that could reach around the
apparatus. With this and a smaller caliper, we were able to measure the six
critical separations with an uncertainty of about 3 $\mu$m. This measurement
contributes a relative uncertainty of 1.4 parts in $10^{5}$ to our combined
uncertainty. The gravitational signal is much less sensitive to uncertainties
in all the other dimensional measurements, but we also invested less effort in
making these other measurements, which contribute a total of 0.8 parts in
$10^{5}$ to the uncertainty budget.
Density variations within the source masses are also a significant contributor
to the uncertainty of our final value. The masses are made of an alloy of
95.5% tungsten sintered with copper and nickel. Because the cylinders were
cast on their sides, our finding a density variation of 1 to 2 parts in
$10^{3}$ across their diameters is not surprising. This density variation was
measured by allowing individual billets to rotate freely in an air-bearing as
well as by cutting apart one of the billets after the experiment was
concluded. The orientation of each source mass stack (as well as the
orientation of the three billets that comprise it) was adjusted to cancel out,
by as much as possible, the effect of this gradient on the total gravity
signal. We also rotated the stacks by $180\,^{\circ}$ halfway through the
experiment to average out the effect of any residual linear component of the
density gradient. Based on the air bearing data, the expected fractional
change in the gravity signal was $(2.4\pm 0.5)\times 10^{-5}$ when the masses
were rotated $180\,^{\circ}$. We actually observed a fractional change of
$(1.3\pm 0.7)\times 10^{-5}$, in reasonable agreement with the calculated
value. The residual nonlinear density variations contribute an uncertainty of
0.8 parts in $10^{5}$ to the final result.
The total mass of the source mass configuration contributes 0.6 parts in
$10^{5}$ to the uncertainty budget which includes both the uncertainty of the
balance used to weigh the masses and the uncertainty in the density of the
displaced air.
The spring constant of each pendulum is obtained by setting the pendulums
swinging (with the damping magnets removed) and recording the period of
oscillation. We make three corrections to the simple pendulum model. The first
correction is from the small, but non-zero, rotational inertia of the wires
and amounts to a relative correction of $(7.5\pm 0.1)\times 10^{-5}$ to the
spring constants. Second, we take into account the fact that the bobs rotate
slightly as they translate. This rotation occurs because the relative loading
on the wires changes as the bob is displaced, causing the wires to stretch
differentially. This rotation results in a correction to the spring constants
of $(5.8\pm 0.4)\times 10^{-5}$. Finally, we account for the force on the
pendulum bobs from the damping magnets due to the diamagnetism of the bobs.
The horizontal force gradient was measured by translating the magnets and
observing the resulting displacement of the pendulum bobs. As copper is
diamagnetic, the bobs were observed to move in the opposite direction from the
magnets (confirming that there was no ferromagnetic contamination on or in the
bobs).
Because they are diamagnetic, there is also a small upward magnetic force on
the bobs that reduces the effective value for $g$ on the bobs. This force was
evaluated by weighing the bobs with and without the magnetic field. We find a
total spring constant correction due to magnetic effects of $(-7.54\pm
0.03)\times 10^{-5}$ for one pendulum and $(-7.34\pm 0.01)\times 10^{-5}$ for
the other.
Corrections due to the finite amplitude of the swing during the pendulum
frequency measurements and corrections due to the finite Q of the pendulums
(with the damping magnets removed) are less than about 1 part in $10^{6}$ and
were ignored. The remainder of the uncertainty in the pendulum spring
constants comes from scatter in the data used to measure the periods (0.5
parts in $10^{5}$) and the measured anelasticity of the pendulum wires (0.2
parts in $10^{5}$).
The pendulum bobs are slightly magnetized by the field of the damping magnets,
and this makes them more sensitive to magnetic gradients than they otherwise
would be. (The field in the vicinity of the pendulum bobs is on the order of
0.01 T, and the susceptibility of the copper bobs is $-1\times 10^{-5}$.)
However, residual fields from the damping magnets are on the order of a few
hundred $\mu$T in the vicinity of the source masses and are too small by more
than an order of magnitude to induce sufficient magnetization in the source
masses to influence the bob position as the tungsten alloy used for the source
masses has a susceptibility of $(6.6\pm 0.3)\times 10^{-4}$. Care was also
taken to eliminate any error due to magnetic fields from the source mass drive
motor.
Because we must move large source masses to generate the gravitational signal,
care must be taken to reduce possible errors due to the change in mass loading
on the apparatus. The vacuum chamber that contains the pendulums straddles,
without touching, the plate upon which the source masses ride. Finally, the
source mass support plate rests kinematically on the floor independently from
the rest of the apparatus. The center of mass of the 480 kg source mass
configuration shifts by 0.2 mm when it is moved from the inner to the outer
position because of a slight deviation from the planned values of the mass
stop locations. Though this is a small shift, the resulting change in floor
tilt translates to a change in the pendulum bob separation because the
pendulums differ in length by 0.3 mm. We evaluated this effect by deliberately
shifting the center of mass position of the source masses by a large amount
and observing the effect on the pendulums (after removing the calculated
gravitational signal). Based on this data, we find a correction of $(-0.4\pm
0.1)\times 10^{-5}$ to our $G$ value.
The compressed air that is fed to the air pucks under the source masses cools
as it is released. Care was taken to ensure that the resulting thermal
gradients did not cause an error in the final results. A vacuum pump connected
to a groove around the outer perimeter of the puck sweeps up the cool air
before it escapes from under the puck. Temperature measurements of different
parts of the apparatus (the source masses, source mass support plate, and
pendulum vacuum chamber) indicated that all parts were at the same temperature
to within $0.1\,^{\circ}\mathrm{C}$. As a check of the temperature sensitivity
of our apparatus, we raised the temperature of the source masses from the
ambient $22\,^{\circ}\mathrm{C}$ to between 30 and $40\,^{\circ}\mathrm{C}$.
With the source masses at this elevated temperature, the pendulum signal
changed by a factor $(4\pm 22)\times 10^{-5}$ after correcting for the mass
position change due to the thermal expansion of the apparatus. We conclude
that temperature effects have a negligible contribution to our uncertainty
budget (aside from a term that we have included in the uncertainty of the
dimensional measurements).
The laser interferometer contributes to the uncertainty budget mainly through
any misalignment of the optical axis with respect to the pendulum bob motion
as well as scatter (due to pendulum motion) in the data used to determine the
free spectral range. We use He-Ne lasers locked to the pendulum and reference
cavities with a Pound-Drever-Hall scheme. About 1 $\mu$W reaches each Fabry-
Perot cavity, and each cavity has a finesse of 4000. Optical effects, such as
stray reflections from the various optical components as well as radiation
pressure on the bob mirrors, are negligible sources of uncertainty.
Figure 2: The calculated $G$ values from data runs in May and June of 2004.
Each value is expressed as the fractional deviation $(\times 10^{-5})$ from
the mean value of $6.672\,34\times
10^{-11}\>\mathrm{m}^{3}\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$. The error bars
include the uncertainty calculated from the scatter within each data set
combined with the $1.5\times 10^{-5}$ relative uncertainty associated with the
observed day-to-day variations in the source mass position. The systematic
components of the uncertainty, as listed in Table 1, are not included in the
error bars.
A summary of the 13 data runs used in this determination of $G$ is shown in
Fig. 2 which gives the calculated G values from data runs in May and June of
2004. Each run consists of between one and a half and seven hours of data like
that shown in the bottom panel of Fig. 1. During the time period covered in
Fig. 2, the six critical source mass dimensions were measured eight times. For
each data point, the value of $G$ was calculated using the average of the
source mass positions that were found before and after that run or that day’s
series of runs. The source mass positions vary slightly from day-to-day
because of movement of the stops as the 120 kg source masses are seated and
variations in the force pressing the masses into the stops. The standard
deviation in these position measurements is 3.6 $\mu$m, which is expected to
cause a standard deviation of $1.5\times 10^{-5}$ in the signal from run to
run. This is very close to the standard deviation of $1.4\times 10^{-5}$
actually seen in Fig. 2.
During the gap between the 5/15 and 6/3 data, there was a large 50 $\mu$m
shift in one mass position that occurred when the source masses slammed into
the stops while we were trying to troubleshoot a faulty motor. This collision
caused a large shift in the raw signal, but no significant shift is seen in
the $G$ values after the new positions were used in the calculations. After
the motor problem, the drive system required constant readjustment and three
data sets were thrown out because the source masses were getting stuck before
they were fully into the mass stops. In addition, four of the data sets (the
5th, 9th, 11th, and 13th points in Fig. 2) were truncated after two hours when
the signal became noticeably unstable towards the end of the run.
Between the data taken on 6/3 and 6/4 (the 6th and 7th data points), each
source mass stack was rotated by $180\,^{\circ}$ to average out the linear
density gradient across the source mass billets. A correction based on the
measured density gradient is included in the data shown in Fig. 2.
Nevertheless, the value for $G$ we give, calculated as the mean of the data
before and after the $180\,^{\circ}$ rotation, does not depend on the value of
this correction.
We have presented here our new determination of the Newtonian constant of
gravitation. Great care was exercised in carrying out the experiment and in
our detailed analysis. Having now completed our measurement, we are reminded
of Cavendish’s description of his 1798 experiment Cavendish (1798): “The
apparatus is very simple.” That statement also applies to the experiment that
we report here. We would add: “The measurement is very hard.”
###### Acknowledgements.
We thank Douglas S. Robertson for writing software to provide an independent
check of our gravity field calculations as well as Hans Green, Blaine Horner,
and Alan Patee for creating the apparatus. We also thank Terry Quinn and
Richard Davis for many helpful discussions. H. Parks is grateful to the
National Research Council for a NIST post-doctoral fellowship. Sandia National
Laboratories is a multiprogram laboratory operated by Sandia Corporation, a
wholly owned subsidiary of Lockheed Martin company, for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-
AC04-94AL85000. This manuscript has been assigned report number SAND
2010-5164J by Sandia National Laboratories.
## References
* Cavendish (1798) H. Cavendish, Philos. Trans. R. Soc. London, 88, 469 (1798).
* Luther and Towler (1982) G. G. Luther and W. R. Towler, Phys. Rev. Lett., 48, 121 (1982).
* Cohen and Taylor (1987) E. R. Cohen and B. N. Taylor, Rev. Mod. Phys., 59, 1121 (1987).
* Kuroda (1995) K. Kuroda, Phys. Rev. Lett., 75, 2796 (1995).
* Gundlach and Merkowitz (2000) J. H. Gundlach and S. M. Merkowitz, Phys. Rev. Lett., 85, 2869 (2000).
* Armstrong and Fitzgerald (2003) T. R. Armstrong and M. P. Fitzgerald, Phys. Rev. Lett., 91, 201101 (2003).
* Quinn _et al._ (2001) T. J. Quinn, C. C. Speake, S. J. Richman, R. S. Davis, and A. Picard, Phys. Rev. Lett., 87, 111101 (2001).
* Luo _et al._ (2009) J. Luo, Q. Liu, L.-C. Tu, C.-G. Shao, L.-X. Liu, S.-Q. Yang, Q. Li, and Y.-T. Zhang, Phys. Rev. Lett., 102, 240801 (2009).
* Bagley and Luther (1997) C. H. Bagley and G. G. Luther, Phys. Rev. Lett., 78, 3047 (1997).
* Karagioz and Izmailov (1996) O. V. Karagioz and V. P. Izmailov, Meas. Tech., 39, 979 (1996).
* Kleinevoß _et al._ (1999) U. Kleinevoß, H. Meyer, A. Schumacher, and S. Hartmann, Meas. Sci. Technol., 10, 492 (1999).
* Kleinevoß _et al._ (2002) U. Kleinevoß, H. Meyer, H. Piel, and S. Hartmann, Precision Electromagnetic Measurements, Conference on, 2002. Conference Digest 2002, 148 (2002).
* Schlamminger _et al._ (2006) S. Schlamminger, E. Holzschuh, W. Kündig, F. Nolting, R. E. Pixley, J. Schurr, and U. Straumann, Phys. Rev. D, 74, 082001 (2006).
* Mohr _et al._ (2008) P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys., 80, 633 (2008).
|
arxiv-papers
| 2010-08-19T02:55:39 |
2024-09-04T02:49:12.253992
|
{
"license": "Public Domain",
"authors": "Harold V. Parks and James E. Faller",
"submitter": "Harold Parks",
"url": "https://arxiv.org/abs/1008.3203"
}
|
1008.3296
|
###### Abstract
We study chemical reactions with complex mechanisms under two assumptions: (i)
intermediates are present in small amounts (this is the quasi-steady-state
hypothesis or QSS) and (ii) they are in equilibrium relations with substrates
(this is the quasiequilibrium hypothesis or QE). Under these assumptions, we
prove the generalized mass action law together with the basic relations
between kinetic factors, which are sufficient for the positivity of the
entropy production but hold even without microreversibility, when the detailed
balance is not applicable. Even though QE and QSS produce useful
approximations by themselves, only the combination of these assumptions can
render the possibility beyond the “rarefied gas” limit or the “molecular
chaos” hypotheses. We do not use any a priori form of the kinetic law for the
chemical reactions and describe their equilibria by thermodynamic relations.
The transformations of the intermediate compounds can be described by the
Markov kinetics because of their low density (low density of elementary
events). This combination of assumptions was introduced by Michaelis and
Menten in 1913\. In 1952, Stueckelberg used the same assumptions for the gas
kinetics and produced the remarkable semi-detailed balance relations between
collision rates in the Boltzmann equation that are weaker than the detailed
balance conditions but are still sufficient for the Boltzmann $H$-theorem to
be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg
conceptual framework.
###### keywords:
chemical kinetics; Lyapunov function; entropy; quasiequilibrium; detailed
balance; complex balance
10.3390/e13050966 13 Received: 25 January 2011; in revised form: 28 March 2011
/ Accepted: 12 May 2011 / Published: 20 May 2011 The Michaelis-Menten-
Stueckelberg Theorem Alexander N. Gorban 1,⋆ and Muhammad Shahzad 1,2 E-Mail:
ag153@le.ac.uk. 05.70.Ln; 82.20.Db
## 1 Introduction
### 1.1 Main Asymptotic Ideas in Chemical Kinetics
There are several essentially different approaches to asymptotic and scale
separation in kinetics, and each of them has its own area of applicability.
In chemical kinetics various fundamental ideas about asymptotical analysis
were developed Klonowski1983 : Quasieqiulibrium asymptotic (QE), quasi steady-
state asymptotic (QSS), lumping, and the idea of limiting step.
Most of the works on nonequilibrium thermodynamics deal with the QE
approximations and corrections to them, or with applications of these
approximations (with or without corrections). There are two basic formulation
of the QE approximation: The thermodynamic approach, based on entropy maximum,
or the kinetic formulation, based on selection of fast reversible reactions.
The very first use of the entropy maximization dates back to the classical
work of Gibbs Gibb , but it was first claimed for a principle of informational
statistical thermodynamics by Jaynes Jaynes1 . A very general discussion of
the maximum entropy principle with applications to dissipative kinetics is
given in the review Bal . Corrections of QE approximation with applications to
physical and chemical kinetics were developed GKIOeNONNEWT2001 ; GorKar .
QSS was proposed by Bodenstein in 1913 Bodenstein1913 , and the important
Michaelis and Menten work MichaelisMenten1913 was published simultaneously.
It appears that no kinetic theory of catalysis is possible without QSS. This
method was elaborated into an important tool for the analysis of chemical
reaction mechanism and kinetics Semenov1939 ; Christiansen1953 ;
Helfferich1989 . The classical QSS is based on the relative smallness of
concentrations of some of the “active” reagents (radicals, substrate-enzyme
complexes or active components on the catalyst surface) BriggsHaldane1925 ;
Aris1965 ; Segel89 .
Lumping analysis aims to combine reagents into “quasicomponents” for dimension
reduction LumpWei1 ; LumpWei2 ; LumpLiRab1 ; LumpLiRab2 . Wei and Prater Wei62
demonstrated that for (pseudo)monomolecular systems there exist linear
combinations of concentrations which evolve in time independently. These
linear combinations (quasicomponents) correspond to the left eigenvectors of
the kinetic matrix: If $lK=\lambda l$ then
${\mathrm{d}}(l,c)/{\mathrm{d}}t=(l,c)\lambda$, where the standard inner
product $(l,c)$ is the concentration of a quasicomponent. They also
demonstrated how to find these quasicomponents in a properly organized
experiment.
This observation gave rise to a question: How to lump components into proper
quasicomponents to guarantee the autonomous dynamics of the quasicomponents
with appropriate accuracy? Wei and Kuo studied conditions for exact LumpWei1
and approximate LumpWei2 lumping in monomolecular and pseudomonomolecular
systems. They demonstrated that under certain conditions a large monomolecular
system could be well-modelled by a lower-order system.
More recently, sensitivity analysis and Lie group approach were applied to
lumping analysis LumpLiRab1 ; LumpLiRab2 , and more general nonlinear forms of
lumped concentrations were used (for example, concentration of quasicomponents
could be a rational function of $c$).
Lumping analysis was placed in the linear systems theory and the relationships
between lumpability and the concepts of observability, controllability and
minimal realization were demonstrated LumpingOBservability . The lumping
procedures were considered also as efficient techniques leading to nonstiff
systems and demonstrated the efficiency of the developed algorithm on kinetic
models of atmospheric chemistry NonstiffAtmospheric2002 . An optimal lumping
problem can be formulated in the framework of a mixed integer nonlinear
programming (MINLP) and can be efficiently solved with a stochastic
optimization method OptimalLumping2008 .
The concept of limiting step gives the limit simplification: The whole network
behaves as a single step. This is the most popular approach for model
simplification in chemical kinetics and in many areas beyond kinetics. In the
form of a bottleneck approach this approximation is very popular from traffic
management to computer programming and communication networks. Recently, the
concept of the limiting step has been extended to the asymptotology of
multiscale reaction networks GorbaRadul2008 ; GorRadZin2010 .
In this paper, we focus on the combination of the QE approximation with the
QSS approach.
### 1.2 The Structure of the Paper
Almost thirty years ago one of us published a book G1 with Chapter 3 entitled
“Quasiequilibrium and Entropy Maximum”. A research program was formulated
there, and now we are in the position to analyze the achievements of these
three decades and formulate the main results, both theoretical and applied,
and the unsolved problems. In this paper, we start this work and combine a
presentation of theory and application of the QE approximation in physical and
chemical kinetics with exposition of some new results.
We start from the formal description of the general idea of QE and its
possible extensions. In Section 2, we briefly introduce main notations and
some general formulas for exclusion of fast variables by the QE approximation.
In Section 3, we present the history of the QE and the classical confusion
between the QE and the quasi steady state (QSS) approximation. Another
surprising confusion is that the famous Michaelis-Menten kinetics was not
proposed by Michaelis and Menten in 1913 MichaelisMenten1913 but by Briggs
and Haldane BriggsHaldane1925 in 1925. It is more important that Michaelis
and Menten proposed another approximation that is very useful in general
theoretical constructions. We described this approximation for general kinetic
systems. Roughly speaking, this approximation states that any reaction goes
through transformation of fast intermediate complexes (compounds), which (i)
are in equilibrium with the input reagents and (ii) exist in a very small
amount.
One of the most important benefits from this approach is the exclusion of
nonlinear kinetic laws and reaction rate constants for nonlinear reactions.
The nonlinear reactions transform into the reactions of the compounds
production. They are in a fast equilibrium and the equilibrium is ruled by
thermodynamics. For example, when Michaelis and Menten discuss the production
of the enzyme-substrate complex ES from enzyme E and substrate S, they do not
discuss reaction rates. These rates may be unknown. They just assume that the
reaction $E+S\rightleftharpoons ES$ is in equilibrium. Briggs and Haldane
involved this reaction into the kinetic model. Their approach is more general
than the Michaelis–Menten approximation but for the Briggs and Haldane model
we need more information, not only the equilibrium of the reaction
$E+S\rightleftharpoons ES$ but also its rates and constants.
When compounds undergo transformations in a linear first order kinetics, there
is no need to include interactions between them because they are present in
very small amounts in the same volume, and their concentrations are also
small. (By the way, this argument is not applicable to the heterogeneous
catalytic reactions. Although the intermediates are in both small amounts and
in a small volume, i.e., in the surface layer, the concentration of the
intermediates is not small, and their interaction does not vanish when their
amount decreases Yablonskii1991 . Therefore, kinetics of intermediates in
heterogeneous catalysis may be nonlinear and demonstrate bifurcations,
oscillations and other complex behavior.)
In 1952, Stueckelberg Stueckelberg1952 used similar approach in his seminal
paper “$H$-theorem and unitarity of the $S$-matrix”. He studied elastic
collisions of particles as the quasi-chemical reactions
$\mathbf{v+w\to v^{\prime}+w^{\prime}}$
($\mathbf{v,w,v^{\prime},w^{\prime}}$ are velocities of particles) and
demonstrated that for the Boltzmann equation the linear Markov kinetics of the
intermediate compounds results in the special relations for the kinetic
coefficients. These relations are sufficient for the $H$-theorem, which was
originally proved by Boltzmann under the stronger assumption of reversibility
of collisions Boltzmann .
First, the idea of such relations was proposed by Boltzmann as an answer to
the Lorentz objections against Boltzmann’s proof of the $H$-theorem. Lorentz
stated the nonexistence of inverse collisions for polyatomic molecules.
Boltzmann did not object to this argument but proposed the “cyclic balance”
condition, which means balancing in cycles of transitions between states
$S_{1}\to S_{2}\to\ldots\to S_{n}\to S_{1}$. Almost 100 years later,
Cercignani and Lampis CercignaniLamp1981 demonstrated that the Lorenz
arguments are wrong and the new Boltzmann relations are not needed for the
polyatomic molecules under the microreversibility conditions. The detailed
balance conditions should hold.
Nevertheless, Boltzmann’s idea is very seminal. It was studied further by
Heitler Heitler1944 and Coester Coester1951 and the results are sometimes
cited as the “Heitler-Coestler theorem of semi-detailed balance”. In 1952,
Stueckelberg Stueckelberg1952 proved these conditions for the Boltzmann
equation. For the micro-description he used the $S$-matrix representation,
which is in this case equivalent for the Markov microkinetics (see also
Watanabe1955 ).
Later, these relations for the chemical mass action kinetics were rediscovered
and called the complex balance conditions HornJackson1972 ; Feinberg1972 . We
generalize the Michaelis-Menten-Stueckelberg approach and study in Section 5
the general kinetics with fast intermediates present in small amount. In
Subsection 5.7 the big Michaelis-Menten-Stueckelberg theorem is formulated as
the overall result of the previous analysis.
Before this general theory, we introduce the formalism of the QE approximation
with all the necessary notations and examples for chemical kinetics in Section
4.
The result of the general kinetics of systems with intermediate compounds can
be used wider than this specific model of an elementary reaction: The
intermediate complexes with fast equilibria and the Markov kinetics can be
considered as the “construction staging” for general kinetics. In Section 6,
we delete the construction staging and start from the general forms of the
obtained kinetic equations as from the basic laws. We study the relations
between the general kinetic law and the thermodynamic condition of the
positivity of the entropy production.
Sometimes the kinetics equations may not respect thermodynamics from the
beginning. To repair this discrepancy, deformation of the entropy may help. In
Section 7, we show when is it possible to deform the entropy by adding a
linear function to provide agreement between given kinetic equations and the
deformed thermodynamics. As a particular case, we got the “deficiency zero
theorem”.
The classical formulation of the principle of detailed balance deals not with
the thermodynamic and global forms we use but just with equilibria: In
equilibrium each process must be equilibrated with its reverse process. In
Section 7, we demonstrate also that for the general kinetic law the existence
of a point of detailed balance is equivalent to the existence of such a linear
deformation of the entropy that the global detailed balance conditions
(Equation (87) below) hold. Analogously, the existence of a point of complex
balance is equivalent to the global condition of complex balance after some
linear deformation of the entropy.
### 1.3 Main Results: One Asymptotic and Two Theorems
Let us follow the ideas of Michaelis-Menten and Stueckelberg and introduce the
asymptotic theory of reaction rates. Let the list of the components $A_{i}$ be
given. The mechanism of reaction is the list of the elementary reactions
represented by their stoichiometric equations:
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}\,$ (1)
The linear combinations $\sum_{i}\alpha_{\rho i}A_{i}$ and
$\sum_{i}\beta_{\rho i}A_{i}$ are the complexes. For each complex
$\sum_{i}y_{ji}A_{i}$ from the reaction mechanism we introduce an intermediate
auxiliary state, a compound $B_{j}$. Each elementary reaction is represented
in the form of the “$2n$-tail scheme” (Figure 1) with two intermediate
compounds:
$\sum_{i}\alpha_{\rho i}A_{i}\rightleftharpoons B_{\rho}^{-}\to
B_{\rho}^{+}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}\,\vspace{-6pt}$ (2)
Figure 1: A $2n$-tail scheme of an extended elementary reaction.
There are two main assumptions in the Michaelis-Menten-Stueckelberg
asymptotic:
* •
The compounds are in fast equilibrium with the corresponding input reagents
(QE);
* •
They exist in very small concentrations compared to other components (QSS).
The smallness of the concentration of the compounds implies that they (i) have
the perfect thermodynamic functions (entropy, internal energy and free energy)
and (ii) the rates of the reactions $B_{i}\to B_{j}$ are linear functions of
their concentrations.
One of the most important benefits from this approach is the exclusion of the
nonlinear reaction kinetics: They are in fast equilibrium and equilibrium is
ruled by thermodynamics.
Under the given smallness assumptions, the reaction rates $r_{\rho}$ for the
elementary reactions have a special form of the generalized mass action law
(see Equation (74) below):
$r_{\rho}=\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})\,$
where $\varphi_{\rho}>0$ is the kinetic factor and
$\exp(\alpha_{\rho},\check{\mu})$ is the Boltzmann factor. Here and further in
the text $(\alpha_{\rho},\check{\mu})=\sum_{i}\alpha_{\rho i}\check{\mu}_{i}$
is the standard inner product, $\exp(\ ,\ )$ is the exponential of the value
of the inner product and $\check{\mu}_{i}$ are chemical potentials $\mu$
divided on $RT$.
For the prefect chemical systems, $\check{\mu}_{i}=\ln(c_{i}/c_{i}^{*})$,
where $c_{i}$ is the concentration of $A_{i}$ and $c_{i}^{*}>0$ are the
positive equilibrium concentrations. For different values of the conservation
laws there are different positive equilibria. The positive equilibrium
$c_{i}^{*}$ is one of them and it is not important which one is it. At this
point, $\check{\mu}_{i}=0$, hence, the kinetic factor for the perfect systems
is just the equilibrium value of the rate of the elementary reaction at the
equilibrium point $c^{*}$: $\varphi_{\rho}=r_{\rho}(c^{*})$.
The linear kinetics of the compound reactions $B_{i}\to B_{j}$ implies the
remarkable identity for the reaction rates, the complex balance condition
(Equation (89) below)
$\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\alpha_{\rho})=\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\beta_{\rho})\,$
for all admissible values of $\check{\mu}$ and given $\varphi$ which may vary
independently. For other and more convenient forms of this condition see
Equation (91) in Section 6. The complex balance condition is sufficient for
the positivity of the entropy production (for decrease of the free energy
under isothermal isochoric conditions). The general formula for the reaction
rate together with the complex balance conditions and the positivity of the
entropy production form the Michaelis-Menten-Stueckelberg theorem (Section
5.7).
The detailed balance conditions (Equation (87) below),
$\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$
for all ${\rho}$, are more restrictive than the complex balance conditions.
For the perfect systems, the detailed balance condition takes the standard
form: $r_{\rho}^{+}(c^{*})=r_{\rho}^{-}(c^{*})$.
We study also some other, less restrictive sufficient conditions for
accordance between thermodynamics and kinetics. For example, we demonstrate
that the $G$-inequality (Equation (92) below)
$\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\alpha_{\rho})\geq\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\beta_{\rho})\,$
is sufficient for the entropy growth and, at the same time, weaker than the
condition of complex balance.
If the reaction rates have the form of the generalized mass action law but do
not satisfy the sufficient condition of the positivity of the entropy
production, the situation may be improved by the deformation of the entropy
via addition of a linear function. Such a deformation is always possible for
the zero deficiency systems. Let $q$ be the number of different complexes in
the reaction mechanism, $d$ be the number of the connected components in the
digraph of the transitions between compounds (vertices are compounds and edges
are reactions). To exclude some degenerated cases a hypothesis of weak
reversibility is accepted: For any two vertices $B_{i}$ and $B_{j}$, the
existence of an oriented path from $B_{i}$ to $B_{j}$ implies the existence of
an oriented path from $B_{j}$ to $B_{i}$.
Deficiency of the system is Feinberg1972
$q-d-{\rm rank}\Gamma\geq 0$
where $\Gamma=(\gamma_{ij})=(\beta_{ij}-\alpha_{ij})$ is the stoichiometric
matrix. If the system has zero deficiency then the entropy production becomes
positive after the deformation of the entropy via addition of a linear
function. The deficiency zero theorem in this form is proved in Section 7.3.
Interrelations between the Michaelis-Menten-Stueckelberg asymptotic and the
transition state theory (which is also referred to as the “activated-complex
theory”, “absolute-rate theory”, and “theory of absolute reaction rates”) are
very intriguing. This theory was developed in 1935 by Eyring Eyring1935 and
by Evans and Polanyi EvansPolanyi1935 .
Basic ideas behind the transition state theory are LaidlerTweedale2007 :
* •
The activated complexes are in a quasi-equilibrium with the reactant
molecules;
* •
Rates of the reactions are studied by studying the activated complexes at the
saddle point of a potential energy surface.
The similarity is obvious but in the Michaelis-Menten-Stueckelberg asymptotic
an elementary reaction is represented by a couple of compounds with the Markov
kinetics of transitions between them versus one transition state, which moves
along the “reaction coordinate”, in the transition state theory. This is not
exactly the same approach (for example, the theory of absolute reaction rates
uses the detailed balance conditions and does not produce anything similar to
the complex balance).
Important technical tools for the analysis of the Michaelis-Menten-
Stueckelberg asymptotic are the theorem about preservation of the entropy
production in the QE approximation (Section 2 and Appendix 1) and the Morimoto
$H$-theorem for the Markov chains (Appendix 2).
## 2 QE and Preservation of Entropy Production
In this section we introduce informally the QE approximation and the important
theorem about the preservation of entropy production in this approximation. In
Appendix 1, this approximation and the theorem are presented with more formal
details.
Let us consider a system in a domain $U$ of a real vector space $E$ given by
differential equations
$\frac{{\mathrm{d}}x}{{\mathrm{d}}t}=F(x)$ (3)
The QE approximation for (3) uses two basic entities: Entropy and slow
variables.
Entropy $S$ is an increasing concave Lyapunov function for (3) with non-
degenerated Hessian $\partial^{2}S/\partial x_{i}\partial x_{j}$:
$\frac{{\mathrm{d}}S}{{\mathrm{d}}t}\geq 0\,$ (4)
In this approach, the increase of entropy in time is exploited (the Second Law
in the form (4)).
The slow variables $M$ are defined as some differentiable functions of
variables $x$: $M=m(x)$. Here we assume that these functions are linear. More
general nonlinear theory was developed in GorKarQE2006 ; GorKarProjector2004
with applications to the Boltzmann equation and polymer physics. Selection of
the slow variables implies a hypothesis about separation of fast and slow
motion. The slow variables (almost) do not change during the fast motion.
After some initial time, the fast variables with high accuracy are functions
of the slow variables: We can write $x\approx x^{*}_{M}$.
The QE approximation defines the functions $x^{*}_{M}$ as solutions to the
following MaxEnt optimization problem:
$S(x)\to\max\;\;\mbox{subject to}\;\;m(x)=M\,$ (5)
The reasoning behind this approximation is simple: During the fast initial
layer motion, entropy increases and $M$ almost does not change. Therefore, it
is natural to assume that $x^{*}_{M}$ is close to the solution to the MaxEnt
optimization problem (5). Further $x^{*}_{M}$ denotes a solution to the MaxEnt
problem.
A solution to (5), $x^{*}_{M}$, is the QE state, the set of the QE states
$x^{*}_{M}$, parameterized by the values of the slow variables $M$ is the QE
manifold, the corresponding value of entropy
$S^{*}(M)=S(x^{*}_{M})$ (6)
is the QE entropy and the equation for the slow variables
$\frac{{\mathrm{d}}M}{{\mathrm{d}}t}=m(F(x^{*}_{M}))$ (7)
represents the QE dynamics.
The crucial property of the QE dynamics is the preservation of entropy
production.
Theorem about preservation of entropy production. Let us calculate
${\mathrm{d}}S^{*}(M)/{\mathrm{d}}t$ at point $M$ according to the QE dynamics
(7) and find ${\mathrm{d}}S(x)/{\mathrm{d}}t$ at point $x=x^{*}_{M}$ due to
the initial system (3). The results always coincide:
$\frac{{\mathrm{d}}S^{*}(M)}{{\mathrm{d}}t}=\frac{{\mathrm{d}}S(x)}{{\mathrm{d}}t}\,$
(8)
The left hand side in (8) is computed due to the QE approximation (7) and the
right hand side corresponds to the initial system (3). The sketch of the proof
is given in Appendix 1.
The preservation of the entropy production leads to the preservation of the
type of dynamics: If for the initial system (3) entropy production is non-
negative, ${\mathrm{d}}S/{\mathrm{d}}t\geq 0$, then for the QE approximation
(7) the production of the QE entropy is also non-negative,
${\mathrm{d}}S^{*}/{\mathrm{d}}t\geq 0$.
In addition, if for the initial system
$({{\mathrm{d}}S}/{{\mathrm{d}}t})|_{x}=0$ if and only if $F(x)=0$ then the
same property holds in the QE approximation.
## 3 The Classics and the Classical Confusion
### 3.1 The Asymptotic of Fast Reactions
It is difficult to find who introduced the QE approximation. It was impossible
before the works of Boltzmann and Gibbs, and it became very well known after
the works of Jaynes Jaynes1 .
Chemical kinetics has been a source for model reduction ideas for decades. The
ideas of QE appear there very naturally: Fast reactions go to their
equilibrium and, after that, remain almost equilibrium all the time. The
general formalization of this idea looks as follows. The kinetic equation has
the form
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=K_{sl}(N)+\frac{1}{\epsilon}K_{fs}(N)$
(9)
Here $N$ is the vector of composition with components $N_{i}>0$, $K_{sl}$
corresponds to the slow reactions, $K_{fs}$ corresponds to fast reaction and
$\epsilon>0$ is a small number. The system of fast reactions has the linear
conservation laws $b_{l}(N)=\sum_{j}b_{lj}N_{j}$: $b_{l}(K_{fs}(N))\equiv 0$.
The fast subsystem
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=K_{fs}(N)$
tends to a stable positive equilibrium $N^{*}$ for any positive initial state
$N(0)$ and this equilibrium is a function of the values of the linear
conservation laws $b_{l}(N(0))$. In the plane $b_{l}(N)=b_{l}(N(0))$ the
equilibrium is asymptotically stable and exponentially attractive.
Vector $b(N)=(b_{l}(N))$ is the vector of slow variables and the QE
approximation is
$\frac{{\mathrm{d}}b}{{\mathrm{d}}t}=b(K_{sl}(N^{*}(b))\,$ (10)
In chemical kinetics, equilibria can be described by conditional entropy
maximum (or conditional extremum of other thermodynamic potentials).
Therefore, for these cases we can apply the formalism of the quasiequilibrium
approximation. The thermodynamic Lyapunov functions serve as tools for
stability analysis and for model reduction Hangos2010 .
The QE approximation, the asymptotic of fast reactions, is well known in
chemical kinetics. Another very important approximation was invented in
chemical kinetics as well. It is the Quasi Steady State (QSS) approximation.
QSS was proposed in Bodenstein1913 and was elaborated into an important tool
for analysis of chemical reaction mechanisms and kinetics Semenov1939 ;
Christiansen1953 ; Helfferich1989 . The classical QSS is based on the relative
smallness of concentrations of some of “active” reagents (radicals, substrate-
enzyme complexes or active components on the catalyst surface) Aris1965 ;
Segel89 . In the enzyme kinetics, its invention was traditionally connected to
the so-called Michaelis-Menten kinetics.
### 3.2 QSS and the Briggs-Haldane Asymptotic
Perhaps the first very clear explanation of the QSS was given by Briggs and
Haldane in 1925 BriggsHaldane1925 . Briggs and Haldane consider the simplest
enzyme reaction $S+E\leftrightharpoons SE\to P+E$ and mention that the total
concentration of enzyme ($[E]+[SE]$) is “negligibly small” compared with the
concentration of substrate $[S]$. After that they conclude that
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}{[SE]}$ is “negligible” compared with
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}{[S]}$ and
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}{[P]}$ and produce the now famous
‘Michaelis-Menten’ formula, which was unknown to Michaelis and Menten:
$k_{1}[E][S]=(k_{-1}+k_{2})[ES]$ or
$[ES]=\frac{[E][S]}{K_{M}+[S]}\;\;{\rm
and}\;\;\frac{{\mathrm{d}}}{{\mathrm{d}}t}{[P]}=k_{2}[ES]=\frac{k_{2}[E][S]}{K_{M}+[S]}\,$
(11)
where the “Michaelis-Menten constant” is
$K_{M}=\frac{k_{-1}+k_{2}}{k_{1}}$
There is plenty of misleading comments in later publications about QSS. Two
most important confusions are:
* •
Enzymes (or catalysts, or radicals) participate in fast reactions and, hence,
relax faster than substrates or stable components. This is obviously wrong for
many QSS systems: For example, in the Michaelis-Menten system all reactions
include enzyme together with substrate or product. There are no separate fast
reactions for enzyme without substrate or product.
* •
Concentrations of intermediates are constant because in QSS we equate their
time derivatives to zero. In general case, this is also wrong: We equate the
kinetic expressions for some time derivatives to zero, indeed, but this just
exploits the fact that the time derivatives of intermediates concentrations
are small together with their values, but not obligatory zero. If we accept
QSS then these derivatives are not zero as well: To prove this we can just
differentiate the Michaelis-Menten formula (11) and find that [ES] in QSS is
almost constant when $[S]\gg K_{M}$, this is an additional condition,
different from the Briggs-Haldane condition $[E]+[AE]\ll[S]$ (for more details
see Segel89 ; Klonowski1983 ; Yablonskii1991 and a simple detailed case study
LiShenLi2008 ).
After a simple transformation of variables the QSS smallness of concentration
transforms into a separation of time scales in a standard singular
perturbation form (see, for example Yablonskii1991 ; GorbKarlinChemEngS2003 ).
Let us demonstrate this on the traditional Michaelis-Menten system:
$\begin{split}&\frac{{\mathrm{d}}[S]}{{\mathrm{d}}t}=-k_{1}[S][E]+k_{-1}[SE]\,\\\
&\frac{{\mathrm{d}}[SE]}{{\mathrm{d}}t}=k_{1}[S][E]-(k_{-1}+k_{2})[SE]\,\\\
&[E]+[SE]=e=const,\,[S]+[P]=s=const\,\end{split}$ (12)
This is a homogeneous system with the isochoric (fixed volume) conditions for
which we write the equations. The Briggs-Haldane condition is $e\ll s$. Let us
use dimensionless variables $x=[S]/s$, $\xi=[SE]/e$:
$\begin{split}&\frac{s}{e}\frac{{\mathrm{d}}x}{{\mathrm{d}}t}=-sk_{1}x(1-\xi)+k_{-1}\xi\,\\\
&\frac{{\mathrm{d}}\xi}{{\mathrm{d}}t}=sk_{1}x(1-\xi)-(k_{-1}+k_{2})\xi\,\end{split}$
(13)
To obtain the standard singularly perturbed system with the small parameter at
the derivative, we need to change the time scale. This means that when $e\to
0$ the reaction goes proportionally slower and to study this limit properly we
have to adjust the time scale: ${\mathrm{d}}\tau=\frac{e}{s}{\mathrm{d}}t$:
$\begin{split}&\frac{{\mathrm{d}}x}{{\mathrm{d}}\tau}=-sk_{1}x(1-\xi)+k_{-1}\xi\,\\\
&\frac{e}{s}\frac{{\mathrm{d}}\xi}{{\mathrm{d}}\tau}=sk_{1}x(1-\xi)-(k_{-1}+k_{2})\xi\,\end{split}$
(14)
For small $e/s$, the second equation is a fast subsystem. According to this
fast equation, for a given constant $x$, the variable $\xi$ relaxes to
$\xi_{\rm QSS}=\frac{sx}{K_{M}+sx}$
exponentially, as $\exp(-(sk_{1}x+k_{-1}+k_{2})t)$. Therefore, the classical
singular perturbation theory based on the Tikhonov theorem Tikhonov1952 ;
Vasil'eva1963 can be applied to the system in the form (14) and the QSS
approximation is applicable even on an infinite time interval Hoppensteadt1966
. This transformation of variables and introduction of slow time is a standard
procedure for rigorous proof of QSS validity in catalysis Yablonskii1991 ,
enzyme kinetics Battelu1986 and other areas of kinetics and chemical
engineering Aris1965 .
It is worth to mention that the smallness of parameter $e/s$ can be easily
controlled in experiments, whereas the time derivatives, transformation rates
and many other quantities just appear as a result of kinetics and cannot be
controlled directly.
### 3.3 The Michaelis and Menten Asymptotic
QSS is not QE but the classical work of Michaelis and Menten
MichaelisMenten1913 was done on the intersection of QSS and QE. After the
brilliantly clear work of Briggs and Haldane, the name “Michaelis-Menten” was
attached to the Briggs and Haldane equation and the original work of Michaelis
and Menten was considered as an important particular case of this approach, an
approximation with additional and not necessary assumptions of QE. From our
point of view, the Michaelis-Menten work includes more and may give rise to an
important general class of kinetic models.
Michaelis and Menten studied the “fermentative splitting of cane sugar”. They
introduced three “compounds”: The sucrose-ferment combination, the fructose-
ferment combination and the glucose-ferment combination. The fundamental
assumption of their work was “that the rate of breakdown at any moment is
proportional to the concentration of the sucrose-invertase compound”.
They started from the assumption that at any moment according to the mass
action law
$[S_{i}][E]=K_{i}[S_{i}E]$ (15)
where $[S_{i}]$ is the concentration of the $i$th sugar (here, $i=0$ for
sucrose, 1 for fructose and 2 for glucose), $[E]$ is the concentration of the
free invertase and $K_{i}$ is the $i$th equilibrium constant.
For simplification, they use the assumption that the concentration of any
sugar in question in free state is practically equal to that of the total
sugar in question.
Finally, they obtain
$[S_{0}E]=\frac{e[S_{0}]}{K_{0}(1+q[P])+[S_{0}]}\,$ (16)
where $e=[E]+\sum_{i}[S_{i}E]$, $[P]=[S_{1}]=[S_{2}]$ and
$q=\frac{1}{K_{1}}+\frac{1}{K_{2}}$.
Of course, this formula may be considered as a particular case of the Briggs-
Haldane formula (11) if we take $k_{-1}\gg k_{2}$ in (11) (i.e., the
equilibration $S+E\leftrightharpoons SE$ is much faster than the reaction
$SE\to P+E$) and assume that $q=0$ in (16) (i.e., fructose-ferment combination
and glucose-ferment combination are practically absent).
This is the truth but may be not the complete truth. The Michaelis-Menten
approach with many compounds which are present in small amounts and satisfy
the QE assumption (15) is a seed of the general kinetic theory for perfect and
non-perfect mixtures.
## 4 Chemical Kinetics and QE Approximation
### 4.1 Stoichiometric Algebra and Kinetic Equations
In this section, we introduce the basic notations of the chemical kinetics
formalism. For more details see, for example, Yablonskii1991 .
The list of components is a finite set of symbols $A_{1},\ldots,A_{n}$.
A reaction mechanism is a finite set of the stoichiometric equations of
elementary reactions:
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}\,$ (17)
where $\rho=1,\ldots,m$ is the reaction number and the stoichiometric
coefficients $\alpha_{\rho i},\beta_{\rho i}$ are nonnegative integers.
A stoichiometric vector $\gamma_{\rho}$ of the reaction (17) is a
$n$-dimensional vector with coordinates
$\gamma_{\rho i}=\beta_{\rho i}-\alpha_{\rho i}\,$ (18)
that is, “gain minus loss” in the $\rho$th elementary reaction.
A nonnegative extensive variable $N_{i}$, the amount of $A_{i}$, corresponds
to each component. We call the vector $N$ with coordinates $N_{i}$ “the
composition vector”. The concentration of $A_{i}$ is an intensive variable
$c_{i}=N_{i}/V$, where $V>0$ is the volume. The vector $c=N/V$ with
coordinates $c_{i}$ is the vector of concentrations.
A non-negative intensive quantity, $r_{\rho}$, the reaction rate, corresponds
to each reaction (17). The kinetic equations in the absence of external fluxes
are
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}r_{\rho}\gamma_{\rho}$ (19)
If the volume is not constant then equations for concentrations include
$\dot{V}$ and have different form (this is typical for the combustion
reactions, for example).
For perfect systems and not so fast reactions, the reaction rates are
functions of concentrations and temperature given by the mass action law for
the dependance on concentrations and by the generalized Arrhenius equation for
the dependance on temperature $T$.
The mass action law states:
$r_{\rho}(c,T)=k_{\rho}(T)\prod_{i}c_{i}^{\alpha_{\rho i}}\,$ (20)
where $k_{\rho}(T)$ is the reaction rate constant.
The generalized Arrhenius equation is:
$k_{\rho}(T)=A_{\rho}\exp\left(\frac{S_{{\rm
a}\rho}}{R}\right)\exp\left(-\frac{E_{{\rm a}\rho}}{RT}\right)\,$ (21)
where $R=8.314\,472~{}\frac{\mathrm{J}}{\mathrm{K~{}mol}}$ is the universal,
or ideal gas constant, $E_{{\rm a}\rho}$ is the activation energy, $S_{{\rm
a}\rho}$ is the activation entropy (i.e., $E_{{\rm a}\rho}-TS_{{\rm a}\rho}$
is the activation free energy), $A_{\rho}$ is the constant pre-exponential
factor. Some authors neglect the $S_{{\rm a}\rho}$ term because it may be less
important than the activation energy, but it is necessary to stress that
without this term it may be impossible to reconcile the kinetic equations with
the classical thermodynamics.
In general, the constants for different reactions are not independent. They
are connected by various conditions that follow from thermodynamics (the
second law, the entropy growth for isolated systems) or microreversibility
assumption (the detailed balance and the Onsager reciprocal relations). In
Section 6.2 we discuss these conditions in more general settings.
For nonideal systems, more general kinetic law is needed. In Section 5 we
produce such a general law following the ideas of the original Michaelis and
Menten paper (this is not the same as the famous “Michaelis-Menten kinetics”).
For this work we need a general formalism of QE approximation for chemical
kinetics.
### 4.2 Formalism of QE Approximation for Chemical Kinetics
#### 4.2.1 4.2.1. QE Manifold
In this section, we describe the general formalism of the QE for chemical
kinetics following GorbKarlinChemEngS2003 .
The general construction of the quasi-equilibrium manifold gives the following
procedure. First, let us consider the chemical reactions in a constant volume
under the isothermal conditions. The free energy $F(N,T)=Vf(c,T)$ should
decrease due to reactions. In the space of concentrations, one defines a
subspace of fast motions $L$. It should be spanned by the stoichiometric
vectors of fast reactions.
Slow coordinates are linear functions that annulate $L$. These functions form
a subspace in the space of linear functions on the concentration space.
Dimension of this space is $s=n-\dim L$. It is necessary to choose any basis
in this subspace. We can use for this purpose a basis $b_{j}$ in $L^{\perp}$,
an orthogonal complement to $L$ and define the basic functionals as
$b_{j}(N)=(b_{j},N)$.
The description of the QE manifold is very simple in the Legendre transform.
The chemical potentials are partial derivatives
$\mu_{i}=\frac{\partial F(N,T)}{\partial N_{i}}=\frac{\partial
f(c,T)}{\partial c_{i}}\,$ (22)
Let us use $\mu_{i}$ as new coordinates. In these new coordinates (the
“conjugated coordinates”), the QE manifold is just an orthogonal complement to
$L$. This subspace, $L^{\perp}$, is defined by equations
$\sum_{i}\mu_{i}\gamma_{i}=0\;\;{\rm for\;any}\;\;\gamma\in L$ (23)
It is sufficient to take in (23) not all $\gamma\in L$ but only elements from
a basis in $L$. In this case, we get the system of $n-\dim L$ linear equations
of the form (23) and their solution does not cause any difficulty. For the
actual computations, one requires the inversion from $\mu$ to $c$.
It is worth to mention that the problems of the selection of the slow
variables and of the description of the QE manifold in the conjugated
variables can be considered as the same problem of description of the
orthogonal complement, $L^{\perp}$.
To finalize the construction of the QE approximation, we should find for any
given values of slow variables (and of conservation laws) $b_{i}$ the
corresponding point on the QE manifold. This means that we have to solve the
system of equations for $c$:
$b(N)=b;\;\;(\mu(c,T),\gamma_{\rho})=0\,$ (24)
where $b$ is the vector of slow variables, $\mu$ is the vector of chemical
potentials and vectors $\gamma_{\rho}$ form a basis in $L$. After that, we
have the QE dependence $c_{QE}(b)$ and for any admissible value of $b$ we can
find all the reaction rates and calculate $\dot{b}$.
Unfortunately, the system (24) can be solved analytically only in some special
cases. In general case, we have to solve it numerically. For this purpose, it
may be convenient to keep the optimization statement of the problem:
$F\to\min$ subject to given $b$. There exists plenty of methods of convex
optimization for solution of this problem.
The standard toy example gives us a fast dissociation reaction. Let a
homogeneous reaction mechanism include a fast reaction of the form
$A+B\rightleftharpoons AB$. We can easily find the QE approximation for this
fast reaction. The slow variables are the quantities $b_{1}=N_{A}-N_{B}$ and
$b_{2}=N_{A}+N_{B}+N_{C}$ which do not change in this reaction. Let the
chemical potentials be $\mu_{A}/RT=\ln c_{A}+\mu_{A0}$, $\mu_{B}/RT=\ln
c_{B}+\mu_{B0}$, $\mu_{AB}/RT=\ln c_{AB}+\mu_{AB0}$. This corresponds to the
free energy $F=VRT\sum_{i}c_{i}(\ln c_{i}+\mu_{i0})$, the correspondent free
entropy (the Massieu-Planck potential) is $-F/T$. The stoichiometric vector is
$\gamma=(-1,-1,1)$ and the equations (24) take the form
$c_{A}-c_{B}=\frac{b_{1}}{V}\,,\;\;c_{A}+c_{B}+c_{AB}=\frac{b_{2}}{V}\,,\;\;\frac{c_{AB}}{c_{A}c_{B}}=K\,$
where $K$ is the equilibrium constant $K=\exp(\mu_{A0}+\mu_{B0}-\mu_{AB0})$.
From these equations we get the expressions for the QE concentrations:
$c_{A}(b_{1},b_{2})=\frac{1}{2}\frac{b_{1}}{V}-\frac{1}{K}+\sqrt{\left(\frac{1}{2}\frac{b_{1}}{V}-\frac{1}{K}\right)^{2}+\frac{b_{1}+b_{2}}{KV}}$
$c_{B}(b_{1},b_{2})=c_{A}(b_{1},b_{2})-\frac{b_{1}}{V}\,,\;\;c_{AB}(b_{1},b_{2})=\frac{b_{1}+b_{2}}{V}-2c_{A}(b_{1},b_{2})$
The QE free entropy is the value of the free entropy at this point,
$c(b_{1},b_{2})$.
#### 4.2.2 4.2.2. QE in Traditional MM System
Let us return to the simplest homogeneous enzyme reaction
$E+S\rightleftharpoons ES\rightarrow P+S$, the traditional Michaelis-Menten
System (12) (it is simpler than the system studied by Michaelis and Menten
MichaelisMenten1913 ). Let us assume that the reaction $E+S\rightleftharpoons
ES$ is fast. This means that both $k_{1}$ and $k_{-1}$ include large
parameters: $k_{1}=\frac{1}{\epsilon}\kappa_{1}$,
$k_{-1}=\frac{1}{\epsilon}\kappa_{-1}$. For small $\epsilon$, we will apply
the QE approximation. Only three components participate in the fast reaction,
$A_{1}=S$, $A_{2}=E$, $A_{3}=ES$. For analysis of the QE manifold we do not
need to involve other components.
The stoichiometric vector of the fast reaction is $\gamma=(-1,-1,1)$. The
space $L$ is one-dimensional and its basis is this vector $\gamma$. The space
$L^{\perp}$ is two-dimensional and one of the convenient bases is
$b_{1}=(1,0,1)$, $b_{2}=(0,1,1)$. The corresponding slow variables are
$b_{1}(N)=N_{1}+N_{3}$, $b_{2}(N)=N_{2}+N_{3}$. The first slow variable is the
sum of the free substrate and the substrate captured in the enzyme-substrate
complex. The second of them is the conserved quantity, the total amount of
enzyme.
The equation for the QE manifold is (15): $k_{1}c_{1}c_{2}=k_{-1}c_{3}$ or
$\frac{c_{1}}{c_{1}^{*}}\frac{c_{2}}{c_{2}^{*}}=\frac{c_{3}}{c_{3}^{*}}$
because $k_{1}c_{1}^{*}c_{2}^{*}=k_{-1}c_{3}^{*}$, where
$c_{i}^{*}=c_{i}^{*}(T)>0$ are the so-called standard equilibrium values and
for perfect systems $\mu_{i}=RT\ln(c_{i}/c_{i}^{*})$,
$F=RTV\sum_{i}c_{i}(\ln(c_{i}/c_{i}^{*})-1)$.
Let us fix the slow variables and find $c_{1,2,3}$. Equations (24) turn into
$c_{1}+c_{3}=b_{1}\,,\;c_{2}+c_{3}=b_{2}\,,\;k_{1}c_{1}c_{2}=k_{-1}c_{3}$
Here we change dynamic variables from $N$ to $c$ because this is a homogeneous
system with constant volume.
If we use $c_{1}=b_{1}-c_{3}$ and $c_{2}=b_{2}-c_{3}$ then we obtain a
quadratic equation for $c_{3}$:
$k_{1}c_{3}^{2}-(k_{1}b_{1}+k_{1}b_{2}+k_{-1})c_{3}+k_{1}b_{1}b_{2}=0\,$ (25)
Therefore,
$c_{3}(b_{1},b_{2})=\frac{1}{2}\left(b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}\right)-\frac{1}{2}\sqrt{\left(b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}\right)^{2}-4b_{1}b_{2}}$
The sign “$-$” is selected to provide positivity of all $c_{i}$. This choice
provides also the proper asymptotic: $c_{3}\to 0$ if any of $b_{i}\to 0$. For
other $c_{1,2}$ we should use $c_{1}=b_{1}-c_{3}$ and $c_{2}=b_{2}-c_{3}$.
The time derivatives of concentrations are:
$\begin{split}&\dot{c}_{1}=-k_{1}c_{1}c_{2}+k_{-1}c_{3}+v_{\rm in}c_{1}^{\rm
in}-v_{\rm out}c_{1}\,\\\
&\dot{c}_{2}=-k_{1}c_{1}c_{2}+(k_{-1}+k_{2})c_{3}+v_{\rm in}c_{2}^{\rm
in}-v_{\rm out}c_{2}\,\\\
&\dot{c}_{3}=k_{1}c_{1}c_{2}-(k_{-1}+k_{2})c_{3}+v_{\rm in}c_{3}^{\rm
in}-v_{\rm out}c_{3}\,\\\ &\dot{c}_{4}=k_{2}c_{3}+v_{\rm in}c_{4}^{\rm
in}-v_{\rm out}c_{4}\,\end{split}$ (26)
here we added external flux with input and output velocities (per unite
volume) $v_{\rm in}$ and $v_{\rm out}$ and input concentrations $c^{\rm in}$.
This is done to stress that the QE approximation holds also for a system with
fluxes if the fast equilibrium subsystem is fast enough. The input and output
velocities are the same for all components because the system is homogeneous.
The slow system is
$\begin{split}&\dot{b}_{1}=\dot{c}_{1}+\dot{c}_{3}=-k_{2}c_{3}+v_{\rm
in}b_{1}^{\rm in}-v_{\rm out}b_{1}\,\\\
&\dot{b}_{2}=\dot{c}_{2}+\dot{c}_{3}=v_{\rm in}b_{2}^{\rm in}-v_{\rm
out}b_{2}\,\\\ &\dot{c}_{4}=k_{2}c_{3}+v_{\rm in}c_{4}^{\rm in}-v_{\rm
out}c_{4}\,\end{split}$ (27)
where $b_{1}^{\rm in}=c_{1}^{\rm in}+c_{3}^{\rm in}$, $b_{2}^{\rm
in}=c_{2}^{\rm in}+c_{3}^{\rm in}$.
Now, we should use the expression for $c_{3}(b_{1},b_{2})$:
$\begin{split}\dot{b}_{1}=&-k_{2}\frac{1}{2}\left[\left(b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}\right)-\frac{1}{2}\sqrt{\left(b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}\right)^{2}-4b_{1}b_{2}}\,\right]+v_{\rm
in}b_{1}^{\rm in}-v_{\rm out}b_{1}\,\\\
\dot{c}_{4}=&k_{2}\frac{1}{2}\left[\left(b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}\right)-\frac{1}{2}\sqrt{\left(b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}\right)^{2}-4b_{1}b_{2}}\,\right]+v_{\rm
in}c_{4}^{\rm in}-v_{\rm out}c_{4}\,\\\ \dot{b}_{2}=&v_{\rm in}b_{2}^{\rm
in}-v_{\rm out}b_{2}\,\end{split}$ (28)
It is obvious here that in the reduced system (28) there exists one reaction
from the lumped component with concentration $b_{1}$ (the total amount of
substrate in free state and in the substrate-enzyme complex) into the
component (product) with concentration $c_{4}$. The rate of this reaction is
$k_{2}c(b_{1}b_{2})$. The lumped component with concentration $b_{2}$ (the
total amount of the enzyme in free state and in the substrate-enzyme complex)
affects the reaction rate but does not change in the reaction.
Let us use for simplification of this system the assumption of the substrate
excess (we follow the logic of the original Michaelis and Menten paper
MichaelisMenten1913 ):
$[S]\gg[SE]\,,\;\;{\it i.e.},\;\;b_{1}\gg c_{3}\,$ (29)
Under this assumption, the quadratic equation (25) transforms into
$\left(1+\frac{b_{2}}{b_{1}}+\frac{k_{-1}}{k_{1}b_{1}}\right)c_{3}=b_{2}+o\left(\frac{c_{3}}{b_{1}}\right)\,$
(30)
and in this approximation
$c_{3}=\frac{b_{2}b_{1}}{b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}}$ (31)
(compare to (16) and (11): This equation includes an additional term $b_{2}$
in denominator because we did not assume formally anything about the smallness
of $b_{2}$ in (29)).
After this simplification, the QE slow equations (28) take the form
$\begin{split}&\dot{b}_{1}=-\frac{k_{2}b_{2}b_{1}}{b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}}+v_{\rm
in}b_{1}^{\rm in}-v_{\rm out}b_{1}\,\\\ &\dot{b}_{2}=v_{\rm in}b_{2}^{\rm
in}-v_{\rm out}b_{2}\,\\\
&\dot{c}_{4}=\frac{k_{2}b_{2}b_{1}}{b_{1}+b_{2}+\frac{k_{-1}}{k_{1}}}+v_{\rm
in}c_{4}^{\rm in}-v_{\rm out}c_{4}\,\end{split}$ (32)
This is the typical form in the reduced equations for catalytic reactions:
Nominator in the reaction rate corresponds to the “brutto reaction” $S+E\to
P+E$ Yablonskii1991 ; YAbLaz1997 .
#### 4.2.3 4.2.3. Heterogeneous Catalytic Reaction
For the second example, let us assume equilibrium with respect to the
adsorption in the CO on Pt oxidation:
CO+Pt$\rightleftharpoons$PtCO; O2+2Pt$\rightleftharpoons$2PtO
(for detailed discussion of the modeling of CO on Pt oxidation, this “Mona
Liza” of catalysis, we address readers to Yablonskii1991 ). The list of
components involved in these 2 reactions is: $A_{1}=$ CO, $A_{2}=$ O2,
$A_{3}=$ Pt, $A_{4}=$ PtO, $A_{5}=$ PtCO (CO2 does not participate in
adsorption and may be excluded at this point).
Subspace $L$ is two-dimensional. It is spanned by the stoichiometric vectors,
$\gamma_{1}=(-1,0,-1,0,1)$, $\gamma_{2}=(0,-1,-2,2,0)$.
The orthogonal complement to $L$ is a three-dimensional subspace spanned by
vectors $(0,2,0,1,0)$, $(1,0,0,0,1)$, $(0,0,1,1,1)$. This basis is not
orthonormal but convenient because of integer coordinates.
The corresponding slow variables are
$\begin{split}&b_{1}=2N_{2}+N_{4}=2N_{\rm O_{2}}+N_{\rm PtO}\,\\\
&b_{2}=N_{1}+N_{5}=N_{\rm CO}+N_{\rm PtCO}\,\\\
&b_{3}=N_{3}+N_{4}+N_{5}=N_{\rm Pt}+N_{\rm PtO}+N_{\rm PtCO}\,\end{split}$
(33)
For heterogeneous systems, caution is needed in transition between $N$ and $c$
variables because there are two “volumes” and we cannot put in (33) $c_{i}$
instead of $N_{i}$: $N_{\rm gas}=V_{\rm gas}c_{\rm gas}$ but $N_{\rm
surf}=V_{\rm surf}c_{\rm surf}$, where where $V_{\rm gas}$ is the volume of
gas, $V_{\rm surf}$ is the area of surface.
There is a law of conservation of the catalyst: $N_{\rm Pt}+N_{\rm PtO}+N_{\rm
PtCO}=b_{3}=const$. Therefore, we have two non-trivial dynamical slow
variables, $b_{1}$ and $b_{2}$. They have a very clear sense: $b_{1}$ is the
amount of atoms of oxygen accumulated in O2 and PtO and $b_{2}$ is the amount
of atoms of carbon accumulated in CO and PtCO.
The free energy for the perfect heterogeneous system has the form
$F=V_{\rm gas}RT\sum_{A_{i}\,{\rm
gas}}c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)-1\right)+V_{\rm
surf}RT\sum_{A_{i}\,{\rm
surf}}c_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)-1\right)\,$ (34)
where $c_{i}$ are the corresponding concentrations and
$c_{i}^{*}=c_{i}^{*}(T)>0$ are the so-called standard equilibrium values. (The
QE free energy is the value of the free energy at the QE point.)
From the expression (34) we get the chemical potentials of the perfect mixture
$\mu_{i}=RT\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)\,$ (35)
The QE manifold in the conjugated variables is given by equations:
$-\mu_{1}-\mu_{3}+\mu_{5}=0\,;\;-\mu_{2}-2\mu_{3}+2\mu_{4}=0$
It is trivial to resolve these equations with respect to $\mu_{3,4}$, for
example:
$\mu_{4}=\frac{1}{2}\mu_{2}+\mu_{3}\,;\;\mu_{5}=\mu_{1}+\mu_{3}$
or with the standard equilibria:
$\frac{c_{4}}{c_{4}^{*}}=\frac{c_{3}}{c_{3}^{*}}\sqrt{\frac{c_{2}}{c_{2}^{*}}}\,,\;\frac{c_{5}}{c_{5}^{*}}=\frac{c_{1}}{c_{1}^{*}}\frac{c_{3}}{c_{3}^{*}}$
or in the kinetic form (we assume that the kinetic constants are in accordance
with thermodynamics and all these forms are equivalent):
$k_{1}c_{1}c_{3}=k_{-1}c_{5}\,,\;k_{2}c_{2}c_{3}^{2}=k_{-2}c_{4}^{2}\,$ (36)
The next task is to solve the system of equations:
$\begin{split}&k_{1}c_{1}c_{3}=k_{-1}c_{5}\,,\;k_{2}c_{2}c_{3}^{2}=k_{-2}c_{4}^{2}\,,2V_{\rm
gas}c_{2}+V_{\rm surf}c_{4}=b_{1}\,,\;\\\ &V_{\rm gas}c_{1}+V_{\rm
surf}c_{5}=b_{2}\,,\;V_{\rm surf}(c_{3}+c_{4}+c_{5})=b_{3}\end{split}$ (37)
This is a system of five equations with respect to five unknown variables,
$c_{1,2,3,4,5}$. We have to solve them and use the solution for calculation of
reaction rates in the QE equations for the slow variables. Let us construct
these equations first, and then return to (37).
We assume the adsorption (the Langmuir-Hinshelwood) mechanism of CO oxidation
(the numbers in parentheses are used below for the numeration of the reaction
rate constants):
$\begin{split}(\pm 1)\;\mbox{CO+Pt$\rightleftharpoons$PtCO}\,\\\ (\pm
2)\;\mbox{O${}_{2}$+2Pt$\rightleftharpoons$2PtO}\,\\\
(3)\;\mbox{PtO+PtCO$\to$CO${}_{2}$+2Pt}\;\end{split}$ (38)
The kinetic equations for this system (including the flux in the gas phase) is
$\displaystyle{\rm CO}\;$ $\displaystyle\dot{N}_{1}=V_{\rm
surf}(-k_{1}c_{1}c_{3}+k_{-1}c_{5})+V_{\rm gas}(v_{\rm in}c_{1}^{\rm
in}-v_{\rm out}c_{1})\,$ $\displaystyle{\rm O_{2}}$
$\displaystyle\dot{N}_{2}=V_{\rm
surf}(-k_{2}c_{2}c_{3}^{2}+k_{-2}c_{4}^{2})+V_{\rm gas}(v_{\rm in}c_{2}^{\rm
in}-v_{\rm out}c_{2})\,$ $\displaystyle{\rm Pt}$
$\displaystyle\dot{N}_{3}=V_{\rm
surf}(-k_{1}c_{1}c_{3}+k_{-1}c_{5}-2k_{2}c_{2}c_{3}^{2}+2k_{-2}c_{4}^{2}+2k_{3}c_{4}c_{5})\,$
$\displaystyle{\rm PtO}$ $\displaystyle\dot{N}_{4}=V_{\rm
surf}(2k_{2}c_{2}c_{3}^{2}-2k_{-2}c_{4}^{2}-k_{3}c_{4}c_{5})\,$
$\displaystyle{\rm PtCO}\;$ $\displaystyle\dot{N}_{5}=V_{\rm
surf}(k_{1}c_{1}c_{3}-k_{-1}c_{5}-k_{3}c_{4}c_{5})\,$ $\displaystyle{\rm
CO_{2}}$ $\displaystyle\dot{N}_{6}=V_{\rm surf}k_{3}c_{4}c_{5}+V_{\rm
gas}(v_{\rm in}c_{6}^{\rm in}-v_{\rm out}c_{6})\,$
Here $v_{\rm in}$ and $v_{\rm out}$ are the flux rates (per unit volume).
For the slow variables this equation gives:
$\begin{split}&\dot{b}_{1}=2\dot{N}_{2}+\dot{N}_{4}=-V_{\rm
surf}k_{3}c_{4}c_{5}+2V_{\rm gas}(v_{\rm in}c_{2}^{\rm in}-v_{\rm
out}c_{2})\\\ &\dot{b}_{2}=\dot{N}_{1}+\dot{N}_{5}=-V_{\rm
surf}k_{3}c_{4}c_{5}+V_{\rm gas}(v_{\rm in}c_{1}^{\rm in}-v_{\rm out}c_{1})\\\
&\dot{b}_{3}=\dot{N}_{3}+\dot{N}_{4}+\dot{N}_{5}=0\\\ &\dot{N}_{6}=V_{\rm
surf}k_{3}c_{4}c_{5}+V_{\rm gas}(v_{\rm in}c_{6}^{\rm in}-v_{\rm
out}c_{6})\end{split}$ (40)
This system looks quite simple. Only one reaction,
PtO+PtCO$\to$CO2+2Pt (41)
is visible. If we know expressions for $c_{3,5}(b)$ then this reaction rate is
also known. In addition, to work with the rates of fluxes, the expressions for
$c_{1,2}(b)$ are needed.
The system of equations (37) is explicitly solvable but the result is quite
cumbersome. Therefore, let us consider its simplification without explicit
analytic solution. We assume the following smallness:
$b_{1}\gg N_{4}\,,\;\;b_{2}\gg N_{5}\,$ (42)
Together with this smallness assumptions equations (37) give:
$\begin{split}&c_{3}=\frac{b_{3}}{V_{\rm
surf}\left(1+\frac{k_{1}}{k_{-1}}\frac{b_{2}}{V_{\rm
gas}}+\sqrt{\frac{1}{2}\frac{k_{2}}{k_{-2}}\frac{b_{1}}{V_{\rm
gas}}}\right)}\\\
&c_{4}=\sqrt{\frac{1}{2}\frac{k_{2}}{k_{-2}}\frac{b_{1}}{V_{\rm
gas}}}\frac{b_{3}}{V_{\rm surf}\left(1+\frac{k_{1}}{k_{-1}}\frac{b_{2}}{V_{\rm
gas}}+\sqrt{\frac{1}{2}\frac{k_{2}}{k_{-2}}\frac{b_{1}}{V_{\rm
gas}}}\right)}\\\ &c_{5}=\frac{k_{1}}{k_{-1}}\frac{b_{2}}{V_{\rm
gas}}\frac{b_{3}}{V_{\rm surf}\left(1+\frac{k_{1}}{k_{-1}}\frac{b_{2}}{V_{\rm
gas}}+\sqrt{\frac{1}{2}\frac{k_{2}}{k_{-2}}\frac{b_{1}}{V_{\rm
gas}}}\right)}\end{split}$ (43)
In this approximation, we have for the reaction (41) rate
$r=k_{3}c_{4}c_{5}=k_{3}\frac{k_{1}}{k_{-1}}\sqrt{\frac{1}{2}\frac{k_{2}}{k_{-2}}}\frac{\sqrt{b_{1}}b_{2}}{V_{\rm
gas}^{3/2}}\frac{b_{3}^{2}}{V_{\rm
surf}^{2}\left(1+\frac{k_{1}}{k_{-1}}\frac{b_{2}}{V_{\rm
gas}}+\sqrt{\frac{1}{2}\frac{k_{2}}{k_{-2}}\frac{b_{1}}{V_{\rm
gas}}}\right)^{2}}$
This expression gives the closure for the slow QE equations (40).
#### 4.2.4 4.2.3. Discussion of the QE procedure for Chemical Kinetics
We finalize here the illustration of the general QE procedure for chemical
kinetics. As we can see, the simple analytic description of the QE
approximation is available when the fast reactions have no joint reagents. In
general case, we need either a numerical solver for (24) or some additional
hypotheses about smallness. Michaelis and Menten used, in addition to the QE
approach, the hypothesis about smallness of the amount of intermediate
complexes. This is the typical QSS hypothesis. The QE approximation was
modified and further developed by many authors. In particular, a computational
optimization approach for the numerical approximation of slow attracting
manifolds based on entropy-related and geometric extremum principles for
reaction trajectories was developed Lebiedz2010 .
Of course, validity of all the simplification hypotheses is a crucial
question. For example, for the CO oxidation, if we accept the hypothesis about
the quasiequilibrium adsorption then we get a simple dynamics which
monotonically tends to the steady state. The state of the surface is
unambiguously presented as a continuous function of the gas composition. The
pure QSS hypothesis results for the Langmuir-Hinshelwood reaction mechanism
(38) without quasiequilibrium adsorption in bifurcations and the multiplicity
of steady states Yablonskii1991 . The problem of validity of simplifications
cannot be solved as a purely theoretical question without the knowledge of
kinetic constants or some additional experimental data.
## 5 General Kinetics with Fast Intermediates Present in Small Amount
### 5.1 Stoichiometry of Complexes
In this Section, we return to the very general reaction network.
Let us call all the formal sums that participate in the stoichiometric
equations (17), the complexes. The set of complexes for a given reaction
mechanism (17) is $\Theta_{1},\ldots,\Theta_{q}$. The number of complexes
$q\leq 2m$ (two complexes per elementary reaction, as the maximum) and it is
possible that $q<2m$ because some complexes may coincide for different
reactions.
A complex $\Theta_{i}$ is a formal sum
$\Theta_{i}=\sum_{j=1}^{n}\nu_{ij}A_{j}=(\nu_{i},A)$, where $\nu_{i}$ is a
vector with coordinates $\nu_{ij}$.
Each elementary reaction (17) may be represented in the form
$\Theta^{-}_{\rho}\to\Theta^{+}_{\rho}$, where $\Theta^{\pm}_{\rho}$ are the
complexes which correspond to the right and the left sides (17). The whole
mechanism is naturally represented as a digraph of transformation of
complexes: Vertices are complexes and edges are reactions. This graph gives a
convenient tool for the reaction representation and is often called the
“reaction graph”.
Let us consider a simple example: 18 elementary reactions (9 pairs of mutually
reverse reactions) from the hydrogen combustion mechanism (see, for example,
Conaireatal2004 ).
$\begin{array}[]{ll}{\rm H+O_{2}\rightleftharpoons O+OH;}&{\rm
O+H_{2}\rightleftharpoons H+OH;}\\\ {\rm OH+H_{2}\rightleftharpoons
H+H_{2}O;}&{\rm O+H_{2}O\rightleftharpoons 2OH;}\\\ {\rm
HO_{2}+H\rightleftharpoons H_{2}+O_{2};}&{\rm HO_{2}+H\rightleftharpoons
2OH;}\\\ {\rm H+OH+M\rightleftharpoons H_{2}O+M;}&{\rm
H+O_{2}+M\rightleftharpoons HO_{2}+M;}\\\ {\rm H_{2}O_{2}+H\rightleftharpoons
H_{2}+HO_{2}}&\end{array}$ (44)
There are 16 different complexes here:
$\displaystyle{\rm\Theta_{1}=H+O_{2},\,\Theta_{2}=O+OH,\,\Theta_{3}=O+H_{2},\,\Theta_{4}=H+OH,}$
$\displaystyle{\rm\Theta_{5}=OH+H_{2},\rm\Theta_{6}=H+H_{2}O,\,\Theta_{7}=O+H_{2}O,\,\Theta_{8}=2OH,\,}$
$\displaystyle{\rm\Theta_{9}=HO_{2}+H,\rm\Theta_{10}=H_{2}+O_{2},\,\Theta_{11}=H+OH+M,\,}$
$\displaystyle{\rm\Theta_{12}=H_{2}O+M,\,\rm\Theta_{13}=H+O_{2}+M,\,\Theta_{14}=HO_{2}+M,}$
$\displaystyle{\rm\Theta_{15}=H_{2}O_{2}+H,\,\Theta_{16}=H_{2}+HO_{2}}\,$
The reaction set (44) can be represented as
$\displaystyle\Theta_{1}\rightleftharpoons\Theta_{2},\,\Theta_{3}\rightleftharpoons\Theta_{4},\,\Theta_{5}\rightleftharpoons\Theta_{6},\,\Theta_{7}\rightleftharpoons\Theta_{8}\rightleftharpoons\Theta_{9}\leftrightharpoons\Theta_{10},$
$\displaystyle\Theta_{11}\rightleftharpoons\Theta_{12},\,\Theta_{13}\rightleftharpoons\Theta_{14},\,\Theta_{15}\rightleftharpoons\Theta_{16}\,$
We can see that this digraph of transformation of complexes has a very simple
structure: There are five isolated pairs of complexes and one connected group
of four complexes.
### 5.2 Stoichiometry of Compounds
For each complex $\Theta_{j}$ we introduce an additional component $B_{j}$, an
intermediate compound and $B^{\pm}_{\rho}$ are those compounds $B_{j}$ ($1\leq
j\leq q$), which correspond to the right and left sides of reaction (17).
We call these components “compounds” following the English translation of the
original Michaelis-Menten paper MichaelisMenten1913 and keep “complexes” for
the formal linear combinations $\Theta_{j}$.
An extended reaction mechanism includes two types of reactions: Equilibration
between a complex and its compound ($q$ reactions, one for each complex)
$\Theta_{j}\rightleftharpoons B_{j}$ (45)
and transformation of compounds $B_{\rho}^{-}\to B_{\rho}^{+}$ ($m$ reactions,
one for each elementary reaction from (17). So, instead of the reaction (17)
we can write
$\sum_{i}\alpha_{\rho i}A_{i}\rightleftharpoons B_{\rho}^{-}\to
B_{\rho}^{+}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}\,$ (46)
Of course, if the input or output complexes coincide for two reactions then
the corresponding equilibration reactions also coincide.
It is useful to visualize the reaction scheme. In Figure 1 we represent the
$2n$-tail scheme of an elementary reaction sequence (46) which is an extension
of the elementary reaction (17).
The reactions between compounds may have several channels (Figure 2): One
complex may transform to several other complexes.
The reaction mechanism is a set of multichannel transformations (Figure 2) for
all input complexes. In Figure 2 we grouped together the reactions with the
same input complex. Another representation of the reaction mechanism is based
on the grouping of reactions with the same output complex. Below, in the
description of the complex balance condition, we use both representations.
The extended list of components includes $n+q$ components: $n$ initial species
$A_{i}$ and $q$ compounds $B_{j}$. The corresponding composition vector
$N^{\oplus}$ is a direct sum of two vectors, the composition vector for
initial species, $N$, with coordinates $N_{i}$ ($i=1,\ldots,n$) and the
composition vector for compounds, $\Upsilon$, with coordinates $\Upsilon_{j}$
($j=1,\ldots,q$): $N^{\oplus}=N\oplus\Upsilon$.
The space of composition vectors $E$ is a direct sum of $n$-dimensional
$E_{A}$ and $q$-dimensional $E_{B}$: $E=E_{A}\oplus E_{B}$.
For concentrations of $A_{i}$ we use the notation $c_{i}$ and for
concentrations of $B_{j}$ we use $\varsigma_{j}$.
The stoichiometric vectors for reactions $\Theta_{j}\rightleftharpoons B_{j}$
(45) are direct sums: $g^{j}=-\nu_{j}\oplus e_{j}$, where $e_{j}$ is the $j$th
standard basis vector of the space $R^{q}=E_{B}$, the coordinates of $e_{j}$
are $e_{jl}=\delta_{jl}$:
$g^{j}=(-\nu_{j1},-\nu_{j2},\ldots,-\nu_{jn},\underbrace{0,\ldots,0,1}_{l},0,\ldots,0)$
(47)
The stoichiometric vectors of equilibration reactions (45) are linearly
independent because there exists exactly one vector for each $l$.
The stoichiometric vectors $\gamma^{jl}$ of reactions $B_{j}\to B_{l}$ belong
entirely to $E_{B}$. They have $j$th coordinate $-1$, $l$th coordinate $+1$
and other coordinates are zeros.
To exclude some degenerated cases a hypothesis of weak reversibility is
accepted. Let us consider a digraph with vertices $\Theta_{i}$ and edges,
which correspond to reactions from (17). The system is weakly reversible if
for any two vertices $\Theta_{i}$ and $\Theta_{j}$, the existence of an
oriented path from $\Theta_{i}$ to $\Theta_{j}$ implies the existence of an
oriented path from $\Theta_{j}$ to $\Theta_{i}$.
Of course, this weak reversibility property is equivalent to weak
reversibility of the reaction network between compounds $B_{j}$.
Figure 2: A multichannel view on the complex transformation. The hidden
reactions between compounds are included in an oval $\mathbf{S}$.
### 5.3 Energy, Entropy and Equilibria of Compounds
In this section, we define the free energy of the system. The basic hypothesis
is that the compounds are the small admixtures to the system, that is, the
amount of compounds $B_{j}$ is much smaller than amount of initial components
$A_{i}$. Following this hypothesis, we neglect the energy of interaction
between compounds, which is quadratic in their concentrations because in the
low density limit we can neglect the correlations between particles if the
potential of their interactions decay sufficiently fast when the distance
between particles goes to $\infty$ Balescu . We take the energy of their
interaction with $A_{i}$ in the linear approximation, and use the perfect
entropy for $B_{i}$. These standard assumptions for a small admixtures give
for the free energy:
$F=Vf(c,T)+VRT\sum_{j=1}^{q}\varsigma_{j}\left(\frac{u_{j}(c,T)}{RT}+\ln\varsigma_{j}-1\right)$
(48)
Let us introduce the standard equilibrium concentrations for $B_{j}$. Due to
the Boltzmann distribution ($\exp(-u/RT)$) and formula (48)
$\varsigma_{j}^{*}(c,T)=\frac{1}{Z}\exp\left(-\frac{u_{j}(c,T)}{RT}\right)$
(49)
where $1/Z$ is the normalization factor. Let us select here the normalization
$Z=1$ and write:
$F=Vf(c,T)+VRT\sum_{j=1}^{q}\varsigma_{j}\left(\ln\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}(c,T)}\right)-1\right)\
$ (50)
We assume that the standard equilibrium concentrations
$\varsigma_{j}^{*}(c,T)$ are much smaller than the concentrations of $A_{i}$.
It is always possible because functions $u_{j}$ are defined up to an additive
constant.
The formula for free energy is necessary to define the fast equilibria (45).
Such an equilibrium is the minimizer of the free energy on the straight line
parameterized by $a$: $c_{i}=c_{i}^{0}-a\nu_{ji}$, $\varsigma_{j}=a$.
If we neglect the products
$\varsigma_{j}\partial\varsigma_{j}^{*}(c,T)/\partial c_{i}$ as the second
order small quantities then the minimizers have the very simple form:
$\vartheta_{j}=\sum_{i}\nu_{ji}\frac{\mu_{i}(c,T)}{RT}$ (51)
or
$\varsigma_{j}=\varsigma^{*}_{j}(c,T)\exp\left(\frac{\sum_{i}\nu_{ji}\mu_{i}(c,T)}{RT}\right)$
(52)
where
$\mu_{i}=\frac{\partial f(c,T)}{\partial c_{i}}$
is the chemical potential of $A_{i}$ and
$\vartheta_{j}=\ln\left(\frac{\varsigma_{j}}{\varsigma^{*}_{j}}\right)$
($RT\vartheta_{j}=\frac{1}{V}\frac{\partial F}{\partial\varsigma_{j}}$ is the
chemical potential of $B_{j}$).
The thermodynamic equilibrium of the system of reactions $B_{j}\to B_{l}$ that
corresponds to the reactions (46) is the free energy minimizer under given
values of the conservation laws.
For the systems with fixed volume, the stoichiometric conservation laws of the
monomolecular system of reaction are sums of the concentrations of $B_{j}$
which belong to the connected components of the reaction graph. Under the
hypothesis of weak reversibility there is no other conservation law. Let the
graph of reactions $B_{j}\to B_{l}$ have $d$ connected components $C_{s}$ and
let $V_{s}$ be the set of indexes of those $B_{j}$ which belong to $C_{s}$:
$B_{j}\in C_{s}$ if and only if $j\in V_{s}$. For each $C_{s}$ there exists a
stoichiometric conservation law
$\beta_{s}=\sum_{j\in V_{s}}\varsigma_{j}=const$ (53)
For any set of positive values of $\beta_{s}$ ($s=1,\ldots,d$) and given $c,T$
there exists a unique conditional maximizer $\varsigma^{\rm eq}_{j}$ of the
free energy (50): For the compound $B_{j}$ from the $s$th connected component
($j\in V_{s}$) this equilibrium concentration is
$\varsigma^{\rm eq}_{j}=\beta_{s}\frac{\varsigma_{j}^{*}(c,T)}{\sum_{l\in
V_{s}}\varsigma_{j}^{*}(c,T)}$ (54)
The positive values of concentrations $\varsigma_{j}$ are the equilibrium
concentrations (54) for some values of $\beta_{s}$ if and only if for any
$s=1,\ldots,d$ and all $j,l\in V_{s}$
$\vartheta_{j}=\vartheta_{l}$ (55)
($\vartheta_{j}=\ln(\varsigma_{j}/\varsigma_{j}^{*})$). This means that
compounds are in equilibrium in every connected component $C_{s}$ the chemical
potentials of compounds coincide in each component $C_{s}$. The system of
equations (55) together with the equilibrium conditions (52) constitute the
equilibrium of the systems. All the equilibria form a linear subspace in the
space with coordinates $\mu_{i}/RT$ ($i=1,\ldots,n$) and $\vartheta_{j}$
($j=1,\ldots,q$).
In the expression for the free energy (50) we do not assume anything special
about free energy of the mixture of $A_{i}$. The density of this free energy,
$f(c,T)$, may be an arbitrary smooth function (later, we will add the standard
assumption about convexity of $f(c,T)$ as a function of $c$). For the
compounds $B_{i}$, we assume that they form a very small addition to the
mixture of $A_{i}$, neglect all quadratic terms in concentrations of $B_{i}$
and use the entropy of the perfect systems, $p\ln p$, for this small
admixture.
This approach results in the explicit expressions for the fast equilibria (52)
and expression of the equilibrium compound concentrations through the values
of the stoichiometric conservation laws (54).
### 5.4 Markov Kinetics of Compounds
For the kinetics of compounds transformations $B_{j}\to B_{l}$, the same
hypothesis of the smallness of concentrations leads to the only reasonable
assumption: The linear (monomolecular) kinetics with the rate constant
$\kappa_{lj}>0$. This “constant” is a function of $c,T$: $\kappa_{lj}(c,T)$.
The order of indexes at $\kappa$ is inverse to the order of them in reaction:
$\kappa_{lj}=\kappa_{l\leftarrow j}$.
The master equation for the concentration of $B_{j}$ gives:
$\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=\sum_{l,\,l\neq
j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)$ (56)
It is necessary to find when this kinetics respect thermodynamics, i.e., when
the free energy decreases due to the system (56). The necessary and sufficient
condition for matching the kinetics and thermodynamics is: The standard
equilibrium $\varsigma^{*}$ (49) should be an equilibrium for (56), that is,
for every $j=1,\ldots,q$
$\sum_{l,\,l\neq j}\kappa_{jl}\varsigma_{l}^{*}=\sum_{l,\,l\neq
j}\kappa_{lj}\varsigma_{j}^{*}$ (57)
This condition is necessary because the standard equilibrium is the free
energy minimizer for given $c,T$ and
$\sum_{j}\varsigma_{j}=\sum_{j}\varsigma_{j}^{*}$. The sum
$\sum_{j}\varsigma_{j}$ conserves due to (56). Therefore, if we assume that
$F$ decreases monotonically due to (56) then the point of conditional minimum
of $F$ on the plane $\sum_{j}\varsigma_{j}=const$ (under given $c,T$) should
be an equilibrium point for this kinetic system. This condition is sufficient
due to the Morimoto $H$-theorem (see Appendix 2).
For a weakly reversible system, the set of the conditional minimizers of the
free energy (54) coincides with with the set of positive equilibria for the
master equations (56) (see Equation (132) in Appendix 2).
### 5.5 Thermodynamics and Kinetics of the Extended System
In this section, we consider the complete extended system, which consists of
species $A_{i}$ ($i=1,\ldots,n$) and compounds $B_{j}$ ($j=1,\ldots,q$) and
includes reaction of equilibration (45) and transformations of compounds
$B_{j}\to B_{l}$ which correspond to the reactions (46).
Thermodynamic properties of the system are summarized in the free energy
function (50). For kinetics of compounds we accept the Markov model (56) with
the equilibrium condition (57), which guarantees matching between
thermodynamics and kinetics.
For the equilibration reactions (45) we select a very general form of the
kinetic law. The only requirement is: This reaction should go to its
equilibrium, which is described as the conditional minimizer of free energy
$F$ (52). For each reaction $\Theta_{j}\rightleftharpoons B_{j}$ (where the
complex is a formal combination: $\Theta_{j}=\sum_{i}\nu_{ji}A_{i}$) we
introduce the reaction rate $w_{j}$. This rate should be positive if
$\vartheta_{j}<\sum_{i}\nu_{ji}\frac{\mu_{i}(c,T)}{RT}$ (58)
and negative if
$\vartheta_{j}>\sum_{i}\nu_{ji}\frac{\mu_{i}(c,T)}{RT}$ (59)
The general way to satisfy these requirement is to select $q$ continuous
function of real variable $w_{j}(x)$, which are negative if $x>0$ and positive
if $x<0$. For the equilibration rates we take
$w_{j}=w_{j}\left(\vartheta_{j}-\sum_{i}\nu_{ji}\frac{\mu_{i}(c,T)}{RT}\right)$
(60)
If several dynamical systems defined by equations $\dot{x}=J_{1}$, …
$\dot{x}=J_{v}$ on the same space have the same Lyapunov function $F$, then
for any conic combination $J=\sum_{k}a_{k}J_{k}$ ($a_{k}\geq 0$,
$\sum_{k}a_{k}>0$) the dynamical system $\dot{x}=J$ also has the Lyapunov
function $F$.
The free energy (50) decreases monotonically due to any reaction
$\Theta_{j}\rightleftharpoons B_{j}$ with reaction rate $w_{j}$ (60) and also
due to the Markov kinetics (56) with the equilibrium condition (57).
Therefore, the free energy decreases monotonically due to the following
kinetic system:
$\begin{split}&\frac{{\mathrm{d}}c_{i}}{{\mathrm{d}}t}=-\sum_{j=1}^{q}\nu_{ji}w_{j}\,\\\
&\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=w_{j}+\sum_{l,\,l\neq
j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)\end{split}$
(61)
where the coefficients $\kappa_{jl}$ satisfy the matching condition (57).
This general system (61) describes kinetics of extended system and satisfies
all the basic conditions (thermodynamics and smallness of compound
concentrations). In the next sections we will study the QE approximations to
this system and exclude the unknown functions $w_{j}$ from it.
### 5.6 QE Elimination of Compounds and the Complex Balance Condition
In this section, we use the QE formalism developed for chemical kinetics in
Section 4 for simplification of the compound kinetics.
First of all, let us describe $L^{\perp}$, where the space $L$ is the subspace
in the extended concentration space spanned by the stoichiometric vectors of
fast equilibration reactions (45). The stoichiometric vector for the
equilibration reactions have a very special structure (47). Dimension of the
space $L$ is equal to the number of complexes: $\dim L=q$. Therefore,
dimension of $L^{\perp}$ is equal to the number of components $A_{i}$: $\dim
L^{\perp}=n$. For each $A_{i}$ we will find a vector $b_{i}\in L^{\perp}$ that
has the following first $n$ coordinates: $b_{ik}=\delta_{ik}$ for
$k=1,\ldots,n$. The condition $(b_{i},g_{j})=0$ gives immediately:
$b_{i,n+j}=\nu_{ji}$. Finally,
$b_{i}=(\overbrace{\underbrace{0,\ldots,0,1}_{i},0,\ldots,0}^{n},\nu_{1i},\nu_{2i},\ldots,\nu_{qi})$
(62)
The corresponding slow variables are
$b_{i}(c,\varsigma)=c_{i}+\sum_{j}\varsigma_{j}\nu_{ji}$ (63)
In the QE approximation all $w_{j}=0$ and the kinetic equations (61) give in
this approximation
$\frac{{\mathrm{d}}b_{i}}{{\mathrm{d}}t}=\sum_{lj,\,l\neq
j}(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j})\nu_{ji}$ (64)
In these equations, we have to use the dependence $\varsigma(b)$. Here we use
the QSS Michaelis and Menten assumption: The compounds are present in small
amounts
$c_{i}\gg\varsigma_{j}$
In this case, we can take $b_{i}$ instead of $c_{i}$ (i.e., take $\mu(b,T)$
instead of $\mu(c,T)$) in the formulas for equilibria (52):
$\varsigma_{j}=\varsigma^{*}_{j}(b,T)\exp\left(\frac{\sum_{i}\nu_{ji}\mu_{i}(b,T)}{RT}\right)$
(65)
In the final form of the QE kinetic equation there remain two “offprints” of
the compound kinetics: Two sets of functions $\varsigma^{*}_{j}(b,T)\geq 0$
and $\kappa_{jl}(b,T)\geq 0$. These functions are connected by the identity
(57). The final form of the equations is
$\begin{split}\frac{{\mathrm{d}}b_{i}}{{\mathrm{d}}t}=&\sum_{lj,\,l\neq
j}\left(\kappa_{jl}\varsigma^{*}_{l}(b,T)\exp\left(\frac{\sum_{i}\nu_{li}\mu_{i}(b,T)}{RT}\right)-\kappa_{lj}\varsigma^{*}_{j}(b,T)\exp\left(\frac{\sum_{i}\nu_{ji}\mu_{i}(b,T)}{RT}\right)\right)\nu_{ji}\end{split}$
(66)
The identity (57), $\sum_{l,\,l\neq
j}\kappa_{jl}\varsigma_{l}^{*}=\sum_{l,\,l\neq
j}\kappa_{lj}\varsigma_{j}^{*}$, provides a sufficient condition for
decreasing of free energy due to the kinetic equations (66). This is a direct
consequence of two theorem: The theorem about the preservation of entropy
production in the QE approximations (see Section 2 and Appendix 1) and the
Morimoto $H$-theorem (see Appendix 2). Indeed, in the QE state the
equilibrated reactions (45) $\Theta_{j}\rightleftharpoons B_{j}$ do not
produce entropy and all changes in the total free energy are caused by the
Markov kinetics $B_{i}\to B_{j}$. Due to the Morimoto $H$-theorem this change
is negative: The Markov kinetics decrease the perfect free energy of compounds
and do not affect the free energy of $A_{i}$. In the QE approximation, the
concentrations of $A_{i}$ are changing together with concentrations of $B_{j}$
because of the equilibrium conditions for reactions
$\Theta_{j}\rightleftharpoons B_{j}$. Due to the theorem of preservation of
the entropy production, the time derivative of the total free energy in this
QE dynamics coincides with the time derivative of the free energy of $B_{j}$
due to Markov kinetics. In addition to this proof, in Section 6 below we give
the explicit formula for entropy production in (66) and direct proof of its
positivity.
Let us stress that the functions $\varsigma^{*}_{j}(b,T)$ and
$\kappa_{jl}(b,T)$ participate in equations (66) and in identity (57) in the
form of the product. Below we use for this product a special notation:
$\varphi_{jl}(b,T)=\kappa_{jl}(b,T)\varsigma^{*}_{l}(b,T)\;(j\neq l)$ (67)
We call this function $\varphi_{jl}(b,T)$ the kinetic factor. The identity
(57) for the kinetic factor is
$\sum_{l,\,l\neq j}\varphi_{jl}(b,T)=\sum_{l,\,l\neq j}\varphi_{lj}(b,T)\mbox{
for all }j$ (68)
We call the thermodynamic factor (or the Boltzmann factor) the second
multiplier in the reaction rates
$\Omega_{l}(b,T)=\exp\left(\frac{\sum_{i}\nu_{li}\mu_{i}(b,T)}{RT}\right)$
(69)
In this notation, the kinetic equations (66) have a simple form
$\frac{{\mathrm{d}}b_{i}}{{\mathrm{d}}t}=\sum_{lj,\,l\neq
j}(\varphi_{jl}(b,T)\Omega_{l}(b,T)-\varphi_{lj}(b,T)\Omega_{j}(b,T))\nu_{ji}$
(70)
The general equations (70) have the form of “sum over complexes”. Let us
return to the more usual “sum over reactions” form. An elementary reaction
corresponds to the pair of complexes $\Theta_{l},\Theta_{j}$ (46). It has the
form $\Theta_{l}\to\Theta_{j}$ and the reaction rate is
$r=\varphi_{jl}\Omega_{l}$. In the right hand side in (70) this reaction
appears twice: first time with sign “$+$” and the vector coefficient $\nu_{j}$
and the second time with sign “$-$” and the vector coefficient $\nu_{l}$. The
stoichiometric vector of this reaction is $\gamma=\nu_{j}-\nu_{l}$. Let us
enumerate the elementary reactions by index $\rho$, which corresponds to the
pair $(j,l)$. Finally, we transform (46) into the sum over reactions form
$\begin{split}\frac{{\mathrm{d}}b_{i}}{{\mathrm{d}}t}&=\sum_{l,j,\,l\neq
j}\varphi_{jl}(b,T)\Omega_{l}(b,T)(\nu_{ji}-\nu_{li})\\\
&=\sum_{\rho}\varphi_{\rho}(b,T)\Omega_{\rho}(b,T)\gamma_{{\rho}i}\end{split}$
(71)
In the vector form it looks as follows:
$\frac{{\mathrm{d}}b}{{\mathrm{d}}t}=\sum_{\rho}\varphi_{\rho}(b,T)\Omega_{\rho}(b,T)\gamma_{{\rho}}$
(72)
### 5.7 The Big Michaelis-Menten-Stueckelberg Theorem
Let us summarize the results of our analysis in one statement.
Let us consider the reaction mechanism illustrated by Figure 2 (46):
$\sum_{i}\alpha_{\rho i}A_{i}\rightleftharpoons B_{\rho}^{-}\to
B_{\rho}^{+}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}$
under the following asymptotic assumptions:
1. 1.
Concentrations of the compounds $B_{\rho}$ are close to their quasiequilibrium
values (65)
$\varsigma_{j}=(1+\delta)\varsigma_{j}^{\rm
QE}=(1+\delta)\varsigma^{*}_{j}(b,T)\exp\left(\frac{\sum_{i}\nu_{ji}\mu_{i}(b,T)}{RT}\right)\,,\;\;\delta\ll
1$
(this may be due to the fast reversible reactions in (46));
2. 2.
Concentrations of the compounds $B_{\rho}$ are much smaller than the
concentrations of the components $A_{i}$: There is a small positive parameter
$\varepsilon\ll 1$, $\varsigma^{*}_{j}=\varepsilon\xi^{*}_{j}$ and
$\xi^{*}_{j}$ do not depend on $\varepsilon$;
3. 3.
Kinetics of transitions between compounds $B_{i}\to B_{j}$ is linear (Markov)
kinetics with reaction rate constants
$k_{ji}=\frac{1}{\varepsilon}\kappa_{ji}$.
Under these assumptions, in the asymptotic $\delta,\varepsilon\to 0$,
$\delta,\varepsilon>0$ kinetics of components $A_{i}$ may be described by the
reaction mechanism
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}$
with the reaction rates
$r_{\rho}=\varphi_{\rho}\exp\left(\frac{(\alpha_{\rho},{\mu})}{RT}\right)$
where the kinetic factors $\varphi_{\rho}$ satisfy the condition (68):
$\sum_{\rho,\,\alpha_{\rho}=\mathbf{v}}\varphi_{\rho}\equiv\sum_{\rho,\,\beta_{\rho}=\mathbf{v}}\varphi_{\rho}$
for any vector $\mathbf{v}$ from the set of all vectors
$\\{\alpha_{\rho},\beta_{\rho}\\}$. This statement includes the generalized
mass action law for $r_{\rho}$ and the balance identity for kinetic factors
that is sufficient for the entropy growth as it is shown in the next Section
6.
## 6 General Kinetics and Thermodynamics
### 6.1 General Formalism
To produce the general kinetic law and the complex balance conditions, we use
“construction staging”: The intermediate complexes with fast equilibria, the
Markov kinetics and other important and interesting physical and chemical
hypothesis.
In this section, we delete these construction staging and start from the forms
(69), (72) as the basic laws. We use also the complex balance conditions (68)
as a hint for the general conditions which guarantee accordance between
kinetics and thermodynamics.
Let us consider a domain $U$ in $n$-dimensional real vector space $E$ with
coordinates $N_{1},\ldots,N_{n}$. For each $N_{i}$ a symbol (component)
$A_{i}$ is given. A dimensionless entropy (or free entropy, for example,
Massieu, Planck, or Massieu-Planck potential which correspond to the selected
conditions Callen1985 ) $S(N)$ is defined in $U$. “Dimensionless” means that
we use $S/R$ instead of physical $S$. This choice of units corresponds to the
informational entropy ($p\ln p$ instead of $k_{\rm B}p\ln p$).
The dual variables, potentials, are defined as the partial derivatives of $S$:
$\check{\mu}_{i}=-\frac{\partial S}{\partial N_{i}}$ (73)
Warning: This definition differs from the chemical potentials (22) by the
factor ${1}/{RT}$: For constant volume the Massieu-Planck potential is $-F/T$
and we, in addition, divide it on $R$. On the other hand, we keep the same
sign as for the chemical potentials, and this differs from the standard
Legendre transform for $S$. (It is the Legendre transform for function $-S$).
The reaction mechanism is defined by the stoichiometric equations (17)
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}$
($\rho=1,\ldots,m$). In general, there is no need to assume that the
stoichiometric coefficients $\alpha_{\rho i},\beta_{\rho i}$ are integers.
The assumption that they are nonnegative, $\alpha_{\rho i}\geq 0,\beta_{\rho
i}\geq 0$, may be needed to prove that the kinetic equations preserve
positivity of $N_{i}$. If $N_{i}$ is the number of particles then it is a
natural assumption but we can use other extensive variables instead, for
example, we included energy in the list of variables to describe the non-
isothermal processes BykGOrYab1982 . In this case, the coefficient
$\alpha_{U}$ for the energy component $A_{U}$ in an exothermic reaction is
negative.
So, for variables that are positive (bounded from below) by their physical
sense, we will use the inequalities $\alpha_{\rho i}\geq 0,\beta_{\rho i}\geq
0$, when necessary, but in general, for arbitrary extensive variables, we do
not assume positivity of stoichiometric coefficients. As it is usually, the
stoichiometric vector of reaction is
$\gamma_{\rho}=\beta_{\rho}-\alpha_{\rho}$ (the “gain minus loss” vector).
For each reaction, a nonnegative quantity, reaction rate $r_{\rho}$ is
defined. We assume that this quantity has the following structure:
$r_{\rho}=\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})$ (74)
where $(\alpha_{\rho},\check{\mu})=\sum_{i}\alpha_{\rho i}\check{\mu}_{i}$.
In the standard formalism of chemical kinetics the reaction rates are
intensive variables and in kinetic equations for $N$ an additional factor—the
volume—appears. For heterogeneous systems, there may be several “volumes”
(including interphase surfaces).
Each reaction has it own “volume”, an extensive variable $V_{\rho}$ (some of
them usually coincide), and we can write
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=\sum_{\rho}V_{\rho}\gamma_{\rho}\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})$
(75)
In these notations, both the kinetic and the Boltzmann factors are intensive
(and local) characteristics of the system.
Let us, for simplicity of notations, consider a system with one volume, $V$
and write
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}\gamma_{\rho}\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})$
(76)
Below we use the form (76). All our results will hold also for the multi-
volume systems (75) under one important assumption: The elementary reaction
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}\,$
goes in the same volume as the reverse reaction
$\sum_{i}\beta_{\rho i}A_{i}\to\sum_{i}\alpha_{\rho i}A_{i}\,$
or symbolically
$V_{\rho}^{+}=V_{\rho}^{-}$ (77)
If this condition (77) holds then the detailed balance conditions and the
complex balance conditions will hold separately in all volumes $V_{\rho}$.
An important particular case of (76) gives us the Mass Action Law. Let us take
the perfect free entropy
$S=-\sum_{i}N_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)-1\right)$ (78)
where $c_{i}=N_{i}/V\geq 0$ are concentrations and $c_{i}^{*}>0$ are the
standard equilibrium concentrations. Under isochoric conditions, $V=const$,
there is no difference between the choice of the main variables, $N$ or $c$.
For the perfect function (78)
$\check{\mu}_{i}=\ln\left(\frac{c_{i}}{c_{i}^{*}}\right)\,,\;\exp(\alpha_{\rho},\check{\mu})=\prod_{i}\left(\frac{c_{i}}{c_{i}^{*}}\right)^{\alpha_{\rho
i}}$ (79)
and for the reaction rate function (74) we get
$r_{\rho}=\varphi_{\rho}\prod_{i}\left(\frac{c_{i}}{c_{i}^{*}}\right)^{\alpha_{\rho
i}}$ (80)
The standard assumption for the Mass Action Law in physics and chemistry is
that $\varphi$ and $c^{*}$ are functions of temperature:
$\varphi_{\rho}=\varphi_{\rho}(T)$ and $c^{*}_{i}=c^{*}_{i}(T)$. To return to
the kinetic constants notation (20) we should write:
$\frac{\varphi_{\rho}}{\prod_{i}{c_{i}^{*}}^{\alpha_{\rho i}}}=k_{\rho}$
Equation (76) is the general form of the kinetic equation we would like to
study. In many senses, this form is too general before we impose restrictions
on the values of the kinetic factors. For physical and chemical systems,
thermodynamics is a source of restrictions:
1. 1.
The energy of the Universe is constant.
2. 2.
The entropy of the Universe tends to a maximum.
(R. Clausius, 1865 Clausius .)
The first sentence should be extended: The kinetic equations should respect
several conservation laws: Energy, amount of atoms of each kind (if there is
no nuclear reactions in the system) conservation of total probability and,
sometimes, some other conservation laws. All of them have the form of
conservation of values of some linear functionals: $l(N)=const$. If the input
and output flows are added to the system then
$\frac{{\mathrm{d}}l(N)}{{\mathrm{d}}t}=Vv^{\rm in}l^{\rm in}-v^{\rm out}l(N)$
where $v^{\rm in,out}$ are the input and output fluxes per unit volume,
$l^{\rm in}$ are the input densities (concentration). The standard requirement
is that every reaction respects all these conservation laws. The formal
expression of this requirement is:
$l(\gamma_{\rho})=0\mbox{ for all }\rho$ (81)
There is a special term for this conservation laws: The stoichiometric
conservation laws. All the main conservation laws are assumed to be the
stoichiometric ones.
Analysis of the stoichiometric conservation laws is a simple linear algebra
task: We have to find the linear functionals that annulate all the
stoichiometric vectors $\gamma_{\rho}$. In contrast, entropy is not a linear
function of $N$ and analysis of entropy production is not so simple.
In the next subsection we discuss various conditions which guarantee the
positivity of entropy production in kinetic equations (76).
### 6.2 Accordance Between Kinetics and Thermodynamics
#### 6.2.1 6.2.1. General Entropy Production Formula
Let us calculate ${\mathrm{d}}S/{\mathrm{d}}t$ due to equations (76):
$\begin{split}\frac{{\mathrm{d}}S}{{\mathrm{d}}t}&=\sum_{i}\frac{\partial
S}{\partial N_{i}}\frac{{\mathrm{d}}N_{i}}{{\mathrm{d}}t}\\\
&=-\sum_{i}\check{\mu}_{i}V\sum_{\rho}\gamma_{\rho
i}\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})\\\
&=-V\sum_{\rho}(\gamma_{\rho},\check{\mu})\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})\end{split}$
(82)
An auxiliary function $\theta(\lambda)$ of one variable $\lambda\in[0,1]$ is
convenient for analysis of ${\mathrm{d}}S/{\mathrm{d}}t$ (it was studied by
Rozonoer and Orlov OrlovRozonoer1984 , see also G1 :
$\theta(\lambda)=\sum_{\rho}\varphi_{\rho}\exp[(\check{\mu},(\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}))]$
(83)
With this function, the entropy production (82) has a very simple form:
$\frac{{\mathrm{d}}S}{{\mathrm{d}}t}=V\left.\frac{{\mathrm{d}}\theta(\lambda)}{{\mathrm{d}}\lambda}\right|_{\lambda=1}$
(84)
The auxiliary function $\theta(\lambda)$ allows the following interpretation.
Let us introduce the deformed stoichiometric mechanism with the stoichiometric
vectors,
$\alpha_{\rho}(\lambda)=\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}\,,\;\beta_{\rho}(\lambda)=\lambda\beta_{\rho}+(1-\lambda)\alpha_{\rho}$
, which is the initial mechanism when $\lambda=1$, the inverted mechanism with
interchange of $\alpha$ and $\beta$ when $\lambda=0$, the trivial mechanism
(the left and right hand sides of the stoichiometric equations coincide) when
$\lambda=1/2$.
For the deformed mechanism, let us take the same kinetic factors and calculate
the Boltzmann factors with $\alpha_{\rho}(\lambda)$:
$r_{\rho}(\lambda)=\varphi_{\rho}\exp(\alpha_{\rho}(\lambda),\check{\mu})$
In this notation, the auxiliary function $\theta(\lambda)$ is a sum of
reaction rates for the deformed reaction mechanism:
$\theta(\lambda)=\sum_{\rho}r_{\rho}(\lambda)$
In particular, $\theta(1)=\sum_{\rho}r_{\rho}$, this is just the sum of
reaction rates.
Function $\theta(\lambda)$ is convex. Indeed
$\frac{{\mathrm{d}}^{2}\theta(\lambda)}{{\mathrm{d}}\lambda^{2}}=\sum_{\rho}\varphi_{\rho}(\gamma_{\rho},\check{\mu})^{2}\exp[(\check{\mu},(\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}))]\geq
0$
This convexity gives the following necessary and sufficient condition for
positivity of entropy production:
$\frac{{\mathrm{d}}S}{{\mathrm{d}}t}>0\mbox{ if and only if
}\theta(\lambda)<\theta(1)\mbox{ for some }\lambda<1$
In several next subsections we study various important particular sufficient
conditions for positivity of entropy production.
#### 6.2.2 6.2.2. Detailed Balance
The most celebrated condition which gives the positivity of entropy production
is the principle of detailed balance. Boltzmann used this principle to prove
his famous $H$-theorem Boltzmann .
Let us join elementary reactions in pairs:
$\sum_{i}\alpha_{\rho i}A_{i}\rightleftharpoons\sum_{i}\beta_{\rho i}A_{i}$
(85)
After this joining, the total amount of stoichiometric equations decreases. If
there is no reverse reaction then we can add it formally, with zero kinetic
factor. We will distinguish the reaction rates and kinetic factors for direct
and inverse reactions by the upper plus or minus:
$r_{\rho}^{+}=\varphi_{\rho}^{+}\exp(\alpha_{\rho},\check{\mu})\,,\;r_{\rho}^{-}=\varphi_{\rho}^{-}\exp(\beta_{\rho},\check{\mu})\,,\;r_{\rho}=r_{\rho}^{+}-r_{\rho}^{-}$
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}\gamma_{\rho}r_{\rho}$ (86)
In this notation, the principle of detailed balance is very simple: The
thermodynamic equilibrium in the direction $\gamma_{\rho}$, given by the
standard condition $(\gamma_{\rho},\check{\mu})=0$, is equilibrium for the
corresponding pair of mutually reverse reactions from (85). For kinetic
factors this transforms into the simple and beautiful condition:
$\varphi_{\rho}^{+}\exp(\alpha_{\rho},\check{\mu})=\varphi_{\rho}^{-}\exp(\beta_{\rho},\check{\mu})\Leftrightarrow(\gamma_{\rho},\check{\mu})=0$
therefore
$\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$ (87)
For the systems with detailed balance we can take
$\varphi_{\rho}=\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$ and write for the
reaction rate:
$r_{\rho}=\varphi_{\rho}(\exp(\alpha_{\rho},\check{\mu})-\exp(\beta_{\rho},\check{\mu}))$
M. Feinberg called this kinetic law the “Marselin-De Donder” kinetics
Feinberg1972_a . This representation of the reaction rates gives for the
auxiliary function $\theta(\lambda)$:
$\theta(\lambda)=\sum_{\rho}\varphi_{\rho}(\exp[(\check{\mu},(\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}))]+\exp[(\check{\mu},(\lambda\beta_{\rho}+(1-\lambda)\alpha_{\rho}))])$
(88)
Each term in this sum is symmetric with respect to change
$\lambda\mapsto(1-\lambda)$. Therefore, $\theta(1)=\theta(0)$ and, because of
convexity of $\theta(\lambda)$, $\theta^{\prime}(1)\geq 0$. This means
positivity of entropy production.
The principle of detailed balance is a sufficient but not a necessary
condition of the positivity of entropy production. This was clearly explained,
for example, by L. Onsager Onsager1931a ; Onsager1931b . Interrelations
between positivity of entropy production, Onsager reciprocal relations and
detailed balance were analyzed in detail by N.G. van Kampen VKampen1973 .
#### 6.2.3 6.2.3. Complex Balance
The principle of detailed balance gives us $\theta(1)=\theta(0)$ and this
equality holds for each pair of mutually reverse reactions.
Let us start now from the equality $\theta(1)=\theta(0)$. We return to the
initial stoichiometric equations (17) without joining the direct and reverse
reactions. The equality reads
$\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\alpha_{\rho})=\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\beta_{\rho})$
(89)
Exponential functions $\exp(\check{\mu},y)$ form linearly independent family
in the space of functions of $\check{\mu}$ for any finite set of pairwise
different vectors $y$. Therefore, the following approach is natural: Let us
equalize in (89) the terms with the same Boltzmann-type factor
$\exp(\check{\mu},y)$. Here we have to return to the complex-based
representation of reactions (see Section 5.1).
Let us consider the family of vectors $\\{\alpha_{\rho},\beta_{\rho}\\}$
($\rho=1,\ldots,m$). Usually, some of these vectors coincide. Assume that
there are $q$ different vectors among them. Let $y_{1},\ldots,y_{q}$ be these
vectors. For each $j=1,\ldots,q$ we take
$R_{j}^{+}=\\{\rho\,|\,\alpha_{\rho}=y_{j}\\}\,,\;R_{j}^{-}=\\{\rho\,|\,\beta_{\rho}=y_{j}\\}$
We can rewrite the equality (89) in the form
$\sum_{j=1}^{q}\exp(\check{\mu},y_{j})\left[\sum_{\rho\in
R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in R_{j}^{-}}\varphi_{\rho}\right]=0$ (90)
The Boltzmann factors $\exp(\check{\mu},y_{j})$ form the linearly independent
set. Therefore the natural way to meet these condition is: For any
$j=1,\ldots,q$
$\sum_{\rho\in R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in
R_{j}^{-}}\varphi_{\rho}=0$ (91)
This is the general complex balance condition. This condition is sufficient
for entropy growth, because it provides the equality $\theta(1)=\theta(0)$.
If we assume that $\varphi_{\rho}$ are constants or, for chemical kinetics,
depend only on temperature, then the conditions (91) give the general solution
to equation (90).
The complex balance condition is more general than the detailed balance.
Indeed, this is obvious: For the master equation (56) the complex balance
condition is trivially valid for all admissible constants. The first order
kinetics always satisfies the complex balance conditions. On the contrary, the
class of the master equations with detailed balance is rather special. The
dimension of the class of all master equations has dimension $n^{2}-n$
(constants for all transitions $A_{i}\to A_{j}$ are independent). For the
time-reversible Markov chains (the master equations with detailed balance)
there is only $n(n+1)/2-1$ independent constants: $n-1$ for equilibrium state
and $n(n-1)/2$ for transitions $A_{i}\to A_{j}$ ($i>j$), because for reverse
transitions the constant can be calculated through the detailed balance.
In general, for nonlinear reaction systems, the complex balance condition is
not necessary for entropy growth. In the next section we will give a more
general condition and demonstrate that there are systems that violate the
complex balance condition but satisfy this more general inequality.
#### 6.2.4 6.2.4. $G$-Inequality
Gorban G1 proposed the following inequality for analysis of accordance
between thermodynamics and kinetics: $\theta(1)\geq\theta(0)$. This means that
for any values of $\check{\mu}$
$\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\alpha_{\rho})\geq\sum_{\rho}\varphi_{\rho}\exp(\check{\mu},\beta_{\rho})$
(92)
In the form of sum over complexes (similarly to (90)) it has the form
$\sum_{j=1}^{q}\exp(\check{\mu},y_{j})\left[\sum_{\rho\in
R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in R_{j}^{-}}\varphi_{\rho}\right]\geq 0$
(93)
Let us call these inequalities, (92), (93), the $G$-inequalities.
Here, two remarks are needed. First, functions $\exp(\check{\mu},y_{j})$ are
linearly independent but this does not allow us to transform inequalities (93)
similarly to (91) even for constant kinetic factors: Inequality between linear
combinations of independent functions may exist and the “simplified system”
$\sum_{\rho\in R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in
R_{j}^{-}}\varphi_{\rho}\geq 0\mbox{ for all }j$
is not equivalent to the $G$-inequality.
Second, this simplified inequality is equivalent to the complex balance
condition (with equality instead of $\geq$). Indeed, for any $\rho=1,\ldots,m$
there exist exactly one $j_{1}$ and one $j_{2}\neq j_{1}$ with properties:
$\rho\in R_{j_{1}}^{+}$, $\rho\in R_{j_{2}}^{-}$. Therefore, for any reaction
mechanism with reaction rates (74) the identity holds:
$\sum_{\rho}\left[\sum_{\rho\in R_{j}^{+}}\varphi_{\rho}-\sum_{\rho\in
R_{j}^{-}}\varphi_{\rho}\right]=0$
If all terms in this sum are non-negative then all of them are zeros.
Nevertheless, if at least one of the vectors $y_{j}$ is a convex combination
of others,
$\sum_{k,\,k\neq j}\lambda_{k}y_{k}=y_{j}\mbox{ for some }\lambda_{k}\geq
0,\,\sum_{k,\,k\neq j}\lambda_{k}=1$
then the $G$ inequality has more solutions than the condition of complex
balance. Let us take a very simple example with two components, $A_{1}$ and
$A_{2}$, three reactions and three complexes:
$2A_{1}\rightleftharpoons A_{1}+A_{2},\,2A_{2}\rightleftharpoons
A_{1}+A_{2},\,2A_{1}\rightleftharpoons 2A_{2}$
$y_{1}=(2,0),\,y_{2}=(0,2),\,y_{3}=(1,1)\,,$
$R_{1}^{+}=\\{1,3\\},\,R_{2}^{+}=\\{2,-3\\},\,R_{3}^{+}=\\{-1,-2\\}$
$R_{1}^{-}=\\{-1,-3\\},\,R_{2}^{-}=\\{-2,3\\},\,R_{3}^{-}=\\{1,2\\}$
The complex balance condition for this system is
$\begin{split}&(\varphi_{1}-\varphi_{-1})+(\varphi_{3}-\varphi_{-3})=0\,\\\
&(\varphi_{2}-\varphi_{-2})-(\varphi_{3}-\varphi_{-3})=0\end{split}$ (94)
The $G$-inequality for this system is
$\begin{split}&(\varphi_{1}+\varphi_{3}-\varphi_{-1}-\varphi_{-3})a^{2}+(\varphi_{2}+\varphi_{-3}-\varphi_{-2}-\varphi_{3})b^{2}\\\
&+(\varphi_{-1}+\varphi_{-2}-\varphi_{1}-\varphi_{2})ab\geq 0\mbox{ for all
}a,b>0\end{split}$ (95)
(here, $a,b$ stand for $\exp(\check{\mu}_{1}),\,\exp(\check{\mu}_{2})$). Let
us use for the coefficients at $a^{2}$ and $b^{2}$ notations $\psi_{a}$ and
$\psi_{b}$. Coefficient at $ab$ in (95) is $-(\psi_{a}+\psi_{b})$, linear
combinations $\psi_{a}=\varphi_{1}+\varphi_{3}-\varphi_{-1}-\varphi_{-3}$ and
$\psi_{b}=\varphi_{2}+\varphi_{-3}-\varphi_{-2}-\varphi_{3}$ are linearly
independent functions of variables $\varphi_{i}$ ($i=\pm 1,\pm 2,\pm 3$) and
we get the following task: To find all pairs of numbers
$(\psi_{a},\psi_{b})\in\mathbb{R}^{2}$ which satisfy the inequality
$\psi_{a}a^{2}+\psi_{b}b^{2}\geq(\psi_{a}+\psi_{b})ab\;\mbox{ for all }a,b>0$
Asymptotics $a\to 0$ and $b\to 0$ give $\psi_{a},\psi_{b}\geq 0$.
Let us use homogeneity of functions in (95), exclude one normalization factor
from $a,b$ and one factor from $\psi_{a},\psi_{b}$ and reduce the number of
variables: $b=1-a$, $\psi_{a}=1-\psi_{b}$: We have to find all such
$\psi_{b}\in[0,1]$ that for all $a\in]0,1[$
$a^{2}(1-\psi_{b})+(1-a)^{2}\psi_{b}-a(1-a)\geq 0$
The minimizer of this quadratic function of $a$ is
$a_{\min}=\frac{1}{4}+\frac{1}{2}\psi_{b}$, $a_{\min}\in]0,1[$ for all
$\psi_{b}\in[0,1]$. The minimal value is
$-2(\frac{1}{2}\psi_{b}-\frac{1}{4})^{2}$. It is nonnegative if and only if
$\psi_{b}=\frac{1}{2}$. When we return to the non-normalized variables
$\psi_{a},\psi_{b}$ then we get the general solution of the $G$-inequality for
this example: $\psi_{a}=\psi_{b}\geq 0$. For the kinetic factors this means:
$\begin{split}(\varphi_{1}-\varphi_{-1})+2(\varphi_{3}-\varphi_{-3})-(\varphi_{2}-\varphi_{-2})=0\,\\\
(\varphi_{1}-\varphi_{-1})+(\varphi_{3}-\varphi_{-3})\geq 0\,\\\
(\varphi_{2}-\varphi_{-2})-(\varphi_{3}-\varphi_{-3})\geq 0\end{split}$ (96)
These conditions are wider (weaker) than the complex balance conditions for
this example (94).
In the Stueckelberg language Stueckelberg1952 , the microscopic reasons for
the $G$-inequality instead of the complex balance (68) can be explained as
follows: Some channels of the scattering are unknown (hidden), hence, instead
of unitarity of $S$-matrix (conservation of the microscopic probability) we
have an inequality (the microscopic probability does not increase).
We can use other values of $\lambda_{0}\in[0,1[$ in inequality
$\theta(1)\geq\theta(\lambda_{0})$ and produce constructive sufficient
conditions of accordance between thermodynamics and kinetics. For example,
condition $\theta(1)\geq\theta(1/2)$ is weaker than $\theta(1)\geq\theta(0)$
because of convexity $\theta(\lambda)$.
One can ask a reasonable question: Why we do not use directly positivity of
entropy production ($\theta^{\prime}(1)\geq 0$) instead of this variety of
sufficient conditions. Of course, this is possible, but inequalities like
$\theta(1)\geq\theta(0)$ or equations like $\theta(1)=\theta(0)$ include
linear combinations of exponents of linear functions and often can be
transformed in algebraic equations or inequalities like in the example above.
Inequality $\theta^{\prime}(1)\geq 0$ includes transcendent functions like
$f\exp f$ (where $f$ is a linear function) which makes its study more
difficult.
## 7 Linear Deformation of Entropy
### 7.1 If Kinetics Does not Respect Thermodynamics then Deformation of
Entropy May Help
Kinetic equations in the general form (76) are very general, indeed. They can
be used for the approximation of any continuous time dynamical system on
compact $U$ Ocherki . In previous sections we demonstrated how to construct
the system in the form (76) with positivity of the entropy production when the
entropy function is given.
Let us consider a reverse problem. Assume that a system in the form (76) is
given but the entropy production is not always positive. How to find a new
entropy function for this system to guarantee the positivity of entropy
production?
Existence of such an entropy is very useful for analysis of stability of the
system. For example, let us take an arbitrary Mass Action Law system (80).
This is a rather general system with the polynomial right hand side. Its
stability or instability is not obvious a priori. It is necessary to check
whether bifurcations of steady states, oscillations and other interesting
effects of dynamics are possible for this system.
With the positivity of entropy productions these questions are much simpler
(for application of thermodynamic potentials to stability analysis see, for
example, Yablonskii1991 ; Ocherki ; VolpertKhudyaev1985 ; HangosAtAl1999 ). If
${\mathrm{d}}S/{\mathrm{d}}t\geq 0$ and it is zero only in steady states, then
any motion in compact $U$ converges to a steady state and all the non-
wandering points are steady states. (A non-wandering point is such a point
$x\in U$ that for any $T>0$ and $\varepsilon>0$ there exists such a motion
$c(t)\in U$ that (i) $\|c(0)-x\|<\varepsilon$ and (ii)
$\|c(T^{\prime})-x\|<\varepsilon$ for some $T^{\prime}>T$: A motion returns in
an arbitrarily small vicinity of $x$ after an arbitrarily long time.)
Moreover, the global maximizer of $S$ in $U$ is an asymptotically stable
steady state (at least, locally). It is a globally asymptotically stable point
if there is no other steady state in $U$.
For the global analysis of an arbitrary system of differential equations, it
is desirable either to construct a general Lyapunov function or to prove that
it does not exist. For the Lyapunov functions of the general form this task
may be quite difficult. Therefore, various finite-dimensional spaces of trial
functions are often in use. For example, quadratic polynomials of several
variables provide a very popular class of trial Lyapunov function.
In this section, we discuss the $n$-parametric families of Lyapunov functions
which are produced by the addition of linear function to the entropy:
$S(N)\mapsto S_{\Delta\check{\mu}}(N)=S(N)-\sum_{i}\Delta\check{\mu}_{i}N_{i}$
(97)
The change in potentials $\check{\mu}$ is simply the addition of
$\Delta\check{\mu}$:
$\check{\mu}_{i}\mapsto\check{\mu}_{i}+\Delta\check{\mu}_{i}$.
Let us take a general kinetic equation (76). We are looking for a
transformation that does not change the reaction rates. The Boltzmann factor
$\Omega_{\rho}=\exp(\check{\mu},\alpha_{\rho})$ transforms due to the change
of the entropy:
$\Omega_{\rho}\mapsto\Omega_{\rho}\exp(\Delta\check{\mu},\alpha_{\rho})$.
Therefore, to preserve the reaction rate, the transformation of the kinetic
factors should be
$\varphi_{\rho}\mapsto\varphi_{\rho}\exp(\Delta\check{\mu},\alpha_{\rho})$ in
order to keep the product $r_{\rho}=\Omega_{\rho}\varphi_{\rho}$ constant.
For the new entropy, $S=S_{\Delta\check{\mu}}$, with the new potential and
kinetic factors, the entropy production is given by (98):
$\begin{split}\frac{{\mathrm{d}}S}{{\mathrm{d}}t}=&-\sum_{\rho}(\gamma_{\rho},\check{\mu})\varphi_{\rho}\exp(\alpha_{\rho},\check{\mu})\\\
=&-\sum_{\rho}(\gamma_{\rho},\check{\mu}^{\rm
old}+\Delta\check{\mu})\varphi_{\rho}^{\rm
old}\exp(\alpha_{\rho},\check{\mu}^{\rm old})\\\ =&\frac{{\mathrm{d}}S^{\rm
old}}{{\mathrm{d}}t}-\sum_{\rho}(\gamma_{\rho},\Delta\check{\mu})\varphi_{\rho}^{\rm
old}\exp(\alpha_{\rho},\check{\mu}^{\rm old})\,\end{split}$ (98)
where the superscript “old” corresponds to the non-deformed quantities.
### 7.2 Entropy Deformation for Restoration of Detailed Balance
It may be very useful to find such a vector $\Delta\check{\mu}$ that in new
variables $\varphi_{\rho}^{+}=\varphi_{\rho}^{-}$. For the analysis of the
detailed balance condition, we group reactions in pairs of mutually inverse
reactions (85). Let us consider an equation of the general form (86) with
$r_{\rho}=r_{\rho}^{+}-r_{\rho}^{-}$, $\varphi_{\rho}^{\pm}>0$.
The problem is: To find such a vector $\Delta\check{\mu}$ that
$\varphi_{\rho}^{+}\exp{(\Delta\check{\mu},\alpha_{\rho})}=\varphi_{\rho}^{-}\exp{(\Delta\check{\mu},\beta_{\rho})}$
(99)
or, in the equivalent form of the linear equation
$(\Delta\check{\mu},\gamma_{\rho})=\ln\left(\frac{\varphi_{\rho}^{+}}{\varphi_{\rho}^{-}}\right)$
(100)
The necessary and sufficient conditions for the existence of such
$\Delta\check{\mu}$ are known from linear algebra: For every set of numbers
$a_{\rho}$ (${\rho}=1,\ldots,m$)
$\sum_{\rho}a_{\rho}\gamma_{\rho}=0\Rightarrow\sum_{\rho}a_{\rho}\ln\left(\frac{\varphi_{\rho}^{+}}{\varphi_{\rho}^{-}}\right)=0$
(101)
To check these conditions, it is sufficient to find a basis of solutions of
the uniform systems of linear equations
$\sum_{\rho}a_{\rho}\gamma_{\rho i}=0\;\;(i=1,\ldots,m)$
(that is, to find a basis of the left kernel of the matrix $\Gamma$, ${\rm
coim}\Gamma$, where $\Gamma=(\gamma_{\rho i})$) and then check for these basis
vectors the condition
$\sum_{\rho}a_{\rho}\ln\left(\frac{\varphi_{\rho}^{+}}{\varphi_{\rho}^{-}}\right)=0$
to prove or disprove that the vector with coordinates
$\ln\left(\frac{\varphi_{\rho}^{+}}{\varphi_{\rho}^{-}}\right)$ belongs to the
image of $\Gamma$, ${\rm im}\Gamma$.
For some of the reaction mechanisms it is possible to restore the detailed
balance condition for the general kinetic equation unconditionally. For these
reactions, for any set of positive kinetic factors, there exists such a vector
$\Delta\check{\mu}$ that the detailed balance condition (100) is valid for the
deformed entropy. According to (101) this means that there is no nonzero
solution $a_{\rho}$ for the equation $\sum_{\rho}a_{\rho}\gamma_{\rho}=0$. In
other words, vectors $\gamma_{\rho}$ are independent.
### 7.3 Entropy Deformation for Restoration of Complex Balance
The complex balance conditions (91) are, in general, weaker than the detailed
balance but they are still sufficient for the entropy growth.
Let us consider an equation of the general form (86). We need to find such a
vector $\Delta\check{\mu}$ that in new variables with the new entropy and
kinetic factors the complex balance conditions $\sum_{\rho\in
R_{j}^{+}}\varphi^{\rm new}_{\rho}-\sum_{\rho\in R_{j}^{-}}\varphi^{\rm
new}_{\rho}=0$ hold.
For our purpose, it is convenient to return to the presentation of reactions
as transitions between complexes. The complexes,
$\Theta_{1},\ldots,\Theta_{q}$ are the linear combinations,
$\Theta_{j}=(y_{j},A)$.
Each elementary reaction (17) with the reaction number $\rho$ may be
represented in the form $\Theta_{j}\to\Theta_{l}$, where $\Theta_{j}=\sum
y_{j}A_{j}$, $\rho\in R_{j}^{+}$ ($\alpha_{\rho}=y_{j}$) and $\rho\in
R_{j}^{-}$ ($\beta_{\rho}=y_{l}$). For this reaction, let us use the notation
$\varphi_{\rho}=\varphi_{lj}$. We used this notation in the analysis of
kinetics of compounds (Section 5.6). The complex balance conditions are
$\sum_{j,\,j\neq l}(\varphi_{lj}-\varphi_{jl})=0$ (102)
To obtain these conditions after the entropy deformation, we have to find such
$\Delta\check{\mu}$ that
$\sum_{j,\,j\neq
l}(\varphi_{lj}\exp{(\Delta\check{\mu},y_{j})}-\varphi_{jl}\exp{(\Delta\check{\mu},y_{l})})=0$
(103)
This is exactly the equation for equilibrium of a Markov chain with transition
coefficients $\varphi_{lj}$. Vector $(\Delta\check{\mu},y_{j})$ should be an
equilibrium state for this chain (without normalization to the unit sum of
coordinates).
For this finite Markov chain a graph representation is useful: Vertices are
complexes and oriented edges are reactions. To provide the existence of a
positive equilibrium we assume weak reversibility of the chain: If there
exists an oriented path from $\Theta_{j}$ to $\Theta_{l}$ then there exists an
oriented path from $\Theta_{l}$ to $\Theta_{j}$.
Let us demonstrate how to transform this problem of entropy deformation into a
linear algebra problem. First of all, let us find any positive equilibrium of
the chain, $\varsigma^{*}_{j}>0$:
$\sum_{j,\,j\neq
l}(\varphi_{lj}\varsigma^{*}_{j}-\varphi_{jl}\varsigma^{*}_{l})=0$ (104)
This is a system of linear equations. If we have already an arbitrary
equilibrium of the chain then other equilibria allow a very simple
description. We already found this description for kinetics of compounds (54)
Let us consider the master equation for the Markov chain with coefficients
$\varphi_{lj}$ and apply the formalism from Appendix 2:
$\frac{{\mathrm{d}}\varsigma}{{\mathrm{d}}t}=\sum_{j,\,j\neq
l}(\varphi_{lj}\varsigma_{j}-\varphi_{jl}\varsigma_{l})=0$ (105)
Let the graph of complex transformations $\Theta_{j}\to\Theta_{l}$ have $d$
connected components $C_{s}$ and let $V_{s}$ be the set of indexes of those
$\Theta_{j}$ which belong to $C_{s}$: $\Theta_{j}\in C_{s}$ if and only if
$j\in V_{s}$. For each $C_{s}$ there exists a conservation law
$\beta_{s}(\varsigma)$ for the master equation (53),
$\beta_{s}(\varsigma)=\sum_{j\in V_{s}}\varsigma_{j}$.
For any set of positive values of $\beta_{s}$ ($s=1,\ldots,q$) there exists a
unique equilibrium vector $\varsigma^{\rm eq}$ for (105) with this values
$\beta_{s}$ (54), (132). The set of equilibria is a linear space with the
natural coordinates $\beta_{s}$ ($s=1,\ldots,d$). We are interested in the
positive orthant of this space, $\beta_{s}>0$. For positive $\beta_{s}$,
logarithms of $\varsigma^{\rm eq}$ form a $d$-dimensional linear manifold in
$R^{q}$ (133). The natural coordinates on this manifold are $\ln\beta_{s}$.
Let us notice that the vector $\varsigma^{\circ}$ with coordinates
$\varsigma^{\circ}_{j}=\left(\frac{\varsigma_{j}^{*}(c,T)}{\sum_{l\in
V_{s}}\varsigma_{l}^{*}(c,T)}\right)\mbox{ for }j\in V_{s}$
is also an equilibrium for (105). This equilibrium is normalized to unit
values of all $\beta_{s}(\varsigma^{\circ})$. In the coordinates
$\ln\beta_{s}$ this is the origin. The equations for $\Delta\check{\mu}$ are
$(\Delta\check{\mu},y_{j})-\ln\beta_{s}=\ln\varsigma^{\circ}_{j}\mbox{ for
}j\in V_{s}$ (106)
This is a system of linear equations with respect to $n+d$ variables
$\Delta\check{\mu}_{i}$ ($i=1,\ldots,n$) and $\ln\beta_{s}$ ($s=1,\ldots,d$).
Let the coefficient matrix of this system be denoted by $\mathbf{M}$.
Analysis of solutions and solvability of such equations is one of the standard
linear algebra tasks. If this system has a solution then the complex balance
in the original system can be restored by the linear deformation of the
entropy. If this system is solvable for any right hand side, then for this
reaction mechanism we always can find the entropy, which provides the complex
balance condition.
Unconditional solvability of (106) means that the left hand side matrix of
this system has rank $q$. Let us express this rank through two important
characteristics: It is ${\rm rank}\\{\gamma_{1},\ldots,\gamma_{m}\\}+d$, where
$d$ is the number of connected components in the graph of transformation of
complexes.
To prove this formula, let us write down the matrix $\mathbf{M}$ of the system
(106). First, we change the enumeration of complexes. We group the complexes
from the same connected component together and arrange these groups in the
order of the connected component number. After this change of enumeration,
$\\{1,\ldots,|V_{1}|\\}=V_{1}$,
$\\{|V_{1}|+1,\ldots,|V_{1}|+|V_{2}|\\}=V_{2}$, …,
$\\{|V_{1}|+|V_{2}|+\ldots+|V_{d-1}|+1,\ldots,|V_{1}|+|V_{2}|+\ldots+|V_{d}|\\}=V_{d}$.
Let $y_{j}$ be here the row vector. The matrix is
$\mathbf{M}=\left[\begin{array}[]{lcccc}y_{1}&1&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ y_{|V_{1}|}&1&0&\ldots&0\\\
y_{|V_{1}|+1}&0&1&\ldots&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
y_{|V_{1}|+\ldots+|V_{d}|}&0&0&\ldots&1\end{array}\right]$ (107)
$\mathbf{M}$ consists of $d$ blocks $\mathbf{M}_{s}$, which correspond to
connected components $C_{s}$ of the graph of transformation of complexes:
$\mathbf{M}_{s}\left[\begin{array}[]{lcccc}y_{|V_{1}|+\ldots+|V_{s-1}|+1}&0&\ldots&1&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
y_{|V_{1}|+\ldots+|V_{s}|}&0&\ldots&1&\ldots\end{array}\right]$ (108)
The first $n$ columns in this matrix are filled by the vectors $y_{j}$ of
complexes, which belong to the component $C_{s}$, then follow $s-1$ columns of
zeros, after that, there is one column of units, and then again zeros. Here,
in (107), (108) we multiplied the last $d$ columns by $-1$. This operation
does not change the rank of the matrix.
Other elementary operations that do not change the rank are: We can add to any
row (column) a linear combination of other rows (columns).
We will use these operations to simplify blocks (108) but first we have to
recall several properties of spanning trees WuChao2004 . Let us consider a
connected, undirected graph $G$ with the set of vertices $\mathcal{V}$ and the
set of edges $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$. A spanning tree
of $G$ is a selection of edges of $G$ that form a tree spanning every vertex.
For a connected graph with $V$ vertices, any spanning tree has $V-1$ edges.
Let for each vertex $\Theta_{j}$ of $G$ a $n$-dimensional vector $y_{i}$ is
given. Then for every edge $(\Theta_{j},\Theta_{l})\in\mathcal{E}$ a vector
$\gamma_{jl}=y_{j}-y_{l}$ is defined. We identify vectors $\gamma$ and
$-\gamma$ and the order of $j,l$ is not important. Let us use $\Gamma_{G}$ for
this set of $\gamma_{jl}$:
$\Gamma_{G}=\\{y_{j}-y_{l}\,|\,(\Theta_{j},\Theta_{l})\in\mathcal{E}\\}$
For any spanning tree $T$ of graph $G$ we have the following property:
${\rm span}\Gamma_{G}={\rm span}\Gamma_{T}$ (109)
in particular, ${\rm rank}\Gamma_{G}={\rm rank}\Gamma_{T}$.
For the digraphs of reactions between complexes, we create undirected graphs
just by neglecting the directions of edges. We keep for them the same
notations as for original digraphs. Let us select any spanning tree $T_{s}$
for the connected component $C_{s}$ in the graph of transformation of
complexes. In $T_{s}$ we select arbitrarily a root complex. After that, any
other complex $\Theta_{j}$ in $C_{s}$ has a unique parent. This is the vertex
connected to it on the path to the root. For the root complex of $C_{s}$ we
use special notation $\Theta_{s}^{\circ}$.
Now, we transform the block (108) without change of rank: For each non-root
complex we subtract from the corresponding row the row which correspond to its
unique parent. After these transformations (and, may be, some permutations of
rows), the block $\mathbf{M}_{s}$ get the following form:
$\left[\begin{array}[]{lcccc}\gamma^{s}_{1}&0&\ldots&0&\ldots\\\
\gamma^{s}_{2}&0&\ldots&0&\ldots\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
\gamma^{s}_{|V_{1}|-1}&0&\ldots&0&\ldots\\\
y_{s}^{\circ}&0&\ldots&1&\ldots\end{array}\right]$ (110)
Here, $\\{\gamma^{s}_{1},\gamma^{s}_{2},\ldots,\gamma^{s}_{|V_{1}|-1}\\}$ is
$\Gamma_{T_{s}}$ for the spanning tree $T_{s}$ and $y_{s}^{\circ}$ is the
coefficient vector for the root complex $\Theta_{s}^{\circ}$.
From the obtained structure of blocks we immediately find that the rank of the
rows with $\gamma$ is ${\rm rank}\\{\gamma_{1},\ldots,\gamma_{m}\\}+d$ due to
(109). Additional $d$ rows with $y_{s}^{\circ}$ are independent due to their
last coordinates and add $d$ to rank. Finally
${\rm rank}\mathbf{M}={\rm rank}\\{\gamma_{1},\ldots,\gamma_{m}\\}+d$ (111)
Obviously, ${\rm rank}\mathbf{M}\leq q$.
In particular, from the formula (111) immediately follows the description of
the reaction mechanisms, for which it is always possible to restore the
thermodynamic properties by the linear deformation of the entropy.
The deficiency zero theorem. If ${\rm rank}\mathbf{M}=q$ then it is always
possible to restore the positivity of the entropy production by the linear
deformation of the entropy.
Feinberg Feinberg1972 called the difference $q-{\rm rank}\mathbf{M}$ the
deficiency of the reaction network. For example, for the “Michaelis-Menten”
reaction mechanism $E+S\rightleftharpoons ES\rightleftharpoons P+S$ ${\rm
rank}\\{\gamma_{1},\gamma_{2}\\}=2$, $d=1$, $q=3$, ${\rm rank}\mathbf{M}=3$
and deficiency is 0.
For the adsorption (the Langmuir-Hinshelwood) mechanism of CO oxidation (38)
${{\rm rank}\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}~{}=~{}3}$, $d=3$, $q=6$,
${\rm rank}\mathbf{M}=6$ and deficiency is 0. To apply the results about the
entropy deformation to this reaction mechanism, it is necessary to introduce
an inverse reaction to the third elementary reaction in (38),
PtO+PtCO$\to$CO2+2Pt with an arbitrarily small but positive constant in order
to make the mechanism weakly reversible.
Let us consider the Langmuir-Hinshelwood mechanism for reduced list of
components. Let us assume that the gas concentrations are constant because of
control or time separation or just as a model “fast” system and just include
them in the reaction rate constants for intermediates. Then the mechanism is
2Pt$\rightleftharpoons$2PtO, Pt$\rightleftharpoons$PtCO, PtO+PtCO$\to$2Pt. For
this system, ${\rm rank}\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}=2$, $d=2$,
$q=5$, ${\rm rank}\mathbf{M}=4$ and deficiency is 1. Bifurcations in this
system are known Yablonskii1991 .
For the fragment of the reaction mechanism of the hydrogen combustion (44),
${\rm rank}\\{\gamma_{1},\ldots,\gamma_{m}\\}=6$, $d=7$, $q=16$, ${\rm
rank}\mathbf{M}=6+7=13$ and deficiency is 3.
### 7.4 Existence of Points of Detailed and Complex Balance
Our formulation of the conditions of detailed and complex balance is not
standard: We formulate them as the identities (87) and (89). These identities
have a global nature and describe the properties of reaction rates for all
states.
The usual approach to the principle of detailed balance is based on
equilibria. The standard formulation is: In all equilibria every process is
balanced with its reverse process. Without special forms of kinetic law this
principle cannot have any consequences for global dynamics. This is a trivial
but not widely known fact. Indeed, let a system $\dot{c}=F(c)$ be given in a
domain $U\subset\mathbb{R}^{n}$ and $\gamma_{1},\ldots,\gamma_{n}$ is an
arbitrary basis in $\mathbb{R}^{n}$. In this basis, we can always write:
$F(c)=\sum_{\rho=1}^{n}r_{\rho}(c)\gamma_{\rho}$. For any equilibrium $c^{*}$,
$r_{\rho}(c^{*})=0$. All the “reaction rates” $r_{\rho}(c)$ vanish
simultaneously. This “detailed balance” means nothing for dynamics because $F$
is an arbitrary vector field. Of course, if the system of vectors
$\\{\gamma_{\rho}\\}$ is not a basis but any complete system of vectors then
such “detailed balance” conditions, $r_{\rho}(c^{*})=0$, also do not imply any
specific features of dynamics without special hypotheses about functions
$r_{\rho}(c)$.
Nevertheless, if we fix the kinetic law then the consequences may be very
important. For example, if kinetics of elementary reactions follow the Mass
Action law then the existence of a positive equilibrium with detailed balance
implies existence of the Lyapunov function in the form of the perfect free
entropy:
$Y=-\sum_{i}c_{i}\left(\ln\left(\frac{c_{i}}{c^{*}_{i}}\right)-1\right)$
where $c_{i}^{*}$ is that positive equilibrium with detailed balance (see, for
example, Yablonskii1991 ).
In this section we demonstrate that for the general kinetic law (74), which
gives the expression of reaction rates through the entropy gradient, if the
kinetic factors are constant (or a function of temperature) then the existence
of the points of detailed (or complex) balance means that the linear
deformation of the entropy exists which restores the global detailed (or
complex) balance conditions (87) (or (89)).
The condition that the kinetic factors are constant means that for a given set
of values $\\{\varphi_{\rho}\\}$ a state with any admissible values of
$\check{\mu}$ is physically possible (admissible). This condition allows us to
vary the potentials $\check{\mu}$ independently of $\\{\varphi_{\rho}\\}$.
Let us assume that for the general kinetic system with the elementary reaction
rates given by (74) a point of detailed balance exists. This means that for
some value of $\check{\mu}=\check{\mu}^{*}$ (the detailed balance point in the
Legendre transform) and for all ${\rho}$ $r_{\rho}^{+}=r_{\rho}^{-}$:
$\varphi_{\rho}^{+}\exp(\alpha_{\rho},\check{\mu}^{*})=\varphi_{\rho}^{-}\exp(\alpha_{\rho},\check{\mu}^{*})$
This formula is exactly the condition (99) of existence of $\Delta\check{\mu}$
which allow us to deform the entropy for restoring the detailed balance in the
global form (87).
If we assume that the point of complex balance exists then there exists such a
value of $\check{\mu}=\check{\mu}^{*}$ (a point of complex balance in the
Legendre transform) that
$\sum_{j,\,j\neq
l}(\varphi_{lj}\exp{(\check{\mu}^{*},y_{j})}-\varphi_{jl}\exp{(\check{\mu}^{*},y_{l})})=0$
This is exactly the deformation condition (103) with
$\Delta\check{\mu}=\check{\mu}^{*}$.
To prove these statements we used an additional condition about possibility to
vary $\check{\mu}$ under given $\\{\varphi_{\rho}\\}$.
So, we demonstrated that for the general kinetic law (74) the existence of a
point of detailed balance is equivalent to the existence of such linear
deformation of the entropy that the global condition (87) holds. Analogously,
the existence of a point of complex balance is equivalent to the global
condition of complex balance after some linear deformation of the entropy.
### 7.5 The Detailed Balance is Needed More Often than the Complex Balance
The complex balance conditions are mathematically nice and more general than
the principle of the detailed balance. They are linked by Stueckelberg to the
Markov models (“$S$-matrix models”) of microscopic kinetics. Many systems
satisfy these conditions (after linear deformation of the entropy) just
because of the algebraic structure of the reaction mechanism (see Section
7.3). Nevertheless, it is used much less than the classical detailed balance.
The reason for the rare use of complex balance is simple: It is less popular
because the stronger condition, the principle of detailed balance, is valid
for most of physical and chemical systems. Onsager revealed the physical
reason for detailed balance Onsager1931a ; Onsager1931b . This is
microreversibility: The microscopic laws of motion are invertible in time: If
we observe the microscopic dynamics of particles in the backward movie then we
cannot find the difference from the real world. This difference occurs in the
macroscopic world.
In microphysics and the $S$-matrix theory this microreversibility property has
the name “$T$-invariance”.
Let us demonstrate how $T$-invariance in micro-world implies detailed balance
in macro-world.
Following Gibbs, we accept the ensemble-based point of view on the macroscopic
states: They are probability distributions in the space of detailed
microscopic states.
First of all, we assume that under given values of conservation laws
equilibrium state exists and is unique.
Second assumption is that the rates of elementary processes are
microscopically observable quantities. This means that somebody (a “demon”),
who observes all the events in the microscopical world can count the rates of
elementary reactions.
Because of $T$-invariance and uniqueness of equilibrium, the equilibrium is
$T$-invariant: If we change all the microscopic time derivatives (velocities)
$v$ to $-v$ then nothing will change.
$T$-transformation changes all reactions to the reverse reactions, just by
reversion of arrows, but the number of the events remains the same: Any
reaction transforms into its reverse reaction but does not change the reaction
rate. This can be formulated also as follows: $T$-transformation maps all
$r_{\rho}^{+}$ into the corresponding $r_{\rho}^{-}$.
Hence, because of the $T$-invariance, the equilibrium rate of each reaction is
equal to the equilibrium rate of the reverse reaction.
The violation of uniqueness of equilibrium for given values of conservation
laws seems improbable. Existence of several equilibria in thermodynamics is
quite unexpected for homogeneous systems but requires more attention for the
systems with phase separation. Nevertheless, if we assume that a multi-phase
system consists of several homogeneous phases, and each of these phases is in
uniform equilibrium, then we return to the previous assumption (with some
white spots for non-uniform interfaces).
$T$-invariance may be violated if the microscopic description is not
reversible in time. Magnetic field and the Coriolis force are the classical
examples for violation of the microscopic reversibility. In a linear
approximation near equilibrium the corresponding modification of the Onsager
relations give the Onsager-Casimir relations Casimir1945 . There are several
attempts for nonlinear formulation of the Onsager-Casimir relations (see
Grmela1993 ).
The principle of detailed balance seems to be still the best nonlinear version
of the Onsager relations for $T$-invariant systems, and the conditions of the
complex balance seem to give the proper relations between kinetic coefficients
in the absence of the microscopic reversibility for nonlinear systems. It is
important to mention here that all these relations are used together with the
general kinetic law (74).
Observability of the rates of elementary reactions deserves a special study.
Two approaches to the reaction rate are possible. If we accept that the
general kinetic law (74) is valid then we can find the kinetic factors by
observation of ${\mathrm{d}}c/{\mathrm{d}}t$ in several points because the
Boltzmann factors are linearly independent. In this sense, they are observable
but one can claim the approximation point of view and state that the general
kinetic law (74) without additional conditions on kinetic factors is very
general and allows to approximate any dynamical system. From this point of
view, kinetic coefficients are just some numbers in the approximation
algorithm and are not observable. This means that there is no such a
microscopic thing as the rate of elementary reaction, and the set of reactions
serves just for the approximation of the right hand side of the kinetic
equation. We cannot fully disprove this point of view but can just say that in
some cases the collision-based approach with physically distinguished
elementary reactions is based on the solid experimental and theoretical
background. If the elementary reactions physically exist then the detailed
balance for $T$-invariant systems is proved.
## 8 Conclusions
We present the general formalism of the Quasiequilibrium approximation (QE)
with the proof of the persistence of entropy production in the QE
approximation (Section 2).
We demonstrate how to apply this formalism to chemical kinetics and give
several examples for the Mass Action law kinetic equation. We discuss the
difference between QE and Quasi-Steady-State (QSS) approximations and analyze
the classical Michaelis-Menten and Briggs-Haldane model reduction approaches
(Section 3). After that, we use ideas of Michaelis, Menten and Stueckelberg to
create a general approach to kinetics.
Let us summarize the main results of our discussion. First of all, we believe
that this is the finish of the Michaelis-Menten-Stueckelberg program. The
approach to modeling of the reaction kinetics proposed by Michaelis and Menten
in 1913 MichaelisMenten1913 for enzyme reactions was independently in 1952
applied by Stueckelberg Stueckelberg1952 to the Boltzmann equation.
The idea of the complex balance (cyclic balance) relations was proposed by
Boltzmann as an answer to the Lorentz objections against Boltzmann’s proof of
the $H$-theorem. Lorentz stated that the collisions of the polyatomic
molecules may have no inverse collisions. Cercignani and Lampis
CercignaniLamp1981 demonstrated that the Boltzmann $H$-theorem based on the
detailed balance conditions is valid for the polyatomic molecules under the
microreversibility conditions and this new Boltzmann’s idea was not needed.
Nevertheless, this seminal idea was studied further by many authors
Heitler1944 ; Coester1951 ; Watanabe1955 mostly for linear systems.
Stueckelberg Stueckelberg1952 proved these conditions for the Boltzmann
equation. He used in his proof the $S$-matrix representation of the micro-
kinetics.
Some consequences of the Stueckelberg approach were rediscovered for the Mass
Action law kinetics by Horn and Jackson in 1972 HornJackson1972 and
supplemented by the “zero deficiency theorem” Feinberg1972 . This is the
history.
In our work, we develop the Michaelis-Menten-Stueckelberg approach to general
kinetics. This is a combination of the QE (fast equilibria) and the QSS (small
amounts) approaches to the real or hypothetical intermediate states. These
intermediate states (compounds) are included in all elementary reactions (46)
as it is illustrated in Figure 1. Because of the small amount, the free energy
for these compounds $B_{i}$ is perfect (48), the kinetics of compounds is the
first order Markov kinetics and satisfies the master equation.
After that, we use the combination of QE and QSS approximations and exclude
the concentrations of compounds. For the general kinetics the main result of
this approach is the general kinetic law (74). Earlier, we just postulated
this law because of its convenient and natural form G1 ; BykGOrYab1982 , now
we have the physical framework where this law can be proved.
We do not assume anything about reaction rates of the main reactions (17). We
use only thermodynamic equilibrium, the hypothesis about fast equilibrium with
compounds and the smallness of concentration of compounds. This smallness
implies the perfect entropy and the first order kinetics for compounds. After
that, we get the reaction rate functions from the qualitative assumptions
about compounds and the equilibrium thermodynamic data.
For example, if we relax the assumption about fast equilibrium and use just
smallness of compound concentrations (the Briggs-Haldane QSS approach
BriggsHaldane1925 ; Aris1965 ; Segel89 ) then we immediately need the formulas
for reaction rates of compound production. Equilibrium data become
insufficient. If we relax the assumption about smallness of concentrations
then we lose the perfect entropy and the first order Markov kinetics. So, only
the combination of QE and QSS gives the desired result.
For the kinetics of rarefied gases the mass action law for elastic collisions
(the Boltzmann equation) or for inelastic processes like chemical reactions
follows from the “molecular chaos” hypothesis and the low density limits. The
Michaelis-Menten-Stueckelberg approach substitutes low density of all
components by low density of the elementary events (or of the correspondent
compounds) together with the QE assumption.
The general kinetic law has a simple form: For an elementary reaction
$\sum_{i}\alpha_{i}A_{i}\to\sum_{i}\beta_{i}A_{i}$
the reaction rate is $r=\varphi\Omega$, where $\Omega>0$ is the Boltzmann
factor, $\Omega=\exp\left(\sum_{i}\alpha_{i}\check{\mu}_{i}\right)$,
$\check{\mu}_{i}=-\partial S/\partial N_{i}$ is the chemical potential $\mu$
divided by $RT$, and $\varphi\geq 0$ is the kinetic factor. Kinetic factors
for different reactions should satisfy some conditions. Two of them are
connected to the basic physics:
* •
The detailed balance: The kinetic factors for mutually reverse reaction should
coincide, $\varphi^{+}=\varphi^{-}$. This identity is proven for systems with
microreversibility (Section 7.5).
* •
The complex balance: The sum of the kinetic factors for all elementary
reactions of the form $\sum_{i}\alpha_{i}A_{i}\to\ldots$ is equal to the sum
of the kinetic factors for all elementary reactions of the form
$\ldots\to\sum_{i}\alpha_{i}A_{i}$ (91). This identity is proven for all
systems under the Michaelis-Menten-Stueckelberg assumptions about existence of
intermediate compounds which are in fast equilibria with other components and
are present in small amounts.
For the general kinetic law we studied several sufficient conditions of
accordance between thermodynamics and kinetics: Detailed balance, complex
balance and $G$-inequality.
In the practice of modeling, a kinetic model may, initially, do not respect
thermodynamic conditions. For these cases, we solved the problem of whether it
is possible to add a linear function to entropy in order to provide agreement
with the given kinetic model and deformed thermodynamics. The answer is
constructive (Section 7) and allows us to prove the general algebraic
conditions for the detailed and complex balance.
Finally, we have to mention that Michaelis, Menten and Stueckelberg did not
prove their “big theorem”. Michaelis and Menten did not recognize that their
beautiful result of mass action law produced from the equilibrium relations
between substrates and compounds, the assumption about smallness of compound
concentrations and the natural hypothesis about linearity of compound kinetics
is a general theorem. Stueckelberg had much more and fully recognized that his
approach decouples the Boltzmann $H$-theorem and the microreversibility
(detailed balance). This is important because for every professional in
theoretical physics it is obvious that the microreversibility cannot be
important necessary condition for the $H$-theorem. Entropy production should
be positive without any relation to detailed balance (the proof of the
$H$-theorem for systems with detailed balance is much simpler but it does not
matter: Just the Markov microkinetics is sufficient for it). Nevertheless,
Stueckelberg did not produce the generalized mass action law and did not
analyze the general kinetic equation. Later, Horn, Jackson and Feinberg
approached the complex balance conditions again and studied the generalized
mass action law but had no significant interest in the microscopic assumptions
behind these properties. Therefore, this paper is the first publication of the
Michaelis-Menten-Stueckelberg theorem.
## References
* (1) Klonowski, W. Simplifying Principles for chemical and enzyme reaction kinetics. Biophys. Chem. 1983, 18, 73–87.
* (2) Gibbs, G.W. Elementary Principles in Statistical Mechanics; Yale University Press: New Haven, CT, USA, 1902.
* (3) Jaynes, E.T. Information theory and statistical mechanics. In Statistical Physics. Brandeis Lectures; Ford, K.W., Ed.; Benjamin: New York, NY, USA, 1963; pp. 160–185.
* (4) Balian, R., Alhassid, Y.; Reinhardt, H. Dissipation in many-body systems: A geometric approach based on information theory. Phys. Rep. 1986, 131, 1–146.
* (5) Gorban, A.N.; Karlin, I.V.; Ilg, P.; Öttinger, H.C. Corrections and enhancements of quasi–equilibrium states. J. Non-Newtonian Fluid Mech. 2001, 96 , 203–219.
* (6) Gorban, A.N.; Karlin, I.V. Invariant Manifolds for Physical and Chemical Kinetics (Lecture Notes in Physics); Springer: Berlin, Germary, 2005.
* (7) Bodenstein, M. Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten. Z. Phys. Chem. 1913, 85, 329–397.
* (8) Michaelis, L.; Menten, M. Die kinetik der intervintwirkung. Biochem. Z. 1913, 49, 333–369.
* (9) Semenov, N.N. On the kinetics of complex reactions. J. Chem. Phys. 1939, 7, 683–699.
* (10) Christiansen, J.A. The elucidation of reaction mechanisms by the method of intermediates in quasi-stationary concentrations. Adv. Catal. 1953, 5, 311–353.
* (11) Helfferich, F.G. Systematic approach to elucidation of multistep reaction networks. J. Phys. Chem. 1989, 93, 6676–6681.
* (12) Briggs, G.E.; Haldane, J.B.S. A note on the kinetics of enzyme action. Biochem. J. 1925, 19, 338–339.
* (13) Aris, R. Introduction to the Analysis of Chemical Reactors; Prentice-Hall, Inc.: Englewood Cliffs, NJ, USA, 1965.
* (14) Segel, L.A.; Slemrod, M. The quasi-steady-state assumption: A case study in perturbation. SIAM Rev. 1989, 31, 446–477.
* (15) Wei, J.; Kuo, J.C. A lumping analysis in monomolecular reaction systems: Analysis of the exactly lumpable system. Ind. Eng. Chem. Fund. 1969, 8, 114–123.
* (16) Wei, J.; Prater, C. The structure and analysis of complex reaction systems. Adv. Catal. 1962, 13, 203–393.
* (17) Kuo, J.C.; Wei, J. A lumping analysis in monomolecular reaction systems. Analysis of the approximately lumpable system. Ind. Eng. Chem. Fund. 1969, 8, 124–133.
* (18) Li, G.; Rabitz, H. A general analysis of exact lumping in chemical kinetics. Chem. Eng. Sci. 1989, 44, 1413–1430.
* (19) Toth, J.; Li, G.; Rabitz, H.; Tomlin, A.S. The effect of lumping and expanding on kinetic differential equations. SIAM J. Appl. Math. 1997, 57, 1531–1556.
* (20) Coxson, P.G.; Bischoff, K.B. Lumping strategy. 2. System theoretic approach. Ind. Eng. Chem. Res. 1987, 26, 2151–2157.
* (21) Djouad, R.; Sportisse, B. Partitioning techniques and lumping computation for reducing chemical kinetics. APLA: An automatic partitioning and lumping algorithm. Appl. Numer. Math. 2002, 43, 383–398.
* (22) Lin, B.; Leibovici, C.F.; Jorgensen, S.B. Optimal component lumping: Problem formulation and solution techniques. Comput. Chem. Eng. 2008, 32, 1167–1172.
* (23) Gorban, A.N.; Radulescu, O. Dynamic and static limitation in reaction networks, revisited. Adv. Chem. Eng. 2008, 34, 103–173.
* (24) Gorban, A.N.; Radulescu, O.; Zinovyev, A.Y. Asymptotology of chemical reaction networks. Chem. Eng. Sci. 2010, 65, 2310–2324.
* (25) Gorban, A.N. Equilibrium encircling. Equations of Chemical Kinetics and Their Thermodynamic Analysis; Nauka: Novosibirsk, USSR, 1984.
* (26) Stueckelberg, E.C.G. Theoreme $H$ et unitarite de $S$. Helv. Phys. Acta 1952, 25, 577–580.
* (27) Boltzmann, L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsber. keis. Akad. Wiss. 1872, 66, 275–370.
* (28) Cercignani, C.; Lampis M. On the $H$-theorem for polyatomic gases. J. Stat. Phys. 1981, 26, 795–801.
* (29) Heitler, W. Quantum Theory of Radiation; Oxford University Press: London, UK, 1944.
* (30) Coester, F. Principle of detailed balance. Phys. Rev. 1951, 84, 1259.
* (31) Watanabe, S. Symmetry of physical laws. Part I. Symmetry in space-time and balance theorems. Rev. Mod. Phys. 1955, 27, 26–39.
* (32) Gorban, A.N.; Gorban, P.A.; Judge, G. Entropy: The Markov ordering approach. Entropy 2010, 12, 1145–1193.
* (33) Yablonskii, G.S.; Bykov, V.I.; Gorban, A.N.; Elokhin, V.I. Kinetic Models of Catalytic Reactions; Elsevier: Amsterdam, The Netherlands, 1991; Volume 32.
* (34) Gorban, A.N.; Karlin, I.V. Method of invariant manifold for chemical kinetics. Chem. Eng. Sci. 2003, 58, 4751–4768.
* (35) Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 1935, 3, 107–115.
* (36) Evans, M.G.; Polanyi, M. Some applications of the transition state method to the calculation of reaction velocities, especially in solution. Trans. Faraday Soc. 1935, 31, 875–894.
* (37) Laidler, K.J. Tweedale, A. The current status of Eyring’s rate theory. In Advances in Chemical Physics: Chemical Dynamics: Papers in Honor of Henry Eyring; Hirschfelder, J.O., Henderson, D., Eds.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2007; Volume 21.
* (38) Gorban, A.N.; Karlin, I.V. Quasi-equilibrium closure hierarchies for the Boltzmann equation. Physica A 2006, 360, 325–364.
* (39) Gorban, A.N.; Karlin, I.V. Uniqueness of thermodynamic projector and kinetic basis of molecular individualism. Physica A 2004, 336, 391–432.
* (40) Hangos, K.M. Engineering model reduction and entropy-based Lyapunov functions in chemical reaction kinetics. Entropy 2010, 12, 772–797.
* (41) Li, B.; Shen, Y.; Li, B. Quasi-steady-state laws in enzyme kinetics. J. Phys. Chem. A 2008, 112, 2311–2321.
* (42) Tikhonov, A.N. Systems of differential equations containing small parameters multiplying some of the derivatives. Mat. Sb. 1952, 31, 575–586.
* (43) Vasil’eva, A.B. Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives. Russ. Math. Survey. 1963, 18, 13–84.
* (44) Hoppensteadt, F.C. Singular perturbations on the infinite interval. Trans. Am. Math. Soc. 1966, 123, 521–535.
* (45) Battelu, F.; Lazzari, C. Singular perturbation theory for open enzyme reaction networks. IMA J. Math. Appl. Med. Biol. 1986, 3, 41–51.
* (46) Feinberg, M. On chemical kinetics of a certain class. Arch. Ration. Mech. Anal. 1972, 46, 1–41.
* (47) Lebiedz, D. Entropy-related extremum principles for model reduction of dissipative dynamical systems. Entropy 2010, 12, 706–719.
* (48) Ò Conaire, M.; Curran, H.J.; Simmie, J.M.; Pitz, W.J.; Westbrook, C.K. A comprehensive modeling study of hydrogen oxidation. Int. J. Chem. Kinet. 2004, 36, 603–622.
* (49) Yablonskii, G.S.; Lazman, M.Z. Non-linear steady-state kinetics of complex catalytic reactions: Theory and applications. Stud. Surf. Sci. Catal. 1997, 109, 371–378.
* (50) Balescu, R. Equilibrium and Nonequilibrium Statistical Mechanics; John Wiley & Sons: New York, NY, USA, 1975.
* (51) Horn, F.; Jackson, R. General mass action kinetics. Arch. Ration. Mech. Anal. 1972, 47, 81–116.
* (52) Callen, H.B. Thermodynamics and an Introduction to Themostatistics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1985.
* (53) Bykov, V.I.; Gorban, A.N.; Yablonskii, G.S. Description of non-isothermal reactions in terms of Marcelin-De-Donder Kinetics and its generalizations. React. Kinet. Catal. Lett. 1982, 20, 261–265.
* (54) Clausius, R. Über vershiedene für die Anwendungen bequeme Formen der Hauptgleichungen der Wärmetheorie. Poggendorffs Annalen der Physic und Chemie 1865, 125, 353–400.
* (55) Orlov, N.N.; Rozonoer, L.I. The macrodynamics of open systems and the variational principle of the local potential, J. Franklin Inst-Eng. Appl. Math. 1984, 318, 283–341.
* (56) Onsager, L. Reciprocal relations in irreversible processes. I. Phys. Rev. 1931, 37, 405–426.
* (57) Onsager, L. Reciprocal relations in irreversible processes. II. Phys. Rev. 1931, 38, 2265–2279.
* (58) Van Kampen, N.G. Nonlinear irreversible processes. Physica 1973, 67, 1–22.
* (59) Gorban, A.N.; Bykov, V.I.; Yablonskii, G.S. Essays on Chemical Relaxation; Nauka: Novosibirsk, USSR, 1986.
* (60) Volpert, A.I.; Khudyaev, S.I. Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics; Nijoff: Dordrecht, The Netherlands, 1985.
* (61) Hangos, K.M.; Alonso, A.A.; Perkins, J.D.; Ydstie, B.E. Thermodynamic approach to the structural stability of process plants. AIChE J. 1999, 45, 802–816.
* (62) Wu, B.Y., Chao, K.-M. Spanning Trees and Optimization Problems; Chapman & Hall/CRC: Boca Raton, FL, USA, 2004.
* (63) Feinberg, M. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 1972, 49, 187–194.
* (64) Casimir, H.B.G. On Onsager’s principle of microscopic reversibility. Rev. Mod. Phys. 1945, 17, 343–350.
* (65) Grmela, M. Thermodynamics of driven systems. Phys. Rev. E 1993, 48, 919–930.
* (66) Gorban, A.N.; Karlin, I.V. Scattering rates versus moments: Alternative Grad equations. Phys. Rev. E 1996, 54, 3109–3112.
* (67) Van Kampen, N.G. Stochastic Processes in Physics and Chemistry; North-Holland: Amsterdam, The Netherlands, 1981.
* (68) Temkin, O.N.; Zeigarnik, A.V.; Bonchev, D.G. Chemical Reaction Networks: A Graph-Theoretical Approach; CRC Press: Boca Raton, FL, USA, 1996.
* (69) Morimoto, T. Markov processes and the $H$-theorem. J Phys. Soc. Jpn. 1963, 12, 328–331.
* (70) Meyn, S.R.; Tweedie, R.L. Markov Chains and Stochastic Stability; Cambridge University Press: Cambridge, UK, 2009.
## Appendix
## 1\. Quasiequilibrium Approximation
### 1.1. Quasiequilibrium Manifold
Let us consider a system in a domain $U$ of a real vector space $E$ given by
differential equations
$\frac{{\mathrm{d}}x}{{\mathrm{d}}t}=F(x)$ (112)
We assume that for any $x_{0}\in U$, solution $x(t;x_{0})$ to the initial
problem $x(0)=x_{0}$ for (112) exists for all $t>0$ and belongs to $U$. Shifts
in time, $x_{0}\mapsto x(t;x_{0})$ ($t>0$), form a semigroup in $U$.
We do not specify the space $E$ here. In general, it may be any Banach or even
more general space. For nonlinear operators we will use the Fréshet
differentials: For an operator $\Psi(x)$ the differential at point $x$ is a
linear operator $(D\Psi)_{x}$:
$(D\Psi)_{x}(y)=\left.\frac{{\mathrm{d}}\Psi(x+\alpha
y)}{{\mathrm{d}}y}\right|_{\alpha=0}$
We use also notation $(D_{x}\Psi)$ when it is necessary to stress the choice
of independent variable. The choice of variables is not obvious.
The QE approximation for (112) uses two basic entities: entropy and slow
variables.
Entropy $S$ is a concave Lyapunov function with non-degenerated Hessian for
(3) which increases in time:
$\frac{{\mathrm{d}}S}{{\mathrm{d}}t}\geq 0\,$ (113)
In this approach, the increase of the entropy in time is exploited (the Second
Law in the form (4)).
Formally, any Lyapunov function may be used. Nevertheless, most of famous
entropies, like the relative Boltzmann-Gibbs-Shannon entropy, the Rényi
entropy, the Burg entropy, the Cressie-Read and the Tsallis entropies could be
defined as universal Lyapunov functions for Markov chains which satisfy some
natural additivity conditions GorbanGorbanJudge2010 .
“Universal” means that they do not depend on kinetic coefficients directly but
only on the equilibrium point. The “natural additivity conditions” require
that these entropies can be represented by sums (or integrals) over states
maybe after some monotonic transformation of the entropy scale, and, at the
same time, are additive with respect to the joining of statistically
independent systems (maybe, after some monotonic rescaling as well).
Slow variables $M$ are defined as some differentiable functions of variables
$x$: $M=m(x)$. We use notation $E_{M}$ for the space of slow variables, $M\in
E_{M}$. Selection of the slow variables implies a hypothesis about separation
of fast and slow motion. In its strongest form it consists of two assumptions:
The slaving assumption and the assumption of small fast-slow projection.
The slaving assumption. For any admissible initial state $x_{0}\in U$ after
some relatively small time $\tau$ (initial layer), solution $x(t;x_{0})$
becomes a function of $M$ (up to a given accuracy $\epsilon$) and can be
represented in a slaving form:
$x(t)=x^{*}_{M(t)}+\delta(t)\;\;{\rm for}\;\;t>\tau\,,\;\;{\rm
where}\;\;M(t)=m(x(t)),\;\|\delta(t)\|<\epsilon\,$ (114)
This means that everything is a function of slow variables, after some initial
time and up to a given accuracy.
The smallness of $\tau$ is essential. If there is no restriction on $\tau$
then every globally stable system will satisfy this assumption because after
some time it will arrive into a small vicinity of equilibrium.
The second assumption requires that the slow variables (almost) do not change
during the fast motion: During the initial layer $\tau$, the state $x$ can
change significantly because of fast motion, but the change in $M=m(x)$ during
$\tau$ are small with $\tau$: The assumption of small fast-slow projection.
The QE approximation defines the functions $x^{*}_{M}$ as solutions to the
following MaxEnt optimization problem:
$S(x)\to\max\;\;\mbox{subject to}\;\;m(x)=M\,$ (115)
The reasoning behind this approximation is simple: During the fast initial
layer motion, entropy increases and $M$ almost does not change. Therefore, it
is natural to assume (and even to prove using smallness of $\tau$ and
$\epsilon$ if the entropy gradient in fast directions is separated from zero)
that $x^{*}_{M}$ in (114) is close to solution to the MaxEnt optimization
problem (115). Further, $x^{*}_{M}$ denotes a solution to the MaxEnt problem.
Some additional conditions on $m$ and $S$ are needed for the regularity of the
dependence $x^{*}_{M}$ on $M$. It is more convenient to discuss these
conditions separately for more specific systems. In general settings, let us
just assume that for given $S$ and $m$ the dependencies $m(x)$ and $x^{*}_{M}$
are differentiable. For their differentials we use the notations $(Dm)_{x}$
and $(Dx^{*}_{M})_{M}$. The differentials are linear operators: $(Dm)_{x}:E\to
E_{M}$ and $(Dx^{*}_{M})_{M}:E_{M}\to E$.
A solution to (115), $x^{*}_{M}$, is the QE state, the corresponding value of
the entropy
$S^{*}(M)=S(x^{*}_{M})$ (116)
is the QE entropy and the equation
$\frac{{\mathrm{d}}M}{{\mathrm{d}}t}=(Dm)_{x^{*}_{M}}(F(x^{*}_{M}))$ (117)
represents the QE dynamics.
Remark. The strong form of the slaving assumption, “everything becomes a
function of the slow variables”, is too strong for practical needs. In
practice, we need just to have a “good” dependence on $M$ for the time
derivative ${\mathrm{d}}M/{\mathrm{d}}t$. Moreover, the short-time
fluctuations of ${\mathrm{d}}M/{\mathrm{d}}t$ do not affect the dependence
$M(t)$ too much, and only the average values
$\langle\dot{M}\rangle_{\theta}(t)=\frac{1}{\theta}\int_{t}^{t+\theta}\frac{{\mathrm{d}}M}{{\mathrm{d}}t}$
for sufficiently small time scale $\theta$ are important.
### 1.2. Preservation of Entropy Production
Theorem about preservation of entropy production. Let us calculate
${\mathrm{d}}S^{*}(M)/{\mathrm{d}}t$ at point $M$ according to the QE dynamics
(117) and find ${\mathrm{d}}S(x)/{\mathrm{d}}t$ at point $x=x^{*}_{M}$ due to
the initial system (3). The results always coincide:
$\frac{{\mathrm{d}}S^{*}(M)}{{\mathrm{d}}t}=\frac{{\mathrm{d}}S(x)}{{\mathrm{d}}t}\,$
(118)
The left hand side in (118) is computed due to the QE approximation (117) and
the right hand side corresponds to the initial system (112). Here, this
theorem is formulated in more general setting than in Section 2: The slow
variables $M=m(x)$ may be nonlinear functions of $x$. For more details about
QE approximation with nonlinear dependencies $M=m(x)$ we refer to papers
GorKarQE2006 ; GorKarProjector2004 ; GorKarPRE1996 . The general theorem about
preservation of entropy production and thermodynamic projector is presented in
GorKarProjector2004 .
###### Proof.
To prove this identity let us mention that
$\frac{{\mathrm{d}}S^{*}(M)}{{\mathrm{d}}t}=(DS^{*})_{M}\left(\frac{{\mathrm{d}}M}{{\mathrm{d}}t}\right)=(DS^{*})_{M}\circ(Dm)_{x^{*}_{M}}(F(x^{*}_{M}))\,$
(119)
where $\circ$ stands for superposition. On the other hand, just from the
definitions of the differential and of the time derivative of a function due
to a system of differential equations, we get
$\frac{{\mathrm{d}}S(x)}{{\mathrm{d}}t}=(DS)_{x}(F(x))\,$ (120)
To finalize the proof, we need an identity
$(DS^{*})_{M}\circ(Dm)_{x^{*}_{M}}=(DS)_{x^{*}_{M}}\,$ (121)
Let us use the Lagrange multipliers representation of the MaxEnt problem:
$(DS)_{x}=\Lambda_{M}\circ(Dm)_{x}\,,\;\;m(x)=M\,$ (122)
This system of two equations has two unknowns: The vector of state $x$ and the
linear functional $\Lambda_{M}$ on the space of slow variables (the Lagrange
multiplier), which depends on $M$ as on a parameter.
By differentiation of the second equation $m(x)=M$, we get an identity
$(Dm)_{x^{*}_{M}}\circ(Dx^{*}_{M})_{M}={\rm id}_{E_{M}}\,$ (123)
where ${\rm id}$ is the unit operator.
Lagrange multiplier $\Lambda_{M}$ is the differential of the QE entropy:
$(DS^{*})_{M}=\Lambda_{M}\,$ (124)
Indeed, due to the chain rule,
$(DS^{*})_{M}=(DS)_{x^{*}_{M}}\circ(Dx^{*}_{M})_{M}$, due to (122),
$(DS)_{x}=\Lambda_{M}(Dm)_{x}$ and, finally
$\begin{split}(DS^{*})_{M}=&(DS)_{x^{*}_{M}}\circ(Dx^{*}_{M})_{M}=\Lambda_{M}\circ(Dm)_{x}\circ(Dx^{*}_{M})_{M}\\\
=&\Lambda_{M}\circ{\rm id}_{E_{M}}=\Lambda_{M}\,\end{split}$
Now we can prove the identity (121):
$(DS^{*})_{M}\circ(Dm)_{x^{*}_{M}}=\Lambda_{M}\circ(Dm)_{x^{*}_{M}}=(DS)_{x^{*}_{M}}$
(here we use the Lagrange multiplier form (122) again). ∎
The preservation of the entropy production leads to the preservation of the
type of dynamics: If for the initial system (112) entropy production is non-
negative, ${\mathrm{d}}S/{\mathrm{d}}t\geq 0$, then for the QE approximation
(117) the production of the QE entropy is also non-negative,
${\mathrm{d}}S^{*}/{\mathrm{d}}t\geq 0$.
In addition, if for the initial system
$({{\mathrm{d}}S}/{{\mathrm{d}}t})_{x}=0$ if and only if $F(x)=0$ then the
same property holds in the QE approximation.
## 2\. First Order Kinetics and Markov Chains
First-order kinetics form the simplest and well-studied class of kinetic
systems. It includes the continuous-time Markov chains (the master equation
VanKampen1981 ), kinetics of monomolecular and pseudomonomolecular reactions
LumpWei2 , and has many other applications.
We consider a general network of linear reactions. This network is represented
as a directed graph (digraph) (Yablonskii1991 ; Temkin1996 ): Vertices
correspond to components $B_{j}$ ($1\leq j\leq q$), edges correspond to
reactions $B_{j}\to B_{l}$. For each vertex $B_{j}$ a positive real variable
$\varsigma_{j}$ (concentration) is defined. For each reaction $B_{j}\to B_{l}$
a rate constant $\kappa_{lj}>0$ is given. To follow the standard notation of
the matrix multiplication, the order of indexes in $\kappa_{ji}$ is always
inverse with respect to reaction: It is $\kappa_{j\leftarrow i}$, where the
arrow shows the direction of the reaction. The kinetic equations for
concentrations $\varsigma_{j}$ have the form
$\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=\sum_{l,\,l\neq
j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)$ (125)
The linear conservation law (for the Markov chains this is the conservation of
the total probability) is:
$\sum_{j}\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=0\;\;\mbox{{\em
i.e.},}\;\;\sum_{j,l,\,l\neq
j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)=0\,.$ (126)
Let a positive vector $\varsigma^{*}$ ($\varsigma^{*}_{j}>0$) be an
equilibrium for the system (125): For every $j=1,\ldots,q$
$\sum_{l,\,l\neq j}\kappa_{jl}\varsigma_{l}^{*}=\sum_{l,\,l\neq
j}\kappa_{lj}\varsigma_{j}^{*}$ (127)
An equivalent form of (125) is convenient. Let us use the equilibrium
condition (127) and write
$\sum_{l,\,l\neq j}\kappa_{lj}\varsigma_{j}=\left(\sum_{l,\,l\neq
j}\kappa_{lj}\varsigma_{j}^{*}\right)\frac{\varsigma_{j}}{\varsigma_{j}^{*}}=\sum_{l,\,l\neq
j}\kappa_{jl}\varsigma_{l}^{*}\frac{\varsigma_{j}}{\varsigma_{j}^{*}}$
Therefore, under condition (127) the master equation (125) has the equivalent
form:
$\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=\sum_{l,\,l\neq
j}\kappa_{jl}\varsigma_{l}^{*}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)$
(128)
The following theorem Morimoto1963 describes the large class of the Lyapunov
functions for the first order kinetics. Let $h(x)$ be a smooth convex function
on the positive real axis. A Csiszár–Morimoto function $H_{h}(\varsigma)$ is
(see the review GorbanGorbanJudge2010 ):
$H_{h}(\varsigma)=\sum_{l}\varsigma_{l}^{*}h\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}\right)$
The Morimoto $H$-theorem. The time derivative of $H_{h}(\varsigma)$ due to
(125) under condition (127) is nonpositive:
$\frac{{\mathrm{d}}H_{h}(\varsigma)}{{\mathrm{d}}t}=\sum_{l,j,\,j\neq
l}h^{\prime}\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\kappa_{jl}\varsigma^{*}_{l}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\leq
0$ (129)
###### Proof.
Let us mention that for any $q$ numbers $h_{i}$, $\sum_{i,j,\,j\neq
i}\kappa_{ij}\varsigma^{*}_{j}(h_{j}-h_{i})=0$. Indeed, for
$h_{i}=p_{i}/p_{i}^{*}$ this is precisely the condition of conservation of the
total probability for equations (128). The extension from a simplex of the
$h_{i}$ values ($\sum_{i}p^{*}_{i}h_{i}=1$, $h_{i}\geq 0$) to the positive
orthant $\mathbb{R}^{q}_{+}$ is trivial because of uniformity of the identity.
Finally, if a linear identity holds in a positive orthant then it holds in the
whole space $\mathbb{R}^{q}$. Therefore,
$\sum_{l,j,\,j\neq
l}h^{\prime}\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\kappa_{jl}\varsigma^{*}_{l}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)=\sum_{l,j,\,j\neq
l}\kappa_{jl}\varsigma^{*}_{l}\left[h\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)-h\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}\right)+h^{\prime}\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\right]\leq
0$
The last inequality holds because of the convexity of $h(x)$:
$h^{\prime}(x)(y-x)\leq h(y)-h(x)$ (Jensen’s inequality).∎
For example, for the convex function $h(x)=x(\ln x-1)$ the Csiszár–Morimoto
function is:
$H_{h}(\varsigma)=\sum_{l}\varsigma_{l}\left(\ln\left(\frac{\varsigma_{l}}{\varsigma^{*}_{l}}\right)-1\right)$
(130)
This expression coincides with the perfect component of the free energy (50)
(to be more precise, $f=RTH_{h}(\varsigma)$).
Each positive equilibrium $\varsigma^{*}$ belongs to the ray of positive
equilibria $\lambda\varsigma^{*}$ ($\lambda>0$). We can select a
$l_{1}$-normalized direction vector and write for an equilibrium
$\varsigma^{\rm eq}$ from this ray:
$\varsigma^{\rm eq}=\beta\frac{\varsigma^{*}}{\sum_{j}\varsigma^{*}_{j}}\,,$
where $\beta=\sum_{j}\varsigma^{\rm eq}_{j}$.
The kinetic equations (125) allow one and only one ray of positive equilibria
if and only if the digraph of reactions is strongly connected: It is possible
to reach any vertex starting from any other vertex by traversing edges in the
directions in which they point. Such continuous-time Markov chains are called
ergodic chains MeynMarkCh2009 .
Let us assume that the system is weakly reversible: For any two vertices
$B_{i}$ and $B_{j}$, the existence of an oriented path from $B_{i}$ to $B_{j}$
implies the existence of an oriented path from $B_{j}$ to $B_{i}$. Under this
assumption the graph of reactions is a unit of strongly connected subgraphs
without connections between them. Let the graph of reactions $B_{j}\to B_{l}$
have $d$ strongly connected components $C_{s}$ and let $V_{s}$ be the set of
indexes of those $B_{j}$ which belong to $C_{s}$: $B_{j}\in C_{s}$ if and only
if $j\in V_{s}$. For each $s=1,\ldots,d$ there exists a conservation law
$\beta_{s}(\varsigma)=\sum_{j\in V_{s}}\varsigma_{j}=const$ (131)
For any set of positive values of $\beta_{s}>0$ ($s=1,\ldots,d$) there exists
a unique equilibrium of (125), $\varsigma^{\rm eq}$, which is positive
($\varsigma^{\rm eq}_{j}>0$). This equilibrium can be expressed through any
positive equilibrium $\varsigma^{*}$:
$\varsigma^{\rm eq}_{j}=\beta_{s}\frac{\varsigma_{j}^{*}(c,T)}{\sum_{l\in
V_{s}}\varsigma_{j}^{*}(c,T)}$ (132)
For positive $\beta_{s}$, logarithms of $\varsigma^{\rm eq}$ form a
$d$-dimensional linear manifold in $R^{q}$:
$\ln\varsigma^{\rm
eq}_{j}=\ln\beta_{s}+\ln\left(\frac{\varsigma_{j}^{*}(c,T)}{\sum_{l\in
V_{s}}\varsigma_{l}^{*}(c,T)}\right)$ (133)
The natural coordinates on this manifold are $\ln\beta_{s}$.
Remark. In the construction of the free energy (130) any positive equilibrium
state $\varsigma^{\rm eq}$ can be used. The correspondent functions differs in
an additive constant. Let us calculate the difference between two “free
energies”:
$\begin{split}&\sum_{j=1}^{q}\varsigma_{j}\left(\ln\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)-1\right)-\sum_{j=1}^{q}\varsigma_{j}\left(\ln\left(\frac{\varsigma_{j}}{\varsigma_{j}^{\rm
eq}}\right)-1\right)\\\
&=\sum_{j=1}^{q}\varsigma_{j}\ln\left(\frac{\varsigma_{j}^{\rm
eq}}{\varsigma_{j}^{*}}\right)=\sum_{s=1}^{d}\beta_{s}(\varsigma)\ln\left(\frac{\beta_{s}(\varsigma^{\rm
eq})}{\beta_{s}(\varsigma^{*})}\right)\end{split}$ (134)
The result is constant in time for the solutions of the master equation (125),
hence, these functions are equivalent: They both are the Lyapunov functions
for (125) and have the same conditional minimizers for given values of
$\beta_{s}>0$ ($s=1,\ldots,d$).
|
arxiv-papers
| 2010-08-19T13:21:10 |
2024-09-04T02:49:12.263530
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban, M. Shahzad",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1008.3296"
}
|
1008.3318
|
# Curvature bounded below:
a definition a la Berg–Nikolaev
N. Lebedeva and A. Petrunin
## 1 Introduction
In this note we give a new characterization of spaces with curvature
$\geqslant 0$ in the sense of Alexandrov. Our work is inspired by
[Berg–Nikolaev] and [Sato], where an analogous definition was given for
curvature $\leqslant 0$.
1.1. Main theorem. Let $\mathcal{X}$ be a complete space with intrinsic
metric. Then $\mathcal{X}$ is an Alexandrov space with curvature $\geqslant 0$
if and only if any quadruple $p,x,y,z\in\mathcal{X}$ satisfies the following
inequality
$|px|^{2}+|py|^{2}+|pz|^{2}\geqslant\tfrac{1}{3}{\hskip 0.5pt\cdot\hskip
0.5pt}\\!\left(|xy|^{2}+|yz|^{2}+|zx|^{2}\right).$ ➊
The inequality ➊ ‣ 1 is quite weak. For example, one can111Say, consider the
metric on $\\{p,x,y,z\\}$ defined as $|px|=py|=|pz|=1$, $|xy|=|xz|=2$ and
$|yz|=\varepsilon$ where $\varepsilon>0$ is sufficiently small; it satisfies ➊
‣ 1 for each relabeling but not ➋ ‣ 1. construct a metric space $\mathcal{F}$
with 4 points which satisfies ➊ ‣ 1 for each relabeling by $p,x,y,z$, such
that $\mathcal{F}$ does not admit an isometric embedding into any Alexandrov
space with curvature $\geqslant 0$.
The similar conditions which were known before simply describe all 4-point
sets in a nonnegatively curved space. For instance the following inequality
for model angles:
$\tilde{\measuredangle}(p\,{}^{x}_{y})+\tilde{\measuredangle}(p\,{}^{y}_{z})+\tilde{\measuredangle}(p\,{}^{z}_{x})\leqslant
2{\hskip 0.5pt\cdot\hskip 0.5pt}\pi.$ ➋
In fact, if a 4-point metric space satisfies ➋ ‣ 1 for each relabeling then it
can be isometrically embedded into Euclidean plane or a $2$-sphere of some
radius $R>0$ (the proof is left to the reader).
Why do we care. Since the condition ➊ ‣ 1 is so weak, it should be useful as a
test to check that a given space has curvature $\geqslant 0$ in the sense of
Alexandrov. However, we are not aware of a single case when it makes life
easier.
To explain the real reason why we are interested in this topic, we need to
reformulate our Main theorem using the language similar to one given in
[Gromov, Section 1.19+].
Let us denote by $\mathbf{M}^{4}$ the set of isometry classes of 4-point
metric spaces. Let $\mathfrak{A}$ and $\mathfrak{B}$ be the sets of isometry
classes in $\mathbf{M}^{4}$ which satisfy correspondingly ➊ ‣ 1 and ➋ ‣ 1 for
all relabeling of points by $p,x,y,z$. (As it mentioned above,
$\mathfrak{B}\subsetneq\mathfrak{A}$.) Further, given a metric space
$\mathcal{X}$, denote by $\mathbf{M}^{4}(\mathcal{X})$ the set of isometry
classes of 4-point subspaces in $\mathcal{X}$.
Main theorem says that if the space $\mathcal{X}$ has intrinsic metric and
$\mathbf{M}^{4}(\mathcal{X})\subset\mathfrak{A}$ then
$\mathbf{M}^{4}(\mathcal{X})\subset\mathfrak{B}$. From above, the set
$\mathfrak{B}$ is the smallest set which satisfies the above property for any
$\mathcal{X}$.
It would be interesting to find a general pattern of such phenomena. Assume
you start with arbitrary $\mathfrak{A}\subset\mathbf{M}^{4}$, can you figure
out what is the corresponding $\mathfrak{B}$, or can one describe the
properties of $\mathfrak{B}$ which might appear this way?
Note that Globalization theorem (see [Burago–Gromov–Perelman]) as well as
Berg–Nikolaev characterization of $\mathrm{CAT}(0)$ spaces both admit
interpretations in the above terms. Also, in [Foertsch–Lytchak–Schroeder], it
was shown that set defined by Ptolemy inequality can appear as $\mathfrak{B}$.
Acknowledgment. We want to thank Alexander Lytchak for an interesting
discussion in the Münster’s botanical garden.
## 2 The proof
The “only if” part follows directly from generalized Kirszbraun’s theorem, see
[Lang–Schroeder]. One only has to check that the inequality ➊ ‣ 1 holds in the
model plane. (Alternatively, one can prove ➋ ‣ 1 $\Rightarrow$ ➊ ‣ 1
directly.)
“if” part. We may assume that $\mathcal{X}$ is geodesic, otherwise pass to its
ultraproduct.
It is sufficient to show that if $z$ is a midpoint of geodesic $[pq]$ in
$\mathcal{X}$ then
$2{\hskip 0.5pt\cdot\hskip
0.5pt}|xz|^{2}\geqslant|xp|^{2}+|xq|^{2}-\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip
0.5pt}|pq|^{2},$ ➌
for any $x\in\mathcal{X}$.
Directly from ➊ ‣ 1 we have the following weaker estimate
$3{\hskip 0.5pt\cdot\hskip
0.5pt}|xz|^{2}\geqslant|xp|^{2}+|xq|^{2}-\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip
0.5pt}|pq|^{2}$ ➍
$p$$q$$x_{0}$$x_{1}$$x_{2}$$z$
Set $x_{0}=x$ and consider a sequence of points $x_{0},x_{1},\dots$ on $[xz]$
such that $|x_{n}z|=\tfrac{1}{3^{n}}{\hskip 0.5pt\cdot\hskip 0.5pt}|xz|$. Let
$\alpha_{n}$ be a sequence of real numbers such that
$\alpha_{n}{\hskip 0.5pt\cdot\hskip
0.5pt}|x_{n}z|^{2}=|x_{n}p|^{2}+|x_{n}q|^{2}-\tfrac{1}{2}{\hskip
0.5pt\cdot\hskip 0.5pt}|pq|^{2}.$
Applying ➊ ‣ 1, we get
$|x_{n+1}p|^{2}+|x_{n+1}q|^{2}+|x_{n+1}x_{n}|^{2}\geqslant\tfrac{1}{3}{\hskip
0.5pt\cdot\hskip 0.5pt}(|x_{n}p|^{2}+|x_{n}q|^{2}+|pq|^{2}).$
Subtract $\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip 0.5pt}|pq|^{2}$ from both
sides; after simplification you get
$\alpha_{n+1}\geqslant 3{\hskip 0.5pt\cdot\hskip 0.5pt}\alpha_{n}-4.$
Now assume ➌ ‣ 2 does not hold; i.e., $\alpha_{0}>2$ then
$\alpha_{n}\to\infty$ as $n\to\infty$. On the other hand, from ➍ ‣ 2, we get
$\alpha_{n}\leqslant 3$, a contradiction. ∎
## P.S.: Arbitrary curvature bound
One can obtain analogous characterization of Alexandrov spaces with curvature
$\geqslant\kappa$ for any $\kappa\in\mathbb{R}$.
Here are the inequalities for cases $\kappa=1$ and $-1$ which correspond to ➊
‣ 1 for quadruple $p,x^{1},x^{2},x^{3}$:
$\left(\sum_{i=1}^{3}\cos|px^{i}|\right)^{2}\leqslant\sum_{i,j=1}^{3}\cos|x^{i}x^{j}|.$
$None$
$\left(\sum_{i=1}^{3}\cosh|px^{i}|\right)^{2}\geqslant\sum_{i,j=1}^{3}\cosh|x^{i}x^{j}|;$
$None$
Note that in both cases we have equality if $p$ is the point of intersections
of medians of the triangle $[x^{1}x^{2}x^{3}]$ in the corresponding model
plane. (In the model planes, medians also pass through incenter.)
The proof goes along the same lines. The case $\kappa=1$ also follows from
Main theorem applied to the cone over the space.
## References
* [Berg–Nikolaev] Berg, I. D.; Nikolaev, I. G. Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata 133 (2008), 195–218.
* [Burago–Gromov–Perelman] Burago, Yu.; Gromov, M.; Perelman, G., A. D. Aleksandrov spaces with curvatures bounded below. (Russian) Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222; translation in Russian Math. Surveys 47 (1992), no. 2, 1–58
* [Foertsch–Lytchak–Schroeder] Foertsch, T.; Lytchak, A.; Schroeder, V. Nonpositive curvature and the Ptolemy inequality. Int. Math. Res. Not. IMRN 2007, no. 22, Art. ID rnm100, 15 pp.
* [Gromov] Gromov, M. Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser Boston, Inc., Boston, MA (1999)
* [Lang–Schroeder] Lang, U.; Schroeder, V., Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. funct. anal. Vol. 7 (1997) 535–560
* [Sato] Sato, Takashi An alternative proof of Berg and Nikolaev’s characterization of $\rm CAT(0)$-spaces via quadrilateral inequality. Arch. Math. (Basel) 93 (2009), no. 5, 487–490.
|
arxiv-papers
| 2010-08-19T14:58:55 |
2024-09-04T02:49:12.280535
|
{
"license": "Public Domain",
"authors": "Nina Lebedeva and Anton Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1008.3318"
}
|
1008.3556
|
# Frequency dependent polarizability of small metallic grains
A. A. Zharov and I. S. Beloborodov Department of Physics and Astronomy,
California State University Northridge, Northridge, CA 91330, USA
###### Abstract
We study the dynamic electronic polarizability of a single nano-scale
spherical metallic grain using quantum mechanical approach. We introduce the
model for interacting electrons bound in the grain allowing us numerically to
calculate the frequency dependence of the polarizability of grains of
different sizes. We show that within this model the main resonance peak
corresponding to the surface plasmon mode is blue-shifted and some minor
secondary resonances above and below the main peak exist. We study the
behavior of blue shift as a function of grain size and compare our findings
with the classical polarizability and with other results in the literature.
###### pacs:
73.20.Mf, 73.22.Lp, 78.67.Bf
## I Introduction
Great efforts in contemporary materials science research focus on properties
of granular materials Beloborodov07 . The interest is motivated by the fact
that granular arrays can be treated as artificial solids with programmable
electronic properties. The ease of adjusting electronic properties of granular
materials is one of their most attractive assets for fundamental studies of
disordered solids and for targeted applications in nanotechnology. The
parameters of granular materials in many ways are determined by the properties
of their building blocks - grains.
In this paper we study the dynamic electronic polarizability of a single nano-
scale spherical metallic grain. This research is important because on one hand
confinement effects can drastically modify the dynamic polarizability of
metallic grains and on the other hand the fundamental question related to the
description of crossover between microscopic and macroscopic behavior of the
grains exists.
The electronic polarizability was first quantum mechanically studied in Ref.
Gorkov . However, the influence of screening effects on static polarizability
was first investigated in Ref. Rice . Later the dynamic polarizability of very
small metallic particles was considered within the jellium-background model
Ekardt ; SpherJ ; Beck ; Koch ; Weick . This model was also used to study the
electronic structure and polarizability of metallic nanoshells Prodan . The
electronic polarizability was also studied in applications for optical
properties of small metallic grains Ruppin ; Halperin ; Wood ; Genzel2 . More
recently the nonperturbative supersymmetry technique was used to consider the
effects of disorder on properties of electronic polarizability Efetov .
Here we study the dynamic electronic polarizability of a single metallic grain
using quantum mechanical approach. We introduce the model for interacting
electrons bound in the grain allowing us numerically to calculate the
frequency dependence of the polarizability of grains of different sizes.
## II Main Results
We numerically calculate the dynamic polarizability $\alpha(\omega)$ of grains
of different sizes. The experimentally measurable quantity is the
photoabsorption cross section $\sigma(\omega)$. It is related to the imaginary
part of the polarizability $\alpha(\omega)$ as follows
$\sigma(\omega)=\frac{4\pi\omega}{c}\textrm{Im}\,{\alpha(\omega)},$ (1)
where $c$ is the speed of light. The photoabsorption cross section
$\sigma(\omega)$ is the main result of this work. It is plotted in Figs. 1 and
2 for two grains of different sizes (the number of electrons $N=16304$ and
$N=1956$, corresponding to the grain sizes of 5.28 nm and 2.6 nm,
respectively).
Our calculations were done at zero temperature. Therefore, due to Fermi-Dirac
statistics all electron states are occupied below the Fermi level and empty
above it. As a result, the total number of electrons $N$, that determines the
size of the grain, is related to the Fermi energy level. This number can be
found by calculating the total number of electron states below the Fermi
level. The Fermi energy can be parameterized by dimensionless quantity
$\alpha_{F}$ as $E_{F}=(\hbar^{2}/2mR^{2})\alpha_{F}^{2}$, where $R$ is a
grain radius. In order to estimate the grain size we need to know the total
number of electrons $N$ and the electron density $r_{s}$. In terms of these
parameters the grain size is given by $R/a_{B}=N^{1/3}r_{s}$, where $a_{B}$ is
Bohr radius. All grain size estimates were done for sodium ($r_{s}=3.93$).
Figure 1: (Color online) Imaginary part of dynamic electronic polarizability
$\alpha(\omega)$ of the metallic grain (in units of $R^{3}$, $R$ being the
radius of a grain) vs dimensionless frequency $\omega/\omega_{cl}$ with
$\omega_{cl}=\omega_{p}/\sqrt{3}$ being the classical resonance frequency,
where $\omega_{p}$ is the plasma resonance frequency. The blue solid line
represents the numerical solution for polarizability $\alpha(\omega)$
corresponding to the number of electrons $N=16304$, $R=5.28$
nm($\alpha_{F}=50$, see Eq. (22)). The red dashed line represents the
classical Drude polarizability $\alpha_{cl}(\omega)$, Eq. (3), for comparison.
Figure 2: (Color online) Imaginary part of dynamic electronic polarizability
$\alpha(\omega)$ of the metallic grain (in the units of $R^{3}$, $R$ being the
radius of a grain) vs dimensionless frequency $\omega/\omega_{cl}$ with
$\omega_{cl}=\omega_{p}/\sqrt{3}$ being the classical resonance frequency. The
blue solid line represents the numerically solution for polarizability
$\alpha(\omega)$ corresponding to the number of electrons $N=1956$, $R=2.6$ nm
($\alpha_{F}=25$, see Eq. (22)). The red dashed line represents the classical
Drude polarizability $\alpha_{cl}(\omega)$, Eq. (3), for comparison.
The behavior of dynamic polarizability $\alpha(\omega)$ presented in Figs. 1
and 2 revels several features: i) the main resonance peak is slightly shifted
(with respect to the classical result) to higher frequency; ii) some smaller
secondary resonances above and below the main resonance frequency exist. Both
of these features have a tendency to decrease as the grain size increases
which is consistent with the fact that macroscopic grain has a fully classical
behavior.
The main resonance peak in Figs. 1 and 2 corresponds to the surface plasmon
excitation. The blue-shift of the main resonance peak in comparison with the
classical Mie resonance (the red dash line in Figs. 1 and 2) can be understood
as follows: In a quantum mechanical problem due to Fermi-Dirac statistics
electron transitions may occur only from an occupied state to an empty state.
For higher external frequency the total number of unrestricted electron
transitions becomes higher. Therefore in a quantum mechanical problem if the
external frequency equals to the classical plasmon frequency,
$\omega_{cl}=\omega_{p}/\sqrt{3}$ in Figs. 1 and 2, the resonance amplitude is
suppressed because of a lack of possible electron transitions. Hence the main
resonance peak has a tendency to move to a higher frequency.
The surface plasmon excitations were also studied quantum mechanically in Ref.
Ovchinnikov , and using extended classical Mie theory in Ref. Ruppin , where
it was shown that the frequency of a surface plasmon for very small grains is
blue-shifted. This is in agreement with our result. Several experimental works
have also observed the blue-shifted surface plasmon resonance peak Duthler ;
Genzel3 .
A different result was obtained in Refs. Ekardt ; Weick , where the main
resonance peak was red-shifted. This discrepancy may be explained by the
difference between the models used. In particular, in Ref. Ekardt the
electronic polarizability was studied within the spherical jellium-background
model. The red-shift in this model is explained by the electron spill-out
effect. Here we use ”electron in a box” approximation. In this model the
spill-out effect is prevented by the infinitely deep potential well.
The width of the plasmon resonance in Figs. 1 and 2 is determined by the
artificially introduced imaginary part of the frequency $\omega$ to avoid
infinite values of polarizability $\alpha(\omega)$ at the resonances. This
width does not correspond to a real width of the photoabsorbtion cross section
$\sigma(\omega)$ in Eq. (1). We discuss this issue in more details in the
”solution” section below.
The secondary resonances shown in Figs. 1 and 2 are significantly lower than
the main peak. These resonances can be divided into two groups: i) low
frequency single particle excitations and ii) high frequency resonances
corresponding to the transitions between the states with higher orbital
quantum numbers. Both of these types of excitations become less pronounced
with increasing grain size. We notice that at low energies the transitions may
occur only in the immediate vicinity to the Fermi surface. Therefore the low
frequency peaks in Figs. 1 and 2 correspond to the resonant transitions
between the states which are located just below and just above the Fermi
surface.
We now comment on the influence of disorder in the grain on the resonance
peaks. First, we notice, that our consideration is based on the fact, that the
disorder is weak. Thus, the conducting electrons are delocalized in our model.
The characteristic energy scale for disorder is the inverse scattering time
$1/\tau$. In the limit of weak disorder the electron propagation within the
grain is almost ballistic and therefore $1/\tau$ is of the order of Thouless
energy, $1/\tau\sim E_{Th}$. However, both of these energy scales are smaller
than the plasma frequency, $\omega_{p}$, which is the characteristic scale for
the resonances. Therefore, we expect that for energies less than $\omega_{p}$
the disorder is irrelevant and can not modify the picture qualitatively.
However, the presence of disorder may lead to the broadening and suppression
of all of the resonance peaks. The broadening effects are beyond our present
consideration. The disorder caused suppression might make the secondary
resonances hard to detect, as they are small compared to the main resonance
peak.
We also mention that our consideration can be generalized for the description
of slightly non-spherical grains. In this case the non ideal grain shape can
be studied within the perturbation theory. It will result in the changes of
the energy levels structure. In particular, the degenerate energy levels will
be split. That will slightly affect the positions of the resonance peaks.
To conclude the analysis of the polarizabilities we compare our findings for
$\alpha(\omega)$ with the known result for classical polarizability
$\alpha_{cl}(\omega)$, Mie1908
$\alpha_{cl}(\omega)=R^{3}\,\frac{\epsilon(\omega)-1}{\epsilon(\omega)+2},$
(2)
where $R$ is the radius of a grain and $\epsilon(\omega)$ is the grain
dielectric constant; within the Drude model
$\epsilon(\omega)=1-\frac{\omega_{p}^{2}}{\omega(\omega+i\delta)}$, where
$\omega_{p}=((4/3)\pi\nu)^{1/2}ev_{F}$ is the plasma resonance frequency and
$\delta$ is the damping factor. The imaginary part of the polarizability
$\alpha_{cl}(\omega)$ in Eq. (2) is given by
$\textrm{Im}\,\alpha_{cl}(\omega^{\prime})=R^{3}\frac{\omega^{\prime}\gamma}{(1-\omega^{\prime})^{2}+\gamma^{2}\omega^{\prime
2}},$ (3)
where we introduce the dimensionless parameter
$\gamma=\delta/(\omega_{p}/\sqrt{3})$ and the dimensionless frequency
$\omega^{\prime}=\omega/(\omega_{p}/\sqrt{3})$. The right hand side of Eq. (3)
has a single resonance peak at frequency $\omega^{\prime}=1$ (or
$\omega=\omega_{p}/\sqrt{3}$) corresponding to the surface plasmon mode
excitation. We plotted $\textrm{Im}\,\alpha_{cl}(\omega)$ in Figs. 1 and 2 for
comparison with our results.
Figure 3: (Color online) Blue shift of the surface plasmon frequency (in the
units of the classic surface plasmon frequency
$\omega_{cl}=\omega_{p}/\sqrt{3}$, with $\omega_{p}$ being the plasma
frequency) as a function of the dimensionless grain radius, $R/\lambda_{F}$,
where $R$ is the grain radius and $\lambda_{F}$ is a Fermi wavelength. Red
dots represent the numerical calculations. The blue solid line represents the
fitting of the numerical calculations with the function $(R/\lambda_{F})^{-1}$
The last important question to discuss is the dependence of the surface
plasmon frequency on the grain size. This dependence is shown in Fig. 3. The
calculated red dots were then fitted with the function that scales as
$(R/\lambda_{F})^{-1}$. On can clearly see that the blue shift has the
tendency to decrease as the grain size increases.
Now we turn to the description of our model and the derivation of electronic
polarizability $\alpha(\omega)$ in Eq. (1).
## III The Model
A single spherical metallic grain is described by the following Hamiltonian
$\hat{H}=\hat{H}_{0}+\hat{H}_{int}.$ (4a) Here $\hat{H}_{0}=\int
d^{3}r\psi^{{\dagger}}(\textbf{r})\left(-\frac{\nabla^{2}}{2m}+U(\textbf{r})\right)\psi(\textbf{r}),$
(4b) is the Hamiltonian of the non-interacting electrons trapped in the
infinitely deep spherical potential well $U(\textbf{r})$:
$U(\textbf{r})=\left\\{\begin{array}[]{lr}0,\hskip 8.5359pt|\textbf{r}|<R\\\
\infty,\hskip 8.5359pt|\textbf{r}|>R\end{array}\right.$ (4c) This form of the
potential allows us simple analytical expansion in eigenfunctions of the
Hamiltonian $\hat{H}_{0}$, which in its turn allows us to use the many-body
approach. The boundary conditions in Eq. 4c are valid if the number of
electrons near the grain surface is small compared to the total number of
electrons in the grain, i. e. $R/\lambda_{F}\gg 1$, where $R$ is the grain
radius and $\lambda_{F}$ is Fermi wavelength. In Eq. (4b)
$\psi^{{\dagger}}(\textbf{r})\,\,[\psi(\textbf{r})]$ is the creation
[annihilation] operator and the integral is performed over the grain volume.
The second term in the r. h. s. of Eq. (4a) is responsible for Coulomb
interaction
$\hat{H}_{int}=\int
d^{3}r\psi^{{\dagger}}(\textbf{r})\psi^{{\dagger}}(\textbf{r}^{\prime})\frac{e^{2}}{|\textbf{r}-\textbf{r}^{\prime}|}\psi(\textbf{r})\psi(\textbf{r}^{\prime}).$
(4d)
The common method of studying the electronic properties of such systems is
based on a mean-field approach. This approach reduces the many-body problem to
a single particle problem with renormalized effective potential, where the
effective potential contains all of the corrections due to the interaction
between the electrons. However, in the present manuscript we use a different
approach. Using the many-body Hamiltonian, Eq. (4a), we consider the Coulomb
interaction, Eq. (4d), perturbatively. This allows us to tackle the problem
without reducing it to a single particle problem with renormalized potential
$U(\textbf{r})$. We perform our calculations in the basis of the
eigenfunctions of the non-interacting Hamiltonian $\hat{H}_{0}$, Eq. (4b) and
take into account interaction effects, Eq. (4d), summing the appropriate
series of Feynman diagrams.
Our consideration is valid for metallic grains meaning that the grain itself
is a good conductor. This means that the grain size $R$ is much larger than
the Fermi wavelength $\lambda_{F}$. In addition, the disorder within the grain
is weak such that electrons within the grain are delocalized. If these
conditions are valid there is a small parameter in the problem - the invers
grain conductance $1/g$, where $g$ is the grain conductance. The perturbation
theory is built then on the expansion with respect to this small parameter.
To calculate the dynamic polarizability $\alpha(\omega)$ we study the
electronic response of a single metallic grain to the external electric
potential $\Phi^{ext}(\textbf{r})e^{-i\omega t}$. The polarizability is
defined as a coefficient between the external electric field E and the induced
dipole moment, $\textbf{p}(\omega)=\alpha(\omega)\textbf{E}$. Thus, the
problem is reduced to find the electric dipole moment $\textbf{p}(\omega)$
excited by the external field
$\textbf{p}(\omega)=\int d^{3}r\,\textbf{r}\rho^{in}(\omega,\textbf{r}),$ (5)
where $\rho^{in}(\omega,\textbf{r})$ is the field-induced electronic charge
density. Within the linear response theory $\rho^{in}(\omega,\textbf{r})$ is
given by the following expression, see e. g. Ref. Efetovbook
$\rho^{in}(\omega,\textbf{r})=-\int
d^{3}r^{\prime}\Phi(\omega,\textbf{r}^{\prime})\chi(\omega,\textbf{r},\textbf{r}^{\prime}).$
(6)
Here $\chi(\omega,\textbf{r},\textbf{r}^{\prime})$ is the electron density-
density correlation function and $\Phi(\omega,\textbf{r})$ is the total
electric potential, which is a sum of the external potential
$\Phi^{ext}(\omega,\textbf{r})$ and the induced potential
$\Phi^{in}(\omega,\textbf{r})$,
$\Phi(\omega,\textbf{r})=\Phi^{ext}(\omega,\textbf{r})+\Phi^{in}(\omega,\textbf{r})$.
Equation (6) corresponds to the random phase approximation Pines .
In the quasi-static limit the potential $\Phi(\omega,\textbf{r})$ can be found
by solving Poisson equation
$\nabla^{2}\Phi(\omega,\textbf{r})=-4\pi\rho^{in}(\omega,\textbf{r})$. Using
this expression in Eq. (6) we obtain the following integral equation for the
potential
$\nabla^{2}\Phi(\omega,\textbf{r})=4\pi\int
d^{3}r^{\prime}\Phi(\omega,\textbf{r}^{\prime})\chi(\omega,\textbf{r},\textbf{r}^{\prime}).$
(7)
In Eq. (7) the electron density-density correlation function
$\chi(\omega,\textbf{r},\textbf{r}^{\prime})$ is expressed in terms of
electron Green’s functions $G(\epsilon,\textbf{r}^{\prime},\textbf{r})$ as
$\chi(\omega,\textbf{r},\textbf{r}^{\prime})=-T\sum_{\epsilon}G(\epsilon+\omega,\textbf{r},\textbf{r}^{\prime})G(\epsilon,\textbf{r}^{\prime},\textbf{r})$.
Using the properties of Green’s functions
$\chi(\omega,\textbf{r},\textbf{r}^{\prime})$ can be written in the form, see
e. g. Ref. Mahanbook
$\chi(\omega,\textbf{r},\textbf{r}^{\prime})=2e^{2}\sum_{n,n^{\prime}}\frac{f(\epsilon_{n})-f(\epsilon_{n^{\prime}})}{\omega-\epsilon_{n}+\epsilon_{n}^{\prime}}\psi^{*}_{n}(r)\psi_{n}(r^{\prime})\psi^{*}_{n^{\prime}}(r^{\prime})\psi_{n^{\prime}}(r),$
(8)
where $f(\epsilon_{n})$ is the Fermi-Dirac distribution function with
$\epsilon_{n}$ being the energy level of the quantum system and $\psi_{n}(r)$
is the eigenfunction of the non-interacting Hamiltonian with $n$ being the
full set of quantum numbers that describe the state. Below we consider the
solution of Eq. (7) in details.
## IV Solution
To solve Eq. (7) we introduce the eigenfunctions $\psi_{nlm}$ and
corresponding eigenvalues $\epsilon_{ln}$ for electrons in the spherical
potential well $U({\textbf{r}})$ of radius $R$
$\psi_{nlm}=\beta_{ln}j_{l}(\alpha_{ln}r/R)Y_{lm}(\theta,\phi),\hskip
5.69046pt\epsilon_{ln}=\alpha_{ln}^{2}/2mR^{2},$ (9)
where $\alpha_{ln}$ is the $n$-th zero of the spherical Bessel function
$j_{l}(x)$, $Y_{lm}(\theta,\phi)$ is the spherical harmonic, and
$\beta_{ln}=(2^{1/2}/R^{3/2})j^{\prime}_{l}(\alpha_{ln})$ is the normalization
coefficient with $j^{\prime}_{l}(\alpha_{ln})$ being the derivative of the
Bessel function taken at $x=\alpha_{ln}$.
The potential $\Phi(\omega,\textbf{r})$ in Eq. (7) consists of two parts:
applied external potential and induced dipole part Rice . Choosing the
direction of the $z$ axis along the external electric field $E_{\omega}$ we
obtain
$\Phi(\omega,\textbf{r})=-E_{\omega}r\cos{\theta}+\Phi^{in}(\omega,\textbf{r}).$
(10)
Here the induced electric field $\Phi^{in}(\omega,\textbf{r})$ vanishes at
infinity,
$\lim_{|\textbf{r}|\rightarrow\infty}\nabla\Phi^{in}(\omega,\textbf{r})=0$. In
addition, the potential $\Phi^{in}(\omega,\textbf{r})$ and its derivative are
continuous everywhere including the grain surface. Outside the grain the
charge density $\rho^{in}(\omega,\textbf{r})$ is zero therefore the potential
$\Phi^{in}(\omega,\textbf{r})$ must satisfy the Laplace’s equation
$\nabla^{2}\Phi^{in}(\omega,\textbf{r})=0,~{}|r|>R.$ (11)
Thus, the potential outside the grain can be found as a classical dipole
potential
$\Phi^{in}(\omega,\textbf{r})=\frac{c_{0}(\omega)E_{\omega}R^{3}\cos\theta}{r^{2}},$
(12)
where $c_{0}(\omega)$ is yet unknown function of frequency $\omega$.
Inside the grain one can seek the solution in the form
$\Phi^{in}(\omega,\textbf{r})=E_{\omega}\left[c_{0}(\omega)r+R\,f(\omega,r)\,\right]\cos\theta,$
(13)
where $f(\omega,r)$ is some unknown function of frequency $\omega$ and
distance $r$. Introducing the function $g(\omega,r)$ as
$\rho^{in}(\omega,\textbf{r})=\frac{E_{\omega}R}{4\pi}g(\omega,r)\cos\theta,$
(14)
we can rewrite Eq. (6) as follows
$g(\omega,r)=\int_{0}^{R}dss^{2}q(\omega,r,s)\left[(c_{0}(\omega)-1)s/R+f(\omega,s)\right],$
(15)
with the kernel $q(\omega,r,s)$ being
$\frac{q(\omega,r,s)}{4\pi}=\int\cos\theta d\phi
d\theta\int\cos\theta^{\prime}d\phi^{\prime}d\theta^{\prime}Y_{10}(\theta)Y_{10}(\theta^{\prime})\chi(\omega,r,s).$
(16)
Since we are interested in the dipole response of the grain we expand the
functions $f(\omega,r)$ and $g(\omega,r)$ in Eq. (15) using the orthogonal set
$\phi_{n}(r)=j_{1}(\alpha_{1n}r/R)$
$\displaystyle f(\omega,r)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}c_{n}(\omega)j_{1}\left(\frac{\alpha_{1n}r}{R}\right),$
$\displaystyle g(\omega,r)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}c_{n}(\omega)\left(\frac{\alpha_{1n}}{R}\right)^{2}j_{1}\left(\frac{\alpha_{1n}r}{R}\right).$
(17)
Thus, the problem of calculating the induced potential
$\Phi^{in}(\omega,\textbf{r})$ inside the grain is reduced to the problem of
calculating the frequency dependent coefficients $c_{n}(\omega)$ in Eq. (IV).
Using these coefficients the dynamic polarizability $\alpha(\omega)$ of a
grain can be written as follows
$\alpha(\omega)=c_{0}(\omega)R^{3},$ (18)
with $R^{3}\equiv\alpha_{cl}(0)$ being the static classical polarizability.
The coefficient $c_{0}(\omega)$ can be considered as a dimensionless dynamic
polarizability and can be found using the boundary conditions
$c_{0}(\omega)=-\frac{1}{3}\sum_{n=1}^{\infty}c_{n}(\omega)j_{1}^{\prime}(\alpha_{1n})\alpha_{1n}.$
(19)
To calculate the frequency dependent coefficients $c_{n}(\omega)$ in Eq. (19)
we substitute Eqs. (IV) into Eq. (15) to obtain the system of linear equations
for coefficients $c_{n}(\omega)$
$A_{mn}(\omega)\,c_{n}(\omega)=a_{m}(\omega),$ (20a) where $\displaystyle
a_{m}(\omega^{\prime})=\frac{3}{2\sqrt{2}\pi}\sqrt{\frac{\lambda_{F}}{a_{B}}}\frac{\lambda_{F}}{R}\sum_{l,n_{1},n_{2}}\left(\frac{f_{ln_{1}}-f_{(l-1)n_{2}}}{\omega^{\prime}-\epsilon_{ln_{1}}+\epsilon_{(l-1)n_{2}}}\frac{l}{\gamma_{ln_{1}}\gamma_{(l-1)n_{2}}}B^{1}_{m}(l,n_{1},n_{2})C^{1}(l,n_{1},n_{2})\right.$
$\displaystyle+\left.\frac{f_{ln_{1}}-f_{(l+1)n_{2}}}{\omega^{\prime}-\epsilon_{ln_{1}}+\epsilon_{(l+1)n_{2}}}\frac{l+1}{\gamma_{ln_{1}}\gamma_{(l+1)n_{2}}}B^{2}_{m}(l,n_{1},n_{2})C^{2}(l,n_{1},n_{2})\right),$
(20b) $\displaystyle
A_{mn}(\omega^{\prime})=\frac{3}{2\sqrt{2}\pi}\sqrt{\frac{\lambda_{F}}{a_{B}}}\frac{\lambda_{F}}{R}\sum_{l,n_{1},n_{2}}\left(\frac{f_{ln_{1}}-f_{(l-1)n_{2}}}{\omega^{\prime}-\epsilon_{ln_{1}}+\epsilon_{(l-1)n_{2}}}\frac{l}{\gamma_{ln_{1}}\gamma_{(l-1)n_{2}}}B^{1}_{m}(l,n_{1},n_{2})B^{1}_{n}(l,n_{1},n_{2})\right.$
$\displaystyle\left.+\frac{f_{ln_{1}}-f_{(l+1)n_{2}}}{\omega^{\prime}-\epsilon_{ln_{1}}+\epsilon_{(l+1)n_{2}}}\frac{l+1}{\gamma_{ln_{1}}\gamma_{(l+1)n_{2}}}B^{2}_{m}(l,n_{1},n_{2})B^{2}_{n}(l,n_{1},n_{2})\right)$
$\displaystyle-\frac{1}{3}a_{m}(\omega^{\prime})\alpha_{1n}j_{1}^{\prime}(\alpha_{1n})-\delta_{mn}\alpha_{1n}^{2}\gamma_{1n}.$
(20c)
Here $a_{B}=1/me^{2}$ is the Bohr radius and we introduce the notation
$\gamma_{ln}=\frac{1}{2}\left[j_{l}^{\prime}(\alpha_{ln})\right]^{2}$, and the
dimensionless frequency $\omega^{\prime}=\omega/(\omega_{p}/\sqrt{3})$, with
$\omega_{p}^{2}=(4/3)\pi\nu e^{2}v_{F}^{2}$ being the plasma frequency.
The quantities $B^{1}_{m}(l,n_{1},n_{2})$, $B^{2}_{m}(l,n_{1},n_{2})$,
$C^{1}(l,n_{1},n_{2})$, and $C^{2}(l,n_{1},n_{2})$ in Eqs. (20) and (20c) are
defined as
$\displaystyle B^{1(2)}_{m}(l,n_{1},n_{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dxx^{2}j_{1}(\alpha_{1n}x)j_{l}(\alpha_{ln_{1}}x)j_{l\mp
1}(\alpha_{(l\mp 1)n_{2}}x),$ $\displaystyle C^{1(2)}(l,n_{1},n_{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{1}dxx^{3}j_{l}(\alpha_{ln_{1}}x)j_{l\mp
1}(\alpha_{(l\mp 1)n_{2}}x),$ (21)
where the upper index $1$ $(2)$ corresponds to $-$ $(+)$. The Fermi-Dirac
functions $f_{ln}$ in Eqs. (20) and (20c) are taken at zero temperature and
defined as
$f_{ln}=\left\\{\begin{array}[]{lr}1,\hskip 8.5359pt\alpha_{ln}<\alpha_{F}\\\
0,\hskip 8.5359pt\alpha_{ln}>\alpha_{F},\end{array}\right.$ (22)
where the dimensionless parameter $\alpha_{F}$ defines the Fermi energy
$E_{F}=(\hbar^{2}/2mR^{2})\alpha_{F}^{2}$. Using Eq. (22) for the distribution
function $f_{ln}$ one can calculate the number of the electrons in the
conducting band as follows
$N=2\sum_{ln}(2l+1)f_{ln}.$ (23)
For numerical calculations we used the complex frequency
$\omega^{\prime}+i\gamma^{\prime}$ with $\gamma^{\prime}=0.006$ being the
dimensionless damping factor. The damping factor $\gamma^{\prime}$ was
introduced to avoid infinite values of the polarizability at the resonances
and does not correspond to the real plasmon resonance broadening. The analysis
of the resonance broadening is beyond the scope of the present work.
Using Eqs. (18) - (20c) we numerically calculate the dynamic polarizability
$\alpha(\omega)$ in Eq. (18) for grains of different sizes. The final results
are shown in Fig. 1 and Fig. 2
## V Conclusions
We studied the dynamic polarizability of spherical metallic grains using
quantum mechanical treatment. We numerically investigated the frequency
behavior of polarizability for relatively large grains. We showed that the
main resonance peak corresponding to the surface plasmon mode is blue-shifted
and some minor secondary resonances above and below the main peak exist. We
studied the dependence of blue shift as a function of grain size and compared
our results with the classical polarizability.
###### Acknowledgements.
We thank G. Weick for helpful discussions. This research was supported by an
award from Research Corporation for Science Advancement.
## References
## References
* (1) I. S. Beloborodov, A. V. Lopatin, V. M. Vinokur, and K. B. Efetov, Rev. Mod. Phys. 79, 469 (2007).
* (2) L. P. Gorkov and G. M. Eliashberg, Sov. Phys. JETP 21, 940 (1965).
* (3) M. J. Rice, W. R. Schneider, and S. Strässler, Phys. Rev. B 8, 474 (1973).
* (4) W. Ekardt, Phys. Rev. Lett. 52, 1925 (1984); W. Ekardt, Phys. Rev. B 31, 6360 (1985).
* (5) W. Ekardt, Phys. Rev. B 29, 1558 (1984).
* (6) D. E. Beck, Solid State Commun. 49, 381 (1984).
* (7) E. Koch and O. Gunnarsson, Phys. Rev. B 54, 5168 (1996).
* (8) G. Weick, R. A. Molina, D. Weinmann and R. A. Jalabert, Phys. Rev. B 72, 115410 (2005).
* (9) E. Prodan and P. Nordlander, Chem. Phys. Lett. 352, 140 (2002).
* (10) R. Ruppin, Phys. Rev. B 11, 2871 (1975).
* (11) W. P. Halperin, Rev. Mod. Phys. 58, 533 (1986).
* (12) D. M. Wood and N. W. Ashcroft, Phys. Rev. B 25, 6255 (1982).
* (13) L. Genzel and U. Kreibig, Z. Phys. B 37, 93 (1980).
* (14) K. B. Efetov, Phys. Rev. Lett. 76, 1908 (1996).
* (15) Y. N. Ovchinnikov, JETP 94, 1188 (2002).
* (16) C. J. Duthler, S. E. Johnson, and H. P. Broida, Phys. Rev. Lett. 26, 1236 (1971).
* (17) L. Genzel, T. P. Martin, and U. Kreibig, Z. Phys. B 21, 339 (1975).
* (18) K. B. Efetov, Supersymmetry in disorder and chaos (Cambridge University Press, New York, 1997).
* (19) D. Pines and Ph. Nozieres, The theory of quantum liquids (Perseus Books, Cambridge, Massachusetts, 1999).
* (20) G. D. Mahan, Many-particle physics (Kluwer, New York, 2000)
* (21) G. Mie, Ann. Phys. 330, 377 (1908).
|
arxiv-papers
| 2010-08-20T19:13:51 |
2024-09-04T02:49:12.288297
|
{
"license": "Public Domain",
"authors": "A. A. Zharov and I. S. Beloborodov",
"submitter": "Alexander Zharov",
"url": "https://arxiv.org/abs/1008.3556"
}
|
1008.3596
|
# Exact Bivariate Polynomial Factorization in Q by Approximation of Roots
††thanks: The work is partially supported by National Basic Research Program
of China 2011CB302400 and the National Natural Science Foundation of
China(Grant NO.10771205).
Yong Feng Wenyuan Wu Jingzhong Zhang
Laboratory of Computer Reasoning and Trustworthy Computing,
University of Electronic Sciences and Technology,
Chengdu, P. R. China 611731
Corresponding author: dr.wenyuanwu@gmail.com
###### Abstract
Factorization of polynomials is one of the foundations of symbolic
computation. Its applications arise in numerous branches of mathematics and
other sciences. However, the present advanced programming languages such as
C++ and J++, do not support symbolic computation directly. Hence, it leads to
difficulties in applying factorization in engineering fields. In this paper,
we present an algorithm which use numerical method to obtain exact factors of
a bivariate polynomial with rational coefficients. Our method can be directly
implemented in efficient programming language such C++ together with the GNU
Multiple-Precision Library. In addition, the numerical computation part often
only requires double precision and is easily parallelizable.
Key words: Factorization of multivariate polynomials, Minimal polynomial,
Interpolation methods, Numerical Continuation.
## 1 Introduction
Polynomial factorization plays a significant role in many problems including
the simplification, primary decomposition, factorized Gröbner basis, solving
polynomial equations and some engineering applications, etc. It has been
studied for a long time and some high efficient algorithms have been proposed.
There are two types of factorization approaches. One is the traditional
polynomial factorization for exact input relying on symbolic computation, and
the other is approximate polynomial factorization for inexact input.
The traditional polynomial factorization methods follow Zassenhaus’ approach
[25][26]. First, Multivariate polynomial factorization is reduced to bivariate
factorization due to Bertini’s theorem and Hensel lifting[6, 7]. Then one of
the two remaining variables is specialized at random. The resulting univariate
polynomial is factored and its factors are lifted up to a high enough
precision. At last, the lifted factors are recombined to get the factors of
the original polynomial.
Approximate factorization is a natural extension of conventional polynomial
factorization. It adapts factorization problem to linear algebra first, then
applies numerical methods to obtain an approximate factorization in complex
which is the exact absolute factorization of a nearby problem. In 1985,
Kaltofen presented an algorithm for computing the absolute irreducible
factorization by floating point arithmetic [13]. Historically, the concept of
approximate factorization appeared first in a paper on control theory[18]. The
algorithm is as follows: 1) represent the two factors $G$ and $H$ of the
polynomials $F$ with unknown coefficients by fixing their terms, 2) determine
the numerical coefficients so as to minimize $\|F-GH\|$. Huang et al [8].
pursued this approach, but it seems to be rarely successful, unless $G$ or $H$
is a polynomial of several terms. In 1991, Sasaki et al. proposed a modern
algorithm[19], which use power-series roots to find approximate factors. This
algorithm is successful for polynomials of small degrees. Subsequently, Sasaki
et al. presented another algorithm[20] which utilizes zero-sum relations. The
zero-sum relations are quite effective for determining approximate factors.
However, computation based on zero-sum relations is practically very time-
consuming. In [21], Sasaki presented an effective method to get as many zero-
sum relations as possible by matrix operations so that approximate
factorization algorithm is improved. Meanwhile, Corless et al proposed an
algorithm for factoring bivariate approximate polynomial based on the idea of
decomposition of affine variety in [4]. However, it is not so efficient to
generalize the algorithm to multivariate case. A major breakthrough is due to
Kaltofen et al. [12, 14] who extended Gao’s work [11] from symbolics to
numerics based on Ruppert matrix and Singular Value Decomposition.
Symbolic factorization has been implemented in many Computer Algebra System.
However, it is difficult to implemented directly in Programming Language such
as C++ and J++, because most of Programming Language standards do not support
symbolic basic operators, and the compilers do not implement the symbolic
computation, on which symbolic factorization is based. It restricts exact
factorization from being applied in many engineering fields. Compared with
symbolic factorization, approximate factorization can be implemented more
easily in the popular programming languages. However it only gives approximate
results even the input is exact. In this paper, we propose an almost
completely numerical algorithm, which is not only implemented directly in the
programming languages, but also achieve exact results.
Except classic symbolic methods, some approaches have been proposed to obtain
exact output by approximation[15][27]. The idea of obtaining exact polynomial
factorization is from the connect between an approximate root of a given
polynomial and its minimal polynomial in $Q$. Certainly, the minimal
polynomial is a factor of the given input. Based on lattice basis reduced
algorithm LLL and Integer Relation algorithm PSLQ of a vectors respectively,
there are two algorithms for finding exact minimal polynomial of an algebraic
number from its approximation. One is a numerical algorithm [15, 5] for
factorization of a univariate polynomial was provided by Transcendental
Evaluation and high-degree evaluation, and the other for factorization of
bivariate polynomial is based on LLL[9, 3]. But they are not efficient.
In this paper, relying on LLL algorithm, we present an almost-completely
numerical method for exact factoring polynomial with rational coefficient in
$\mathbb{Q}$. First, we choose a sample point in $\mathbb{Q}^{n-1}$ at random.
After specialization (i.e. substitution) at the point, the roots of the
resulting univariate polynomial can be found very efficiently up to
arbitrarily high accuracy. Then applying minimal polynomial algorithm to these
roots yields an exact factorization of the univariate polynomial in
$\mathbb{Q}$. Next we shall move the sample point in “good direction” to
generate enough number of points by using numerical continuation. Especially,
for the rest sample points, the corresponding exact factorization can be found
by using the same combination of roots as found in the first step. And these
roots give more univariate polynomials for next step. Finally, the
multivariate factorization can be obtained by interpolation.
The paper is organized as follows. Section 2 gives a brief introduction of the
preparation knowledge. Minimal polynomial algorithm will be discussed in
Section 3. Then we present our method in Section 4,5 and 6.
## 2 Preparation
In this section, we briefly introduce the background knowledge and related
topics to our article.
### 2.1 Homotopy Continuation Methods
Homotopy continuation methods play a fundamental role in Numerical Algebraic
Geometry and provide an efficient and stable way to compute all isolated roots
of polynomial systems. These methods have been implemented in many software
packages e.g. Hom4PS [16], Bertini [1], PHCpack [24].
The basic idea is to embed the target system into a family of systems
continuously depending on parameters. Then each point in the parameter space
corresponds to a set of solutions. Suppose we know the solutions at a point.
Then we can track the solutions from this starting point to the point
representing the target system we want to solve.
First let us look at the simplest case: a univariate polynomial $f(z)$ with
degree $d$. We know that $f(z)$ has $d$ roots in $\mathbb{C}$ (counting
multiplicities). Of course we can embed $f(z)$ into the family
$a_{d}z^{d}+a_{d-1}z^{d-1}+\cdots+a_{0}$, where the $a_{i}$ are parameters.
Now choose a start point corresponding to $z^{d}-1$ in this parameter space,
whose roots are
$z^{0}_{k}=e^{2k\pi\sqrt{-1}/d},\;k=0,1,...,d-1$ (1)
Then we use a real straight line in the parameter space to connect $z^{d}-1$
with $f(z)$:
$H(z,t):=tf(z)+(1-t)(z^{d}-1).$ (2)
This form is a subclass of the family depending on only one real parameter
$t\in[0,1]$.
When $t=0$ we have the start system $H(z,0)=z^{d}-1$ and when $t=1$ we have
our target system $H(z,1)=f(z)$. An important question is to show how to track
individual solutions as $t$ changes from $0$ to $1$. Let us look at the
tracking of the solution $z_{k}$ (the $k$-th root of $f(z)$). When $t$ changes
from $0$ to $1$, it describes a curve, which is function of $t$, denoted by
$z_{k}=z_{k}(t)$. So $H(z_{k}(t),t)\equiv 0$ for all $t\in[0,1]$.
Consequently, we have
$0\equiv\frac{dH(z_{k}(t),t)}{dt}=\frac{\partial H(z,t)}{\partial
z}\frac{dz_{k}(t)}{dt}+\frac{\partial H(z,t)}{\partial t}.$ (3)
This problem is reduced to an ode for the unknown function $z_{k}(t)$ together
with an algebraic constraint $H(z_{k}(t),t)\equiv 0$. The initial condition is
the start solution $z_{k}(0)=z^{0}_{k}$ and $z_{k}(1)$ is a solution of our
target problem $f(z)=0$.
###### Remark 1
In the book [2], Blum, Smale et al. show that on average an approximate root
of a generic polynomial system can be found in polynomial time. Also
application of the polynomial cost method for numerically solving differential
algebraic equations [10] gives polynomial cost method for solving homotopies.
But there is a prerequisite for the continuous tracking: $\frac{\partial
H(z,t)}{\partial z}\neq 0$ along the curve $z=z_{k}(t)$. If the equations
$z-z_{k}(t)=0$ and $tf^{\prime}(z)+d(1-t)z^{d-1}=0$ have intersection at some
point $(t,z_{k}(t))$, then we cannot continue the tracking. There is way to
avoid this singular case, called the “gamma trick” that was first introduced
in [17]. We know two complex curves almost always have intersections at
complex points, but here $t$ must be real. So if we introduce a random complex
transformation to the second curve, the intersection points will become
complex points and such a singularity will not appear when $t\in[0,1)$. Let us
introduce a random angle $\theta\in[-\pi,\pi]$ and modify the homotopy (2) to
$H(z,t):=tf(z)+e^{i\theta}(1-t)(z^{d}-1).$ (4)
It is easy to show that the $k$-th starting solution is still $z^{0}_{k}$ in
(1) and that $z_{k}(1)$ is still a root of $f(z)$.
### 2.2 Genericity and Probability One
In an idealized model where paths are tracked exactly and the random angle can
be generated to infinite precision, the homotopy (4) can be proved to succeed
“with probability one”. To clarify this statement, it is necessary to use a
fundamental concept in algebraic geometry: genericity.
###### Definition 1 (Generic)
Let $X$ be an irreducible algebraic variety. We say a property $P$ holds
generically on $X$, if the set of points of $X$ that do not satisfy $P$ are
contained in a proper subvariety $Y$ of $X$. The points in $X\backslash Y$ are
called generic points.
The set $X\backslash Y$ is called a Zariski open set of $X$. Roughly speaking,
if $Y$ is a proper subvariety of an irreducible variety $X$ and $p$ is a
random point on $X$ with uniform probability distribution, then the
probability that $p\notin Y$ is one. So we can consider a random point as a
generic point on $X$ without a precise description of $Y$. Many of the
desirable behaviors of homotopy continuation methods rely on this fact.
### 2.3 Coefficient-Parameter Homotopy
There are several versions of the Coefficient-Parameter theorem in [22]. Here
we only state the basic one.
###### Theorem 1
Let $F(z;q)=\\{f_{1}(z;q),...,f_{n}(z;q)\\}$ be a polynomial system in $n$
variables $z$ and $m$ parameters $q$. Let $\mathcal{N}(q)$ denote the number
of nonsingular solutions as a function of $q$:
$\mathcal{N}(q):=\\#\left\\{z\in\mathbb{C}^{n}:F(z;q)=0,\;\det\left(\frac{\partial
F}{\partial z}(z;q)\right)\neq 0\right\\}$ (5)
Then,
1. 1.
There exist $N$, such that $\mathcal{N}(q)\leq N$ for any
$q\in\mathbb{C}^{m}$. Also $\\{q\in\mathbb{C}^{m}:\mathcal{N}(q)=N\\}$ is a
Zariski open set of $\mathbb{C}^{m}$. The exceptional set
$Y=\\{q:\mathcal{N}(q)<N\\}$ is an affine variety contained in a variety with
dimension $m-1$.
2. 2.
The homotopy $F(z;\phi(t))=0$ with
$\phi(t):[0,1)\rightarrow\mathbb{C}^{m}\backslash Y$ has $N$ continuous non-
singular solution paths $z(t)$.
3. 3.
When $t\rightarrow 1^{-}$, the limit of $z_{k}(t),\;k=1,...,N$ includes all
the non-singular roots of $F(z;\phi(1))$.
An important question is how to choose a homotopy path $\phi(t)$ which can
avoid the exceptional set $Y$. The following lemma [22] gives an easy way to
address this problem.
###### Lemma 1
Fix a point $q$ and a proper algebraic set $Y$ in $\mathbb{C}^{m}$. For a
generic point $p\in\mathbb{C}^{m}$, the one-real-dimensional open line segment
$\phi(t):=(1-t)\;p+t\;q,t\in[0,1)$ is contained in $\mathbb{C}^{m}\backslash
Y$.
### 2.4 Reductions
Before factorization of a given polynomial, we shall first apply certain
reductions to the input to obtain a square-free polynomial over $\mathbb{Q}$,
which can remove multiplicities and ease the computation of the roots. Also we
can assume each factor involves all the variables and has more than one term.
Otherwise, we can compute the GCD to reduce the problem. For example, let
$F=f(x,y)g(y)$. Then $F_{x}=f_{x}g$ and $\gcd(F,F_{x})=g$ which gives us the
factor $g(y)$.
By the Hilbert Irreducibility Theorem, we can further reduce the problem to
univariate case by generic (random) specialization of one variable to a
rational number. More precisely, if $f(x,y)$ is irreducible in
$\mathbb{Q}[x,y]$, then for a generic rational number $y_{0}$, $f(x,y_{0})$ is
also irreducible in the ring $\mathbb{Q}[x]$. It means that the factorization
is commutable with generic specialization.
For univariate polynomial, there are symbolic methods to preform exact
factorization in $\mathbb{Q}$. Here we are more interested in numerical
methods, namely from approximate roots to exact factors.
## 3 Minimal Polynomial by Approximation
There are two methods to compute the minimal polynomial of an algebraic number
from its approximation. One is based on the LLL algorithm of the basis
reduction[15], and another is based on PSLQ[5]. The later one is more
efficient than the former one. However, it can only compute the minimal
polynomial of a real algebraic number while the former one can find minimal
polynomial of a complex algebraic number. Hence, we introduce the former
algorithm which is more suitable for this paper here. We refer the reader to
the paper [15] for more details.
Let $p(x)=\sum_{i=0}^{i=n}p_{i}x^{i}$ be a polynomial. The length $|p|$ of
$p(x)$ is defined as the Euclidean norm of the vector
$(p_{0},p_{1},\cdots,p_{n})$, and the height $|p|_{\infty}$ as the
$L_{\infty}$-norm of the vector $(p_{0},p_{1},\cdots,p_{n})$. The degree and
height of an algebraic number are defined as the degree and height,
respectively, of its minimal polynomial.
Suppose that we have upper bound $d$ and $H$ on the degree and height
respectively of an algebraic number with $|\alpha|\leq 1$, and a complex
rational number $\bar{\alpha}$ approximating $\alpha$ such that
$|\bar{\alpha}|\leq 1$ and $|\alpha-\bar{\alpha}|<2^{-s}/(4d)$, where $s$ is
the smallest positive integer such that
$2^{s}>2^{d^{2}/2}(d+1)^{(3d+4)/2}H^{2d}$
###### Algorithm 1
[miniPoly]
For $n=1,2,\cdots,d$ in succession, do the following steps
* Step 1:
construct
$\left(\begin{array}[]{ccccccc}1&0&0&\cdots&0&2^{s}\cdot
Re(\bar{\alpha}_{0})&2^{s}\cdot Im(\bar{\alpha}_{0})\\\
0&1&0&\cdots&0&2^{s}\cdot Re(\bar{\alpha}_{1})&2^{s}\cdot
Im(\bar{\alpha}_{1})\\\ 0&0&1&\cdots&0&2^{s}\cdot
Re(\bar{\alpha}_{2})&2^{s}\cdot Im(\bar{\alpha}_{2})\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&0&0&\cdots&1&2^{s}\cdot
Re(\bar{\alpha}_{n})&2^{s}\cdot Im(\bar{\alpha}_{n})\end{array}\right)$ (6)
where $Re(a)$ and $Im(a)$ stand for the real part and imaginary part,
respectively, of complex $a$, $\alpha_{0}=1$ and
$|\bar{\alpha_{i}}-\bar{\alpha}^{i}|\leq 2^{-s-1/2}$ for $i=1,2,\cdots,n$.
Note $\bar{\alpha_{i}}$ can be computed by rounding the powers of
$\bar{\alpha}$ to $s$ bits after the binary points.
* Step 2:
Denote by $b_{i}$ the row $i+1$ of the matrix in (6). Apply the basic
reduction algorithm to lattice $L_{s}=(b_{0},b_{1},\cdots,b_{n})$, and obtain
the reduced basis of the lattice.
* Step 3:
If the first basis vector
$\tilde{v}=(v_{0},v_{1},\cdots,v_{n},v_{n+1},v_{n+2})$ in the reduced basis
satisfies $|\tilde{v}|\leq 2^{d/2}(d+1)H$, then return polynomial
$v(x)=\sum_{i=0}^{n}v_{i}x^{i}$ as the minimal polynomial of algebraic number
$\alpha$.
Note: It is no major restriction to consider $\alpha$ with $|\alpha|\leq 1$
only. In fact, if $|\alpha|>1$ satisfies the polynomial
$h(x)=\sum_{i=0}^{d}h_{i}x^{i}$, then $1/\alpha$ satisfies the polynomial
$\sum_{i=0}^{d}h_{d-i}x^{i}$. Therefore, if $\sum_{i=0}^{d}h_{d-i}x^{i}$ is
computed, the $h(x)$ is obtained. Furthermore, an $\varepsilon$-approximation
$\bar{\alpha}$ to $\alpha$ with $|\alpha|>1$ easily yields a
$3\varepsilon$-approximation $\bar{\beta}$ to $\beta=1/\alpha$. This can be
easily verified.
The following theorem shows the computation amount of calculating the minimal
polynomial of an algebraic number[15]:
###### Theorem 2
Let $\alpha$ be an algebraic number, and let $d$ and $H$ be upper bounds on
the degree and height, respectively, of $\alpha$. Suppose that we are given an
approximation $\bar{\alpha}$ to $\alpha$ such that $|\alpha-\bar{\alpha}|\leq
2^{-s}/(12d)$, where $s$ is the smallest positive integer such that
$2^{s}>2^{d^{2}/2}(d+1)^{(3d+4)/2}H^{2d}$
Then the minimal polynomial of $\alpha$ can be determined in $O(n_{0}\cdot
d^{4}(d+\log H))$ arithmetic operations on integers having
$0(d^{2}\cdot(d+\log H))$ binary bits, where $n_{0}$ is the degree of
$\alpha$.
## 4 Finding More Polynomials by Continuation
In the previous stage, we have the factors after specialization , which are
univariate polynomials. To construct the factor of two variables by using
interpolation, we need more information, i.e. specializations at more points.
The main tool is the homotopy continuation method.
### 4.1 Applying numerical continuation to factorization
Suppose an input polynomial $F(x,y)$ is reducible. Geometrically, if
$\mathcal{C}$ denotes the zero set of $f$ i.e. the union of many curves in
$\mathbb{C}^{2}$, removing the singular locus of $\mathcal{C}$ from each curve
$\mathcal{C}_{i}$, the regular sets $\mathcal{S}_{i}$ are connected in
$\mathbb{C}^{2}$. Moreover, the singular locus has lower dimension,
consequently it is a set of isolated points.
Suppose $f(x,y)$ is an irreducible factor of $F$ in $\mathbb{Q}$. Let
$y_{0},y_{1}$ be random rational numbers. By the Hilbert Irreducibility
Theorem the univariate polynomials $f_{0}=f(x,y_{0})$ and $f_{1}=f(x,y_{1})$
are irreducible as well. Suppose we know the roots of $f_{0}$. Then we can
choose a path to connect $y_{0}$ and $y_{1}$ avoiding the singular locus which
has measure zero. By the Coefficient-Parameter Theorem, all the roots of
$f_{1}$ can be obtained by the following homotopy continuations:
$\left\\{\begin{array}[]{cl}f(x,y)=0&\\\ (1-t)(y-y_{0})+t(y-y_{1})\gamma=0&\\\
\end{array}\right.$ (7)
Moreover any generic complex number $\gamma$ implies that the homotopy path
can avoid the singular locus by Lemma 1 when we track the path.
### 4.2 Control of the precision
Let $\\{x_{1},..,x_{m}\\}$ be the exact roots of $f_{1}$ and $g$ be the
primitive polynomial of $f_{1}$. Then
$g=\alpha\prod_{i=1}^{m}(x-x_{i})\in\mathbb{Z}[x],$ (8)
for some integer number $\alpha$.
Note that we only have the approximate roots
$\\{\tilde{x}_{1},..,\tilde{x}_{m}\\}$.
###### Proposition 1
Let $p=\prod_{i=1}^{m}(x-x_{i})$ and
$\tilde{p}=\prod_{i=1}^{m}(x-\tilde{x}_{i})$. Let
$\delta=\max_{i=1,..,m}\\{|x_{i}-\tilde{x}_{i}|\\}$ and
$r=\max_{i=1,..,m}\\{|\tilde{x}_{i}|\\}$. If $\delta$ is sufficiently small.
Then
$||\tilde{\textbf{p}}-\textbf{p}||_{\infty}\leq\left(\max_{i=1,..,m}\\{\;r^{i-1}{{m-1}\choose{i-1}}\\}\;m+1\right)\;\delta$
(9)
Proof. Let $x_{i}=\tilde{x}_{i}+\delta_{i}$. Thus, $|\delta_{i}|\leq\delta$.
The left hand side
$||\tilde{\textbf{p}}-\textbf{p}||_{\infty}=||\prod_{i=1}^{m}(x-x_{i}+\delta_{i})-\prod_{i=1}^{m}(x-x_{i})||=||\sum_{j=1}^{m}\prod_{i\neq
j}(x-x_{j})\delta_{j}||+o(\delta)$. An upper bound of the coefficients of
$\prod_{i\neq j}(x-x_{j})$ with respect to $x^{m-i}$ is
${{m-1}\choose{i-1}}r^{i-1}$. Hence,
$||\tilde{\textbf{p}}-\textbf{p}||_{\infty}\leq\max_{i=1,...,m}{{m-1}\choose{i-1}}r^{i-1}\cdot
m\delta+\delta$ $\square$
Now let us consider how to find $\alpha$. Suppose the input polynomial is
$F(x,y)$ and $f$ is a factor of $F$. The primitive polynomial of $f(x,y_{1})$,
which is $g$, must be a factor of the primitive polynomial of $F(x,y_{1})$.
Thus, the leading coefficient of $g$ must be a factor of the leading
coefficient of the primitive polynomial of $F(x,y_{1})$. Let $\alpha$ be the
leading coefficient of the primitive polynomial of $F(x,y_{1})$. Then let
$p=\alpha\prod_{i=1}^{m}(x-x_{i})\in\mathbb{Z}[x]$. Note that itself may not
be primitive, but its primitive polynomial is $g$.
Let $M=\max_{i=1,..,m}\\{i\;r^{i-1}{{m}\choose{i}}\\}+1$.
$\tilde{p}=\alpha\prod_{i=1}^{m}(x-\tilde{x}_{i})$. Thus, if
$\delta<\frac{1}{2\alpha M}$ then $||p-\tilde{p}||_{\infty}<0.5$. It means
that we can round each coefficient of $\tilde{p}$ to the nearest integer to
obtain exact polynomial $p$ which gives $g$.
### 4.3 Detecting the degrees of factors
After specialization at $y=y_{0}$, we obtain the information about the number
of factors and the degree of each factor with respect to $x$. The degrees with
respect to $y$ of factors provide the bound of the number of interpolation
nodes. Certainly, we can use the degree of the input $\deg_{y}(F)$ as the
bound. However, the degrees with respect to $y$ of factors are usually much
less than $\deg_{y}(F)$, especially when there are many factors. Therefore,
for high efficiency, it is better to know the degree with respect to $y$ of
each factor. Now we will apply numerical continuation to detect such degree
information.
We define the notation of $2$-tuple degree to be
$\deg(f)=[\deg_{x}(f),\deg_{y}(f)].$
Suppose $\deg(f)=[m,n]$ and $f$ has $r$ factors. Applying an approach of
univariate polynomial solving to $f(x,y_{0})$ and $f(x_{0},y)$ yields points
on the curve $A=\\{(x_{1},y_{0}),(x_{2},y_{0}),...,(x_{m},y_{0})\\}$ and
$B=\\{(x_{0},y_{1}),(x_{0},y_{2}),...,(x_{0},y_{n})\\}$ respectively. In
addition, we also know the decomposition of two points sets in $r$ groups with
cardinalities $\\{a_{1},...,a_{r}\\}$ and $\\{b_{1},...,b_{r}\\}$. Moreover
$\sum a_{i}=m$ and $\sum b_{i}=n$.
Choose one point from each group of the first set $A$. Starting from these
points, we track the homotopy path
$\left\\{\begin{array}[]{cl}f(x,y)=0&\\\ (1-t)(y-y_{0})+t(x-x_{0})\gamma=0&\\\
\end{array}\right.$ (10)
Because of the genericity of the choice of $y_{0}$, $x_{0}$ and $\gamma$, the
path avoids the singular locus. When $t=1$, the endpoint must belong to the
second set $B$. For example if the starting point of the first group of $A$
and its end point belongs to the $i$th group of $B$. Then we know the first
factor has degree $[a_{1},b_{i}]$. Similarly, the degrees of other factors can
be detected in the same way.
## 5 Interpolation
Polynomial interpolation is a classical numerical method. It is studied very
well for univariate polynomials in numerical computation. Polynomial
interpolation problem is to determine a polynomial $f(x)\in F[x]$ with degree
not greater than $n\in\mathbb{N}$ for a given pairs
$\\{(x_{i},f_{i}),i=0,\cdots,n\\}$ satisfying $f(x_{i})=f_{i}$ for
$i=0,\cdots,n$, where $F$ is a field and $x_{i},f_{i}\in F$. In general, there
are four types of polynomial interpolation method: Lagrange Interpolation,
Neville’s Interpolation, Newton’s Interpolation and Hermite Interpolation.
Lagrange interpolation and Newton’s Interpolation formula are suited for
obtaining interpolation polynomial for a given set
$\\{(x_{i},f_{i}),i=0,\cdots,n\\}$. Neville’s interpolation method aims at
determining the value of the interpolating polynomial at some point. If the
interpolating problem prescribes at each interpolation point
$\\{x_{i},i=0,\cdots,n\\}$ not only the value but also the derivatives of
desired polynomial, then the Hermite formula is preferred.
Different from the traditional interpolation problem above, our problem is to
construct a bivariate polynomial from a sequence of univariate polynomials at
chosen nodes. It is important to point out that the univariate polynomials are
constructed by roots, which may not be equal to the polynomials by
substitutions. But the only difference for each polynomial is just a scaling
constant.
More precisely, in this paper, we aim to solve a special polynomial
interpolation problem: given a set of nodes and square free polynomials
$\\{(y_{i}\in F,f_{i}(x)\in F[x]),i=0,\cdots,k\\}$, compute a square free
polynomial $f(x,y)\in F[x,y]$ of degree with respect to $x$ not greater than
$n$, where $F$ is a field, such that $f(x,y_{i})$ and $f_{i}(x)$ have the same
roots.
### 5.1 Illustrative examples
###### EXAMPLE 5.1
Let $f=x^{2}+y^{2}-1$. Since its degree with respect to $y$ is two, we need
three interpolation nodes which are $y=-1/2,0,1/2$. Suppose we know the roots
at each node, then the interpolating polynomials are
$\\{f_{0}=x^{2}-3/4,f_{1}=x^{2}-1,f_{2}=x^{2}-3/4\\}$. To construct original
polynomial $f$, we can use Lagrange method.
Let $\ell_{1}=\frac{y(y-1/2)}{(-1/2-0)(-1/2-1/2)}=2y^{2}-y$. Similarly,
$\ell_{2}=-4y^{2}+1$ and $\ell_{3}=2y^{2}+y$. It is easy to check that
$(x^{2}-3/4)\ell_{1}+(x^{2}-1)\ell_{2}+(x^{2}-3/4)\ell_{3}=f$.
In the example above, the coefficient of $f$ with respect to $x^{2}$ is a
constant $1$. Making the interpolating polynomials given by (8) monic, we can
construct $f$ correctly by Lagrange basis. However, if the coefficient is
nonconstant, i.e. a polynomial of $y$, then it is not straightforward to find
$f$. The example below shows this problem.
###### EXAMPLE 5.2
Let $f=xy-1$. The nodes are $y=2,3$. We know the roots are $1/2,1/3$
respectively at the nodes. Then the monic interpolating polynomials are
$\\{x-1/2,x-1/3\\}$. If we still apply Lagrange basis
$\ell_{1}=-y+3,\;\ell_{2}=y-2$, it gives
$(x-1/2)(-y+3)+(x-1/3)(y-2)=x+1/6\;y-5/6$ which is totally different from the
target polynomial $xy-1$.
The basic reason is that the interpolating polynomials are not the polynomials
after specialization s, and the only difference is certain scaling constants.
To find these constants, we need more information. Now we use one more node:
when $y=4$, the monic interpolating polynomial is $x-1/4$. By multiplying a
scaling constant to $f$ we can assume $f(x,4)=x-1/4$, then there exist $a,b$
such that $f(x,2)=a(x-1/2)$ and $f(x,3)=b(x-1/3)$. The corresponding Lagrange
bases are
$\ell_{1}=(y-3)(y-4)/2,\;\ell_{2}=-(y-2)(y-4),\;\ell_{3}=(y-2)(y-3)/2$. Then
$f=a(x-1/2)\ell_{1}+b(x-1/3)\ell_{2}+(x-1/4)\ell_{3}$. The coefficient of $f$
with respect to $y^{2}$ must be zero. Consequently we have
$1/2\,\left(x-1/2\right)a+\left(x-1/3\right)b+1/2\,x-1/8=0$ (11)
which implies a linear system
$1/2\;a-b+1/2=0,-1/4\;a+1/3\;b-1/8=0$
The solution is $a=1/2,b=3/4$. Substituting them back to two nodes
interpolation formula yields the polynomial we need, up to a constant $1/4$
$1/2(x-1/2)(-y+3)+3/4(x-1/3)(y-2)=(xy-1)/4$
### 5.2 Interpolation with indeterminates
To extend the idea in example 5.2, we present a method to construct desired
bivariate polynomial by using monic univariate interpolating polynomials.
Suppose $f$ is irreducible and its degrees with respect to $x$ and $y$ are $m$
and $n$ respectively. Consider $x$ as the main variable, we can express this
polynomial by $f=\sum_{i=0}^{m}c_{i}(y)x^{i}$, where $c_{i}$ are polynomials
of $y$ of degree less than or equal to $n$. We can consider each $c_{i}$ as a
vector in monomial basis. Suppose there are $r$ linearly independent
coefficients. If $r=1$, then $c_{i}(y)=a_{i}c_{0}(y)$ for some constant
$a_{i}$ and $f=(\sum_{i=0}^{m}a_{i}x^{i})\cdot c_{0}(y)$. It contradicts the
assumption that $f$ is irreducible. Hence, $r\geq 2$.
Now we consider how to construct $f$ by using the interpolating polynomials
$\\{f_{0}(x),f_{1}(x),...,f_{k}(x)\\}$ at $k+1$ nodes
$\\{y_{0},y_{1},...,y_{k}\\}$ respectively chosen at random.
Let $C$ be a $(k+1)\times(m+1)$ matrix $[\textbf{c}_{0},...,\textbf{c}_{m}]$
where $\textbf{c}_{i}$ is the column vector in monomial basis
$\\{y^{k},y^{k-1},...,1\\}$ of the polynomial $c_{i}$. Let $V$ be the
Vandermonde matrix
$\left(\begin{array}[]{cccc}y_{0}^{k}&y_{0}^{k-1}&\cdots&1\\\
y_{1}^{k}&y_{1}^{k-1}&\cdots&1\\\ \vdots&\vdots&\vdots&\vdots\\\
y_{k}^{k}&y_{k}^{k-1}&\cdots&1\\\ \end{array}\right)$. Let $A$ be a
$(k+1)\times(m+1)$ matrix where $A_{ij}$ is the coefficient of the $i$th
interpolating polynomial with respect to $x^{j}$. To make the solution unique,
we may fix $f(x,y_{0})=f_{0}$ and suppose $f(x,y_{i})=\lambda_{i}f_{i}$ and
$\lambda_{i}\neq 0$ for $i=1,..,k$. Let
$\Lambda=\left(\begin{array}[]{cccc}1&&&\\\ &\lambda_{1}&&\\\ &&\ddots&\\\
&&&\lambda_{k}\\\ \end{array}\right)$.
Therefore,
$V\cdot C=\Lambda\cdot A.$ (12)
Since $\\{y_{i}\\}$ are distinct, the Vandermonde matrix has inverse and
consequently $C=V^{-1}\cdot\Lambda\cdot A$. By our assumption, the degree with
respect to $y$ is $n$. It means that the first $k-n$ rows of $C$ must be zero.
The zero at the $i$th row and $j$th column corresponds an equation. Thus, it
leads to a linear system
$\mathrm{Row}(V^{-1},i)\cdot\Lambda\cdot\mathrm{Col}(A,j)=0,$ (13)
for $1\leq i\leq k-n$ and $1\leq j\leq m+1$ with $k$ unknowns.
Only $r$ linearly independent columns in $A$, so there are $(k-n)\;r$
equations and $k$ unknowns. The existence of the solution is due to the
origination of the interpolating polynomials $f(x,y_{i})=\lambda_{i}f_{i}$ for
$i=1,...,k$. The linear system has unique solution implies that $(k-n)r\geq
k$. Thus, $k\geq rn/(r-1)$. Let $\mu$ be the smallest integer greater than or
equal to $\frac{rn}{r-1}$, namely
$\mu=\lceil\frac{rn}{r-1}\rceil.$ (14)
Thus, to determine the scaling constants $\\{\lambda_{i}\\}$, we need at least
$\mu$ more interpolation nodes.
To find an upper bound for the number of nodes, let us consider $f$ as a monic
polynomial with rational function coefficients. All the coefficients
$\\{c_{m},...,c_{0}\\}$ can be uniquely determined by rational function
interpolation of $x^{m}+c_{m-1}/c_{m}x^{m-1}+\cdots+c_{0}/c_{m}$ at $2n+1$
nodes. Therefore, it requires $2n$ nodes except the initial one. Thus, we have
$\mu\leq k\leq 2n$.
But this upper bound is often overestimated, and for some special case the
polynomial $f$ can be constructed by using less nodes.
###### Proposition 2
Let $f$ be a polynomial in $\mathbb{Q}[x,y]$ and $\deg(f)=[m,n]$. Suppose
$m\geq n$ and f has $n+1$ linearly independent coefficients. Then $f$ can be
uniquely determined by $n+2$ monic interpolating polynomials
$\\{f_{0}(x),...,f_{n+1}(x)\\}$ up to a scaling constant.
Proof. Suppose the first $n+1$ columns of $A$ are linearly independent. By
Equation (12), we construct $n+1$ equations:
$\mathrm{Row}(V^{-1},1)\cdot\Lambda\cdot\mathrm{Col}(A,j)=0$, for
$j=1,...,n+1$. Let $B$ be the transpose of the submatrix consisting of the
first $n+1$ columns of $A$ and $\textbf{v}=(v_{1},...,v_{n+2})^{t}$ be the
transpose of $\mathrm{Row}(V^{-1},1)$. Thus,
$\textbf{0}=B\cdot\left(\begin{array}[]{cccc}\lambda_{1}&&&\\\
&\lambda_{2}&&\\\ &&\ddots&\\\ &&&\lambda_{n+2}\\\
\end{array}\right)\cdot\textbf{v}=B\cdot\left(\begin{array}[]{cccc}v_{1}&&&\\\
&v_{2}&&\\\ &&\ddots&\\\ &&&v_{n+2}\\\
\end{array}\right)\cdot(\lambda_{1},...,\lambda_{n+2})^{t}$
Because the first $n+1$ coefficients of $f$ are linearly independent, the
evaluations of them at $n+2$ random points must be linearly independent. So
the rank of $B$ is $n+1$. Here $v$ can be expressed by explicit form of the
Vandermonde matrix [23] which is a vector of polynomials of
$\\{y_{0},...,y_{n+1}\\}$. For generic choice of $\\{y_{0},...,y_{n+1}\\}$,
each $v_{i}\neq 0$. Hence, the rank of
$B\cdot\left(\begin{array}[]{cccc}v_{1}&&&\\\ &v_{2}&&\\\ &&\ddots&\\\
&&&v_{n+2}\\\ \end{array}\right)$ is still $n+1$ and its nullity equals one.
We can choose any solution $\\{\lambda_{i}\\}$ to construct $f$ by Lagrange
basis: $\sum_{i=0}^{n}\lambda_{i}f_{i}\ell_{i}$. $\square$
###### Remark 2
In our algorithm, we compute $\\{\lambda_{i}\\}$ starting from $\mu$ more
nodes (together with the initial node $y_{0}$), and we add incrementally more
nodes if necessary. But interestingly, the experimental results show that the
lower bound $\mu$ is often enough. This fact deserves further study.
###### Algorithm 2
[Interpolation]
| Input : a set of polynomials
$\\{f_{0}(x),...,f_{k}(x)\\}\subset\mathbb{Z}[x,y]$
---|---
a set of rational numbers $\\{y_{0},...,y_{k}\\}$
an integer $n$ the degree of $f$ with respect to $y$
| Output: $f\in\mathbb{Z}[x,y]$, such that $f(x,y_{i})=f_{i}(x)$.
1. 1.
Let $A$ be the matrix consisting of the coefficient row vectors of the input
univariate polynomials.
2. 2.
Let $r=\mathrm{Rank}(A)$. If $k<\mu$, then it needs more interpolation nodes.
3. 3.
Solve the homogenous linear system (13) to obtain the scaling constants
$\\{\lambda_{1},..,\lambda_{k}\\}$
4. 4.
If the solution is not unique, then it needs more interpolation nodes.
5. 5.
Else $f=\sum_{i=0}^{k}\lambda_{i}f_{i}\ell_{i}\in\mathbb{Q}[x,y]$
6. 6.
Return the primitive polynomial of $f$
## 6 Combination of Tools
Now we combine the tools introduced in previous sections to obtain a new
factorization algorithm. A preliminary version of the algorithm is implemented
in Maple. For the efficiency, it requires a more sophisticate version in C++,
even parallel program.
### 6.1 Main steps of the algorithm
###### Algorithm 3
[Factorization]
$F$ = Fac$(f)$
Input : | $f$, a primitive polynomial $f\in\mathbb{Z}[x,y]$ such that $\gcd(f,f_{x})=1$
---|---
Output: | $F$, a set of primitive polynomials $\\{f_{1},...,f_{r}\\}\subset\mathbb{Z}[x,y]$, such that $f=\prod f_{i}$.
1. 1.
Apply a numerical solver to approximate the roots of $f(x,y_{0})=0$ and
$f(x_{0},y)=0$ at generic points $x_{0},y_{0}\in\mathbb{Q}$.
2. 2.
Apply miniPoly to roots above and decompose the solutions. And generate the
minimal polynomials for them and we have grouping information of roots. In
this step, it needs Newton iteration to refine the roots up to desired
accuracy.
3. 3.
Apply homotopy (10) to obtain the degrees of each factor.
4. 4.
For group $i$ (corresponding to the factor $f_{i}$), $i=1,...,r$, use homotopy
(7) to generate its approximate roots at random rational numbers
$\\{y_{1},...,y_{k}\\}$.
5. 5.
For each set of roots at $y_{j}$, refine the roots to the accuracy given by
Proposition 1, then make the product and construct the polynomial
$f_{i}(x,y_{j})$.
6. 6.
Call interpolate with the interpolating polynomials
$\\{f_{i}(x,y_{0}),...,f_{i}(x,y_{k})\\}$ to construct $f_{i}(x,y)$.
### 6.2 A simple example
Let us consider a polynomial $f=(x\,y-2)\,(x^{2}+y^{2}-1)$. First, we choose a
sequence of random rational numbers
$\\{97/101,1,104/101,123/101,129/101,...\\}$. Substituting $y=97/101$ into $f$
yields Mignotte bound of the coefficients of factors $9170981$ and the digits
required to produce the minimal polynomial is $110$ by Theorem 2. Then compute
the approximate roots of $f(x,97/101)$ up to $110$ digits accuracy. The
miniPoly subroutine gives two groups of points: $[[1,2],[3]]$ and the
corresponding minimal polynomials $[-792+10201\,x^{2},-202+97\,x]$. By Hilbert
Irreducibility theorem, there are two factors. On the other hand, fix the
value of $x$ and obtain the univariate polynomials
$[-202+97\,y,-792+10201\,y^{2}]$ and $[[3],[1,2]]$.
Starting from the first point of group one, the Homotopy (10) path ends at a
point which satisfies $y,-792+10201\,y^{2}$. It implies that
$-792+10201\,x^{2}$ and $y,-792+10201\,y^{2}$ are from the same factor of
degree $[2,2]$. By Equation (14), we need $\mu=\lceil\frac{rn}{r-1}\rceil=4$
more interpolating polynomials which are produced by Homotopy (7). Thus, there
are five polynomials
$[-792+10201\,x^{2},x^{2},615+10201\,x^{2},4928+10201\,x^{2},6440+10201\,x^{2}]$.
The scaling constants
$[\lambda_{1}=1,\lambda_{2}=10201,\lambda_{3}=1,\lambda_{4}=1,\lambda_{5}=1]$
are obtained by system (13). Consequently, the Lagrange interpolation formula
gives the correct factor $-1+x^{2}+y^{2}$.
Since the degree of the other factor is $[1,1]$, it needs $\mu=2$ more
polynomials and they are $[-202+97\,x,x-2,-101+52\,x]$. The corresponding
scaling constants are $[\lambda_{1}=1/2,\lambda_{2}=101/2,\lambda_{3}=1]$ and
the resulting factor is $x\,y-2$.
## 7 Conclusion
A new numerical method to factorize bivariate polynomials exactly is presented
in this article. We implemented our algorithm in Maple to verify the
correctness. More importantly, the main components of our algorithm, miniPoly
and numerical homotopy continuation can be implemented directly in C++ or J++
with existing multi-precision packages, e.g. GNU MP library. Furthermore,
these two numerical components are naturally parallelizable. Therefore, it
gives an alternative way to exact factorization which can take the advantages
of standard programming languages and parallel computation techniques widely
used by industries.
In this article, we mainly focus on bivariate case. It is quite
straightforward to extend to multivariate case. However, the number of the
interpolation nodes grows exponentially as the increasing of the number of
monomials. A more practical way to deal with such difficulty is to exploit the
sparsity if the factors are sparse. It desires the further study in our future
work.
## References
* [1] Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, Charles W. Wampler. Software for Numerical Algebraic Geometry available at http://www.nd.edu/~sommese/bertini/
* [2] L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer–Verlag, 1998.
* [3] Jingwei Chen, Yong Feng, Xiaolin Qin, Jingzhong Zhang. Exact Polynomial Factorization by Approximate High Degree Algebraic Numbers. in: SNC’09, ACM, Kyoto, Japan, P. 21-28,2009.
* [4] Robert M. Corless, Mark W. Giesbrecht, et al, Towards factoring bivariate approximate polynomials, In Proc. ISSAC 2001, ACM press, pp.85-92
* [5] Xiaolin Qin, Yong Feng, Jingwei Chen, Jingzhong Zhang. Finding Exact Minimal Polynomial by Approximations. in: SNC’09, ACM, Kyoto, Japan, P. 125-131,2009. (EI 20094812515100)
* [6] J. Von Zur Gathen, Irreducibility of multivariate polynomials, J. Comput. System Sci., 31 (1985), pp. 225-264.
* [7] Joachim von zur Gathen et. al. Modern computer Algebra, Cambridge University Press.(1999)
* [8] Huang, Y., Wu, W.,Stetter, H., and Zhi, L. Pseudofactors of multivariate polynomials. In Proc. ISSAC’00(2000), ACM Press, pp.161-168.
* [9] M.-P. van der Hulst and A. Lenstra. Factorization of Polynomials by Transcendental Evaluation. In EUROCAL ’85, pages 138–145, 1985.
* [10] S. Ilie, R. M. Corless and G. Reid. Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time. Numerical Algorithms, Vol 42(2), pages 161-171, 2006.
* [11] Shuhong Gao. Factoring multivariate polynomials via partial differential equations Mathematics of Computation, Volume 72, Issue 242, pages: 801 - 822, 2003.
* [12] Shuhong Gao, Erich Kaltofen, John May, Zhengfeng Yang and Lihong Zhi. Approximate factorization of multivariate polynomials via differential equations. Processings of ISSAC’04, pages: 167 - 174, ACM 2004.
* [13] Erich Kaltofen, Lakshman Yagati, Improved Sparse Mutivariate Polynomial Interpolation Algorithms. In Symbolic Algebraic Comput. Internat. Symp. ISSAC ’88 Proc. [-17], pages 467-474, 1988.
* [14] Erich Kaltofen, John May, Zhengfeng Yang, and Lihong Zhi. Approximate factorization of multivariate polynomials using singular value decomposition. J. Symbolic Comput., 43(5):359-376, 2008.
* [15] R.Kannan, A.K.Lenstra, and L.Lovász. Polynomial Factorization and Nonrandomness of Bits of Algebraic and Some Transcendental Numbers, Mathematics of Computation. Vol.30, No.181, Jan. 1988, P.235-250.
* [16] T. Y. Li and Tsung-Lin Lee. HOMotopy method for solving Polynomial Systems, software available at https://www.msu.edu/~leetsung/.
* [17] A. Morgan and A. Sommese. A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput., Vol 24(2), pages 101-113, 1987.
* [18] Mou-Yan, Z., and Unbehausen, R. Approximate factorization of multivariable polynomials. Signal Proces. 14(1988), 141-152.
* [19] Sasaki, T., Suzuki, M., et al., Approximate factorization of multivariate polynomials and absolute irreducibility testing. Japan J. Indust. Appl. Math. 8 (1991),357-375.
* [20] Sasaki, T., Saito T., and Hilano, T., Analysis of approximate factorization algorithm. Japan J. Indust. Appl. Math.9 (1992),351-368.
* [21] Tateaki Sasaki, Approximate multivariate polynomial factorization based on zero-sum relations. In Proc. ISSAC’2001, ACM Press, pp.284-291.
* [22] A.J. Sommese and C.W. Wampler. The Numerical solution of systems of polynomials arising in engineering and science. World Scientific Press, Singapore, 2005.
* [23] Turner, L. Richard. Inverse of the Vandermonde matrix with applications.
* [24] J. Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software 25(2): 251–276, 1999. Software available at http://www.math.uic.edu/~jan.
* [25] H.Zassenhaus. On Hensel Factorization I. J. Numer Theory, 1(1):291-311, 1969.
* [26] H. Zassenhaus. A Remark on the hensel factorization method. Math. Comp., 32(141):287-292, 1978.
* [27] Jingzhong Zhang, Yong Feng, Obtaining Exact Value by Approximate Computations, Science in China Series A: Mathematics Vol. 50, No.9, 1361-1368, Sept. 2007.
|
arxiv-papers
| 2010-08-21T03:28:20 |
2024-09-04T02:49:12.294249
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yong Feng, Wenyuan Wu, Jingzhong Zhang",
"submitter": "Yong Feng",
"url": "https://arxiv.org/abs/1008.3596"
}
|
1008.3622
|
# Color-Magnitude Relations of Early-type Dwarf Galaxies in the Virgo Cluster:
An Ultraviolet Perspective
Suk Kim11affiliation: Department of Astronomy and Space Science, Chungnam
National University, Daejeon 305-764, Korea; screy@cnu.ac.kr , Soo-Chang
Rey11affiliation: Department of Astronomy and Space Science, Chungnam National
University, Daejeon 305-764, Korea; screy@cnu.ac.kr 2 2affiliationmark: ,
Thorsten Lisker33affiliation: Astronomisches Rechen-Institut, Zentrum für
Astronomie der Universität Heidelberg (ZAH), Mönchhofstraße 12-14, D-69120
Heidelberg, Germany , and Sangmo Tony Sohn44affiliation: Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
###### Abstract
We present ultraviolet (UV) color-magnitude relations (CMRs) of early-type
dwarf galaxies in the Virgo cluster, based on Galaxy Evolution Explorer
(GALEX) UV and Sloan Digital Sky Survey (SDSS) optical imaging data. We find
that dwarf lenticular galaxies (dS0s), including peculiar dwarf elliptical
galaxies (dEs) with disk substructures and blue centers, show a surprisingly
distinct and tight locus separated from that of ordinary dEs, which is not
clearly seen in previous CMRs. The dS0s in UV CMRs follow a steeper sequence
than dEs and show bluer UV$-$optical color at a given magnitude. We also find
that the UV CMRs of dEs in the outer cluster region are slightly steeper than
that of their counterparts in the inner region, due to the existence of faint,
blue dEs in the outer region. We explore the observed CMRs with population
models of a luminosity-dependent delayed exponential star formation history.
We confirm that the feature of delayed star formation of early-type dwarf
galaxies in Virgo cluster is strongly correlated with their morphology and
environment. The observed CMR of dS0s is well matched by models with
relatively long delayed star formation. Our results suggest that dS0s are most
likely transitional objects at the stage of subsequent transformation of late-
type progenitors to ordinary red dEs in the cluster environment. In any case,
UV photometry provides a powerful tool to disentangle the diverse
subpopulations of early-type dwarf galaxies and uncover their evolutionary
histories.
###### Subject headings:
galaxies: clusters: individual (Virgo) — galaxies: dwarf — galaxies: star
formation — ultraviolet: galaxies
††slugcomment: 22affiliationtext: Department of Physics and Astronomy, Johns
Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA
## 1\. Introduction
It is well established that early-type galaxies in clusters form a tight and
well-defined color-magnitude relation (CMR) in the optical bands (Visvanathan
& Sandage 1977; Sandage & Visvanathan 1978a, b) such that brighter galaxies
are generally redder. The physical origin (e.g., metallicity vs. age) of the
CMR is still a matter of debate. Nonetheless, it is believed that the CMR is
an important tool for understanding star formation histories (SFHs) of early-
type galaxies and their links to the galaxy formation scenarios.
Since the ultraviolet (UV) flux of an integrated population is a good tracer
of recent star formation activities, the CMR in the UV band provides an
important constraint on SFH in galaxies. In recent years, UV CMRs for early-
type galaxies in the local Universe based on Galaxy Evolution Explorer (GALEX)
observations have shown a substantially larger scatter in the bluer
UV$-$optical colors than optical CMRs. This indicates that a large fraction
(10 $\sim$ 30% of the early-type galaxies examined with GALEX) of early-type
galaxies have experienced low level ($\lesssim$ 1 M⊙yr-1) residual star
formation over the last 8 billion years (Yi et al. 2005; Kaviraj et al. 2007,
2008; Schawinski et al. 2007; Haines et al. 2008).
The SFHs of galaxies are strongly dependent on their masses (Ferreras & Silk
2000; Caldwell et al. 2003; Nelan et al. 2005; Thomas et al. 2005) and low-
mass dwarf galaxies have much more extended SFHs with longer timescales of gas
consumption for star formation compared to the massive galaxies (e.g., Mateo
1998; Grebel & Gallagher 2004 for Local Group and Gavazzi et al. 2002 for
Virgo cluster). Moreover, low-mass galaxies are more susceptible to external
processes such as galaxy harassment, tidal interaction, and ram-pressure
stripping that ultimately affect their star formation and evolutionary
histories. In this context, it is important to compare the UV CMR of early-
type dwarf galaxies with that of the massive counterparts.
Boselli et al. (2005) first studied the UV properties of early-type galaxies
in the Virgo cluster using GALEX data. They found that UV CMRs show a
discontinuity between massive and dwarf early-type galaxies in contrast to
what is observed at optical wavelengths. They suggested that the residual star
formation activity is more important in low mass early-type dwarf galaxies,
while the UV flux in the massive counterparts is dominated by hot, evolved old
stellar populations. However, their results are based on a sample restricted
to bright dwarf galaxies detected from GALEX observations of its early
operation. In order to understand the SFH in galaxies related to their masses
using UV CMRs, a larger sample is required to reach into the fainter and lower
mass dwarf regime.
Recently, several subclasses of dwarf ellipticals (dEs) with morphological
substructures such as disks, spiral arms, bars, and blue centers, have been
discovered (Jerjen, Kalnajs, & Binggeli 2000; Barazza, Binggeli, & Jerjen
2002; Lisker et al. 2006a, b). These galaxies constitute a significant
fraction of bright dEs in the Virgo cluster (Lisker et al. 2007). Their
properties might be related to recent or ongoing star formation activities and
distinct environmental effects. The discovery of these objects hints that the
early-type dwarf galaxies are heterogeneous objects, which originated from
various channels of evolutionary scenarios (see Lisker 2009 for a review).
Here, we present new UV CMRs of early-type dwarf galaxies in the Virgo cluster
using extensive GALEX UV photometric data in combination with SDSS data. Our
goal is to study whether the various subclasses of early-type dwarf galaxies
show different sequences in UV CMRs related to their star formation and
evolutionary histories.
## 2\. Data and Analysis
We used UV images from the GALEX Release 3 (GR3) dataset. GALEX observed the
Virgo cluster as part of the All-sky Imaging Survey (AIS), Nearby Galaxy
Survey (NGS), and Deep Imaging Survey (DIS) in two UV bands : far-ultraviolet
(FUV; 1350$-$1750Å) and near-ultraviolet (NUV; 1750$-$2750Å). GALEX imaged 97
fields of Virgo cluster covering a total $\sim$ 82 deg2. The depth of each
field varies in accordance with its survey mode: 16 NGS fields
(NUV$\sim$3,000s, FUV$\sim$1,500s), 80 AIS fields (NUV, FUV$\sim$100s), and 1
DIS field (NUV$\sim$22,000s). Most NGS fields (12 of 16) cover the regions
within angular distance of 2 degree from the M87. Using SExtractor (Bertin &
Arnouts 1996), we performed photometry for all detected objects. For this, we
required fluxes at least 1$\sigma$ above the sky noise. We adopted
MAG$\\_$AUTO(total) as the source magnitude. Flux calibrations were applied to
bring the final photometry into the AB magnitude system (Oke 1990). The
typical errors are 0.10 mag and 0.14 mag in the NUV and FUV, respectively.
We take advantage of the comprehensive sample of certain or possible cluster
members classified as dE and dwarf lenticular (dS0) galaxies in the Virgo
Cluster Catalog (VCC) of Binggeli et al. (1985). In addition, we include 11
lenticular galaxies (S0s) of the VCC with optical magnitudes similar to dS0.
The cross-identification between 774 VCC early-type dwarf galaxies and GALEX
photometry results in 193 and 59 galaxies in the NUV and FUV band,
respectively. All matched objects were visually inspected and we retained
objects with clear detection. NGS fields reach limiting magnitudes of $\sim$
23.0 mag in the NUV and FUV, while AIS ones reach $\sim$ 22.0 mag. All FUV-
detected galaxies are also detected in the NUV. The resulting sample includes
fainter dwarf galaxies compared to previous UV studies using the GALEX
Internal Release (IR1.0) (Boselli et al. 2005). We secured galaxies down to mB
$\sim$ 20 mag. GALEX UV data have been combined with SDSS r-band data from
SDSS Data Release 5 (DR5). The SDSS photometric pipeline fails to measure
accurately the local sky flux around Virgo dEs and thus the total magnitude
(see Lisker et al. 2007 for the details). Therefore, we performed our own sky
subtraction and photometric measurement, following the procedure of Lisker et
al. (2007). Only foreground Galactic extinction correction for each galaxy is
applied (Schlegel et al. 1998). We use the reddening law of Cardelli et al.
(1989) to derive the following : RNUV=8.90, RFUV=8.16, and Rr=2.72 . We adopt
a Virgo cluster distance of 15.9 Mpc, i.e. a distance modulus m$-$M=31.01 mag
(Graham et al. 1999).
## 3\. Results
### 3.1. Ultraviolet Color-Magnitude Relations of Early-Type Dwarf Galaxies
In Figure 1, we present optical (Fig. 1a), NUV (Fig. 1b), and FUV (Fig 1c)
CMRs for dEs (red circles) and dS0s (yellow circles). Of galaxies classified
as dS0s, a substantial fraction corresponds to dEs with disk substructures
(stars, Lisker et al. 2006a) or blue centers (triangles; Lisker et al. 2006b).
Note that, in what follows, we refer to dS0s and peculiar dEs (disk
substructure or blue center) collectively as dS0s. We also include blue
compact dwarf galaxies (BCDs, squares) drawn from the VCC, for comparison
purposes. UV CMRs follow the general trend of the optical CMR, i.e., early-
type dwarf galaxies become progressively bluer with decreasing optical
luminosity. However, the UV colors span a much wider range than the optical
CMR, owing to the wide baseline of UV to optical colors: while $g-r$ only
spans a range of $\sim$0.6 mag, NUV$-r$ and FUV$-r$ varies up to 4.5 mag and
6.0 mag, respectively.
The most interesting feature in our UV CMRs is that dS0s form a tight sequence
which is clearly distinct from that of normal dEs. In UV CMRs, dS0s follow a
steeper sequence than dEs (dotted line in Fig. 1a-c gives the mean of dS0s).
Meanwhile, the optical CMR of dS0s (see Fig. 1a) is not much different from
that of normal dEs. We note that Boselli et al. (2005, 2008) were not able to
observe such features in their UV CMRs mainly due to their limited sample of
dwarf galaxies. In addition, the faint end of the dS0 sequence in UV CMRs
appears to be linked to the BCDs. Note that some galaxies originally
classified as BCDs in the VCC have a similar appearance like dEs with blue
centers (Lisker et al. 2006b): the visual classification between dE with blue
center and BCD appears to have a smooth transition. This is now confirmed by
our UV CMRs. Furthermore, several studies claimed that BCDs might be potential
progenitors of dEs (see Lisker 2009 and references therein). Our UV CMRs shown
in Fig. 1 indicate that dS0s evidently have different stellar population
properties as compared to normal dEs.
Since the UV flux is sensitive to young ($\lesssim$ 1 Gyr) stellar
populations, the bluer UV colors of dS0s at fixed luminosity implies that dS0s
have experienced recent or ongoing star formation activities whereas dEs have
been relatively quiescent in the past few Gyrs. To confirm this, we examined
SDSS spectra and available literature (Boselli et al. 2008; Michielsen et al.
2008; Paudel et al. 2010) and found that the majority of dS0s show relatively
strong H$\alpha$ emission and/or H$\beta$ absorption lines. Interestingly,
dS0s showing H$\alpha$ emission lines are systematically less luminous and
have strong NUV and FUV fluxes. We found that 85% of 13 faint ($M_{r}$ $>$
-16.9) dS0s exhibit H$\alpha$ emission lines with EW $>$ 2 Å. Meanwhile, dS0s
that show strong H$\beta$ absorptions are preferentially located in the
luminous part of the sequence (63% of 16 dS0s with $M_{r}$ $<$ -16.9 show
H$\beta$ EW $>$ 2.5 Å). On the other hand, 6% and 36% of normal dEs show hints
of ongoing and post star formation, respectively, according to the SDSS
spectra of 83 sample with $M_{r}<$-13.7. Therefore, spectroscopic results
confirm the systematically distinct sequence of dS0s in UV CMRs, implying
evidence of ongoing or post star formation (see also Boselli et al. 2008).
The environment in a cluster plays an important role in the formation and
evolution of the member galaxies. Our large sample size allows us to examine
this effect for early-type dwarf galaxies in the Virgo cluster. In order to
investigate the environmental dependence of the CMRs, we divide the sample
into two based on their distance from the giant elliptical galaxy M87. Given
that dwarf galaxies with angular distances of less than 2 degree from the
center of M87 are dynamically connected with M87 (Binggeli et al. 1987), we
adopt this angular distance for dividing our sample into inner and outer dwarf
galaxies. While handful outer dwarf galaxies (three in the NUV) might be
associated with other giant galaxy, M49, those galaxies do not change our
results. In Fig. 1d and 1e, we present the spatial distribution of early-type
dwarf galaxies detected in the GALEX NUV and FUV, respectively. In all panels
of Fig. 1, large and small symbols denote dwarf galaxies in the inner and
outer region, respectively. As suggested by Lisker et al. (2007), dS0s are not
centrally clustered around M87.
We have fitted CMRs of dEs in two different regions using a first-order least
squares method. In the optical band (Fig. 1a), CMRs for dEs in the inner and
outer regions are nearly identical. However, in the UV (Fig. 1b, c), dEs in
the outer region (dashed line) have a steeper relation than those in the inner
region (solid line). In Fig. 1a-c, we also plot color distributions of dEs in
the inner (solid histogram) and outer (dotted histogram) region. Again, $g-r$
color distributions between inner and outer dEs are indistinguishable
(histograms in Fig. 1a). Yet, the UV$-$optical color distributions of dEs in
two different regions are very different; while inner dEs are confined to the
redder side (NUV$-r$ $\gtrsim$ 3.5, FUV$-r$ $\gtrsim$ 5), those in the outer
region exhibit wider distributions extending to bluer colors (histograms in
Fig 1b, c). This mainly results from the contribution of faint ( Mr $\gtrsim$
-14) dEs in the outer region showing bluer UV$-$optical colors. Our results
are in good agreement with those from other studies where early-type dwarf
galaxies with hints of star formation are found to be preferentially located
in the outskirts of clusters (Drinkwater et al. 2001; Conselice et al. 2003;
Smith et al. 2008, 2009). In Table 1, we present the main relations of CMRs
for early-type dwarf galaxies.
### 3.2. Comparison with Population Models
We now compare observed optical and UV CMRs with evolutionary stellar
population models. The models of Bruzual & Charlot (2003) are used for the
young ($<$ 1 Gyr) stellar populations. We combine them with the models of Yi
(2003) in order to represent flux of old ($>$ 1 Gyr) stellar populations. We
assume a delayed exponential SFH given by
$SFR(T,\tau)=(T/\tau^{2})\times exp(-T^{2}/2\tau^{2}),$
where T is a time from the onset of star formation and $\tau$ is the star
formation timescale regulating the delay of the maximum star formation rate
(SFR) and the steepness of its decay (Gavazzi et al. 2002).
In Figure 2, we show population models overlaid on the observed CMRs. Model
lines are computed for an epoch of T = 13 Gyr for three cases: dEs in the
outer region (left columns), dEs in the inner region (middle columns), and
dS0s (right columns). For each case, we adopt different ranges of $\tau$ that
best match the observed CMRs. The $\tau$ values increase with decreasing
luminosity of the galaxy. The different model lines (dotted lines) in each
panel refer to those obtained for six different metallicities (Z = 0.0001,
0.0004, 0.001, 0.004, 0.01, 0.02 from bottom to top), which cover the observed
metallicity range of dwarf galaxies (e.g., Barazza & Binggeli 2002; Jerjen,
Binggeli, & Barazza 2004). Three large rectangles in each panel denote the
model predictions adopting an empirical luminosity-metallicity relation for
dEs (Barazza & Binggeli 2002). Since the UV flux is very sensitive to even
small variations on the SFR, NUV$-r$ and FUV$-r$ colors show higher dependence
on $\tau$ than $g-r$ color.
The observed CMRs of dEs in the inner region (Fig. 2a-c) are well matched by
the model lines with small $\tau$ range (2 $<$ $\tau$ $<$ 3.5 Gyr). Since a
small $\tau$ range implies a burst of star formation with a short timescale at
an early epoch, the resulting CMR is expected be relatively flat. This is
consistent with our observed CMRs as shown in Fig. 2a-c.
Unlike dEs in the inner region, dEs in the outer region (Fig. 2d-f) extend to
fainter ($M_{r}$ $\gtrsim$ -14) magnitudes in the CMRs. These faint dEs show
on average predominantly bluer UV$-r$ colors than the extrapolated mean line
of dEs in the inner region (see Fig. 1). That is, fainter dEs become bluer
more rapidly than bright dEs do with decreasing magnitude. Overall, this makes
the CMRs of dEs in the outer region different from that of the inner region
dEs. Such environmental dependences of CMRs were also observed in other
studies (Tanaka et al. 2005; Baldry et al. 2006; Haines et al. 2007; Gavazzi
et al. 2010). In both optical and UV CMRs, the distribution of luminous
($M_{r}$ $\lesssim$ -14) dEs in the outer region is similar to that of the
counterparts in the inner region, and are well matched by models with small
$\tau$ range (2 $<$ $\tau$ $<$ 3.5 Gyr, two solid lines for Z=0.0001 and 0.02
in Fig. 2d-f). On the other hand, fainter dEs in the outer region lie
significantly to the bluer side of these model predictions. In order to match
the distribution of these faint and blue dEs, models with wider $\tau$ range
(2 $<$ $\tau$ $<$ 6 Gyr, dotted lines) are required, which translates into a
relatively higher SFR at the current epoch, owing to a delayed star formation.
As for the dS0s, models with large $\tau$ range (2 $<$ $\tau$ $<$ 7 Gyr) are
good matches to the observed CMRs (Fig. 2g-i), describing well the steep
sequence of dS0s. Therefore, we conclude that dS0s have likely experienced
relatively long delayed star formations similar to dEs found in the outer
region. This is consistent with observational results indicating residual or
ongoing star formation activities (Lisker et al. 2006a, b, 2007; Boselli et
al. 2008).
## 4\. Discussion
The dS0s seem to share properties with both normal dEs and late-type dwarf
galaxies. While their overall appearance is closer to that of normal dEs,
their various characteristics are distinct in many aspects. First, in the
optical images, they exhibit substructures such as disks, blue centers, or
irregular central features (Lisker et al. 2006a, b, 2007). Second, a number of
them show Balmer emissions and/or absorptions as well as HI gas content
(Lisker et al. 2006a, b, 2007, 2008; Boselli et al. 2008). Third, they are not
clustered around the center of the potential well and have an asymmetric
velocity distribution with multiple peaks, indicative of an unrelaxed
population from recent infall (Lisker et al. 2007).
If we assume that a fraction of early-type dwarf galaxies originated from
galaxies that fell into the cluster’s potential well, one could imagine that a
galaxy in the denser regions of the cluster would have experienced an earlier
destruction of substructures, accompanied with rapid truncation of its star
formation, due to strong environmental effects (e.g., galaxy harassment, Moore
et al. 1999 and ram-pressure stripping, Gunn & Gott 1972, see Boselli &
Gavazzi 2006; Boselli et al. 2008; Smith et al. 2010 for the details). Our UV
CMR results support this view, in the sense that bluer UV$-$optical colors of
dS0s are interpreted as contribution from young stellar populations. Of
course, this assumes that the progenitor galaxies had experienced star
formation activities before the infall into the cluster’s potential well. The
hypothesis is also consistent with recent results that rotationally supported
dEs with disk features are on average younger than pressure supported normal
dEs (e.g., Toloba et al. 2009; Michielsen et al. 2008). Consequently, dS0s are
most likely subclasses of transitional objects between red dEs and late-type
bluer dwarf galaxies, and will eventually migrate to red-sequence early-type
dwarf galaxies (Boselli et al. 2008; Haines et al. 2008; Gavazzi et al. 2010).
In this regard, the study of the diverse subpopulations of early-type dwarf
galaxies might play an important role in the issue of growth of the low-mass
part of the red sequence via the evolution of the blue sequence and green
valley, under the downsizing paradigm (Cowie et al. 1996) at the current
epoch. A comparative study of the UV CMRs for dwarf galaxies in various
clusters and groups with different densities and dynamical conditions will
provide additional insight into the physical processes of galaxy
transformation via cluster environmental effects. We defer presenting our
results on such topics to our forthcoming papers.
We are grateful for the clarifications and improvements suggested by an
anonymous referee. This research was supported by Basic Science Research
Program through the National Research Foundation of Korea (NRF) funded by the
Ministry of Education, Science and Technology (No. 2009-0070263) and the NRF
grant funded by the Korea government (MEST) (No. 2009-0062863). T.L. is
supported within the framework of the Excellence Initiative by the German
Research Foundation (DFG) through the Heidelberg Graduate School of
Fundamental Physics (grant number GSC 129/1). Facilities: GALEX
## References
* Baldry et al. (2006) Baldry, I. K., Balogh, M. L., Bower, R. G., Glazebrook, K., Nichol, R. C., Bamford, S. P., & Budavari, T. 2006, MNRAS, 373, 469
* Barazza & Binggeli (2002) Barazza, F. D., & Binggeli, B. 2002, A&A, 394, 15
* Barazza et al. (2002) Barazza, F. D., Binggeli, B., & Jerjen, H. 2002, A&A, 391, 823
* Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393
* Binggeli et al. (1985) Binggeli, B., Sandage, A., & Tammann, G. A. 1985, AJ, 90, 1681
* Binggeli et al. (1987) Binggeli, B., Tammann, G. A., & Sandage, A. 1987, AJ, 94, 251
* Boselli et al. (2005) Boselli, A., Cortese, L., Deharveng, J. M., et al. 2005, ApJ, 629, L29
* Boselli & Gavazzi (2006) Boselli, A., & Gavazzi, G. 2006, PASP, 118, 517
* Boselli et al. (2008) Boselli, A., Boissier, S., Cortese, L., & Gavazzi, G. 2008, ApJ, 674, 742
* Bruzual & Charlot (2003) Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000
* Caldwell et al. (2003) Caldwell, N., Rose, J. A., & Concannon, K. D. 2003, AJ, 125, 2891
* Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* Conselice et al. (2003) Conselice, C. J., Gallagher, J. S., III, & Wyse, R. F. G. 2003, AJ, 125, 66
* Cowie et al. (1996) Cowie, L. L., Songaila, A., Hu, E. M., & Cohen, J. G. 1996, AJ, 112, 839
* Drinkwater et al. (2001) Drinkwater, M. J., Gregg, M. D., Holman, B. A., & Brown, M. J. I. 2001, MNRAS, 326, 1076
* Ferreras & Silk (2000) Ferreras, I., & Silk, J. 2000, MNRAS, 316, 786
* Gavazzi et al. (2002) Gavazzi, G., Bonfanti, C., Sanvito, G., Boselli, A., & Scodeggio, M. 2002, ApJ, 576, 135
* Gavazzi et al. (2010) Gavazzi, G., Fumagalli, M., Cucciati, O., & Boselli, A. 2010, astro-ph:1003.3795
* Graham et al. (1999) Graham, J. A., Ferrarese, L., Freedman, W. L., et al. 1999, ApJ, 516, 626
* Grebel & Gallagher (2004) Grebel, E. K., & Gallagher, J. S., III 2004, ApJ, 610, L89
* Gunn & Gott (1972) Gunn, J. E., & Gott, J. R., III 1972, ApJ, 176, 1
* Haines et al. (2007) Haines, C. P., Gargiulo, A., La Barbera, F., Mercurio, A., Merluzzi, P., & Busarello, G. 2007, MNRAS, 381, 7
* Haines et al. (2008) Haines, C. P., Gargiulo, A., & Merluzzi, P. 2008, MNRAS, 385, 1201
* Jerjen et al. (2000) Jerjen, H., Kalnajs, A., & Binggeli, B. 2000, A&A, 358, 845
* Jerjen et al. (2004) Jerjen, H., Binggeli, B., & Barazza, F. D. 2004, AJ, 127, 771
* Kaviraj et al. (2007) Kaviraj, S., Schawinski, K., Devriendt, J. E. G., et al. 2007, ApJS, 173, 619
* Kaviraj et al. (2008) Kaviraj, S., Khochfar, S., Schawinski, K., Yi, S. K., Gawiser, E., Silk, J., Virani, S. N., Cardamone, C. N., van Dokkum, P. G., & Urry, C. M. 2008, MNRAS, 388, 67
* Lisker et al. (2006) Lisker, T., Grebel, E. K., & Binggeli, B. 2006a, AJ, 132, 497
* Lisker et al. (2006) Lisker, T., Glatt, K., Westera, P., & Grebel, E. K. 2006b, AJ, 132, 2432
* Lisker et al. (2007) Lisker, T., Grebel, E. K., Binggeli, B., & Glatt, K. 2007, ApJ, 660, 1186
* Lisker (2009) Lisker, T. 2009, Astronomische Nachrichten, 330, 1043
* Mateo (1998) Mateo, M. L. 1998, ARA&A, 36, 435
* Michielsen et al. (2008) Michielsen, D., Boselli, A., Conselice, C. J., Toloba, E., Whiley, I. M., Aragon-Salamanca, A., Balcells, M., Cardiel, N., Cenarro, A. J., Gorgas, J., Peletier, R. F., & Vazdekis, A. 2008, MNRAS, 385, 1374
* Moore et al. (1999) Moore, B., Lake, G., Quinn, T., & Stadel, J. 1999, MNRAS, 304, 465
* Nelan et al. (2005) Nelan, J. E., Smith, R. J., Hudson, M. J., Wegner, G. A., Lucey, J. R., Moore, S. A. W., Quinney, S. J., & Suntzeff, N. B. 2005, ApJ, 632, 137
* Oke (1990) Oke, J. B. 1990, AJ, 99, 1621
* Paudel et al. (2010) Paudel, S., Lisker, T., Kuntschner, H., Grebel, E., & Glatt, K. 2010, MNRAS, 405, 800
* Sandage & Visvanathan (1978a) Sandage, A., & Visvanathan, N. 1978a, ApJ, 223, 707
* Sandage & Visvanathan (1978b) Sandage, A., & Visvanathan, N. 1978b, ApJ, 225, 742
* Schawinski et al. (2007) Schawinski, K., Kaviraj, S., Khochfar, S., et al. 2007, ApJS, 173, 512
* Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525
* Smith et al. (2009) Smith, R. J., Lucey, J. R., & Hudson, M. J. 2009, MNRAS, 400, 1690
* Smith et al. (2010) Smith, R. J., Davies, J. I., & Nelson, A. H. 2010, MNRAS, 619
* Smith et al. (2008) Smith, R. J., Marzke, Ronald O., Hornschemeier, Ann E., Bridges, Terry J., Hudson, Michael J., Miller, Neal A., Lucey, John R., Vazquez, Gerardo A., & Carter, David 2008, MNRAS, 386, L96
* Tanaka et al. (2005) Tanaka, M., Kodama, T., Arimoto, N., Okamura, S., Umetsu, K., Shimasaku, K., Tanaka, I., & Yamada, T. 2005, MNRAS, 362, 268
* Thomas et al. (2005) Thomas, D., Maraston, C., Bender, R., & Mendes de Oliveira, C. 2005, ApJ, 621, 673
* Toloba et al. (2009) Toloba, E., Boselli, A., Gorgas, J., Peletier, R. F., Cenarro, A. J., Gadotti, D. A., Gil de Paz, A., Pedraz, S., & Yildiz, U. 2009, ApJ, 707, L17
* Visvanathan & Sandage (1977) Visvanathan, N., & Sandage, A. 1977, ApJ, 216, 214
* Yi (2003) Yi, S. K. 2003, ApJ, 582, 202
* Yi et al. (2005) Yi, S. K., Yoon, S.-J., Kaviraj, S., et al. 2005, ApJ, 619, L111
Figure 1.— Color-magnitude relation (CMR) and spatial distribution of early-
type dwarf galaxies. (a) Optical CMR for various early-type dwarf galaxies;
dEs (red filled circles), dS0s (yellow filled circles), dEs with disk (stars),
dEs with blue center (triangles). Additionally, we plot blue compact dwarf
galaxies as squares. We divide the dwarf galaxies into two subsamples
according to their spatial distribution; inner region (large symbols) and
outer region (small symbols). The solid and dashed line represents the linear
least squares fit to the observed means for dEs in the inner and outer region,
respectively, while dotted line is for dS0s. Mean errors for the sample are
shown as error bars in the upper left corner. Right to the CMR, we present
color distributions of dEs in the inner (solid histogram) and outer (dotted
histogram) region normalized to the total number of sample. (b, c) Same as
(a), but for UV CMRs of galaxies detected in the NUV and FUV, respectively.
(d, e) Spatial distribution of galaxies detected in the NUV and FUV,
respectively. Large circle indicates boundary of 2 deg from the M87 for
dividing galaxies in the inner and outer region. Figure 2.— Observed color-
magnitude relations in comparison with population models for three cases of
galaxies; dEs in the inner region (left column), dEs in the outer region
(middle column), and dS0s (right column). The model lines are produced by
assuming a delayed exponential star formation history. For each case, we adopt
different relations of the characteristic star formation timescale $\tau$
(Gyr) with magnitude, to match the sequence of observed CMRs (see main text
for the details): $\tau$ = 0.19M${}_{r}+$5.56 for the inner region dEs, $\tau$
= 0.50M${}_{r}+$11.50 for the outer region dEs, and $\tau$ =
0.63M${}_{r}+$13.88 for dS0s. Different model lines in each panel are those
with different metallicities of Z = 0.0001, 0.0004, 0.001, 0.004, 0.01, 0.02
(from bottom to top). In each panel, the gray symbols in the background
represent the whole sample. The symbols of the galaxies in each case are same
as Fig. 1. Three large rectangles in each panel denote the model predictions
adopting empirical luminosity-metallicity relation and mean error of [Fe/H] at
given magnitudes for dEs (Barazza & Binggeli 2002). Table 1Relations for
early-type dwarf galaxies
$\begin{array}[]{ccrrrr}\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
x&y&a{}{}{}{}{}{}&b{}{}{}{}{}{}{}&R{}{}&rms\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\lx@intercol\hfil$Dwarf Ellipticals in the Inner
Region$\hfil\lx@intercol\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
M_{r}&g-r&~{}~{}~{}-0.04\pm 0.01&0.04\pm 0.14&-0.40&0.10\\\
M_{r}&NUV-r&~{}~{}~{}-0.20\pm 0.04&1.45\pm 0.58&-0.52&0.41\\\
M_{r}&FUV-r&~{}~{}~{}-0.46\pm 0.18&-0.54\pm 2.90&-0.52&0.76\\\ \hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr\lx@intercol\hfil$Dwarf Ellipticals in the
Outer Region$\hfil\lx@intercol\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr M_{r}&g-r&-0.03\pm
0.01&0.05\pm 0.13&-0.43&0.11\\\ M_{r}&NUV-r&-0.63\pm 0.06&-5.27\pm
0.88&-0.80&0.75\\\ M_{r}&FUV-r&-1.04\pm 0.13&-10.79\pm 1.82&-0.90&0.88\\\
\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\lx@intercol\hfil$Dwarf
Lenticulars$\hfil\lx@intercol\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr M_{r}&g-r&-0.05\pm
0.02&-0.28\pm 0.27&-0.47&0.07\\\ M_{r}&NUV-r&-0.99\pm 0.15&-12.66\pm
2.52&-0.75&0.67\\\ M_{r}&FUV-r&-2.03\pm 0.24&-29.04\pm 4.06&-0.87&0.81\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$
Note: Cols. (1) and (2): $x$ and $y$ variables. Cols. (3) and (4): Slope $a$
and intercept $b$ of the linear fit. Col. (5): Pearson correlation
coefficient. Col. (6): Mean dispersion around the best fit.
|
arxiv-papers
| 2010-08-21T10:00:46 |
2024-09-04T02:49:12.301416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Suk Kim, Soo-Chang Rey, Thorsten Lisker, and Sangmo Tony Sohn",
"submitter": "Suk Kim",
"url": "https://arxiv.org/abs/1008.3622"
}
|
1008.3634
|
# A note on hyperbolic flows in sub-Riemannian Structures
Chengbo Li School of science, Tianjin University, Tianjin, 300072, P.R.China;
email: chengboli@gmail.com
###### Abstract.
The _curvature_ and the _reduced curvature_ are basic differential invariants
of the pair (Hamiltonian system, Lagrange distribution) on the symplectic
manifold. It is shown in [4] that the negativity of the reduced curvature
implies the hyperbolicity of any compact invariant set of the Hamiltonian flow
restricted to a prescribed energy level. We consider the Hamiltonian flows of
the curve of least action of natural mechanical systems in sub-Riemannian
structures with symmetries.We give sufficient conditions for the reduced flows
(after reduction of the first integrals induced from the symmetries) to be
hyperbolic and show new examples of Anosov flows. This result is a
generalization of [8] and a partial generalization of [13] on magnetic flows.
## 1\. introduction
A prime example of Anosov flow is the geodesic flow on a compact Riemannian
manifold with negative sectional curvature ([6]). It describes inertial motion
of a point particle confined to the manifold. In this context, magnetic flows,
the flows generated by special forces, were discussed more than 30 years ago
by Anosov and Sinai [7]. They were studied recently by Gouda [8], Grognet
[12], M. and P. Paternain [9] and M. P. Wojtkowski [13]. In the last
reference, the potential and the so-called Gaussian thermostats of external
fields were also considered. .
In the present note, we focus on the hyperbolicity of the flows associated
with a natural mechanical system in a sub-Riemannian structure with
multidimensional symmetries. In this case, the sub-Riemannian structures are
reduced to a Riemannian manifold with a (vector-valued) magnetic field. We
give sufficient conditions for the reduced Hamiltonian flows (after the
reduction of the first integrals) to be hyperbolic in terms of the Riemannian
curvature tensor and the magnetic field. As a consequence, a class of Anosov
flows are also given.
In the second section, we formulate the main results of the note. We firstly
introduce the notion of a dynamical Lagrangian distribution and then discuss
the reduction after the first integrals. The key point is that we can
construct the (reduced) curvature maps (forms) for the (reduced) dynamical
Lagrangian distribution based on the work [14] and [3]. The negativity of
reduced curvature forms implies the hyperbolicity of the Hamiltonian flows
([4]). Applying this criteria we give sufficient conditions for the reduced
Hamiltonian flows to be hyperbolic, based on an expression of the reduced
curvature forms via the Riemannian curvature tensor and the magnetic field.
The last section is devoted to the proofs of the main results. We apply the
similar technique as in [10] to give the proof of the expression of the
reduced curvature forms and the sufficient conditions of hyperbolic flows then
easily follows.
## 2\. main results
### 2.1. Dynamical Lagrangian distributions
Let $M$ be an even dimensional symplectic manifold endowed with a symplectic
form $\sigma$. A _Lagrange distribution_ $\Delta\subset TM$ is a smooth vector
sub-bundle of $TM$ such that each fiber $\Delta_{x}=\Delta\cap T_{x}M,\ x\in
M$ is a Lagrangian subspace of the symplectic space $T_{x}M$. Basic examples
are cotangent bundles endowed with the standard symplectic structure and the
“vertical”distribution:
(1) $M=T^{*}N,\ \Pi_{x}=T_{x}(T^{*}_{q}N),\quad\forall x=(p,q)\in T^{*}M,p\in
T^{*}_{q}M,q\in M.$
Let $h$ be a Hamiltonian function on $M$ and denote by $\vec{h}$ the
corresponding Hamiltonian vector field: $i_{\vec{h}}\sigma=dh$. We will assume
that $\vec{h}$ is a complete vector field without loss of generality since we
will study the dynamics of the Hamiltonian systems on a compact set. The pair
$(\vec{h},\Delta)$ will be said to be _a dynamical Lagrangian distribution_ of
the symplectic manifold $(M,\sigma).$
Dynamical Lagrangian distributions appear naturally in Differential Geometry,
Calculus of Variations and Rational Mechanics. The model example can be
described as follows:
Example 1 On a manifold $M$ for a given smooth function
$L:TM\rightarrow\mathbb{R}$, which is convex on each fiber, we consider the
following standard problem of Calculus of Variation with fixed endpoints
$q_{0}$ and $q_{1}$ and fixed time $T$:
(2) $\displaystyle A(q(\cdot))=\int^{T}_{0}L(q(t),\dot{q}(t))dt\mapsto{\rm
min}$ (3) $\displaystyle q(0)=q_{0},\quad q(T)=q_{1}.$
Suppose that the Legendre transform $h:T^{*}M\rightarrow\mathbb{R}$ of the
function $L$,
(4) $h(p,q)=\max_{X\in T_{q}M}(p(X)-L(q,X)),\ q\in M,p\in T^{*}_{q}M$
is well defined and smooth on $T^{*}M$. We will say that the dynamical
Lagrangian distributions $(\vec{h},\Pi)$ is associated with the problem
(2)-(3), where $\Pi$ is as in (1). $\square$
To describe the dynamical property of a dynamical Lagrangian distribution
$(\vec{h},\Delta)$, we define the Jacobi curve (at point $x\in M$) of the pair
$(\vec{h},\Delta)$ as follows:
(5) $J_{x}(t):=e^{-t\vec{h}}_{*}\left(\Delta_{e^{t\vec{h}}x}\right),$
where $e^{t\vec{h}},\ t\in\mathbb{R}$ denotes the Hamiltonian flow generated
by the vector field $\vec{h}$.
It is clear that the Jacobi curves $J_{x}(t)$ are curves in the Lagrange
Grassmannian of the symplectic space $T_{x}^{*}M$. They are not arbitrary
curves of the Lagrangian Grassmannian but inherit special features of the pair
$(\vec{h},\mathcal{D})$. To specify these features recall that the tangent
space $T_{\Lambda}L(W)$ to the Lagrangian Grassmannian $L(W)$ of a linear
symplectic space $W$ (endowed with a symplectic form $\omega$) at the point
$\Lambda$ can be naturally identified with the space ${\rm Quad}(\Lambda)$ of
all quadratic forms on linear space $\Lambda\subset W$. Namely, given
$\mathfrak{V}\in T_{\Lambda}L(W)$ take a curve $\Lambda(t)\in L(W)$ with
$\Lambda(0)=\Lambda$ and $\dot{\Lambda}=\mathfrak{V}$. Given some vector
$l\in\Lambda$, take a curve $\ell(\cdot)$ in $W$ such that
$\ell(t)\in\Lambda(t)$ for all $t$ and $\ell(0)=l$. Define the quadratic form
(6) $Q_{\mathfrak{V}}(l)=\omega(l,\frac{d}{dt}\ell(0)).$
Using the fact that the spaces $\Lambda(t)$ are Lagrangian, it is easy to see
that $Q_{\mathfrak{V}}(l)$ does not depend on the choice of the curves
$\ell(\cdot)$ and $\Lambda(\cdot)$ with the above properties, but depends only
on $\mathfrak{V}$. So, we have the linear mapping from $T_{\Lambda}L(W)$ to
the spaces ${\rm Quad}(\Lambda)$, $\mathfrak{V}\mapsto Q_{\mathfrak{V}}$. A
simple counting of dimensions shows that this mapping is a bijection and it
defines the required identification. A curve $\Lambda(\cdot)$ in a Lagrange
Grassmannian is called _regular_ , if its velocity is a nondegenerated
quadratic form at every $\tau$. A curve $\Lambda(\cdot)$ is called _monotone_
(monotonically nondecreasing or monotonically nonincreasing) if the velocity
is sign definite (nonnegative or nonpositive) at any point. For later
convenience, a dynamical Lagrangian distribution is said to be regular
(monotone) if the associated Jacobi curves are regular (monotone).
The group of symplectomorphisms of the ambient space acts naturally on
Lagrangian distribution and Hamiltonian vector fields, therefore it acts also
on dynamical Lagrangian distributions. It turns out ([14]) that one can
construct the canonical bundle of moving frames and the complete system of
symplectic invariants for parametrized curves in Lagrange Grassmannians
satisfying very general assumptions (including monotone curves as a particular
case). The complete system of symplectic invariants (value at $t=0$) for the
Jacobi curve $J_{x}(\cdot),x\in M$ is called the curvature maps of
$(\vec{h},\Delta)$ and it is the basic differential invariants of the pair
$(\vec{h},\Delta)$ w.r.t. the action of symplectic group of $M$. In this
section, we will restrict us to the curvature maps for monotone regular
dynamical Lagrangian distribution since our goal is to obtain a sufficient
condition for hyperbolicity of the reduced Hamiltonian flows after the
reduction of first integrals, while the reduced dynamical Lagrangian
distributions are monotone regular (see Lemma 1 below). Note also that the
curvature maps for regular curves in Lagrangian Grassmannians are constructed
in earlier work [3].
More precisely, let $\mathfrak{R}_{x}(t)$ be the curvature map for the Jacobi
curve $J_{x}(t),x\in M$. Then the linear map
$\mathfrak{R}_{x}^{(\vec{h},\Delta)}:=\mathfrak{R}_{x}(0)=\mathfrak{R}_{x}(t)|_{t=0}:\Delta_{x}\rightarrow\Delta_{x}$
is said to be the _curvature map_ (at $x$) of the dynamical Lagrangian
distribution $(\vec{h},\Delta)$. It gives a symmetric bilinear forms (at $x$)
$r^{(\vec{h},\Delta)}_{x}(v,w):=\sigma(R_{x}^{(\vec{h},\Delta)}w,[\vec{h},V]),\quad
v,w\in\Delta_{x}$
where $V$ is a smooth section of the sub-bundle $\Delta$ with $V(x)=v$. The
corresponding quadratic form will be called the _curvature form_ of the
dynamical Lagrangian distribution $(\vec{h},\Delta)$.
Example 2 (Natural mechanical system) In Example 1, let
$M=R^{n},\Pi_{(p,q)}=(R^{n},0),L(q,X)=\frac{1}{2}|X|^{2}-W(q)$
(in this case the function $A(q(t),\dot{q}(t))$ is the Action functional of
the natural mechanical system with potential energy $W(q)$). Then the
curvature forms can be written as follows:
(7)
$r^{(\vec{h},\Pi)}_{(p,q)}(\partial_{p_{i}},\partial_{p_{j}})=\frac{\partial^{2}W}{\partial
q_{i}\partial q_{j}}(q),\quad\forall 1\leq i,j\leq n.$
In other words, in this case the curvature forms are naturally identified with
the Hessian of the potential $W$. $\square$
Example 3 (Riemannian manifold) Let $(M,g)$ be a Riemannian manifold. Let
$L(q,X)=\frac{1}{2}g(X,X)$. The inner product $g(\cdot,\cdot)$ defines the
canonical isomorphism between $T_{q}M$ and $T^{*}_{q}M$. For any $q\in M$ and
$p\in T^{*}_{q}M$ we will denote by $p^{h}$ the image of $p$ under this
isomorphism, namely, the vector $p^{h}\in T_{q}M$, satisfying
(8) $p(\cdot)=g(p^{h},\cdot)$
Since the fibers $T_{q}^{*}M$ are linear spaces, one can identify
$\Pi_{\lambda}(=T_{\lambda}T^{*}_{q}M)$ with $T_{\pi(\lambda)}^{*}M$, i.e. the
operation $p^{h}$ is defined also on each $p\in\Pi_{\lambda}$ with values in
$T_{\pi(\lambda)}M$. For any given $\lambda=(p,q)\in T^{*}M,p\in M,p\in
T^{*}_{q}M$, it turns out ([3]) that
(9)
$(\mathfrak{R}^{(\vec{h},\Pi)}_{\lambda}v)^{h}=R^{\nabla}(p^{h},v^{h})p^{h},\quad
v\in\Pi_{\lambda},$
where $R^{\nabla}$ is the Riemannian curvature tensor of the metric $g$.
$\square$
Example 4 (Natural mechanical system on a Riemannian manifold) We add the
potential in the action functional in the previous example, i.e.
$L(q,X)=\frac{1}{2}g(X,X)-W(q)$ . Then the curvature maps satisfies
(10)
$(\mathfrak{R}^{(\vec{h},\Pi)}_{\lambda}v)^{h}=R^{\nabla}(p^{h},v^{h})p^{h}+\nabla_{v^{h}}(\nabla
W)(q),\quad v\in\Pi_{\lambda},$
where $\nabla W$ is the gradient of $W$ w.r.t. the Riemannian metric $g$.
$\square$
### 2.2. Reduced curvature forms and hyperbolicity
Assume that the dynamical Lagrange distribution $(\vec{h},\Delta)$ have
arbitrary $s$ first integrals $g_{1},...,g_{s}$ in involution with the
Hamiltonian $h$, i.e. $s$ functions on $M$ such that
$\\{h,g_{i}\\}=0,\quad\\{g_{i},g_{j}\\}=0,\quad\forall 1\leq i,j\leq s,$
where $\\{,\\}$ is the Poisson bracket. This problem appear naturally in the
framework of mechanical systems and variational problems with symmetries. Let
$\mathcal{G}=(g_{1},...,g_{s})$ and let
(11) $\Delta^{\mathcal{G}}_{x}=(\cap_{i=1}^{s}{\rm ker}\
d_{x}g_{i})\cap\Delta_{x}+{\rm span}\\{\vec{g}_{1}(x),...,\vec{g}_{s}(x)\\}$
Clearly, the distribution
$\Delta^{\mathcal{G}}=\\{\Delta^{\mathcal{G}}_{x},x\in M\\}$ is a Lagrangian
distribution. Hence, we get a reduced dynamical Lagrangian distribution
$(\vec{h},\Delta^{\mathcal{G}})$ after the reduction by first integrals
$\mathcal{G}$. Its curvature maps (forms) will be called _the reduced
curvature maps (forms)_ after the reduction by first integrals $\mathcal{G}$.
Example 5 Assume that we have one first integral $g$ of $h$ such that the
Hamiltonian vector field $\vec{g}$ preserves the distribution $\Delta$, i.e.
$(e^{t\vec{g}})_{*}\Delta=\Delta$. Fixing some value $c$ of $g$, one can
define (at least locally) the following quotient manifold:
$M_{g,c}=g^{-1}(c)/{\mathcal{C}}$ , where $\mathcal{C}$ is the line foliation
of the integral curves of the vector field $\vec{g}$. The manifold $M_{g,c}$
naturally inherits a symplectic form from the original symplectic structure
$(M,\sigma)$. Furthermore, if we denote by $\Phi:g^{-1}(c)\rightarrow M_{g,c}$
the canonical projection on the quotient set, the vector field
$\Phi_{*}(\vec{h})$ is well defined Hamiltonian vector field on $M_{g,c}$ due
to the fact that the vector fields $\vec{h}$ and $\vec{g}$ commute. For
simplicity, we still denote $\Phi_{*}(\vec{h})$ by $\vec{h}$. Actually we have
simply described the standard reduction of the Hamiltonian systems on the
level set of the first integrals in Mechanics (see e.g. [1]). In this way, to
any dynamical Lagrangian distribution $(\vec{h},\Delta)$ on $M$ one can
associate the dynamical Lagrangian distribution $(\vec{h},\Phi_{*}\Delta)$ on
the symplectic manifold $M_{g,c}$ of smaller dimension. $\square$
It is well known that the geodesic flows on a compact Riemannian manifold with
negative sectional curvature is Anosov ([6]). On the other hand, the reduced
curvature maps (forms) of the dynamical Lagrangian distributions associated
with the geodesic problem on a Riemannian manifold are naturally identified
with the sectional curvature tensor (See Example 6 below). Hence, we could
roughly formulate the result of Anosov as follows: negativity of the reduced
curvature forms implies the hyperbolicity of the geodesic flows. To go further
from this viewpoint, one can obtain a natural generalization ([4]) in the
framework of dynamical Lagrangian distributions. It will serve as a criteria
in the study of the hyperbolic flows in sub-Riemannian structures.
###### Definition 1.
Let $e^{tX},\ t\in\mathbb{R}$ be the flow generated by the vector field $X$ on
a manifold $P$. A compact invariant set $A\subset P$ of the flow $e^{tX}$ is
called a hyperbolic set if there exists a Riemannian structure in a
neighborhood of $A$, a positive constant $\delta$, and a splitting:
$T_{z}P=E_{z}^{+}\oplus E_{z}^{-}\oplus\mathbb{R}X(z),\ z\in A$ such that
$X(z)\neq 0$ and
1. (1)
$e^{tX}_{*}E^{+}_{z}=E^{+}_{e^{tX}z},\ e^{tX}_{*}E^{-}_{z}=E^{-}_{e^{tX}z},$
2. (2)
$\|e^{tX}_{*}\zeta^{+}\|\geq e^{\delta t}\|\zeta^{+}\|,\ \forall
t>0,\forall\zeta^{+}\in E^{+}_{z},$
3. (3)
$\|e^{tX}_{*}\zeta^{-}\|\leq e^{-\delta t}\|\zeta^{-}\|,\ \forall
t>0,\forall\zeta^{-}\in E^{-}_{z}.$
If the entire manifold $P$ is a hyperbolic set, then the flow $e^{tX}$ is
called a flow of Anosov type.
###### Theorem 1.
Let $c=(c_{0},c_{1},...,c_{s})$ be constants. Let $S$ be a compact invariant
set of the flow $e^{t\vec{h}}$ contained in a fixed level of
$h^{-1}(c_{0})\cap_{i=1}^{s}g_{i}^{-1}(c_{i})$ and
$\vec{h}(x),\vec{g}_{i}(x)\notin\mathcal{D}_{x},\forall x\in S,i=1,...,s$. If
the reduced curvature form $r_{x}^{(\mathcal{G},h)}$ of the dynamical
Lagrangian distribution $(\vec{h},\mathcal{D})$ (after the reduction by first
integrals $(\mathcal{G},h$)) is negative at every point $x$ of $S$, then $S$
is a hyperbolic set of the flow
$e^{t\vec{h}}|_{h^{-1}(c_{0})\cap_{i=1}^{s}g_{i}^{-1}(c_{i})}$.
### 2.3. Descriptions of main results
We now specialize to the study of a natural mechanical system on a sub-
Riemannian manifold with symmetries.
Let $M$ be a connected smooth manifold. A distribution $\mathcal{D}$ on $M$ is
a sub-bundle of the tangent bundle $TM$. It is said to be _completely
nonholonomic_ if any local frame $\\{X_{i}:1\leq i\leq n\\}$ for
$\mathcal{D}$, together with all its iterated Lie brackets
$[X_{i},X_{j}],[X_{i},[X_{j},X_{k}]]$, … , spans the tangent bundle $TM$. A
Lipschitzian curve $\gamma:[0,T]\longrightarrow M$ is said to be admissible if
$\dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)}$ for a.e. $t\in[0,T]$. From the
Rashevskii-Chow theorem ([5]) it follows that there is an admissible curve
joining any two points of $M$. _A sub-Riemannian metric_ is a smoothly varying
positive definite inner product $\left\langle\cdot,\cdot\right\rangle$ on
$\mathcal{D}$. In particular, when $\mathcal{D}$ is equal to the tangent
bundle, $\left\langle\cdot,\cdot\right\rangle$ gives a Riemannian metric.
_A sub-Riemannian structure_ , denoted by the triple
$(M,\mathcal{D},\left\langle\cdot,\cdot\right\rangle)$, is a smooth
$n$-dimensional connected manifold $M$ equipped with a sub-Riemannian metric
$\left\langle\cdot,\cdot\right\rangle$ on a completely nonholonomic
distribution $\mathcal{D}$. In this case, we call the manifold $M$ _a sub-
Riemannian manifold_. In the present note we consider sub-Riemannian metrics
$\left\langle\cdot,\cdot\right\rangle$ on distribution $\mathcal{D}$ of corank
$s$, having $s$ transversal infinitesimal symmetries, i.e. $s$ vector fields
$X_{1},\ldots,X_{s}$ on $M$ such that
(12) $e^{tX_{i}}_{*}\mathcal{D}=\mathcal{D}\ ,\
(e^{tX_{i}})^{*}\left\langle\cdot,\cdot\right\rangle=\left\langle\cdot,\cdot\right\rangle,\quad
1\leq i\leq s,$
and $TM=\mathcal{D}\oplus\text{span}\\{X_{1},\ldots,X_{s}\\}$. Suppose further
that the symmetries $\\{X_{i}:\ 1\leq i\leq s\\}$ are commutative (see Remark
1 for noncommutative case), i.e.
(13) $[X_{i},X_{j}]=0,\quad\forall 1\leq i,j\leq s.$
We consider the natural mechanical system on a sub-Riemannian manifold (ASR):
(14) $\displaystyle
A(\gamma(\cdot))=\int^{T}_{0}(\frac{1}{2}\|\dot{\gamma}\|^{2}-W(\gamma))dt\mapsto{\rm
min}$ (15) $\displaystyle\gamma(\cdot)\ {\rm is\
admissible},\quad\gamma(0)=q_{0},\quad\gamma(T)=q_{1}.$
where $\|\cdot\|$ is the norm w.r.t. the metric
$\left\langle\cdot,\cdot\right\rangle$. We assume further that the potential
$W$ in (14) is constant along the integral curves of any $X_{i}$, or,
equivalently,
(16) $X_{i}(W)=0,\ i=1,...,s.$
It is more convenient to regard it as an optimal control problem and its
extremals can be described by the Pontryagin Maximum Principle of Optimal
Control Theory ([11]). There are two different types of extremals: abnormal
and normal, according to vanishing or nonvanishing of Lagrange multiplier near
the functional, respectively. The minimizers of the problem are the
projections of either normal extremals or abnormal extremals.
In the present note we will focus on normal extremals only. To describe them
let us introduce some notations. Let $T^{*}M$ be the cotangent bundle of $M$
and $\sigma$ be the canonical symplectic form on $T^{*}M$, i.e.
$\sigma=-d\varsigma$, where $\varsigma$ is the tautological (Liouville) 1-form
on $T^{*}M$. Let
(17) $h(p,q)=\max_{u\in\mathcal{D}}(p\cdot
u-\frac{1}{2}\|u\|^{2}+W(q))=\frac{1}{2}\|p|_{\mathcal{D}_{q}}\|^{2}+W(q),\
q\in M,\ p\in T^{*}_{q}M,$
where $p|_{\mathcal{D}_{q}}$ is the restriction of the linear functional $p$
to $\mathcal{D}_{q}$ and the norm $\|p|_{\mathcal{D}_{q}}\|$ is defined w.r.t.
the Euclidean structure on $\mathcal{D}_{q}.$ It is well defined and smooth in
the open set $O=T^{*}M\backslash\mathcal{D}^{\perp}$, where
$\mathcal{D}^{\perp}$ is the annihilator of $\mathcal{D}$, that is,
(18) $\mathcal{D}^{\perp}=\\{(p,q)\in T^{*}M:p(v)=0\,\,\forall
v\in\mathcal{D}_{q}\\}.$
For any vector field $X_{i}$ define the “quasiimpluses”
$u_{i}:T^{*}M\rightarrow\mathbb{R}$ by
$\quad u_{i}(p,q)=p(X_{i}(q)),\ q\in T^{*}_{q}M,q\in M,\ \forall 1\leq i\leq
s.$
Let $h$ be the sub-Riemannian Hamiltonian as in (17). Then it follows from
(12) and (16) that
(19) $\\{h,u_{i}\\}=0,\forall 1\leq i\leq s.$
and from (13) it follows that
(20) $\\{u_{i},u_{j}\\}=0,\ \forall 1\leq i,j\leq s,$
where $\\{\ ,\ \\}$ is the Poisson bracket. In other words, $u_{i}(1\leq i\leq
s)$ are first integrals in involution of the Hamiltonian system
$e^{t\vec{h}}$.
As before, let $\Pi$ be the “vertical”distribution, i.e.
$\Pi_{\lambda}=T_{\lambda}T_{\pi(\lambda)}^{*}M$, where $\pi:T^{*}M\to M$ is
the canonical projection. Now we can apply the reduction to the dynamical
Lagrangian distributions $(\vec{h},\Pi)$ after the first integrals
$u_{i}(1\leq i\leq s)$ (c.f. Example 5). For this, fix constants
$c_{0},c_{1},...,c_{s}$, where $c_{0}>0$ is sufficient large. Take a common
level set
$\mathcal{H}_{c}:=\\{h=c_{0}\\}\cap\\{u_{i}=c_{i},1\leq i\leq s\\}.$
Then
$W^{c}_{\lambda}=T_{\lambda}\mathcal{H}_{c}/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),\ 1\leq i\leq s\\}$
is a linear symplectic space with the symplectic form $\sigma^{c}$ naturally
inherited from the symplectic form $\sigma$. Moreover,
$\Pi_{\lambda}^{c}=(T_{\lambda}\mathcal{H}_{c}\cap\Pi_{\lambda})/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),\ 1\leq i\leq s\\}$
is a Lagrangian subspace in $W^{c}_{\lambda}$. Hence, we get the reduced
dynamical Lagrangian distribution $(\vec{h},\Pi^{c})$ in the linear symplectic
space $W^{c}$.
###### Remark 1.
The reduction procedure above also applies for the case that the symmetries
$\\{X_{i}:\ 1\leq i\leq s\\}$ are not commutative but still satisfy that
$g=\rm{span}_{\mathbb{R}}\\{X_{i},\ 1\leq i\leq s\\}$ is a Lie algebra and the
derived Lie algebra $g^{2}=[g,g]$ is a proper Lie subalgebra. Indeed, take a
basis of $g^{2}:(g_{1},...,g_{k})$ and complete it to a basis of
$g:(g_{1},...,g_{k},g_{k+1},...,g_{s})$. Then if we select a level set
$c=(c_{1},...,c_{s})$ such that
(21) $c_{i}=0,\ 1\leq i\leq k,$
then one can see that $g$ is commutative on this level set. Therefore, it
reduces to commutative case and then the Poission reduction can be applied.
Actually, it is equivalent to considering a sub-Riemannian structure on the
manifold obtained by reduction of the original one by $g_{1},...,g_{k}$ on
which the symmetries consist of a commutative Lie algebra $g/g^{2}$.
The (reduced) curvature maps (forms) of $(\vec{h},\Pi^{c})$ is naturally
related to the ambient sub-Riemannian structures (with symmetries), while the
later can be reduced to a Riemannian manifold equipped with a
$\mathbb{R}^{s}$-valued magnetic field. Denote by $\widetilde{M}$ the quotient
of $M$ by the leaves of the integral manifold of the involutive distribution
spanned by $X_{1},\ldots X_{s}$ and denote the factorization map by ${\rm
pr}:M\rightarrow\widetilde{M}$. Then $\widetilde{M}$ is (at least locally) a
Riemannian manifold equipped with the Riemannian metric $g$ induced from the
sub-Riemannian metric. Furthermore, let $\omega=(\omega_{i})_{1\leq i\leq s}$
be the $\mathbb{R}^{s}$-valued 1-form defined by $\omega_{i}|_{\mathcal{D}}=0$
and $\omega_{i}(X_{j})=\delta_{ij},\ \forall 1\leq i,j\leq s.$ Then
$d\omega=(d\omega_{i})_{1\leq i\leq s}$ induces a $\mathbb{R}^{s}$-valued
2-form on $\widetilde{M}$ (still denoted by $d\omega=(d\omega_{i})$) and one
can define a $\mathbb{R}^{s}$-valued tensor
$J=(J_{i}(\tilde{q}),\tilde{q}\in\widetilde{M})$ of type $(1,1)$ on
$\widetilde{M}$ satisfying
$g_{\tilde{q}}(J_{i}(\tilde{q})v,w)=d\omega_{i}(\tilde{q})(v,w),\ v,w\in
T_{\tilde{q}}\widetilde{M},\tilde{q}\in\widetilde{M},\ \forall 1\leq i\leq s.$
Let $\Xi^{c}$ be the $s$-foliation such that its leaves are integral curves of
$\\{\vec{u}_{i},1\leq i\leq s\\}$. Let ${\rm PR}^{c}:T^{*}M\to T^{*}M/\Xi^{c}$
be the canonical projection to the quotient manifold. Now we show that the
quotient manifold $N^{c}=\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}$ can be
naturally identified with $T^{*}\widetilde{M}$. Indeed, a point
$\tilde{\lambda}$ in $\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}$ can be
identified with a leaf $(\rm{PR}^{c})^{-1}(\tilde{\lambda})$ of $\Xi^{c}$
which has a form
$((e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}p,e^{\sum_{i=1}^{s}t_{i}X_{i}}q),$
where $\lambda=(p,q)\in({\rm PR}^{c})^{-1}(\tilde{\lambda})$, $q\in M$ and
$p\in T_{q}^{*}M$. On the other hand, any element in $T^{*}\widetilde{M}$ can
be identified with a one-parametric family of pairs
$((e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}(p|_{\mathcal{D}}),e^{\sum_{i=1}^{s}t_{i}X_{i}}q)$.
The mapping $I^{c}:\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}\to
T^{*}\widetilde{M}$ defined by
$I^{c}:(e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}p,e^{\sum_{i=1}^{s}t_{i}X_{i}}q)\mapsto(e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}(p|_{\mathcal{D}}),e^{\sum_{i=1}^{s}t_{i}X_{i}}q)$
is one-to-one ($p(X_{i})=\hbox{const.}$ is already prescribed) and it defines
the required identification.
Before the statement of the main result of the note, let us introduce some
notations. Let $D^{\bot}$ be as in (18). Denote
$\mathcal{D}_{q}^{\bot}=\mathcal{D}^{\bot}\cap T^{*}_{q}M.$ Then one has the
following series of natural identifications:
(22) $\Pi^{c}_{\lambda}\sim
T^{*}_{q}M/\mathcal{D}_{q}^{\bot}\sim\mathcal{D}^{*}_{q}\stackrel{{\scriptstyle\left\langle\cdot,\cdot\right\rangle}}{{\sim}}\mathcal{D}_{q}\sim
T_{{\rm pr}(q)}\widetilde{M}$
where $\mathcal{D}_{q}^{*}\subseteq T^{*}_{q}M$ is the dual space of
$\mathcal{D}_{q}$. Given $v\in T_{\lambda}T^{*}_{q}M$ ($\sim T^{*}_{q}M$),
where $q=\pi(\lambda)$, we can assign a unique vector $v^{h}\in T_{{\rm
pr}(q)}\widetilde{M}$ to its equivalence class in
$T_{q}^{*}M/\mathcal{D}_{q}^{\bot}$ by using the identifications (22).
Conversely, to any $X\in T_{\hbox{pr}(q)}\widetilde{M}$ one can assign an
equivalence class of $T_{\lambda}(T^{*}_{q}M)/\mathcal{D}_{q}^{\bot}$. Denote
by $X^{v}\in T_{\lambda}T^{*}_{q}M$ the unique representative of this
equivalence class such that $du_{i}(X^{v})=0,\,\forall 1\leq i\leq s$.
For simplicity, we henceforth denote: $J^{c}=\sum_{i=1}^{s}c_{i}J_{i}$.
###### Theorem 2.
The curvature forms $r_{\lambda}^{c}$ of the dynamical Lagrangian distribution
$(\vec{h},\Pi^{c})$ is expressed as follows. For any $v\in\Pi^{c}_{\lambda}$,
$\displaystyle r^{c}_{\lambda}(v)$ $\displaystyle=$ $\displaystyle
g(R^{\nabla}(p^{h},v^{h})p^{h},v^{h})+g(\nabla
J^{c}(p^{h},v^{h}),v^{h})+\frac{1}{4}g(J^{c}v^{h},J^{c}v^{h})$
$\displaystyle+$
$\displaystyle\frac{3}{8(c_{0}+W)}\left(g(J^{c}p^{h},v^{h})\right)^{2}+\frac{3}{2(c_{0}+W)}g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h})$ $\displaystyle+$
$\displaystyle\frac{3}{2(c_{0}+W)}\left(g(v^{h},\nabla W)\right)^{2}+{\rm
Hess}\ W(v^{h},v^{h}).$
It follows from relations (16),(19) and (20) that $e^{t\vec{h}}$ induces a
(reduced) Hamiltonian flow $\Phi_{t}$ on $N^{c}$, where
$N^{c}=\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}$, as before. Then following
theorem is a direct consequence Theorem 1.
###### Theorem 3.
Assume that $K^{c}\subset N^{c}$ is a compact invariant set of the flow
$\Phi_{t}$ on $N^{c}$. If the curvature form $r_{\lambda}^{c}$ is negative at
every point of $K^{c}$, then $K^{c}$ is a hyperbolic set of the flow
$\Phi_{t}$ on $N^{c}$.
As mentioned, the manifold $N^{c}$ is naturally identified $T^{*}M$. Now
denote by $S_{1}\widetilde{M}$ the unit tangent bundle. Combining the previous
theorem with Theorem 2, we get the following
###### Theorem 4.
Assume that the reduced Riemannian manifold $(\widetilde{M},g)$ is compact and
has sectional curvature bounded from above by $k_{\rm max}$. If the constants
$c_{0},c_{1},...,c_{s}$ satisfy
$\displaystyle\max_{v,w\in S_{1}\widetilde{M},v\perp w}g(v,\nabla
J^{c}(w;v))+\frac{1}{4}g(J^{c}v,J^{c}v)+\frac{3}{8(c_{0}+W)}g(w,J^{c}v)g(w,J^{c}v)$
$\displaystyle+\frac{3}{2(c_{0}+W)}g(v,\nabla
W)g(J^{c}w,v)+3\left(\frac{\|\nabla W\|}{2(c_{0}+W)}\right)^{2}+\frac{\|{\rm
Hess}\ W\|}{2(c_{0}+W)}<-k_{\rm max},$
then the flow $\Phi_{t}$ is an Anosov flow.
###### Corollary 1.
Pure potential flows, i.e. $J_{i}=0(1\leq i\leq s)$,
(23) $\max_{\tilde{q}\in\widetilde{M}}\left(3\left(\frac{\|\nabla
W\|}{2(c_{0}+W)}\right)^{2}+\frac{\|{\rm Hess}\
W\|}{2(c_{0}+W)}\right)<-k_{\rm max}.$
###### Corollary 2.
Pure magnetic flows, i.e. $W=0$,
(24) $\max_{v,w\in S_{1}\widetilde{M},v\perp w}{cg(v,\nabla
J(w;v))+c^{2}g(Jv,Jv))}<-k_{\rm{max}}.$
###### Remark 2.
The left-hand side of the inequality in Theorem 4 is always positive, because
the second term inside the max is positive and the first term can be made
nonnegative, if necessary, by changing the sign of $w$. Hence, Theorem 4 makes
sense only if $k_{\rm{max}}<0$.
The flow $\Phi_{t}$ on $N^{c}$ can be considered as a perturbation of the
Riemannian geodesic flow: the flow $\Phi_{t}$ on $N^{c}$ remains to be an
Anosov flow for sufficient small constants $c_{i}(1\leq i\leq s)$ and for
proper potential function $W$ (sufficient small norm of $W$ and its
derivatives). When $s=1$, it coincides with Theorem 4.1 (the case of Gaussian
thermostats of external fields $E=0$ there) in [13]; when $s=1$ and $W=0$, the
condition of Anosov magnetic flows (24) coincides with the main results in
[8].
## 3\. proof of the main results
The rest of the note is devoted to the proof of Theorem 2.
### 3.1. Reduced curvature maps
As before, fix constants $c_{0},c_{1},...,c_{s}$, where $c_{0}>0$ is
sufficient large. Let $J^{c}_{\lambda}(t)$ be the Jacobi curves associated
with the reduced dynamical Lagrangian distribution $(\vec{h},\Pi^{c})$,
namely,
(25) $J^{c}_{\lambda}(t):=e^{-t\vec{h}}_{*}\Pi^{c}_{e^{t\vec{h}}\lambda}.$
###### Lemma 1.
The reduced Jacobi curve $J^{c}_{\lambda}(\cdot)$ is a regular monotone
nondecreasing curve in Lagrange Grassmannian $L(W^{c}_{\lambda})$.
###### Proof.
First note that if $\bar{\lambda}=e^{\bar{t}\vec{h}}\lambda$ and
$\phi:W^{c}_{\lambda}\rightarrow W^{c}_{\bar{\lambda}}$ is a symplectic
transformation induced in the natural way by a linear mapping
$e^{t\vec{h}}_{*}:T_{\lambda}\mathcal{H}_{c}\rightarrow
T_{\bar{\lambda}}\mathcal{H}_{c}$, where, as before,
$\mathcal{H}_{c}:=\\{h=c_{0}\\}\cap\\{u_{i}=c_{i},1\leq i\leq s\\}.$
Then by (25) we have
(26)
$J^{c}_{\bar{\lambda}}(t)=\phi\bigl{(}J^{c}_{\lambda}(t-\bar{t})\bigr{)}.$
Further, it turns out (see, for example, [2, Proposition 1]) that the velocity
of the Jacobi curve $J_{\lambda}^{c}(\cdot)$ at $t=0$ is equal to the
restriction of the Hessian of $h$ to the tangent space to $\Pi_{\lambda}^{c}$
at the point $\lambda$. This together with the relation (26) and the
construction of $\Pi_{\lambda}^{c}$ implies easily that $J^{c}_{\lambda}(t)$
is a regular monotone nondecreasing curve. ∎
###### Theorem 5.
Let $\Lambda(\cdot)$ be a regular curve in the Lagrange Grassmannian $L(G)$ of
a $2n$-dimensional linear symplectic space $G$. Then there exists a moving
Darboux frame $(E(t),F(t))$ of $G$:
$E(t)=(e_{1}(t),...,e_{n}(t)),\ F(t)=(f_{1}(t),...,f_{n}(t))$
such that $\Lambda(t)=\rm{span}\\{E(t)\\}$ and there exists a one-parametric
family of linear self-adjoint operators
$\mathfrak{R}(t):\Lambda(t)\rightarrow\Lambda(t)$ satisfying
(27) $\begin{cases}E^{\prime}(t)=F(t),\\\ F^{\prime}(t)=-R(t)E(t).\end{cases}$
The moving frame $(E(t),F(t))$ is a called _a normal moving frame_ of
$\Lambda(t)$ and the linear operator $\mathfrak{R}(t)$ is called _the
curvature map_ of $\Lambda(t)$. A moving frame
$(\widetilde{E}(t),\widetilde{F}(t))$ is a normal moving frame of $\Lambda(t)$
if and only there exists a constant orthogonal matrix $U$ of size $n\times n$
such that
(28) $\widetilde{E}(t)=E(t)U,\ \widetilde{F}(t)=F(t)U.$
###### Remark 3.
Note that from (27) it follows that if
$\bigl{(}\widetilde{E}(t),\widetilde{F}(t)\bigr{)}$ is a Darboux moving frame
such that $\widetilde{E}(t)$ is an orthonormal frame of $\Lambda(t)$ and ${\rm
span}\,\\{\widetilde{F}(t)\\}=\Lambda^{\rm trans}(t)$. Then there exists a
curve of antisymmetric matrices $B(t)$ such that
(29)
$\left\\{\begin{array}[]{l}\widetilde{E}^{\prime}(t)=\widetilde{E}(t)B(t)+\widetilde{F}_{a}(t)\\\
\widetilde{F}^{\prime}(t)=-\widetilde{E}(t)\widetilde{\mathcal{R}}(t)+\widetilde{F}(t)B(t),\end{array}\right.$
where $\widetilde{\mathcal{R}}(t)$ is the matrix of the curvature map
$\mathfrak{R}(t)$ on $\Lambda(t)$ w.r.t. the basis $\widetilde{E}(t)$.
As a matter of fact, normal moving frames define a principal $O(n)$-bundle of
symplectic frame in $G$ endowed with a canonical connection. Also, relations
(28) imply that the following $n$-dimensional subspaces
(30) $\Lambda^{\rm trans}(t)={\rm span}\\{F(t)\\}$
of $G$ does not depend on the choice of the normal moving frame. It is called
the _canonical complement_ of $\Lambda(t)$ in $G$. Moreover, the subspaces
$\Lambda(t)$ and $\Lambda^{\rm trans}(t)$ are endowed with the _canonical
Euclidean structure_ such that the tuple of vectors $E(t)$ and $F(t)$
constitute an orthonormal frame w.r.t. to it, respectively.
Finally, the linear map from $\Lambda(t)$ to $\Lambda(t)$ with the matrix
$R(t)$ from (27) in the basis $\\{E(t)\\}$, is independent of the choice of
normal moving frames. It will be denoted by $\mathfrak{R}(t)$ and it is called
the _curvature map_ of the curve $\Lambda(t)$.
Now we apply the above results for curves in Lagrange Grassmannians to sub-
Riemannian structures. Since $\mathfrak{J}^{c}_{\lambda}(0)$ and
$\Pi^{c}_{\lambda}$ can be naturally identified, there is a canonical
splitting of $W^{c}_{\lambda}:$
(31)
$W^{c}_{\lambda}=\Pi^{c}_{\lambda}\oplus\widetilde{\mathfrak{J}}^{c}(\lambda),$
where $\widetilde{\mathfrak{J}}^{c}(\lambda)={\rm span}(F^{\lambda}(0))$ is
the canonical complement. In other words,
$\widetilde{\mathfrak{J}}^{c}(\lambda)$ is actually a (nonlinear) Ehresmann
connection of $\Pi^{c}_{\lambda}$ in $W^{c}_{\lambda}$. It also follows that
the subspaces $\Pi_{\lambda}^{c}$ and $\widetilde{\mathfrak{J}}^{c}(\lambda)$
are equipped with a canonical Euclidean structure. Moreover, one can define
the curvature map of the dynamical Lagrangian distribution
$(\vec{h},\Pi^{c})$, i.e.
$\mathfrak{R}^{c}_{\lambda}:\Pi^{c}_{\lambda}\rightarrow\Pi^{c}_{\lambda}$
such that $\mathfrak{R}^{c}_{\lambda}=\mathfrak{R}_{\lambda}(0)$ of the
curvature maps of the Jacobi curve $\mathfrak{J}^{c}_{\lambda}(\cdot)$ at
$t=0$. This curvature maps are intrinsically related to the sub-Riemannian
structure and will be called _the reduced curvature map_ of the sub-Riemannian
structure.
Let $\lambda\in T^{*}M$ and let $\lambda(t)=e^{t\vec{h}}\lambda$. Assume that
$(E^{\lambda}(t),F^{\lambda}(t))$ is a normal moving frame of the Jacobi curve
$\mathfrak{J}^{c}_{\lambda}(t)$ attached at point $\lambda$. Let
$\mathfrak{E}$ be the Euler field on $T^{*}M$, i.e. the infinitesimal
generator of the homotheties on its fibers. Clearly
$T_{\lambda}(T^{*}M)=T_{\lambda}\mathcal{H}_{c}\oplus\mathbb{R}\mathfrak{E}(\lambda)\oplus{\rm
span}\\{\partial_{u_{i}}(\lambda),1\leq i\leq s\\}$. The flow $e^{t\vec{h}}$
on $T^{*}M$ induces the push-forward maps $e^{t\vec{h}}_{*}$ between the
corresponding tangent spaces $T_{\lambda}T^{*}M$ and $T_{\lambda(t)}T^{*}M$,
which in turn induce naturally the maps between the spaces
$T_{\lambda}(T^{*}M)/{\rm span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),1\leq
i\leq s\\}$ and $T_{\lambda(t)}T^{*}M/{\rm
span}\\{\vec{h}(\lambda(t)),\vec{u}_{i}(\lambda(t)),1\leq i\leq s\\}$. The map
$\mathcal{K}^{t}$ between $T_{\lambda}(T^{*}M)/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),1\leq i\leq s\\}$ and
$T_{\lambda(t)}T^{*}M/{\rm
span}\\{\vec{h}(\lambda(t)),\vec{u}_{i}(\lambda(t)),1\leq i\leq s\\}$, sending
$E^{\lambda}(0)$ to $e^{t\vec{h}}_{*}E^{\lambda}(t)$, $F^{\lambda}(0)$ to
$e^{t\vec{h}}_{*}F^{\lambda}(t)$, and the equivalence class of
$\mathfrak{E}(\lambda),\partial_{u_{i}}(\lambda)(1\leq i\leq s)$ to the
equivalence class of
$\mathfrak{E}(e^{t\vec{h}}\lambda),\partial_{u_{i}}(\lambda(t))(1\leq i\leq
s)$, is independent of the choice of normal moving frames. The map
$\mathcal{K}^{t}$ is called _the parallel transport_ along the extremal
$e^{t\vec{h}}\lambda$ at time $t$. For any $v\in T_{\lambda}(T^{*}M)/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),1\leq i\leq s\\}$, its image
$v(t)=\mathcal{K}^{t}(v)$ is called _the parallel transport of $v$ at time
$t$_. Note that from the definition of the reduced Jacobi curves and the
construction of normal moving frames it follows that the restriction of the
parallel transport $\mathcal{K}_{t}$ to the vertical subspace
$T_{\lambda}(T_{\pi(\lambda)}^{*}M)$ of $T_{\lambda}(T^{*}M)$ can be
considered as a map onto the vertical subspace
$T_{\lambda(t)}(T_{\pi(\lambda(t))}^{*}M)$ of $T_{\lambda(t)}(T^{*}M)$. A
vertical vector field $V$ is called _parallel_ if
$V(e^{t\vec{h}}\lambda)=\mathcal{K}^{t}\bigl{(}V(\lambda)\bigr{)}$.
Example 6 (Riemannian geodesic flow) In this case, $\mathcal{D}=TM,W=0$ and
there is no symmetries at all ($s=0$). In [3] the reduced curvature map was
expressed by the Riemannian curvature tensor. If we adopt the notations from
Example 3 and take the constants $c_{0}=\frac{1}{2},c_{i}=0(1\leq i\leq s)$.
Then
(32)
$\mathfrak{R}_{\lambda}^{c}(v)=R^{\nabla}(p^{h},v^{h})p^{h},\quad\forall\lambda=(q,p)\in\mathcal{H}_{c},q\in
M,p\in T^{*}_{q}M,v\in\Pi^{c}_{\lambda}.$
Given a vector $X\in T_{q}M$ denote by $\nabla_{X}$ its lift to the Levi-
Civita connection, considered as an Ehresmann connection on $T^{*}M$. Then by
constructions the Hamiltonian vector field $\vec{h}$ is horizontal and
satisfies $\vec{h}=\nabla_{p^{h}}$. Take any $v,w\in\Pi_{\lambda}^{c}$ and let
$V$ be a vertical vector field such that $V(\lambda)=v$. From (32) , structure
equation (27), and the fact that the Levi-Civita connection (as an Eheresmann
connection on $T^{*}M$) is a Lagrangian distribution (c.f. [3]) it follows
that the Riemannian curvature tensor satisfies the following identity:
(33) $\langle
R^{\nabla}(p^{h},v^{h})p^{h},w^{h}\rangle=-\sigma\left([\nabla_{p^{h}},\nabla_{V^{h}}](\lambda),\nabla_{w^{h}}\right).\quad\square$
### 3.2. Proof of Theorem 2
We first express the canonical complement in terms of the Levi-Civita
connection of the Riemannian metric and the tensor $J_{i}^{c}$ and then we can
give the proof of Theorem 2 using some calclus formulae which is developped in
[10].
#### 3.2.1. The canonical complement $\widetilde{\mathfrak{J}}^{c}(\lambda)$
The restriction of the parallel transport $\mathcal{K}^{t}$ to
$\Pi^{c}_{\lambda}$ is characterized by the following two properties:
1. (1)
$\mathcal{K}^{t}$ is an orthogonal transformation of spaces
$\Pi_{\lambda}^{c}$ and $\Pi_{e^{t\vec{h}}\lambda}^{c}$;
2. (2)
The space ${\rm
span}\\{\frac{d}{dt}\bigl{(}(e^{-t\vec{h}})_{*}(\mathcal{K}^{t}v)\bigr{)}|_{{}_{t=0}}:v\in\Pi_{\lambda}^{c}\\}$
is isotropic.
Then $\widetilde{\mathfrak{J}}^{c}(\lambda)={\rm
span}\\{\frac{d}{dt}\bigl{(}(e^{-t\vec{h}})_{*}(\mathcal{K}^{t}v)\bigr{)}|_{{}_{t=0}}:v\in\Pi_{\lambda}^{c}\\}$.
To express $\widetilde{\mathfrak{J}}^{c}(\lambda)$ in terms of the Riemannian
manifold and the magnetic field, we show the decomposition of the symplectic
form $\sigma$ (the standard symplectic form on $T^{*}\widetilde{M}$) and the
Hamiltonian field $\vec{h}$. One can see that the diffeomorphism $I^{c}$,
defined as before, are not in general symplectic. Indeed, each level set
inherits a symplectic structure depending on the choice of the level
$\\{c_{i}:1\leq i\leq s\\}$.
By the construction of the map $I^{c}$, for any vector field $X$ on
$T^{*}\widetilde{M}$, we can assign the vector field $\underline{X}$ on
$T^{*}M$ s.t. $PR^{c}_{*}\underline{X}=((I^{c})^{-1})_{*}X$ and
$\pi_{*}\underline{X}\in\mathcal{D}$. In the following, denote by
$\nabla_{p^{h}}$ the lift of $p^{h}$ to $T^{*}\widetilde{M}$ with respect to
the Levi-Civita connection and denote
$\Omega^{c}=\sum_{i=1}^{s}c_{i}d\omega_{i}$ and
$\bar{\sigma}=(I^{c}\circ\rm{PR}^{c})^{*}\sigma$. We will denote by
$\tilde{\sigma}$ the standard symplectic form on $T^{*}\widetilde{M}$. The
proof of the following lemma is complete similar to that of Lemma 3.1-3.3 in
[10] and thus is omitted.
###### Lemma 2.
The following decomposition formulae hold.
1. (1)
On the level set
$\\{u_{i}=c_{i},i=1,...,s\\},\quad\sigma=\bar{\sigma}-(\pi\circ{\rm
pr})^{*}(\Omega^{c})$;
2. (2)
For any vectors $X,V\in T_{\lambda}T^{*}M$ with $\pi_{*}V=0$ we have
$\sigma(X,v)=g(\pi_{*}X,V^{h});$
3. (3)
$\vec{h}(p,q)=\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v}+\overrightarrow{W}$.
Comparing with the sub-Riemannian geodesic problem, we will develop some
additional calculus formulae about the potential $W$.
###### Lemma 3.
Let $V_{1},V_{2}$ be the vector fields on $T^{*}M$ with
$\pi_{*}V_{1}=\pi_{*}V_{2}=0$. Then
1. (1)
$\overrightarrow{W}=-(\nabla W)^{v};$
2. (2)
$\bar{\sigma}([\overrightarrow{W},\underline{\nabla_{V_{1}^{h}}}],\underline{\nabla_{V_{2}^{h}}}]=-{\rm
Hess}\ W(V_{1}^{h},V_{2}^{h});$
3. (3)
$\overrightarrow{W}\left(g(V_{1}^{h},V_{2}^{h})\right)-g\left(([\overrightarrow{W},V_{1}])^{h},V_{2}^{h}\right)-g\left(V_{1}^{h},([\overrightarrow{W},V_{2}])^{h}\right)=0.$
###### Proof.
(1) From the second item of the last lemma it follows that
$\sigma(\overrightarrow{W},V_{1}^{h})=V_{1}^{h}(W)=dW(V_{1}^{h})=g(\nabla
W,V_{1}^{h})=-\sigma((\nabla W)^{v},V_{1}^{h}).$
Taking into account that $\pi_{*}\overrightarrow{W}=0$, we get
$\overrightarrow{W}=-(\nabla W)^{v},\quad{\rm
span}\\{\partial_{u_{i}},i=1,...s\\}.$
On the other hand, it follows from (50) and item (1) of the present lemma that
$\sigma(\vec{u}_{i},\overrightarrow{W})=-\vec{u}_{i}(W)=-X_{i}(W)=0,\
i=1,...,s.$
Thus, we get the required identity $\overrightarrow{W}=-(\nabla W)^{v}$.
(2) Both sides of the required identity are linear w.r.t. $V_{1},V_{2}$,
respectively, thus it is sufficient to prove it for the case that
$V_{1}^{h},V_{2}^{h}$ are both vector fields on $T^{*}\widetilde{M}$. But for
this case the required identity is a direct consequence of the definition of
the Hessian.
(3) Left-hand side is linear w.r.t. $V_{1},V_{2}$, respectively, thus it is
sufficient to prove it for the case that $V_{1}^{h},V_{2}^{h}$ are both vector
fields on $T^{*}\widetilde{M}$. In this case, the vector fields
$(V_{1}^{h})^{v},(V_{2}^{h})^{v}$, together with $(\nabla W)^{v}$ are all
constants on the fibers of $T^{*}M$ and then the required identity become
trivial. ∎
Given any $X\in\Pi_{\lambda}^{c}$ denote by $\widetilde{\nabla}_{X^{h}}$ the
lift of $X$ to $\widetilde{\mathfrak{J}}^{c}(\lambda)$: the unique vector
$\widetilde{\nabla}_{X^{h}}\in\widetilde{\mathfrak{J}}^{c}(\lambda)$ such that
$({\rm pr}\circ\pi)_{*}\widetilde{\nabla}_{X^{h}}=X^{h}$. Then there exist the
unique $B\in{\rm End}(\Pi^{c}_{\lambda})$ and
$\tilde{A}\in(\Pi_{\lambda}^{c})^{*}$ such that
(34) $\widetilde{\nabla}_{v^{h}}=\underline{\nabla_{v^{h}}}+Bv,\quad\forall
v\in\Pi_{\lambda}^{c},$
where $\nabla$ stands for the lifts to the Levi-Civita connection on
$T^{*}\widetilde{M}$, as before.
###### Lemma 4.
The linear operator $B$ is antisymmetric w.r.t. the canonical Euclidean
structure in $\Pi_{\lambda}^{c}$.
###### Proof.
Fix a point $\bar{\lambda}\in T^{*}M$ and consider a small neighborhood $U$ of
$\bar{\lambda}$. Let $\mathcal{E}=\\{\mathcal{E}^{i}\\}_{i=1}^{m-1}$ be a
frame of $\Pi_{\lambda}^{c}$ (i.e. $\Pi_{\lambda}^{c}={\rm
span}\,\mathcal{E}(\lambda)$) for any $\lambda\in U$ such that the following
four conditions hold
1. (1)
$\mathcal{E}$ is orthogonal w.r.t. the canonical Euclidean structure on
$\Pi_{\lambda}^{c}$;
2. (2)
Each vector field $\mathcal{E}^{i}$ is parallel w.r.t the canonical parallel
transport $\mathcal{K}_{t}$, i.e.
$\mathcal{E}^{i}(e^{t}\vec{h}\lambda)=\mathcal{K}^{t}\mathcal{E}^{i}(\lambda)$
for any $\lambda$ and $t$ such that $\lambda,e^{t\vec{h}}\lambda\in U$;
3. (3)
The vector fields $(J^{c}p^{h})^{v}$ and $\mathcal{E}^{i}$ commute on $U\cap
T_{\pi(\bar{\lambda})}^{*}M$;
4. (4)
The vector fields $\vec{u}_{i},\ \forall 1\leq i\leq s$ and $\mathcal{E}^{i}$
commute on $U\cap T_{\pi(\bar{\lambda})}^{*}M$.
Note that the frame $\mathcal{E}$ with properties above exists, because the
Hamiltonian vector field $\vec{h}$ is transversal to the fibers of $T^{*}M$
and it commutes with $\vec{u}_{i},\ \forall 1\leq i\leq s$.
From the property (2) of the parallel transport $\mathcal{K}^{t}$ in this
subsection it follows that
(35) $\widetilde{\nabla}_{(\mathcal{E}^{i})^{h}}=-{\rm
ad}\vec{h}\,\mathcal{E}^{i}$
Using the above defined identification $I^{c}:N^{c}\to T^{*}\widetilde{M}$,
one can look on the restriction of the tuple of vector fields $\mathcal{E}$ to
the submanifold $\\{u_{i}=c_{i},i=1,...,s\\}$ as on the tuple of the vertical
vector fields of $T^{*}\widetilde{M}$ (which actually span the tangent to the
intersection of the fiber of $T^{*}\widetilde{M}$ with the level to the
corresponding Riemannian Hamiltonian). Then first the tuple $\mathcal{E}$ is
the tuple of orthonormal vector fields (w.r.t. the canonical Euclidean
structure on the fibers of $T^{*}\widetilde{M}$, induced by the Riemannian
metric $g$). Further, by the equations (29) the Levi-Civita connection of $g$
is characterized by the fact that there exists a field of antisymmetric
operators $\widetilde{B}\in{\rm End}(\Pi^{c}_{\lambda})$ such that
(36)
$[\nabla_{p^{h}},\widetilde{\mathcal{E}}^{i}(\lambda)]=-\nabla_{\bigl{(}\widetilde{\mathcal{E}}^{i}(\lambda)\bigr{)}^{h}}-\widetilde{B}\widetilde{\mathcal{E}}^{i}(\lambda)$
On the other hand, from (35),(36), using the second item of Lemma 2 and the
property (3) of $\mathcal{E}^{i}$, one has
(37) $\begin{split}\widetilde{\nabla}_{(\mathcal{E}^{i})^{h}}=-{\rm
ad}\vec{h}\,\mathcal{E}^{i}=-\bigl{[}\underline{\nabla_{p^{h}}}-\sum_{i=1}^{s}c_{i}(J_{i}p^{h})^{v}+\overrightarrow{W},\mathcal{E}^{i}]=\underline{\nabla_{\bigl{(}\mathcal{E}^{i}(\lambda)\bigr{)}^{h}}}+\widetilde{B}\,\mathcal{E}^{i}(\lambda)+[\overrightarrow{W},\mathcal{E}^{i}(\lambda)].\end{split}$
From item (3) of Lemma 3 and property (1) of $\mathcal{E}^{i}$ it follows
(38)
$g(([\overrightarrow{W},\mathcal{E}^{i}])^{h},(\mathcal{E}^{j})^{h})+g((\mathcal{E}^{i})^{h},([\overrightarrow{W},\mathcal{E}^{j}])^{h})=\overrightarrow{W}\left(g((\mathcal{E}^{i})^{h},(\mathcal{E}^{j})^{h})\right)=0.$
Therefore, from (37) and (38) we conclude that $B$ is antisymmetric. ∎
###### Lemma 5.
The operator $B$ satisfies
(39) $(Bv)^{h}=-\frac{1}{2}J^{c}v^{h}\quad{\rm mod}\ p^{h},\quad\forall
v\in\Pi^{c}_{\lambda}.$
###### Proof.
Since $\widetilde{\mathfrak{J}}^{c}(\lambda)$ is an isotropic subspace, we
have
$\sigma(\widetilde{\nabla}_{v_{1}^{h}},\widetilde{\nabla}_{v_{2}^{h}})=0,\quad\forall\,v_{1},v_{2}\in\Pi^{c}_{\lambda}.$
On the other hand, using Proposition 2, the fact that the Levi-Civita
connection (as an Ehresmann connection) is a Lagrangian distribution in
$T^{*}\widetilde{M}$ and Lemma 2, we get
$\displaystyle
0=\sigma(\widetilde{\nabla}_{v_{1}^{h}},\widetilde{\nabla}^{c}_{v_{2}^{h}})$
$\displaystyle=$
$\displaystyle\Bigl{(}(I^{c}\circ\hbox{PR}^{c})^{*}\tilde{\sigma}-({\rm
pr}\circ\pi)^{*}\Omega^{c}\Bigr{)}\Bigl{(}\underline{\nabla_{v^{h}_{1}}}+Bv_{1},\underline{\nabla_{v^{h}_{2}}}+Bv_{2}\Bigr{)}$
$\displaystyle=$
$\displaystyle-\Omega^{c}(v_{1}^{h},v_{2}^{h})-g\big{(}(Bv_{1})^{h},v_{2}^{h}\big{)}+g\big{(}(Bv_{2})^{h},v_{1}^{h})$
$\displaystyle=$
$\displaystyle-g(J^{c}v_{1}^{h},v_{2}^{h})-g\big{(}(Bv_{1})^{h},v_{2}^{h}\big{)}+g\big{(}(B^{*}v_{1})^{h},v_{2}^{h}).$
where $B^{*}$ is the dual of $B$ w.r.t. the Euclidean structure in
$\Pi_{\lambda}^{c}$. Taking into account that $B$ is antisymmetric, we get
(40) $(Bv)^{h}=-\frac{1}{2}J^{c}v^{h}\quad{\rm mod}\ p^{h}.$
∎
###### Corollary 3.
The canonical complement $\widetilde{\mathfrak{J}}^{c}(\lambda)$ can be
expressed as follows:
$\widetilde{\mathfrak{J}}^{c}(\lambda)=\\{\underline{\nabla_{v^{h}}}-\frac{1}{2}(J^{c}v^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g\left(v^{h},J^{c}p^{h}+2\nabla W\right)(p^{h})^{v},\
v\in\Pi^{c}_{\lambda}\\}.$
###### Proof.
It follows from (34) and (39) that there exist $A\in(\Pi^{c}_{\lambda})^{*}$
such that
$\widetilde{\nabla}_{v^{h}}=\underline{\nabla_{v^{h}}}-\frac{1}{2}(J^{c}v^{h})^{v}+A(v)(p^{h})^{v}.$
Note that $\sigma(\vec{h},(p^{h})^{v})=g(p^{h},p^{h})=\|p^{h}\|^{2}$. Hence,
from the fact that $\widetilde{\nabla}_{v^{h}}$ is tangent to the Hamiltonian
vector field $\vec{h}$, we get easily that
$A(v)=-\frac{1}{2\|p^{h}\|^{2}}\cdot g\left(v^{h},J^{c}p^{h}+2\nabla
W\right),$
which completes the proof of the corollary. ∎
#### 3.2.2. The reduced curvature map
As a direct consequence of structure equation (27), we get the following
preliminary descriptions of the reduced curvature map:
###### Proposition 1.
Let $v\in\Pi^{c}_{\lambda}$. Let $V$ be a parallel vector field such that
$V(\lambda)=v$. Then the curvature maps satisfy the following identities:
(41) $\displaystyle
g\big{(}(\mathfrak{R}_{\lambda}^{c}v)^{h},v^{h}\big{)}=-\sigma(\hbox{ad}\vec{h}\
(\widetilde{\nabla}_{V^{h}}),\widetilde{\nabla}_{v^{h}}).$
It follows that in order to calculate the reduced curvature map it is
sufficient to know how to express the Lie bracket of vector fields on the
cotangent bundle $T^{*}M$ via the covariant derivatives of Levi-Civita
connection on $T^{*}\widetilde{M}$.
###### Proposition 2.
For any tensors $A,B$ of type $(1,1)$ on $\widetilde{M}$, the following
identity holds:
1. (1)
$[(Ap^{h})^{v},(Bp^{h})^{v}]=(B(Ap^{h}))^{v}-(A(Bp^{h}))^{v},$
2. (2)
$[\underline{\nabla_{p^{h}}},(Ap^{h})^{v}]=-\underline{\nabla_{Ap^{h}}}+((\nabla_{p^{h}}A)p^{h})^{v}.$
For simplicity, denote $\bar{\sigma}=(I^{c}\circ\rm{PR}^{c})^{*}\sigma$ and
$\Omega^{c}=\sum_{i=1}^{s}c_{i}d\omega_{i},$ as before. As in the proof of
Lemma 4, we can take a parallel vector field $V$ such that $V(\lambda)=v$ and
(42) $[(J^{c}p^{h})^{v},V](\bar{\lambda})=0,\quad\bar{\lambda}\in U\cap
T_{q}^{*}M,$
where $U$ is a neighborhood of $\lambda.$ Similar to Proposition 4 of [10], we
have
###### Lemma 6.
Let $V,V_{1},V_{2}$ be vector fields on $T^{*}M$ with
$\pi_{*}V=\pi_{*}V_{1}=\pi_{*}V_{2}=0$. Then
1. (1)
$([(J^{c}p^{h})^{v},(J^{c}V^{h})^{v}])^{h}=J^{c}([(J^{c}p^{h})^{v},(V^{h})^{v}])^{h}$,
2. (2)
$\bar{\sigma}([(J^{c}p^{h})^{v},\underline{\nabla_{V_{1}^{h}}}],\underline{\nabla_{V_{2}^{h}}})=g(\nabla
J^{c}(p^{h},V_{1}^{h}),V_{2}^{h}),$
3. (3)
$({\rm
pr}\circ\pi)_{*}([(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}])=J^{c}V^{h},$
4. (4)
$({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}])=\frac{1}{2}J^{c}V^{h}-\frac{1}{2\|p^{h}\|^{2}}g(J^{c}V^{h},p^{h})p^{h}$.
Let us simplify the right-hand side of the identity (41). First, from the last
line of the structural equations (27) it follows that
(43) $({\rm
pr}\circ\pi)_{*}(\hbox{ad}\vec{h}(\widetilde{\nabla}_{V^{h}}))\in\mathbb{R}p^{h}.$
Besides,
$\sigma\left(\vec{h},\frac{1}{2}(J^{c}v^{h})^{v}+\frac{1}{2\|p^{h}\|^{2}}g\left(v^{h},J^{c}p^{h}\right)(p^{h})^{v}\right)=\frac{1}{2}g(p^{h},J^{c}v^{h})+\frac{1}{2\|p^{h}\|^{2}}g(v^{h},J^{c}p^{h})g(p^{h},p^{h})=0.$
Hence from Lemma 2 and Corollary 3 it follows that
$\displaystyle\sigma(\hbox{ad}\vec{h}(\nabla_{V^{h}}),\nabla_{v^{h}})=\sigma\left(\hbox{ad}\vec{h}(\nabla_{V^{h}}),\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}-\frac{1}{2}(J^{c}V^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h})(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)$
$\displaystyle+$
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}-\frac{1}{2}(J^{c}V^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h})(p^{h})^{v}],-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle+$
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v},-\frac{1}{\|p^{h}\|^{2}}\cdot
g(V^{h},\nabla
W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle+$
$\displaystyle\sigma\left([\overrightarrow{W},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle+$
$\displaystyle\sigma\left([\overrightarrow{W},-\frac{1}{2}(J^{c}V^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h}+2\nabla W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)$
$\displaystyle=:$ $\displaystyle T_{1}+T_{2}+T_{3}+T_{4}+T_{5}$
We will deal with the terms $T_{i}(1\leq i\leq 5)$ in steps.
Step 1 It follows identity (33) that
(44)
$\bar{\sigma}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}})=-g(R^{\nabla}(p^{h},v^{h})p^{h},v^{h}).$
Also it follows from item (3) of Lemma 6 and item (4) of Lemma 6 that
$\displaystyle\Omega^{c}(({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}]),v^{h})+\frac{1}{2}\bar{\sigma}([\underline{\nabla_{p^{h}}},(J^{c}V^{h})^{v}],\underline{\nabla_{v^{h}}})$
$\displaystyle=$ $\displaystyle-g\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}]),J^{c}v^{h}\bigr{)}+\frac{1}{2}\bar{\sigma}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}],(J^{c}v^{h})^{v})$
$\displaystyle=$ $\displaystyle-\frac{1}{2}g\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}]),J^{c}v^{h}\bigr{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\|J^{c}v^{h}\|^{2}+\frac{1}{4\|p^{h}\|^{2}}\left(g(J^{c}v^{h},p^{h})\right)^{2}$
Also it follows from straightforward computations that
(45) $\Omega^{c}\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},(J^{c}v^{h})^{v}]),v^{h}\bigr{)}=-\Omega^{c}\bigl{(}J^{c}v^{h},v^{h}\bigr{)}=\|J^{c}v^{h}\|^{2}$
Also it follows from item (2) of Proposition 2 that
(46)
$\bar{\sigma}([\underline{\nabla_{p^{h}}},(p^{h})^{v}],\underline{\nabla_{v^{h}}})=\bar{\sigma}(-\underline{\nabla_{p^{h}}},\underline{\nabla_{v^{h}}})=0.$
and
(47) $\Omega^{c}\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},(p^{h})^{v}]),v^{h}\bigr{)}=-\Omega^{c}(p^{h},v^{h}\bigr{)}=g(p^{h},J^{c}v^{h}).$
It follows from item (2) of Lemma 6 that
(48)
$\bar{\sigma}([(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}})=g(\nabla
J^{c}(p^{h},v^{h}),v^{h}).$
Applying item (3) of Lemma 6, we get
(49) $\Omega^{c}(({\rm
pr}\circ\pi)_{*}([(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}]),v^{h})=\Omega^{c}(J^{c}v^{h},v^{h})=-\|J^{c}v^{h}\|^{2}.$
And it follows from (42) and item (1) of Lemma 6 that
(50) $[(J^{c}p^{h})^{v},(J^{c}V^{h})^{v}]=0.$
And it follows from item (1) of Proposition 2 that
(51) $[(J^{c}p^{h})^{v},(p^{h})^{v}]=0.$
Then it follows that
$\sigma\left([-(J^{c}p^{h})^{v},-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h}+2\nabla W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=0$
Summarizing all the calculations above, we have
(52) $T_{1}=-g(R^{\nabla}(p^{h},v^{h})p^{h},v^{h})-g(\nabla
J^{c}(p^{h},v^{h}),v^{h})-\frac{1}{4}\|J^{c}v^{h}\|^{2}-\frac{3}{4\|p^{h}\|^{2}}\left(g(p^{h},J^{c}v^{h})\right)^{2}$
Step 2 Again, it follows from item (4) of Lemma 6 that
(53)
$\sigma([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}],(p^{h})^{v})=-\frac{1}{2}g(J^{c}v^{h},p^{h}).$
And it follows from straightforward computations that
(54)
$\sigma\left([\underline{\nabla_{p^{h}}},\frac{1}{2}(J^{c}V^{h})^{v}+\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h})(p^{h})^{v}],(p^{h})^{v}\right)=0.$
And it follows from item (3) of Lemma 6 that
(55)
$\sigma\left([(J^{c}V^{h})^{v},\underline{\nabla_{V^{h}}}],(p^{h})^{v}\right)=g(J^{c}v^{h},p^{h})).$
Hence, we have
(56) $T_{2}=-\frac{1}{\|p^{h}\|^{2}}g(J^{c}p^{h},v^{h})g(\nabla W,v^{h}).$
Step 3 It follows from item (2) of Proposition 2 that
$[\underline{\nabla_{p^{h}}},(p^{h})^{v}]=-\underline{\nabla_{p^{h}}}$, hence
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}},-\frac{1}{\|p^{h}\|^{2}}\cdot
g(V^{h},\nabla
W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)\sigma\left([\underline{\nabla_{p^{h}}},(p^{h})^{v}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)\sigma\left(-\underline{\nabla_{p^{h}}},\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h})-\frac{1}{\|p^{h}\|^{2}}\cdot\left(g(V^{h},\nabla
W)\right)^{2}.$
And it follows from (51) that
(57) $\sigma\left([(J^{c}p^{h})^{v},-\frac{1}{\|p^{h}\|^{2}}\cdot
g(V^{h},\nabla W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=0.$
Hence,
(58) $T_{3}=-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h})-\frac{1}{\|p^{h}\|^{2}}\cdot\left(g(V^{h},\nabla
W)\right)^{2}$
Step 4 From item (2) of Lemma 3, we have
(59)
$\bar{\sigma}\left([\overrightarrow{W},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}}\right)=-{\rm
Hess}\ W(v^{h},v^{h}).$
###### Lemma 7.
The following identity holds.
$({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}])=\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)p^{h}.$
###### Proof.
First of all, from (43),item (3) of Lemma 2 and Lemma 3 it follows
$({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v}+\overrightarrow{W},\underline{\nabla_{V^{h}}}-\frac{1}{2}(J^{c}V^{h})^{v}])=0,\quad{\rm
mod}\ p^{h}.$
Then together with item (3)-(4) of Lemma 6 and item (2) of Proposition 2 we
have
$({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}])=0,\quad{\rm
mod}\ p^{h}.$
Furthermore, from the classical Cartan’s formula we have
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
d\sigma(\overrightarrow{W},\underline{\nabla_{V^{h}}},(p^{h})^{v})=\overrightarrow{W}(\sigma(\underline{\nabla_{V^{h}}},(p^{h})^{v}))-\underline{\nabla_{V^{h}}}(\sigma(\overrightarrow{W},(p^{h})^{v}))+(p^{h})^{v}(\sigma(\overrightarrow{W},\underline{\nabla_{V^{h}}}))$
$\displaystyle-$
$\displaystyle\sigma([\overrightarrow{W},\underline{\nabla_{V^{h}}}],(p^{h})^{v})+\sigma([\overrightarrow{W},(p^{h})^{v}],\underline{\nabla_{V^{h}}})-\sigma([\underline{\nabla_{V^{h}}},(p^{h})^{v}],\overrightarrow{W}).$
Since $\overrightarrow{W}$ is constant on the fiber of $T^{*}M$, one can
easily show
$(p^{h})^{v}(\sigma(\overrightarrow{W},\underline{\nabla_{V^{h}}}))-\sigma([\underline{\nabla_{V^{h}}},(p^{h})^{v}],\overrightarrow{W})=0.$
Then,
$\sigma([\overrightarrow{W},\underline{\nabla_{V^{h}}}],(p^{h})^{v})=\sigma([\overrightarrow{W},(p^{h})^{v}],\underline{\nabla_{V^{h}}})=g(\nabla
W,v^{h}).$
Hence, the required identity follows and the lemma is proved. ∎
As a direct consequence of the last lemma, we have
$\displaystyle\Omega^{c}\bigl{(}({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}]),v^{h}\bigr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\|p^{h}\|^{2}}g(J^{c}p^{h},v^{h})g(v^{h},\nabla W),$
$\displaystyle\sigma\left([\overrightarrow{W},\underline{\nabla_{V^{h}}}],(p^{h})^{v}\right)$
$\displaystyle=$ $\displaystyle g(v^{h},\nabla W).$
As a result of above calculations, we get
$T_{4}=-{\rm Hess}\
W(v^{h},v^{h})-\frac{1}{\|p^{h}\|^{2}}g(J^{c}p^{h},v^{h})g(v^{h},\nabla
W)-\frac{1}{\|p^{h}\|^{2}}(g(v^{h},\nabla W))^{2}.$
Step 5 First of all, we show the following
###### Lemma 8.
The following identity holds.
$[\overrightarrow{W},(J^{c}V^{h})^{v}]=\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(Jp^{h})^{v}.$
###### Proof.
As $\overrightarrow{W}=-(\nabla W)^{v}$ is constant on the fiber of $T^{*}M$,
we can proceed with the following calculations
(61) $([\overrightarrow{W},(J^{c}V^{h})^{v}])^{h}=({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{J^{c}V^{h}}}])=J^{c}\left(({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}])\right).$
Substituting the identity of Lemma 7 into the last identity, we get
$([\overrightarrow{W},(J^{c}V^{h})^{v}])^{h}=\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)J^{c}p^{h},$
and then the required identity follows. ∎
As a direct consequence, we have
$\sigma\left([\overrightarrow{W},(J^{c}V^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h}).$
Furthermore, since
$[\overrightarrow{W},(p^{h})^{v}]=\overrightarrow{W}=-(\nabla W)^{v}$,then
$\sigma\left([\overrightarrow{W},(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=g(\nabla
W,v^{h}).$
Therefore,
$T_{5}=-\frac{1}{\|p^{h}\|^{2}}(g(v^{h},\nabla W))^{2}.$
Combining the results of Step 1-5 and using the fact
$\|p^{h}\|^{2}=2(c_{0}+W)$, we get the expression of the reduced curvature
maps, as shown in Theorem 2.
## References
* [1] Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. Westview Press, 2nd edition, 1994.
* [2] A. Agrachev and I. Zelenko. Geometry of Jacobi curves. I. J. Dynamical and Control Systems, 8(1):93–140, 2002.
* [3] A. A. Agrachev and R. V. Gamkrelidze. Feedback-invariant optimal control theory - i. regular extremals. J. Dynamical and Control Systems, 3:343–389, 1997.
* [4] Andrei A. Agrachev and N. Chtcherbakova. Hamiltonian systems of negative curvature are hyperbolic. Russian Math. Dokl., 400:295–298, 2005.
* [5] Andrei A. Agrachev and Yuri L. Sachkov. Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences, Control Theory and Optimization(87). Springer-Verlag, 2004.
* [6] D. V. Anosov. Geodesic flows on the closed Riemannian manifold of negative curvature. Proceedings of the Steklov Institute of Mathematics, 90:3–209. AMS, Providence, RI, 1967.
* [7] D. V. Anosov and Ya. G. Sinai. Certain smooth ergodic systems. Russian Math. Surveys, 22:103–167, 1967.
* [8] N. Gouda. Magnetic flows of Anosov type. Tôhoku Math. J., 49:165–183, 1997.
* [9] G.P.Paternain and M.Paternain. Anosov geodesic flows and twisted symplectic structures. in: Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math., Longman, 362:132–145, 1996.
* [10] C. Li and I. Zelenko. Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries. arXiv: 0908.4397v1 [math. DG], SISSA preprint 53/2009/M, submitted to Journal of Geometry and Physics.
* [11] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko. The Mathematical Theory of Optimal Processes. Wiley, New York, 1962.
* [12] S.Grognet. Flots magn$\acute{e}$tiques en courbure n$\acute{e}$gative. Ergodic Theory Dynam. Systems, 19:413–436, 1999.
* [13] Maciej P. Wojtkowski. Magnetic flows and gaussian thermostats on manifolds of negative curvature. FUNDAMENTA MATHEMATICAE, 163, 2000.
* [14] I. Zelenko and C. Li. Differential geometry of curves in Lagrange Grassmannians with given Young diagram. Differential Geometry and its Applications, doi: 10.1016/j.difgeo.2009.07.002.
|
arxiv-papers
| 2010-08-21T14:14:20 |
2024-09-04T02:49:12.308571
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chengbo Li",
"submitter": "Li Chengbo",
"url": "https://arxiv.org/abs/1008.3634"
}
|
1008.3640
|
11institutetext: Yale University, Physics Department 11email:
steve.lamoreaux@yale.edu
# Progress in Experimental Measurements of the Surface-Surface Casimir Force:
Electrostatic Calibrations and Limitations to Accuracy
Steve K. Lamoreaux
###### Abstract
Several new experiments have extended studies of the Casimir force into new
and interesting regimes. This recent work will be briefly reviewed. With this
recent progress, new issues with background electrostatic effects have been
uncovered. The myriad of problems associated with both patch potentials and
electrostatic calibrations are discussed and the remaining open questions are
brought forward.
## 1 Introduction
Nowadays, it is unclear what it means to write a review article, or a review
chapter for a book, on a particular subject. This unclarity results simply
from the ease with which modern digital reference and citation resources can
be used; with a mere typing of a keyword or two into a computer hooked up to
the internet, one has an instant review any field of interest. As such, at the
present time, review articles tend to be op-ed pieces that tend to be less
than scientifically enlightening. Rather than continue in the tradition of
collecting up a series of electronic database searches, I will give an
overview of some recent experiments and also describe how anomalous
electrostatic effects might have affected the results of these experiments.
This Chapter is not meant to be a review of every paper in the Casimir force
experimental measurement field, but a review of what I consider are the
credible experiments, that have carried the field forward, that were performed
over the last decade or so. As such, there will be little mention of
experimental studies that have claimed 1% or better agreement, simply because
it is unclear to me what these experiments really mean. If the reader is
interested, a recent review of this 1% level work is presented in mohideenrev
. Of course I admit freely that my review presented here reflects my own
opinions, however I hope the reader accepts or rejects my points based on
verifiable facts and an independent scientific analysis. It must be remembered
that simply because a paper appears in print, in a credible and leading
journal, it is not necessarily scientifically correct or accepted by the
community at large. Neither does the fact that work is funded by the DOE, NSF,
or DARPA (or other funding agencies beyond the realm of the U.S.A.) guarantee
its validity or broad acceptance in the scientific community. An perhaps most
interestingly as a remark on the general history of science, the “consensus
opinion” is not necessarily correct either. In particular, in the surface-
surface Casimir force measurement field, there have been more than a few
“Comments” on various papers; the interested reader would do well to ignore
most, but not all, of these “Comments” as they are confusing, if not bogus,
but certainly inflammatory.
Watching the field develop since my 1997 experimental result skl97prl , which
served as a watershed for new interest in surface-surface Casimir force
measurements, has been fascinating. I had no preconceived notions as to how
large or small the effect should be relative to the case of assumed simple
perfect conductors (e.g., ignoring effects like surface plasmons), but I had
no illusions as to the accuracy of my work, hence the words “Demonstration of
the Casimir force” in the title of my paper. I simply did not have the time or
resources to perform a study of possible systematic effects that likely
limited the accuracy of my result; the precision was at the 5% level, at the
point of closest approach. Again the accuracy of my result was, and remains,
an open question, as it does for any experiment.
At the time the work reported in skl97prl was performed, there were no
precision calculations of the Casimir force for real materials. Describing the
metal plates with the simplest plasma model, for parallel plates, the
correction to the force compared to the perfect conducting case is schwinger
$\eta(d)=1-{16\over 3}{c\over\omega_{p}d},$ (1)
where $\eta(d)$ is a force correction factor which varies with plate
separation $d$, $c$ is the velocity of light, and $\omega_{p}$ is the plasma
frequency, where the form of the permittivity of the metal is
$\epsilon(\omega)=1-{\omega_{p}^{2}\over\omega^{2}},$ (2)
which is valid at high frequency. As $\omega$ approaches zero, Eq. (2) become
invalid, and in addition the effect of static conductivity must be included
also. Equation (1) can be easily modified for a sphere-plane geometry skl97prl
. However, the magnitude of this correction was certainly outside what was
reasonable based on the precision of my experiment, which appeared to be best
described by plates with perfect conductivity. There was some skepticism
regarding the lack of a finite conductivity correction in my result, and
although several theorists expressed interest in performing a more accurate
calculation, none did. Eventually I attempted the calculations myself, with
mixed results. My calculations were based on published optical properties of
Au and Cu, with the Cu calculations intended as a test case. These
calculations showed roughly 10-15% (for Cu) and 20-30% (for Au) reductions in
force, compared to perfect conductors, for distances of order one micron; I
eventually found an error in the radius of curvature of the spherical-surface
plate used in my experiment sklcalc ; sklerratum that lowered the
experimentally measured force by 10%, but did not bring the experimental
result into agreement with my Au calculation. Later work showed that Au and Cu
are nearly identical, with my Cu result being the more accurate; the
discrepancy was due to the way I interpolated between data points in the
tabulated optical data lambrecht1 . With the refined calculation, my
experiment and theory appeared to be in agreement, however by this time I was
skeptical of my results, as stated in the ensuing discussion, in lambrecht2 .
Interestingly enough, I had spent considerable effort trying to find
corrections that would bring my experimental result into agreement with my
original inaccurate calculation, so I felt that I was prepared to comment
against a new theoretical result, obtained by Boström and Sernelius bossern ,
that leads to a major correction to the Casimir force between real, non-
superconducting materials. This correction reduces the force by a full factor
of two at large separations. More will be said of this correction later in
this review; in particular, in light of new electrostatic systematic effects
that have recently been discovered, the rhetoric against the result of Boström
and Sernelius no longer appears as certain. In addition, all of the 1% work
that was reported before bossern does not show the predicted correction, nor
does subsequent 1% level work. So we are faced with the possibility that the
degree of precision isn’t as high as stated in the 1% work, or that the theory
is not at all understood. Instead of questioning experimental accuracy, new
fantastic theoretical suggestions have been made, regarding the low frequency
permittivity of metals, that eliminate the new correction. This remains a
major open topic in the field.
There is a tendency among workers in this field to confuse precision with
accuracy, of which I am guilty myself. Precision relates to the number of
significant figures a measurement device or system provides; lots of digits
can be useful for detecting small changes in some “large” parameter, assuming
that the system is stable. Accuracy is the assignment of meaning to precision,
it is the connection between accepted definitions of, for example, lengths,
voltages, and forces, and the measurements that come out of an experimental
apparatus. As an example, for Casimir force measurements using the sphere-
plane geometry, an essential parameter is the radius of curvature of the
sphere. A radius of curvature accuracy of 0.5% for a sphere of 0.2 mm diameter
corresponds to 1000 nm, a bit larger than the wavelength of visible light.
Thus optical measurements of adequate accuracy appear as hopeless; can
electron microscopy attain this level of precision? The answer is not obvious.
Of course, an experiment can be designed that does not require a high accuracy
radius of curvature measurement, e.g., when the ratio of Casimir to
electrostatic force is measured. Nonetheless, the attention to this problem in
those works reporting 1% or better accuracy does not appear as sufficient to
warrant such accuracy claims. The precision might be that level, but the cross
checks required for accurate work are missing.
In general, to attain a given experimental accuracy, say 1%, requires that the
calibrations and force measurements must be done to much better than 1%
accuracy, particularly for comparisons between theory and experiment with no
adjustable parameters. As there are possibly five or more absolute
measurements that must be made to interpret an experiment, a reasonable
requirement for the average calibration accuracy is 0.5%, assuming that the
uncertainties can be added in quadrature (this point is open to debate; many
precision measurement experts insist that the uncertainties be simply added,
which bring the required average accuracy to the 0.2% level). Some of the
required calibrations are as follows: The optical properties of the surfaces
must be adequately characterized to allow calculation of the force to 0.5%
accuracy; The radius of curvature of the spherical surface (for a sphere-plane
experiment) needs to be measured to 0.5% accuracy; The absolute separation
must be determined to high accuracy. This last point is perhaps the most
difficult, as
$\left|{\delta F\over F}\right|=\left|n{\delta d\over d}\right|,$ (3)
where $n$ is the exponent in the power law. For a sphere-plane geometry where
$n\approx-3$ we see immediately that if we want 0.5% force accuracy as limited
by the distance measurement, at the point of closest approach, say 100 nm,
then the fractional error must be $0.5\%/3$ or about 0.17%, and when $d=100$
nm this corresponds to $\delta d=0.17$ nm $=1.7$ Å. This is at the level
where, in the atomic force microscopy (AFM) community, the definition of the
surface location is agreed as controversial. So we see immediately that it
pointless to include any discussion of experiments that claim 1% accuracy as
the radius measurement is not discussed in sufficient detail in any of the
papers making such claims. My statements here should be considered as a call
for details.
The general experimental techniques used in all Casimir experiments to date
are rather straightforward. Many experiments employ AFM or micromechanical
techniques drawn from fields that enjoy tremendous engineering support. The
trick of Casimir force measurements lies in the attainment of very high force
measurement sensitivity subjected to precise and rigorous calibrations, and in
the elimination of long-range background electrostatic effects that can mask
or distort the now-well-studied AFM signals extrapolated to very large
distances. At large distances, the attractive force between two surfaces,
“the” Casimir force, becomes a property of the bulk material(s) that the
plates comprise, and is viewed as a fundamental physical effect arising from
the quantum vacuum, as opposed to AFM signals used to detect surface
roughness, for example. Experimental rigor is required to transform precision
into accuracy on the fundamental vacuum effect.
Because the measurement techniques are largely borrowed from other fields, I
will not give a nuts and bolts discussion of measurements in this review, for
the simple reason that I know nothing about AFM techniques. Nowadays one can
simply buy an AFM system from Veeco, for example, and adapt it to the samples
and longer distance ranges required for Casimir measurement. There are
companies that commercially produce bare cantilevers, and most engineering
schools have fabrication facilities where NEMs and MEMs systems can be
produced with just about any desired properties in configurations limited only
by the imagination. Alternatively, my own work employs torsion balances, and
the interested reader can refer to Cavendish’s experiment for most details of
such systems. An analysis of the force sensitivity of a torsion pendulum can
be found in buttlam .
The principle advantage to AFM type or torsion pendulum type measurements (in
fact there is no fundamental difference between them, it’s a matter of scale)
is elimination of stiction associated with the fulcrum type balances used in
practically all earlier experiments. The proliferation of high accuracy
mechanical and optomechanical translation stages, together with high quality
digital data acquisition systems has made precision Casimir force measurement
possible; the questions of accuracy are now the central theme, not the simple
detection of the force.
This is not to say that the experiments are easy or simple; again, the art of
the experiments lies in the attainment of high force measurement sensitivity,
reliable calibrations, the production of well-characterized optical surfaces,
and the elimination of background effects due to, for example, electrostatic
effects. The electrostatic effects are common to all experiments, either in
regard to system calibrations or systematic background effect, or both. Given
the importance of electrostatic effects, I will discuss them at length in this
review.
It is often said that the Casimir force is simply the retarded van der Waals
potential. This view strikes me as fundamentally flawed, as the Casimir force
does not depend on the properties of the individual atoms of the plates, but
on their bulk properties. Indeed, the non-additivity of the van der Waals
effect has been discussed at length in the literature (see milonni for a
discussion and references). It is more profitable to think of the Casimir
force as the zero point electromagnetic field stress on a parallel plate
waveguide. This force is apparently largest when the waveguide is constructed
from perfectly conducting material(s). The effects of imperfect conductivity
can be calculated provided the optical constants of the material(s) are known
over an adequate wavelength range. Furthermore, most of the surface-surface
Casimir effect is due to conduction electrons. It is meaningless to assign a
retarded van der Waals force between the individual electrons in a conductor.
Likewise, if the Casimir force was simply the retarded van der Waals force, it
would make little sense consider modifying the Casimir force, in a fundamental
way, by altering the mode structure imposed by specially tailored boundaries.
## 2 Motivation for the Experimental Study of the Casimir Force: Some Recent
Results
The Casimir force is of fundamental interest in that it is taken as evidence
for the existence of the fluctuations associated with the quantum vacuum cas .
One can almost as easily derive the Casimir force by treating the
electromagnetic field classically, with the field fluctuation due to
dissipation in the material bodies; this is the Lifshitz approach lifshitz . A
principal controversy associated with the quantum vacuum interpretation lies
in the fact that the zero point electromagnetic field energy, when integrated
to the Planck scale (which is the natural cutoff), leads to a cosmological
energy density some 130 orders of magnitude larger than observed. This is an
open problem in modern physics.
There are three principal motivations for studying the Casimir force. One
question is how well do we understand the basic underlying physics? This
relates to second motivation which lies in the testing for the existence of
short range corrections to gravity, or a new force associated with axion
exchange, for example. For such tests, the Casimir force represents a
systematic background effect that must be characterized or physically
eliminated by employing a shield. The third motivation comes from interest in
modifying the Casimir force to eliminate stiction, for example, or make it
useful in nanodevices. These categories are not mutually exclusive, and of
course overlap considerably as the questions all have a fundamental element.
### 2.1 Progress in Understanding the Fundamental Casimir Force
In 2000, Bostöm and Sernelius bossern put forward the first fundamentally new
idea relating to the surface-surface Casimir effect in over 40 years, since
Lifshitz’s paper lifshitz , which lies in the treatment of material
permittivities in the zero-frequency limit. The problem of finite conductivity
was addressed earlier by Hargreaves and later by Schwinger et al. schwinger
who proposed a possible means to deal with it, that is, to let the surface
material permittivity diverge before setting the frequency to zero. The point
is that in calculating the Casimir force at finite temperature, the integral
includes a Boltzmann’s factor which accounts for the thermal population of the
electromagnetic modes,
$N(\omega)+{1\over 2}={1\over e^{\hbar\omega/k_{b}T}-1}+{1\over 2}={1\over
2}\coth{\hbar\omega\over 2k_{b}T},$ (4)
where $\hbar\omega$ is the energy of a photon, $k_{b}$ is Boltzmann’s
constant, and $T$ is the absolute temperature. Because $\coth x$ has simple
poles at $x=\pm in\pi$, the integral over frequency in calculating the Casimir
force can be replaced by a sum of the residues at the poles of Eq. (4), or
Matsubara frequencies,
$\omega_{n}={n\pi k_{b}T\over\hbar}.$ (5)
Analytic continuation of the permittivity function allows the transformation
of the integral from over real frequencies to a contour integral on the
complex frequency plane, and it is valid to replace the integral over
frequency with a sum over the poles. The upshot is that the transverse
electric ($TE$) mode with $n=0$ does not contribute to the force at all if the
permittivity diverges slower than $\omega^{-2}$ in the limit as $\omega$ goes
to zero. It is generally assumed that for metals with a finite conductivity,
at zero frequency the permittivity goes as
$\epsilon(\omega)={4\pi i\sigma\over c\omega},$ (6)
in which case the $TE$ $n=0$ mode does not contribute at all to the force.
This is important because at room temperature, at distances greater than about
10 microns, this mode accounts for roughly half of the force. The implied
correction at separations of 1 micron is about 30%. This appears to be at odds
with a number of experiments, including my own. In particular, I had spent
much effort in finding a correction to my experiment that would bring the
results into agreement with my own incorrect calculation for Au. Thus I was
well-equipped to reject this result outright, as did a number of others.
One possible solution is that the permittivity diverges as $\omega^{-2}$ as
the frequency goes to zero. This has led to the proposal of a generalized
plasma model mosplas ,
$\epsilon_{gp}(i\xi)=\epsilon(i\xi)+{\omega_{p}^{2}\over\xi^{2}},$ (7)
where $i\xi$ represents the frequency along the imaginary axis, $\epsilon$ is
the usual Drude model permittivity, for example, and $\omega_{p}$ is the so-
called plasma frequency due to free electrons. Normally this expansion is
assumed to be valid at very high frequencies, much above the resonances in the
system of atoms and charges that comprise the plates. However assuming the
permittivity of this form brings back the contribution of the $TE$ $n=0$ mode,
and apparently improves the agreement between theory and experiment.
There are consequences in a broader complex of phenomena when this generalized
plasma model is introduced. In particular, if we consider the interaction of a
low-frequency magnetic field with a material surface, by use of Maxwell’s
equation, it is straightforward to show that jackson
$-\nabla^{2}\vec{H}={\omega^{2}\over c}\epsilon(\omega)\vec{H},$ (8)
which represents so-called eddy current effects, and can be easily extended to
the complex frequency plane. We see immediately that if $\epsilon$ diverges as
$\omega^{-2}$ that at zero frequency,
$-\nabla^{2}\vec{H}\propto\vec{H},$ (9)
which predicts that a static magnetic field will interact with an ordinary
conductor in a manner different from universal diamagnetism. Such an extra
effect is not experimentally observed, as Eq. (8) together with Eq. (6) is
known to describe the non-diamagnetic interaction of low frequency fields with
conductors. So we are faced with discarding over a century of electrical
engineering knowledge in order to explain a few 1% level Casimir force
experiments of questionable accuracy, and my own. This is not acceptable.
The crux of the problem lies in the fact that at equilibrium, all electric
fields at a surface of a conductor must terminate normal to the surface landl
. An electric field parallel to a surface implies a flowing current; such
currents can exist in a transitory fashion as associated with a fluctuation as
required for generating the Casimir force, but such fluctuations cannot occur
with zero frequency. For the $TE$ modes, the electric field is parallel to the
surface, so at zero frequency $TE$ modes simply cannot be supported, assuming
that equilibrium and zero frequency are equivalent. We will return to this
problem later in this review in relation to electrostatic calibrations.
This issue is, however, not yet settled as new precise experiments are
required. It is interesting that this effect becomes less pronounced at
smaller separations, simply because the $n=0$ modes contribute a relatively
smaller fraction to the total force. For my own experiment skl97prl the
possibility of a systematic error is becoming more and more apparent. It
should be emphasized, however, that AFM type experiments probe an order of
magnitude smaller distance scale that the torsion pendulum experiments, and
the relative contributions of various effects are rapidly varying.
Work with AFMs and MEM type systems have demonstrated the difficulty of
producing metal and other films, together with their characterization, that
allows a comparison between experiment and theory at a level of better than
10%. For example, Svetovoy et al. optprops show that the prediction of the
Casimir force between metals with a precision better than 10% must be based on
the material optical response measured from visible to mid-infrared range,
that the tabulated data is generally not good enough for precision work better
than 10% accuracy. The issues of roughness are well-discussed in mohideenrev ,
however, additional new work by de Zwol et al. dezwol amplifies the problems
of surface roughness particularly in determining the absolute separation. It
appears that the best prospect for determining the correct form of the
permittivity function at zero frequency is to do a measurement at very large
separations. Indeed, problems of surface roughness correction virtually
disappear for typical optical finishes at distances about 500 nm. Above 2-3
microns, the difference between the force with and without the $TE$ $n=0$ mode
approaches a factor of two. Recent experimental work on Au films at Yale show
that the Boström-Sernelius analysis is likely correct, but this work is at a
very preliminary stage.
### 2.2 The Detection of New Long Range Forces
In the mid-1980’s, the question of the possible existence of a new so-called
fifth force was suggested based on data from Eötvos-type experiments fishbach
. Presently, interest in such forces is greater than ever due to possible
modification of gravity as allowed by String Theory, and due to the
observation of dark energy in the Universe which might be due to particles
associated with new long range forces that could manifest themselves on many
different length scales mostepdark . The basic idea is that our four
dimensional Universe is embedded in a space of more than 10 dimensions.
Leakage of lines of force between the larger space and our four dimensional
world could lead to a modification of the inverse square law, for example.
Although there is no specific prediction from a String theory, the possibility
does exist in its context.
With the publication of my 1997 experimental result, I received many
suggestions to analyze my experiment in light of an additional force that
would appear along with the Casimir force, however I rejected these
suggestions because my experiment was intended as a demonstration and any
limit would be at the level of 100% of the Casimir force. Taken as a fraction
of the gravitational field, my result was not particularly spectacular.
Nonetheless, other analyzed my experiment. Among the first to do so, in the
context of a general review of limits on sub-centimeter forces, was Long et
al. longprice and earlier, with a more detailed analysis, was Klimchitskaya
et al. klimfifth .
The most ambitious recent work on this subject is by Decca et al. deccafifth
who achieved an astounding accuracy without observing any anomalous effects.
Use of the proximity force theorem, to be discussed later in this review, to
calculate the limits on a possible new force has been criticized. The issue is
that the proximity force theorem really only applies to a force that depends
on the location of the body surfaces; the approximation is not valid for the
volume integral required for calculating the anomalous force. The
applicability is addressed by Dalvit and Onofrio dalono where corrections to
the calculation in deccacalc are pointed out.
Earlier work by Decca et al. deccaiso appears as more reliable at
constraining new forces. The technique developed here, a so-called
isoelectronic method, relied on the properties of an Au film being independent
of the substrate. For different materials coated with Au films of identical
optical characteristics and of sufficient thickness, the Casimir force should
be the same. In this work Au/Au and Au/Ge composites are compared, and the
result is “Casimir-less.” Techniques such as this appear as the most likely
way to achieve the best sensitivity to new forces, however, unfortunately the
minimum separation is limited by the Au film thickness, hence the later work
deccafifth . It should be noted that use of a screening film to eliminate
electrostatic forces and other background effect have been used in other
“fifth force” experiments for separations at the mm scale, but clearly the
trick can be scaled down to distances limited only by the skill of the
experimenter luther .
### 2.3 Modification of the Casimir Force
The possibility of modification of the Casimir force is a topic of current
great interest. With the rising of nanotechnology, the need to control,
modify, or make good use of the Casimir force is imperative as it is among the
dominant forces affecting MEMs and NEMs. At very short distances, at the
atomic scale, the large-scale geometrical aspects of the surfaces become
irrelevant, and the force becomes dominated by the van der Waals force between
atoms comprising the plates; the atom-atom force along with roughness leads to
stiction and friction. At such short distances, the treatment of the plates in
a continuum fashion fails. Any possibility to control either the short range
or long range force can have enormous technological benefits. These issues
have generated renewed interest in measuring the Casimir force with improved
precision, in applying it to nano-mechanical devices, and in controlling it.
In many instances, the attractive nature of the force leads to more problems
that to solutions because, for example, it leads to irreversible sticking of
the components in a nano-device. There have been proposals to develop
“metamaterials” which provide a boundary condition that makes the force
repulsive, but the extremely large frequency range of electromagnetic field
modes that contribute to the force suggests that this is not possible
dalvitmilonni .
The internal sticking problem of MEMs, however, might be slightly overstated.
Recent commentary relating to this possible problem has been based on the work
of Buks and Roukes bandr where irreversible stiction was observed in MEMs
devices. In this work, the mechanical motion was monitored by use of an
electron beam which caused the components of the MEMs to become highly
charged. Whether the irreversibility is really due to the Casimir force, or if
it is due to charge surface interactions, remains an open question.
Nonetheless, it is agreed that a full understanding of the Casimir force, and
its possible control, are central to the future of MEMs and NEMs engineering.
The prospects of engineering a coating that can significantly modify the
Casimir force appear as dismal. This is because the Casimir force is a “broad-
band” phenomenon. Use of magnetic films has been suggested, but unfortunately
ferromagnetic response does not extended into the near-infrared and visible
spectrum that would be required to modify the Casimir force.
Recently, it has been demonstrated experimentally that a conductive oxide
film, Indium-Tin Oxide (ITO) produces a Casimir force about half of that due
to metals ito . ITO has a number of interesting features, including
transparency over the optical spectrum and chemical inertness. Thus it appears
as an interesting material from a nanoengineering viewpoint.
Casimir himself attempted to apply his namesake force to the electron,
specifically to calculate the fine structure constant. Casimir modelled the
electron as a conducting ball of uniform charge that would contract due to the
zero point energy of the external electromagnetic modes. This force would be
balanced by the space charge repulsion of the uniform charge density, when the
conducting sphere of constant total charge was just the right diameter. The
fine structure constant $\alpha\approx 1/137$ constant, which relates to the
electron diameter, could then be determined from fundamental parameters along
with a calculation of how the electromagnetic mode zero point energy changes
as the sphere contracts milonni . However, Boyer subsequently found that the
exterior spherical modes cause the sphere to expand boyer . Boyer’s result was
interesting enough that it led to the exploration of the effects of geometry
on the Casimir force.
The change in boundary conditions that had been considered cannot be realized
experimentally; for example, if one cuts a conducting sphere in half and tries
to measure the force between the hemispheres, the force is different from the
stress outside the continuous conducting sphere– simply because the two halves
are now separated by a vacuum gap and there will be an attraction there, and
because the structure of the surface modes is altered by the gap. Nonetheless,
several experiments aimed at directly modifying the Casimir force have been
performed in the last decade or so, and are continuing.
### 2.4 Hydrogen Switchable Mirror
An experiment with a surprising result employed a hydrogen switchable mirror,
and a change in the Casimir force was sought when the mirror was switched
between its low reflectivity and high reflectivity states capasso . The
surprise was that no significant change in the Casimir force was observed with
the switching, despite the rather dramatic change in the mirror from nearly
transparent to highly reflecting.
The explanation of the null result likely lies in the construction of the
mirror which has a very thin (5 nm) palladium layer to protect the underlying
sensitive structure. This layer tends to dominate the Casimir effect, even
though the layer is about one-half of a skin depth for the frequencies that
are affect by the hydrogen switching. Other complications include the narrow
spectral width of the mirror state which reduces the effect further, and the
layered structure of the mirror–it is possible that the principal activity
occurs in the deeper layers. In spite of these problems, hope remains that an
effect on the Casimir force will be detectable demanh .
### 2.5 Geometrical Boundary Effects
Until now, no significant or non-trivial corrections to the Casimir force due
to boundary modifications have been observed experimentally. As mentioned
above, for the systems that had previously been considered such as the
conducting sphere, it is not clear that an experimental measurement of the
external stress is even possible. Cutting a sphere in half clearly changes the
boundary value problem; it is unlikely that the two halves of such a sliced
sphere will be repelled with a force that is given by the external stress on
the sphere.
However, there are other possible ways to generate a geometrical influence on
the Casimir force. A conceptually straightforward way is to contour the
surfaces of the plates at a length scale comparable to the mode wavelengths
that contribute most to the net Casimir force. For a plate separation $d$, the
wavelengths that contribute most are $\approx\pi d$. This means that a surface
nano-patterned at 400 nm length scale should show a significant geometrical
effects for separations below 1 $\mu$m. Using such a system, Chan et al. have
produced a convincing measurement of a non-trivial geometrical influence on
the Casimir force chan .
[scale=.65]chanexpcopy.eps
Figure 1: An approximately scaled schematic representation of the experiment
of Chan et. al. The trench arrays, of varying width and depth, were made from
the same doped p-type Si substrate. (Public Domain, by S.K. Lamoreaux)
These measurements, between a nanostructured silcon surface and a Au coated
sphere, were made using a micromechanical torsional oscillator. The change in
resonant frequency of the oscillator, as a function of separation between the
Au sphere and the surface, provided a measure of the gradient of the Casimir
force. The sphere, of radius 50 $\mu$m coated with 400 nm of gold, was
attached to one side of the oscillator that comprised a 3.5 $\mu$m thick, 500
$\mu$m square silicon plate suspended by two tiny torsion rods. The sphere and
oscillator was moved toward the nanostructured surface by use of a
piezoelectric actuator.
Two different nanostructured plates, compared with a smooth plate, were
measured in this work. The geometry of the nanostructures, rectangular
trenches etched in the surface of highly p-doped silicon, were chosen because
the effects are expected to be large in such a geometry. Emig and Büscher had
previously calculated the effective modification of the Casimir force due to
such a geometry, but for the case of perfect conductors emig . Even though the
calculations were not for real materials, these theoretical results appeared
as a reasonable starting point for a comparison with an experiment.
Although much progress has recently been made toward a realistic and
believable accuracy and precision with which the Casimir force can be
calculated for real materials optprops , the problems associated with the
well-known experimental variability of sputtered or evaporated films were
avoided in the work of Chan et al. by comparing two different nanostructured
plates with a smooth plate, all made from the same silicon substrate, and all
using the same Au coated sphere. The trick is comparable to the Isoelectronic
method described in Sec. 2.2. So even though ab initio calculations of the
Casimir force for real material using tabulated optical properties cannot be
accurate to better than 10%, this problem was simply circumvented by the
comparison technique.
The geometric modification of the Casimir force was detected by measuring a
deviation from that expected by use of the Proximity Force Approximation
(PFA), or the Pairwise Additive Approximation (PAA), both of which will be
described later in this review. The success of the PFA is so good that it
suggests a means of detecting a geometrical effect. Basically, the surface is
divided into infinitesimal units, and it is assumed that the total Casimir
force can be determined by adding the Casmir force, appropriately scaled by
area, between surface unit pairs in opposite surfaces; this is the PAA. Thus,
for the nanostructured surfaces, a 50% reduction in force would be expected by
the PAA, because the very deep trenches (depth $t=2a\approx 1\mu$m), etched as
a regular array, were designed to remove half of the surface. As mentioned,
two different trench spacings $\lambda$ were fabricated and measured, such
that $\lambda/a=1.87$ (sample A) and $0.82$ (sample B), and compared to a
smooth surface. The Casimir force between the gold sphere and the smooth
plate, as calculated from the tabulated properties of gold and silicon, taking
into account the conductivity due to the doping, agree with the experimental
results to about 10% accuracy. For sample A, the force is 10% larger than
expected by the PAA, using the measured smooth surface force, and for sample
B, it is 20% larger, in the range $150<z<250$ nm. The deviation increases as
$\lambda/a$ decreases, as expected.
The theory of Emig and Büscher predicts deviations from the PAA twice as large
as were observed. Nonetheless, the results of Chan et al. indicate a clear
effect of geometry on the Casimir force. However, much theoretical work
remains to be done toward gaining a complete understanding of the experimental
observations. The already difficult calculations are made more so by the
finite conductivity effects of the plates, and the sharp features of the
trenches as opposed to the smooth simple sinusoidal corrugations. New
calculational techniques have been developed that will allow reasonable
accuracy calculations. Also a number of possible systematics associated with
electrostatic effects were not fully investigated.
### 2.6 Repulsive Casimir Effect
The generalized Liftshitz formulation of the Casimir force allows for a
material between the plates. The force is thus altered from the case of a
vacuum between the plates, and the effect can be calculated. It is easy to
envision filling the space between the sphere and plate of a Casimir setup
with a liquid and measuring the effects of replacing the vacuum. A first
experiment using alcohol between the plates was done by Munday et al. alcohol
where a substantial reduction in the force was observed compared to what is
expected with vacuum between the plates. The effects of Debye screening and
other electrostatic effects were also thoroughly studied screening .
Munday et al. extended their studies to a very interesting situation where the
Casimir force becomes repulsive, by suitably choosing the permittivities of
the plates and liquids. If the plates’ material dielectric permittivities are
$\epsilon_{1}$ and $\epsilon_{2}$, and the liquid between has $\epsilon_{3}$,
the force will be repulsive when $\epsilon_{1}>\epsilon_{3}>\epsilon_{2}$. Of
course, the permittivities are frequency dependent, so this relationship must
hold over a sufficiently broad range of frequencies.
Perhaps a more familiar problem is the wetting of a material surface by a
liquid. In this case, one plate is replaced by air or vacuum so
$\epsilon_{2}=1$, and if the liquid permittivity is less than that of the
remaining plate, the liquid spreads out in a thin film rather than forming
droplets. For example, liquid helium, which has a very small permittivity,
readily forms a thin film because it is “repelled” by the vacuum
$(\epsilon_{1}>\epsilon_{3}>\epsilon_{2}=1)$, and we say that the liquid wets
the surface. On the other hand, liquid mercury which has a high effective
permittivity does not wet glass $(\epsilon_{1}<\epsilon_{3}>\epsilon_{2}=1)$.
Although there are many liquids that wet glass or fused silica, there are only
a few sets of materials that will satisfy the requirement for a repulsive
force between material plates. The set employed by Munday et al. was fused
silica and gold, with bromobenzene as the liquid. The experimental setup was
based on an atomic force microscope (AFM) that was modified slightly for the
detection of average surface forces rather than atomic-scale point forces. For
measuring the Casimir force, the sharp tip was replaced by a gold coated
microsphere (diameter = 39.8 microns) which serves as the gold plate. Using a
spherical surface for one plate simplifies the system geometry, which is
completely defined by the sphere radius and distance of closest approach from
the flat fused silica plate.
A problem that all Casimir force experiments face is the system force
calibration. For this work and related work, a most clever calibration
technique was devised. Because the fluid produces a hydrodynamic force when
the sphere/plate separation is changed, and this force is linear with
velocity, subtracting the force when the separation is changed at two
different speeds produces the hydrodynamic force without any contribution from
the Casimir force. The hydrodynamic force thus measured, which can be
calculated to high accuracy, provided the calibration. In addition, this
force, scaled to the appropriate velocity, was then subtracted from the force
vs. distance measurement, yielding a clean measurement of the Casimir force.
The measurements spanned a range of 20 nm to several hundred nm, with the
minimum distance limited by surface roughness, and the maximum distance
limited by system sensitivity. Various spurious effects were accounted for and
shown to have no significant contribution within the statistical accuracy of
the measurement.
Showing that it is indeed possible to produce and measure a repulsive Casimir
force is important to both fundamental physics and to nanodevice engineering.
There has been much discussion of such forces as they will provide a means of
quantum levitation of one material above another. Even in a fluid, it will be
possible to suppress mechanical stiction and make ultra-low friction sensors
and devices. It might be possible to “tune” the liquid (e.g., by use of a
mixture) so that at sufficiently large distances, the force becomes
attractive, while being repulsive at short distances. This would allow objects
to levitate above a liquid covered surface, for example.
## 3 Approximations, Electrostatic Calibrations, and Background Effects
Wittingly or unwittingly, many approximations have been included in all
Casimir force experiments to date. For example, most experiments employ the
use an electrostatic force from accurately measured applied voltage for
calibrations and the detection of spurious contact potentials between the
plates. The force is assumed to follow the form
$F(d)={1\over 2}{\partial C(d)\over\partial d}V^{2},$ (10)
where $C(d)$ is the capacitance between the Casimir plates, as a function of
distance $d$ between them. An exact calculation exists between a sphere and a
plane, however, for most situations the so-called Proximity Force
Approximation (PFA) can be used. In the case of a plate with spherical surface
with curvature $R$, with a distance $d$ at the point of closest approach to a
plane surface, the force between the two plates is
$F(d)=2\pi R{\cal E}(d),$ (11)
where ${\cal E}(d)$ is the energy per unit area between plane parallel
surfaces that leads to the attractive force.
Briefly, the PFA was introduced by Deryagiun Deryagiun to describe the
Casimir force between a curved surfaces, and this approximation is known to be
extremely accurate when the curvature is much less than the separation between
the surfaces. The PFA can be used beyond the Casimir force and has quite
general applicability PFA . The PFA is a special case of the Pairwise Additive
Approximation (PAA) where the plate surfaces are divided into infinitesimal
area elements, and the force is determined through a pairwise addition of
corresponding elements. The PFA and PAA work very well for electrostatic
effects because, for a conductor (even poor) in equilibrium, the electric
lines of force must be normal to the surface, otherwise currents would flow in
contradiction to the assumption that the system is in equilibrium.
The use of a sphere and a flat plate vastly simplifies an experiment because
the system is fully mechanically defined in terms of the point of closest
approach and the radius of curvature of the sphere. For two flat plates the
system is specified by two tilt angles, the areas, long-scale smoothness, and
a separation, which all need to be defined, measured, and controlled. It is
interesting to note that if the force is measured as a function of applied
voltage in the sphere-plane configuration that the result should be
$F(d)={\pi\epsilon_{0}R\over d}V^{2}=\alpha V^{2},$ (12)
where $\epsilon_{0}$ is the permittivity of free space, and $R$ is the radius
of curvature of the spherical surface. The absolute distance between the
sphere and the plane surface is proportional to $\alpha^{-1}$ and this
provides a means of determining the distance.
Even when the full form of the sphere-plane capacitance is used in Eq. (10),
approximations still exist. Specifically, there are additional terms to the
force given by Eq. (10) because the capacitance is in fact a tensor. This can
be easily seen, as when a charged sphere is bisected, the two halves repel
each other, with a force
$F={q^{2}\over 8R},$
where $q$ is the charge on the sphere landl (Prob. 2, Sec. 5). Note that this
is the force for a fixed charge, which must be modified for a fixed voltage.
The point is that the two halves experience a force, even though their
potential difference is zero; there are apparently additional terms that need
to be added to Eq. (10). As the geometry is not critical in this argument, we
can conclude that if the two plates of a Casimir experiment are at the same
non-zero potential, there will be an additional force repulsive force between
them. This sort of effect has not been considered at all.
The other problem that has received significant attention only recently is the
effect of patch potentials on a conducting surface. The effect is well-known,
and is largest with clean samples because when dirt is present, ions tend to
accumulate at the boundaries between the patches, shielding the effect landl
(Sec. 23).
To date, every Casimir effect that has bothered measuring the contact
potential as a function of distance has shown an apparent distance dependence
of that potential. Various experiments are nicely reviewed in staticsurvey .
The basic essential problem manifests itself in anomalous behavior in the
electrostatic calibration of an experiment, for example, as experienced in
kimanom . It was suggested that the anomalous effects that were observed are
due to irregularities of the spherical surface. Roughness effects rough
certainly can cause problems at short distances, but the possibility that the
anomalous effects are due to simple geometrical effects is credibly discarded
in staticsurvey .
The contact potential is simply measured by finding a voltage potential
difference $V_{m}$ between the two plates that minimizes the force given by
Eq. (10). $V_{m}$ is manifest as an asymmetry in the force between $\pm V$
applied between the plates.
For our measurements using Gemanium (Ge) plates ourge , we were initially
confused because a $1/d^{1.2}$ to $1/d^{1.5}$ force persisted when the
electrostatic force was minimized at each distance. Our initial conclusion was
that there was a distance offset, as described in the next section, together
with an uncompensated voltage offset. de Man et al. iannuzzi2009 have also
observed a distance dependence of the contact potential, and concluded that it
did not lead to any anomalies in their electrostatic calibrations, however,
the measurements are at shorter distances than were used in the Ge experiment.
In general, the relative electrostatic effect, compared to the Casimir force,
should scale roughly as $(1/d)/(1/d^{3})=d^{2}$. I will now tell the story of
how we came to understand the results of our measurements using Ge plates.
[scale=.65]casphoto.eps
Figure 2: A photograph of the apparatus, in operation, used to measure the
attractive force between Ge plates. The glass bell jar introduces some
distortion; visible are the “compensating plates” on the left of the torsion
pendulum, and the plates (2.54 cm diameter) between which the Casimir force is
measured, on the right. A ThorLab T25XYZ translation stage is used to position
the “fixed” plate. The fine tungsten torsion wire is not visible. (Public
Domain, by S.K. Lamoreaux)
### 3.1 Inclusion of the Debye Screening Length?
In the early calibrations of our Ge plate Casimir experiment ourge , we had a
long-range background force that depended on distance not quite as $1/d$, as
described above. Our initial guess was that there was a distance offset in our
calibrations due to penetration into the plates of the calibration electric
field. The problem is that a quasi-static electric field can propagate a
finite distance into a semiconductor (see, e.g., 5 ); this distance is
determined by the combined consideration of diffusion and field driven
electric currents, leading to an effective field penetration length (Debye-
Hückel length)
$\lambda=\sqrt{\epsilon\epsilon_{0}kT\over e^{2}c_{t}},$ (13)
where $c_{t}=c_{h}+c_{e}$ is the total carrier concentration, which for an
intrinsic semiconductor, $c_{e}=c_{h}$. For intrinsic Ge $\lambda\approx 0.6\
\mu$m, while for a good conductor, it is less than 1 nm. $\lambda$ is
independent of the applied field so long as the applied field $E$ times
$\lambda$ is less than the thermal energy, $k_{b}T$ where $k_{b}$ is
Boltzmann’s constant. In this limit, and at sufficiently low frequencies and
wavenumbers, thermal diffusion dominates the field penetration into the
material. A sufficiently low frequency for Ge would be $v_{c}/\lambda\sim 10$
GHz, where $v_{c}$ is a typical thermal velocity of a carrier.
The potential in a plane semiconductor, if the potential is defined on a
surface $x=0$ is
$V(x)=V(0)e^{-|x|/\lambda},$ (14)
where $\lambda$ is the Debye-Hückel screening length, defined previously.
We are interested in finding the electrostatic energy between two thick Ge
plates separated by a distance $d$, with a voltages $+V/2$ and $-V/2$ applied
to the back surfaces of the plates. In this case, the field is normal to the
surface. After we find the energy per unit area, we can use the proximity
force approximation to get the attractive force between a spherical and flat
plate.
Let $x=0$ refer to the surface of the plate 1, and $x=d$ refer to the surface
of plate 2. By symmetry, the potential at the center position between the
plates is zero. The potential in plate 1 can be written as
$V_{1}(x)=V/2-(V/2-V_{s})e^{-|x|/\lambda},$ (15)
and for the space between the plates
$V_{0}(x)=-2V_{s}x/d+V_{s},$
where we assume the field is uniform. $V_{s}$, the surface potential, is to be
determined.
We need only consider the boundary conditions in plate 1, which are
$V_{1}(-\infty)=V/2,$ $V_{0}(0)=V_{1}(0),$
(which has already been used)
$\epsilon{dV_{1}(x)\over dx}|_{x=0}={dV_{0}(x)\over dx}|_{x=0},$
where the last two imply that $D=\epsilon E$ is continuous across the
boundary.
The solution is
$V_{s}={V\over 2}\left({1\over 1+2\lambda/\epsilon d}\right).$ (16)
With this result, it is straightforward to calculate the total field energy
per unit area in both plates and in the space between the plates. The result
is
${\cal E}={1\over 2}{\epsilon_{0}V^{2}\over
d}\left[{y+y^{2}\over(y+2)^{2}}\right],$ (17)
where the dimensionless length $y=\epsilon d/\lambda$ has been introduced. By
expanding this result for small $y$, it can be easily seen that the effects
appears as an apparent offset in the distance that is determined by measuring
the capacitance between the plates. For small voltages, this offset is
approximately $\lambda/\epsilon=0.68/16\approx 0.05\ \mu$m.
If $V-V_{s}$ is large compared to $k_{b}T$, the effective penetration depth
increases because the charge density is modified in the vicinity of the
surface. The potential in the plates is no longer a simple exponential,
however one can define an effective shielding length 5
${\lambda^{\prime}\over\lambda}={|\Phi|\over\sqrt{e^{\Phi}+e^{-\Phi}-2}},$
(18)
where
$\Phi={V-V_{s}\over k_{b}T}.$ (19)
Given that $k_{b}T=30$ meV, at plate separations of order 1 $\mu$m for Ge this
begins to be a large correction when voltages larger than 60 mV are applied
between the plates, however, the potentials used in our experiment were far
smaller.
We eventually realized that this effect is not present at very low
frequencies; the lifetime of Ge surface states is on the order of
milliseconds. The lack of penetration of quasi-static fields into
semiconductors was first observed in the development of the field effect
transistor, and explained by Bardeen bardeen as shielding due to surface
states. Again, in equilibrium, the electric field must enter normal to the
plate surfaces, otherwise a current would be flowing in contradiction to the
assumption of equilibrium. Therefore, even on very poor conductors, charges
rearrange to force any applied field to be perpendicular to the surface; when
this situation is attained, the electric field terminates at the surface. The
boundary condition is that of a perfect conductor.
The presence of time-dependent surface states might be responsible for some of
the anomalous electrostatic calibration effects observed by Kim et al. kimanom
. Particulary if there is a slight oxide coating on a metal surface, the
surface states might not have enough time to reach equilibrium in the dynamic
measurement system that was employed. The relaxation times for trapped surface
states can be many milliseconds. However, the possibility that these sorts of
states contribute to the anomalous effect is very speculative, and it is
difficult to come up with an experiment to check this hypothesis.
As an aside, our consideration of this effect led us to the realization that
the usual permittivity treatment of materials with non-degenerate conduction
electrons is not correct, but must be solved in a different way than simply
assigning a conductivity to the material carrierpaper . The discussion of this
theoretical point is beyond the scope of this review.
### 3.2 Variable Contact Potential
It was recognized that a distance dependence of the minimizing potential would
lead to extra electrostatic forces that are not necessarily zero at the
minimizing potential myarxiv . The force at the voltage which minimizes the
force at each separation was thought to represent the pure “Casimir” force
between the plates. However, the applied voltage $V_{a}(d)$ required to
minimize the (electrostatic) force is observed to depend on $d$, and is of the
form (in the 1-50 $\mu$m range)
$V_{a}(d)=a\log d+b,$ (20)
where $a$ and $b$ are constants with magnitude of a few mV. This variation
leads to a long-range $1/d$-like potential for the minimized force. An
analysis suggests that this force is better described as $1/d^{m}$ where
$m\approx 1.2-1.4$.
As we show here, the variation in $V_{a}(d)$ implies an additional force that
increases as $1/d^{1.25}$, assuming that the voltage variation is due to the
potential of the plates actually changing with distance. Such changes could
come about due to external fixed fields or potential variations associated
with the plate translation mechanism, and is equivalent to having an
adjustable battery in series with the plates. We were unable to come up with a
model that can give a sufficiently large effect based on interactions between,
for example, the charge carriers. in the plates. However, at sufficient
sensitivity, it is likely that such effects will be important.
This analysis, while it predicts the correct form of the extra force, predicts
that this force is negative or repulsive. However, it is enlightening to go
through the analysis, and this work will never be published elsewhere. An
understanding of the specific origin of the variation of applied minimizing
potential $V_{a}(d)$ is not necessary to correct for the additional force that
it causes, we simply need the experimentally determined $V_{a}(d)$, and assume
it is tied to the plate positions.
We note further that $V_{a}(d)$ is not a measure of the contact potential, but
the voltage which minimizes the force. We call the “true” contact potential
$V_{c}(d)$, which might depend on distance.
In performing our experiment, at each separation $d$, $V_{a}$ is varied and
its value that minimizes the attractive force is determined.
${\cal E}(d)={1\over 2}C(d)(V_{a}+V_{c}(d))^{2},$ (21)
where $C(d)$ is the capacitance between the plates, $V_{a}$ is the applied
potential and is an independent variable, and $V_{c}(d)$ is the average
weighted contact potential between the plates.
The force between the plates is given by the derivative of $\cal E$,
$F(d)={\partial{\cal E}(d)\over\partial d}={1\over 2}{\partial
C(d)\over\partial d}(V_{a}+V_{c}(d))^{2}+C(d)(V_{a}+V_{c}(d)){\partial
V_{c}(d)\over\partial d},$ (22)
Now the minimum in the force is determined by the derivative with $V_{a}$:
${\partial F(d)\over\partial V_{a}}={\partial C(d)\over\partial
d}(V_{a}+V_{c}(d))+C(d){\partial V_{c}(d)\over\partial d}=0,$ (23)
which determines $V_{a}(d)$, no longer an independent variable. Thus,
${\partial V_{c}(d)\over\partial d}=-{1\over C(d)}{\partial C(d)\over\partial
d}(V_{a}(d)+V_{c}(d)),$ (24)
which allows the determination of $V_{c}(d)$ when $V_{a}(d)$ is known. The
differential equation can be solved numerically, noting that at long distances
$V_{a}(d)=-V_{c}(d)$, and that $V_{c}(d)$ become constant.
The electrostatic force between the plates at the minimized potential is given
by
$F(d)=-{1\over 2}{\partial\over\partial
d}\left[C(d)(V_{a}(d)+V_{c}(d))^{2}\right].$ (25)
There are some nice features to this result. First, if we apply a constant
offset $V_{0}$ to $V_{c}(d)$, this effect is compensated by $V_{a}(d)-V_{0}$
which is easily seen as the relationship is linear.
Unfortunately, the sign of the effect indicates that it is repulsive, and thus
is not the explanation of the long range force that persists at the minimizing
potential as observed in our Ge experiment.
However, it should be emphasized that any precision measurement of the Casimir
force requires verification that the contact potential is not changing as a
function of distance, and if it is, a correction to the force as described
here might very well exist.
### 3.3 Patch Potential Effects
It is often assumed that the surface of a conductor is an equipotential. While
this would be true for a perfectly clean surface of a homogeneous conductor
cut along one of its crystalline planes, it is not the case for any real
surface which can be polycrystalline, stressed, or chemically contaminated.
Experiments show that even with precautions for extreme cleanliness, typical
surface potential variations are on the order of at least a few millivolts
LIGO . This is most likely due to local variations in surface crystalline
structure, giving rise to varying work functions and hence varying-potential
patches. It is well known that the work function of a metal surface depends on
the crystallographic plane along which it lies; as an example, for gold the
work functions are 5.47 eV, 5.37 eV, and 5.31 eV for surfaces in the $\langle
100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ directions
respectively. This variation is most likely due to different effective
electron masses, hence Fermi energies, for the different axes.
The means by which surface potential patches form is described in landl , Sec.
22. Briefly, When two conductors, A and B, of different work functions are
brought into contact, electrons flow until the chemical potential (i.e., the
Fermi energy) in both conductors equalizes. If we consider moving an electron
in a closed path that moves from inside conductor A, across the boundary to
inside conductor B, through the surface of B into the vacuum, back though
surface A, and to the starting point, the total work must be zero in
equilibrium. If we take the contact potential difference between the
conductors as $\phi_{ab}$, and the surface work functions as $W_{a}$ and
$W_{b}$, for the total work to be zero we must have
$\phi_{ab}=W_{b}-W_{a},$
implying that the contact potential is simply the difference in the surface
work functions.
It is straightforward to calculate the electric field energy of random
patches, as has been done by Speake and Trenkel Speake2003 . Consider two
plane and parallel surfaces separated by a distance $d$. Assume a potential
$V=0$ at $x=0$, while at $x=d$, $V=V_{0}\cos ky$. It is easy to show that, in
the region between the plates,
$V(x,y)=V_{0}\cos ky{e^{kx}-e^{-kx}\over e^{kd}-e^{-kd}}.$
The field energy, per unit area is given by
${\cal E}=\int_{0}^{d}\left[\left({\partial V\over\partial
x}\right)^{2}+\left({\partial V\over\partial y}\right)^{2}\right]dx,$
where we have used the fact that
$\langle\cos^{2}ky\rangle=\langle\sin^{2}ky\rangle=1/2$ to do the $y$
integral. Letting
$u=e^{kx}-e^{-kx}\ \ \ \ dv=e^{kx}-e^{-kx},$
so
$du=k[e^{kx}+e^{-kx}]\ \ \ \ \ v={1\over k}[e^{kx}+e^{-kx}],$
and integrating by parts
$\int_{0}^{d}[e^{kx}-e^{-kx}]^{2}dx={1\over
k}[e^{2kx}-e^{-2kx}]|_{0}^{d}-\int_{0}^{d}[e^{kx}+e^{-kx}]^{2}dx.$
The LHS is proportional to the field energy for $E_{y}$ while the last term on
the RHS is proportional to (minus) the field energy for $E_{y}$. We thus have
${\cal E}=k{V_{0}^{2}\over 2}{e^{2kd}-e^{-2kd}\over[e^{kd}-e^{-kd}]^{2}}.$
By use of the proximity force approximation, the (attractive) force between a
flat surface and spherical surface is $F(d)=2\pi R{\cal E}(d)$ where $R$ is
the radius of curvature, where $d$ is the point of closest approach between
the surfaces. In the limit $kd\rightarrow 0$,
$F=2\pi R{V_{0}^{2}\over 4d}\propto{1\over d}.$
This shows that when $kd\ll 1$ or $d\ll\lambda/2\pi$ where $\lambda$ is a
characteristic length of a potential patch, the force goes as $1/d$. This is
what we expect from the PAA when the surfaces are very close.
There is an intermediate range where the force transforms from $1/d$ to
exponential variation; at further distances, the force becomes a constant, as
$\cal E$ does not vary with $d$. Between parallel plates, at long distances,
the force is zero because the field energy does not change with separation. It
is interesting to note this significant difference between the PFA result for
a spherical surface and the result for parallel plates. As a constant force is
in reality unobservable, this long distance force should be subtracted from
the PFA result.
It should be noted that the field equations are linear, so we can add other
$\cos(k^{\prime}y),\ \cos(k^{\prime}z)$ fluctuations, and the integral over
$z,y$ leads to delta functions of $k-k^{\prime}$. We can therefore rewrite the
attractive force as an integral over $k_{y},k_{z}$ where we have
$V_{k_{y}}(y)+V_{k_{z}}(z)$ representing the amplitude spectrum in $k$ space
of the surface fluctuations. If we take $V_{k_{y}}(y)\sim V_{k_{z}}(z)$ and
assume they are uncorrelated, the integral over $k_{y},k_{z}$ leads to
$F=\pi RV_{rms}^{2}\int_{0}^{\infty}(2\pi k\
dk)(kS(k)){e^{2kd}-e^{-2kd}\over[e^{kd}-e^{-kd}]^{2}},$
where, by use of the Wiener-Khinchine theorem, $S(k)$ is the normalized cosine
Fourier transform (in polar coordinates) of the surface potential spatial
correlation function.
In order to compute the patch effect on the force in the sphere-plane
configuration we make use of the proximity force approximation. Just as in the
case of roughness in Casimir physics rough , one must distinguish between two
PFAs: one is for the treatment of the curvature of the sphere (valid when
$d\ll R$, where $R$ is the radius of curvature), and the other one is the PFA
applied to the surface patch distribution (valid when $kd\ll 1$). We assume
that we are in the conditions for PFA for the curvature, but we keep $kd$
arbitrary. Then, the electrostatic force in the sphere-plane case is
$F_{sp}(d)=2\pi R{\cal E}(d)$, implying
$F_{sp}=2\pi\epsilon_{0}R\int_{0}^{\infty}dk\frac{k^{2}e^{-kd}}{\sinh(kd)}S(k).$
(26)
There are a number of models that can be used to describe the surface
fluctuations. The simplest is to say that the potential autocorrelation
function is, for a distance $r$ along a plate surface,
${\cal R}(r)=\left\\{\begin{array}[]{ll}{V_{0}^{2}}&{\rm for}\
r\leq\lambda,\\\ 0&{\rm for}\ r>\lambda.\end{array}\right.$ (27)
Then, by the Wiener-Khinchin theorem, the power spectral density $S(k)$ can be
evaluated as the cosine two-dimensional Fourier transform of the
autocorrelation function, which in our notation is stein
$S(k)={V_{0}^{2}}\lambda^{2}{J_{1}(\lambda k)\over\lambda k},$ (28)
with $J_{1}$ the Bessel function of first kind. The plane-sphere force is then
given by, using $k=u/\lambda$,
$F_{sp}=2\pi\epsilon_{0}R\int_{0}^{\infty}du\ u{J_{1}(u)\over
e^{2ud/\lambda}-1}.$ (29)
A numerical calculation shows that, for $d<.01\lambda$,
$F_{sp}\approx{\pi\epsilon_{0}RV_{0}^{2}\over d},$ (30)
suggesting that $V^{2}_{\rm rms}=V_{0}^{2}$, as expected. For
$50\lambda>d>\lambda$, the force falls with distance as $1/d^{3}$.
We see immediately that at short distances, there is a residual force due to
patches that varies as $1/d$, and there is no minimizing potential that can
compensate this effect. It is, in a restricted sense, equivalent to having an
oscillating potential between the plates; there is no way for a static field
to compensate the oscillating field energy.
[scale=.65]patchmodelcopy.eps
Figure 3: A toy model illustrating the mechanism for the generation of a
distance-dependent minimizing electrostatic potential $V_{m}(d)$ and
electrostatic residual force $F^{\rm el}_{\rm res}(d)$. (Public Domain, by
S.K. Lamoreaux)
As described in the last Section, in our own work ourge and in a number of
other experiments kimanom ; iannuzzi2009 , a distance-variation in the
electrical potential minimizing the force between the plates has been
observed. It had been suggested already that this variation in contact
potential can cause an additional electrostatic force, and an estimate was
made for the possible size of the effect myarxiv . However, further
experimental work shows that the model used in myarxiv , where the varying
contact potential is considered to be a varying voltage in series with the
plates, does not reproduce our experimental results ourwork2009 .
A model that produces a residual electrostatic force consistent with our
observations ourge is shown in Fig. 3. In this figure, the two capacitors
(short distance, $C_{a}(d)$, long distance $C_{b}(d+\Delta)$) create a net
force on the lower continuous plate (setting $V_{1}=0$ initially),
$F(d,V_{0})=-\frac{1}{2}C_{a}^{\prime}V_{0}^{2}-\frac{1}{2}C_{b}^{\prime}(V_{0}+V_{c})^{2},$
(31)
where
$C_{a}^{\prime}={\partial C_{a}(d)\over\partial d};\ \ \
C_{b}^{\prime}={\partial C_{b}(d+\Delta)\over\partial d},$ (32)
and $V_{0}$ can be varied, with $V_{c}$ a fixed property of the plates. The
force is minimized when
$\displaystyle\left.{\partial F(d,V_{0})\over\partial
V_{0}}\right|_{V_{0}=V_{m}}=0$ $\displaystyle\Rightarrow$ $\displaystyle
V_{m}(d)=-{C_{b}^{\prime}V_{c}\over C_{a}^{\prime}+C_{b}^{\prime}},$ (33)
implying a residual electrostatic force
$\displaystyle F^{\rm el}_{\rm res}(d)$ $\displaystyle=$ $\displaystyle
F(d,V_{0}=V_{m}(d))$ (34) $\displaystyle=$
$\displaystyle-\left[C_{a}^{\prime}+{C_{a}^{\prime 2}\over
C_{b}^{\prime}}\right]{V_{m}^{2}(d)\over
2}=-\left[\frac{C_{a}^{\prime}C_{b}^{\prime}}{C_{a}^{\prime}+C_{b}^{\prime}}\right]{V_{c}^{2}\over
2}.$
It is easy to take a case of parallel plate capacitors
($C_{a}^{\prime}=-\epsilon_{0}A/d^{2}$ and
$C_{b}^{\prime}=-\epsilon_{0}A/(d+\Delta)^{2}$, where $A$ is the area of each
of the upper plates in Fig. 3, assumed to be equal; hence, the lower
continuous plate has area $2A$) and to show that there is a residual
electrostatic force at the minimizing potential. Indeed, in such case,
$\displaystyle V_{m}(d)$ $\displaystyle=$ $\displaystyle-
V_{c}\frac{d^{2}}{d^{2}+(d+\Delta)^{2}},$ (35) $\displaystyle F^{\rm el}_{\rm
res}(d)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}A}{2}\;\frac{V_{c}^{2}}{d^{2}+(d+\Delta)^{2}}.$
(36)
Alternatively, in terms of $V_{m}(d)$ (up to $V_{1}$, see below), the force is
$F^{\rm el}_{\rm
res}(d)=\frac{\epsilon_{0}A}{2}\;\frac{V_{m}^{2}(d)[d^{2}+(d+\Delta)^{2}]}{d^{4}}.$
(37)
Experimentally, $V_{m}(d)$ cannot be directly measured; measurements can only
determine it up to an overall offset $V_{1}$ which arbitrarily depends on the
sum of contact potentials in the complete circuit between the plates.
Therefore the force should be written at proportional to
$(V_{m}(d)+V_{1})^{2}$ instead of simply $V_{m}^{2}(d)$, where $V_{1}$ is
determined by a fit to experimental data, for example. In the limit $\Delta\gg
d$, the residual force is proportional to $1/d^{4}$ in the plane-offset plane
case here considered.
If we now consider the sphere-plane case,
$C_{a}^{\prime}(d)=-2\pi\epsilon_{0}R/d$, and the denominator of Eq. (37)
becomes $d^{2}$. If we further consider the surface divided up into
infinitesimal areas, each with a random potential, and integrate over the
surface to get the net force, there is a further reduction of the power of $d$
in the denominator (just as in the proximity force approximation), leaving the
sphere-plane force proportional to $1/d$. This motivates writing the residual
force as
$F_{res}(d)={\pi\epsilon_{0}R\left[(V_{m}(d)+V_{1})^{2}+V_{rms}^{2}\right]\over
d},$ (38)
where it is understood that $V_{m}(d)$ is experimentally measured, and $V_{1}$
is a fit parameter that represents a sort of surface average potential, plus
circuit offsets (this equation is supported both by numerical studies and by
our experimental results ourge ; ourwork2009 , and is valid when
$|V_{1}|>>|V_{m}(d)|$) as observed. The last term in Eq. (38) is the expected
random (i.e., does not contribute to $V_{m}(d)$) patch potential force, but
here should be thought as a fit parameter that reflects the magnitude of
$V_{rms}$. With this result, the long range force observed in our experiment
could be explained, and our work with Ge was completed. The agreement with
theory is excellent, however, there is very little difference in theoretical
prediction of the force with and without the $TE$ $n=0$ mode, so this work was
not able to help with that controversy.
As a final note, the variations in surface potential could be a simple
function of position on the conducting surface, for example, due to stresses
or impurities within the samples. Alternatively, if there is a slight
roughness to the surface, the peaks could have a different potentials that the
valleys associated with surface irregularities. This latter possibility
appears to be a better model as we were unable to detect a variation in
$V_{m}$ when the plates were moved relative to each other, which might be
expected for positional surface patches. However, the level of the surface
fluctuations is quite small, and for example is beyond the range of state of
the art Kelvin probes kelvinprobe . These issues need further investigation.
## 4 Conclusions and Outlooks
In many respects, we can consider the measurement of the Casimir force between
surfaces as a mature field. However, many open issues remain, particularly in
the limits of accuracy that can be expected. In recent years, we have seen a
number of experiments claiming 1% precision, but many counter claims that such
accuracy is beyond what is possible due to finite knowledge of a plethora of
corrections and required absolute calibrations. Some open issues include the
effects of finite conductivity on the contribution of the $TE$ $n=0$ surface
mode; the usual Drude model of the permittivity of a metal suggests that this
mode does not contribute at all to the force, reducing the force by a factor
of two at large separations. It is unclear whether additional short-range AFM
type measurements will clear this problem up, as at short distances, the
correction is relatively small. Improved measurements at distances above a few
microns would appear to offer the best prospects for bringing these issues to
closure. Recent work with our torsion pendulum system at Yale seems to be in
favor of the no-$TE$ $n=0$ mode, although the precision is not yet sufficient
to make a strong claim. Over the next few months we hope to have new higher
accuracy data analyzed.
The effects of patch potentials has not been fully investigated in all
experiments to date. For example, in my 1997 experiment skl97prl , an
anomalous component to the $1/d$ force would result in an error in the
distance determination, which only needed to be 0.1 micron to bring my
experiment into agreement with the Bostöm and Sernelius calculation. Likewise,
the boundary modification experiment of Chan et al. chan did not consider in
any obvious way excess forces due to electrostatic patch effects, which might
be expected as substantial due to the sharp features of the etched silicon
trenches, and will vary as $1/d^{3}$ in the limit of the separation much large
than the trench spacing. It is hard to imagine that such an effect is more
than 10% of the Casimir force, but some analysis and additional experiments
are necessary to eliminate the possibility of such a systematic effect.
In any case, a reasonable ultimate experimental goal is the attainment of 1%
agreement between theory and experiment, in terms of true accuracy; this is
not a question of simple precision. Hopefully the readers of this review will
realize the complexity and difficulty of the challenge presented by this goal.
## 5 Acknowledgements
I thank my colleagues and collaborators W.-J. Kim, A.O. Sushkov, H.X. Tang,
and D.A.R. Dalvit for many fruitful discussions that led to the understanding
of our Ge experiment, and to deeper understanding of the Casimir force in
general. I also thank R. Onofrio and S. de Man to a number of discussions over
the last few years that were helpful in clarifying a number of issues. SKL was
supported by the DARPA/MTO Casimir Effect Enhancement project under SPAWAR
contract number N66001-09-1-2071.
## References
* (1) G.L. Klimchitskaya, U. Mohideen, and V.M. Mostepanenko: The Casimir force between real materials: Experiment and theory. Rev. of Mod. Phys. 81, 1827 (2009).
* (2) S. K. Lamoreaux: Demonstration of the Casimir force in the 0.6 to 6 $\mu$m range. Phys. Rev. Lett. 78, 5 (1997).
* (3) C.M. Hargreaves: Corrections to the retarded dispersion force between metal bodies. Proc. Kon. Ned. Akad. Wetensh. 68B, 231 (1965); J. Schwinger, L.L. De Raad, Jr., and K.A. Milton: Casimir effect in dielectrics. Ann. Phys. (New York) 115, 1 (1978).
* (4) S.K. Lamoreaux: Calculation of the Casimir force between imperfectly conducting plates . Phys. Rev. A bf 59, R3149 (1999).
* (5) S.K. Lamoreaux: Erratum: Demonstration of the Casimir force in the 0.6 to 6 $\mu$m range. Phys. Rev. Lett. 81, 5475 (1998).
* (6) A. Lambrecht and S. Reynaud: Casimir force between metallic mirrors. Euro. Phys. Jour. D 8, 309 (2000).
* (7) A. Lambrecht, S. Reynaud, and S.K. Lamoreaux: Comment on Demonstration of the Casimir force in the 0.6 to 6 $\mu$m range. Phys. Rev. Lett. 84, 5672 (2000), and references therein.
* (8) M. Boström and B. Sernlius: Thermal effects on the Casimir force in the 0.1-5 mu m range. Phys. Rev. Lett. 84, 4757 (2000); M. Boström and B.E. Sernelius: Entropy of the Casimir effect between real metal plates. Physica A 339, 53 (2004).
* (9) S.K. Lamoreaux and W.T. Buttler: Thermal noise limitations to force measurements with torsion pendulums: Applications to the measurement of the Casimir force and its thermal correction. Phys. Rev. E 71, 036109 (2005).
* (10) Peter W. Milonni, The Quantum Vacuum (Academic Press, San Diego, 1994), pp 286-288.
* (11) H.B.G. Casimir: On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Akad. Wetenschap 51, 793 (1948).
* (12) E.M. Lifshitz: The theory of molecular attractive forces between solids. Sov. Phys. JETP 2, 73 (1956).
* (13) See, e.g., V. Mostepanenko: Experiment, theory, and the Casimir effect. J. Phys. Conf. Ser. 161, 012003 (2009).
* (14) J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975). See Sec. 7.5.
* (15) L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).
* (16) V.B. Svetovoy, P.J. van Zwol, G. Palasantzas, and J. Th. M. De Hosson: Optical properties of gold films and the Casimir force. Phys.Rev. B 77, 035439 (2008).
* (17) P.J. van Zwol, V.B. Svetovoy, and G. Palasantzas: Distance upon contact: Determination from roughness profile. Phys. Rev. B 80, 235401 (2009).
* (18) E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and S.H. Aronson: Reanalysis of the Eotvos experiment. Phys. Rev. Lett. 56, 3 (1986).
* (19) V.M.Mostepanenko and I. Yu. Sokolov: Laboratory tests for the constituents of dark matter. Astronoische Nachrichten 311, 197 (1990).
* (20) J. C. Long, H. W. Chan, and J. C. Price: Experimental status of gravitational-strength forces in the sub-centimeter regime. Nucl. Phys. B 539, 23 (1999).
* (21) G. L. Klimchitskaya, E. R. Bezerra de Mello, and V. M. Mostepanenko: Constraints on the parameters of degree-type hypothetical forces following from the new Casimir force measurement. Phys. Lett. A 236, 280 (1997).
* (22) R.S. Decca, D. Lopez, E. Fischbach, G.L. Klimchitskaya, D. E. Kraus, and V.M. Mostepanenko: Tests of new physics from precise measurements of the Casimir pressure between two gold-coated plates. Phys. Rev. D 75, 77101 (2007).
* (23) D.A.R. Dalvit and R. Onofrio: On the use of the proximity force approximation for deriving limits to short-range gravitational-like interactions from sphere-plane Casimir force experiments. Phys. Rev. D 80, 064025 (2009).
* (24) R.S. Decca, E. Fischbach, G.L. Klimchitskaya, D.E. Krause, D. Lopez, D. and V.M. Mostepanenko: Comment on Anomalies in electrostatic calibrations for the measurement of the Casimir force in a sphere-plane geometry. Phys. Rev. D 79, 124021 (2009).
* (25) R. S. Decca, D. Lopez, H. B. Chan, E. Fischbach, D. E. Krause, and C. R. Jamell: Constraining new forces in the Casimir regime using the isoelectronic technique. Phys. Rev. Lett. 94, 240401 (2005).
* (26) G.Luther, Private Communication, 1997.
* (27) F.S.S. Rosa, D.A.R. Dalvit, P.W. Milonni: Casimir interactions for anisotropic magnetodielectric metamaterials. Phys. Rev. A 78, 032117 (2008).
* (28) E. Buks and M.L. Roukes: Metastability and the Casimir effect in micromechanical systems. Europhys. Lett. 54, 220 (2001).
* (29) S. de Man, K. Heeck, R.J. Wijngaarden, and D. Iannuzzi: Halving the casimir force with conductive oxides. Phys. Rev. Lett. 103, 040402 (2009).
* (30) T.H. Boyer: Quantum electromagnetic zero-point energy of a conducting spherical shell and the Casimir model for a charged particle. Phys. Rev. 174, 1764 (1968).
* (31) D. Iannuzzi, M. Lisanti, and F. Capasso: Effect of hydrogen-switchable mirrors on the Casimir force, PNAS 101, 4019 (2004).
* (32) S. de Man and D. Iannuzzi: On the use of hydrogen switchable mirrors in Casimir force experiments, New Jour. Phys. 8, 235 (2006).
* (33) H.B. Chan, Y. Bao, J. Zou, R.A. Cirelli, F. Klemens, W.M. Mansfield, and C.S. Pai: H.B. Chan, Y. Bao, J. Zou, R.A. Cirelli, F. Klemens, W.M. Mansfield, and C.S. Pai. Phys. Rev. Lett. 101, 03040 (2008).
* (34) R. Büscher and T. Emig: Nonperturbative approach to Casimir interactions in periodic geometries. Phys. Rev. A 69, 062101 (2004).
* (35) J.N. Munday and F. Capasso: Precision measurement of the Casimir-Lifshitz force in a fluid. Phys. Rev. A 75, 060102 (2007).
* (36) J.N. Munday, J.N. F. Capasso, V.A. Parsegian, and S.M. Bezrukov: Measurements of the Casimir-Lifshitz force in fluids: The effect of electrostatic forces and Debye screening. Phys. Rev. A 78, 032109 (2008).
* (37) J.N. Munday, F. Capasso, and V.A. Parsegian: Measured long-range repulsive Casimir-Lifshitz forces. Nature 457, 170 (2008).
* (38) B.V. Derjaguin and I.I. Abrikosova: Direct measurement of the molecular attraction of solid bodies. 1. Statement of the problem and method of measuring forces by using negative feedback. Sov. Phys. JETP 3, 819 (1957).
* (39) Proximity forces J. Blocki, J. Randrup, J., W.J. Swiatecki, and C.F. Tsang: Proximity forces. Annals of Physics 105, 427 (1977); J. Blocki, and W.J. Swiatecki: Generalization of the proximity force theorem. Annals of Physics, 132, 53 (1981).
* (40) W.J. Kim, M. Brown-Hayes, D.A.R. Dalvit, J. H. Brownell, and R. Onofrio, Phy. Rev. A, 79, 026102 (2009).
* (41) W.J. Kim, M. Brown-Hayes, D.A.R. Dalvit, J.H. Brownell, and R. Onofrio: Anomalies in electrostatic calibrations for the measurement of the Casimir force in a sphere-plane geometry. Phys. Rev. A 78, 020101 (2008).
* (42) P. A. Maia Neto, A. Lambrecht, and S. Reynaud: Casimir effect with rough metallic mirrors. Phys. Rev. A 72, 012115 (2005).
* (43) S. de Man, K. Heeck, D. Iannuzzi: No anomalous scaling in electrostatic calibrations for Casimir force measurements. Phys. Rev. A 79, 024102 (2009).
* (44) W.J. Kim, A.O. Sushkov, D.A.R. Dalvit, and S.K. Lamoreaux: Measurement of the short-range attractive force between Ge plates using a torsion balance. Phy. Rev. Lett. 103, 060401 (2009).
* (45) A.I. Spitsyn and V.M. Vanstan: Potential gradient along semiconductor-surface projections in an external electric field. Radiophysics and Quantum Electronics 36, 752 (1993).
* (46) J. Bardeen: Surface states and rectification at metal semi-conductor contact. Phys. Rev. 71, 717 (1947).
* (47) D.A.R. Dalvit and S.K. Lamoreaux: Computation of casimir forces for dielectrics or intrinsic semiconductors based on the boltzmann transport equation. Journal of Physics: Conference Series, 161, 012009 (2009); D.A.R. Dalvit and S.K. Lamoreaux: Contribution of drifting carriers to the casimir-lifshitz and casimir-polder interactions with semiconductor materials. Phys. Rev. Lett. 101, 163203 (2008).
* (48) S. K. Lamoreaux: Electrostatic background forces due to varying contact potentials in Casimir experiments. arXiv:0808.0885.
* (49) N. A. Robertson, “Kelvin Probe Measurements of the Patch Effect” Report LIGO-G070481-00-R (available at http://www.ligo.caltech.edu/docs/G/G070481-00.pdf).
* (50) C. C. Speake and C. Trenkel: Forces between conducting surfaces due to spatial variations of surface potential. Phys. Rev. Lett. 90, 160403 (2003).
* (51) W. J. Kim, A. O. Sushkov, D. A. R. Dalvit, and S. K. Lamoreaux: Measurement of the short-range attractive force between Ge plates using a torsion balance. Phy. Rev. Lett. 103, 060401 (2009); W.J. Kim, A.O Sushkov, D.A.R. Dalvit, and S.K. Lamoreaux: Surface contact potential patches and Casimir force measurements. Phys. Rev. A 81, 022505 (2010).
* (52) E. Stein and G. Weiss, Introduction to Fourier Transforms on Euclidean Spaces(Princeton University Press, Princeton, 1971).
* (53) N. Nonnemacher, M.P. OBoyle, and H.K. Wickramasinghe: Kelvin Probe Microscopy. Applied Physics Letters 58, 2921 (1991); H.O. Jacobs, P. Leuchtmann, O.J. Homan, and A. Stemmer: Resolution and contrast in Kelvin probe force microscopy. Journal of Applied Physics 84, 1168 (1998).
|
arxiv-papers
| 2010-08-21T15:32:28 |
2024-09-04T02:49:12.317907
|
{
"license": "Public Domain",
"authors": "Steve K. Lamoreaux",
"submitter": "Steve K. Lamoreaux",
"url": "https://arxiv.org/abs/1008.3640"
}
|
1008.3641
|
# Capacity Limits of Multiuser Multiantenna Cognitive Networks
Yang Li and Aria Nosratinia
The University of Texas at Dallas, Richardson, TX 75080, USA
Email: liyang@student.utdallas.edu, aria@utdallas.edu
###### Abstract
Unlike point-to-point cognitive radio, where the constraint imposed by the
primary rigidly curbs the secondary throughput, multiple secondary users have
the potential to more efficiently harvest the spectrum and share it among
themselves. This paper analyzes the sum throughput of a multiuser cognitive
radio system with multi-antenna base stations, either in the uplink or
downlink mode. The primary and secondary have $N$ and $n$ users, respectively,
and their base stations have $M$ and $m$ antennas, respectively. We show that
an uplink secondary throughput grows with $\frac{m}{N+1}\log n$ if the primary
is a downlink system, and grows with $\frac{m}{M+1}\log n$ if the primary is
an uplink system. These growth rates are shown to be optimal and can be
obtained with a simple threshold-based user selection rule. Furthermore, we
show that the secondary throughput can grow proportional to $\log n$ while
simultaneously pushing the interference on the primary down to zero,
asymptotically. Furthermore, we show that a downlink secondary throughput
grows with $m\log\log n$ in the presence of either an uplink or downlink
primary system. In addition, the interference on the primary can be made to go
to zero asymptotically while the secondary throughput increases proportionally
to $\log\log n$. Thus, unlike the point-to-point case, multiuser cognitive
radios can achieve non-trivial sum throughput despite stringent primary
interference constraints.
## I Introduction
Currently, the spectrum assigned to licensed (primary) users is heavily under-
utilized [1]. Cognitive radio aims to improve the utilization of spectrum by
allowing cognitive (secondary) users to access the same spectrum as primary
users, as long as any performance degradation of the primary users is
tolerable.
In general, secondary users can access the spectrum via methods known as
overlay, interweave, and underlay [2]. In the overlay technique the secondary
user not only transmits its own signal, but also acts as a relay to compensate
for its interference on the primary user. The overlay method depends on the
secondary transmitter having access to primary’s message [3].111Sometimes,
this is referred to as an interference channel with degraded message sets. In
the interweave technique [4], the secondary user first senses spectrum holes
and then transmits in the detected holes. Reliable sensing in the presence of
fading and shadowing has proved to be challenging [5]. Finally, in the
underlay technique [6], the secondary can transmit as long as the interference
caused on the primary is less than a pre-defined threshold. The secondary user
in this case is neither required to know the primary user’s message nor
restricted to transmit in spectrum holes.
This paper studies performance limits of an underlay cognitive network
consisting of multi-user and multi-antenna primary and secondary systems. The
primary and secondary systems are subject to mutual interference, where the
secondary has to comply with a set of interference constraints imposed by the
primary. We are interested in the average sum rate (throughput) of the
secondary system as the number of secondary users grows. Moreover, we study
how the secondary throughput is affected by the size of primary network as
well as the severity of the interference constraints, which is one of the key
issues in the design of an underlay cognitive network.
A summary of the results of this paper is as follows. We assume that the
primary and secondary have $N$ and $n$ users, respectively, and their base
stations have $M$ and $m$ antennas, respectively.
* •
Secondary uplink (MAC): the secondary average throughput is shown to grow as
$\Theta(\log n)$, which is achieved by a threshold-based user selection rule.
More precisely, the average throughput of the secondary MAC channel grows as
$\frac{m}{N+1}\log n+O(1)$ when it coexists with the primary broadcast
channel, and grows as $\frac{m}{M+1}\log n+O(1)$ when it coexists with the
primary MAC channel. By developing asymptotically tight upper bounds, these
growth rates are further proven to be optimal. Moreover, the interference on
the primary system can be asymptotically forced to zero, while the secondary
throughput still grows as $\Theta(\log n)$. Specifically, for some non-
negative exponent $q$, the interference on the primary can be made to decline
as $\Theta(n^{-q})$, while the throughput of a secondary MAC grows as
$\frac{m-qN}{N+1}\log n+O(1)$ and $\frac{m-qM}{M+1}\log n+O(1)$, respectively
in cases of primary broadcast and MAC channel. The above results imply that
asymptotically the secondary system can attain a non-trivial throughput
without degrading the performance of the primary system.
* •
Secondary downlink (broadcast): the secondary average throughput is shown to
scale with $m\log\log n+O(1)$ in the presence of either the primary broadcast
or MAC channel. Hence, the growth rate of throughput is unaffected (thus
optimal) by the presence of the primary system. In addition, the interference
on the primary can be asymptotically forced to zero, while maintaining the
secondary throughput as $\Theta(\log\log n)$. Specifically, for an arbitrary
exponent $0<q<1$, the interference can be made to decline as
$\Theta\big{(}(\log n)^{-q}\big{)}$, while the secondary average throughput
grows as $m(1-q)\log\log n+O(1)$.
Some of the related earlier work is as follows. Much of the past work in the
underlay cognitive radio involves point-to-point primary and secondary
systems. Ghasemi et al [6] studies the ergodic capacity of a point-to-point
secondary link under various fading channels. Multiple antennas at the
secondary transmitter are exploited by [7] to manage the tradeoff between the
secondary throughput and the interference on the primary. In the context of
multi-user cognitive radios, Zhang et al [8] studies the power allocation of a
single-antenna secondary system under various transmit power constraints as
well as interference constraints. Gastpar [9] studies the secondary capacity
via translating a receive power constraint into a transmit power constraint.
Recently, ideas from opportunistic communication [10] were used in underlay
cognitive radios by selectively activating one or more secondary users to
maximize the secondary throughput while satisfying interference constraints.
The user selection in cognitive radio is complicated because the secondary
system must be mindful of two criteria: the interference on the primary and
the rate provided to the secondary. Karama et al [11] selects secondary users
with channels almost orthogonal to a single primary user, so that the
interference on the primary is reduced. Jamal et al [12, 13] obtains
interesting scaling results for the sum rate by selecting users causing the
least interference. Some distinctions of our work and [12, 13] are worth
noting. First, Jamal et al [12, 13] studies the hardening of sum rate via
convergence in probability, while we analyze the average throughput, which
requires a very different approach.222In general, convergence in probability
does not imply convergence in any moment (thus average throughput) [14]. For
example, consider a sequence of rates $R_{n}=\log(1+X_{n})$, where
$X_{n}=\left\\{\begin{array}[c]{ll}1&\mbox{with probability
$1-\frac{1}{n}$}\\\ \exp(n^{2})&\mbox{with probability $\frac{1}{n}$
}\end{array}\right.$ Then, $\lim_{n\uparrow\infty}R_{n}=\log 2$ in
probability, however, $\lim_{n\uparrow\infty}\mathbb{E}[R_{n}]=\infty$ in
probability. Therefore, the average rate $\mathbb{E}[R_{n}]$ cannot be
predicted based on the hardening (in probability) of $R_{n}$. Second, we study
a multi-antenna cognitive network whereas [12, 13] considers a single antenna
network. Third, we study the effect of the primary network size (number of
constraints) on the secondary throughput, while [12, 13] considers a single
primary constraint.
We use the following notation: $[\,\cdot\,]_{i,j}$ refers to the $(i,j)$
element in a matrix, $|\cdot|$ refers to the cardinality of a set or the
Euclidean norm of a vector, $\text{diag}(\cdot)$ refers to a diagonal matrix,
$\text{tr}(\cdot)$ refers to the trace of a matrix, and $I_{k\times k}$ refers
to the $k\times k$ identity matrix. All $\log(\cdot)$ is natural base. For any
$\epsilon>0$, some positive $c_{1}$ and $c_{2}$, and sufficiently large $n$:
$\displaystyle f(n)=O\big{(}g(n)\big{)}:$ $\displaystyle|f(n)|$
$\displaystyle<c_{1}\,|g(n)|$ $\displaystyle f(n)=\Theta\big{(}g(n)\big{)}:$
$\displaystyle c_{2}\,|g(n)|<|f(n)|$ $\displaystyle<c_{1}\,|g(n)|$
$\displaystyle f(n)=o\big{(}g(n)\big{)}:$ $\displaystyle|f(n)|$
$\displaystyle<\epsilon\,|g(n)|$
We let $\mathcal{R}_{mac,w/o}^{opt}$ and $\mathcal{R}_{bc,w/o}^{opt}$ be the
maximum average throughput achieved by the secondary MAC and broadcast channel
in the absence of the primary, respectively. In this case, we have regular MAC
and broadcast channels, and it is well known that
$\mathcal{R}_{mac,w/o}^{opt}$ scales as $m\log n$, and
$\mathcal{R}_{bc,w/o}^{opt}$ scales as $m\log\log n$.
The remainder of this paper is organized as follows. Section II describes the
system model. The average throughput of the secondary MAC channel is studied
in Section III, where in Section III-C we prove the achieved throughout is
asymptotically optimal. The average throughput of the secondary broadcast
channel is investigated in Section IV. Numerical results are shown in Section
V. Finally, Section VI concludes this paper.
## II System Model
Figure 1: Coexistence of the secondary MAC channel and the primary system
Figure 2: Coexistence of the secondary broadcast channel and the primary
system
We consider a cognitive network consisting of a primary and a secondary, each
being either a MAC or broadcast channel (Figure 1 and Figure 2). The primary
system has one base station with $M$ antennas and $N$ users, while the
secondary system consists of one base station with $m$ antennas and $n$ users.
The primary and secondary are subject to mutual interference, which is treated
as noise. The secondary system must comply with a set of interference power
constraints imposed by the primary. For simplicity of exposition, at the
beginning primary and secondary users (except base stations) are assumed to
have one antenna, however, as shown in the sequel, most of the results can be
directly extended to a scenario where each user has multiple antennas.
A block-fading channel model is assumed. All channel coefficients are fixed
throughout each transmission block, and are independent, identically
distributed (i.i.d.) circularly-symmetric-complex-Gaussian with zero mean and
unit variance, denoted by $\mathcal{CN}(0,1)$. The secondary base station acts
as a scheduler: For each transmission block, a subset of the secondary users
is selected to transmit to (or receive from) the secondary base station. We
denote the collection of selected (active) secondary users as $\mathcal{S}$.
We begin by introducing a system model that applies to all four scenarios in
Figures 1 and 2, thus simplifying notation in the remainder of the paper. The
secondary received signal is given by:
$\mathbf{y}=\mathbf{H}(\mathcal{S})\,\mathbf{x}_{s}+\mathbf{G}_{s}\,\mathbf{x}_{p}+\mathbf{w}$
(1)
where $\mathbf{y}$ represents the received signal vector, either signals at a
multi-antenna base station (uplink) or at different users (downlink).
$\mathbf{H}(\mathcal{S})$ is the channel coefficient matrix between the active
secondary users and their base station. $\mathbf{G}_{s}$ represents the cross
channel coefficient matrix from the primary transmitter(s) to the secondary
receiver(s). The primary and secondary transmit signal vectors are
$\mathbf{x}_{p}$ and $\mathbf{x}_{x}$. The variable $\mathbf{w}$ is the
received noise vector, where each entry of $\mathbf{w}$ is i.i.d.
$\mathcal{CN}(0,1)$.
We assume both primary and secondary systems use Gaussian signaling, subject
to short-term power constraints. The transmit covariance matrices of the
primary and secondary systems are
$Q_{p}=\mathbb{E}\big{[}\mathbf{x}_{p}\mathbf{x}^{{\dagger}}_{p}\big{]}$ (2)
and
$Q_{s}=\mathbb{E}\big{[}\mathbf{x}_{s}\mathbf{x}^{{\dagger}}_{s}\big{]}$ (3)
When the secondary is a MAC channel, each secondary user is subject to an
individual short term power constraint $\rho_{s}$. The users do not cooperate,
therefore $Q_{s}$ is diagonal:
$\displaystyle
Q_{s}=\text{diag}\big{(}\rho_{1},\cdots,\rho_{|\mathcal{S}|}\big{)}$ (4)
where $\rho_{\ell}\leq\rho_{s}$, for $\ell=1,\cdots,|\mathcal{S}|$. In this
case, $\mathbf{H}(\mathcal{S})$ has dimension $m\times|\mathcal{S}|$.
When the secondary is a broadcast channel, we assume the secondary base
station is subject to a short term power constraint $P_{s}$:
$\text{tr}(Q_{s})\leq P_{s}$ (5)
In this case, $\mathbf{H}(\mathcal{S})$ has dimension $|\mathcal{S}|\times m$.
When the primary is a MAC channel, each primary user transmits with power
$\rho_{p}$ without user cooperation:
$Q_{p}=\rho_{p}\,I_{N\times N}$ (6)
Furthermore, each receive antenna at the primary base station can tolerate
interference with power $\Gamma$ from the secondary system,333If each primary
antenna or user tolerates a different interference power, the results of this
paper still hold, as seen later. that is
$\big{[}\mathbf{G}_{p}\,Q_{s}\,\mathbf{G}_{p}^{{\dagger}}\big{]}_{\ell,\ell}\leq\Gamma$
(7)
for $\ell=1,\cdots,M$, where $\mathbf{G}_{p}$ represents the cross channel
coefficient matrix from the secondary base station (or active users) to the
primary base station.
When the primary is a broadcast channel, the power constraint at the primary
base station is $\text{tr}(Q_{p})\leq P_{p}$. For simplicity, we assume444The
asymptotic results remain the same, even if we allow $Q_{p}$ to be an
arbitrary covariance matrix.
$Q_{p}=\frac{P_{p}}{M}\,I_{M\times M}$ (8)
Furthermore, each primary user tolerates interference with power $\Gamma$:
$\big{[}\mathbf{G}_{p}\,Q_{s}\,\mathbf{G}_{p}^{{\dagger}}\big{]}_{\ell,\ell}\leq\Gamma$
(9)
for $\ell=1,\cdots,N$, where $\mathbf{G}_{p}$ is the cross channel coefficient
matrix from the secondary base station (or active users) to the primary users.
## III Cognitive MAC Channel
Consider a MAC secondary in the presence of either a broadcast or MAC primary.
We wish to find how much throughput is available to the secondary subject to
rigid constraints on the secondary-on-primary interference. We first construct
a transmission strategy and find the corresponding (achievable) average
throughput. Then, we develop upper bounds that are tight with respect to the
throughput achieved.
The framework for the transmission strategy is as follows: For each
transmission block, the secondary base station determines an active user set
$\mathcal{S}$ as well as transmit power for all active users $Q_{s}$. For each
transmission, from (1), the sum rate (throughput) of the secondary system is:
$\displaystyle R_{mac}$
$\displaystyle=\log\text{det}\bigg{(}I+\mathbf{H}(\mathcal{S})Q_{s}\mathbf{H}^{{\dagger}}(\mathcal{S})+\mathbf{G}_{s}Q_{p}\mathbf{G}_{s}^{{\dagger}}\bigg{)}-\log\text{det}\bigg{(}I+\mathbf{G}_{s}Q_{p}\mathbf{G}_{s}^{{\dagger}}\bigg{)}$
(10)
subject to the interference constraints (9) and (7) for the primary broadcast
and MAC channel respectively.
The secondary average throughput is given by
$\mathcal{R}_{mac}=\mathbb{E}[R_{mac}]$ (11)
For the development of upper bounds, we assume the secondary base station
knows all the channels. This is a genie-like argument that is used solely for
development of upper bounds. For the achievable scheme, the requirement is
more modest and is outlined after the description of the achievable scheme
(see Remark 1).
### III-A Achievable Scheme
The objective is to choose $\mathcal{S}$ and $Q_{s}$, i.e., the secondary
active transmitters and their power, such that secondary throughput is
maximized subject to interference constraints on the primary.
The choice of $\mathcal{S}$ and $Q_{s}$ is coupled through the interference
constraints: either more secondary users can transmit with smaller power, or
fewer of them with higher power. We focus on a simple power policy: All active
secondary users transmit with the maximum allowed power $\rho_{s}$. Hence,
given an active user set $\mathcal{S}$, we have
$Q_{s}=\rho_{s}I_{|\mathcal{S}|\times|\mathcal{S}|}$ (12)
It will be shown that the on-off transmission (without any further power
adaptation) suffices to (asymptotically) achieve the maximum average
throughput. Furthermore, its simplicity facilitates analysis.
Recall that each primary user can tolerate interference with power $\Gamma$.
The interference on a primary user is guaranteed to be below this level if
$k_{s}$ secondary users are active, each causing interference no more than
$\alpha=\frac{\Gamma}{k_{s}}$. This bound allows us to honor the interference
constraints on the primary while decoupling the action of different secondary
users. Based on this observation, we construct a user selection rule as
follows. First, we define an eligible secondary user set that disqualifies
users that cause too much interference on the primary:
$\mathcal{A}=\left\\{\begin{array}[c]{ll}\big{\\{}i:\rho_{s}\big{|}[\mathbf{G}_{p}]_{ji}\big{|}^{2}<\alpha,\
\text{for}\ j=1,\cdots,N\big{\\}}&\mbox{\text{primary broadcast}}\\\
\big{\\{}i:\rho_{s}\big{|}[\mathbf{G}_{p}]_{ji}\big{|}^{2}<\alpha,\
\text{for}\ j=1,\cdots,M\big{\\}}&\mbox{\text{primary MAC}}\end{array}\right.$
(13)
where $[\mathbf{G}_{p}]_{ji}$ is the channel coefficient from the secondary
user $i$ to the primary user (antenna) $j$, and $\alpha$ is a pre-designed
interference quota. A secondary user is eligible if its interference on each
primary user (antenna) is less than $\alpha$. Now, to satisfy the interference
bound, we limit the number of secondary transmitters to no more than $k_{s}$,
where
$k_{s}=\frac{\Gamma}{\alpha}$ (14)
If $|\mathcal{A}|\leq k_{s}$, then all eligible users can transmit. If
$|\mathcal{A}|>k_{s}$, then $k_{s}$ users will be chosen randomly from among
the eligible users to transmit.555Naturally the number of active users must be
an integer, i.e., $\lfloor k_{s}\rfloor$. We do not carry the floor operation
in the following developments for simplicity, noting that due to the
asymptotic nature of the analysis, the floor operation has no effect on the
final results. The number of eligible users, $|\mathcal{A}|$, is a random
variable; the number of active users is
$|\mathcal{S}|=\min\big{(}k_{s},|\mathcal{A}|\big{)}$ (15)
The transmission of $|\mathcal{S}|$ eligible users induces interference no
more than $\Gamma$ on any primary user or antenna. Notice that the manner of
user selection guarantees that the channel coefficients in
$\mathbf{H}(\mathcal{S})$ remain independent and distributed as
$\mathcal{CN}(0,1)$.
Now we want to design an interference quota $\alpha$ to maximize the secondary
average throughput. Neither very small nor very large values of $\alpha$ are
useful within our framework: If $\alpha$ is very small, for most transmissions
few (if any) secondary users will be eligible, thus the secondary throughput
will be small. If $\alpha>\Gamma$, any transmitting user might violate the
interference constraint, so the secondary must shut down (equivalently, we
have $k_{s}<1$). The value of individual interference constraint $\alpha$, or
equivalently $k_{s}$, must be set somewhere between these extremes.
Clearly, a desirable outcome would be to allow exactly the number of users
that are indeed eligible for transmission, i.e., $k_{s}\approx|\mathcal{A}|$.
But one cannot guarantee this in advance, because $|\mathcal{A}|$ is a random
variable. Motivated by this general insight, we choose $\alpha$ such that
$k_{s}=\mathbb{E}[|\mathcal{A}|]$ (16)
In Section III-C, we will verify that this choice of $\alpha$ is enough to
asymptotically achieve the maximum throughput.
###### Remark 1
The above scheme does not require the secondary users to have full channel
knowledge. Each secondary user can compare its own cross channel gains with a
pre-defined interference quota $\alpha$, and then decide its eligibility.
After this, each eligible user can inform the secondary base station via
$1$-bit, so that the secondary base station can determine $\mathcal{A}$
without knowing the cross channels from the secondary users to the primary
system. The secondary channels $\mathbf{H}(\mathcal{S})$ and the cross
channels $\mathbf{G}_{s}$ can be estimated at the secondary base station.
Therefore, this scheme can be implemented with little exchange of channel
knowledge.
### III-B Throughput Calculation
#### III-B1 Secondary MAC with Primary Broadcast
The primary base station transmits to $N$ primary users, where each user
tolerates interference with power $\Gamma$. Notice that in (13),
$[\mathbf{G}_{p}]_{ji}$ is the channel coefficient from the secondary user $i$
to the primary user $j$ which is i.i.d. $\mathcal{CN}(0,1)$. Thus,
$\big{|}[\mathbf{G}_{p}]_{ji}\big{|}^{2}$ is i.i.d. exponential. Therefore,
$|\mathcal{A}|$ is binomially distributed with parameter $(n,p)$, where
$p=\big{(}1-e^{-\frac{\alpha}{\rho_{s}}}\big{)}^{N}$ (17)
For small $\frac{\alpha}{\rho_{s}}$, we have
$p\approx\bigg{(}\frac{\alpha}{\rho_{s}}\bigg{)}^{N}$ (18)
From (16), the interference quota $\alpha$ is chosen such that
$k_{s}=np\approx n\bigg{(}\frac{\alpha}{\rho_{s}}\bigg{)}^{N}$ (19)
Substitute $\alpha=\frac{\Gamma}{k_{s}}$ into the above equation, and denote
the associated solution for $k_{s}$ as $\bar{k}_{s}$:
$\bar{k}_{s}=\bigg{(}\frac{\Gamma}{\rho_{s}}\bigg{)}^{\frac{N}{N+1}}(n)^{\frac{1}{N+1}}$
(20)
Thus, we can see $\Theta(n^{\frac{1}{N+1}})$ secondary users are allowed to
transmit, and the interference quota is on the order of
$\Theta(n^{-\frac{1}{N+1}})$. With the above choice of interference quota, or
the number of allowable active users, we state one of the main results of this
paper as follows.
###### Theorem 1
Consider a secondary MAC with a $m$-antenna base station and $n$ users each
with power constraint $\rho_{s}$. The secondary MAC operates in the presence
of a primary broadcast channel transmitting with power $P_{p}$ to $N$ users
each with interference tolerance $\Gamma$. The secondary average throughput
satisfies:
$\displaystyle\mathcal{R}_{mac}$ $\displaystyle\geq\frac{m}{N+1}\log
n+\frac{1}{N+1}\log\big{(}\rho_{s}\Gamma^{N}\big{)}-m\log(1+P_{p})+O\big{(}n^{-\frac{1}{N+1}}\log
n\big{)}$ (21) $\displaystyle\mathcal{R}_{mac}$
$\displaystyle\leq\frac{m}{N+1}\log
n+\frac{1}{N+1}\log\big{(}\rho_{s}\Gamma^{N}\big{)}-\mathcal{R}_{I}+O\big{(}n^{-\frac{1}{N+1}}\big{)}$
(22)
with
$\mathcal{R}_{I}=m_{\mathsf{min}}\log\bigg{(}1+\frac{P_{p}}{M}\exp\bigg{(}\frac{1}{m_{\mathsf{min}}}\sum_{j=1}^{m_{\mathsf{min}}}\sum_{i=1}^{m_{\mathsf{max}}-j}\frac{1}{i}-\gamma\bigg{)}\bigg{)}$
(23)
where $m_{\mathsf{min}}=\min(m,M)$ and $m_{\mathsf{max}}=\max(m,M)$. This
throughput is achieved under the threshold-based user selection with the
choice of $\bar{k}_{s}$ given by (20).
Proof: See Appendix A. $\,\Box$
###### Remark 2
The essence of the above result is that the secondary average throughput grows
as $\frac{m}{N+1}\log n+O(1)$, i.e., inversely proportional to the number of
primary users. A noteworthy special case is when the primary base station
chooses to transmit to a number of users equal to the number of its transmit
antennas ($N=M$), a strategy which is known to be near-optimum in terms of
sum-rate [15]. Under this condition:
$\mathcal{R}_{mac}=\frac{m}{M+1}\log n+O(1)$
Therefore, we have
$\lim_{n\rightarrow\infty}\frac{\mathcal{R}_{mac}}{\mathcal{R}_{mac,w/o}^{opt}}=\frac{1}{M+1}$
(24)
where $\mathcal{R}_{mac,w/o}^{opt}$ is the maximum average throughput of the
secondary MAC in the absence of the primary system. This ratio shows that the
compliance penalty of the secondary MAC system and its relationship with the
characteristics of the primary network.
###### Remark 3
The results in Theorem 1 can be directly extended to a scenario where each
primary user tolerates a different level of interference. As long as all
primary users allow non-zero interference (no matter how small), we can let
$\Gamma$ be the minimum allowable interference, and the theorem still holds.
So far we have analyzed the effect of small but constant primary interference
constraints, and shown that the secondary throughput improves with increasing
the number of secondary users. However, the flexibility provided by the
increasing number of secondary users can be exploited not only to increase
secondary throughput, but also to reduce the primary interference. In fact, it
is possible to simultaneously suppress the interference on the primary down to
zero while increasing the secondary throughput proportional to $\log n$. The
following corollary makes this idea precise:
###### Corollary 1
Assuming the interference on each primary user is bounded as $\Theta(n^{-q})$,
the average secondary throughput satisfies
$\mathcal{R}_{mac}=\frac{m-qN}{N+1}\log n+O(1)$ (25)
where $0<q<\frac{m}{N}$.
Proof: Because the proof of Theorem 1 holds for $\Gamma=\Theta(n^{-q})$, the
corollary follows by substituting $\Gamma=\Theta(n^{-q})$ into the lower and
upper bounds given by Theorem 1. $\,\Box$
###### Remark 4
The corollary above explores a tradeoff where primary interference is made to
decrease polynomially, i.e., proportional to $n^{-q}$. We saw that this leads
to a secondary sum rate that decreases linearly in $q$. If we reduce the
primary interference more slowly, i.e., decreasing as $\Theta(\frac{1}{\log
n})$, the growth rate of secondary sum-rate will behave as though the primary
interference constraint is fixed. Conversely, if we try to suppress the
primary interference faster than $\Theta(n^{-q})$, the secondary throughput
will asymptotically remain stagnant or will go to zero.
#### III-B2 Secondary MAC with a Primary MAC
Recall that each antenna at the primary base station allows interference with
power $\Gamma$. By regarding each antenna of the primary base station as a
virtual user, we can re-use most of the analysis that was developed in the
previous section. Thus, the steps leading to Eq. (20) can be repeated to
obtain the number of allowable active secondary users:
$\bar{k}_{s}=\bigg{(}\frac{\Gamma}{\rho_{s}}\bigg{)}^{\frac{M}{M+1}}(n)^{\frac{1}{M+1}}$
(26)
With this allowable active users $\bar{k}_{s}$ and slight modifications, we
obtain a result that parallels Theorem 1.
###### Theorem 2
Consider a secondary MAC with a $m$-antenna base station and $n$ users each
with power constraint $\rho_{s}$. The secondary MAC operates in the presence
of a primary MAC channel where each user transmits with power $\rho_{p}$ to a
$M$-antenna base station with interference tolerance $\Gamma$ on each antenna.
The secondary average throughput satisfies:
$\displaystyle\mathcal{R}_{mac}$ $\displaystyle\geq\frac{m}{M+1}\log
n+\frac{1}{M+1}\log\big{(}\rho_{s}\Gamma^{M}\big{)}-m\log(1+\rho_{p}N)+O\big{(}n^{-\frac{1}{M+1}}\log
n\big{)}$ (27) $\displaystyle\mathcal{R}_{mac}$
$\displaystyle\leq\frac{m}{M+1}\log
n+\frac{1}{M+1}\log\big{(}\rho_{s}\Gamma^{M}\big{)}-\mathcal{R}_{I}+O\big{(}n^{-\frac{1}{M+1}}\big{)}$
(28)
with
$\mathcal{R}_{I}=m_{\mathsf{min}}\log\bigg{(}1+\rho_{p}\exp\bigg{(}\frac{1}{m_{\mathsf{min}}}\sum_{j=1}^{m_{\mathsf{min}}}\sum_{i=1}^{m_{\mathsf{max}}-j}\frac{1}{i}-\gamma\bigg{)}\bigg{)}$
(29)
where $m_{\mathsf{min}}=\min(m,N)$ and $m_{\mathsf{max}}=\max(m,N)$. This
throughput is achieved under the threshold-based user selection with the
choice of $\bar{k}_{s}$ given by (26).
A tradeoff exists between the primary interference reduction and the secondary
throughput enhancement, which is stated by the following corollary. All the
remarks made after Corollary 1 are applicable here.
###### Corollary 2
Assuming the interference on each antenna of the primary base station is
bounded as $\Theta(n^{-q})$, the average secondary throughput satisfies
$\mathcal{R}_{mac}=\frac{m-qM}{M+1}\log n+O(1)$ (30)
where $0<q<\frac{m}{M}$.
### III-C Upper Bounds for Secondary Throughput
So far we have seen achievable rates of a cognitive MAC channel in the
presence of either a primary broadcast or MAC. We now develop corresponding
upper bounds.
###### Theorem 3
Consider a secondary MAC with a $m$-antenna base station and $n$ users. The
maximum average throughput of the secondary, $\mathcal{R}_{mac}^{opt}$,
satisfies
$\mathcal{R}_{mac}^{opt}\leq\frac{m}{N+1}\log n+O(\log\log n)$ (31)
in the presence of a primary broadcast channel transmitting to $N$ users.
Similarly, $\mathcal{R}_{mac}^{opt}$ satisfies
$\mathcal{R}_{mac}^{opt}\leq\frac{m}{M+1}\log n+O(\log\log n)$ (32)
in the presence of a primary MAC, where each user transmits to a $M$-antenna
base station.
Proof: See Appendix B. $\,\Box$
###### Remark 5
By comparing the upper bounds with the achievable rates obtained by the
thresholding strategy, we see that the achievable rates are at most
$O(\log\log n)$ away from the upper bounds, a difference which is negligible
relative to the dominant term $\Theta(\log n)$. Thus, the growth of the
maximum average throughput of a cognitive MAC is $\frac{m}{N+1}\log n$ in the
presence of the primary broadcast channel, and $\frac{m}{M+1}\log n$ in the
presence of the primary MAC channel. Both the achievable rates and the upper
bounds show that the average cognitive sum-rate is inversely proportional to
the number of primary-imposed constraints, asymptotically.
### III-D Discussion
Recall that our method determines eligible cognitive MAC users based on their
cross channel gains. To satisfy the interference constraints, our selection
rule then allows $\Theta(n^{\frac{1}{N+1}})$, or $\Theta(n^{\frac{1}{M+1}})$,
of these users to be active simultaneously, in the presence of either the
primary broadcast or MAC. If there are more eligible users than the allowed
number, we choose from among the eligible users randomly. In this process, the
forward channel gain of the cognitive users does not come into play, and still
an optimal growth rate is achieved. This can be intuitively explained as
follows. The total received signal power at the cognitive base station grows
linearly with the number of active users, and the total received signal power
determines the sum rate. On the other hand, selecting good cognitive users
according to their secondary channel strengths can only offer logarithmic
power gains (with respect to $n$) [10], which is negligible compared to the
linear gains due to increasing the number of active users. Therefore the cross
channel gains are more important in this case.666In a somewhat different
context, the work of Jamal et al. [13] also indicates that cross channels can
be more important than the forward channels. Note that we do not imply that
knowledge of the cognitive forward channel is useless; our conclusion only
says that once the cross channels are taken into account, the asymptotic
growth of the secondary throughput cannot be improved by any use of the
cognitive forward channel.
Although we have allowed the base stations to have multiple antennas, so far
the users have been assumed to have only one antenna. We now consider a
generalization to the case where all users have multiple antennas. Consider a
secondary MAC in the presence of a primary broadcast, where each primary and
secondary user have $t_{p}$ and $t_{s}$ antennas respectively. We apply a
separate interference constraint on each antenna of each primary user, which
guarantees the satisfaction of the overall interference constraint on any
primary user. On each of the $t_{s}$-antenna secondary users, we shall
allocate $t_{s}-1$ degrees of freedom for zero-forcing and only one degree of
freedom for cognitive transmission. Using this strategy, we can ensure that
$t_{s}-1$ of the receive antennas on the primary are exempt from interference.
Thus, the total number of interference constraints will reduce from $t_{p}N$
to $t_{p}N+1-t_{s}$. By using an analysis similar to the development of
Theorem 1, one can show that the growth rate $\frac{m\log
n}{\max(1,\,t_{p}N+2-t_{s})}$ is achievable. For the converse, the situation
is more complicated, because here the correlation among the antennas of the
secondary users must be accounted for. Nevertheless, in some cases it is
possible to show without much difficulty that the above achieved throughput is
indeed asymptotically optimal. For example, in the presence of the primary
MAC, if $t_{s}>M$, the secondary MAC channel can have a throughput that grows
as $m\log n$ by letting each active secondary user completely eliminate the
interference on the primary. Similarly, in the presence of a primary broadcast
channel, if $t_{s}>t_{p}N$, the secondary MAC channel can also have a
throughput that grows as $m\log n$. The achieved growth rate is optimal
because it coincides with the the growth rate of
$\mathcal{R}_{mac,w/o}^{opt}$, which is always an upper bound.
## IV Cognitive Broadcast Channel
### IV-A Achievable Scheme
We consider a random beam-forming technique where the secondary base station
opportunistically transmits to $m$ secondary users simultaneously [16].
Specifically, the secondary base station constructs $m$ orthonormal beams,
denoted by $\\{\mathbf{\phi}_{j}\\}_{j=1}^{m}$, and assigns each beam to a
secondary user. Then, the secondary base station broadcasts to $m$ selected
users. The selection of users and beam assignment will be addressed shortly.
Considering an equal power allocation among $m$ users, the transmitted signal
from the secondary base station is given by:
$\mathbf{x}_{s}=\sum_{j=1}^{m}\sqrt{\frac{P}{m}}\;\mathbf{\phi}_{j}\;x_{j}$
(33)
where $\mathbf{\phi}_{j}$ is the beam-forming vector $j$ with dimension
$m\times 1$, $x_{j}$ is the signal transmitted along the beam $j$, and $P$ is
the total transmit power. In this case, we have
$Q_{s}=\frac{P}{m}I_{m\times m}$ (34)
Notice that $P$ is subject to the power constraint $P_{s}$ as well as a set of
interference constraints imposed by the primary. Thus, the value of $P$
depends on the cross channels from the secondary base station to the primary
system.
Assuming the beam $j$ is assigned to user $i$. From (1) and (33), the received
signal at the secondary user $i$ is given by
$y_{i}=\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}x_{j}+\sum_{k\neq
j}\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{k}x_{k}+\mathbf{g}_{s,i}^{{\dagger}}\mathbf{x}_{p}+w_{i}$
(35)
where $\mathbf{h}^{{\dagger}}_{i}$ is the $1\times m$ vector of channel
coefficient from the secondary base station to the secondary user $i$, and
$\mathbf{g}_{s,i}^{{\dagger}}$ is the $1\times M$ (or $1\times N$) vector of
channel coefficients from the primary base station (or users) to the secondary
user $i$. The received signal-to-noise-plus-interference-ratio (SINR) at the
secondary user $i$ (with respect to beam $j$) is
$\mathsf{SINR}_{i,j}=\frac{\frac{P}{m}|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}|^{2}}{1+\frac{P}{m}\sum_{k\neq
j}|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{k}|^{2}+\mathbf{g}^{{\dagger}}_{s,i}\,Q_{p}\,\mathbf{g}_{s,i}}$
(36)
The random beam technique assigns each beam to the secondary user that results
in the highest SINR. Because the probability of more than two beams being
assigned to the same secondary user is negligible, we have [16]
$\displaystyle\mathcal{R}_{bc}$
$\displaystyle\approx\mathbb{E}\bigg{[}\sum_{j=1}^{m}\log\big{(}1+\max_{1\leq
i\leq n}\mathsf{SINR}_{i,j}\big{)}\bigg{]}$ (37)
$\displaystyle=m\mathbb{E}\bigg{[}\log\big{(}1+\max_{1\leq i\leq
n}\mathsf{SINR}_{i,j}\big{)}\bigg{]}$ (38)
The above analysis holds in the presence of either the primary broadcast or
MAC channel; the only difference is the constraints on $P$ and $Q_{p}$. Since
the SINR is symmetric across all beams, the subscript $j$ will be omitted in
the following analysis.
###### Remark 6
We briefly address the issue of channel state information. All users are
assumed to have receiver side channel state information. On the transmit side,
the secondary base station does not need to have full channel knowledge; only
the SINR is needed. Each secondary user can estimate its own SINR with respect
to each beam, and feed it back to the secondary base station [16]. Based on
collected SINR, the secondary base station performs user selection. The
secondary base station needs to know $\mathbf{G}_{p}$ to adjust $P$ such that
the interference constraints on the primary are satisfied.
### IV-B Throughput Calculation
#### IV-B1 Secondary Broadcast with Primary Broadcast
The secondary system has to comply with the constraints on $N$ primary users.
To maximize the throughput, the secondary base station transmits at the
maximum allowable power. From (9) and (34), we have
$P=\min\big{(}\frac{m\Gamma}{|\mathbf{g}^{{\dagger}}_{p,1}|^{2}},\cdots,\frac{m\Gamma}{|\mathbf{g}^{{\dagger}}_{p,N}|^{2}},P_{s}\big{)}$
(39)
where $\mathbf{g}^{{\dagger}}_{p,\ell}$ is the row $\ell$ of $\mathbf{G}_{p}$.
Then, we substitute $Q_{p}$ given by (8) into (36), and obtain the SINR at the
secondary user $i$ with respect to the beam $j$:
$\mathsf{SINR}_{i}=\frac{|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}|^{2}}{\frac{m}{P}+\sum_{k\neq
j}|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{k}|^{2}+\frac{mP_{p}}{MP}|\mathbf{g}_{s,i}|^{2}}$
(40)
Our analysis of $\max_{i}\mathsf{SINR}_{i}$, which is required to evaluate the
throughput in Eq. (38), does not follow [16] because the denominator involves
a sum of two Gamma distributions with different scale parameters: $\sum_{k\neq
j}|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{k}|^{2}$ has Gamma$(m-1,1)$ and
$\frac{mP_{p}}{MP}|\mathbf{g}_{s,i}|^{2}$ has Gamma$(M,\frac{mP_{p}}{MP})$.
Fortunately, lower and upper bounds can be leveraged to simplify the analysis.
We define:
$\theta=\frac{mP_{p}}{MP}$ (41)
We consider the case when $\frac{mP_{p}}{MP_{s}}\geq 1$. The techniques can
then be generalized to the case of $\frac{mP_{p}}{MP_{s}}<1$.777When
$\frac{mP_{p}}{MP_{s}}<1$, one can define $\theta=\max(\frac{mP_{p}}{MP},1)$.
Then, we can use Bayesian expansion via conditioning on
$\\{P<\frac{mP_{p}}{M}\\}$ and its complement, where both conditional terms
can be shown to have the same growth rate. When $\frac{mP_{p}}{MP_{s}}\geq 1$,
we have $\theta\geq 1$ for all $P$. We define:
$L_{i}=\frac{|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}|^{2}}{\frac{m}{P}+\theta\big{(}\sum_{k\neq
j}|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{k}|^{2}+|\mathbf{g}_{s,i}|^{2}\big{)}}$
(42)
and
$U_{i}=\frac{|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}|^{2}}{\frac{m}{P}+\theta|\mathbf{g}_{s,i}|^{2}}$
(43)
where $L_{i}$ and $U_{i}$ are random variables that depend on channel
realizations. Conditioned on $P$, the denominators of $L_{i}$ and $U_{i}$ have
Gamma distributions, which simplifies the analysis.
For $1\leq i\leq n$, we have
$L_{i}\leq\mathsf{SINR}_{i}\leq U_{i}$ (44)
Hence,
$L_{max}\leq\max_{1\leq i\leq n}\mathsf{SINR}_{i}\leq U_{max}$ (45)
where $L_{max}=\max_{i}L_{i}$ and $U_{max}=\max_{i}U_{i}$. Therefore for any
$x$, we have
$\mathbb{P}(L_{max}>x)\leq\mathbb{P}(\max_{1\leq i\leq
n}\mathsf{SINR}_{i}>x)\leq\mathbb{P}(U_{max}>x)$ (46)
which implies [17] that $\max_{i}\mathsf{SINR}_{i}$ is stochastically greater
than $L_{max}$, but stochastically smaller than $U_{max}$. We now use the
following fact about stochastic ordering:
###### Lemma 1 ([17])
If random variable $X$ is stochastically smaller than $Y$ and $h(\cdot)$ is an
increasing function, assuming $h(X)$ and $h(Y)$ are measurable according to
their distributions:
$\mathbb{E}[h(X)]\leq\mathbb{E}[h(Y)]$ (47)
Based on the above lemma, the secondary average throughput is bounded as
follows:
$m\mathbb{E}\big{[}\log(1+L_{max})\big{]}\leq\mathcal{R}_{bc}\leq
m\mathbb{E}\big{[}\log(1+U_{max})\big{]}$ (48)
We study the lower and upper bounds given by (48), instead of directly
analyzing $\mathcal{R}_{bc}$. Some useful properties of $L_{max}$ and
$U_{max}$ are as follows.
###### Lemma 2
Conditioned on $P=\rho$,
$\displaystyle\mathbb{P}\bigg{(}L_{max}\geq b_{n}-\frac{\rho}{m}\log\log
n\,\bigg{|}\,P=\rho\bigg{)}=1-\Theta\bigg{(}\frac{1}{n}\bigg{)}$ (49)
$\displaystyle\mathbb{P}\bigg{(}U_{max}<d_{n}+\frac{\rho}{m}\log\log
n\,\bigg{|}\,P=\rho\bigg{)}=1-\Theta\bigg{(}\frac{1}{\log n}\bigg{)}$ (50)
$\displaystyle\mathbb{E}\bigg{[}U_{max}\,\bigg{|}\,U_{max}>d_{n}+\frac{\rho}{m}\log\log
n,P=\rho\bigg{]}<O(n\log n)$ (51)
where $b_{n}=\frac{\rho}{m}\log n-\frac{\rho(m+M-1)}{m}\log\log
n+O\big{(}\log\log\log n\big{)}$ and $d_{n}=\frac{\rho}{m}\log n-\frac{\rho
M}{m}\log\log n+O\big{(}\log\log\log n\big{)}$.
Proof: See Appendix C. $\,\Box$
Based on the above two lemmas, we obtain the following results for the
secondary throughput:
###### Theorem 4
Consider a secondary broadcast channel with $n$ users and a $m$-antenna base
station with power constraint $P_{s}$. The secondary broadcast operates in the
presence of a primary broadcast channel transmitting with power $P_{p}$ to $N$
users each with interference tolerance $\Gamma$. The secondary average
throughput satisfies:
$\displaystyle\mathcal{R}_{bc}$ $\displaystyle>m\log\big{(}\Gamma\log
n\big{)}-m\log\big{(}\tilde{\mu}_{1}+\frac{m\Gamma}{P_{s}}\big{)}+O\big{(}\frac{\log\log
n}{\log n}\big{)}$ $\displaystyle\mathcal{R}_{bc}$
$\displaystyle<m\log(\Gamma\log n)-m\log\tilde{\mu}_{2}+O(1)$
where $\tilde{\mu}_{1}=\mathbb{E}[\max_{1\leq i\leq
N}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}]$ and
$\tilde{\mu}_{2}=\big{(}\mathbb{E}\big{[}1/\max_{1\leq i\leq
N}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}\big{]}\big{)}^{-1}$.
Proof: See Appendix D. $\,\Box$
###### Remark 7
The result above states that $\mathcal{R}_{bc}=m\log\log n+O(1)$, thus
$\lim_{n\rightarrow\infty}\frac{\mathcal{R}_{bc}}{\mathcal{R}_{bc,w/o}^{opt}}=1$
(52)
where $\mathcal{R}_{bc,w/o}^{opt}$ is the maximum average throughput of the
secondary broadcast channel in the absence of the primary system. Therefore,
the achieved average throughput is asymptotically optimal, because we always
have $\mathcal{R}_{bc}\leq\mathcal{R}_{bc,w/o}^{opt}$. Thus, we have a
positive result: The growth rate of the secondary average throughput is
unaffected by the constraints and interference imposed by the primary, as long
as each primary user tolerates some small but fixed interference.
The above results naturally lead to the question: How small can we make the
interference on the primary, while still having a secondary average throughput
that grows as $\Theta(\log\log n)$. We find that $\Gamma$, the interference on
each primary user, can asymptotically go to zero, as shown by the next
corollary.
###### Corollary 3
Assuming the interference on each primary user is bounded as
$\Theta\big{(}(\log n)^{-q}\big{)}$, the average secondary throughput
satisfies:
$\mathcal{R}_{bc}=(1-q)m\log\log n+O(1)$ (53)
where $0<q<1$.
###### Remark 8
Reducing the interference on the order of $\Theta\big{(}(\log n)^{-q}\big{)}$
sheds lights on how fast the interference can be reduced on the primary, while
having a non-trivial secondary throughout. For $q>1$, it does not imply
$\mathcal{R}_{bc}$ is zero or negative; it only means that $\mathcal{R}_{bc}$
is on the order of $o(\log\log n)$. Slower interference reduction, e.g.
proportional to $\Theta\big{(}(\log\log n)^{-1}\big{)}$, will give maximal
asymptotic growth of secondary throughput, i.e., $m\log\log n$.
#### IV-B2 Secondary Broadcast with Primary MAC
The analysis of this case closely parallels the analysis of the primary
broadcast. The secondary transmit power is given by
$P=\min\big{(}\frac{m\Gamma}{|\mathbf{g}^{{\dagger}}_{p,1}|^{2}},\cdots,\frac{m\Gamma}{|\mathbf{g}^{{\dagger}}_{p,M}|^{2}},P_{s}\big{)}$
(54)
where $\mathbf{g}^{{\dagger}}_{p,\ell}$ is the row $\ell$ of $\mathbf{G}_{p}$.
The MAC primary system produces power $N\rho_{p}$ and has $M$ interference
constraints. From the viewpoint of the secondary, this is all the information
that is needed. Therefore the analysis of Theorem 4 can be essentially
repeated to obtain the following result.
###### Theorem 5
Consider a secondary broadcast channel with $n$ users and a $m$-antenna base
station with power constraint $P_{s}$. The secondary broadcast operates in the
presence of a primary MAC where each user transmits with power $\rho_{p}$ to a
$M$-antenna base station with interference tolerance $\Gamma$ on each antenna.
The secondary average throughput satisfies:
$\displaystyle\mathcal{R}_{bc}$ $\displaystyle>m\log\big{(}\Gamma\log
n\big{)}-m\log\big{(}\tilde{\mu}_{3}+\frac{m\Gamma}{P_{s}}\big{)}+O\big{(}\frac{\log\log
n}{\log n}\big{)}$ $\displaystyle\mathcal{R}_{bc}$
$\displaystyle<m\log(\Gamma\log n)-m\log\tilde{\mu}_{4}+O(1)$
where $\tilde{\mu}_{3}=\mathbb{E}[\max_{1\leq i\leq
M}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}]$ and
$\tilde{\mu}_{4}=\big{(}\mathbb{E}\big{[}1/\max_{1\leq i\leq
M}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}\big{]}\big{)}^{-1}$.
###### Remark 9
Theorem 4 and Theorem 5 can be extended to a scenario where each primary and
secondary user has multiple antennas. A straightforward way is to regard each
primary and secondary antenna as a virtual user. Using an analysis similar to
the single-antenna case, the secondary broadcast channel can be shown to
achieve a throughput scaling as $m\log\log n$ (thus optimal). The details are
straight forward and are therefore omitted for brevity.
Similar to Corollary 3, we can also obtain the tradeoff between the primary
interference reduction and the secondary throughput enhancement as follows.
All the remarks following Corollary 3 apply to the present case as well.
###### Corollary 4
Assuming the interference on each antenna of the primary base station is
bounded as $\Theta\big{(}(\log n)^{-q}\big{)}$, the average secondary
throughput satisfies:
$\mathcal{R}_{bc}=(1-q)m\log\log n+O(1)$ (55)
where $0<q<1$.
## V Numerical Results
In this section, we concentrate on numerical results in the presence of the
primary broadcast channel; the results in the presence of the primary MAC
channel are similar thus omitted. For all simulations, we consider:
$P_{p}=P_{s}=\rho_{s}=5$, the secondary base station has $m=4$ antennas, and
the primary base station has $M=2$ antennas and the number of primary users is
$N=2$.
Figure 3 illustrates the secondary average throughput given by Theorem 1. The
allowable interference power on each primary user is $\Gamma=2$. The slope of
the throughput curve is discontinuous at some points, because the allowable
number of active secondary users must be an integer $\lfloor k_{s}\rfloor$
(also see Eq.(19)). As mentioned earlier, the floor operation does not affect
the asymptotic results. Figure 4 presents the tradeoff between the tightness
of the primary constraints and the secondary throughput, as shown by Corollary
1. The interference power constraint $\Gamma$ is $2n^{-q}$ for $q=0.1$ and
$0.2$ respectively. As expected, for $q=0.2$ the interference on primary
decreases faster than $q=0.1$ and the secondary throughput increases more
slowly.
Figure 3: Secondary MAC: Throughput versus user number ($\Gamma=2$) Figure 4:
Secondary MAC: Throughput versus user number ($\Gamma=2n^{-q}$)
Figure 5 shows the secondary throughput versus the number of secondary users
in the presence of the primary broadcast channel (Theorem 4), where the
interference power is $\Gamma=2$. In Figure 6, we show the tradeoff between
the secondary throughput and the interference on the primary, as described in
Corollary 3. We set $\Gamma$ to decline as $2(\log n)^{-q}$, for $q=0.5$ and
$q=0.8$, respectively. Clearly, for $q=0.5$, the interference power decreases
faster than $q=0.8$, while the secondary throughput increases more slowly.
Figure 5: Secondary broadcast: Throughput versus user number ($\Gamma=2$)
Figure 6: Secondary broadcast: Throughput versus user number ( $\Gamma=2(\log
n)^{-q}$)
## VI Conclusion
In this paper, we study the performance limits of an underlay cognitive
network consisting of a multi-user and multi-antenna primary and secondary
systems. We find the average throughput limits of the secondary system as well
as the tradeoff between this throughput and the tightness of constraints
imposed by the primary system. Given a set of interference power constraints
on the primary, the maximum average throughput of the secondary MAC grows as
$\frac{m}{N+1}\log n$ (primary MAC), and $\frac{m}{M+1}\log n$ (primary
broadcast). These growth rates are attained by the simple threshold-based user
selection rule. Interestingly, the secondary system can force its interference
on the primary to zero while maintaining a growth rate of $\Theta(\log n)$.
For the secondary broadcast channel, the secondary average throughput can grow
as $m\log\log n$ in the presence of either the primary broadcast or MAC
channel. Hence, the growth rate of the throughput is unaffected by the
presence of the primary (thus optimal). Furthermore, the interference on the
primary can also be made to decline to zero, while maintaining the secondary
average throughput to grow as $\Theta(\log\log n)$.
## Appendix A Proof of Theorem 1
Proof: We rewrite (10) as
$R_{mac}=\log\text{det}\bigg{(}I+\mathbf{H}(\mathcal{S})Q_{s}\mathbf{H}^{{\dagger}}(\mathcal{S})\big{(}I+\mathbf{G}_{s}Q_{p}\mathbf{G}_{s}^{{\dagger}}\big{)}^{-1}\bigg{)}$
(56)
Because for any positive definite matrix $A$ and $B$, the function
$\log\text{det}(I+AB^{-1})$ is convex in $B$ [18, Lemma ii@.3], we have
$\displaystyle\mathcal{R}_{mac}$
$\displaystyle=\mathbb{E}_{\mathbf{H}}\big{[}\mathbb{E}_{\mathbf{G}_{s}}[R_{mac}\,|\,\mathbf{H}]\big{]}$
(57)
$\displaystyle>\mathbb{E}_{\mathbf{H}}\bigg{[}\log\text{det}\bigg{(}I+\mathbf{H}(\mathcal{S})Q_{s}\mathbf{H}^{{\dagger}}(\mathcal{S})\big{(}I+\mathbb{E}[\mathbf{G}_{s}Q_{p}\mathbf{G}_{s}^{{\dagger}}]\big{)}^{-1}\bigg{)}\bigg{]}$
(58)
$\displaystyle=\mathbb{E}_{\mathbf{H}}\bigg{[}\log\text{det}\bigg{(}I+\frac{\rho_{s}}{1+P_{p}}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})\bigg{)}\bigg{]}$
(59)
where (58) uses the Jensen inequality and the fact that
$\mathbf{H}(\mathcal{S})$ and $\mathbf{G}_{s}$ are independent. Substituting
$Q_{p}$ from (8) and noting that
$\mathbb{E}[\mathbf{G}_{s}\mathbf{G}_{s}^{{\dagger}}]=MI_{m\times m}$, we have
(59).
Now we bound the right hand side of (59). Recall that $|\mathcal{A}|$ and
$|\mathcal{S}|$ are the random number of eligible users and active users,
respectively. By the Chebychev inequality, for any $\epsilon>0$, we have
$\displaystyle\mathbb{P}\bigg{(}|\mathcal{A}|>(1-\epsilon)\bar{k}_{s}\bigg{)}$
$\displaystyle>1-\frac{1-p}{\epsilon^{2}np}$ (60)
$\displaystyle=1-O\big{(}\bar{k}_{s}^{-1}\big{)}$ (61)
where in the above we use the fact $\bar{k}_{s}=np$. Then, we expand (59)
based the event $\\{|\mathcal{A}|>(1-\epsilon)\bar{k}_{s}\\}$ and its
complement, and discard the non-negative term associated with its complement:
$\displaystyle\mathcal{R}_{mac}$
$\displaystyle>\mathbb{E}\bigg{[}\log\text{det}\bigg{(}I+\frac{\rho_{s}}{1+P_{p}}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})\bigg{)}\,\bigg{|}\,|\mathcal{A}|>(1-\epsilon)\bar{k}_{s}\bigg{]}\mathbb{P}\bigg{(}|\mathcal{A}|>(1-\epsilon)\bar{k}_{s}\bigg{)}$
(62)
$\displaystyle\geq\mathbb{E}\bigg{[}\log\text{det}\bigg{(}I+\frac{\rho_{s}}{1+P_{p}}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})\bigg{)}\,\bigg{|}\,|\mathcal{A}|=(1-\epsilon)\bar{k}_{s}\bigg{]}\bigg{(}1-O\big{(}\bar{k}_{s}^{-1}\big{)}\bigg{)}$
(63)
$\displaystyle=\mathbb{E}\bigg{[}\log\text{det}\bigg{(}I+\frac{\rho_{s}}{1+P_{p}}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})\bigg{)}\,\bigg{|}\,|\mathcal{S}|=(1-\epsilon)\bar{k}_{s}\bigg{]}\bigg{(}1-O\big{(}\bar{k}_{s}^{-1}\big{)}\bigg{)}$
(64)
where in the inequality (63), we apply the result in (61) and the fact that
the conditional expectation of the right hand side of (62) is non-decreasing
in $|\mathcal{A}|$. Since $|\mathcal{S}|=(1-\epsilon)\bar{k}_{s}$ in case of
$|\mathcal{A}|=(1-\epsilon)\bar{k}_{s}$, then we obtain (64) due to the
average throughput depending on $|\mathcal{A}|$ via the size of $\mathcal{S}$.
Recall that each entry of $\mathbf{H}(\mathcal{S})$ is i.i.d.
$\mathcal{CN}(0,1)$. Conditioned on $|\mathcal{S}|=(1-\epsilon)\bar{k}_{s}$,
$\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})$ is a Wishart
Matrix with degrees of freedom $(1-\epsilon)\bar{k}_{s}$, we have [19, Theorem
1]
$\displaystyle\mathcal{R}_{mac}$
$\displaystyle>\bigg{(}m\log\big{(}1+\frac{(1-\epsilon)\rho_{s}\bar{k}_{s}}{1+P_{p}}\big{)}+O\big{(}\bar{k}_{s}^{-1}\big{)}\bigg{)}\bigg{(}1-O\big{(}\bar{k}_{s}^{-1}\big{)}\bigg{)}$
(65)
$\displaystyle=m\log\big{(}1+\frac{(1-\epsilon)\rho_{s}\bar{k}_{s}}{1+P_{p}}\big{)}+O\big{(}\frac{\log\bar{k}_{s}}{\bar{k}_{s}}\big{)}$
(66)
$\displaystyle=m\log\rho_{s}\bar{k}_{s}+m\log(1-\epsilon)-m\log(1+P_{p})+O\big{(}\frac{\log\bar{k}_{s}}{\bar{k}_{s}}\big{)}$
(67)
Since the above inequality holds for any $\epsilon>0$, we have
$\mathcal{R}_{mac}\geq
m\log\rho_{s}\bar{k}_{s}-m\log(1+P_{p})+O\big{(}\frac{\log\bar{k}_{s}}{\bar{k}_{s}}\big{)}$
(68)
Now we find an upper bound for $\mathcal{R}_{mac}$. For convenience, we denote
$R_{mac,0}=\log\text{det}\bigg{(}I+\rho_{s}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})+\mathbf{G}_{s}\,Q_{p}\,\mathbf{G}_{s}^{{\dagger}}\bigg{)}$
(69)
and
$R_{I}=\log\text{det}\bigg{(}I+\mathbf{G}_{s}\,Q_{p}\,\mathbf{G}_{s}^{{\dagger}}\bigg{)}$
(70)
So the average throughput can be written as
$\displaystyle\mathcal{R}_{mac}$
$\displaystyle=\mathbb{E}\big{[}R_{mac,0}\big{]}-\mathbb{E}\big{[}R_{I}\big{]}$
(71)
Using the inequality $\text{det}(A)\leq\big{(}\text{tr}(A)/k)\big{)}^{k}$
[20], where $A$ is a $k\times k$ positive definite matrix, $R_{mac,0}$ is
bounded by
$R_{mac,0}\leq
m\log\bigg{(}1+\frac{1}{m}\text{tr}\bigg{(}\rho_{s}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})+\mathbf{G}_{s}\,Q_{p}\,\mathbf{G}_{s}^{{\dagger}}\bigg{)}\bigg{)}$
(72)
Therefore,
$\displaystyle\mathbb{E}[R_{mac,0}]$ $\displaystyle\leq
m\mathbb{E}\bigg{[}\log\bigg{(}1+\frac{1}{m}\text{tr}\bigg{(}\rho_{s}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})+\mathbf{G}_{s}\,Q_{p}\,\mathbf{G}_{s}^{{\dagger}}\bigg{)}\bigg{)}\bigg{]}$
(73) $\displaystyle\leq
m\log\bigg{(}1+\frac{\rho_{s}}{m}\mathbb{E}\big{[}\text{tr}\big{(}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})\big{)}\big{]}+\frac{1}{m}\mathbb{E}\big{[}\text{tr}\big{(}\mathbf{G}_{s}Q_{p}\mathbf{G}_{s}^{{\dagger}}\big{)}\big{]}\bigg{)}$
(74) $\displaystyle\leq m\log\big{(}1+\rho_{s}\bar{k}_{s}+P_{p}\big{)}$ (75)
where (74) uses the Jensen inequality. To obtain the inequality (75), we use
the facts that
$\mathbb{E}\big{[}\text{tr}\big{(}\mathbf{G}_{s}Q_{p}\mathbf{G}_{s}^{{\dagger}}\big{)}\big{]}=P_{p}$
by substituting $Q_{p}$ given by (8) as well as
$\mathbb{E}\big{[}\text{tr}\big{(}\mathbf{H}(\mathcal{S})\mathbf{H}^{{\dagger}}(\mathcal{S})\big{)}\big{]}\leq
m\bar{k}_{s}$ due to $|\mathcal{S}|\leq\bar{k}_{s}$.
Now we lower bound the second term in (71). From [21, Theorem 1], we have
$\displaystyle\mathbb{E}[R_{I}]$ $\displaystyle\geq
m_{\mathsf{min}}\log\bigg{(}1+\frac{P_{p}}{M}\exp\bigg{(}\frac{1}{m_{\mathsf{min}}}\sum_{j=1}^{m_{\mathsf{min}}}\sum_{i=1}^{m_{\mathsf{max}}-j}\frac{1}{i}-\gamma\bigg{)}\bigg{)}$
(76) $\displaystyle\stackrel{{\scriptstyle\Delta}}{{=}}\mathcal{R}_{I}$ (77)
where $m_{\mathsf{min}}=\min(m,M)$, $m_{\mathsf{max}}=\max(m,M)$ and $\gamma$
is the Euler’s constant. Notice that $\mathcal{R}_{I}$ is a finite constant
independent of $n$ and $\Gamma$.
Combining (75) and (77), we have
$\displaystyle\mathcal{R}_{mac}$ $\displaystyle\leq
m\log(1+\rho_{s}\bar{k}_{s}+P_{p})-\mathcal{R}_{I}$ (78)
Finally, substituting $\bar{k}_{s}$ given by (20) and noting that
$\bar{k}_{s}=\Theta(n^{\frac{1}{N+1}})$, we have
$\displaystyle\mathcal{R}_{mac}$ $\displaystyle\geq\frac{m}{N+1}\log
n+\frac{1}{N+1}\log\big{(}\rho_{s}\Gamma^{N}\big{)}-m\log(1+P_{p})+O\big{(}n^{-\frac{1}{N+1}}\log
n\big{)}$ (79) $\displaystyle\mathcal{R}_{mac}$
$\displaystyle\leq\frac{m}{N+1}\log
n+\frac{1}{N+1}\log\big{(}\rho_{s}\Gamma^{N}\big{)}-\mathcal{R}_{I}+O\big{(}n^{-\frac{1}{N+1}}\big{)}$
(80)
where we use the identity $\log(x+y)=\log x+\log(1+x/y)$ in the above
inequalities. This completes the proof. $\,\Box$
## Appendix B Proof of Theorem 3
Proof: We develop an upper bound for the secondary throughput in the presence
of the primary broadcast only; the development is similar in the presence of
the primary MAC and thus is omitted. We consider an arbitrary active user set
$\mathcal{S}$ and transmit covariance matrix given by (4), such that the
interference constraints on the primary are satisfied.
By removing the interference from the primary to the secondary, the secondary
throughput is enlarged. Then, using the inequality
$\text{det}(A)\leq\big{(}\text{tr}(A)/k\big{)}^{k}$ [20], where $A_{k\times
k}$ is a positive definite matrix, we have
$\displaystyle R_{mac}$ $\displaystyle\leq
m\log\bigg{(}1+\frac{1}{m}\text{tr}\big{(}\mathbf{H}(\mathcal{S})Q_{s}\mathbf{H}^{{\dagger}}(\mathcal{S})\big{)}\bigg{)}$
(81)
Let $\mathbf{h}_{i}$ be the $m\times 1$ vector of channel coefficients from
the secondary user $i$ ($i\in\mathcal{S}$) to the secondary base station,
corresponding to a certain column of $\mathbf{H}(\mathcal{S})$. Since $Q_{s}$
is diagonal, we have
$\displaystyle\text{tr}\big{(}\mathbf{H}(\mathcal{S})Q_{s}\mathbf{H}^{{\dagger}}(\mathcal{S})\big{)}$
$\displaystyle=\sum_{i\in\mathcal{S}}\rho_{i}\,\text{tr}\big{(}\mathbf{h}_{i}\mathbf{h}_{i}^{{\dagger}}\big{)}$
(82) $\displaystyle=\sum_{i\in\mathcal{S}}\rho_{i}\,|\mathbf{h}_{i}|^{2}$ (83)
$\displaystyle\leq\max_{i\in\mathcal{S}}|\mathbf{h}_{i}|^{2}\,\sum_{i\in\mathcal{S}}\rho_{i}$
(84) $\displaystyle\leq\max_{1\leq i\leq
n}|\mathbf{h}_{i}|^{2}\,\sum_{i\in\mathcal{S}}\rho_{i}$ (85)
where $\rho_{i}$ is the transmit power of the secondary user $i$. Let
$P_{sum}=\sum_{i\in\mathcal{S}}\rho_{i}$ (86)
and
$h_{max}=\max_{1\leq i\leq n}|\mathbf{h}_{i}|^{2}$ (87)
We can rewrite the right hand side of (81) as
$R_{mac}\leq m\log\big{(}1+\frac{1}{m}h_{max}P_{sum}\big{)}$ (88)
We first bound $P_{sum}$ and formulate an optimization as:
$\displaystyle\max_{\mathcal{S},\,\\{\rho_{i}\\}}\,P_{sum}$ $\displaystyle
s.t.:$ $\displaystyle\ \rho_{i}\leq\rho_{s}\ \text{for}\ i\in\mathcal{S},$
$\displaystyle\big{[}\mathbf{G}_{p}\,Q_{s}\,\mathbf{G}_{p}^{{\dagger}}\big{]}_{\ell,\ell}\leq\Gamma\
\text{for}\ 1\leq\ell\leq N$ (89)
which is a standard linear programming, and the solution is denoted by
$P_{sum}^{*}$. Then, $P_{sum}^{*}$ is the maximum total transmit power,
depending on the channel realizations for each transmission.
Subject to the interference constraints on the primary, the user selection and
power allocation are coupled, and a direct analysis is difficult. Instead, we
will find an upper bound for $P_{sum}^{*}$. Notice that the total interference
(on all primary users) caused by the secondary user $i$ is
$\rho_{i}|\mathbf{g}_{p,i}|^{2}$, where $\mathbf{g}_{p,i}$ is the vector of
channel coefficients from the secondary $i$ to all $N$ primary users. We relax
the set of individual interference constraints in (89) with a single sum
interference constraint:
$\displaystyle\sum_{i\in\mathcal{S}}\rho_{i}|\mathbf{g}_{p,i}|^{2}\leq
N\Gamma$ (90)
Notice that $\mathbf{g}_{p,i}$ corresponds to a certain column in
$\mathbf{G}_{p}$.
Order the cross channel gains $\\{|\mathbf{g}_{p,i}|^{2}\\}_{i=1}^{n}$ of all
the secondary users and denote the ordered cross channel gains by
$|\tilde{\mathbf{g}}_{p,1}|^{2}\leq|\tilde{\mathbf{g}}_{p,2}|^{2}\leq\cdots\leq|\tilde{\mathbf{g}}_{p,n}|^{2}$
(91)
Then, we further relax the sum interference constraint (90) by replacing
$\\{|\mathbf{g}_{p,i}|^{2}\\}_{i\in\mathcal{S}}$ with the first
$|\mathcal{S}|$ smallest cross channel gains
$\\{|\tilde{\mathbf{g}}_{p,i}|^{2}\\}_{i=1}^{|\mathcal{S}|}$. Thus, we have:
$\displaystyle\max_{\mathcal{S},\,\\{\rho_{i}\\}}\,P_{sum}$ s.t.:
$\displaystyle\sum_{i=1}^{|\mathcal{S}|}\rho_{i}|\tilde{\mathbf{g}}_{p,i}|^{2}\leq
N\Gamma$ $\displaystyle\rho_{i}\leq\rho_{s}\ \text{for}\ 1\leq
i\leq|\mathcal{S}|$ (92)
For any channel realizations, the solution for the above problem, denoted by
$P_{sum,1}^{*}$, is always greater than, or equal to $P_{sum}^{*}$. Notice
that $P_{sum,1}^{*}$ is also a random variable. Since
$\\{|\tilde{\mathbf{g}}_{p,i}|^{2}\\}$ is in non-decreasing in $i$, the set of
$\\{\rho_{i}\\}$ that achieves $P_{sum,1}^{*}$ satisfies
$\rho_{i}\geq\rho_{j}$, for $i\leq j$. In other words, we have
$\rho_{i}=\rho_{s}$, for $i=1$ to $|\mathcal{S}|-1$, and
$\rho_{i}\leq\rho_{s}$, for $i=|\mathcal{S}|$.
Let $S_{max}$ be the maximum value of $|\mathcal{S}|$ that satisfies the
constraint
$\rho_{s}\sum_{i=1}^{|\mathcal{S}|-1}|\tilde{\mathbf{g}}_{p,i}|^{2}\leq
N\Gamma$ (93)
We have
$P_{sum,1}^{*}\leq\rho_{s}S_{max}$ (94)
where in (94) we have an inequality, because the constraint (93) is relaxed by
discarding $\rho_{|\mathcal{S}|}$ compared to the interference constraint in
(92) .
Now, we focus on bounding $\rho_{s}S_{max}$. For any positive integer $k$, we
have
$\mathbb{P}\big{(}S_{max}<k\big{)}\geq\mathbb{P}\big{(}\sum_{i=1}^{k-1}|\tilde{\mathbf{g}}_{p,i}|^{2}>\frac{N\Gamma}{\rho_{s}}\big{)}$
(95)
which comes from the fact that the event of the right hand side implies the
event of the left hand side. Notice that
$\sum_{i=1}^{k-1}|\tilde{\mathbf{g}}_{p,i}|^{2}$ is a sum of least order
statistics out of $\\{|\mathbf{g}_{p,i}|^{2}\\}_{i=1}^{n}$ with i.i.d.
Gamma$(N,1)$ distributions. We apply some results in the development of [13,
Proposition 12], and obtain888For our case, $\frac{1}{\lambda}=\gamma=N$.
$\mathbb{P}\big{(}\sum_{i=1}^{f(n)-1}|\tilde{\mathbf{g}}_{p,i}|^{2}>\frac{N\Gamma}{\rho_{s}}\big{)}>1-O\big{(}\frac{1}{f(n)}\big{)}$
(96)
where $f(n)=c_{0}\,n^{\frac{1}{N+1}}$, and
$c_{0}=\big{(}\frac{\Gamma(N+1)}{(1-\epsilon)\rho_{s}}N^{-\frac{1}{N}}\big{)}^{\frac{N}{N+1}}$.
For large $N$ and small $\epsilon$,
$c_{0}\approx\frac{\Gamma}{\rho_{s}}(N+1)$.
Let $k=f(n)$ in (95) and combine with (96):
$\displaystyle\mathbb{P}\bigg{(}\rho_{s}S_{max}<\rho_{s}\,f(n)\bigg{)}>1-O\big{(}n^{-\frac{1}{N+1}}\big{)}$
(97)
After characterizing $\rho_{s}S_{max}$, now we return to $P_{sum}^{*}$. To
simplify notation, we denote
$\bar{p}_{sum}=\rho_{s}\,f(n)$ (98)
Because $P_{sum}^{*}\leq P_{sum,1}^{*}\leq\rho_{s}S_{max}$ for any channel
realizations, from (97), we have
$\displaystyle\mathbb{P}\bigg{(}P_{sum}^{*}\geq\bar{p}_{sum}\bigg{)}$
$\displaystyle=1-\mathbb{P}\bigg{(}P_{sum}^{*}<\bar{p}_{sum}\bigg{)}$
$\displaystyle<1-\mathbb{P}\bigg{(}\rho_{s}S_{max}<\bar{p}_{sum}\bigg{)}$
$\displaystyle<O\big{(}n^{-\frac{1}{N+1}}\big{)}$ (99)
Now, we complete the analysis of $P_{sum}^{*}$, and move to $h_{max}$. Because
$\\{|\mathbf{h}_{i}|^{2}\\}_{i=1}^{n}$ have i.i.d. Gamma$(m,1)$ distributions,
using the similar arguments developed in Lemma 2, we obtain
$\displaystyle\mathbb{P}\bigg{(}h_{max}>\zeta_{n}\bigg{)}=O\big{(}\frac{1}{\log
n}\big{)}$ (100)
$\displaystyle\mathbb{E}\big{[}h_{max}\,\big{|}\,h_{max}>\zeta_{n}\big{]}<O(n\log
n)$ (101)
where $\zeta_{n}$ is a deterministic sequence satisfying
$\zeta_{n}=\log n+m\log\log n+O(\log\log\log n)$ (102)
Now we are ready to develop the upper bound for the secondary throughput.
Since $P_{sum}\leq P_{sum}^{*}$, from (88), we have
$\displaystyle\mathcal{R}_{mac}$ $\displaystyle\leq
m\mathbb{E}_{\mathbf{H},P}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}P_{sum}^{*}\bigg{)}\bigg{]}$
(103) $\displaystyle\leq
m\mathbb{E}_{\mathbf{H},P}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}P_{sum}^{*}\bigg{)}\,\bigg{|}\,P_{sum}^{*}<\bar{p}_{sum}\bigg{]}\mathbb{P}\big{(}P_{sum}^{*}<\bar{p}_{sum}\big{)}$
$\displaystyle\quad+m\mathbb{E}_{\mathbf{H},P}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}P_{sum}^{*}\bigg{)}\,\bigg{|}\,P_{sum}^{*}\geq\bar{p}_{sum}\bigg{]}\mathbb{P}\big{(}P_{sum}^{*}\geq\bar{p}_{sum}\big{)}$
(104) $\displaystyle\leq
m\mathbb{E}_{\mathbf{H}}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}\bar{p}_{sum}\bigg{)}\bigg{]}\cdot
1$
$\displaystyle\quad+m\mathbb{E}_{\mathbf{H}}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}\rho_{s}n\bigg{)}\bigg{]}\cdot
O\big{(}n^{-\frac{1}{N+1}}\big{)}$ (105) $\displaystyle\leq
m\mathbb{E}_{\mathbf{H}}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}\bar{p}_{sum}\bigg{)}\,\bigg{|}\,h_{max}\leq\zeta_{n}\bigg{]}\mathbb{P}\big{(}h_{max}\leq\zeta_{n}\big{)}$
$\displaystyle\quad+m\mathbb{E}_{\mathbf{H}}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}\bar{p}_{sum}\bigg{)}\,\bigg{|}\,h_{max}>\zeta_{n}\bigg{]}\mathbb{P}\big{(}h_{max}>\zeta_{n}\big{)}$
$\displaystyle\quad+m\mathbb{E}_{\mathbf{H}}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}\rho_{s}n\bigg{)}\,\bigg{|}\,h_{max}\leq\zeta_{n}\bigg{]}\mathbb{P}\big{(}h_{max}\leq\zeta_{n}\big{)}O\big{(}n^{-\frac{1}{N+1}}\big{)}$
$\displaystyle\quad+m\mathbb{E}_{\mathbf{H}}\bigg{[}\log\bigg{(}1+\frac{1}{m}h_{max}\rho_{s}n\bigg{)}\,\bigg{|}\,h_{max}>\zeta_{n}\bigg{]}\mathbb{P}\big{(}h_{max}>\zeta_{n}\big{)}O\big{(}n^{-\frac{1}{N+1}}\big{)}$
(106) $\displaystyle\leq
m\log\bigg{(}1+\frac{1}{m}\,\zeta_{n}\,\bar{p}_{sum}\bigg{)}\cdot 1$
$\displaystyle\quad+m\log\bigg{(}1+\frac{\bar{p}_{sum}}{m}\,\mathbb{E}\big{[}h_{max}\,\big{|}\,h_{max}>\zeta_{n}\big{]}\bigg{)}\mathbb{P}\big{(}h_{max}>\zeta_{n}\big{)}$
$\displaystyle\quad+m\log\bigg{(}1+\frac{1}{m}\zeta_{n}\,\rho_{s}n\bigg{)}\cdot
1\cdot O\big{(}n^{-\frac{1}{N+1}}\big{)}$
$\displaystyle\quad+m\log\bigg{(}1+\frac{\rho_{s}n}{m}\mathbb{E}\big{[}h_{max}\,\big{|}\,h_{max}>\zeta_{n}\big{]}\bigg{)}\mathbb{P}\big{(}h_{max}>\zeta_{n}\big{)}O\big{(}n^{-\frac{1}{N+1}}\big{)}$
(107) $\displaystyle\leq
m\log\bigg{(}1+\frac{1}{m}\,\zeta_{n}\,\bar{p}_{sum}\bigg{)}$
$\displaystyle\quad+m\log\bigg{(}1+\frac{\bar{p}_{sum}}{m}O(n\log
n)\bigg{)}O(\frac{1}{\log n})$
$\displaystyle\quad+m\log\bigg{(}1+\frac{1}{m}\zeta_{n}\rho_{s}n\bigg{)}\,O\big{(}n^{-\frac{1}{N+1}}\big{)}$
$\displaystyle\quad+m\log\bigg{(}1+\frac{\rho_{s}n}{m}O(n\log
n)\bigg{)}O(\frac{1}{\log n})O\big{(}n^{-\frac{1}{N+1}}\big{)}$ (108)
where the second term in (105) comes from using (99) as well as the fact that
$P_{sum}^{*}$ is upper bounded by $\rho_{s}n$. In (107), we apply the Jensen
inequality to obtain the second and fourth terms. Using (100) and (101), we
have the second and fourth terms in (108). Finally, by substituting
$\bar{p}_{sum}$ and $\zeta_{n}$, we obtain
$\mathcal{R}_{mac}\leq\frac{m}{N+1}\log n+O(\log\log n)$ (109)
This concludes the proof of this theorem. $\,\Box$
## Appendix C Proof of Lemma 2
Proof: First, we prove (49). Let
$Z=|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}|^{2}$ and
$Y=\theta\big{(}\sum_{k\neq
j}|\mathbf{h}^{{\dagger}}_{i}\mathbf{\phi}_{j}|^{2}+|\mathbf{g}_{s,i}|^{2}\big{)}$.
Then, $Z$ has the exponential distribution, and $Y$ has the
Gamma$\big{(}(m+M-1),\theta\big{)}$ distribution. We can write
$L_{i}=\frac{Z}{c+Y}$ (110)
where $c=\frac{m}{\rho}$. Conditioned on $Y$, the pdf of $L_{i}$ is given by
$\displaystyle f_{L}(x)$
$\displaystyle=\int_{0}^{\infty}f_{L|Y}(x|y)f_{Y}(y)dy$ (111)
$\displaystyle=\int_{0}^{\infty}(c+y)e^{-(c+y)x}\times\frac{y^{m+M-1}e^{-y/\theta}}{(m+M-1)!\,\theta^{m+M}}dy$
(112) $\displaystyle=\frac{e^{-cx}}{(1+\theta x)^{m+M}}\big{(}c(1+\theta
x)+\theta(m+M-1)\big{)}$ (113)
So the cdf of $L_{i}$ is
$\displaystyle F_{L}(x)$ $\displaystyle=1-\int_{x}^{\infty}f_{L}(t)dt$ (114)
$\displaystyle=1-\frac{e^{-cx}}{(1+\theta x)^{m+M-1}}$ (115)
We define a grow function as
$\displaystyle g_{L}(x)$ $\displaystyle=\frac{1-F_{L}(x)}{f_{L}(x)}$ (116)
$\displaystyle=\frac{1+\theta x}{c(1+\theta x)+\theta(m+M-1)}$ (117)
Since $\lim_{x\rightarrow\infty}g_{L}^{\prime}(x)=0$, the limiting
distribution of $L_{max}=\max_{1\leq i\leq n}L_{i}$ exists [22]:
$\lim_{n\rightarrow\infty}\big{(}F_{L}(b_{n}+a_{n}x)\big{)}^{n}=e^{-e^{-x}}$
(118)
where $b_{n}=F_{L}^{-1}(1-1/n)$ and $a_{n}=g_{L}(b_{n})$. In general, an exact
closed-form solution for $a_{n}$ and $b_{n}$ is intractable, but an
approximation can be obtained, which is sufficient for asymptotic analysis.
After manipulating (115), we have
$b_{n}=\frac{1}{c}\log n-\frac{m+M-1}{c}\log\log n+O\big{(}\log\log\log
n\big{)}$ (119)
and thus
$a_{n}=\frac{1}{c}+O\big{(}\frac{1}{\log n}\big{)}$ (120)
It is straightforward to verify
$\lim_{n\rightarrow\infty}\big{(}ng_{L}^{\prime}(b_{n})\big{)}=\infty$, so we
apply the expansion developed in [23, Eq. (22)]
$\big{(}F_{L}(b_{n}+a_{n}x)\big{)}^{n}=\exp\bigg{(}-\exp(-x+\Theta(\frac{x^{2}}{\log^{2}n})\big{)}\bigg{)}$
(121)
Let $x_{1}=-\log\log n$ and substitute $x_{1}$ into (121), we obtain (49).
Now, we prove (50) and (51). Since $U_{i}$ is similar to $L_{i}$, except that
the denominator now has the Gamma$\big{(}M,\theta\big{)}$ distribution.
Following the same steps of obtaining (121), we have the expansion of the cdf
of $U_{max}$:
$\big{(}F_{U}(d_{n}+c_{n}x)\big{)}^{n}=\exp\bigg{(}-\exp(-x+\Theta(\frac{x^{2}}{\log^{2}n})\big{)}\bigg{)}$
(122)
where
$d_{n}=\frac{1}{c}\log n-\frac{M}{c}\log\log n+O\big{(}\log\log\log n\big{)}$
(123)
and
$c_{n}=\frac{1}{c}+O\big{(}\frac{1}{\log n}\big{)}$ (124)
(50) follows by substituting $x_{2}=\log\log n$ into (122).
Finally, because $\mathbb{E}[U_{max}]<n\mathbb{E}[U_{i}]$ [22], we have
$\displaystyle\mathbb{E}\bigg{[}U_{max}\,\bigg{|}\,U_{max}>d_{n}+\frac{1}{c}\log\log
n\bigg{]}$
$\displaystyle\leq\frac{n\mathbb{E}[U_{i}]}{\mathbb{P}\big{(}U_{max}>d_{n}+\frac{1}{c}\log\log
n\big{)}}$ (125) $\displaystyle=\Theta(n\log n)$ (126)
where we use (50) in the last equality. $\,\Box$
## Appendix D Proof of Theorem 4
Proof: We first find a lower bound for the secondary average throughput
$\mathcal{R}_{bc}$. We condition on $P=\rho$ and let
$l_{n}=b_{n}-\frac{\rho}{m}\log\log n$, where $b_{n}$ is given by Lemma 2.
Using (48) and Lemma 1, the conditional throughput $\mathcal{R}_{bc|P}(\rho)$
can be bounded as
$\displaystyle\mathcal{R}_{bc|P}(\rho)$ $\displaystyle\geq
m\mathbb{E}\bigg{[}\log\big{(}1+L_{max}\big{)}\,\bigg{|}\,P=\rho\bigg{]}$
(127) $\displaystyle\geq
m\mathbb{E}\bigg{[}\log\big{(}1+L_{max}\big{)}\,\bigg{|}\,L_{max}\geq
l_{n},\,P=\rho\bigg{]}\mathbb{P}\big{(}L_{max}\geq
l_{n}\,\big{|}\,P=\rho\big{)}$ (128)
$\displaystyle>m\bigg{(}\log\big{(}\frac{\rho}{m}\log
n\big{)}+O\big{(}\frac{\log\log n}{\log
n}\big{)}\bigg{)}\bigg{(}1-\Theta\big{(}n^{-1}\big{)}\bigg{)}$ (129)
$\displaystyle=m\log\big{(}\frac{\rho}{m}\log n\big{)}+O\big{(}\frac{\log\log
n}{\log n}\big{)}$ (130)
From (127) to (128), we discard the non-negative term associated with the
event $\\{L_{max}<l_{n}\\}$. Using (49) from Lemma 2 and the identity
$\log(x+y)=\log x+\log(1+y/x)$, we have (129).
Now we take the expectation with respect to $P$. From (39), we have
$P>\frac{m\Gamma}{\max_{1\leq i\leq
N}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}+m\Gamma/P_{s}}$ (131)
where $\mathbf{g}_{p,i}^{{\dagger}}$ is the $1\times m$ vector of channel
coefficients from the secondary base station to the primary user $i$. Let the
pdf of $\max_{1\leq i\leq N}|\mathbf{g}_{p}(i)|^{2}$ be $f_{g_{p}}(x)$.
Because the random variable $P$ is (stochastically) greater than the right
hand side of (131), from Lemma 1 and (130), we have
$\displaystyle\mathcal{R}_{bc}$
$\displaystyle>\int_{0}^{\infty}m\log\bigg{(}\frac{\Gamma\log
n}{x+m\Gamma/P_{s}}\bigg{)}f_{g_{p}}(x)\;dx+O\bigg{(}\frac{\log\log n}{\log
n}\bigg{)}$ (132) $\displaystyle\geq m\log\bigg{(}\frac{\Gamma\log
n}{\tilde{\mu}_{1}+m\Gamma/P_{s}}\bigg{)}+O\bigg{(}\frac{\log\log n}{\log
n}\bigg{)}$ (133) $\displaystyle=m\log\big{(}\Gamma\log
n\big{)}-m\log\big{(}\tilde{\mu}_{1}+m\Gamma/P_{s}\big{)}+O\bigg{(}\frac{\log\log
n}{\log n}\bigg{)}$ (134)
where (133) comes from the convexity of $\log(a+\frac{b}{x+c})$ and
$\tilde{\mu}_{1}=\mathbb{E}[\max_{1\leq i\leq N}|\mathbf{g}_{p}(i)|^{2}]$
(135)
To find an upper bound, we still begin with the conditional throughput
$\mathcal{R}_{bc|P}(\rho)$. Let $u_{n}=d_{n}+\frac{\rho}{m}\log\log n$, where
$d_{n}$ is given by Lemma 2. Then
$\displaystyle\mathcal{R}_{bc|P}(\rho)$ $\displaystyle\leq
m\mathbb{E}\bigg{[}\log\big{(}1+U_{max}\big{)}\,\bigg{|}\,P=\rho\bigg{]}$
(136) $\displaystyle\leq
m\mathbb{E}\bigg{[}\log\big{(}1+U_{max}\big{)}\,\bigg{|}\,U_{max}<u_{n},\,P=\rho\bigg{]}\mathbb{P}\big{(}U_{max}<u_{n}\big{|}P=\rho\big{)}$
(137)
$\displaystyle\quad+m\mathbb{E}\bigg{[}\log\big{(}1+U_{max}\big{)}\,\bigg{|}\,U_{max}\geq
u_{n},\,P=\rho\bigg{]}\mathbb{P}\big{(}U_{max}\geq u_{n}\big{|}P=\rho\big{)}$
(138) $\displaystyle<m\log(1+u_{n})\big{(}1-\Theta\big{(}\frac{1}{\log
n}\big{)}\big{)}$
$\displaystyle\quad+m\log\big{(}1+\mathbb{E}[U_{max}\,|\,U_{max}\geq
u_{n},\,P=\rho\big{]}\big{)}\Theta\big{(}\frac{1}{\log n}\big{)}$ (139)
$\displaystyle<m\log(1+\frac{\rho}{m}\log n)+O(1)$ (140)
where (136) comes from (48). We apply (50) in Lemma 2 and the Jensen
inequality to obtain (139). Using (51) in Lemma 2 and substituting $u_{n}$, we
obtain (140).
After calculating an upper bound for the conditional throughput, we average
over $P$. From (39), we have
$P\leq\frac{m\Gamma}{\max_{1\leq i\leq N}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}}$
(141)
We denote
$\frac{1}{\tilde{\mu}_{2}}=\mathbb{E}\big{[}1/\max_{1\leq i\leq
N}|\mathbf{g}_{p,i}^{{\dagger}}|^{2}\big{]}$ (142)
Then, by the Jensen inequality, we have
$\displaystyle\mathcal{R}_{bc}$ $\displaystyle<m\log\big{(}1+\frac{\log
n}{m}\mathbb{E}[P]\big{)}+O(1)$ (143)
$\displaystyle<m\log\big{(}1+\frac{\Gamma}{\tilde{\mu}_{2}}\log n\big{)}+O(1)$
(144) $\displaystyle=m\log(\Gamma\log n)-m\log\tilde{\mu}_{2}+O(1)$ (145)
where (144) holds since $\mathbb{E}[P]\leq\frac{m\Gamma}{\tilde{\mu}_{2}}$.
The theorem follows. $\,\Box$
## References
* [1] F. C. Commission, “Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies,” Dec. 2003.
* [2] S. A. Jafar, S. Srinivasa, I. Maric, and A. Goldsmith, “Breaking spectrum gridlock with cognitive radios: An information theoretic perspective,” _Proceedings of the IEEE_ , vol. 97, no. 5, pp. 894–914, May 2009.
* [3] N. Devroye, P. Mitran, and V. Tarokh, “Achievable rates in cognitive radio channels,” _IEEE Trans. Inform. Theory_ , vol. 52, no. 5, pp. 1813–1827, May 2006.
* [4] “Cognitive radio: an integrated agent architecture for software defined radio,” PhD Dissertation, KTH, Stockholm, Sweden, Dec. 2000.
* [5] A. Sahai, N. Hoven, and R. Tandra, “Some fundamental limits on cognitive radio,” in _Allerton Conf. Communiction, Control, and Computing_ , Oct. 2004\.
* [6] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum-sharing in fading environments,” _IEEE Trans. Wireless Commun._ , vol. 6, no. 2, pp. 649–658, Feb. 2007.
* [7] R. Zhang and Y.-C. Liang, “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,” _Selected Topics in Signal Processing, IEEE Journal of_ , vol. 2, no. 1, pp. 88 –102, Feb. 2008.
* [8] R. Zhang, S. Cui, and Y.-C. Liang, “On ergodic sum capacity of fading cognitive multiple-access and broadcast channels,” _IEEE Trans. Inform. Theory_ , vol. 55, no. 11, pp. 5161–5178, Nov. 2009.
* [9] M. Gastpar, “On capacity under receive and spatial spectrum-sharing constraints,” _IEEE Trans. Inform. Theory_ , vol. 53, no. 2, pp. 471–487, Feb. 2007.
* [10] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” _IEEE Trans. Inform. Theory_ , vol. 48, no. 6, pp. 1277–1294, June 2002.
* [11] K. Hamdi, W. Zhang, and K. B. Letaief, “Opportunistic spectrum sharing in cognitive MIMO wireless networks,” _IEEE Trans. Wireless Commun._ , vol. 8, no. 8, pp. 4098–4109, Aug. 2009.
* [12] N. Jamal, H. E. Saffar, and P. Mitran, “Throughput enhancements in point-to-multipoint cognitive systems,” in _IEEE ISIT_ , June/July 2009, pp. 2742–2746.
* [13] ——, “Asymptotic scheduling gains in point-to-multipoint cognitive networks.” [Online]. Available: http://arxiv.org/pdf/1001.3365
* [14] R. J. Serfling, _Approximation theorems of mathematical statistics_. New York: Wiley, 1980.
* [15] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna gaussian broadcast channel,” _Information Theory, IEEE Transactions on_ , vol. 49, no. 7, pp. 1691 – 1706, July 2003.
* [16] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” _IEEE Trans. Inform. Theory_ , vol. 51, no. 2, pp. 506–522, Feb. 2005.
* [17] M. Shaked and J. G. Shanthikumar, _Stochastic orders and their applications_. Boston: Academic Press, 1994\.
* [18] S. Diggavi and T. Cover, “The worst additive noise under a covariance constraint,” _Information Theory, IEEE Transactions on_ , vol. 47, no. 7, pp. 3072 –3081, Nov. 2001.
* [19] B. Hochwald, T. Marzetta, and V. Tarokh, “Multiple-antenna channel hardening and its implications for rate feedback and scheduling,” _Information Theory, IEEE Transactions on_ , vol. 50, no. 9, pp. 1893 – 1909, Sept. 2004.
* [20] T. M. Cover and J. A. Thomas, _Elements of Information Theory_. John Wiley and Sons, 1991.
* [21] O. Oyman, R. Nabar, H. Bolcskei, and A. Paulraj, “Characterizing the statistical properties of mutual information in mimo channels,” _Signal Processing, IEEE Transactions on_ , vol. 51, no. 11, pp. 2784 – 2795, Nov. 2003\.
* [22] H. A. David and H. N. Nagaraja, _Order statistics_. Wiley, 2003.
* [23] N. T. Uzgoren, “The asymptotic development of the distribution of the extreme values of a sample,” in Studies in Mathematics and Mechanics Presented to Richard von Mise. New York: Academic, 1954, pp. 346-353.
|
arxiv-papers
| 2010-08-21T15:33:36 |
2024-09-04T02:49:12.329344
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Li and Aria Nosratinia",
"submitter": "Yang Li",
"url": "https://arxiv.org/abs/1008.3641"
}
|
1008.3674
|
# Explicit Factorization of Prime Integers in Quartic Number Fields defined by
$X^{4}+aX+b$
L houssain El Fadil
###### Abstract.
For every prime integer $p$, an explicit factorization of the principal ideal
$p\mathbb{Z}_{K}$ into prime ideals of $\mathbb{Z}_{K}$ is given, where $K$ is
a quartic number field defined by an irreducible polynomial
$X^{4}+aX+b\in\mathbb{Z}[X]$.
Supported by ERCIM
Key words: Prime ideal factorization, Newton polygons, Quartic number fields.
AMS classification: 11Y40.
## Introduction
Let $K$ be a quartic number field defined by an irreducible polynomial
$P(X)=X^{4}+aX+b\in\mathbb{Z}[X]$, $\mathbb{Z}_{K}$ its ring of integers,
$\triangle$ the discriminant of $P$, $d_{K}$ its discriminant, $\alpha$ a
complex root of $P(X)$ and $ind(P)=[\mathbb{Z}_{K}:\mathbb{Z}[\alpha]]$ the
index of $\mathbb{Z}[\alpha]$. In this paper, the goal is to give an explicit
factorization of $p\mathbb{Z}_{K}$ into prime ideals of $\mathbb{Z}_{K}$; the
form $p\mathbb{Z}_{K}=\prod_{i=1}^{r}P_{i}^{e_{i}}$ and for every prime factor
$P_{i}$, an integral element $w_{i}$ such that $P_{i}=(p,w_{i})$ are given.
Let $p$ be a prime integer. It is well known that if $p$ does not divide
$ind(P)$, then the Dedekind’s theorem gives us an explicit factorization of
the principal ideal $p\mathbb{Z}_{K}$ into prime ideals of $\mathbb{Z}_{K}$:
$p\mathbb{Z}_{K}$ is $p$-analogous to the factorization of $\bar{P}(X)$ modulo
$p$. (see, for example [3, page 257]).
###### Theorem 0.1.
Let $p$ be a prime integer. Denote by $\bar{}$ the canonical map of
$\mathbb{Z}[X]$ into ${F_{p}}[X]$, and let
$\bar{P}(X)=\prod_{i=1}^{r}g_{i}(X)^{e_{i}}$, where $g_{1}(X)$,..,$g_{r}(X)$
are distinct irreducible in ${F_{p}}[X]$ and $e_{1}$,..$e_{r}$ are positive
integers. For every $i$, let $P_{i}=(p,f_{i}(\alpha))$, where
$f_{i}\in\mathbb{Z}[X]$ is a monic lifting over $g_{i}(X)$. Then
If $p$ does not divides $ind(P)$, then
$p\mathbb{Z}_{K}=\prod_{i=1}^{r}P_{i}^{e_{i}}$ and for every $i$,
$e(P_{i}/p)=e_{i}$ and $f(P_{i}/p)=f_{i}=deg(g_{i})$.
If $p$ is not a common index divisor of $K$, then there exist an element
$\phi\in\mathbb{Z}_{K}$ which generates $\mathbb{Z}_{K}$ and
$v_{p}(ind(\pi_{\phi})=0$, where $\pi_{\phi}$ is the minimal polynomial of
$\phi$, and then we can apply Dedekind’s theorem to obtain the prime ideal
decomposition explicitly. However given $\alpha$, it is not easy to determine
such an element $\phi$ in general. The construction of $\phi$ was based on the
$p$-integral bases of $\mathbb{Z}_{K}$ given in [1]. If $p$ is common index
divisor of $K$, then for every prime $P$ factor of $p\mathbb{Z}_{K}$, an
element $\beta\in\mathbb{Z}_{K}$ such that $v_{P}(\beta)=1$ and for every
prime ideal $Q\neq P$ above $p$, $v_{Q}(\beta)=0$ will be constructed. A such
$\beta\in\mathbb{Z}_{K}$ satisfies $P=(p,\beta)$.
For every prime $p$, let $v_{p}$ be the $p$-adic discrete valuation defined in
${\mathbb{Q}}_{p}$ by $v_{p}(p)=1$. $v_{p}$ is extended to
${\mathbb{Q}}_{p}[X]$ by $v_{p}(A(X)):=Min\\{v_{p}(a_{i})\,,\,0\leq i\leq
r\\}$, where $A(X)=\sum_{i=0}^{r}a_{i}X^{i}$. For every
$(x,m)\in\mathbb{Z}^{2}$, denote $x_{p}=\frac{x}{p^{v_{p}(x)}}$, $x\,\
\mbox{\rm(mod }{m})$ the remainder of the Euclidean division of $x$ by $m$.
For every odd prime $p$, denote $(\frac{-}{p})$ the Legendre symbol. If $p$
does not divide $x$ and there exists $t\in\mathbb{Z}$ such that $t^{n}=x\
\mbox{\rm(mod }{p})$, then let $(\frac{x}{p})_{n}=1$. Otherwise,
$(\frac{x}{p})_{n}\neq 1$.
## 1\. Newton polygons
Let $F(X)$ be an irreducible polynomial in $\mathbb{Z}[X]$,
$\phi\in\mathbb{Z}[X]$ a monic polynomial of degree at least $1$. Let
$F(X)=\sum_{i=0}^{n}a_{i}(X)\phi(X)^{i}=a_{0}(X)+..+a_{n-1}(X)\phi(X)^{n-1}+a_{n}(X)\phi(X)^{n}$
be the $\phi$-adic development of $F(X)$ ( for every $i$, $deg(a_{i}(X))\leq
deg(\phi)-1$). Let $p$ be a prime integer. The $\phi$-Newton polygon of
$F(x)$, with respect to $p$, is the lower convex envelope of the set of points
$(i,u_{i})$, $u_{i}<\infty$, in the Euclidian plane, where
$u_{i}=v_{p}(a_{i}(X))$.
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$N_{\phi}(F)$
The $\phi$-Newton polygon is the union of different adjacent sides
$S_{0},\dots,S_{g}$ with increasing slope
$\lambda_{0}<\lambda_{1}<\cdots<\lambda_{g}$. We shall write
$N_{\phi}(F)=S_{0}+\cdots+S_{g}$. The principal part of $N_{\phi}(F)$, denoted
$N^{+}_{\phi}(F)$, is the polygon determined by the sides of negative slope of
$N_{\phi}(F)$.
For every $0\leq i\leq n$, we attach to any abscissa the following residual
coefficient $c_{i}\in\mathbb{F}_{\phi}$:
$c_{i}=\left\\{\begin{array}[]{ll}0,&\mbox{ if $(i,u_{i})$ lies strictly above
$N$ or }u_{i}=\infty,\\\
\operatorname{red}\left(\dfrac{a_{i}(X)}{p^{u_{i}}}\right),&\mbox{ if
$(i,u_{i})$ lies on }N.\end{array}\right.$
Let $S$ be one of the sides of $N$, with slope $\lambda$, and let
$\lambda=-h/e$, with $h,e$ positive coprime integers. Let $l=\ell(S)$ be the
length of the projection of $S$ to the $x$-axis, $d(S):=\ell(S)/e$: the degree
of $S$. Note that $S$ is divided into $d(S)$ segments by the points of integer
coordinates that lie on $S$. Let $s$ be the initial abscissa of $S$ and $d$
the degree of $S$. If $\phi$ is a monic polynomial such that $\bar{\phi}$ is
irreducible factor of $\bar{F}(X)$ modulo $p$, then let
$\mathbb{F}_{\phi}:=\frac{\mathbb{F}_{p}[X]}{(\phi)}$ and
$F_{S}(Y):=c_{s}+c_{s+e}\,Y+\cdots+c_{s+(d-1)e}\,Y^{d-1}+c_{s+de}\,Y^{d}\in\mathbb{F}_{\phi}[Y]$,
the residual polynomial of $f(X)$ attached to $S$.
In the remainder, $F(X)$ is a monic irreducible polynomial, $\theta$ is a
complex root of $F(X)$ and $\phi$ is a monic polynomial such that $\bar{\phi}$
is irreducible factor of $\bar{F}(X)$ modulo $p$. Let $N=S_{k}+..+S_{1}+S_{0}$
be the $X\phi$-Newton polygon of $F(X)$ with respective slopes
$\lambda_{k}<..<\lambda_{1}<\lambda_{0}$. $F(X)$ is said to be $\phi$-regular
if for every $i$, $F_{S_{i}}(Y)$ is square free.
## 2\. Main results
In this section, for every prime integer $p$, an explicit factorization of the
principal ideal $p\mathbb{Z}_{K}$ into prime ideals of $\mathbb{Z}_{K}$ is
given (Note only the form is given, but two element generators of each prime
ideal factor are given). Recall the celebrated Theorem of Hensel:
###### Theorem 2.1 (Hensel).
Let $P(X)\in\mathbb{Z}[X]$ be a monic irreducible polynomial, $p$ a prime
integer, ${\mathbb{Q}}_{p}$ the $p$-adic completion of ${\mathbb{Q}}$ and
$\mathbb{Z}_{p}$ its ring of integers. Let $P(X)=F_{1}(X)\cdots F_{t}(X)$ be
the factorization into a product of monic irreducible polynomials in
${\mathbb{Q}}_{p}[X]$ and consider the local fields
$K_{i}={\mathbb{Q}}_{p}[X]/(F_{i}(X))$, for $i=1,\dots,t$. The factorization
of $p\mathbb{Z}_{K}$ into a product of prime ideals of $\mathbb{Z}_{K}$ is:
$p\mathbb{Z}_{K}=\mathfrak{p}_{1}^{e_{1}}\cdots\mathfrak{p}_{t}^{e_{t}}$,
where $f(\mathfrak{p}_{i})=f(K_{i}/{\mathbb{Q}}_{p})$ and
$e_{i}=e(K_{i}/{\mathbb{Q}}_{p}).$
###### Lemma 2.2.
Assume that $P(X)=\bar{\phi}^{l}(X)\,\ \mbox{\rm(mod }{p})$ such that
$N_{\phi}{P}=S$ is one side and $P_{S}(Y)$ is irreducible in
$\mathbb{F}_{\phi}[Y]$, where
$\mathbb{F}_{\phi}=\frac{\mathbb{F}_{p}[X]}{(\phi(X))}$. Let $\lambda=-h/e$ be
the slope of $S$ such that $h$ and $e$ are positive coprime integers. Define
$\alpha_{S}:=\phi(\alpha)^{e}/p^{h}$. Then $v_{p}(\alpha_{S})=0$ and
$P_{S}(\alpha_{S})=0$.
###### Proof.
Let $P(X)=\sum_{k=0}^{l}a_{k}(X)\phi(X)^{k}$ be the $\phi$-adic development of
$P(X)$. Since $N_{\phi}(P)$ is one side of slope $-\lambda$, we have for every
$i$, $v(a_{i}(X))\geq(i-l)\lambda$ and $v(a_{0}(X))=-l\lambda$. Thus,
$v_{p}(\phi(\alpha))=-\lambda$. Moreover, since for every $0\leq j\leq d$ and
for every $je<k<(j+1)e$, $\operatorname{red}\frac{a_{k}(X)}{p^{u_{k}}}=0$ and
$\frac{P(\alpha)}{p^{l}}=0$, then
$\overline{(\frac{\phi(\alpha)^{e}}{p^{h}})^{d}+\frac{a_{l-e}(\alpha)}{p^{h}}\frac{\phi(\alpha)^{e}}{p^{h}})^{d-1}+\cdots+\frac{a_{e}(\alpha)}{p^{h(d-1)}}\frac{\phi(\alpha)^{e}}{p^{h}})+\frac{a_{le}(\alpha)}{p^{l}}}=0$
in $\mathbb{F}_{\phi}$. Therefore, $P_{S}(\alpha_{S}))=0$ in
$\mathbb{F}_{\phi}$. ∎
Let $P(X)\in\mathbb{Z}[X]$ be a monic irreducible polynomial and
$\bar{P}(X)=\prod_{i=1}^{r}\phi_{i}^{l_{i}}\,\ \mbox{\rm(mod }{p})$ its
factorization into irreducible polynomials of $\mathbb{F}_{p}[X]$. For every
$i:=1..r$, let $N_{i}:=N^{+}_{\phi_{i}}(P)=S_{1}^{i}+\cdots+S_{k_{i}}^{i}$ and
for every $j:=1..k_{i}$, let
$P_{S_{j}}(Y)=\prod_{s=1}^{r_{ij}}\psi_{s}(Y)^{n_{s}}$ be the factorization of
$P_{S_{j}}(Y)$ into irreducible polynomials of $\mathbb{F}_{\phi}[Y]$, where
$\mathbb{F}_{\phi}=\frac{\mathbb{F}_{p}[X]}{(\phi(X))}$.
###### Theorem 2.3.
Under these hypothesis, if every $P_{S_{j}}(Y)$ is square free (every
$n_{s}=1$), then
$p\mathbb{Z}_{K}=\prod_{i=1}^{r}\prod_{j=1}^{k_{i}}\prod_{s=1}^{r_{ij}}\mathfrak{p}_{s}^{e(S_{j}^{i})},$
and for every $i:=1..r$, $j:=1..k_{i}$ and $s:=1..r_{ij}$,
$v_{\mathfrak{p}_{s}}({\phi_{i}}(\alpha))=-e(S_{j}^{i})\lambda_{j}$, where
$e(S_{j}^{i})$ is the ramification index of the side $S_{j}^{i}$ and
$-\lambda_{j}$ is its slope.
###### Proof.
By Hensel Theorem, it suffices to factorize $P(X)$ into irreducible
polynomials of ${\mathbb{Q}}[X]$. By Hensel Lemma, it suffices to show this
result for $r=1$. By Theorem of the polygon we can assume that $N_{1}=S$ is
one side. Since $P_{S}(Y)$ is square free, by Theorem of the residual
polynomial, we can assume that $P_{S}(Y)$ is irreducible in
$\mathbb{F}_{\phi}[Y]$. By Theorem of the product, if $P_{S}(Y)$ is
irreducible in $\mathbb{F}_{\phi}[Y]$, then $P(X)$ is irreducible in
${\mathbb{Q}}[X]$.
Now, assume that $P(X)\in\mathbb{Z}_{p}[X]$ is a monic irreducible polynomial
in ${\mathbb{Q}}[X]$ such that $\bar{P}(X)=\phi^{l}(X)\,\ \mbox{\rm(mod
}{p})$, $N_{\phi}(P)=S$ is one side and $P_{S}(Y)$ is irreducible in
$\mathbb{F}_{\phi}[Y]$ of degree $d(S)$. Denote
${\mathbb{K}}={\mathbb{Q}}_{p}[\alpha]$, $\mathbb{Z}_{{\mathbb{K}}}$ its ring
of integers over $\mathbb{Z}_{p}$ and $\mathfrak{p}$ the maximal ideal of
$\mathbb{Z}_{{\mathbb{K}}}$. By Hensel Theorem,
$p\mathbb{Z}_{{\mathbb{K}}}=\mathfrak{p}^{e(\mathfrak{p})}$, where
$e(\mathfrak{p})=\frac{deg(P)}{f(\mathfrak{p})}$.
Let $\lambda=-\frac{e}{h}$ be the slope of $S$ such $e$ and $h$ are positive
coprime. Let $\alpha_{S}=\frac{\phi^{e}(\alpha)}{p^{h}}$ and
$\iota\colon\mathbb{Z}_{p}[X,Y]\longrightarrow\mathbb{Z}_{p},\qquad
a(X,Y)\mapsto a(\alpha,\alpha_{S}).$ Then the ideal
$\mathfrak{m}=\iota^{-1}(\mathfrak{p})$ is a maximal ideal of
$\mathbb{Z}_{p}[X,Y]$ generated by $(p,\phi(X),\varphi(X,Y))$, where
$\varphi(X,Y)\in\mathbb{Z}_{p}[X,Y]$ is a monic polynomial such that
$\psi(Y):=\operatorname{red}(\varphi(X,Y))$ is irreducible in
$\mathbb{F}_{\phi}[Y]$. Thus,
$\frac{\mathbb{Z}_{{\mathbb{K}}}}{\mathfrak{p}}\simeq\frac{\mathbb{F}_{\phi}[Y]}{(\psi(Y))}$
and $f(\mathfrak{p})=d(S).m$. Thus,
$e(\mathfrak{p})=\frac{deg(P)}{d(S).m}=\frac{l.m}{d(S).m}=e(S)$, where
$m=deg(\phi)$.
On the other hand, since $N_{1}=S$ is one side of slope $\lambda$, we have
$v_{p}(\phi(\alpha)^{l})=v_{p}(a_{0}(\alpha))=-l\lambda$,
$v_{p}(\phi(\alpha))=-\lambda$ and
$v_{\mathfrak{p}}(\phi(\alpha))=-e(\mathfrak{p})\lambda$. ∎
Let $P(X)=X^{4}+aX+b$. Note that if there exists a prime $p$ such that
$v_{p}(a)\geq 3$ and $v_{p}(b)\geq 4$, then let $q_{1}$ and $q_{2}$ be
respectively the quotient of $v_{p}(a)$ by $3$ and of $v_{p}(b)$ by $4$. Let
$\theta:=\frac{\alpha}{p^{q}}$, where $q:=Min(q_{1},q_{2})$. Then $\theta$ is
integral with minimal polynomial $F(X)=X^{4}+AX+B\in\mathbb{Z}[X]$, where
$A=\frac{a}{p^{3q}}$ and $B=\frac{b}{p^{4q}}$. As
$K={\mathbb{Q}}[\theta]={\mathbb{Q}}[\alpha]$, then up to replace $\alpha$ by
$\frac{\alpha}{p^{q}}$, we can assume that for every prime $p$, $v_{p}(a)\leq
2$ or $v_{p}(b)\leq 3$.
###### Theorem 2.4.
Let $p$ be a prime integer. In the following tables, the form of
$p\mathbb{Z}_{K}$ as a product of prime ideals of $\mathbb{Z}_{K}$, and for
every prime factor $P$ of $p\mathbb{Z}_{K}$ an integral element $\phi$ of $K$
such that $P=(p,\phi)$ are given: If $p\mathbb{Z}_{K}=\prod_{i}P_{i}^{e_{i}}$,
then for every $i\neq j$, an element $\beta_{i}\in K$ such that
$v_{P_{i}}=(\beta_{i})=1$ and $v_{P_{j}}=(\beta_{i})=0$ is given.
$TableA:v_{p}(a)\geq 1\,and\,v_{p}(b)\geq 1$ Case | Conditions | $p$ | $p\mathbb{Z}_{K}$ | Generators
---|---|---|---|---
A1 | $v_{p}(b)=3$, $v_{p}(a)\geq 3$ | | $P^{4}$ | $P=(p,\frac{\alpha^{3}}{p^{2}})$
A2 | $v_{p}(b)\geq 3$, $v_{p}(a)=2$ | | $P_{1}P_{2}^{3}$ | $P_{1}=(p,\frac{\alpha^{3}}{p^{2}})$, $P_{2}=(p,a_{p}+\frac{\alpha^{3}}{p^{2}})$
A3 | $v_{p}(b)\geq 2$, $v_{p}(a)=1$ | | $P_{1}P_{2}^{3}$ | $P_{1}=(p,\frac{\alpha^{3}}{p})$, $P_{2}=(p,a_{p}+\frac{\alpha^{3}}{p})$
A4 | $v_{p}(b)=2$, $v_{p}(a)\geq 2$ | $\neq 2$ | $P^{2}$ | $P=(p,\frac{\alpha^{3}}{p})$
| $(\frac{-b_{p}}{p})=-1$ | | |
A5 | $v_{p}(b)=2$, $v_{p}(a)=2$ | $\neq 2$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(p,t+\frac{\alpha^{2}}{p})$
| $(\frac{-b_{p}}{p})=1$ | | | $P_{2}=(p,-t+\frac{\alpha^{2}}{p})$, ($v_{p}(t^{2}+b_{p})=1$)
A6 | $v_{p}(b)=2$, $v_{p}(a)\geq 3$ | $\neq 2$ | $P_{1}^{2}P_{2}^{2}$ | $P_{2}=(p,t+\frac{\alpha^{3}+\alpha^{2}}{p})$
| $(\frac{-b_{p}}{p})=1$ | | | $P_{2}=(p,-t+\frac{\alpha^{3}+\alpha^{2}}{p})$, ($t^{2}+b_{p}=0\,\ \mbox{\rm(mod }{p})$)
A7 | $v_{p}(b)=1$, $v_{p}(a)\geq 1$ | | $P^{4}$ | $P=(p,\alpha)$
A8 | $v_{p}(b)=2$, $v_{p}(a)\geq 2$ | $2$ | | go to $TableA8$
$TableA8:\,v_{2}(b)=2\,and\,v_{2}(a)\geq 2$ Case | conditions | $\phi_{2}$ | $2\mathbb{Z}_{K}$ | Generators
---|---|---|---|---
A8.1 | $v_{2}(b)=2$, $v_{2}(a)=2$ | $\frac{\alpha^{2}+2}{2}$ | $P^{4}$ | $P=(2,\phi_{2})$
A8.2 | $b=12\ \mbox{\rm(mod }{16})$, $v_{2}(a)=3$ | $\frac{\alpha^{3}+2\alpha}{4}$ | $P^{4}$ | $P=(2,\phi_{2})$
A8.3 | $b=4\ \mbox{\rm(mod }{16})$, $v_{2}(a)=3$ | $\frac{\alpha^{3}+2\alpha^{2}+2\alpha}{4}$ | $P^{4}$ | $P=(2,\phi_{2})$
A8.4 | $b=4\ \mbox{\rm(mod }{32}),v_{2}(a)=4$ | $\frac{\alpha^{2}+2\alpha+2}{4}$ | $P^{4}$ | $P=(2,\phi_{2})$
A8.5 | $b=20\ \mbox{\rm(mod }{64}),v_{2}(a)=4$ | $\frac{\alpha^{2}+2\alpha-2}{4}$ | $P^{2}$ | $P=(2,\phi_{2})$
A8.6 | $b=52\ \mbox{\rm(mod }{64}),a=16\ \mbox{\rm(mod }{64})$ | $\frac{\alpha^{3}-2\alpha^{2}-2\alpha}{8}$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(2,\phi_{2})$, $P_{2}=(2,\phi_{2}+1)$
A8.7 | $b=52\ \mbox{\rm(mod }{64}),a=48\ \mbox{\rm(mod }{64})$ | $\frac{\alpha^{3}-2\alpha^{2}+6\alpha}{8}$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(2,\phi_{2})$, $P_{2}=(2,\phi_{2}+1)$
A8.8 | $b=12\ \mbox{\rm(mod }{32}),v_{2}(a)\geq 4$ | $\frac{\alpha^{3}+2\alpha}{4}$ | $P^{2}$ | $P=(2,\phi_{2})$
A8.9 | $b=28\ \mbox{\rm(mod }{32}),v_{2}(a)=4$ | $\frac{\alpha^{2}+2}{4}$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(2,\phi_{2})$, $P_{2}=(2,\phi_{2}+1)$
A8.10 | $b=28\ \mbox{\rm(mod }{32}),v_{2}(a)\geq 5$ | $\frac{\alpha^{2}+12\alpha+2}{4}$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(2,\phi_{2})$, $P_{2}=(2,\phi_{2}+1)$
A8.11 | $b=20\ \mbox{\rm(mod }{32}),v_{2}(a)\geq 5$ | $\frac{\alpha^{2}+2\alpha-2}{4}$ | $P^{4}$ | $P=(2,\phi_{2})$
A8.12 | $b=36\ \mbox{\rm(mod }{64}),v_{2}(a)\geq 5$ | $\frac{\alpha^{2}+2\alpha+2}{4}$ | $P^{2}$ | $P=(2,\phi_{2})$
A8.13 | $b=4\ \mbox{\rm(mod }{64}),v_{2}(a)=5$ | $\frac{\alpha^{3}-2\alpha-4}{8}$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(2,\phi_{2})$, $P_{2}=(2,\phi_{2}+1)$
A8.14 | $b=4\ \mbox{\rm(mod }{64}),v_{2}(a)\geq 6$ | $\frac{\alpha^{3}+4\alpha^{2}-2\alpha+4}{8}$ | $P_{1}^{2}P_{2}^{2}$ | $P_{1}=(2,\phi_{2})$, $P_{2}=(2,\phi_{2}+1)$
$TableB:\,v_{p}(b)\geq 1\,and\,v_{p}(a)=0$ Case | Conditions | $p$ | $\phi$ | $p\mathbb{Z}_{K}$ | Generators
---|---|---|---|---|---
B1 | | $2$ | $\alpha$ | $P_{1}P_{2}P_{3}$ | $P_{1}=(2,\alpha)$, $P_{2}=(2,1+\alpha)$
| | | | $P_{3}=(2,1+\alpha+\alpha^{2})$ |
B2 | $(\frac{-a}{p})_{3}\neq 1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(p,\alpha)$, $P_{2}=(p,a+\alpha^{3})$
B3 | $(\frac{-3}{p})=-1$, $(\frac{-a}{p})_{3}=1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}P_{3}$ | $P_{1}=(p,\alpha)$, $P_{2}=(p,-u+\alpha)$
| | | | | $P_{3}=(p,u^{2}+u\alpha+\alpha^{2})$
| | | | | $u^{3}=-a\,\ \mbox{\rm(mod }{p})$
B4 | $(\frac{-3}{p})=1$, $(\frac{-a}{p})_{3}=1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}P_{3}P_{4}$ | $P_{1}=(p,\alpha)$, $P_{2}=(p,-u+\alpha)$
| | | | | $u^{3}=-a\,\ \mbox{\rm(mod }{p})$, $P_{3}=(p,v_{1}+\alpha)$
| | | | | $P_{4}=(p,-(u+v_{1})+\alpha)$
| | | | | $v^{2}=-3\,\ \mbox{\rm(mod }{p}),2v_{1}=-u(1+v)\,\ \mbox{\rm(mod }{p})$
B5 | $v_{3}(b)\geq 2,v_{3}(a)=0$ | 3 | $\alpha$ | $P_{1}P_{2}^{3}$ | $P_{1}=(3,\alpha)$, $P_{2}=(3,\alpha-a)$
| $a^{2}\neq 1\ \mbox{\rm(mod }{9})$ | | | |
B6 | $v_{3}(b)\geq 2$, $a^{2}=1\ \mbox{\rm(mod }{9})$ | 3 | | $P_{1}P_{2}^{2}P_{3}$ | go to $TableB6$
B7 | $v_{3}(a)=0,b=6\ \mbox{\rm(mod }{9})$ | 3 | $\alpha$ | $P_{1}^{3}P_{2}$ | $P_{1}=(3,\alpha-a)$, $P_{2}=(3,\alpha)$
| $a^{2}\neq 4\ \mbox{\rm(mod }{9})$ | | | |
B8 | $b=6\ \mbox{\rm(mod }{9}),a^{2}=4\ \mbox{\rm(mod }{9})$ | 3 | $\alpha$ | $P_{1}P_{2}^{2}P_{3}$ | go to $TableB6$
B9 | $v_{3}(a)=0,b=3\ \mbox{\rm(mod }{9})$ | 3 | $\alpha$ | $P_{1}^{3}P_{2}$ | $P_{1}=(3,\alpha-a)$, $P_{2}=(3,\alpha)$
| $a^{2}\neq 7\ \mbox{\rm(mod }{9})$ | | | |
B10 | $b=3\ \mbox{\rm(mod }{9}),a^{2}=7\ \mbox{\rm(mod }{9})$ | 3 | | $P_{1}P_{2}^{3}$ | $P_{1}=(3,\theta-4a)$, $P_{2}=(3,\frac{\theta^{3}-4a\theta^{2}}{3})$
| $v_{3}(b+a^{4}-a^{2})=2$ | | | | $\theta=\alpha-a$
B11 | $b=3\ \mbox{\rm(mod }{9}),a^{2}=7\ \mbox{\rm(mod }{9})$ | 3 | | | go to $TableB11$
| $v_{3}(b+a^{4}-a^{2})\geq 3$ | | | |
$TableB11$
Let $s\in\mathbb{Z}$ such that $as=-4b_{3}\ \mbox{\rm(mod
}{3^{v_{3}(\triangle)+1}})$, and let $\theta=\alpha-s$, $A=4s^{3}+a$ and
$B=s^{4}+as+b$.
Case | Conditions | $\phi$ | $3\mathbb{Z}_{K}$ | Generators
---|---|---|---|---
B11.1 | $v_{3}(\triangle)=6$, $B_{3}=1\,\ \mbox{\rm(mod }{3})$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{9}$ | $P_{1}P_{2}$ | $P_{1}=(3,\phi)$, $P_{2}=(3,\phi^{3}-\phi-s)$
B11.2 | $v_{3}(\triangle)=6$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{9}$ | $P_{1}P_{2}P_{3}$ | $P_{1}=(3,\phi)$, $P_{2}=(3,\phi^{2}-\phi-1)$
| $B_{3}=-1\,\ \mbox{\rm(mod }{3})$, $s=1\,\ \mbox{\rm(mod }{3})$ | | | $P_{3}=(3,\phi+1)$
B11.3 | $v_{3}(\triangle)=6$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{9}$ | $P_{1}P_{2}P_{3}$ | $P_{1}=(3,\phi)$, $P_{2}=(3,\phi^{2}+\phi-1)$
| $B_{3}=-1\,\ \mbox{\rm(mod }{3})$, $s=-1\,\ \mbox{\rm(mod }{3})$ | | | $P_{3}=(3,\phi-1)$
B11.4 | otherwise | | go to $TableB11.4$ |
$TableC:v_{p}(b)=0,v_{p}(a)\geq 1$ | Conditions | $p$ | $\phi$ | $p\mathbb{Z}_{K}$ | Generators
---|---|---|---|---|---
1 | $(\frac{2}{p})=1$, $(\frac{b}{p})_{4}=1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(p,\alpha^{2}+u\alpha+t)$, $t^{2}=b\,\ \mbox{\rm(mod }{p})$
| | | | | $P_{2}=(p,\alpha^{2}-u\alpha+t)$, $2t=u^{2}\,\ \mbox{\rm(mod }{p})$
2 | $(\frac{-1}{p})=1$,$(\frac{-b}{p})_{4}=1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}P_{3}P_{4}$ | $P_{1}=(p,\alpha+t)$, $P_{2}=(p,\alpha-t)$, $t^{4}=-b\,\ \mbox{\rm(mod }{p})$
| | | | | $P_{3}=(p,\alpha+ut)$, $P_{4}=(p,\alpha-ut)$ , $u^{2}=-1\,\ \mbox{\rm(mod }{p})$
3 | $(\frac{-1}{p})=-1$, $(\frac{-b}{p})_{4}=1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}P_{3}$ | $P_{1}=(p,\alpha+t)$, $P_{2}=(p,\alpha-t)$, $t^{4}=-b\,\ \mbox{\rm(mod }{p})$
| | | | | $P_{3}=(p,\alpha^{2}+t^{2})$
4 | $(\frac{-b}{p})=1$, $(\frac{-b}{p})_{4}\neq 1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(p,\alpha^{2}+t)$, $P_{2}=(p,\alpha^{2}-t)$, $t^{2}=-b\,\ \mbox{\rm(mod }{p})$
5 | $(\frac{4b}{p})_{4}=1$ ,$(\frac{-b}{p})_{4}\neq 1$ | $\geq 5$ | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(p,\alpha^{2}+u\alpha+t)$, $P_{2}=(p,\alpha^{2}-u\alpha+t)$
| | | | | $u^{4}=4b\,\ \mbox{\rm(mod }{p})$, $t^{2}=b\,\ \mbox{\rm(mod }{p})$
6 | $(\frac{-b}{p})_{4}\neq 1$ | $\geq 5$ | $\alpha$ | $P$ | $P=(p)$
| $(\frac{4b}{p})_{4}\neq 1$, $(\frac{-b}{p})=-1$ | | | |
7 | $b=1\,\ \mbox{\rm(mod }{3})$ | 3 | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(3,\alpha^{2}+\alpha-1)$
| | | | | $P_{1}=(3,\alpha^{2}-\alpha-1)$
8 | $b=-1\,\ \mbox{\rm(mod }{3})$ | 3 | $\alpha$ | $P_{1}^{2}P_{2}P_{3}$ | $P_{1}=(3,\alpha^{2}+1)$
| | | | | $P_{2}=(3,\alpha-1)$, $P_{3}=(3,\alpha+1)$
9 | $b=1\ \mbox{\rm(mod }{4})$, $a=0\ \mbox{\rm(mod }{4})$ | 2 | $\alpha$ | $P^{4}$ | $P=(2,\alpha-1)$
10 | $b=3\ \mbox{\rm(mod }{4})$, $a=2\ \mbox{\rm(mod }{4})$ | 2 | $\alpha$ | $P^{4}$ | $P=(2,\alpha-1)$
11 | $b=1\ \mbox{\rm(mod }{4})$, $a=2\ \mbox{\rm(mod }{4})$ | 2 | | $P_{1}^{3}P_{2}$ | go to $table8$
12 | $b=7\ \mbox{\rm(mod }{8})$, $a=4\ \mbox{\rm(mod }{8})$ | 2 | | $P^{2}$ | $P=(2,\alpha-1)$
13 | $b=7\ \mbox{\rm(mod }{8})$, $a=0\ \mbox{\rm(mod }{8})$ | 2 | $\frac{\theta^{3}+4\theta^{2}+6\theta}{4}$ | $P_{1}^{2}P_{2}$ | $P_{1}=(2,\phi+1)$
| $1+b+a=8\ \mbox{\rm(mod }{16})$ | | $\theta=\alpha-1$ | | $P_{2}=(2,\phi^{2}+\phi+1)$
14 | $b=7\ \mbox{\rm(mod }{8})$, $a=0\ \mbox{\rm(mod }{8})$ | 2 | | $P_{1}^{2}P_{2}P_{3}$ | go to $TableC14$
| $1+b+a=0\ \mbox{\rm(mod }{16})$ | | | |
15 | $b=3\ \mbox{\rm(mod }{8})$, $a=4\ \mbox{\rm(mod }{8})$ | 2 | | go to $TableC14$ |
$TableD:\,v_{p}(ab)=0$
For $p\geq 5$, let $s\in\mathbb{Z}$ such that $3as+4b=0\ \mbox{\rm(mod
}{p^{v_{p}(\triangle)}})$, and let $\theta=\alpha-s$.
Case | Conditions | $p$ | $\phi$ | $p\mathbb{Z}_{K}$ | Generators
---|---|---|---|---|---
D1 | | 2 | $\alpha$ | $P$ | $P=(2)$
D2 | $b=-1\,\ \mbox{\rm(mod }{3})$ | 3 | $\alpha$ | $P$ | $P=(3)$
D3 | $a=b=1\,\ \mbox{\rm(mod }{3})$ | 3 | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(3,\alpha-1)$, $P_{2}=(3,\alpha^{3}+\alpha^{2}+\alpha-1)$
D4 | $a=b=1\,\ \mbox{\rm(mod }{3})$ | 3 | $\alpha$ | $P_{1}P_{2}$ | $P_{1}=(3,\alpha+1)$, $P_{2}=(3,\alpha^{3}-\alpha^{2}+\alpha+1)$
D5 | $v_{p}(\triangle)=0$ | $p\geq 5$ | $\alpha$ | $p$-analogous to$\bar{P}(X)$ |
D6 | $v_{p}(\triangle)=1$ | $p\geq 5$ | $\theta$ | $P_{1}P_{2}P_{3}^{2}$ | $P_{3}=(p,\theta)$, $P_{1}=(p,\theta-v_{1})$, $P_{2}=(p,\theta-v_{2})$
| $(\frac{-2}{p})=1$ | | | | $u^{2}=-2\,\ \mbox{\rm(mod }{p}),v_{1}=-s(2+u),v_{2}=-s(2-u)\,\ \mbox{\rm(mod }{p})$
D7 | $v_{p}(\triangle)=1$ | $p\geq 5$ | $\theta$ | $P_{1}P_{2}^{2}$ | $P_{2}=(p,\theta)$,
| $(\frac{-2}{p})=-1$ | | | | $P_{1}=(p,\theta^{2}+4s\theta+6s^{2})$
D8 | $v_{p}(\triangle)\geq 2$ | $p\geq 5$ | | | go to $TableD8$
$TableB6:B=a^{4}-a^{2}+b,\,A=-4a^{3}+a$ Case | Conditions | $3\mathbb{Z}_{K}$ | $\beta_{3}$ | $\beta_{2}$ | $\beta_{1}$
---|---|---|---|---|---
B6.1 | $v_{3}(B)=2$, $v_{3}(A)=1$ | $P_{1}P_{2}^{2}P_{3}$ | $\frac{\theta^{3}-4a\theta^{2}}{3}$ | $\frac{\theta^{3}-4a\theta^{2}+6sa^{2}\theta+A}{3}+\theta^{2}$ | $\theta-4a$
B6.2 | $v_{3}(B)=3+k$, $v_{3}(A)=1$ | $P_{1}P_{2}^{2}P_{3}$ | $\frac{\theta^{3}-4a\theta^{2}}{3}+3$ | $\frac{\theta^{3}-4a\theta^{2}+6a^{2}\theta+A}{3}+\theta$ | $\theta-4a$
$TableB11.4:B=s^{4}+as+b,\,A=4s^{3}+a,\,as=-4b_{3}\ \mbox{\rm(mod }{3^{v_{3}(\triangle)+1}})$ Case | Conditions | $3\mathbb{Z}_{K}$ | $\beta_{4}$ | $\beta_{3}$ | $\beta_{2}$ | $\beta_{1}$
---|---|---|---|---|---|---
1 | $v_{3}(b)\geq 2$, $a^{2}\neq 7\ \mbox{\rm(mod }{9})$ | $P_{1}P_{2}^{2}P_{3}$ | | $\frac{\theta^{3}-4a\theta^{2}}{3}$ | $\frac{\theta^{3}-4a\theta^{2}+6sa^{2}\theta+A}{3}+\theta^{2}$ | $\theta-4a$
| $v_{3}(a^{4}-a^{2}+b)=2$ | | | | |
2 | $v_{3}(b)\geq 2$, $a^{2}\neq 7\ \mbox{\rm(mod }{9})$, | $P_{1}P_{2}^{2}P_{3}$ | | $\frac{\theta^{3}-4a\theta^{2}}{3}+3$ | $\frac{\theta^{3}-4a\theta^{2}+6a^{2}\theta+A}{3}+\theta$ | $\theta-4a$
| $v_{3}(a^{4}-a^{2}+b)\geq 3$ | | | | |
3 | $v_{3}(b)\geq 2$, $a^{2}=7\ \mbox{\rm(mod }{9})$, | $P_{1}P_{2}^{3}$ | | | $\frac{\theta^{3}+4s\theta^{2}}{3}+\theta$ | $\theta+4s$
| $v_{3}(a^{4}-a^{2}+b)=2$ | | | | |
4 | $v_{3}(\triangle)=2r+1$, $r\geq 4$ | $P_{1}P_{2}P_{3}^{2}$ | | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{3^{r-1}}+\frac{\theta^{2}+4s\theta}{3}$ | $\frac{\theta^{3}+4s\theta^{2}}{3}+\theta$ | $\theta+4s$
5 | $v_{3}(\triangle)=7$, | $P_{1}P_{2}P_{3}^{2}$ | | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{3^{r-1}}+\frac{\theta^{2}+4s\theta}{3}$ | $\frac{\theta^{3}+4s\theta^{2}}{3}+\theta^{2}$ | $\theta+4s$
6 | $v_{3}(\triangle)=2r$, $r\geq 5$ | $P_{1}P_{2}P_{3}$ | | $\frac{\theta^{2}+4s\theta}{3}+\theta$ | $\frac{\theta^{2}+4s\theta+6s^{2}}{3}+\theta$ | $\theta+4s$
| $(\frac{-2B_{3}}{3})=-1$ | | | | |
7 | $v_{3}(\triangle)=2r\geq 8$ | $P_{1}P_{2}P_{3}P_{4}$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}(\theta+3^{r-2}t)}{3^{r-1}}$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}(\theta-3^{r-2}t)}{3^{r-1}}$ | $\frac{\theta^{2}+4s\theta+6s^{2}}{3}+\theta$ | $\theta+4s$
| $(\frac{-2B_{3}}{3})=1$ | | $2s^{2}t^{2}+B_{3}=3\ \mbox{\rm(mod }{9})$ | | |
If ($b=7\ \mbox{\rm(mod }{8})$, $a=0\ \mbox{\rm(mod }{8})$ and $1+b+a=0\
\mbox{\rm(mod }{16})$) or $b=3\ \mbox{\rm(mod }{8})$ and $a=4\ \mbox{\rm(mod
}{8})$, then let $s\in\mathbb{Z}$ such that $P(X)$ is $(X+s)$-regular, and let
$A=4s+a$, $B=s^{4}+as+b$ and $\theta=\alpha-s$.
$TableC14:v_{2}(B)\geq 3,v_{2}(A)\geq 2$ Case | Conditions | $2\mathbb{Z}_{K}$ | $\beta_{3}$ | $\beta_{2}$ | $\beta_{1}$
---|---|---|---|---|---
1 | $v_{2}(B)\geq 2$ | $P_{1}^{3}P_{2}$ | | $\frac{\theta^{3}}{2}+2$ | $\frac{\theta^{3}+A}{2}+\theta$
| $v_{2}(A)=1$ | | | |
2 | $v_{2}(B)=2r-1$ | $P_{1}^{2}P_{2}$ | | $\frac{\theta^{2}}{2}+2$ | $\frac{\theta^{2}+6}{2}+\theta$
| $v_{2}(A)=r\geq 2$ | | | |
3 | $v_{2}(B)>2r$ | $P_{1}^{2}P_{2}P_{3}$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{2^{r}}$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+2^{r-1}t)}{2^{r}}+2$ | $\frac{\theta^{2}+6}{2}+\theta$
| $v_{2}(A)=r\geq 2$ | | | $A_{2}t=1\ \mbox{\rm(mod }{4})$ |
4 | $v_{2}(B)=2r$ | $P_{1}^{2}P_{2}P_{3}$ | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{2^{r}}$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+2^{r-1}t)}{2^{r}}$ | $\frac{\theta^{2}+6}{2}+\theta$
| $v_{2}(A)=r\geq 2$ | | | $A_{2}t=3\ \mbox{\rm(mod }{4})$ |
5 | $v_{2}(B)=2r$ | $P_{1}^{2}P_{2}^{2}$ | | $\frac{\theta^{2}}{2}+\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{2^{r}}$ | $\frac{\theta^{2}+6}{2}+\theta$
| $v_{2}(A)\geq r+1$ | | | |
$TableD8:p\geq 5,\,v_{p}(ab)=0{\mbox{a}nd}v_{p}(\triangle)\geq 2$
Let $s\in\mathbb{Z}$ such that $3as+4b=0\ \mbox{\rm(mod
}{p^{v_{p}(\triangle)+1}})$, $\theta=\alpha-s$, $B=s^{4}+as+b$ and
$A=4s^{3}+a$.
Conditions | $p\mathbb{Z}_{K}$ | $\beta_{4}$ | $\beta_{3}$ | $\beta_{2}$ | $\beta_{1}$
---|---|---|---|---|---
$v_{p}(\triangle)=2r+1$ | $P_{1}P_{2}P_{3}^{2}$ | | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{p^{r}}$ | $\theta-v_{1}$ | $\theta-v_{2}$
$(\frac{-2}{p})=1$ | | | $B_{p}=6s^{2}u\ \mbox{\rm(mod }{p})$ | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$ | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$
| | | | $v_{1}=-s(2+u)\ \mbox{\rm(mod }{p^{2}})$ | $v_{2}=-s(2-u)\ \mbox{\rm(mod }{p^{2}})$
$v_{p}(\triangle)=2r+1$ | $P_{1}P_{2}^{2}$ | | | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{p^{r}}$ | $\theta^{2}+4s\theta+6s^{2}+p$
$(\frac{-2}{p})=-1$ | | | | $B_{p}=6s^{2}u\,\ \mbox{\rm(mod }{p})$ |
$v_{p}(\triangle)=2r$ | $P_{1}P_{2}P_{3}$ | | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{p^{r-1}}$ | $\theta-v_{1}$ | $\theta-v_{2}$
$(\frac{-2}{p})=1$ | | | $v_{p}(6s^{2}+B_{p})=1$ | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$ | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$
$(\frac{-6B_{p}}{p})=-1$ | | | | $v_{1}=-s(2+u)\ \mbox{\rm(mod }{p^{2}})$ | $v_{2}=-s(2-u)\ \mbox{\rm(mod }{p^{2}})$
$v_{p}(\triangle)=2$ | $P_{1}P_{2}P_{3}P_{4}$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+pt)}{p}+\theta^{2}$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta-pt)}{p}+\theta^{2}$ | $\theta-v_{1}$ | $\theta-v_{2}$
$(\frac{-2}{p})=1$ | | $v_{p}(6s^{2}t^{2}+B_{p})\geq 2$ | | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$ | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$
$(\frac{-6B_{p}}{p})=1$ | | | | $v_{1}=-s(2+u)\ \mbox{\rm(mod }{p^{2}})$ | $v_{2}=-s(2-u)\ \mbox{\rm(mod }{p^{2}})$
$v_{p}(\triangle)=2r\geq 4$ | $P_{1}P_{2}P_{3}P_{4}$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+p^{r}t)}{p^{r}}+\theta$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta-p^{r}t)}{p^{r}}+\theta$ | $\theta-v_{1}$ | $\theta-v_{2}$
$(\frac{-2}{p})=1$ | | $v_{p}(6s^{2}t^{2}+B_{p})=1$ | | $u^{2}+2=p\ \mbox{\rm(mod }{p^{2}})$ | $v_{p}(u^{2}+2)=1$
$(\frac{-6B_{p}}{p})=1$ | | | | $v_{1}=-s(2+u)\ \mbox{\rm(mod }{p^{2}})$ | $v_{p}(v_{2}+s(2-u))\geq 2$
$v_{p}(\triangle)=2r$ | $P_{1}P_{2}$ | | | $\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{p^{r-1}}$ | $\theta^{2}+4s\theta+6s^{2}+p$
$(\frac{-2}{p})=-1$ | | | | |
$(\frac{-6B_{p}}{p})=-1$ | | | | |
$v_{p}(\triangle)=2$ | $P_{1}P_{2}P_{3}$ | | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+pt)}{p}+\theta^{2}$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta-pt)}{p}+\theta^{2}$ | $\theta^{2}+4s\theta+6s^{2}+p$
$(\frac{-2}{p})=-1$ | | | $v_{p}(6s^{2}t^{2}+B_{p})\geq 2$ | | $v_{p}(6s^{2}u+B_{p})=1$
$(\frac{-6B_{p}}{p})=1$ | | | | |
$v_{p}(\triangle)=2r\geq 4$ | $P_{1}P_{2}P_{3}$ | | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+p^{r}t)}{p^{r}}+\theta$ | $\frac{(\theta^{2}+4s\theta+6s^{2})(\theta-p^{r}t)}{p^{r}}+\theta$ | $\theta^{2}+4s\theta+6s^{2}+p$
$(\frac{-2}{p})=-1$ | | | $v_{p}(6s^{2}t^{2}+B_{p})=1$ | |
$(\frac{-6B_{p}}{p})=1$ | | | | |
Proof of Theorem. All cases, except ($p=2$, $v_{p}(b)=2$ and $v_{p}(a)\geq
2$), correspond to a situation where $P(X)$ is $p$-regular. The case : $p=2$,
$v_{p}(b)=2$ and $v_{p}(a)\geq 2$ is handled in $TableA8$ by using technics of
Newton polygons of second order.
Denote $C(X)$ the minimal polynomial of a possible $\phi$ such that
$v_{p}([\mathbb{Z}_{K}:\mathbb{Z}[\phi]])=0$.
$v_{p}(a)\geq 1$ and $v_{p}(b)\geq 1$.
1. (1)
$v_{p}(a)\geq 3$ and $v_{p}(b)=3$. Then $\bar{P}(X)=X^{4}\,\ \mbox{\rm(mod
}{p})$ and the $X$-Newton polygon of $P(X)$ is one side of slope $3/4$. Thus,
$p\mathbb{Z}_{K}=P^{4}$. Since $v_{P}(\alpha)=4\times 3/4=3$, then
$v_{P}(\frac{\alpha^{3}}{p^{2}})=1$ and $P=(p,\frac{\alpha^{3}}{p^{2}})$.
2. (2)
Let $\phi=\frac{\alpha^{3}}{p^{h}}$. Then
$C(X)=X^{4}+3\frac{a}{p^{h}}X^{3}+3\frac{a^{2}}{p^{2h}}X^{2}+\frac{a^{3}}{p^{3h}}X+\frac{b^{3}}{p^{4h}}$.
It follows that if $v_{p}(a)=1$ and $v_{p}(b)\geq 2$, then
$\phi=\frac{\alpha^{3}}{p}$ is a $p$-generator of $\mathbb{Z}_{K}$,
$\bar{C}(X)=X(X+a_{p})^{3}\,\ \mbox{\rm(mod }{p})$ and
$p\mathbb{Z}_{K}=(p,\alpha)(p,\alpha+a_{p})^{3}$. If $v_{p}(a)=2$ and
$v_{p}(b)\geq 3$, then $\phi=\frac{\alpha^{3}}{p^{2}}$ is a $p$-generator of
$\mathbb{Z}_{K}$, $\bar{C}(X)=X(X+a_{p})^{3}\,\ \mbox{\rm(mod }{p})$ and
$p\mathbb{Z}_{K}=(p,\phi)(p,\phi+a_{p})^{3}$.
3. (3)
If $v_{p}(b)=1$ and $v_{p}(a)\geq 1$, then $v_{p}(ind(P))=0$, then $\alpha$ is
a $p$-generator of $\mathbb{Z}_{K}$ and $p\mathbb{Z}_{K}=(p,\alpha)^{4}$.
4. (4)
If $v_{p}(b)=2$ and $v_{p}(a)\geq 2$, then $N_{X}(P)=S$ is one side such that
$P_{S}(y)=Y^{2}+b_{p}$. It follows that:
1. (a)
If $(\frac{-b_{p}}{p})=-1$, then $p\mathbb{Z}_{K}=P^{2}$ such that
$v_{P}(\alpha)=1$. Thus, $P=(p,\frac{\alpha^{3}}{p})$.
2. (b)
If $(\frac{-b_{p}}{p})=1$ and $v_{p}(a)=2$, then
$p\mathbb{Z}_{K}=P_{1}^{2}P_{2}^{2}$. For $\phi=\frac{\alpha^{2}}{p}$, we have
$C(X)=X^{4}-Xpa_{p}^{2}+2b_{p}X^{2}+b_{p}^{2}$ and
$\bar{C}(X)=(X^{2}+b_{p})^{2}\,\ \mbox{\rm(mod }{p})$. Let $t\in\mathbb{Z}$
such that $v_{p}(t^{2}+b_{p})=1$. Let $\phi(X)=X+t$ and
$\phi(X)^{4}-4t\phi(X)^{3}+(6t^{2}+2B)\phi(X)^{2}+(-pa_{p}^{2}-4tb_{p}-4t^{3})\phi(X)+(b_{p}^{2}+tpa_{p}^{2}+2t^{2}b_{p}+t^{4})$
be the $\phi(X)$-adic development of $C(X)$. Since
$v_{p}(b_{p}^{2}+tpa_{p}^{2}+2t^{2}b_{p}+t^{4})=1$, $\frac{\alpha^{2}}{p}$ is
a $p$-generator of $\mathbb{Z}_{K}$. As $\bar{C}(X)=(X^{2}+b_{p})^{2}\,\
\mbox{\rm(mod }{p})$, we have $P_{1}=(p,t+\frac{\alpha^{2}}{p})$ and
$P_{2}=(p,-t+\frac{\alpha^{2}}{p})$ $v_{p}(t^{2}+b_{p})=1$.
3. (c)
If $(\frac{-b_{p}}{p})=1$ and $v_{p}(a)=\geq 3$, then
$p\mathbb{Z}_{K}=P_{1}^{2}P_{2}^{2}$. Let
$\phi=\frac{\alpha^{3}+\alpha^{2}}{p}$. Since
$\alpha^{3}+\alpha^{2}\not\in{\mathbb{Q}}$,
$K={\mathbb{Q}}[\alpha^{3}+\alpha^{2}]$ and
$\bar{C}(X)=X^{4}+2b_{p}X^{2}-4pb_{p}^{2}X+b_{p}^{2}\ \mbox{\rm(mod }{p^{2}})$
and $\bar{C}(X)=(X^{2}+b_{p})^{2}\,\ \mbox{\rm(mod }{p})$. Let $\phi(X)=X+t$
such that $t^{2}+b_{p}=0\,\ \mbox{\rm(mod }{p})$ and
$C(X)=\phi^{4}(X)-4t\phi^{3}(X)+(6t^{2}+2b_{p})\phi^{2}(X)+(-4pb_{p}^{2}-4tb_{p}-4t^{3})\phi(X)+(b_{p}^{2}+4tpb_{p}^{2}+2t^{2}b_{p}+t^{4})$
be the $\phi(X)$-adic development of $C(X)$. Since
$v_{p}(b_{p}^{2}+4tpb_{p}^{2}+2t^{2}b_{p}+t^{4})=1$,
$\frac{\alpha^{3}+\alpha^{2}}{p}$ is a $p$-generator of $\mathbb{Z}_{K}$,
$P_{1}=(p,t+\frac{\alpha^{3}+\alpha^{2}}{p})$ and
$P_{2}=(p,-t+\frac{\alpha^{3}+\alpha^{2}}{p})$.
5. (5)
$v_{2}(b)=2$ and $v_{2}(a)\geq 2$. In that case, using technics of Newton
polygons of second order, for every subcase, a monic polynomial
$\phi_{2}\in\mathbb{Z}[X]$ such that $t=(X,1/2,\phi_{2})$ is $P(X)$-complet is
given (see [5, Def 3.9, p 38]). Let $N_{2}$ be the $\phi_{2}$-Newton polygon
of second order of $P(X)$. From [5, Cor 3.8, p 38], if $N_{2}$ is one side,
then $2\mathbb{Z}_{K}=P^{e}$, where $e=2e_{2}$ and $e_{2}$ is the ramification
index of $N_{2}$. If $N_{2}$ is two sides, then for every side, $e_{2}=1$ and
then $2\mathbb{Z}_{K}=P_{1}^{2}P_{2}^{2}$.
1. (a)
If $v_{2}(a)=2$, then for $\phi_{2}=X^{2}+2$ and $N_{2}$ is one side. Thus,
$2\mathbb{Z}_{K}=P^{4}$, $v_{P}({\alpha^{2}+2})=5$ and
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2}{2})$.
2. (b)
If $v_{2}(a)=3$ and $b=4\ \mbox{\rm(mod }{16})$ (resp. $v_{2}(a)=3$ and $b=12\
\mbox{\rm(mod }{16})$), then for $\phi_{2}=X^{2}+2X+2$ (resp.
$\phi_{2}=X^{2}+2$) and $N_{2}$ is one side. Then $2\mathbb{Z}_{K}=P^{4}$,
$v_{P}({\alpha^{2}+2\alpha+2})=7$ (resp. $v_{P}({\alpha^{2}+2})=7$) and
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{3}+2\alpha^{2}+2\alpha}{4})$ (resp.
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{3}+2\alpha}{4})$).
3. (c)
If $v_{2}(a)=4$ and $b=4\ \mbox{\rm(mod }{32})$, then for
$\phi_{2}=X^{2}+2X+2$, $N_{2}$ is one side, $2\mathbb{Z}_{K}=P^{4}$ and
$v_{P}({\alpha^{2}+2\alpha+2})=9$. Thus,
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2\alpha+2}{4})$.
4. (d)
If $v_{2}(a)=4$ and $b=20\ \mbox{\rm(mod }{64})$, then for
$\phi_{2}=X^{2}+2X-2$, $N_{2}$ is one side, $2\mathbb{Z}_{K}=P^{2}$ and
$v_{P}({\alpha^{2}+2\alpha-2})=5$. Thus,
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2\alpha-2\alpha}{4})$.
5. (e)
If $v_{2}(a)\geq 4$ and $b=12\ \mbox{\rm(mod }{32})$, then for
$\phi_{2}=X^{2}+2$, $N_{2}$ is one side, $2\mathbb{Z}_{K}=P^{2}$ and
$v_{P}({\alpha^{2}+2})=4$. Thus,
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{3}+2\alpha}{4})$.
6. (f)
If $v_{2}(a)\geq 5$ and $b=20\ \mbox{\rm(mod }{32})$, then for
$\phi_{2}=X^{2}+2X-2$, $N_{2}$ is one side, $2\mathbb{Z}_{K}=P^{4}$ and
$v_{P}({\alpha^{2}+2\alpha-2})=9$. Thus,
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2\alpha-2\alpha}{4})$.
7. (g)
If $v_{2}(a)\geq 5$ and $b=36\ \mbox{\rm(mod }{64})$, then for
$\phi_{2}=X^{2}+2X+2$, $N_{2}$ is one side, $2\mathbb{Z}_{K}=P^{2}$ and
$v_{P}({\alpha^{2}+2\alpha+2})=5$. Thus,
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2\alpha-2\alpha}{4})$.
6. (6)
$b=4+64B$ and $a=32+64A$. For $\phi=\frac{\alpha^{3}-2\alpha-4}{8}$, we have
$\bar{C}(X)=X^{4}+2X^{3}+3X^{2}+2X+2\ \mbox{\rm(mod }{4})$. Thus,
$v_{2}(C(1))=v_{2}(C(0))=1$, $\phi$ is a $2$-generator of $\mathbb{Z}_{K}$ and
$2\mathbb{Z}_{K}=(2,\phi)^{2}(2,\phi+1)^{2}$.
7. (7)
$b=4+64B$ and $a=64A$. For $\phi=\frac{\alpha^{3}+4\alpha^{2}-2\alpha+4}{8}$,
we have $\bar{C}(X)=X^{4}+2X^{3}+3X^{2}+2X+2\ \mbox{\rm(mod }{4})$. It follows
that $v_{2}(C(1))=v_{2}(C(0))=1$, $\phi$ is a $2$-generator of
$\mathbb{Z}_{K}$ and $2\mathbb{Z}_{K}=(2,\phi)^{2}(2,\phi+1)^{2}$.
8. (8)
$b=28+32B$ and $a=16+32A$. For $\phi=\frac{\alpha^{2}+2}{4}$, we have
$C(X)=X^{4}-2X^{3}+(5+4B)X^{2}+(-16A^{2}-8-16A-4B)X+6+8A+8A^{2}+8B+4B^{2}=(X+1)^{2}X^{2}\,\
\mbox{\rm(mod }{2})$. Since $v_{2}(C(0))=v_{2}(C(1))=1$, we have $\phi$ is a
$2$-generator of $\mathbb{Z}_{K}$ and
$2\mathbb{Z}_{K}=(2,\phi+1)^{2}(2,\phi)^{2}$.
9. (9)
$b=52+64B$ and $a=16+64A$. For
$\phi=\frac{\alpha^{3}-2\alpha^{2}-2\alpha}{8}$,
$C(X)=X^{4}+2X^{3}+3X^{2}+2X+2\ \mbox{\rm(mod }{4})$. Thus,
$v_{2}(C(0))=v_{2}(C(1))=1$, $\phi$ is a $2$-generator of $\mathbb{Z}_{K}$ and
$2\mathbb{Z}_{K}=(2,\phi+1)^{2}(2,\phi)^{2}$.
10. (10)
$b=52+64B$ and $a=48+64A$. For
$\phi=\frac{\alpha^{3}-2\alpha^{2}+6\alpha}{8}$,
$C(X)=X^{4}+2X^{3}+3X^{2}+2X+2\ \mbox{\rm(mod }{4})$. Thus,
$v_{2}(C(0))=v_{2}(C(1))=1$, $\phi$ is a $2$-generator of $\mathbb{Z}_{K}$ and
$2\mathbb{Z}_{K}=(2,\phi+1)^{2}(2,\phi)^{2}$.
11. (11)
$b=28+32B$ and $a=32A$. For $\phi=\frac{\alpha^{2}+12\alpha+2}{4}$,
$C(X)=X^{4}+2X^{3}+X^{2}+2\ \mbox{\rm(mod }{4})$. Thus,
$v_{2}(C(0))=v_{2}(C(1))=1$, $\phi$ is a $2$-generator of $\mathbb{Z}_{K}$ and
$2\mathbb{Z}_{K}=(2,\phi+1)^{2}(2,\phi)^{2}$.
$v_{p}(a)=0$ and $v_{p}(b)\geq 1$.
If $p\neq 3$, then $v_{p}(ind(P))=0$, $\alpha$ is a $p$-genrator of
$\mathbb{Z}_{K}$ and $p\mathbb{Z}_{K}$ is $p$-analogous to $\bar{P}(X)$.
For $p=3$, let $F(X)=P(X-a)=X^{4}-4aX^{3}+6a^{2}X^{2}+AX+B$ and
$\theta=\alpha+a$ ($A=-a(4a^{2}-1)$ and $B=(a^{4}-a^{2}+b)$).
1. (1)
If $v_{3}(B)=1$, then $v_{3}(ind(P))=0$ and
$3\mathbb{Z}_{K}=(3,\alpha+a)^{3}(3,\alpha)$.
2. (2)
If $v_{3}(B)=2$ and $v_{3}(A)\geq 2$, then $3\mathbb{Z}_{K}=P_{1}P_{2}^{3}$,
$v_{P_{1}}(\theta-4a)=1$, $v_{P_{1}}(\theta)=0$, $v_{P_{2}}(\theta-4a)=0$ and
$v_{P_{2}}(\theta)=2$. Thus, $P_{1}=(3,\theta-4a)$,
$P_{2}=(3,\frac{\theta^{3}-4a\theta^{2}}{3})$.
3. (3)
If $v_{3}(A)=1$ and $v_{3}(B)\geq 2$, then
$3\mathbb{Z}_{K}=P_{1}P_{2}^{2}P_{3}$.
Since $\theta^{4}-4a\theta^{3}=-6a^{2}\theta^{2}+A\theta+B$ and
$v_{P_{1}}(\theta)=0$, we have $v_{P_{1}}(\theta^{3}-4a\theta^{2})=1$ and
$P_{1}=(3,\theta-4a)$.
If $v_{3}(B)=2$, then let $\beta_{3}=\frac{\theta^{3}-4a\theta^{2}}{3}$ and
$\beta_{2}=\frac{\theta^{3}-4a\theta^{2}+6sa^{2}\theta+A}{3}+\theta^{2}$.
Since $v_{P_{3}}(\theta)=v_{P_{2}}(\theta)=1$ and $v_{P_{1}}(\theta)=0$, we
have $v_{P_{3}}(\beta_{3})=1$ and $v_{P_{i}}(\beta_{3})=0$ for $i\neq 3$. We
have also, $v_{P_{2}}(\beta_{2})=1$ and $v_{P_{i}}(\beta_{3})=0$ for $i\neq
2$.
The case $v_{3}(B)\geq 3$ is similar to the previous case.
4. (4)
If $v_{3}(A)\geq 2$ and $v_{3}(B)\geq 3$ ($v_{3}(\triangle)\geq 6$), then let
$s\in\mathbb{Z}$ such that $as=-4b_{3}\ \mbox{\rm(mod
}{3^{v_{3}(\triangle)+1}})$. Let $F(X)=P(X+s)=X^{4}+4sX^{3}+6s^{2}X^{2}+AX+B$
and $\theta=\alpha-s$. Since $A=4s^{3}+a$ and $B=s^{4}+as+b$,
$v_{3}(A)=v_{3}(B)=v_{3}(\triangle)-3\geq 3$. It follows that:
If $v_{3}(\triangle)=6$, then for
$\phi=\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta+A}{9}$, we have
$\bar{C}(X)=X(X^{3}+2s^{2}B_{3}X-4sB_{3}^{2})\,\ \mbox{\rm(mod }{3})$. So, if
$B_{3}=1\,\ \mbox{\rm(mod }{3})$, then $\bar{C}(X)=X(X^{3}-X-s)\,\
\mbox{\rm(mod }{3})$ and $3\mathbb{Z}_{K}=(3,\phi)(3,\phi^{3}-\phi-s)$. If
$B_{3}=-1\,\ \mbox{\rm(mod }{3})$ and $s=1\,\ \mbox{\rm(mod }{3})$, then
$\bar{C}(X)=X(X^{2}-X-1)(X+1)\,\ \mbox{\rm(mod }{3})$. If $B_{3}=-1\,\
\mbox{\rm(mod }{3})$ and $s=-1\,\ \mbox{\rm(mod }{3})$, then
$\bar{C}(X)=X(X^{2}+X-1)(X-1)\,\ \mbox{\rm(mod }{3})$
5. (5)
If $v_{3}(\triangle)\geq 7$, then $N_{X}(F)=S_{3}+S_{2}+S_{1}$,
$F_{S_{1}}(Y)=Y+4s$ and $F_{S_{2}}(Y)=sY-1$. Since
$\theta^{3}(\theta^{2}+4s)=-(6s^{2}\theta^{2}+A\theta+B)$, then
$P_{1}=(3,\theta+4s)$.
1. (a)
If $v_{3}(\triangle)=2r+1$ ($r\geq 3$), then $v_{3}(A)=v_{3}(B)=2r-2$,
$F_{S_{3}}(Y)=2s^{2}Y+B_{3}$, $3\mathbb{Z}_{K}=P_{1}P_{2}P_{3}^{2}$,
$v_{P_{3}}(\theta)=2r-3$, $v_{P_{2}}(\theta)=1$ and $v_{P_{1}}(\theta)=0$. For
every $i\neq j$, $v_{P_{i}}(\beta_{i})=1$ and $v_{P_{i}}(\beta_{j})=0$.
2. (b)
If $v_{3}(\triangle)=2r$, then $F_{S_{3}}(Y)=6s^{2}Y^{2}+B_{3}$. Thus, if
$(\frac{-2B_{3}}{3})=-1$, then $3\mathbb{Z}_{K}=P_{1}P_{2}P_{3}$. If
$(\frac{-2B_{3}}{3})=-1$, then $3\mathbb{Z}_{K}=P_{1}P_{2}P_{3}P_{4}$.
If $(\frac{-2B_{3}}{3})=-1$, then let
$\beta_{3}=\frac{\theta^{2}+4s\theta}{3}+\theta+3$ and
$\beta_{2}=\frac{\theta^{2}+4s\theta+6s^{2}}{3}+\theta$.
If $(\frac{-2B_{3}}{3})=1$, then $F_{S_{3}}(Y)=6s^{2}(Y+t)(Y-t),$ where
$6s^{2}t^{2}+B_{3}=0\,\ \mbox{\rm(mod }{3})$. Thus,
$3\mathbb{Z}_{K}=P_{1}P_{2}P_{3}P_{4}$, where $P_{3}$ and $P_{4}$ are
respectively attached to $Y+t$ and $Y-t$. Let $\phi=X+3^{r-2}t$ and consider
the $\phi$-adic development of $F(X)$, where $2s^{2}t^{2}+B_{3}=3\
\mbox{\rm(mod }{9})$. We have $v_{P_{3}}(\phi(\theta))=r-1$ and
$v_{P_{4}}(\phi(\theta))=r-2$. Therefore,
$\beta_{4}=\frac{\theta^{3}+4s\theta^{2}+6s^{2}(\theta+3^{r-2}t)}{3^{r-1}}$,
$\beta_{3}=\frac{\theta^{3}+4s\theta^{2}+6s^{2}(\theta-3^{r-2}t)}{3^{r-1}}$
and $\beta_{2}=\frac{\theta^{2}+4s\theta+6s^{2}}{3}+\theta$, where
$2s^{2}t^{2}+B_{3}=3\ \mbox{\rm(mod }{9})$ satisfy:
$v_{P_{3}}(\theta^{3}+4s\theta^{2})=v_{P_{4}}(\theta^{3}+4s\theta^{2})=2(r-1)$,
$v_{P_{3}}(\theta+3^{r-2}t)=r-1$, $v_{P_{4}}(\theta+3^{r-2}t)=r-2$,
$v_{P_{3}}(\theta-3^{r-2}t)=r-2$ and $v_{P_{4}}(\theta-3^{r-2}t)=r-1$. So,
$v_{P_{i}}(\beta_{i})=1$ and $v_{P_{j}}(\beta_{i})=0$ for every $i\neq j$.
$v_{p}(a)\geq 1$ and $v_{p}(b)=0$.
If $p\neq 2$, then $v_{p}(ind(P))=0$. Thus, $\alpha$ ia $p$-generator of
$\mathbb{Z}_{K}$, and then $p\mathbb{Z}_{K}$ is $p$-analogous to
$\bar{P}(X)=X^{4}+aX+b\,\ \mbox{\rm(mod }{p})$. For $p\geq 5$, if
$(\frac{-b}{p})_{4}=1$, then $\bar{P}(X)=(X-u)(X+u)(X^{2}+u^{2})$, where
$u^{4}=-b\,\ \mbox{\rm(mod }{p})$.
$(\frac{-b}{p})_{4}\neq 1$, then $\bar{P}(X)$ is irreducible in
$\mathbb{F}_{p}[X]$ or $\bar{P}(X)=(X^{2}+rX+s)(X^{2}-rX+t)\,\ \mbox{\rm(mod
}{p})$. In the last case, ($r=0\,\ \mbox{\rm(mod }{p})$, $s=-t\,\
\mbox{\rm(mod }{p})$ and $s^{2}=-b\,\ \mbox{\rm(mod }{p})$) or ($r\neq 0\,\
\mbox{\rm(mod }{p})$, $s=t\,\ \mbox{\rm(mod }{p})$, $2s=r^{2}\,\ \mbox{\rm(mod
}{p})$ and $s^{2}=b\,\ \mbox{\rm(mod }{p})$); i.e.,
$\bar{P}(X)=(X^{2}+s)(X^{2}-s)\,\ \mbox{\rm(mod }{p})$, where $s^{2}=b\,\
\mbox{\rm(mod }{p})$ or $\bar{P}(X)=(X^{2}+uX+s)(X^{2}-uX+s)\,\ \mbox{\rm(mod
}{p})$, where $u^{4}=4b\,\ \mbox{\rm(mod }{p})$ and $2s=u^{2}\,\ \mbox{\rm(mod
}{p})$.
For $p=2$, let $F(X)=P(X+1)=X^{4}+4x^{3}+6X^{2}+AX+B$ and $\theta=\alpha-1$.
1. (a)
If $v_{2}(B)=3$, $A=4\ \mbox{\rm(mod }{8})$, then for
$\phi=\frac{\theta^{3}+4\theta^{2}+6\theta}{4}$, we have
$C(X)=X^{4}+(3+2K)X^{3}+(2+2L)X^{2}+(3+2K)X+1+2L\ \mbox{\rm(mod }{4})$, where
$A=4+8K$ and $B=8+16L$. Since $C(1)=2\ \mbox{\rm(mod }{4})$, $\phi$ is a
$2$-generator of $\mathbb{Z}_{K}$, $2\mathbb{Z}_{K}=P_{1}^{2}P_{2}$, where
$P_{1}=(2,\phi+1)$ and $P_{2}=(2,\phi^{2}+\phi+1)$.
2. (b)
$b=3\ \mbox{\rm(mod }{8})$ and $a=4\ \mbox{\rm(mod }{8})$ or ($b=7\
\mbox{\rm(mod }{8})$, $a=0\ \mbox{\rm(mod }{8})$ and $1+b+a=0\ \mbox{\rm(mod
}{16})$), then let $s\in\mathbb{Z}$ such that $P(X)$ is $X+s$-regular. Let
$F(X)=P(X-s)=X^{4}+4sX^{3}+6s^{2}X^{2}+AX+B$ and $\theta=\alpha-s$. It follows
that:
1. (i)
$v_{2}(A)=1$ and $v_{2}(B)\geq 2$. Then $2\mathbb{Z}_{K}=P_{1}^{3}P_{2}$.
2. (ii)
If $v_{2}(A)=r$ and $v_{2}(B)=2r-1$, then $N_{X}(F)=S_{1}+S_{2}$ such that
$F_{S_{1}}(Y)=Y+1$ and $F_{S_{2}}(Y)=Y^{2}+Y+1$. Thus,
$2\mathbb{Z}_{K}=P_{1}^{2}P_{2}$, where $v_{P_{1}}(\theta)=1$ and
$v_{P_{2}}(\theta)=r-1$. Hence, for every $i\neq j$, $v_{P_{i}}(\beta_{i})=1$
and $v_{P_{i}}(\beta_{j})=0$.
3. (iii)
If $v_{2}(A)=r$ and $v_{2}(B)\geq 2r$, then $N_{X}(F)=S_{1}+S_{2}+S_{3}$ such
that $F_{S_{1}}(Y)=Y+1$ and $F_{S_{3}}(Y)=F_{S_{2}}(Y)=Y+1$. Thus,
$2\mathbb{Z}_{K}=P_{1}^{2}P_{2}P_{3}$. Let
$\beta_{3}=\frac{\theta^{3}+4s\theta^{2}+6s^{2}\theta}{2^{r}}$,
$\beta_{2}=\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+2^{r-1}t)}{2^{r}}$ and
$\beta_{1}=\frac{\theta^{2}+6}{2}+\theta$, where $t\in\mathbb{Z}$ is choused
such that $v_{P_{2}}(\theta+2^{r-1}t)=r$: Let $\phi(X)=X+2^{r-1}t$ and
consider the $\phi(X)$-adic of $F(X)$. If $v_{2}(B)=2r$, then $A_{2}t=3\
\mbox{\rm(mod }{4})$ and f $v_{2}(B)>2r$, then $A_{2}t=1\ \mbox{\rm(mod
}{4})$.
4. (iv)
If $v_{2}(A)=r+1+k$ and $v_{2}(B)=2r$, then $N_{X}(F)=S_{1}+S_{2}$ such that
$F_{S_{1}}(Y)=Y+1$ and $F_{S_{3}}(Y)=Y+1$. Thus,
$2\mathbb{Z}_{K}=P_{1}^{2}P_{2}^{2}$, where $v_{P_{1}}(\theta)=1$ and
$v_{P_{2}}(\theta)=2r-1$.
$v_{p}(ab)=0$.
If $p\in\\{2,3\\}$, then $v_{p}(ind(P))=0$ and then $p\mathbb{Z}_{K}$ is
$p$-analogous to $\bar{P}(X)$. For $p\geq 5$, let $s\in\mathbb{Z}$ such that
$3as+4b=0\ \mbox{\rm(mod }{p^{v_{p}(\triangle)+1}})$, $\theta=\alpha-s$ and
$F(X)=P(X+s)=X^{4}+4sX^{3}+6s^{2}X^{2}+AX+B$. Then
$v_{p}(A)=v_{p}(B)=v_{p}(\triangle)$.
If $v_{p}(\triangle)=0$, then $p\mathbb{Z}_{K}$ is $p$-analogous to
$\bar{P}(X)$. If $v_{p}(\triangle)=1$ and $(\frac{-2}{p})=1$, then
$p\mathbb{Z}_{K}=P_{1}^{2}P_{2}P_{3}$, where $P_{1}=(p,\theta)$,
$P_{2}=(p,\theta+u)$, $P_{3}=(p,\theta-u)$, $\theta=\alpha-s$ and $3at+4b=0\
\mbox{\rm(mod }{p^{2}})$.
If $v_{p}(\triangle)=1$ and $(\frac{-2}{p})=-1$, then
$p\mathbb{Z}_{K}=P_{1}^{2}P_{2}$, where $P_{1}=(p,\theta)$ and
$P_{2}=(p,\theta^{2}+4s\theta+6s^{2})$. If $v_{p}(\triangle)\geq 2$, then let
$s\in\mathbb{Z}$ such that $3as+4b=0\ \mbox{\rm(mod
}{p^{v_{p}(\triangle)+1}})$, $\theta=\alpha-s$ and
$F(X)=P(X+s)=X^{4}+4sX^{3}+6s^{2}X^{2}+AX+B$, where $B=s^{4}+as+b$ and
$A=4s^{3}+a$. Then $v_{p}(A)=v_{p}(B)=v_{p}(\triangle)$ and
$N_{X}(F)=S_{0}+S_{1}$ with respective slopes $0$ and
$\frac{v_{p}(\triangle)}{2}$. It follows that:
1. (a)
If $v_{p}(\triangle)=2r+1$, then $N_{X}(F)=S_{1}+S_{2}$ such that
$F_{S_{1}}(Y)=Y^{2}+4sY+6s^{2}$ and $F_{S_{2}}(Y)=6s^{2}Y+B_{p}$. Thus, if
$(\frac{-2}{p})=1$, then $p\mathbb{Z}_{K}=P_{1}P_{2}P_{3}^{2}$. If
$(\frac{-2}{p})=-1$, then $p\mathbb{Z}_{K}=P_{1}P_{2}^{2}$.
2. (b)
If $v_{p}(\triangle)=2r$, then $N_{X}(F)=S_{1}+S_{2}$ such that
$F_{S_{1}}(Y)=Y^{2}+4sY+6s^{2}$ and $F_{S_{2}}(Y)=6s^{2}Y^{2}+B_{p}$. Thus,
$(\frac{-2}{p})=1$ and $(\frac{-6B_{p}}{p})=-1$, then
$p\mathbb{Z}_{K}=P_{1}P_{2}P_{3}$.
If $(\frac{-2}{p})=-1$ and $(\frac{-6B_{p}}{p})=-1$, then
$p\mathbb{Z}_{K}=P_{1}P_{2}$.
If $(\frac{-2}{p})=-1$ and $(\frac{-6B_{p}}{p})=1$, then
$F_{S_{2}}(Y)=6s^{2}(Y-t)(Y+t)$, where $t$ is a root of $F_{S_{2}}(Y)$ in
$\mathbb{F}_{3}$. Thus, $p\mathbb{Z}_{K}=P_{1}P_{2}P_{3}$, where $P_{1}$,
$P_{2}$ and $P_{3}$ are respectively attached to $S_{1}$, $Y+t$ and $Y-t$. Let
$\beta_{2}=\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+p^{r}t)}{p^{r}}+\theta$
and
$\beta_{3}=\frac{(\theta^{2}+4s\theta+6s^{2})(\theta-p^{r}t)}{p^{r}}+\theta$,
where $t\in\mathbb{Z}$ is choused such that $v_{P_{2}}(\theta+p^{r}t)=r+1$,
$v_{P_{3}}(\theta+p^{r}t)=r$, $v_{P_{2}}(\theta+-p^{r}t)=r$ and
$v_{P_{3}}(\theta-p^{r}t)=r+1$: If $r\geq 2$, then $6s^{2}t^{2}+B_{p}=p\
\mbox{\rm(mod }{p^{2}})$. If $r=1$, then $6s^{2}t^{2}+B_{p}=0\ \mbox{\rm(mod
}{p^{2}})$. In that way, we have
$P_{2}=(p,\frac{(\theta^{2}+4s\theta+6s^{2})(\theta+p^{r}t)}{p^{r}}+\theta)$
and
$P_{3}=(p,\frac{(\theta^{2}+4s\theta+6s^{2})(\theta-p^{r}t)}{p^{r}}+\theta)$.
If $(\frac{-2}{p})=1$ and $(\frac{-6B_{p}}{p})=1$, then
$p\mathbb{Z}_{K}=P_{1}P_{2}P_{3}P_{4}$.
###### Examples 2.5.
Let $P(X)=X^{4}+aX+b\in\mathbb{Z}[X]$ be an irreducible polynomial, $\alpha$ a
complex root of $P(X)$ and $K={\mathbb{Q}}[\alpha]$.
1. (1)
$a=2^{10}.5$ and $b=2^{9}.3.5$. Let $\theta=\frac{\alpha}{4}$. Then $\theta$
is integral with minimal polynomial $F(X)=X^{4}+80X+30$ and
$K={\mathbb{Q}}[\theta]$. Thus $2\mathbb{Z}_{K}=(2,\theta)^{4}$,
$3\mathbb{Z}_{K}=(3,\theta)(3,\theta-1)^{3}$, $5\mathbb{Z}_{K}=(5,\theta)^{4}$
and for every $p\not\in\\{2,3,5\\}$, $p\mathbb{Z}_{K}$ is $p$-analogous to
$\bar{F}(X)$.
2. (2)
$a=48$ and $b=188$. From $TableA8$, row $A8.9$, we have
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2}{4})^{2}(2,\frac{\alpha^{2}+6}{4})^{2}$.
3. (3)
$a=144$ and $b=36$. From $TableA8$, row $A8.4$, we have
$2\mathbb{Z}_{K}=(2,\frac{\alpha^{2}+2\alpha+2}{4})^{4}$. Since
$(\frac{-b_{3}}{3})=-1$, from $TableA$, row $A4$ $3\mathbb{Z}_{K}=P^{2}$,
where $P=(3,\frac{\alpha^{3}}{3})$. Since $\triangle=-2^{14}.3^{6}.971$, for
every $p\not\in\\{2,3,5\\}$, $p\mathbb{Z}_{K}$ is $p$-analogous to
$\bar{P}(X)$.
4. (4)
$a=28$, $b=189$ and $p=2$. Let $F(X)=P(X+1)=X^{4}+4X^{3}+6X^{2}+32X+218$.
Since $B=218=2.109$, we have $2\mathbb{Z}_{K}=(2,\alpha-1)^{4}$.
5. (5)
$a=22$ and $b=66$. Then $\triangle=2^{4}.3^{5}.11^{3}.13$. We have
$2\mathbb{Z}_{K}=(2,\alpha)^{4}$, $11\mathbb{Z}_{K}=(11,\alpha)^{4}$,
$13\mathbb{Z}_{K}=(13,\alpha)(13,\alpha^{3}+9)$ and for every prime
$p\not\in\\{2,3,11,13\\}$, $p\mathbb{Z}_{K}$ is $p$-analogous to $\bar{P}(X)$.
For $p=3$, since $a^{2}=7\ \mbox{\rm(mod }{9}),\,b=3\ \mbox{\rm(mod }{9})$ and
$b+a^{4}-a^{2}=9.25982$, from $TableB$, row $B10$,
$3\mathbb{Z}_{K}=P_{1}P_{2}^{3}$, where $P_{1}=(3,\alpha-5a)$ and
$P_{2}=(3,\frac{\theta^{3}-4a\theta^{2}}{3})$.
6. (6)
$a=3^{6}.5^{5}.139$ and $b=2^{2}.3^{5}.5^{5}.139$. Let
$\theta=\frac{\alpha}{15}$. Then $\theta$ is integral with minimal polynomial
$F(X)=X^{4}+AX+B$ and $K={\mathbb{Q}}[\theta]$, where $A=3^{3}.5^{2}.139$ and
$B=2^{2}.3.5.139$. Thus
$2\mathbb{Z}_{K}=(2,\theta+1)(2,\theta^{2}+\theta+1)(2,\theta)$,
$3\mathbb{Z}_{K}=(3,\theta)^{4}$, $5\mathbb{Z}_{K}=(5,\theta)^{4}$,
$139\mathbb{Z}_{K}=(139,\theta)^{4}$. Since
$\triangle=2092367789117959822875=202317851.7.3^{3}.5^{3}.139^{3}.163$, for
every $p\not\in\\{3,5,139\\}$, $p\mathbb{Z}_{K}$ is $p$-analogous to
$\bar{F}(X)$.
## References
* [1] S. Alaca and K. S. Williams, $p$-integral basis of a quartic field defined by a trinomial $X^{4}+aX+b$, Far East J. Math. Sci. 12 (2004), 137-168.
* [2] Blair K. Spearman and Knneth S. Williams, The prime ideal factorization of 2 in pure quartic fields with index 2, Math. J. Okayama Univ 48(2006), 43-46.
* [3] H. Cohen, A course in computational algebraic number theory, GTM 138, Springer-Verlag Berlin Heidelberg, New York, Paris, Tokyo, second correction (1995).
* [4] J. Guardia, J. Montes and E. Nart, Newton Polygons of Higher Order in Algebraic Number Theory, arxiv.org/abs/0807.2620v2 [Math.NT] 31 October 2008\.
* [5] J. Guardia, J. Montes and E. Nart, Higher Newton polygons in the computation of discriminant and prime ideal decomposition in number fields, arXiv:0807.4065v3 [math.NT] 3 November 2008.
* [6] P. LLorente, E. Nart and N. Vila, Decompositin of primes in number fields defined by trinomials, J. Theorie des nombres de Bourdeaux, tome 1, 1(1991), 27-41.
Lhoussain El Fadil FPO, P.O. Box 638-Ouarzazte 45000, Morocco
lhouelfadil@hotmail.com
|
arxiv-papers
| 2010-08-22T01:34:41 |
2024-09-04T02:49:12.340935
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lhoussain El Fadil",
"submitter": "Lhoussain El fadil",
"url": "https://arxiv.org/abs/1008.3674"
}
|
1008.3726
|
# Hyers–Ulam stability of second-order linear dynamic equations on time scales
Douglas R. Anderson Department of Mathematics and Computer Science, Concordia
College, Moorhead, MN 56562 USA andersod@cord.edu
http://www.cord.edu/faculty/andersod/bib.html
###### Abstract.
We establish the stability of second-order linear dynamic equations on time
scales in the sense of Hyers and Ulam. To wit, if an approximate solution of
the second-order linear equation exists, then there exists an exact solution
to the dynamic equation that is close to the approximate one.
###### Key words and phrases:
Ordinary difference equations; ordinary dynamic equations; inhomogeneous
equations; time scales; reduction of order
###### 2000 Mathematics Subject Classification:
34N05, 26E70, 39A10
## 1\. introduction
In 1940, Ulam posed the following problem concerning the stability of
functional equations: give conditions in order for a linear mapping near an
approximately linear mapping to exist. The problem for the case of
approximately additive mappings was solved by Hyers who proved that the Cauchy
equation is stable in Banach spaces, and the result of Hyers was generalized
by Rassias.
Throughout this work we assume the reader has a working knowledge of time
scales.
###### Definition 1.1.
For real constants $\alpha$ and $\beta$, consider the second-order linear
dynamic equation
$x^{\Delta\Delta}(t)+\alpha x^{\Delta}(t)+\beta x(t)=0,\quad
t\in[a,b]_{\mathbb{T}}.$ (1.1)
If whenever $y\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]{{}_{\mathbb{T}}}$
satisfies
$\left|y^{\Delta\Delta}+\alpha y^{\Delta}+\beta y\right|\leq\varepsilon$
on $[a,b]_{\mathbb{T}}$, there exists a solution
$u\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of (1.1) such
that $|y-u|\leq K\varepsilon$ on $[a,b]_{\mathbb{T}}$ for some constant $K>0$,
then (1.1) has Hyers-Ulam stability $[a,b]_{\mathbb{T}}$.
###### Theorem 1.2 (Constant Coefficients).
If the characteristic equation $\lambda^{2}+\alpha\lambda+\beta=0$ has two
distinct positive roots, then (1.1) has Hyers-Ulam stability on
$[a,b]_{\mathbb{T}}$.
###### Proof.
Let $\varepsilon>0$ be given, and let
$y\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]_{\mathbb{T}}$ such that
$\left|y^{\Delta\Delta}+\alpha y^{\Delta}+\beta y\right|\leq\varepsilon$ on
$[a,b]_{\mathbb{T}}$. Let $\lambda_{1},\lambda_{2}$ be the distinct positive
roots of $\lambda^{2}+\alpha\lambda+\beta=0$. On $[a,b]_{\mathbb{T}}$ define
$g:=y^{\Delta}-\lambda_{1}y;$
then $g^{\Delta}=y^{\Delta\Delta}-\lambda_{1}y^{\Delta}$, so that
$\displaystyle|g^{\Delta}-\lambda_{2}g|=|y^{\Delta\Delta}-\lambda_{1}y^{\Delta}-\lambda_{2}y^{\Delta}+\lambda_{1}\lambda_{2}y|=|y^{\Delta\Delta}+\alpha
y^{\Delta}+\beta y|\leq\varepsilon$
on $[a,b]_{\mathbb{T}}$. Thus $-\varepsilon\leq
g^{\Delta}-\lambda_{2}g\leq\varepsilon$, or rewritten,
$\frac{-\varepsilon}{1+\mu\lambda_{2}}\leq
g^{\Delta}+(\ominus\lambda_{2})g^{\sigma}\leq\frac{\varepsilon}{1+\mu\lambda_{2}}.$
For the case $0<\lambda_{2}\leq 1$ there exists $M>0$ such that
$M\lambda_{2}>1$, so without loss of generality we can assume that
$\lambda_{2}>1$. Then
$\varepsilon(\ominus\lambda_{2})\leq
g^{\Delta}+(\ominus\lambda_{2})g^{\sigma}\leq-\varepsilon(\ominus\lambda_{2}).$
Multiply by $e_{\ominus\lambda_{2}}(\cdot,a)$ to see that
$\varepsilon\left(e_{\ominus\lambda_{2}}(\cdot,a)\right)^{\Delta}(t)\leq\left(ge_{\ominus\lambda_{2}}(\cdot,a)\right)^{\Delta}(t)\leq-\varepsilon\left(e_{\ominus\lambda_{2}}(\cdot,a)\right)^{\Delta}(t),$
so that delta integrating from $t$ to $b$ yields
$-\varepsilon\left(e_{\ominus\lambda_{2}}(t,a)-e_{\ominus\lambda_{2}}(b,a)\right)\leq
g(b)e_{\ominus\lambda_{2}}(b,a)-g(t)e_{\ominus\lambda_{2}}(t,a)\leq\varepsilon\left(e_{\ominus\lambda_{2}}(t,a)-e_{\ominus\lambda_{2}}(b,a)\right).$
Then
$-\varepsilon
e_{\ominus\lambda_{2}}(t,a)\leq\left(g(b)-\varepsilon\right)e_{\ominus\lambda_{2}}(b,a)-g(t)e_{\ominus\lambda_{2}}(t,a)\leq\varepsilon
e_{\ominus\lambda_{2}}(t,a)-2e_{\ominus\lambda_{2}}(b,a),$
whence
$-\varepsilon
e_{\ominus\lambda_{2}}(t,a)\leq\left(g(b)-\varepsilon\right)e_{\ominus\lambda_{2}}(b,a)-g(t)e_{\ominus\lambda_{2}}(t,a)\leq\varepsilon
e_{\ominus\lambda_{2}}(t,a).$
Multiplying the above inequality by $e_{\lambda_{2}}(t,a)$ results in
$-\varepsilon\leq\left(g(b)-\varepsilon\right)e_{\ominus\lambda_{2}}(b,t)-g(t)\leq\varepsilon.$
If we let
$z(t):=\left(g(b)-\varepsilon\right)e_{\ominus\lambda_{2}}(b,t)=\left(g(b)-\varepsilon\right)e_{\lambda_{2}}(t,b),\quad
t\in[a,b]_{\mathbb{T}},$
then clearly $z^{\Delta}(t)=\lambda_{2}z(t)$ and $|g(t)-z(t)|\leq\varepsilon$
for $t\in[a,b]_{\mathbb{T}}$. Since $g(t)=y^{\Delta}(t)-\lambda_{1}y(t)$ for
$t\in[a,b]_{\mathbb{T}}$, we have
$-\varepsilon\leq y^{\Delta}(t)-\lambda_{1}y(t)-z(t)\leq\varepsilon.$
By an argument similar to the one given above, we can show that there exists
$u(t):=(y(b)-\varepsilon)e_{\lambda_{1}}(t,b)-e_{\lambda_{1}}(t,a)\int_{t}^{b}\frac{z(s)}{1+\mu(s)\lambda_{1}}e_{\ominus\lambda_{1}}(s,a)\Delta
s$
such that $|y(t)-u(t)|\leq\varepsilon$ for $t\in[a,b]_{\mathbb{T}}$ and
$u\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]_{\mathbb{T}}$ satisfies
$u^{\Delta}-\lambda_{1}u-z=0$ on $[a,b]_{\mathbb{T}}$. Consequently
$u^{\Delta\Delta}(t)-(\lambda_{1}+\lambda_{2})u^{\Delta}(t)+\lambda_{1}\lambda_{2}u(t)=0,\quad
t\in[a,b]_{\mathbb{T}}$
that is
$u^{\Delta\Delta}+\alpha u^{\Delta}+\beta u=0$
on $[a,b]_{\mathbb{T}}$, completing the proof. ∎
For the next result consider the inhomogeneous second-order linear dynamic
equation
$x^{\Delta\Delta}(t)+\alpha x^{\Delta}(t)+\beta x(t)=f(t),\quad
t\in[a,b]_{\mathbb{T}}.$ (1.2)
###### Theorem 1.3 (Inhomogeneous with Constant Coefficients).
Assume the characteristic equation $\lambda^{2}+\alpha\lambda+\beta=0$ has two
distinct positive roots. For every $\varepsilon>0$,
$f\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$, and
$y\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]_{\mathbb{T}}$, if
$|y^{\Delta\Delta}+\alpha y^{\Delta}+\beta y-f|\leq\varepsilon$ (1.3)
on $[a,b]_{\mathbb{T}}$, then there exists a solution
$u\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of (1.2) such
that $|y-u|\leq K\varepsilon$ on $[a,b]_{\mathbb{T}}$ for some constant $K>0$,
that is to say (1.2) has Hyers-Ulam stability on $[a,b]_{\mathbb{T}}$.
The next theorem considers the inhomogeneous second-order linear dynamic
equation with variable coefficients
$x^{\Delta\Delta}(t)+p(t)x^{\Delta}(t)+q(t)x(t)=f(t),\quad
t\in[a,b]_{\mathbb{T}}.$ (1.4)
First we will need the following lemma.
###### Lemma 1.4.
Let $d,f\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$ such that
$1+\mu(t)d(t)\neq 0$ for all $t\in[a,b]_{\mathbb{T}}$ and
$\sup_{t\in[a,b]_{\mathbb{T}}}\left|e_{d}(t,a)\right|\int_{a}^{t}\left|e_{d}(a,\sigma(s))\right|\Delta
s<\infty.$
Let $x\in\operatorname{C^{\Delta}_{rd}}[a,b]_{\mathbb{T}}$. Then the first-
order dynamic equation
$x^{\Delta}(t)-d(t)x(t)-f(t)=0,\quad t\in[a,b]_{\mathbb{T}}$ (1.5)
has Hyers-Ulam stability, that is whenever
$g\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ satisfies
$\left|g^{\Delta}(t)-d(t)g(t)-f(t)\right|\leq\varepsilon,\quad
t\in[a,b]_{\mathbb{T}}$
there exists a solution
$w\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of (1.5) such that
$|g-w|\leq L\varepsilon$ on $[a,b]_{\mathbb{T}}$ for some constant $L>0$.
###### Proof.
Given $\varepsilon>0$, suppose there exists
$g\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ that satisfies
$\left|g^{\Delta}(t)-d(t)g(t)-f(t)\right|\leq\varepsilon,\quad
t\in[a,b]_{\mathbb{T}}.$
Set
$\ell:=g^{\Delta}-dg-f;$
by [1, Theorem 2.77] we have that $g$ is given by
$g(t)=e_{d}(t,a)g(a)+\int_{a}^{t}e_{d}(t,\sigma(s))\left(f(s)+\ell(s)\right)\Delta
s.$
Let $w$ be the unique solution of the initial value problem
$w^{\Delta}-dw-f=0,\quad w(a)=g(a).$
Then
$w(t)=e_{d}(t,a)g(a)+\int_{a}^{t}e_{d}(t,\sigma(s))f(s)\Delta s,$
and
$\displaystyle|g(t)-w(t)|$ $\displaystyle=$
$\displaystyle\left|\int_{a}^{t}e_{d}(t,\sigma(s))\ell(s)\Delta s\right|$
$\displaystyle\leq$
$\displaystyle\left|e_{d}(t,a)\int_{a}^{t}e_{d}(a,\sigma(s))\ell(s)\Delta
s\right|$ $\displaystyle\leq$
$\displaystyle\varepsilon\sup_{t\in[a,b]_{\mathbb{T}}}\left|e_{d}(t,a)\right|\int_{a}^{t}\left|e_{d}(a,\sigma(s))\right|\Delta
s$ $\displaystyle\leq$ $\displaystyle L\varepsilon$
for all $t\in[a,b]_{\mathbb{T}}$, where
$L:=\sup_{t\in[a,b]_{\mathbb{T}}}\left|e_{d}(t,a)\right|\int_{a}^{t}\left|e_{d}(a,\sigma(s))\right|\Delta
s$ is a constant independent of $g$ and $\varepsilon$. Since $w$ solves (1.5)
by construction, the proof in complete. ∎
###### Theorem 1.5 (Inhomogeneous with Variable Coefficients).
Let $p,q,f\in\operatorname{C_{rd}}[a,b]_{\mathbb{T}}$ and consider (1.4).
Assume the related dynamic Riccati equation
$z^{\Delta}(t)+p(t)z(t)-z(t)z^{\sigma}(t)=q(t),\quad t\in[a,b]_{\mathbb{T}}$
has a particular solution $z$ with both $1+\mu(t)(z^{\sigma}(t)-p(t))\neq 0$
and $1-\mu(t)z(t)\neq 0$ for all $t\in[a,b]_{\mathbb{T}}$. Furthermore assume
that
$\sup_{t\in[a,b]_{\mathbb{T}}}\left|e_{(z^{\sigma}-p)}(t,a)\right|\int_{a}^{t}\left|e_{(z^{\sigma}-p)}(a,\sigma(s))\right|\Delta
s<\infty$ (1.6)
and
$\sup_{t\in[a,b]_{\mathbb{T}}}\left|e_{-z}(t,a)\right|\int_{a}^{t}\left|e_{-z}(a,\sigma(s))\right|\Delta
s<\infty.$ (1.7)
Then (1.4) has Hyers-Ulam stability on $[a,b]_{\mathbb{T}}$.
###### Proof.
We need to show that if there exists a
$y\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]_{\mathbb{T}}$ that satisfies
$|y^{\Delta\Delta}(t)+p(t)y^{\Delta}(t)+q(t)y(t)-f(t)|\leq\varepsilon$ (1.8)
for $t\in[a,b]_{\mathbb{T}}$, and the dynamic Riccati equation
$z^{\Delta}(t)+p(t)z(t)-z(t)z^{\sigma}(t)=q(t)$
has a particular solution $z$ with both $1+\mu(t)(z^{\sigma}(t)-p(t))\neq 0$
and $1-\mu(t)z(t)\neq 0$ for $t\in[a,b]_{\mathbb{T}}$ such that (1.6) and
(1.7) hold, then there exists a solution
$u\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of (1.4) such
that $|y-u|\leq K\varepsilon$ on $[a,b]_{\mathbb{T}}$ for some constant $K>0$.
Let $\varepsilon>0$ be given, and let
$y\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]_{\mathbb{T}}$ such that
$\left|y^{\Delta\Delta}+py^{\Delta}+qy-f\right|\leq\varepsilon$ on
$[a,b]_{\mathbb{T}}$. Assume $z$ is a particular solution of the Riccati
equation $z^{\Delta}+pz-zz^{\sigma}=q$ on $[a,b]_{\mathbb{T}}$, and set
$g:=y^{\Delta}+zy,\qquad d:=z^{\sigma}-p.$
Then $g^{\Delta}=y^{\Delta\Delta}+z^{\sigma}y^{\Delta}+z^{\Delta}y$, so that
$\displaystyle|g^{\Delta}-dg-f|=|y^{\Delta\Delta}+z^{\sigma}y^{\Delta}+z^{\Delta}y-(z^{\sigma}-p)(y^{\Delta}+zy)-f|=|y^{\Delta\Delta}+py^{\Delta}+qy-f|\leq\varepsilon$
on $[a,b]_{\mathbb{T}}$. As all of the hypotheses of Lemma 1.4 hold, equation
(1.5) has Hyers-Ulam stability, and there exists a solution
$w\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of
$w^{\Delta}(t)-d(t)w(t)-f(t)=0,\quad t\in[a,b]_{\mathbb{T}}$ (1.9)
where $w$ is given by
$w(t)=e_{d}(t,a)g(a)+\int_{a}^{t}e_{d}(t,\sigma(s))f(s)\Delta s,$
and there exists an $L>0$ such that
$|g(t)-w(t)|\leq L\varepsilon,\quad t\in[a,b]_{\mathbb{T}}.$
Since $g=y^{\Delta}+zy$, we have that
$|y^{\Delta}(t)+z(t)y(t)-w(t)|\leq L\varepsilon,\quad t\in[a,b]_{\mathbb{T}}.$
Again apply Lemma 1.4 to see that there exists a solution
$u\in\operatorname{C^{\Delta}_{rd}}[a,b]{{}_{\mathbb{T}}}$ of
$u^{\Delta}(t)+z(t)u(t)-w(t)=0,\quad t\in[a,b]_{\mathbb{T}}$ (1.10)
given by
$u(t)=e_{-z}(t,a)y(a)+\int_{a}^{t}e_{-z}(t,\sigma(s))w(s)\Delta s,$
and there exists an $K>0$ such that
$|y(t)-u(t)|\leq KL\varepsilon,\quad t\in[a,b]_{\mathbb{T}}.$
Moreover,
$\displaystyle u^{\Delta\Delta}+pu^{\Delta}+qu-f$ $\displaystyle=$
$\displaystyle w^{\Delta}-z^{\sigma}u^{\Delta}-z^{\Delta}u+pu^{\Delta}+qu-f$
$\displaystyle=$
$\displaystyle(dw+f)-(d+p)u^{\Delta}+(q-z^{\Delta})u+pu^{\Delta}-f$
$\displaystyle=$ $\displaystyle d(w-u^{\Delta}-zu)$ $\displaystyle=$
$\displaystyle 0$
on $[a,b]_{\mathbb{T}}$, so that $u$ is a solution of (1.4), and actually
$u\in\operatorname{C^{\Delta^{2}}_{rd}}[a,b]{{}_{\mathbb{T}}}$. ∎
## References
* [1] M. Bohner and A. Peterson, _Dynamic Equations on Time Scales, An Introduction with Applications_ , Birkhäuser, Boston, 2001.
* [2] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, _Results Math._ 18 (1990) 18–56.
* [3] S. M. Jung, Hyers–Ulam Stability of Linear Differential Equations of First Order, I, _International Journal of Applied Mathematics & Statistics_ Vol. 7, No. Fe07 (2007) 96–100.
* [4] Y. J. Li and Y. Shen, Hyers–Ulam stability of linear differential equations of second order, _Applied Mathematics Letters_ 23 (2010) 306–309.
* [5] Y. J. Li and Y. Shen, Hyers–Ulam stability of nonhomogeneous linear differential equations of second order, _International Journal of Mathematics and Mathematical Sciences_ Vol. 2009, Article ID 576852, 7 pages.
* [6] T. Miura, S. Miyajima, and S. Takahasi, A characterization of Hyers-Ulam stability of first order linear differential operators, _Journal of Mathematical Analysis and Applications_ Vol. 286, Issue 1 (2003) 136–146.
* [7] D. Popa, Hyers–Ulam stability of the linear recurrence with constant coefficients, _Advances in Difference Equations_ 2005:2 (2005) 101–107.
|
arxiv-papers
| 2010-08-22T22:43:39 |
2024-09-04T02:49:12.348932
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Douglas R. Anderson",
"submitter": "Douglas R. Anderson",
"url": "https://arxiv.org/abs/1008.3726"
}
|
1008.3763
|
# RIGHT AND LEFT MODULES OVER THE FROBENIUS SKEW POLYNOMIAL RING IN THE
$F$-FINITE CASE
RODNEY Y. SHARP Department of Pure Mathematics, University of Sheffield,
Hicks Building, Sheffield S3 7RH, United Kingdom R.Y.Sharp@sheffield.ac.uk
and YUJI YOSHINO Department of Mathematics, Faculty of Science, Okayama
University, Tsushima-Naka 3-1-1, Okayama 700-8530, Japan
yoshino@math.okayama-u.ac.jp
###### Abstract.
The main purposes of this paper are to establish and exploit the result that,
over a complete (Noetherian) local ring $R$ of prime characteristic for which
the Frobenius homomorphism $f$ is finite, the appropriate restrictions of the
Matlis-duality functor provide an equivalence between the category of left
modules over the Frobenius skew polynomial ring $R[x,f]$ that are Artinian as
$R$-modules and the category of right $R[x,f]$-modules that are Noetherian as
$R$-modules.
###### Key words and phrases:
Commutative Noetherian ring, prime characteristic, Frobenius homomorphism,
skew polynomial ring, Matlis duality.
###### 2000 Mathematics Subject Classification:
Primary 13A35, 16S36, 13E05, 13E10, 13J10
The first author was partially supported by the Engineering and Physical
Sciences Research Council of the United Kingdom (Overseas Travel Grant Number
EP/C538803/1), and also by the Foundation for International Exchange Program
of Okayama University. The second author was partially supported by Japan
Society for the Promotion of Science (Grant-in-Aid (B) 21340008).
## 0\. Introduction
Throughout the paper, $R$ will denote a commutative Noetherian ring of prime
characteristic $p$. We shall only assume that $R$ is local when this is
explicitly stated; then, the notation ‘$(R,{\mathfrak{m}})$’ will denote that
${\mathfrak{m}}$ is the maximal ideal of $R$. We shall always denote by
$f:R\longrightarrow R$ the Frobenius homomorphism, for which $f(r)=r^{p}$ for
all $r\in R$. We shall work with the skew polynomial ring $R[x,f]$ associated
to $R$ and $f$ in the indeterminate $x$ over $R$. Recall that $R[x,f]$ is, as
a left $R$-module, freely generated by $(x^{i})_{i\in{\mathbb{N}_{0}}}$ (we
use $\mathbb{N}$ and ${\mathbb{N}_{0}}$ to denote the set of positive integers
and the set of non-negative integers, respectively), and so consists of all
polynomials $\sum_{i=0}^{n}r_{i}x^{i}$, where $n\in{\mathbb{N}_{0}}$ and
$r_{0},\ldots,r_{n}\in R$; however, its multiplication is subject to the rule
$xr=f(r)x=r^{p}x\quad\mbox{~{}for all~{}}r\in R\/.$
Note that $R[x,f]$ can be considered as a positively-graded ring
$R[x,f]=\bigoplus_{n=0}^{\infty}R[x,f]_{n}$, with $R[x,f]_{n}=Rx^{n}$ for all
$n\in{\mathbb{N}_{0}}$. The ring $R[x,f]$ will be referred to as the Frobenius
skew polynomial ring over $R$.
In the case when $(R,{\mathfrak{m}})$ is local, several authors have used,
often as an aid to the study of tight closure, the natural Frobenius action on
the top local cohomology module $H^{\dim R}_{{\mathfrak{m}}}(R)$ of $R$: see,
for example, R. Fedder [5], Fedder and K.-i. Watanabe [6], K. E. Smith [23],
N. Hara and Watanabe [8] and F. Enescu [2], [3]. The natural Frobenius action
provides the top local cohomology module of $R$ with a natural structure as a
left module over $R[x,f]$. The top local cohomology module of $R$ is Artinian
as $R$-module, and so the papers cited above studied one example of a left
$R[x,f]$-module that is Artinian as $R$-module. In recent years there have
been studies of more general left $R[x,f]$-modules that are Artinian as
$R$-modules: see, for example, M. Katzman [12] and the first author’s [18],
[20] and [21] (the authors are listed alphabetically).
On the other hand, the second author showed in [24, Proposition 3.5] that, if
$R$ is $F$-finite, that is, the Frobenius map $f:R\longrightarrow R$ is a
finite homomorphism, then each non-zero injective $R$-module $I$ has a non-
trivial structure as a right $R[x,f]$-module. The main purpose of this paper
is to build on that work to show that, when $R$ is $F$-finite, whenever $M$ is
a left $R[x,f]$-module, then $\operatorname{Hom}_{R}(M,I)$ can be given a
structure as right $R[x,f]$-module that extends its $R$-module structure, and,
furthermore, whenever $N$ is a right $R[x,f]$-module, then
$\operatorname{Hom}_{R}(N,I)$ can be given a structure as left $R[x,f]$-module
that extends its $R$-module structure. Special attention is given to the case
where $(R,{\mathfrak{m}})$ is local, complete and $F$-finite, and $I$ is taken
to be $E:=E_{R}(R/{\mathfrak{m}})$, the injective envelope of the simple
$R$-module. Classical Matlis duality yields that whenever $G$ is an $R$-module
that is Artinian (respectively Noetherian), then the natural ‘evaluation’
$R$-homomorphism
$G\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$ is an
isomorphism, and the ‘Matlis dual’ $\operatorname{Hom}_{R}(G,E)$ of $G$ is
Noetherian (respectively Artinian). Our results, when combined with Matlis
duality, lead to the conclusion that the appropriate restrictions of the
functor $\operatorname{Hom}_{R}(-,E)$ provide an equivalence between the
category of left $R[x,f]$-modules that are Artinian as $R$-modules (and all
$R[x,f]$-homomorphisms between them) and the category of right
$R[x,f]$-modules that are Noetherian as $R$-modules (and all
$R[x,f]$-homomorphisms between them).
We can then use this equivalence to translate (in this complete, local,
$F$-finite case) known results about left $R[x,f]$-modules that are Artinian
as $R$-modules into results about right $R[x,f]$-modules that are Noetherian
as $R$-modules. One example of this concerns the Hartshorne–Speiser–Lyubeznik
Theorem, which we now recall.
###### 0.1 Theorem (G. Lyubeznik [14, Proposition 4.4]).
(Compare Hartshorne–Speiser [9, Proposition 1.11].) Suppose that
$(R,{\mathfrak{m}})$ is local, and let $G$ be a left $R[x,f]$-module that is
Artinian as $R$-module. Then there exists $e\in{\mathbb{N}_{0}}$ with the
following property: whenever $g\in G$ is such that $x^{n}g=0$ for some
$n\in\mathbb{N}$, then $x^{e}g=0$.
Hartshorne and Speiser first proved this result in the case where $R$ is local
and contains its residue field which is perfect. Lyubeznik applied his theory
of $F$-modules to obtain the result without restriction on the local ring $R$
of characteristic $p$. There is a short proof of the
Hartshorne–Speiser–Lyubeznik Theorem in [19]. It was shown in [17, Corollary
1.8] that the result is still valid if the hypothesis that $R$ be local is
dropped.
The Hartshorne–Speiser–Lyubeznik Theorem has been used to establish the
existence of uniform test exponents for Frobenius closures of parameter ideals
in local rings in certain circumstances. Let ${\mathfrak{a}}$ be an ideal of
$R$; let $n\in{\mathbb{N}_{0}}$. Recall that the $n$-th Frobenius power
${\mathfrak{a}}^{[p^{n}]}$ of ${\mathfrak{a}}$ is the ideal of $R$ generated
by all $p^{n}$-th powers of elements of ${\mathfrak{a}}$. The Frobenius
closure ${\mathfrak{a}}^{F}$ of ${\mathfrak{a}}$ is defined by
${\mathfrak{a}}^{F}:=\big{\\{}r\in R\ |\ \mbox{there
exists~{}}n\in{\mathbb{N}_{0}}\mbox{~{}such
that~{}}r^{p^{n}}\in{\mathfrak{a}}^{[p^{n}]}\big{\\}}.$
This is an ideal of $R$, and so is finitely generated; therefore there exists
a power $Q_{0}$ of $p$ such that
$({\mathfrak{a}}^{F})^{[Q_{0}]}={\mathfrak{a}}^{[Q_{0}]}$, and we define
$Q({\mathfrak{a}})$ to be the smallest power of $p$ with this property. In
[13, Theorem 2.5], M. Katzman and Sharp used the Hartshorne–Speiser–Lyubeznik
Theorem to show that, when $(R,{\mathfrak{m}})$ is local and Cohen–Macaulay,
the set
$\\{Q({\mathfrak{a}}):{\mathfrak{a}}\mbox{~{}is an ideal generated by part of
a system of parameters of~{}}R\\}$
is bounded; in [11], C. Huneke, Katzman, Sharp and Y. Yao again used the
Hartshorne–Speiser–Lyubeznik Theorem (and quite a few other techniques) to
establish the same conclusion in a generalized Cohen–Macaulay local ring.
We are able to use our above-mentioned equivalence of categories to prove the
following result (as Theorem 3.1), in the case where $R$ is $F$-finite, local
and complete.
Theorem. Assume that $(R,{\mathfrak{m}})$ is $F$-finite, local and complete.
Let $N$ be a right $R[x,f]$-module that is Noetherian as $R$-module. Then
there exists $e\in{\mathbb{N}_{0}}$ such that $Nx^{e}=Nx^{e+1}$.
This result can be viewed as a dual of the Hartshorne–Speiser–Lyubeznik
Theorem. A natural question is whether this ‘dual Hartshorne–Speiser–Lyubeznik
Theorem’ is still valid if all the hypotheses about $R$, except the one that
it (is a commutative Noetherian ring and) has characteristic $p$, are dropped:
we shall show, in the final section of the paper, that this question has an
affirmative answer.
Another useful result about left $R[x,f]$-modules that are Artinian as
$R$-modules concerns graded annihilators: the graded annihilator of a (left or
right) $R[x,f]$-module $T$ is the largest graded two-sided ideal of $R[x,f]$
that annihilates $T$.
###### 0.2 Theorem (R. Y. Sharp [18, Corollary 3.11]).
Let $G$ be a left $R[x,f]$-module that is Artinian as $R$-module. Suppose that
$G$ is $x$-torsion-free, that is, $xg=0$ for $g\in G$ implies that $g=0$. Then
there are only finitely many graded annihilators of $R[x,f]$-submodules of
$G$.
The first author has been able to use this result to prove existence theorems
about tight closure test elements: see [21, Theorem 4.16].
We are able to use our above-mentioned equivalence of categories to prove the
following result (as Theorem 3.5), in the case where $R$ is $F$-finite, local
and complete.
Theorem. Assume that $(R,{\mathfrak{m}})$ is $F$-finite, local and complete.
Let $M$ be a right $R[x,f]$-module that is Noetherian as $R$-module. Suppose
that $M$ is $x$-divisible, that is $M=Mx$. Then there are only finitely many
graded annihilators of $R[x,f]$-homomorphic images of $M$.
Again, it is natural to ask whether this result is still valid if all the
hypotheses about $R$, except the one that it (is a commutative Noetherian ring
and) has characteristic $p$, are dropped. At the time of writing, we have not
been able to answer to this question.
Note. Most of the research reported in this paper was carried out during a
visit by Sharp to the University of Okayama in March 2008\. After the paper
had been accepted, it was pointed out to us that some of its results have been
independently obtained by M. Blickle and G. Boeckle in their paper [1]. In
detail, Theorem 1.20 below appears in [1, Section 5.1], and the result of
Theorem 3.4 below follows from [1, Proposition 2.14] (which Blickle and
Boeckle prove via an argument of O. Gabber from [7, Section 13]).
## 1\. Right and left modules over the Frobenius skew polynomial ring
The notation and terminology used in the Introduction will be used throughout
the paper.
First of all, let us recall some of the basic facts about bimodules, which we
shall use in the rest of the paper. See, for example, Rotman [16, Lemma 8.80,
Theorem 8.99].
###### 1.1 Remark.
Let $A$, $B$, $C$ and $D$ be commutative rings.
* (i)
An Abelian group $M$ is an $(A,B)$-bimodule if $M$ is a left $A$-module, a
right $B$-module and the two actions of the rings are related by the following
rule:
$(am)b=a(mb)\quad\text{for all}\ a\in A,\ b\in B\ \text{and}\ m\in M.$
* (ii)
If $M$ is an $(A,B)$-bimodule and $N$ is a $(B,C)$-bimodule, then
$M\otimes_{B}N$ is naturally an $(A,C)$-bimodule, where the bimodule structure
is given by
$a(m\otimes n)c=(am)\otimes(nc)\quad\text{for all}\ a\in A,\ c\in C,\ m\in M\
\text{and}\ n\in N.$
* (iii)
If $M$ is an $(A,B)$-bimodule and $N$ is an $(A,C)$-bimodule, then the set of
all left $A$-homomorphisms from $M$ to $N$, denoted by
$\operatorname{Hom}_{lA}(M,N)$, is naturally a $(B,C)$-bimodule, where
$(b\varphi c)(m)=(\varphi(mb))c\quad\text{for all}\ b\in B,\ c\in C,\ m\in M\
\text{and}\ \varphi\in\operatorname{Hom}_{lA}(M,N).$
Similarly if $M$ is an $(A,B)$-bimodule and $N$ is a $(C,B)$-bimodule, then
the set of all right $B$-homomorphisms from $M$ to $N$, denoted by
$\operatorname{Hom}_{rB}(M,N)$, is naturally a $(C,A)$-bimodule, where
$(c\psi a)(m)=c(\psi(am))\quad\text{for all}\ c\in C,\ a\in A,\ m\in M\
\text{and}\ \psi\in\operatorname{Hom}_{rB}(M,N).$
* (iv)
If $M$ is an $(A,B)$-bimodule, $N$ is a $(B,C)$-bimodule, and $L$ is an
$(A,D)$-bimodule, then there exists a $(C,D)$-bimodule isomorphism, the so-
called adjoint isomorphism,
$\Xi:\operatorname{Hom}_{lA}(M\otimes_{B}N,L)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{lB}(N,\operatorname{Hom}_{lA}(M,L))$
which is such that
$((\Xi(\phi))(n))(m)=\phi(m\otimes n)\quad\text{for all}\ n\in N,\ m\in M\
\text{and}\ \phi\in\operatorname{Hom}_{lA}(M\otimes_{B}N,L).$
* (v)
Similarly, if $M$ is an $(A,B)$-bimodule, $N$ is a $(B,C)$-bimodule, and $L$
is a $(D,C)$-bimodule, then there exists an ‘adjoint’ $(D,A)$-bimodule
isomorphism
$\Theta:\operatorname{Hom}_{rC}(M\otimes_{B}N,L)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rB}(M,\operatorname{Hom}_{rC}(N,L))$
which is such that
$((\Theta(\varphi))(m))(n)=\varphi(m\otimes n)\quad\text{for all}\ m\in M,\
n\in N\ \text{and}\ \varphi\in\operatorname{Hom}_{rC}(M\otimes_{B}N,L).$
* (vi)
If $M$ is an $(A,B)$-bimodule, $N$ is a $(C,D)$-bimodule, and $L$ is an
$(A,D)$-bimodule, then there exists a $(B,C)$-bimodule isomorphism
$\Omega:\operatorname{Hom}_{lA}(M,\operatorname{Hom}_{rD}(N,L))\cong\operatorname{Hom}_{rD}(N,\operatorname{Hom}_{lA}(M,L))$
for which
$((\Omega(\psi))(n))(m)=(\psi(m))(n)\quad\text{for all}\ m\in M,\ n\in N\
\text{and}\ \psi\in\operatorname{Hom}_{lA}(M,\operatorname{Hom}_{rD}(N,L)).$
Recall that $R$ denotes a commutative Noetherian ring of prime characteristic
$p$ and that $f:R\longrightarrow R$ denotes the Frobenius homomorphism. We
shall only assume that $R$ is $F$-finite when this is explicitly stated.
Let $M$ be an $R$-module. We always regard $M$ as an $(R,R)$-bimodule by
$r\cdot m\cdot s=rsm$ for $r,s\in R$ and $m\in M$. On the other hand, we
define the $(R,R)$-bimodule $M_{f}$ to be $M_{f}=M$ as Abelian group with
$(R,R)$-bimodule structure defined by
$r\cdot m\cdot s=rs^{p}m\quad\text{for all~{}}r,s\in R\ \text{and}\ m\in M.$
Note that the Frobenius map $f:R\to R_{f}$ is a right $R$-module homomorphism.
Similarly, we define the $(R,R)$-bimodule ${}_{f}M$ to be ${}_{f}M=M$ as
Abelian group with $(R,R)$-bimodule structure defined by
$r\cdot m\cdot s=r^{p}sm\quad\text{for all~{}}r,s\in R\ \text{and}\ m\in M.$
###### 1.2 Remark.
Let $M$ be an $R$-module.
* (i)
By 1.1(ii), $M\otimes_{R}R_{f}$ has naturally a structure of $(R,R)$-bimodule.
The action of $R$ is given by
$s\cdot(m\otimes r)\cdot s^{\prime}=srs^{\prime p}m\otimes 1\quad\text{for
all~{}}r,s,s^{\prime}\in R\ \text{and}\ m\in M.$
Thus there is an isomorphism $M_{f}\cong M\otimes_{R}R_{f}$ as
$(R,R)$-bimodules.
* (ii)
Similarly, ${}_{f}R\otimes_{R}M$ is an $(R,R)$-bimodule with action given by
$s\cdot(r\otimes m)\cdot s^{\prime}=1\otimes s^{p}rs^{\prime}m\quad\text{for
all~{}}r,s,s^{\prime}\in R\ \text{and}\ m\in M.$
There is an isomorphism ${}_{f}M\cong{}_{f}R\otimes_{R}M$ of
$(R,R)$-bimodules.
* (iii)
By 1.1(iii), the Abelian group $\operatorname{Hom}_{lR}(R_{f},M)$ consisting
of all left $R$-homomorphisms from $R_{f}$ to $M$ is an $(R,R)$-bimodule with
action of $R$ given by
$(s\varphi s^{\prime})(r)=(\varphi(r\cdot
s))s^{\prime}=rs^{p}s^{\prime}\varphi(1)\quad\text{for
all~{}}r,s,s^{\prime}\in R\ \text{and}\
\varphi\in\operatorname{Hom}_{lR}(R_{f},M).$
It is easy to see that $\operatorname{Hom}_{lR}(R_{f},M)\cong{}_{f}M$ as
$(R,R)$-bimodules.
* (iv)
Similarly, the set $\operatorname{Hom}_{rR}(R_{f},M)$ of all right
$R$-homomorphisms from $R_{f}$ to $M$ is an $(R,R)$-bimodule with
$(s\psi s^{\prime})(r)=s\psi(s^{\prime}r)\quad\text{for
all~{}}r,s,s^{\prime}\in R\ \text{and}\
\psi\in\operatorname{Hom}_{rR}(R_{f},M).$
We shall use a refinement of the following result.
###### 1.3 Lemma (Y. Yoshino [24, Lemma 3.6]).
Suppose that $(R,{\mathfrak{m}})$ is local and $F$-finite. Denote by $E$ the
injective envelope $E_{R}(R/{\mathfrak{m}})$ of the simple $R$-module, which
we regard as a right $R$-module. Then there is a right $R$-module isomorphism
$E\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rR}(R_{f},E)$,
where $\operatorname{Hom}_{rR}(R_{f},E)$ carries the right $R$-module
structure described in Remark 1.2(iv).
We shall use the following refinement, in which it is not assumed that $R$ is
local.
###### 1.4 Lemma.
Suppose that $R$ is $F$-finite, and let $I$ be an injective $R$-module. Then
there is an $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{lR}(R_{f},I)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rR}(R_{f},I).$
###### Proof.
It is a consequence of the adjoint isomorphism of Remark 1.1(v) that
$\operatorname{Hom}_{rR}(R_{f},I)$ is injective as right $R$-module. On the
other hand, by 1.2(iii), we have an isomorphism of $(R,R)$-bimodules
$\operatorname{Hom}_{lR}(R_{f},I)\cong{}_{f}I$.
We can use the well-known decomposition theory for injective $R$-modules due
to E. Matlis (reviewed in, for example, [15, §18]) to see that it is enough
for us to prove the result when $I=E_{R}(R/{\mathfrak{p}})$ for a prime ideal
${\mathfrak{p}}$ of $R$, and so we assume that this is so in the rest of the
proof.
Since, for a prime ideal ${\mathfrak{q}}$ of $R$, each element of
$E_{R}(R/{\mathfrak{q}})$ is annihilated by some power of ${\mathfrak{q}}$,
and multiplication by an element $r\in R\setminus{\mathfrak{q}}$ provides an
automorphism of $E_{R}(R/{\mathfrak{q}})$, it follows that
$\operatorname{Hom}_{rR}(R_{f},E_{R}(R/{\mathfrak{p}}))$ (with the right
$R$-module structure described in Remark 1.2(iv)) is isomorphic to a direct
sum of $\mu$ copies of $E_{R}(R/{\mathfrak{p}})$. First we prove that the
cardinal $\mu$ is exactly $1$. Thus
$\operatorname{Hom}_{rR}(R_{f},E_{R}(R/{\mathfrak{p}}))\cong\bigoplus\mu\left(E_{R}(R/{\mathfrak{p}})\right),$
as right $R$-modules. We consider $(R_{f})_{{\mathfrak{p}}}$ as the
localization of the right $R$-module $R_{f}$ at ${\mathfrak{p}}$ and write the
resulting action of $R_{{\mathfrak{p}}}$ on the right. (Thus
$(r/s)\cdot(a/t)=ra^{p}/st$ for $r\in R_{f},\ a\in R$ and $s,t\in
R\setminus{\mathfrak{p}}$.) We can also endow this $(R_{f})_{{\mathfrak{p}}}$
with a left $R_{{\mathfrak{p}}}$-module structure under which
$\left(\frac{a}{t}\right)\cdot\left(\frac{r}{s}\right)=\frac{t^{p-1}ar}{st}\quad\text{for
all~{}}r\in R_{f},\ a\in R\ \text{and}\ s,t\in R\setminus{\mathfrak{p}}.$
These two structures turn $(R_{f})_{{\mathfrak{p}}}$ into an
$(R_{{\mathfrak{p}}},R_{{\mathfrak{p}}})$-bimodule, and then there is an
$(R_{{\mathfrak{p}}},R_{{\mathfrak{p}}})$-bimodule isomorphism
$\beta:(R_{f})_{{\mathfrak{p}}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(R_{{\mathfrak{p}}})_{f}$
for which $\beta(r/s)=r/s^{p}$ for all $r\in R_{f}$ and $s\in
R\setminus{\mathfrak{p}}$. Since $R_{f}$ is finitely generated as right
$R$-module, there is a right $R_{{\mathfrak{p}}}$-module isomorphism
$\left(\operatorname{Hom}_{rR}(R_{f},E_{R}(R/{\mathfrak{p}}))\right)_{{\mathfrak{p}}}\cong\operatorname{Hom}_{rR_{{\mathfrak{p}}}}\\!\left((R_{f})_{{\mathfrak{p}}},(E_{R}(R/{\mathfrak{p}}))_{{\mathfrak{p}}}\right)$
when $\operatorname{Hom}_{rR}(R_{f},E_{R}(R/{\mathfrak{p}}))$ (respectively
$\operatorname{Hom}_{rR_{{\mathfrak{p}}}}\\!\left((R_{f})_{{\mathfrak{p}}},(E_{R}(R/{\mathfrak{p}}))_{{\mathfrak{p}}}\right)$)
is considered as a right $R$-module (respectively a right
$R_{{\mathfrak{p}}}$-module) via Remark 1.1(iii). One can use this
isomorphism, and the isomorphism $\beta$ above, to see that there is a right
$R_{{\mathfrak{p}}}$-module isomorphism
$\left(\operatorname{Hom}_{rR}(R_{f},E_{R}(R/{\mathfrak{p}}))\right)_{{\mathfrak{p}}}\cong\operatorname{Hom}_{rR_{{\mathfrak{p}}}}\\!\left((R_{{\mathfrak{p}}})_{f},E_{R_{{\mathfrak{p}}}}(R_{{\mathfrak{p}}}/{\mathfrak{p}}R_{{\mathfrak{p}}})\right).$
The last module is right $R_{{\mathfrak{p}}}$-isomorphic to
$E_{R_{{\mathfrak{p}}}}(R_{{\mathfrak{p}}}/{\mathfrak{p}}R_{{\mathfrak{p}}})$
by Lemma 1.3. Therefore $\mu=1$.
We have thus shown that there is a right $R$-module isomorphism
$\varphi:I\to\operatorname{Hom}_{rR}(R_{f},I)$. To finish the proof, we show
that this mapping $\varphi$, regarded as a mapping
${}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$, is actually a left $R$-module
homomorphism, and therefore an $(R,R)$-bimodule isomorphism. For $z\in{}_{f}I$
and $a\in R$, we have, for all $r\in R_{f}$,
$\displaystyle\varphi(a\cdot z)(r)$
$\displaystyle=\varphi(za^{p})(r)=(\varphi(z)a^{p})(r)=\varphi(z)(a^{p}r)$
$\displaystyle=\varphi(z)(r\cdot
a)=(\varphi(z)(r))a=a(\varphi(z)(r))=(a\varphi(z))(r),$
so that $\varphi(a\cdot z)=a\varphi(z)$. Therefore $\varphi$ is a left
$R$-homomorphism. ∎
###### 1.5 Remark.
If, in Lemma 1.4, we drop the hypothesis that $R$ is $F$-finite, then the
conclusion is no longer always true. For one example, let $K$ be a countable
field of characteristic $p$ with $[K:K^{p}]$ infinite but countable, and set
$R=K$. We show now that $\operatorname{Hom}_{rK}(K_{f},K)\not\cong{}_{f}K$ as
right $K$-modules. Assume that $\operatorname{Hom}_{rK}(K_{f},K)\cong{}_{f}K$
as right $K$-modules and seek a contradiction.
Let $\overline{K}$ be an algebraic closure of $K$, and let $K^{1/p}$ denote
the subfield of $\overline{K}$ consisting of all $p$th roots of elements of
$K$. The assumption implies that $\operatorname{Hom}_{K}(K^{1/p},K)\cong
K^{1/p}$ as $K^{1/p}$-modules. In particular,
$\operatorname{Hom}_{K}(K^{1/p},K)$ has countable dimension as a vector space
over $K$. Let $(\alpha_{n})_{n\in\mathbb{N}}$ be a $K$-basis of $K^{1/p}$, so
that $K^{1/p}=\bigoplus_{n\in\mathbb{N}}K\alpha_{n}$. Then
$\operatorname{Hom}_{K}(K^{1/p},K)=\operatorname{Hom}_{K}\left(\bigoplus_{n\in\mathbb{N}}K\alpha_{n},K\right)\cong\prod_{n\in\mathbb{N}}\operatorname{Hom}_{K}(K\alpha_{n},K)$,
and this has uncountable dimension as a vector space over $K$, and this is a
contradiction.
###### 1.6 Discussion.
The Frobenius skew polynomial ring $R[x,f]$ was defined in the Introduction.
It follows from [13, Lemma 1.3] that extension of the $R$-module structure on
an $R$-module $H$ to a structure of left $R[x,f]$-module is equivalent to the
provision of an Abelian group homomorphism $\xi:H\longrightarrow H$ for which
$\xi(rh)=r^{p}\xi(h)$ for all $r\in R$ and $h\in H$. (In fact, $\xi$ and the
action of $x$ are related by the formula $\xi(h)=xh$ for all $h\in H$.)
There is a bijective correspondence between
$\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,H)$ and
$\left\\{\xi\in\operatorname{End}_{\mathbb{Z}}(H)\ |\
\xi(rh)=r^{p}\xi(h)\text{~{}for all~{}}r\in R\text{~{}and~{}}h\in H\right\\}$
under which $\alpha\in\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,H)$
corresponds to $h\mapsto\alpha(1\otimes h)$. In view of this, we are going to
use the notation $(H,\alpha)$ to describe a left $R[x,f]$-module ${\bf H}$,
where $H$ is the underlying $R$-module and
$\alpha\in\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,H)$ is such that
$xh=\alpha(1\otimes h)$ for all $h\in H$.
Under the adjoint isomorphism of Remark 1.1(iv), an
$\alpha\in\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,H)$ corresponds to an
$\widetilde{\alpha}\in\operatorname{Hom}_{lR}(H,\operatorname{Hom}_{lR}(R_{f},H))$.
Note that $xh=(\widetilde{\alpha}(h))(1)$ for all $h\in H$. We write ${\bf
H}=(H,\alpha)=[H,\widetilde{\alpha}]$.
With such notation, a left $R[x,f]$-homomorphism ${\bf H}=(H,\alpha)\to{\bf
H}^{\prime}=(H^{\prime},\alpha^{\prime})$ of left $R[x,f]$-modules is an
$R$-homomorphism $\varphi:H\to H^{\prime}$ for which the diagram
$\begin{CD}R_{f}\otimes_{R}H@>{\alpha}>{}>H\\\
@V{1\otimes\varphi}V{}V@V{\varphi}V{}V\\\
R_{f}\otimes_{R}H^{\prime}@>{\alpha^{\prime}}>{}>H^{\prime}\\\ \end{CD}$
commutes.
###### 1.7 Discussion.
Similarly, extension of the $R$-module structure on an $R$-module $M$ to a
structure of right $R[x,f]$-module is equivalent to the provision of an
Abelian group homomorphism $\xi:M\longrightarrow M$ for which
$\xi(mr^{p})=\xi(m)r$ for all $r\in R$ and $m\in M$. The map $\xi$ and the
action of $x$ are related by the formula $\xi(m)=mx$ for all $m\in M$.
There is a bijective correspondence between
$\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},M)$ and
$\left\\{\xi\in\operatorname{End}_{\mathbb{Z}}(M)\ |\
\xi(mr^{p})=\xi(m)r\text{~{}for all~{}}r\in R\text{~{}and~{}}m\in M\right\\}$
under which $\beta\in\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},M)$ corresponds
to $m\mapsto\beta(m\otimes 1)$. In view of this, we are going to use the
notation $(M,\beta)$ to describe a right $R[x,f]$-module ${\bf M}$, where $M$
is the underlying $R$-module and
$\beta\in\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},M)$ is such that
$mx=\beta(m\otimes 1)$ for all $m\in M$.
Under the adjoint isomorphism of Remark 1.1(v), a
$\beta\in\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},M)$ corresponds to a
$\widetilde{\beta}\in\operatorname{Hom}_{rR}(M,\operatorname{Hom}_{rR}(R_{f},M))$.
Note that $mx=(\widetilde{\beta}(m))(1)$ for all $m\in M$. We write ${\bf
M}=(M,\beta)=[M,\widetilde{\beta}]$.
###### 1.8 Notation.
We shall use ${}_{R[x,f]}\operatorname{Mod}$ to denote the category of all
left $R[x,f]$-modules and left $R[x,f]$-homomorphisms between them, and
$\operatorname{Mod}_{R[x,f]}$ to denote the category of all right
$R[x,f]$-modules and right $R[x,f]$-homomorphisms between them.
###### 1.9 Examples.
(i) The Frobenius endomorphism $f:R\to R$ induces a left $R$-module
homomorphism $\alpha:R_{f}\otimes_{R}R\to R$ for which $\alpha(a\otimes
b)=af(b)=ab^{p}$ for all $a\in R_{f}$ and $b\in R$. This therefore yields the
left $R[x,f]$-module $(R,\alpha)$, in which we have $xr=r^{p}$ for all $r\in
R$.
Let $c\in R$ be any element. Then there is a left $R$-module homomorphism
$\alpha_{c}:R_{f}\otimes R\to R$ such that $\alpha_{c}(a\otimes b)=cab^{p}$
for $a\in R_{f}$ and $b\in R$. Thus we obtain a left $R[x,f]$-module
$(R,\alpha_{c})$, in which $xr=cr^{p}$ for all $r\in R$. It is straightforward
to check that $(R,\alpha)\cong(R,\alpha_{c})$ as left $R[x,f]$-modules if and
only if $c$ is a unit in $R$ possessing a $(p-1)$th root in $R$. Thus it is
possible for there to be many left $R[x,f]$-modules with the same underlying
$R$-module.
(ii) Suppose that our ring $R$ is reduced and that we are given a non-trivial
$R^{p}$-homomorphism $\pi:R\to R^{p}$. (In the case where $R$ is $F$-finite
and $F$-pure, we can find such a $\pi$ that is a surjective mapping, because
$R^{p}$ is a direct summand of $R$ as an $R^{p}$-module: see [10, Corollary
5.3].) In this situation, we have a right $R$-module homomorphism
$\beta:R\otimes_{R}R_{f}\to R$ for which $\beta(a\otimes b)=\pi(ab)^{1/p}$ for
all $a\in R$ and $b\in R_{f}$. This yields a right $R[x,f]$-module
$(R,\beta)$, in which we have $rx=\pi(r)^{1/p}$ for all $r\in R$.
We have shown in 1.4 that whenever $R$ is $F$-finite and $I$ is an injective
$R$-module, there is an $(R,R)$-bimodule isomorphism
$\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$; of course, $\Psi$ is, in
particular, a right $R$-module homomorphism. Therefore we have the following
as a corollary to 1.4.
###### 1.10 Corollary.
Suppose that $R$ is $F$-finite, and let $I$ be an injective $R$-module. Then
there is a right $R[x,f]$-module ${\bf I}=[I,\Psi]$ which has $I$ as
underlying $R$-module, and is such that
$\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$ is an $(R,R)$-bimodule
isomorphism. Note that $zx=(\Psi(z))(1)$ for all $z\in I$.
###### 1.11 Lemma.
Let the situation and notation be as in Corollary 1.10, and consider the right
$R[x,f]$-module ${\bf I}=[I,\Psi]$. Then ${\bf I}$ has the following property:
if $z\in{\bf I}$ is such that, for a fixed $n\in{\mathbb{N}_{0}}$, we have
$zrx^{n}=0$ for all $r\in R$, then $z=0$.
###### Proof.
The claim is clear when $n=0$, and we deal now with the case where $n=1$. We
have
$0=(zr)x=\left(\Psi(zr)\right)(1)=\left((\Psi(z))r\right)(1)=(\Psi(z))(r)\quad\text{for
all $r\in R$}.$
Therefore $\Psi(z)=0$, so that $z=0$ because $\Psi$ is an isomorphism.
Now suppose, inductively, that $n\in\mathbb{N}$ with $n>1$, and that the claim
has been proved for all smaller values of $n$. Suppose that $zrx^{n}=0$ for
all $r\in R$. Then $(zrx^{n-1})sx=zrs^{p^{n-1}}x^{n}=0$ for all $r,s\in R$. It
follows from the case where $n=1$ that $zrx^{n-1}=0$ for all $r\in R$; it then
follows from the inductive hypothesis that $z=0$. ∎
###### 1.12 Discussion.
Throughout the rest of this section, assume that our ring $R$ is $F$-finite,
and let $I$ be an injective $R$-module. We fix a right $R[x,f]$-module
structure on $I$ as in Corollary 1.10, so that ${\bf I}=[I,\Psi]$ is a right
$R[x,f]$-module with $\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$ an
$(R,R)$-bimodule isomorphism. We denote by $(-)^{\vee}$ the duality functor
determined by $I$, so that $X^{\vee}=\operatorname{Hom}_{R}(X,I)$ for each
$R$-module $X$.
Now suppose we are given a left $R[x,f]$-module ${\bf H}=(H,\alpha)$ with
$\alpha\in\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,H)$.
1. (i)
Here we produce a right $R[x,f]$-module structure on $H^{\vee}$.
First apply the functor $(-)^{\vee}$ to the left $R$-homomorphism
$\alpha:R_{f}\otimes_{R}H\to H$: the result is a right $R$-homomorphism
$\alpha^{\vee}:H^{\vee}\to\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,I).$ But
there is an $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,I)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{lR}(H,\operatorname{Hom}_{lR}(R_{f},I))$
given by Remark 1.1(iv), and use of the $(R,R)$-bimodule isomorphism $\Psi$
produces a further $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{lR}(H,\operatorname{Hom}_{lR}(R_{f},I))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{lR}(H,\operatorname{Hom}_{rR}(R_{f},I)).$
In addition, Remark 1.1(vi) provides an $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{lR}(H,\operatorname{Hom}_{rR}(R_{f},I))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rR}(R_{f},\operatorname{Hom}_{lR}(H,I)).$
Composition of these therefore yields a right $R$-homomorphism
$\gamma:H^{\vee}\longrightarrow\operatorname{Hom}_{rR}(R_{f},H^{\vee})$, and
we shall denote by $D(\alpha)$ the right $R$-homomorphism
$H^{\vee}\otimes_{R}R_{f}\to H^{\vee}$ that corresponds to $\gamma$ under the
adjoint isomorphism of Remark 1.1(v). (Note that
$H^{\vee}=\operatorname{Hom}_{lR}(H,I)=\operatorname{Hom}_{rR}(H,I)$.)
Thus $D(\alpha)$ makes $H^{\vee}$ into a right $R[x,f]$-module. We define
${\bf D}({\bf H})={\bf D}(H,\alpha):=(H^{\vee},D(\alpha))=[H^{\vee},\gamma].$
It is straightforward to use the above definition of $\gamma$ to check that
(1) $(D(\alpha)(m\otimes r))(h)=\left(\Psi(m(\alpha(1\otimes
h)))\right)(r)\quad\mbox{for all~{}}m\in H^{\vee},\ r\in
R_{f}\mbox{~{}and~{}}h\in H.$
2. (ii)
Now let ${\bf H}^{\prime}=(H^{\prime},\alpha^{\prime})$ be a second left
$R[x,f]$-module and let $\varphi:{\bf H}\to{\bf H}^{\prime}$ be a left
$R[x,f]$-homomorphism. Thus $\varphi$ is an $R$-homomorphism $H\to H^{\prime}$
which makes the diagram
$\begin{CD}R_{f}\otimes_{R}H@>{\alpha}>{}>H\\\
@V{\operatorname{Id}\otimes\varphi}V{}V@V{}V{\varphi}V\\\
R_{f}\otimes_{R}H^{\prime}@>{\alpha^{\prime}}>{}>H^{\prime}\end{CD}$
commute. It is straightforward to check that the diagram
$\begin{CD}H^{\vee}@>{\alpha^{\vee}}>{}>\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H,I)@>{\cong}>{}>\operatorname{Hom}_{rR}(R_{f},\operatorname{Hom}_{lR}(H,I))\\\
@A{\varphi^{\vee}}A{}A@A{}A{(\operatorname{Id}\otimes\varphi)^{\vee}}A@A{\operatorname{Hom}(R_{f},\varphi^{\vee})}A{}A\\\
H^{\prime\vee}@>{\alpha^{\prime\vee}}>{}>\operatorname{Hom}_{lR}(R_{f}\otimes_{R}H^{\prime},I)@>{\cong}>{}>\operatorname{Hom}_{rR}(R_{f},\operatorname{Hom}_{lR}(H^{\prime},I))~{},\end{CD}$
in which the upper horizontal isomorphism is the one used in the construction
in part (i) and the lower horizontal isomorphism is the corresponding one for
${\bf H}^{\prime}$, commutes. Therefore $\varphi^{\vee}:H^{\prime\vee}\to
H^{\vee}$ defines a right $R[x,f]$-homomorphism ${\bf D}({\bf
H}^{\prime})\to{\bf D}({\bf H})$, which we denote by ${\bf D}(\varphi)$.
###### 1.13 Proposition.
Let the situation and notation be as in Discussion 1.12. There is a
contravariant functor ${\bf
D}:{}_{R[x,f]}{\mathrm{Mod}}\to{\mathrm{Mod}}_{R[x,f]}$ which maps a left
$R[x,f]$-module $(H,\alpha)$ to $(H^{\vee},D(\alpha))$ where $D(\alpha)$ is
given by (1) in Discussion 1.12(i).
###### 1.14 Proposition.
Let the situation and notation be as in Discussion 1.12 and let ${\bf
H}=(H,\alpha)$ be a left $R[x,f]$-module. The right $R[x,f]$-module structures
on ${\bf I}$ and ${\bf D}({\bf H})=(\operatorname{Hom}_{lR}(H,I),D(\alpha))$
are such that
(2) $(mx)(h)=(m(xh))x\quad\text{for all $m\in
H^{\vee}=\operatorname{Hom}_{lR}(H,I)$ and $h\in H$}.$
###### Proof.
Recall that the right $R[x,f]$-module structure on $I$ is given by
$\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$, so that $zx=(\Psi(z))(1)$
for all $z\in I$.
Let $m\in H^{\vee}=\operatorname{Hom}_{lR}(H,I)$ and $h\in H$. By (1) in
Discussion 1.12(i), we have
$(mx)(h)=(D(\alpha)(m\otimes 1))(h)=\Psi(m(\alpha(1\otimes
h)))(1)=(m(\alpha(1\otimes h)))x=(m(xh))x.$
∎
We now provide the right $R[x,f]$-module analogue of Discussion 1.12.
###### 1.15 Discussion.
The hypotheses and notation are as in Discussion 1.12. Let ${\bf M}=(M,\beta)$
be a right $R[x,f]$-module, where
$\beta\in\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},M)$.
1. (i)
Here we produce a left $R[x,f]$-module structure on $M^{\vee}$.
First apply the functor $(-)^{\vee}$ to the right $R$-homomorphism
$\beta:M\otimes_{R}R_{f}\to M$: the result is a left $R$-homomorphism
$\beta^{\vee}:M^{\vee}\to\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},I).$ But
there is an $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{rR}(M\otimes_{R}R_{f},I))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rR}(M,\operatorname{Hom}_{rR}(R_{f},I))$
given by Remark 1.1(v), and use of the $(R,R)$-bimodule isomorphism
$\Psi^{-1}$ produces a further $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{rR}(M,\operatorname{Hom}_{rR}(R_{f},I))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rR}(M,\operatorname{Hom}_{lR}(R_{f},I)).$
In addition, Remark 1.1(vi) provides an $(R,R)$-bimodule isomorphism
$\operatorname{Hom}_{rR}(M,\operatorname{Hom}_{lR}(R_{f},I))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{lR}(R_{f},\operatorname{Hom}_{rR}(M,I)).$
Composition of these therefore yields a left $R$-homomorphism
$\delta:M^{\vee}\longrightarrow\operatorname{Hom}_{lR}(R_{f},M^{\vee})$, and
we shall denote by $D^{\prime}(\beta)$ the left $R$-homomorphism
$R_{f}\otimes_{R}M^{\vee}\to M^{\vee}$ that corresponds to $\delta$ under the
adjoint isomorphism of Remark 1.1(iv). (Note that
$M^{\vee}=\operatorname{Hom}_{lR}(M,I)=\operatorname{Hom}_{rR}(M,I)$.) Thus
$D^{\prime}(\beta)$ makes $M^{\vee}$ into a left $R[x,f]$-module. We define
${\bf D}^{\prime}({\bf M})={\bf
D}(M,\beta):=(M^{\vee},D^{\prime}(\beta))=[M^{\vee},\delta].$
It is straightforward to use the above definition of $\delta$ to check that
(3) $(D^{\prime}(\beta)(r\otimes h))(m)=\left(\Psi^{-1}(r^{\prime}\mapsto
h(\beta(m\otimes r^{\prime})))\right)r\quad\mbox{for all~{}}h\in M^{\vee},\
r\in R_{f}\mbox{~{}and~{}}m\in M.$
2. (ii)
Now let ${\bf M}^{\prime}=(M^{\prime},\beta^{\prime})$ be a second right
$R[x,f]$-module and let $\psi:{\bf M}\to{\bf M}^{\prime}$ be a right
$R[x,f]$-homomorphism. An argument similar to that in Discussion 1.12(ii)
shows that $\psi^{\vee}:M^{\prime\vee}\to M^{\vee}$ defines a left
$R[x,f]$-homomorphism ${\bf D}^{\prime}({\bf M}^{\prime})\to{\bf
D}^{\prime}({\bf M})$, which we denote by ${\bf D}^{\prime}(\psi)$.
###### 1.16 Proposition.
Let the situation and notation be as in Discussion 1.15. There is a
contravariant functor ${\bf
D}^{\prime}:{\mathrm{Mod}}_{R[x,f]}\to{}_{R[x,f]}{\mathrm{Mod}}$ which maps a
right $R[x,f]$-module $(M,\beta)$ to $(M^{\vee},D^{\prime}(\beta))$ where
$D^{\prime}(\beta)$ is given by (3) in Discussion 1.15(i).
###### 1.17 Proposition.
Let the situation and notation be as in Discussion 1.15. Let ${\bf
M}=(M,\beta)$ be a right $R[x,f]$-module, so that ${\bf D}^{\prime}({\bf
M})=(M^{\vee},D^{\prime}(\beta))$ is a left $R[x,f]$-module by Proposition
1.16. The left action of $x$ on $M^{\vee}$ can be described as follows: for
$h\in M^{\vee}$, the result $xh$ of multiplying $h$ on the left by $x$ is the
unique $h^{\prime}\in M^{\vee}$ for which
(4) $(h^{\prime}(m))rx=h(mrx)\quad\text{for all $m\in M$ and $r\in R$}.$
###### Proof.
First of all,
$\displaystyle((xh)(m))rx$ $\displaystyle=\left((D^{\prime}(\beta)(1\otimes
h))(m)\right)rx=\left(\Psi^{-1}(r^{\prime}\mapsto h(\beta(m\otimes
r^{\prime})))\right)rx$
$\displaystyle=\left(\Psi\left(\left(\Psi^{-1}(r^{\prime}\mapsto
h(\beta(m\otimes
r^{\prime})))\right)r\right)\right)(1)=\left((r^{\prime}\mapsto
h(\beta(m\otimes r^{\prime})))r\right)(1)=h(\beta(m\otimes r))$
$\displaystyle=h(\beta(mr\otimes 1))=h(mrx).$
It therefore remains for us to show that if $h^{\prime}\in M^{\vee}$ is such
that $(h^{\prime}(m))rx=h(mrx)$ for all $m\in M$ and $r\in R$, then
$h^{\prime}=xh$. It is therefore enough for us to show that if
$h^{\prime\prime}\in M^{\vee}$ is such that $(h^{\prime\prime}(m))rx=0$ for
all $m\in M$ and $r\in R$, then $h^{\prime\prime}=0$. However, this is easy,
because Lemma 1.11 shows that $h^{\prime\prime}(m)=0$ for all $m\in M$. ∎
Propositions 1.13 and 1.16 prepare the ground for several subsequent results
in this paper.
###### 1.18 Proposition.
Let the situation and notation be as in Propositions 1.13 and 1.16, so that
$R$ is $F$-finite and $I$ is an injective $R$-module with fixed
$(R,R)$-bimodule isomorphism
$\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$.
For each $R$-module $G$, we write $G^{\vee}=\operatorname{Hom}_{R}(G,I)$ as
before. Let $\omega_{G}:G\longrightarrow(G^{\vee})^{\vee}$ be the natural
‘evaluation’ $R$-homomorphism for which $\omega_{G}(g)(h)=h(g)$ for all $h\in
G^{\vee}$ and $g\in G$. Recall that, as $G$ varies through the category
${}_{R}\operatorname{Mod}$ of all $R$-modules and $R$-homomorphisms, the
$\omega_{G}$ constitute a natural transformation from the identity functor on
${}_{R}\operatorname{Mod}$ to the functor $((-)^{\vee})^{\vee}$.
1. (i)
If ${\bf H}=(H,\alpha)$ is a left $R[x,f]$-module, then $\omega_{H}$ is a left
$R[x,f]$-module homomorphism from ${\bf H}$ to ${\bf D}^{\prime}({\bf D}({\bf
H}))$. As ${\bf H}$ varies through ${}_{R[x,f]}\operatorname{Mod}$, the
$\omega_{H}$ constitute a natural transformation from the identity functor on
that category to the functor ${\bf D}^{\prime}\circ{\bf D}$.
2. (ii)
If ${\bf M}=(M,\beta)$ is a right $R[x,f]$-module, then $\omega_{M}$ is a
right $R[x,f]$-module homomorphism from ${\bf M}$ to ${\bf D}({\bf
D}^{\prime}({\bf M}))$. As ${\bf M}$ varies through
$\operatorname{Mod}_{R[x,f]}$, the $\omega_{M}$ constitute a natural
transformation from the identity functor on that category to the functor ${\bf
D}\circ{\bf D}^{\prime}$.
###### Proof.
(i) In view of Propositions 1.13 and 1.16, it only remains for us to show
that, for a left $R[x,f]$-module ${\bf H}=(H,\alpha)$, the $R$-homomorphism
$\omega_{H}:H\to(H^{\vee})^{\vee}$ is actually a left $R[x,f]$-module
homomorphism. To this end, we compare, for an $h\in H$, the elements
$\omega_{H}(xh)$ and $x(\omega_{H}(h))$. Now, $x\omega_{H}(h)$ is, by (4) in
Proposition 1.17, the unique element $h^{\prime}\in(H^{\vee})^{\vee}$ that
satisfies
$\left(h^{\prime}(m)\right)rx=\omega_{H}(h)(mrx)\quad\text{for all $m\in
H^{\vee}$ and $r\in R$}.$
It is enough to show that $h^{\prime}=\omega_{H}(xh)$ satisfies this. But
$(\omega_{H}(xh)(m))rx=(m(xh))rx=(r(m(xh)))x=(m(rxh))x=((mr)(xh))x=(mrx)(h),$
where we have used (2) in 1.14 for the last equality. Since
$(mrx)(h)=\omega_{H}(h)(mrx)$, the proof of part (i) is complete.
(ii) In view of Propositions 1.13 and 1.16, it only remains for us to show
that, for a right $R[x,f]$-module ${\bf M}=(M,\beta)$, the $R$-homomorphism
$\omega_{M}:M\to(M^{\vee})^{\vee}$ is actually an $R[x,f]$-homomorphism. To
this end, we compare, for an $m\in M$, the elements $\omega_{M}(mx)$ and
$(\omega_{M}(m))x$.
Now, for all $h\in M^{\vee}$, we have
$\displaystyle\left((\omega_{M}(m))x\right)(h)$
$\displaystyle=\left(\omega_{M}(m)(xh)\right)x$ (by (2) in 1.14)
$\displaystyle=\left((xh)(m))\right)x$ $\displaystyle=h(mx)$ (by (4) in 1.17)
$\displaystyle=\left(\omega_{M}(mx)\right)(h).$
Hence $\omega_{M}(mx)=(\omega_{M}(m))x$. ∎
###### 1.19 Remark.
Let $R^{\prime}$ be a general commutative Noetherian ring and let $I$ be an
injective $R^{\prime}$-module. For each $R^{\prime}$-module $M$ we write
$M^{\vee}:=\operatorname{Hom}_{R^{\prime}}(M,I)$ and denote by $\omega_{M}$
the natural evaluation mapping $M\to(M^{\vee})^{\vee}$ defined by
$\omega_{M}(m)(h)=h(m)$ for all $h\in M^{\vee}$ and $m\in M$. We say that $M$
is $I$-reflexive if $\omega_{M}$ is an isomorphism. It is routine to check
that, for an $R^{\prime}$-module $M$, the composition
$(\omega_{M})^{\vee}\circ\omega_{M^{\vee}}:M^{\vee}\longrightarrow M^{\vee}$
is the identity map. Therefore, if $M$ is $I$-reflexive, then so too is
$M^{\vee}$. It is easily verified that the full subcategory of
${}_{R^{\prime}}\operatorname{Mod}$ consisting of all $I$-reflexive modules is
closed under finite direct sums, direct summands and extensions. But in
general it is not a Serre subcategory of ${}_{R^{\prime}}\operatorname{Mod}$,
as can be seen by consideration of the case where $R^{\prime}$ is a Noetherian
integral domain that is not a field and $I$ is taken to be the quotient field
of $R^{\prime}$.
Suppose, in addition, that $(R^{\prime},{\mathfrak{m}})$ is (Noetherian) local
and complete. Choose $I=E:=E_{R^{\prime}}(R^{\prime}/{\mathfrak{m}})$, so that
$(-)^{\vee}$ becomes the Matlis-duality functor
$\operatorname{Hom}_{R^{\prime}}(-,E)$. In this case, $E$-reflexive modules
are called Matlis-reflexive. It is well known that all Noetherian
$R^{\prime}$-modules and all Artinian $R^{\prime}$-modules are Matlis-
reflexive, and that $(-)^{\vee}$ provides a duality between the category of
all Noetherian $R^{\prime}$-modules (and all $R^{\prime}$-homomorphisms
between them) and the category of all Artinian $R^{\prime}$-modules (and all
$R^{\prime}$-homomorphisms between them). Furthermore it was proved by E.
Enochs [4, Proposition 1.3] that an $R^{\prime}$-module $M$ is Matlis-
reflexive if and only if it can be embedded into a short exact sequence
$\begin{CD}0@>{}>{}>N@>{}>{}>M@>{}>{}>A@>{}>{}>0,\end{CD}$
in which $A$ is an Artinian $R^{\prime}$-module and $N$ is a Noetherian
$R^{\prime}$-module. Therefore, the full subcategory of
${}_{R^{\prime}}\operatorname{Mod}$ consisting of all Matlis-reflexive modules
is an Abelian category itself, and is actually the smallest Serre subcategory
of ${}_{R^{\prime}}\operatorname{Mod}$ that contains all Noetherian modules
and all Artinian modules.
Let the situation and notation be as in Proposition 1.18, so that $R$ is
$F$-finite and $I$ is an injective $R$-module with a fixed $(R,R)$-bimodule
isomorphism $\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$. Let
$\mathcal{L}_{I}$ be the category of all left $R[x,f]$-modules which are
$I$-reflexive as $R$-modules, and all left $R[x,f]$-homomorphisms between
them. Similarly let $\mathcal{R}_{I}$ be the category of right
$R[x,f]$-modules which are $I$-reflexive as $R$-modules, and all right
$R[x,f]$-homomorphisms between them.
In general, $\mathcal{L}_{I}\subseteq{}_{R[x,f]}\operatorname{Mod}$ and
$\mathcal{R}_{I}\subseteq\operatorname{Mod}_{R[x,f]}$ are full subcategories,
which are closed under finite direct sums, direct summands and extensions. If,
in addition, $(R,{\mathfrak{m}})$ is local and complete and
$I=E_{R}(R/{\mathfrak{m}})$, then $\mathcal{L}_{I}$ and $\mathcal{R}_{I}$ are
Abelian categories by Remark 1.19.
From the definitions of the functors ${\bf D}$ and ${\bf D}^{\prime}$, it is
easy to use Remark 1.19 to see that they induce functors ${\bf
D}:\mathcal{L}_{I}\to(\mathcal{R}_{I})^{\text{op}}$ and ${\bf
D}^{\prime}:\mathcal{R}_{I}\to(\mathcal{L}_{I})^{\text{op}}$.
The following theorem is the main result of this paper.
###### 1.20 Theorem.
Let the situation and notation be as in 1.18, so that $R$ is $F$-finite and
$I$ is an injective $R$-module with a fixed $(R,R)$-bimodule isomorphism
$\Psi:{}_{f}I\to\operatorname{Hom}_{rR}(R_{f},I)$. Then the functors ${\bf
D}:\mathcal{L}_{I}\to(\mathcal{R}_{I})^{\text{{\rm op}}}$ and ${\bf
D}^{\prime}:\mathcal{R}_{I}\to(\mathcal{L}_{I})^{\text{{\rm op}}}$ are inverse
equivalences of categories.
###### Proof.
For any ${\bf H}=(H,\alpha)\in\mathcal{L}_{I}$, the evaluation mapping
$\omega_{H}:H\to(H^{\vee})^{\vee}$ is an $R[x,f]$-isomorphism, by Proposition
1.18(i). Therefore the natural transformation $\omega:\operatorname{Id}\to{\bf
D}^{\prime}\circ{\bf D}$ of 1.18(i) is a natural equivalence of functors on
$\mathcal{L}_{I}$. Similarly, $\omega:\operatorname{Id}\to{\bf D}\circ{\bf
D}^{\prime}$ is a natural equivalence of functors on $\mathcal{R}_{I}$. ∎
From this theorem we have the following corollary in the complete local case.
###### 1.21 Corollary.
Assume that $(R,{\mathfrak{m}})$ is $F$-finite, complete and local, and let
$I=E:=E_{R}(R/{\mathfrak{m}})$. We fix an $(R,R)$-bimodule isomorphism
$\Psi:{}_{f}E\to\operatorname{Hom}_{rR}(R_{f},E)$.
1. (i)
It follows from Theorem 1.20 that ${\bf D}$ and ${\bf D}^{\prime}$ are inverse
equivalences between the category of left $R[x,f]$-modules that are Artinian
as $R$-modules and the category of right $R[x,f]$-modules that are Noetherian
as $R$-modules.
2. (ii)
Similarly, ${\bf D}$ and ${\bf D}^{\prime}$ are inverse equivalences between
the category of right $R[x,f]$-modules that are Artinian as $R$-modules and
the category of left $R[x,f]$-modules that are Noetherian as $R$-modules.
## 2\. Graded annihilators
Let ${\mathfrak{B}}$ be a subset of $R[x,f]$. It is easy to see that
${\mathfrak{B}}$ is a graded two-sided ideal of $R[x,f]$ if and only if there
is an ascending chain $({\mathfrak{b}}_{n})_{n\in{\mathbb{N}_{0}}}$ of ideals
of $R$ (which must, of course, be eventually stationary) such that
${\mathfrak{B}}=\bigoplus_{n\in{\mathbb{N}_{0}}}{\mathfrak{b}}_{n}x^{n}$. In
particular, note that $R[x,f]x^{t}=\bigoplus_{i\geq t}Rx^{i}$ is a graded two-
sided ideal of $R[x,f]$, for each $t\in{\mathbb{N}_{0}}$.
###### 2.1 Definitions.
Let ${\bf H}=(H,\alpha)$ denote a left $R[x,f]$-module and let ${\bf
M}=(M,\beta)$ denote a right $R[x,f]$-module; let ${\mathfrak{B}}$ be a two-
sided ideal of $R[x,f]$.
The annihilator of ${\bf M}$ will be denoted by $\operatorname{ann}{\bf
M}_{R[x,f]}$. Thus
$\operatorname{ann}{\bf M}_{R[x,f]}=\\{\theta\in R[x,f]\ |\
g\theta=0\mbox{~{}for all~{}}g\in M\\},$
and this is a two-sided ideal of $R[x,f]$. The annihilator
$\operatorname{ann}_{R[x,f]}{\bf H}$ of ${\bf H}$ is defined similarly; it is
also a two-sided ideal of $R[x,f]$.
We define the graded annihilator $\operatorname{gr-ann}{\bf M}_{R[x,f]}$ of
the right $R[x,f]$-module ${\bf M}$ by
$\operatorname{gr-ann}{\bf M}_{R[x,f]}=\left\\{\sum_{i=0}^{n}r_{i}x^{i}\in
R[x,f]\ \Big{|}\ n\in{\mathbb{N}_{0}}\mbox{~{}and~{}}r_{i}\in
R,\,r_{i}x^{i}\in\operatorname{ann}{\bf M}_{R[x,f]}\mbox{~{}for
all~{}}i=0,\ldots,n\right\\}.$
Thus $\operatorname{gr-ann}{\bf M}_{R[x,f]}$ is the largest graded two-sided
ideal of $R[x,f]$ contained in $\operatorname{ann}{\bf M}_{R[x,f]}$.
The graded annihilator of ${\bf H}$ is defined similarly: it is the largest
graded two-sided ideal of $R[x,f]$ that annihilates ${\bf H}$. See [18, 1.5].
Recall also that $\operatorname{ann}_{{\bf H}}{\mathfrak{B}}$ denotes the
$R[x,f]$-submodule of ${\bf H}$ given by
$\operatorname{ann}_{{\bf H}}{\mathfrak{B}}=\\{h\in{\bf H}\ |\ \theta
h=0\mbox{~{}for all~{}}\theta\in{\mathfrak{B}}\\}.$
Observe that $M{\mathfrak{B}}$ is an $R[x,f]$-submodule of ${\bf M}$. For
$t\in{\mathbb{N}_{0}}$, we have $MR[x,f]x^{t}=\\{mx^{t}:m\in M\\}$, and we
shall therefore denote this $R[x,f]$-submodule of ${\bf M}$ by ${\bf M}x^{t}$.
We shall say that ${\bf M}$ is $x$-divisible precisely when ${\bf M}={\bf
M}x$.
Recall (from [18, 1.2]) that ${\bf H}=(H,\alpha)$ is said to be $x$-torsion-
free if $xh=0$, for $h\in H$, only when $h=0$. The set
$\Gamma_{x}(H):=\left\\{h\in H\ |\ x^{j}h=0\mbox{~{}for
some~{}}j\in\mathbb{N}\right\\}$ is an $R[x,f]$-submodule of ${\bf H}$, called
the $x$-torsion submodule of ${\bf H}$.
We are now going to compare, in the situation of Theorem 1.20, the graded
annihilators of a left $R[x,f]$-module ${\bf H}$ and the right $R[x,f]$-module
${\bf D}({\bf H})$, and also the graded annihilators of a right
$R[x,f]$-module ${\bf M}$ and the left $R[x,f]$-module ${\bf D}^{\prime}({\bf
M})$.
Recall that an injective cogenerator for $R$ is an injective $R$-module $I$
such that $\operatorname{Hom}_{R}(G,I)\neq 0$ for every non-zero $R$-module
$G$. See [22, p. 46]. It should be remarked that if $I$ is an injective
cogenerator, then the evaluation map
$\omega_{G}:G\longrightarrow(G^{\vee})^{\vee}$ is a monomorphism for all
$R$-modules $G$.
###### 2.2 Proposition.
Let the situation and notation be as in Theorem 1.20. Let ${\bf H}=(H,\alpha)$
be a left $R[x,f]$-module and ${\bf M}=(M,\beta)$ be a right $R[x,f]$-module.
Then
1. (i)
$\operatorname{gr-ann}_{R[x,f]}{\bf H}\subseteq\operatorname{gr-ann}{\bf
D}({\bf H})_{R[x,f]}$;
2. (ii)
$\operatorname{gr-ann}{\bf M}_{R[x,f]}\subseteq\operatorname{gr-
ann}_{R[x,f]}{\bf D}^{\prime}({\bf M})$;
3. (iii)
if $I$ is an injective cogenerator for $R$, we have
$\operatorname{gr-ann}_{R[x,f]}{\bf H}=\operatorname{gr-ann}{\bf D}({\bf
H})_{R[x,f]}\quad\mbox{and}\quad\operatorname{gr-ann}{\bf
M}_{R[x,f]}=\operatorname{gr-ann}_{R[x,f]}{\bf D}^{\prime}({\bf M})\mbox{;}$
4. (iv)
in particular, in the special case in which $(R,{\mathfrak{m}})$ is local, and
$I$ is taken to be $E_{R}(R/{\mathfrak{m}})$, we have
$\operatorname{gr-ann}_{R[x,f]}{\bf H}=\operatorname{gr-ann}{\bf D}({\bf
H})_{R[x,f]}\quad\mbox{and}\quad\operatorname{gr-ann}{\bf
M}_{R[x,f]}=\operatorname{gr-ann}_{R[x,f]}{\bf D}^{\prime}({\bf M}).$
###### Proof.
(i) Recall from (2) in 1.14 that the right action of $R[x,f]$ on ${\bf D}({\bf
H})=(H^{\vee},D(\alpha))$ is such that $(mx)(h)=\left(m(xh)\right)x$ for all
$m\in H^{\vee}$ and all $h\in H$; an easy inductive argument shows that
$(mx^{n})(h)=\left(m(x^{n}h)\right)x^{n}$ for all $n\in\mathbb{N}$.
Now let $r\in R$ and $n\in{\mathbb{N}_{0}}$ be such that $rx^{n}H=0$. We show
that $rx^{n}$ annihilates the right $R[x,f]$-module ${\bf D}({\bf
H})=(H^{\vee},D(\alpha))$. Let $m\in H^{\vee}$ and $h\in H$. Then, by the
preceding paragraph,
$(mrx^{n})(h)=((mr)x^{n})(h)=\left((mr)(x^{n}h)\right)x^{n}=\left(m(rx^{n}h)\right)x^{n}=0.$
It follows that $\operatorname{gr-ann}_{R[x,f]}{\bf
H}\subseteq\operatorname{gr-ann}{\bf D}({\bf H})_{R[x,f]}$.
(ii) Let $r\in R$ and $n\in{\mathbb{N}_{0}}$ be such that $Mrx^{n}=0$. We show
that $rx^{n}$ annihilates the left $R[x,f]$-module ${\bf D}^{\prime}({\bf
M})=(M^{\vee},D(\beta))$. This is clear when $n=0$, and so we suppose that
$n>0$. Let $h\in M^{\vee}$ and $m\in M$. Then recall from (4) in 1.17 that
$((xh)(m))r^{\prime}x=h(mr^{\prime}x)$ for all $m\in M$ and $r^{\prime}\in R$.
An easy inductive argument shows that
$((x^{n}h)(m))r^{\prime}x^{n}=h(mr^{\prime}x^{n})\quad\text{for all $m\in M$
and $r^{\prime}\in R$.}$
It follows from this that, for $r\in R$,
$((rx^{n}h)(m))r^{\prime}x^{n}=h(mrr^{\prime}x^{n})\quad\text{for all $m\in M$
and $r^{\prime}\in R$.}$
But, since $mrr^{\prime}x^{n}=mr^{\prime}rx^{n}=0$, we have $rx^{n}h(m)=0$ for
all $m\in M$, by Lemma 1.11. Therefore $rx^{n}h=0$ and $rx^{n}$ annihilates
the left $R[x,f]$-module ${\bf D}^{\prime}({\bf M})$. Hence
$\operatorname{gr-ann}{\bf M}_{R[x,f]}\subseteq\operatorname{gr-
ann}_{R[x,f]}{\bf D}^{\prime}({\bf M}).$
(iii) By parts (i) and (ii), we have $\operatorname{gr-ann}_{R[x,f]}{\bf
H}\subseteq\operatorname{gr-ann}{\bf D}({\bf
H})_{R[x,f]}\subseteq\operatorname{gr-ann}_{R[x,f]}{\bf D}^{\prime}\circ{\bf
D}({\bf H})$. However, since $I$ is an injective cogenerator for $R$, the
homomorphism of left $R[x,f]$-modules
$\omega_{H}:H\longrightarrow(H^{\vee})^{\vee}$ is actually an
$R[x,f]$-monomorphism, and so it follows that
$\operatorname{gr-ann}_{R[x,f]}{\bf D}^{\prime}\circ{\bf D}({\bf
H})\subseteq\operatorname{gr-ann}_{R[x,f]}{\bf H}.$
The first equality is therefore proved. The second is proved similarly.
(iv) This is a special case of part (iii), because, when $(R,{\mathfrak{m}})$
is local, $E_{R}(R/{\mathfrak{m}})$ is an injective cogenerator for $R$. ∎
###### 2.3 Proposition.
Let the situation and notation be as in 1.20, and assume in addition that
$(R,{\mathfrak{m}})$ is local and complete and that $I$ is taken to be
$E:=E_{R}(R/{\mathfrak{m}})$. Let ${\mathfrak{B}}$ be a graded two-sided ideal
of $R[x,f]$.
1. (i)
Let ${\bf H}=(H,\alpha)$ be a left $R[x,f]$-module that is Matlis-reflexive as
$R$-module. Let $\iota:\operatorname{ann}_{{\bf
H}}{\mathfrak{B}}\longrightarrow{\bf H}$ denote the inclusion
$R[x,f]$-monomorphism. Then the induced homomorphism of right $R[x,f]$-modules
${\bf D}(\iota):{\bf D}({\bf H})\longrightarrow{\bf
D}(\operatorname{ann}_{{\bf H}}{\mathfrak{B}})$ has kernel ${\bf D}({\bf
H}){\mathfrak{B}}$.
2. (ii)
Let ${\bf M}=(M,\beta)$ be a right $R[x,f]$-module that is Noetherian as
$R$-module. Let $\sigma:{\bf M}{\mathfrak{B}}\longrightarrow{\bf M}$ denote
the inclusion $R[x,f]$-monomorphism. Then the induced homomorphism of left
$R[x,f]$-modules ${\bf D}^{\prime}(\sigma):{\bf D}^{\prime}({\bf
M})\longrightarrow{\bf D}^{\prime}({\bf M}{\mathfrak{B}})$ has kernel
$\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}}$, so that
$\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}}\cong{\bf
D}^{\prime}({\bf M}/{\bf M}{\mathfrak{B}})$ as left $R[x,f]$-modules.
###### Proof.
(i) Since ${\mathfrak{B}}\subseteq\operatorname{gr-
ann}_{R[x,f]}(\operatorname{ann}_{{\bf
H}}{\mathfrak{B}})\subseteq\operatorname{gr-ann}({\bf
D}(\operatorname{ann}_{{\bf H}}{\mathfrak{B}}))_{R[x,f]}$ (by Proposition
2.2(i)), it follows from the fact that ${\bf D}(\iota)$ is an
$R[x,f]$-homomorphism that ${\bf D}({\bf
H}){\mathfrak{B}}\subseteq\operatorname{Ker}{\bf D}(\iota)$. There is
therefore an induced $R[x,f]$-epimorphism $\phi:{\bf D}({\bf H})/{\bf D}({\bf
H}){\mathfrak{B}}\longrightarrow{\bf D}(\operatorname{ann}_{{\bf
H}}{\mathfrak{B}})$ for which $\phi(m+{\bf D}({\bf H}){\mathfrak{B}})={\bf
D}(\iota)(m)$ for all $m\in{\bf D}({\bf H})$. Let $\lambda:{\bf D}({\bf
H})\longrightarrow{\bf D}({\bf H})/{\bf D}({\bf H}){\mathfrak{B}}$ denote the
canonical $R[x,f]$-epimorphism, and note that $\phi\circ\lambda={\bf
D}(\iota)$. We therefore have a commutative diagram
$\begin{CD}\operatorname{ann}_{{\bf H}}{\mathfrak{B}}@>{}>{\iota}>{\bf
H}@={\bf H}\\\ @V{}V{\omega_{\operatorname{ann}_{{\bf
H}}{\mathfrak{B}}}}V@V{\cong}V{\omega_{H}}V\\\ {\bf D}^{\prime}\circ{\bf
D}(\operatorname{ann}_{{\bf H}}{\mathfrak{B}})@>{}>{{\bf
D}^{\prime}(\phi)}>{\bf D}^{\prime}({\bf D}({\bf H})/{\bf D}({\bf
H}){\mathfrak{B}})@>{}>{{\bf D}^{\prime}(\lambda)}>{\bf D}^{\prime}\circ{\bf
D}({\bf H})\\\ \end{CD}$
in the category $\\!\phantom{i}{}_{R[x,f]}\operatorname{Mod}$. By Proposition
2.2(ii), we have $\operatorname{Im}{\bf
D}^{\prime}(\lambda)\subseteq\operatorname{ann}_{{\bf D}^{\prime}\circ{\bf
D}({\bf H})}{\mathfrak{B}}$; since $\omega_{H}$ is an $R[x,f]$-isomorphism and
${\bf D}^{\prime}(\lambda)$ and ${\bf D}^{\prime}(\phi)$ are monomorphisms, it
follows from the above commutative diagram that ${\bf D}^{\prime}(\phi)$ is an
isomorphism. Since $I=E_{R}(R/{\mathfrak{m}})$ is an injective cogenerator for
$R$, we can therefore deduce that $\phi$ is an isomorphism, so that ${\bf
D}({\bf H}){\mathfrak{B}}=\operatorname{Ker}{\bf D}(\iota)$.
(ii) Let $j:\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}{\mathfrak{B}}\longrightarrow{\bf D}^{\prime}({\bf M})$ denote the
inclusion map and $k:{\bf M}\longrightarrow{\bf M}/{\bf M}{\mathfrak{B}}$
denote the natural epimorphism. Apply part (i) to the left $R[x,f]$-module
${\bf D}^{\prime}({\bf M})$ to obtain an exact sequence
$\begin{CD}0@>{}>{}>({\bf D}\circ{\bf D}^{\prime}({\bf
M})){\mathfrak{B}}@>{}>{}>{\bf D}\circ{\bf D}^{\prime}({\bf M})@>{{\bf
D}(j)}>{}>{\bf D}(\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}{\mathfrak{B}})@>{}>{}>0\\\ \end{CD}$
in $\operatorname{Mod}_{R[x,f]}$. Since $M$ is Noetherian as $R$-module, the
$R[x,f]$-homomorphism $\omega_{M}:{\bf M}\longrightarrow{\bf
D}^{\prime}\circ{\bf D}({\bf M})$ is an isomorphism. There is therefore a
commutative diagram
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{\bf
M}{\mathfrak{B}}@>{}>{}>{\bf M}@>{k}>{}>{\bf M}/{\bf
M}{\mathfrak{B}}@>{}>{}>0\\\ @V{\cong}V{}V@V{\cong}V{\omega_{M}}V\\\
0@>{}>{}>({\bf D}\circ{\bf D}^{\prime}({\bf M})){\mathfrak{B}}@>{}>{}>{\bf
D}\circ{\bf D}^{\prime}({\bf M})@>{{\bf D}(j)}>{}>{\bf
D}(\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}})@>{}>{}>0\\\
\end{CD}$
with exact rows in the category $\operatorname{Mod}_{R[x,f]}$. This induces an
$R[x,f]$-isomorphism $\gamma:{\bf M}/{\bf
M}{\mathfrak{B}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\bf
D}(\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}})$ which, when
inserted into the above diagram, is such that the extended diagram is still
commutative. Now apply the functor ${\bf D}^{\prime}$ to the right-most square
(involving $\gamma$) in that extended diagram: the result is the right-most
square in the commutative diagram
$\begin{CD}\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}{\mathfrak{B}}@>{\omega_{\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}{\mathfrak{B}}}}>{\cong}>{\bf D}^{\prime}\circ{\bf
D}(\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}})@>{{\bf
D}^{\prime}(\gamma)}>{\cong}>{\bf D}^{\prime}({\bf M}/{\bf
M}{\mathfrak{B}})\\\ @V{\subseteq}V{j}V@V{}V{{\bf D}^{\prime}({\bf
D}(j))}V@V{}V{{\bf D}^{\prime}(k)}V\\\ {\bf D}^{\prime}({\bf
M})@>{\omega_{M^{\vee}}}>{\cong}>{\bf D}^{\prime}\circ{\bf D}\circ{\bf
D}^{\prime}({\bf M})@>{{\bf D}^{\prime}(\omega_{M})}>{\cong}>{\bf
D}^{\prime}({\bf M})~{}~{}~{}\qquad.\\\ \end{CD}$
Note that ${\bf D}^{\prime}({\bf M})$ is Artinian as $R$-module, so that
$\omega_{M^{\vee}}$ and $\omega_{\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}({\mathfrak{B}})}$ are both isomorphisms. Since ${\bf
D}^{\prime}(\omega_{M})\circ\omega_{M^{\vee}}=\operatorname{Id}_{M^{\vee}}$
(as noted in Remark 1.19), it follows from this commutative diagram that the
kernel of the induced $R[x,f]$-homomorphism ${\bf D}^{\prime}(\sigma):{\bf
D}^{\prime}({\bf M})\longrightarrow{\bf D}^{\prime}({\bf M}{\mathfrak{B}})$,
which is equal to the image of the $R[x,f]$-homomorphism ${\bf
D}^{\prime}(k):{\bf D}^{\prime}({\bf M}/{\bf
M}{\mathfrak{B}})\longrightarrow{\bf D}^{\prime}({\bf M})$, is precisely
$\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}}$. ∎
###### 2.4 Corollary.
Let the situation and notation be as in 1.20, and assume in addition that
$(R,{\mathfrak{m}})$ is local and complete and that $I$ is taken to be
$E:=E_{R}(R/{\mathfrak{m}})$. Let ${\bf M}$ be a right $R[x,f]$-module that is
Noetherian as $R$-module. Then ${\bf M}$ is $x$-divisible if and only if ${\bf
D}^{\prime}({\bf M})$ is $x$-torsion-free.
###### Proof.
Let $j:{\bf M}x\longrightarrow{\bf M}$ be the inclusion $R[x,f]$-monomorphism.
By Proposition 2.3(ii), the kernel of ${\bf D}^{\prime}(j):{\bf
D}^{\prime}({\bf M})\longrightarrow{\bf D}^{\prime}({\bf M}x)$ is
$\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}R[x,f]x$.
Now ${\bf M}$ is $x$-divisible if and only if $j$ is an isomorphism; since $E$
is an injective cogenerator for $R$, this is the case if and only if ${\bf
D}^{\prime}(j)$ is an isomorphism; and, by the above comment (and the fact
that ${\bf D}^{\prime}(j)$ must always be an $R[x,f]$-epimorphism), ${\bf
D}^{\prime}(j)$ is an isomorphism if and only if $\operatorname{ann}_{{\bf
D}^{\prime}({\bf M})}R[x,f]x=0$, that is, if and only if ${\bf
D}^{\prime}({\bf M})$ is $x$-torsion-free. ∎
## 3\. Some applications
As was mentioned in the Introduction, the Hartshorne–Speiser–Lyubeznik Theorem
has been applied to establish the existence of a uniform Frobenius test
exponent for Frobenius closures of parameter ideals in a local ring
$(R,{\mathfrak{m}})$ that is Cohen–Macaulay, or just generalized
Cohen–Macaulay. The non-local version of the Hartshorne–Speiser–Lyubeznik
Theorem given in [17, Corollary 1.8] can be reformulated as follows: if ($R$
is not necessarily local and) $H$ is a left $R[x,f]$-module that is Artinian
as $R$-module, then there exists an $e\in{\mathbb{N}_{0}}$ such that
$\operatorname{ann}_{H}R[x,f]x^{e}=\operatorname{ann}_{H}R[x,f]x^{e+1}$. With
this in mind, one can regard the following result as a ‘dual
Hartshorne–Speiser–Lyubeznik Theorem’ for the case where $(R,{\mathfrak{m}})$
is $F$-finite, local and complete.
###### 3.1 Theorem.
Assume that $(R,{\mathfrak{m}})$ is $F$-finite, local and complete. Let ${\bf
M}=(M,\beta)$ be a right $R[x,f]$-module that is Noetherian as $R$-module.
Then there exists $e\in{\mathbb{N}_{0}}$ such that ${\bf M}x^{e}={\bf
M}x^{e+1}$, that is, such that $MR[x,f]x^{e}=MR[x,f]x^{e+1}$.
###### Proof.
Let $E:=E_{R}(R/{\mathfrak{m}})$. Select an $(R,R)$-bimodule isomorphism
$\Psi:{}_{f}E\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{rR}(R_{f},E)$:
recall that Lemma 1.4 ensures that there is such a $\Psi$. By Proposition 1.16
and Remark 1.19, we know that ${\bf D}^{\prime}({\bf M})=(M^{\vee},D(\beta))$
is a left $R[x,f]$-module; by Matlis duality, as $R$-module,
$M^{\vee}=\operatorname{Hom}_{R}(M,E)$ is Artinian. Therefore, by the
Hartshorne–Speiser–Lyubeznik Theorem, there exists an $e\in{\mathbb{N}_{0}}$
such that $\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}R[x,f]x^{e}=\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}R[x,f]x^{e+1}$.
Let $i:{\bf M}x^{e}\stackrel{{\scriptstyle\subseteq}}{{\longrightarrow}}{\bf
M}$, $j:{\bf
M}x^{e+1}\stackrel{{\scriptstyle\subseteq}}{{\longrightarrow}}{\bf M}$ and
$k:{\bf M}x^{e+1}\stackrel{{\scriptstyle\subseteq}}{{\longrightarrow}}{\bf
M}x^{e}$ be the inclusion $R[x,f]$-homomorphisms, so that $i\circ k=j$. In
view of Proposition 2.3(ii), there is a commutative diagram
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>\operatorname{ann}_{{\bf
D}^{\prime}({\bf M})}R[x,f]x^{e}@>{}>{\subseteq}>{\bf D}^{\prime}({\bf
M})@>{}>{{\bf D}^{\prime}(i)}>{\bf D}^{\prime}({\bf M}x^{e})@>{}>{}>0\\\
@V{\subseteq}V{}V\Big{\|}@V{{\bf D}^{\prime}(k)}V{}V\\\
0@>{}>{}>\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}R[x,f]x^{e+1}@>{}>{\subseteq}>{\bf D}^{\prime}({\bf M})@>{}>{{\bf
D}^{\prime}(j)}>{\bf D}^{\prime}({\bf M}x^{e+1})@>{}>{}>0\\\ \end{CD}$
with exact rows in the category $\\!\phantom{i}{}_{R[x,f]}\operatorname{Mod}$.
Since $\operatorname{ann}_{{\bf D}^{\prime}({\bf
M})}R[x,f]x^{e}=\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}R[x,f]x^{e+1}$,
we see that ${\bf D}^{\prime}(k)$ must be an isomorphism; therefore, $k$ must
be an isomorphism, because $E$ is an injective cogenerator for $R$. Therefore
${\bf M}x^{e}={\bf M}x^{e+1}$. ∎
It is natural to ask whether the conclusion of Theorem 3.1 is still valid if
we drop the assumptions about $R$ (except the one that $R$ has characteristic
$p$). In Theorem 3.4 below, we shall show that this is indeed the case. We
first present two preparatory lemmas, in which we assume only that $R$ is a
commutative Noetherian ring of characteristic $p$.
###### 3.2 Lemma.
Let ${\bf M}=(M,\beta)$ be a right $R[x,f]$-module. Suppose that there is an
element $s\in R$ such that $Ms\subseteq Mx$. Then, for all $k\in\mathbb{N}$,
we have $Ms^{2}\subseteq Mx^{k}$.
###### Proof.
We prove the lemma by induction on $k$. When $k=1$, there is nothing to prove.
Assume that $Ms^{2}\subseteq Mx^{k}$ for a $k\in\mathbb{N}$. Then, since
$Ms^{p}\subseteq Ms^{2}\subseteq Mx^{k}$, we have $Ms^{p}x\subseteq Mx^{k+1}$.
Thus $Ms^{2}=(Ms)s\subseteq(Mx)s=Ms^{p}x\subseteq Mx^{k+1}$. The lemma is
therefore proved by induction. ∎
###### 3.3 Lemma.
Let ${\bf M}=(M,\beta)$ be a right $R[x,f]$-module and $S$ be a
multiplicatively closed subset of $R$. Then the module of fractions $S^{-1}M$
has a natural right $(S^{-1}R)[x,f]$-module structure in which
$\left(\frac{m}{s}\right)x=\frac{ms^{p-1}x}{s}\quad\text{for all $m\in M$ and
$s\in S$.}$
This structure is such that $S^{-1}(Mx^{k})=(S^{-1}M)x^{k}$ for all
$k\in\mathbb{N}$.
###### Proof.
It is straightforward to construct a right $(S^{-1}R)[x,f]$-module structure
on $S^{-1}M$ with the specified properties. An easy inductive argument shows
that
$\left(\frac{m}{s}\right)x^{k}=\frac{ms^{p^{k}-1}x^{k}}{s}\quad\text{for all
$k\in\mathbb{N}$, $m\in M$ and $s\in S$.}$
It is clear from this that $S^{-1}(Mx^{k})\supseteq(S^{-1}M)x^{k}$ for a
$k\in\mathbb{N}$.
To establish the reverse inclusion, let $\alpha\in S^{-1}(Mx^{k})$, so that
$\alpha=(mx^{k})/s$ for some $m\in M$ and $s\in S$. Then
$\alpha=\frac{mx^{k}}{s}=\frac{mx^{k}s^{p^{k}-1}}{s^{p^{k}}}=\frac{m(s^{p^{k}-1})^{p^{k}}x^{k}}{s^{p^{k}}}=\frac{m(s^{p^{k}})^{p^{k}-1}x^{k}}{s^{p^{k}}}=\left(\frac{m}{s^{p^{k}}}\right)x^{k}\in(S^{-1}M)x^{k}.$
∎
###### 3.4 Theorem.
Assume only that $R$ is a commutative Noetherian ring of characteristic $p$.
Let ${\bf M}=(M,\alpha)$ be a right $R[x,f]$-module that is Noetherian as
$R$-module. Then there exists $k\in{\mathbb{N}_{0}}$ such that ${\bf
M}x^{k}={\bf M}x^{k+1}$.
###### Proof.
It is straightforward to check that $(0:_{M}Rx^{k}):=\\{m\in M\ |\
mRx^{k}=0\\}$ is an $R[x,f]$-submodule of $M$ for each $k\in\mathbb{N}$.
Define $(0:_{M}Rx^{\infty}):=\bigcup_{k\in\mathbb{N}}(0:_{M}Rx^{k})$; this is
also an $R[x,f]$-submodule of $M$; therefore
$M^{\gamma}:=M/(0:_{M}Rx^{\infty})$ is again a right $R[x,f]$-module.
Since $M$ is Noetherian as $R$-module, the ascending chain
$(0:_{M}Rx)\subseteq(0:_{M}Rx^{2})\subseteq\cdots\subseteq(0:_{M}Rx^{k})\subseteq\cdots$
must eventually be stationary, say at $(0:_{M}Rx^{\ell})$; then
$(0:_{M}Rx^{\infty})=(0:_{M}Rx^{\ell})$. We point out that if there exists
$k\in{\mathbb{N}_{0}}$ such that $M^{\gamma}x^{k}=M^{\gamma}x^{k+1}$, then
$Mx^{k}\subseteq Mx^{k+1}+(0:_{M}Rx^{\ell})$, so that multiplication on the
right by $x^{\ell}$ yields that $Mx^{k+\ell}\subseteq Mx^{k+\ell+1}$. Thus, if
the conclusion of the theorem is true for $M^{\gamma}$, then it is true for
$M$.
Next, we define $M\cdot x^{\infty}:=\bigcap_{k\in\mathbb{N}}Mx^{k}$; this is
an $R[x,f]$-submodule of $M$, and so $M^{\sigma}:=M/M\cdot x^{\infty}$ is
again a right $R[x,f]$-module. Note also that, if there exists
$k\in{\mathbb{N}_{0}}$ such that $M^{\sigma}x^{k}=M^{\sigma}x^{k+1}$, then
$Mx^{k}\subseteq Mx^{k+1}+M\cdot x^{\infty}=Mx^{k+1}$. Thus, if the conclusion
of the theorem is true for $M^{\sigma}$, then it is true for $M$.
Consider the sequence of right $R[x,f]$-modules
$M\to
M^{\sigma}\to(M^{\sigma})^{\gamma}\to((M^{\sigma})^{\gamma})^{\sigma}\to(((M^{\sigma})^{\gamma})^{\sigma})^{\gamma}\to\cdots,$
where, at each stage, the arrow denotes the appropriate natural
$R[x,f]$-epimorphism. The first three paragraphs of this proof show that, if
the claim in the theorem holds for any module $M^{\prime}$ in this sequence,
then it holds for all modules to the left of $M^{\prime}$, including $M$
itself. If, for each $n\in\mathbb{N}$, we let $K_{n}$ denote the kernel of the
composition of the first $n$ epimorphisms in this sequence, then
$K_{1}\subseteq K_{2}\subseteq\cdots\subseteq K_{n}\subseteq\cdots$ is an
ascending chain of $R$-submodules of $M$, and therefore eventually stationary.
This means that, in the above displayed sequence, there is a term to the right
of which all the epimorphisms are isomorphisms. Therefore it is enough for us
to prove the theorem under the additional assumptions that
(5) $(0:_{M}Rx^{\infty})=0\quad\text{and}\quad M\cdot x^{\infty}=0.$
We shall show now that these additional assumptions force $M$ to be zero (in
which case $Mx=Mx^{2}$). Suppose that $M\not=0$, and seek a contradiction.
Denote by $\mathrm{Min}_{R}(M)$ the set of minimal prime ideals in
$\mathrm{Supp}_{R}(M)$. This is a non-empty set, because $M\not=0$. We set
$S:=R\setminus\bigcup_{{\mathfrak{p}}\in\mathrm{Min}_{R}(M)}{\mathfrak{p}}$, a
multiplicatively closed subset of $R$. Then, the $S^{-1}R$-module $S^{-1}M$ is
a non-trivial Artinian module, since $\dim_{S^{-1}R}S^{-1}M=0$. By Lemma 3.3,
$S^{-1}M$ has a natural structure as a right $(S^{-1}R)[x,f]$-module. Consider
the descending sequence of right $(S^{-1}R)[x,f]$-submodules
$S^{-1}M\supseteq(S^{-1}M)x\supseteq(S^{-1}M)x^{2}\supseteq(S^{-1}M)x^{3}\supseteq(S^{-1}M)x^{4}\supseteq\cdots$
of $S^{-1}M$. Since $S^{-1}M$ is Artinian as $S^{-1}R$-module, we must have
$(S^{-1}M)x^{k}=(S^{-1}M)x^{k+1}$ for some $k\in\mathbb{N}$. Then,
$S^{-1}(Mx^{k})=S^{-1}(Mx^{k+1})$ by virtue of Lemma 3.3. Since $Mx^{k}$ is
finitely generated as $R$-module, it follows that there is an element $s\in S$
such that $(Mx^{k})s\subseteq Mx^{k+1}$. Now, applying Lemma 3.2 to the right
$R[x,f]$-module $Mx^{k}$, we have that $(Mx^{k})s^{2}\subseteq
Mx^{k+k^{\prime}}$ for all $k^{\prime}\in\mathbb{N}$. Therefore
$(Mx^{k})s^{2}\subseteq M\cdot x^{\infty}=0$ by the assumption (5). Since
$(Mx^{k})s^{2}=Ms^{2p^{k}}x^{k}$, we have shown that $Ms^{2p^{k}}x^{k}=0$;
hence $Ms^{2p^{k}}\subseteq(0:_{M}Rx^{k})\subseteq(0:_{M}Rx^{\infty})=0$.
Consequently we have $Ms^{2p^{k}}=0$. Therefore $s$ belongs to
$\sqrt{\mathrm{ann}_{R}(M)}$; therefore $s\in{\mathfrak{p}}$ for all
${\mathfrak{p}}\in\mathrm{Min}_{R}(M)$. But this contradicts the fact that $s$
is an element of $S$. ∎
One of the main results of [18] is that, if ${\bf H}$ is an $x$-torsion-free
left $R[x,f]$-module that is Artinian as $R$-module, then there are only
finitely many graded annihilators of $R[x,f]$-submodules of ${\bf H}$. See
[18, Corollary 3.11]. This result has relevance to the existence of tight
closure test elements in certain circumstances: see [18, Corollary 4.7] and
[20, Theorem 3.5]. We can use our work in §1 and §2 to obtain a dual result in
the special case where $R$ is $F$-finite, local and complete.
###### 3.5 Theorem.
Assume that $(R,{\mathfrak{m}})$ is $F$-finite, local and complete. Let ${\bf
M}=(M,\beta)$ be an $x$-divisible right $R[x,f]$-module that is Noetherian as
$R$-module. Then there are only finitely many graded annihilators of
$R[x,f]$-homomorphic images of ${\bf M}$.
###### Proof.
Select an $(R,R)$-bimodule isomorphism
$\Psi:{}_{f}E_{R}(R/{\mathfrak{m}})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}_{R}(R_{f},E_{R}(R/{\mathfrak{m}}))$:
recall that Lemma 1.4 ensures that there is such a $\Psi$. Use the notation of
1.20, but take $I$ to be $E:=E_{R}(R/{\mathfrak{m}})$. By Proposition 1.16, we
know that ${\bf D}^{\prime}({\bf M})$ is a left $R[x,f]$-module; as such, it
is $x$-torsion-free, by Corollary 2.4. By Matlis duality, as $R$-module,
$M^{\vee}=\operatorname{Hom}_{R}(M,E)$ is Artinian. By [18, Lemma 1.9,
Definition 1.10 and Corollary 3.11], there are only finitely many graded
annihilators of $R[x,f]$-submodules of ${\bf D}^{\prime}({\bf M})$. It is
therefore enough for us to show that, if ${\mathfrak{B}}$ is the graded-
annihilator of some $R[x,f]$-homomorphic image of ${\bf M}$, then
${\mathfrak{B}}$ is the graded-annihilator of some $R[x,f]$-submodule of ${\bf
D}^{\prime}({\bf M})$. This we do.
Thus there is an $R[x,f]$-submodule $\mathbf{L}$ of ${\bf M}$ such that
${\mathfrak{B}}=\operatorname{gr-ann}({\bf M}/\mathbf{L})_{R[x,f]}$. Therefore
${\bf M}{\mathfrak{B}}\subseteq\mathbf{L}$, so that there is an
$R[x,f]$-epimorphism ${\bf M}/{\bf M}{\mathfrak{B}}\longrightarrow{\bf
M}/\mathbf{L}$. Therefore
${\mathfrak{B}}\subseteq\operatorname{gr-ann}({\bf M}/{\bf
M}{\mathfrak{B}})_{R[x,f]}\subseteq\operatorname{gr-ann}({\bf
M}/\mathbf{L})_{R[x,f]}={\mathfrak{B}},$
so that ${\mathfrak{B}}=\operatorname{gr-ann}({\bf M}/{\bf
M}{\mathfrak{B}})_{R[x,f]}$. It now follows from Proposition 2.2(iv) that
${\mathfrak{B}}=\operatorname{gr-ann}_{R[x,f]}{\bf D}^{\prime}({\bf M}/{\bf
M}{\mathfrak{B}}).$
However, Proposition 2.3(ii) shows that there is an isomorphism ${\bf
D}^{\prime}({\bf M}/{\bf M}{\mathfrak{B}})\cong\operatorname{ann}_{{\bf
D}^{\prime}({\bf M})}{\mathfrak{B}}$ of left $R[x,f]$-modules; therefore
${\mathfrak{B}}$ is the graded annihilator of the $R[x,f]$-submodule
$\operatorname{ann}_{{\bf D}^{\prime}({\bf M})}{\mathfrak{B}}$ of ${\bf
D}^{\prime}({\bf M})$. This completes the proof. ∎
It is natural to ask whether the conclusion of Theorem 3.5 is still valid if
we drop the assumptions about $R$ (except the one that $R$ has characteristic
$p$).
###### 3.6 Question.
Assume only that $R$ is (a commutative Noetherian ring) of characteristic $p$.
Let ${\bf M}$ be an $x$-divisible right $R[x,f]$-module that is Noetherian as
$R$-module. Is the set of graded annihilators of $R[x,f]$-homomorphic images
of ${\bf M}$ finite?
At the time of writing, we are not able to answer Question 3.6.
## References
* [1] M. Blickle and G. Boeckle, Cartier modules: finiteness results, J. reine angew. Math., to appear; arXiv math.AC 0909.2531v1.
* [2] F. Enescu, $F$-injective rings and $F$-stable primes, Proc. Amer. Math. Soc. 131 (2003) 3379–3386.
* [3] F. Enescu, Local cohomology and $F$-stability, J. Algebra 322 (2009) 3063–3077.
* [4] E. Enochs, Flat Covers and Flat Cotorsion Modules, Proc. Amer. Math. Soc., 92 (1984) 179–184.
* [5] R. Fedder, $F$-purity and rational singularity in graded complete intersection rings, Transactions Amer. Math. Soc. 301 (1987) 47–62.
* [6] R. Fedder and K-i. Watanabe, A characterization of $F$-regularity in terms of $F$-purity, in: M. Hochster, C. Huneke and J. D. Sally (Eds.), Commutative algebra: proceedings of a microprogram held June 15 — July 2, 1987, Mathematical Sciences Research Institute Publications 15, Springer, New York, 1989, pp. 227–245.
* [7] O. Gabber, Notes on some $t$-structures, Geometric aspects of Dwork theory, Vol. I,II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 711–734.
* [8] N. Hara and K-i. Watanabe, The injectivity of Frobenius acting on cohomology and local cohomology modules, Manuscripta Math. 90 (1996) 301–315.
* [9] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic $p$, Annals of Math. 105 (1977) 45–79.
* [10] M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976) 117–172.
* [11] C. Huneke, M. Katzman, R. Y. Sharp and Y. Yao, Frobenius test exponents for parameter ideals in generalized Cohen–Macaulay local rings, J. Algebra 305 (2006) 516–539.
* [12] M. Katzman, Parameter-test-ideals of Cohen–Macaulay rings, Compositio Math. 144 (2008) 933–948.
* [13] M. Katzman and R. Y. Sharp, Uniform behaviour of the Frobenius closures of ideals generated by regular sequences, J. Algebra 295 (2006) 231–246.
* [14] G. Lyubeznik, $F$-modules: applications to local cohomology and $D$-modules in characteristic $p>0$, J. reine angew. Math. 491 (1997) 65–130.
* [15] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986\.
* [16] J. J. Rotman, Advanced modern algebra, Prentice Hall, Inc., Upper Saddle River, NJ, 2002.
* [17] R. Y. Sharp, Tight closure test exponents for certain parameter ideals, Michigan Math. J. 54 (2006) 307–317.
* [18] R. Y. Sharp, Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure, Transactions Amer. Math. Soc. 359 (2007) 4237–4258.
* [19] R.Y.Sharp, On the Hartshorne–Speiser–Lyubeznik Theorem about Artinian modules with a Frobenius action, Proc. Amer. Math. Soc. 135 (2007) 665–670.
* [20] R. Y. Sharp, Graded annihilators and tight closure test ideals, J. Algebra 322 (2009) 3410–3426.
* [21] R. Y. Sharp, An excellent $F$-pure ring of prime characteristic has a big tight closure test element, Transactions Amer. Math. Soc., to appear; arXiv math.AC 0907.2857.
* [22] D. W. Sharpe and P. Vámos, Injective modules, Cambridge Tracts in Mathematics and Mathematical Physics 62, Cambridge University Press, 1972.
* [23] K. E. Smith, Tight closure of parameter ideals, Inventiones mathematicae 115 (1994) 41–60.
* [24] Y. Yoshino, Skew-polynomial rings of Frobenius type and the theory of tight closure, Communications in Algebra 22 (1994) 2473–2502.
|
arxiv-papers
| 2010-08-23T07:26:22 |
2024-09-04T02:49:12.354633
|
{
"license": "Public Domain",
"authors": "Rodney Y. Sharp, Yuji Yoshino",
"submitter": "Yuji Yoshino",
"url": "https://arxiv.org/abs/1008.3763"
}
|
1008.3788
|
# Doubly Exponential Solution for Randomized Load Balancing Models with
General Service Times
Quan-Lin Li
School of Economics and Management Sciences
Yanshan University, Qinhuangdao 066004, P.R. China
(September 5, 2010)
###### Abstract
The randomized load balancing model (also called _supermarket model_) is now
being applied to the study of load balancing in data centers and multi-core
servers systems. It is very interesting to analyze the general service times
in the supermarket model, and specifically understand influence of the heavy-
tailed service times on the doubly exponential solution. Since the supermarket
model is a complex queueing system, the general service times make its
analysis more challenging than the exponential or PH service case. Up to now,
it still is an open problem whether or how the heavy-tailed service times can
disrupt the doubly exponential structure of the fixed point in the supermarket
model.
In this paper, we provide a novel and simple approach to study the supermarket
model with general service times. This approach is based on the supplementary
variable method used in analyzing stochastic models extensively. We organize
an infinite-size system of integral-differential equations by means of the
density dependent jump Markov process, and obtain a close-form solution:
doubly exponential structure, for the fixed point satisfying the system of
nonlinear equations, which is always a key in the study of supermarket models.
The fixed point is decomposited into two groups of information under a product
form: the arrival information and the service information. Based on this, we
indicate two important observations: the fixed point for the supermarket model
is different from the tail of stationary queue length distribution for the
ordinary M/G/1 queue, and the doubly exponential solution to the fixed point
can extensively exist even if the service time distribution is heavy-tailed.
Furthermore, we analyze the exponential convergence of the current location of
the supermarket model to its fixed point, and study the Lipschitz condition in
the Kurtz Theorem under general service times. Based on these analysis, one
can gain a new understanding how workload probing can help in load balancing
jobs with general service times such as heavy-tailed service.
Keywords: Randomized load balancing, supermarket model, density dependent jump
Markov process, fixed point, doubly exponential solution, heavy-tailed
distribution, exponential convergence, Lipschitz condition.
## 1 Introduction
Randomized load balancing, where a job is assigned to a server from a small
subset of randomly chosen servers, is very simple to implement, and can
surprisingly deliver better performance (for example reducing collisions,
waiting times, backlogs) in a number of applications, such as, data center,
hash tables, distributed memory machines, path selection in networks, and task
assignment at web servers. One useful model that has been extensively used to
study the randomized load balancing schemes is the supermarket model. In the
supermarket model, a key result by Vvedenskaya, Dobrushin and Karpelevich [31]
indicated that when each Poisson arriving job is assigned to the shortest one
of $d\geq 2$ randomly chosen queues with exponential service times, the
equilibrium queue length can decay doubly exponentially in the limit as the
population number $n\rightarrow\infty$, and the stationary fraction of queues
with at least $k$ customers is $\rho^{\frac{d^{k}-1}{d-1}}$, which is a
substantially exponential improvement over the case for $d=1$, where the tail
of stationary queue length distribution in the corresponding M/M/1 queue is
$\rho^{k}$.
The distributed load balancing strategies, in which individual job decisions
are based on information on a limited number of other processors, have been
studied analytically by Eager, Lazokwska and Zahorjan [5, 6, 7] and through
trace-driven simulations by Zhou [33]. Based on this, the supermarket model is
developed by queueing theory and Markov processes. Most of recent research
applied the density dependent jump Markov processes to deal with a simple
supermarket model with Poisson arrival processes and exponential service
times, a key result of which illustrates that there exists a unique fixed
point which is decreasing doubly exponentially. That approach used in the
literature relies on determining the behavior of the supermarket model as its
size grows to infinity, and its behavior is naturally described as a system of
differential equations, which leads to a closed form solution: doubly
exponential structure, of the fixed point. Readers may refer to, such as,
analyzing a basic and simple supermarket model by Azar, Broder, Karlin and
Upfal [2], Vvedenskaya, Dobrushin and Karpelevich [31], Mitzenmacher [19, 20].
Certain generalization of the supermarket model have been explored, for
example, simple variations by Mitzenmacher and Vöcking [27], Mitzenmacher [21,
22, 25], Vöcking [30], Mitzenmacher, Richa, and Sitaraman [26] and Vvedenskaya
and Suhov [32]; and analyzing load information by Mirchandaney, Towsley, and
Stankovic [28], Dahlin [3], Mitzenmacher [24, 26]. Furthermore, Martin and
Suhov [18], Martin [17], Suhov and Vvedenskaya [29] studied the supermarket
mall model by means of the fast Jackson network, where each node in a Jackson
network is replaced by $N$ parallel servers, and a job joins the shortest of
$d$ randomly chosen queues at the node to which it is directed. Luczak and
McDiamid [15, 16] studied the maximum queue length of the original supermarket
model with exponential service times when the service speed scales linearly
with the number of jobs in the queue. Li, Lui and Wang [11, 12] discussed the
supermarket model with PH service times and the supermarket model with
Markovian arrival processes, respectively. Readers may refer to an excellent
overview by Mitzenmacher, Richa, and Sitaraman [26].
This paper is interested in analyzing the supermarket model with general
service times, which is an open problem for determining whether or how the
heavy-tailed service times can disrupt the doubly exponential structure of the
fixed point. On the other hand, note that the supermarket model is a complex
queueing system and has much different characteristics from the ordinary
queueing systems, thus the general service times make its analysis more
challenging than the exponential or PH service case. Up to now, there has not
been an effective method to be able to deal with the supermarket model with
general service times.
The main contributions of the paper are threefold. The first one is to provide
a novel and simple approach to study the supermarket model with general
service times. This approach is based on the supplementary variable method but
is described as a new integral-differential structure for expressing and
computing the fraction of queues efficiently. Using the new approach, we setup
an infinite-size system of integral-differential equations, which makes
applications of the density dependent jump Markov processes to be able to deal
with the general distributions, such as general service times, involved in the
supermarket model. The second one is to obtain a close-form solution: doubly
exponential structure, for the fixed point satisfying the system of nonlinear
equations, which is always a key in the study of supermarket models.
Furthermore, this paper analyzes the exponential convergence of the current
location of the supermarket model to its fixed point, and studies the
Lipschitz condition in the Kurtz Theorem under general service times. Also,
this paper provides numerical examples to illustrate the effectiveness of our
approach in analyzing the randomized load balancing schemes with the non-
exponential service requirements. The third one is to obtain that the fixed
point is decomposited into two groups of information under a product form: the
arrival information and the service information. Based on this, we indicate
three important observations:
(a)
The fixed point for the supermarket model is different from the tail of
stationary queue length distribution for the ordinary M/G/1 queue, because the
fixed point is light-tailed but the stationary queue length is heavy-tail if
the service times are heavy-tailed. Note that such a difference is illustrated
in this paper for the first time, while it can not be observed in the
literature for the supermarket model with Poisson arrivals and exponential
service times, e.g., see Mitzenmacher, Richa, and Sitaraman [26].
(b)
The doubly exponential solution to the fixed point can extensively exist even
if the service time distribution is heavy-tailed. This is an answer of the
above open problem to illustrate the role played by the heavy-tailed service
time distribution in the doubly exponential solution to the fixed point.
(c)
The doubly exponential solution to the fixed point is not unique for a more
general supermarket model. Note that we give three different doubly
exponential solutions in the supermarket model with Poisson arrivals and PH
service times, thus it is very interesting to provide all the doubly
exponential solutions for a more general supermarket model.
Based on this, one can gain the new and important understanding how the
workload probing can help in load balancing jobs with general service times
such as heavy-tailed service.
The remainder of this paper is organized as follows. In Section 2, we first
describe a supermarket model with general service times, which is always
useful in the study of randomized load balancing schemes. Then the supermarket
model is expressed as a systems of integral-differential equations in terms of
the density dependent jump Markov processes. In Section 3, we first introduce
a fixed point of the system of integral-differential equations, and set up a
system of nonlinear equations satisfied by the fixed point. Then we provide a
close-form solution: doubly exponential structure, to the system of nonlinear
equations. In Section 4, we provide a necessary discussion on the key
parameter $\theta$ used in the doubly exponential structure, and indicate that
the doubly exponential solution to the fixed point extensively exists even if
the service time distribution is heavy-tailed. In Section 5, we give three
methods to analyze the supermarket model with Poisson arrivals and PH service
times, and provide three different ways to determine the key parameter
$\theta$ and compute the doubly exponential solution to the fixed point. We
show that the doubly exponential solution to the fixed point is not unique for
a more general supermarket. In Section 6, we study the exponential convergence
of the current location of the supermarket model to its fixed point. Not only
does the exponential convergence indicates the existence of the fixed point,
but it also explains such a convergent process is very fast. In Section 7, we
apply the Kurtz Theorem to study the supermarket model with the general
service times, and analyze the Lipschitz condition with respect to general
service times. Some concluding remarks are given in Section 8.
## 2 Supermarket Model
In this section, we first describe a supermarket model with general service
times. Then we provide a novel and simple approach to setup an infinite-size
system of integral-differential equations based on the density dependent jump
Markov processes. Note that this approach is based on the supplementary
variable method but is described as a new integral-differential structure so
that the corresponding boundary conditions are written in a different version.
Let us formally describe the supermarket model, which is abstracted as a
multi-server multi-queue stochastic system. Customers arrive at a queueing
system of $n>1$ servers as a Poisson process with an average arrival rate
$n\lambda$ for $\lambda>0$. The service time $\chi_{k}$ of the $k$th customer
is general with the distribution function
$G\left(x\right)=P\left\\{\chi_{k}\leq
x\right\\}=1-\exp\left\\{-\int_{0}^{x}\mu\left(y\right)dy\right\\},$
where all the random variables $\chi_{k}$ for $k\geq 1$ are i.i.d. with the
mean $E\left[\chi_{k}\right]=1/\mu$. Each arriving customer chooses $d\geq 1$
servers independently and uniformly at random from these $n$ servers, and
waits for service at the server which currently contains the fewest number of
customers. If there is a tie, servers with the fewest number of customers will
be chosen randomly. All customers in any service center will be served in the
First-Come-First-Served (FCFS) manner. Figure 1 simply shows such a
supermarket model.
Figure 1: working structure of the supermarket model
In the study of supermarket models, it is necessary for us to study general
service time distributions, for example, heavy-tailed distributions. Not only
because the general distribution makes analysis of the supermarket models more
difficult and challenging than those in the literature for the exponential or
PH service case, but it also allows us to model more realistic systems and
understand their performance implication under the randomized load balancing
strategy. As indicated in [8], the process times of many parallel jobs, in
particular, jobs to data centers, tend to be non-exponential. Unless we state
otherwise, we assume that all the random variables defined above are
independent, and that the system is operating under the condition:
$\rho=\lambda/\mu<1$.
###### Lemma 1
The supermarket model with general service times is stable if
$\rho=\lambda/\mu<1.$
Proof: Let $Q_{O}\left(t\right)$ and $Q_{S}\left(t\right)$ be the queue
lengths of the ordinary M/G/1 queue and of an arbitrary server in the
supermarket model at time $t$, respectively. Note that in the supermarket
model, each customer chooses $d$ servers independently and uniformily at
random, and the queue length of the entering server is currently shorten, it
is easy to see that for each $t\geq 0$,
$0\leq
Q_{S}\left(t\right)\underset{\text{st}}{\leq}Q_{O}\left(t\right)\text{.}$ (1)
Since the ordinary M/G/1 queue is stable if $\rho=\lambda/\mu<1$, it follows
from (1) that the supermarket model with general service times is stable if
$\rho=\lambda/\mu<1$. This completes the proof.
For $k\geq 1$, we define $n_{k}\left(t,x\right)$d$x$ as the number of queues
with at least $k$ customers and the residual service time of each server be in
the interval $[x,x+$d$x)$ at time $t\geq 0$. Clearly, $0\leq
n_{k}\left(t,x\right)\leq n$ for $x\geq 0$ and $1\leq k\leq n$. Let
$s_{k,n}\left(t,x\right)=\frac{n_{k}\left(t,x\right)}{n},$
which is the density function of the fraction of queues with at least $k$
customers and the residual service time of each server be $x$. We write
$S_{k}\left(t,x\right)=\lim_{n\rightarrow\infty}s_{k,n}\left(t,x\right),\text{
\ for }k\geq 1.$
We define $n_{0,n}\left(t\right)$ as the number of queues with at least $0$
customers at time $t\geq 0$. Clearly, $n_{0,n}\left(t\right)=n$. Let
$s_{0,n}\left(t\right)=\frac{n_{0,n}\left(t\right)}{n}.$
Then $s_{0,n}\left(t\right)=1$ for all $t\geq 0$ and
$S_{0}\left(t\right)=\lim_{n\rightarrow\infty}s_{0,n}\left(t\right)=1.$
Let $V\left(t\right)$ be the fraction of queues with zero customer at time
$t$. Then
$S_{0}\left(t\right)=V\left(t\right)+\int_{0}^{+\infty}S_{1}\left(t,x\right)\text{d}x.$
Thus we have
$S\left(t,x\right)=\left(S_{0}\left(t\right),S_{1}\left(t,x\right),S_{2}\left(t,x\right),\ldots\right).$
The following proposition shows that the sequence
$\left\\{S_{k}\left(t,x\right)\right\\}$ is monotone increasing for $k\geq 1$,
while its proof is easily by means of the definition of
$S_{k}\left(t,x\right)$ for $k\geq 1$.
###### Proposition 1
For $1\leq k<l$
$S_{l}\left(t,x\right)<S_{k}\left(t,x\right)<S_{0}\left(t\right)=1$
and
$\int_{0}^{+\infty}S_{l}\left(t,x\right)\text{d}x<\int_{0}^{+\infty}S_{k}\left(t,x\right)\text{d}x<S_{0}\left(t\right)=1.$
Now, we setup a system of integral-differential equations by means of the
density dependent jump Markov process. To that end, we provide an example with
$k\geq 2$ to indicate how to derive the system of integral-differential
equations.
Consider the supermarket model with $n$ queues, and determine the expected
change in the number of servers with at least $k$ customers and the residual
service time of each server be $x$ over a small time period of length d$t$.
The probability that a customer arriving during this time period is
$n\lambda$d$t$, and the probability that an arriving customer joins a queue of
size $k-1$ is given by
$\int_{0}^{+\infty}s_{k-1,n}^{d}\left(t,x\right)$d$x-\int_{0}^{+\infty}s_{k,n}^{d}\left(t,x\right)$d$x$.
Thus, the probability that during this time period, any arriving customer
joins a queue of size $k-1$ is given by
$n\lambda\text{d}t\cdot\left[\int_{0}^{+\infty}s_{k-1,n}^{d}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}s_{k,n}^{d}\left(t,x\right)\text{d}x\right].$
Similarly, the probability that a customer leaves a server of size $k$ is
given by
$ndt\cdot\left[\int_{0}^{+\infty}\mu\left(x\right)s_{k,n}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}\mu\left(x\right)s_{k+1,n}\left(t,x\right)\text{d}x\right].$
Therefore we can obtain
$\displaystyle\frac{\text{d}\int_{0}^{+\infty}n_{k}\left(t,x\right)\text{d}x}{\text{d}t}=$
$\displaystyle
n\lambda\left[\int_{0}^{+\infty}s_{k-1,n}^{d}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}s_{k,n}^{d}\left(t,x\right)\text{d}x\right]$
$\displaystyle+n\left[\int_{0}^{+\infty}\mu\left(x\right)s_{k,n}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}\mu\left(x\right)s_{k+1,n}\left(t,x\right)\text{d}x\right],$
which leads to
$\displaystyle\frac{\text{d}\int_{0}^{+\infty}s_{k,n}\left(t,x\right)\text{d}x}{\text{d}t}=$
$\displaystyle\lambda\left[\int_{0}^{+\infty}s_{k-1,n}^{d}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}s_{k,n}^{d}\left(t,x\right)\text{d}x\right]$
$\displaystyle+\left[\int_{0}^{+\infty}\mu\left(x\right)s_{k,n}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}\mu\left(x\right)s_{k+1,n}\left(t,x\right)\text{d}x\right].$
(2)
Taking $n\rightarrow\infty$ in the both sides of (2), we have
$\displaystyle\frac{\text{d}\int_{0}^{+\infty}S_{k}\left(t\right)\text{d}x}{\text{d}t}=$
$\displaystyle\lambda\left[\int_{0}^{+\infty}S_{k-1}^{d}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}S_{k}^{d}\left(t,x\right)\text{d}x\right]$
$\displaystyle+\left[\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(t,x\right)\text{d}x-\int_{0}^{+\infty}\mu\left(x\right)S_{k+1}\left(t,x\right)\text{d}x\right].$
(3)
Using a similar analysis to that for deriving Equation (3), we can easily
obtain a system of integral-differential equations for the fraction density
vector $S\left(t,x\right)$ as follows:
$S_{0}\left(t\right)=1\text{ for all }t\geq 0,$ (4)
$\frac{\mathtt{d}}{\text{d}t}S_{0}\left(t\right)=-\lambda
S_{0}^{d}\left(t\right)+\int_{0}^{+\infty}\mu\left(x\right)S_{1}\left(t,x\right)\text{d}x,$
(5)
$\displaystyle\frac{\mathtt{d}\int_{0}^{+\infty}S_{1}\left(t,x\right)\text{d}x}{\text{d}t}=$
$\displaystyle\lambda
S_{0}^{d}\left(t\right)-\lambda\int_{0}^{+\infty}S_{1}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)S_{1}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{2}\left(t,x\right)\text{d}x,$
(6)
and for $k\geq 2$,
$\displaystyle\frac{\mathtt{d}\int_{0}^{+\infty}S_{k}\left(t,x\right)\text{d}x}{\text{d}t}=$
$\displaystyle\lambda\int_{0}^{+\infty}S_{k-1}^{d}\left(t,x\right)\text{d}x-\lambda\int_{0}^{+\infty}S_{k}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{k+1}\left(t,x\right)\text{d}x.$
(7)
###### Remark 1
When there are $n$ servers in the supermarket model, it is necessary to give a
finite-size system of integral-differential equations for the fraction density
vector
$S^{\left(n\right)}\left(t,x\right)=\left(s_{0,n}\left(t\right),s_{1,n}\left(t,x\right),\ldots,s_{n,n}\left(t,x\right)\right)$
as follows:
$s_{0,n}\left(t\right)=1\text{ for all }t\geq 0,$
$\frac{\mathtt{d}}{\text{d}t}s_{0,n}\left(t\right)=-\lambda
s_{0,n}^{d}\left(t\right)+\int_{0}^{+\infty}\mu\left(x\right)s_{1,n}\left(t,x\right)\text{d}x,$
$\displaystyle\frac{\mathtt{d}\int_{0}^{+\infty}s_{1,n}\left(t,x\right)\text{d}x}{\text{d}t}=$
$\displaystyle\lambda
s_{0,n}^{d}\left(t\right)-\lambda\int_{0}^{+\infty}s_{1,n}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)s_{1,n}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)s_{2,n}\left(t,x\right)\text{d}x,$
and for $n\geq k\geq 2$,
$\displaystyle\frac{\mathtt{d}\int_{0}^{+\infty}s_{k,n}\left(t,x\right)\text{d}x}{\text{d}t}=$
$\displaystyle\lambda\int_{0}^{+\infty}s_{k-1,n}^{d}\left(t,x\right)\text{d}x-\lambda\int_{0}^{+\infty}s_{k,n}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)s_{k,n}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)s_{k+1,n}\left(t,x\right)\text{d}x.$
## 3 Doubly Exponential Solution
In this section, we discuss the fixed point of the system of integral-
differential equations in Equations (4) to (7), and set up a system of
nonlinear equations satisfied by the fixed point. Also, we provide a closed-
form solution: doubly exponential structure, to the system of nonlinear
equations.
A row vector
$\pi\left(x\right)=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$
is called a fixed point of the fraction density vector
$S\left(t,x\right)=\left(S_{0}\left(t\right),S_{1}\left(t,x\right),S_{2}\left(t,x\right),\ldots\right)$
if there exists a $t_{0}\geq 0$ such that $S_{0}\left(t\right)=\pi_{0}$ and
$S_{k}\left(t,x\right)=\pi_{k}\left(x\right)$ for all $t\geq t_{0}$ and $k\geq
1$. It is easy to see that if $\pi\left(x\right)$ is a fixed point of the
fraction density vector $S\left(t,x\right)$ for all $t\geq t_{0}$, then
$\frac{\mathtt{d}}{\text{d}t}S_{0}\left(t\right)_{|t\geq t_{0}}=0$
and for $k\geq 1$
$\frac{\mathtt{d}}{\text{d}t}S_{k}\left(t,x\right)_{|t\geq t_{0}}=0$
which leads to
$\int_{0}^{+\infty}\frac{\mathtt{d}}{\text{d}t}S_{k}\left(t,x\right)_{|t\geq
t_{0}}\text{d}x=0.$ (8)
Since for $k\geq 1$
$0\leq S_{k}\left(t,x\right)\leq 1,$
using the Dominated Convergence Theorem we obtain
$\frac{\mathtt{d}}{\text{d}t}\int_{0}^{+\infty}S_{k}\left(t,x\right)_{|t\geq
t_{0}}\text{d}x=0.$
Therefore, if
$\pi\left(x\right)=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$
is a fixed point of the fraction density vector
$S\left(t,x\right)=\left(S_{0}\left(t\right),S_{1}\left(t,x\right),S_{2}\left(t,x\right),\ldots\right)$
for all $t\geq t_{0}$, then the system of integral-differential equations (4)
to (7) can be simplified as
$\pi_{0}=1$ (9)
$-\lambda\pi_{0}^{d}+\int_{0}^{+\infty}\mu\left(x\right)\pi_{1}\left(x\right)\text{d}x=0,$
(10)
$\lambda\pi_{0}^{d}\left(t\right)-\lambda\int_{0}^{+\infty}\pi_{1}^{d}\left(x\right)\text{d}x-\int_{0}^{+\infty}\mu\left(x\right)\pi_{1}\left(x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)\pi_{2}\left(x\right)\text{d}x=0,$
(11)
and for $k\geq 2$,
$\lambda\int_{0}^{+\infty}\pi_{k-1}^{d}\left(x\right)\text{d}x-\lambda\int_{0}^{+\infty}\pi_{k}^{d}\left(x\right)\text{d}x-\int_{0}^{+\infty}\mu\left(x\right)\pi_{k}\left(x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)\pi_{k+1}\left(x\right)\text{d}x=0.$
(12)
In what follows we derive a closed-form expression for
$\pi\left(x\right)=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$.
It follows from Equations (9) and (10) that
$\int_{0}^{+\infty}\mu\left(x\right)\pi_{1}\left(x\right)\text{d}x=\lambda.$
(13)
To solve Equation (13), using the fact that
$\int_{0}^{+\infty}\mu\left(x\right)\overline{G}\left(x\right)$d$x=1$ we have
$\pi_{1}\left(x\right)=\lambda\overline{G}\left(x\right)=\rho\cdot\mu\overline{G}\left(x\right).$
(14)
Based on the fact that $\pi_{0}=1$ and
$\pi_{1}\left(x\right)=\rho\cdot\mu\overline{G}\left(x\right)$, it follows
from Equations (11) and (13) that
$-\lambda\rho^{d}\cdot\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)\pi_{2}\left(x\right)\text{d}x=0.$
Let
$\theta=\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}$d$x$,
and we assume that $0<\theta<+\infty$. Then
$\int_{0}^{+\infty}\mu\left(x\right)\pi_{2}\left(x\right)\text{d}x=\lambda\theta\rho^{d}.$
(15)
Using a similar analysis on Equation (15), we have
$\pi_{2}\left(x\right)=\lambda\theta\rho^{d}\overline{G}\left(x\right)=\theta\rho^{d+1}\cdot\mu\overline{G}\left(x\right).$
(16)
Based on $\pi_{1}\left(x\right)=\rho\cdot\mu\overline{G}\left(x\right)$ and
$\pi_{2}\left(x\right)=\theta\rho^{d+1}\cdot\mu\overline{G}\left(x\right)$, we
can compute
$\lambda\int_{0}^{+\infty}\pi_{1}^{d}\left(x\right)\text{d}x=\lambda\theta\rho^{d},$
$\int_{0}^{+\infty}\mu\left(x\right)\pi_{2}\left(x\right)\text{d}x=\theta\rho^{d+1}\mu\int_{0}^{+\infty}\mu\left(x\right)\overline{G}\left(x\right)\text{d}x=\lambda\theta\rho^{d}$
and
$\lambda\int_{0}^{+\infty}\pi_{2}^{d}\left(x\right)\text{d}x=\lambda\theta^{d}\rho^{d^{2}+d}\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}\text{d}x=\lambda\theta^{d+1}\rho^{d^{2}+d},$
thus it follows from Equation (12) that for $k=2$,
$\int_{0}^{+\infty}\mu\left(x\right)\pi_{3}\left(x\right)\text{d}x=\lambda\theta^{d+1}\rho^{d^{2}+d},$
which leads to
$\pi_{3}\left(x\right)=\theta^{d+1}\rho^{d^{2}+d}\overline{G}\left(x\right)=\theta^{d+1}\rho^{d^{2}+d+1}\cdot\mu\overline{G}\left(x\right).$
(17)
Based on the above analysis for the simple expressions $\pi_{k}\left(x\right)$
for $k=1,2$ and $3$, we can summarize the following theorem.
###### Theorem 1
The fixed point
$\pi=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$
is given by
$\pi_{0}=1,$ $\pi_{1}\left(x\right)=\rho\cdot\mu\overline{G}\left(x\right)$
and for $k\geq 2,$
$\pi_{k}\left(x\right)=\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\mu\overline{G}\left(x\right),$
(18)
or
$\pi_{k}\left(x\right)=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\mu\overline{G}\left(x\right).$
(19)
Proof By induction, one can easily derive the above result.
It is clear from (16) and (17) that Equation (18) or (19) is correct for the
cases with $l=2,3$. Now, we assume that Equation (19) is correct for the cases
with $l=k$. Then
$\lambda\int_{0}^{+\infty}\pi_{k-1}^{d}\left(x\right)\text{d}x=\lambda\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-d}{d-1}},$
$\int_{0}^{+\infty}\mu\left(x\right)\pi_{k}\left(x\right)\text{d}x=\lambda\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-d}{d-1}}$
and
$\lambda\int_{0}^{+\infty}\pi_{k}^{d}\left(x\right)\text{d}x=\lambda\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k+1}-d}{d-1}},$
it follows from Equation (12) that
$\int_{0}^{+\infty}\mu\left(x\right)\pi_{k+1}\left(x\right)\text{d}x=\lambda\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k+1}-d}{d-1}}.$
Thus, for $l=k+1$ we have
$\pi_{k+1}\left(x\right)=\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k+1}-1}{d-1}}\cdot\mu\overline{G}\left(x\right).$
This completes the proof.
Let
$\widetilde{\theta}=\int_{0}^{+\infty}\left[\overline{G}\left(x\right)\right]^{d}$d$x$.
Then $\theta=\mu^{d}\widetilde{\theta}$. The following corollary provides
another expression for the fixed point.
###### Corollary 2
$\pi_{0}=1$
and for $k\geq 1$
$\pi_{k}\left(x\right)=\lambda^{\frac{d^{k}-1}{d-1}}\cdot\left\\{\widetilde{\theta}^{\frac{d^{k-1}-1}{d-1}}\overline{G}\left(x\right)\right\\}.$
It is easy to see from Corollary 2 that the fixed point is decomposited into
two groups of information under a product form: the arrival information and
the service information. At the same time, the service information indicates
that the doubly exponential solution to the fixed point must exist for
$0<\mu<+\infty$, even if the service times are heavy-tailed.
The following corollary provides an upper bound for the fixed point.
###### Corollary 3
For $k\geq 1$ and $x\geq 0,$
$\pi_{k}\left(x\right)<\int_{0}^{+\infty}\pi_{k}\left(x\right)\text{d}x<\rho^{\frac{d^{k-1}-1}{d-1}}\frac{\lambda^{d^{k}}}{\mu}.$
Proof: Note that $0\leq\overline{G}\left(x\right)\leq 1$, we have
$\widetilde{\theta}=\int_{0}^{+\infty}\left[\overline{G}\left(x\right)\right]^{d}\text{d}x<\int_{0}^{+\infty}\overline{G}\left(x\right)\text{d}x=\frac{1}{\mu}.$
It follows from Corollary 2 that
$\pi_{k}\left(x\right)<\int_{0}^{+\infty}\pi_{k}\left(x\right)\text{d}x<\int_{0}^{+\infty}\lambda^{\frac{d^{k}-1}{d-1}}\widetilde{\theta}^{\frac{d^{k-1}-1}{d-1}}\overline{G}\left(x\right)\text{d}x=\rho^{\frac{d^{k-1}-1}{d-1}}\frac{\lambda^{d^{k}}}{\mu}.$
This completes the proof.
Now, we compute the expected sojourn time $T_{d}$ which a tagged arriving
customer spends in the supermarket model. For the general service times, a
tagged arriving customer is the $k$th customer in the corresponding queue with
the following probability
$\int_{0}^{+\infty}\pi_{k-1}^{d}\left(x\right)\text{d}x-\int_{0}^{+\infty}\pi_{k}^{d}\left(x\right)\text{d}x=\theta^{\frac{d^{k-2}-1}{d-1}}\rho^{\frac{d^{k-1}-1}{d-1}}-\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}.$
When $k\geq 1$, the head customer in the queue has been served, and so its
service time is residual and is denoted as $X_{R}$. Under the stationary
setting, we have
$P\left\\{X_{R}\leq
x\right\\}=\int_{0}^{x}\left[\mu\overline{G}\left(y\right)\right]\text{d}y$
with
$E\left[X_{R}\right]=\int_{0}^{+\infty}\int_{x}^{+\infty}\left[\mu\overline{G}\left(y\right)\right]\text{d}y\text{d}x.$
Thus it is easy to see that the expected sojourn time of the tagged arriving
customer is given by
$\displaystyle E\left[T_{d}\right]=$ $\displaystyle\left[\pi_{0}^{\odot
d}-\int_{0}^{+\infty}\pi_{1}^{d}\left(x\right)\text{d}x\right]E\left[X\right]$
$\displaystyle+\sum_{k=1}^{\infty}\left[\int_{0}^{+\infty}\pi_{k}^{d}\left(x\right)\text{d}x-\int_{0}^{+\infty}\pi_{k+1}^{d}\left(x\right)\text{d}x\right]\left[E\left[X_{R}\right]+kE\left[X\right]\right]$
$\displaystyle=$
$\displaystyle\left[1-\int_{0}^{+\infty}\pi_{1}^{d}\left(x\right)\text{d}x\right]E\left[X\right]+\int_{0}^{+\infty}\pi_{1}^{d}\left(x\right)\text{d}xE\left[X_{R}\right]$
$\displaystyle+E\left[X\right]\sum_{k=1}^{\infty}k\left[\int_{0}^{+\infty}\pi_{k}^{d}\left(x\right)\text{d}x-\int_{0}^{+\infty}\pi_{k+1}^{d}\left(x\right)\text{d}x\right]$
$\displaystyle=$
$\displaystyle\left\\{E\left[X_{R}\right]-E\left[X\right]\right\\}\int_{0}^{+\infty}\pi_{1}^{d}\left(x\right)\text{d}x+E\left[X\right]\left\\{1+\sum_{k=1}^{\infty}\int_{0}^{+\infty}\pi_{k}^{d}\left(x\right)\text{d}x\right\\}$
$\displaystyle=$
$\displaystyle\theta\rho^{d}\left\\{E\left[X_{R}\right]-E\left[X\right]\right\\}+E\left[X\right]\left[\sum_{k=1}^{\infty}\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k}-d}{d-1}}\right].$
If the service times are exponential, then
$E\left[X_{R}\right]=E\left[X\right]$, thus we obtain
$E\left[T_{d}\right]=\frac{1}{\mu}\left[\sum_{k=1}^{\infty}\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k}-d}{d-1}}\right],$
which is the same as Corollary 3.8 in Mitzenmacher [20].
We consider a computational example for the expected sojourn time in the
supermarket model with an Erlang service time distribution
$E\left(m,\mu\right)$, where $m=2,\mu=1,d=2$. Figure 3 shows how the the
expected sojourn time depends on the arrival rate.
Figure 2: the expected sojourn time $E\left[T_{d}\right]$ depends on the
arrival rate $\lambda$
With the results from Equation (18) or (19), let us now provide some useful
discussions on the _asymptotic behavior_ of the fixed point
$\pi\left(x\right)=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$.
Note that we express $a_{k}\backsim O\left(b_{k}\right)$ if
$\lim_{k\rightarrow\infty}a_{k}/b_{k}=c\in\left(-\infty,0\right)\cup\left(0,+\infty\right)$.
###### Remark 2
If the general distribution $G\left(x\right)$ and its mean $1/\mu$ are given,
then
$\theta=\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}$d$x$
is a deterministic factor. We have
$\frac{\pi_{k}\left(x\right)}{\theta^{\frac{d^{k-1}-1}{d-1}}}\sim
O\left(\rho^{\frac{d^{k}-1}{d-1}}\right)\mu\overline{G}\left(x\right),\text{ \
as }k\rightarrow\infty.$
In this case, the heavy traffic should have a bigger influence on the
asymptotic behavior of the sequence
$\left\\{\frac{\pi_{k}\left(x\right)}{\theta^{\frac{d^{k-1}-1}{d-1}}}\right\\}$.
###### Remark 3
If $\rho$ is given, then
$\frac{\pi_{k}\left(x\right)}{\rho^{\frac{d^{k}-1}{d-1}}}\sim
O\left(\theta^{\frac{d^{k-1}-1}{d-1}}\right)\mu\overline{G}\left(x\right),\text{
\ as }k\rightarrow\infty.$
In this case, the maximal value $\theta_{\max}$ of the positive number
$\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}$d$x$ should
have a bigger influence on the asymptotic behavior of the sequence
$\left\\{\frac{\pi_{k}\left(x\right)}{\rho^{\frac{d^{k}-1}{d-1}}}\right\\}$.
## 4 A discussion for the key parameter $\theta$
In this section, we provide a necessary discussion for the key parameter
$\theta$ in the doubly exponential solution of Theorem 1. Based on this, for
the fixed point we give a new and important observation: the doubly
exponential solution to the fixed point can extensively exist for
$0<\mu<+\infty$, even if the service time distribution is heavy-tailed.
Note that
$\displaystyle\theta$
$\displaystyle=\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}\text{d}x$
$\displaystyle=\frac{\int_{0}^{+\infty}\left[\overline{G}\left(x\right)\right]^{d}\text{d}x}{\left[\int_{0}^{+\infty}\overline{G}\left(x\right)\text{d}x\right]^{d}},$
(20)
it is easy to see that $\theta=1$ if $d=1$. Thus, we need to analyze the case
for $k\geq 2$ as follows.
Since $0\leq\overline{G}\left(x\right)\leq 1$, we get that
$0\leq\left[\overline{G}\left(x\right)\right]^{d}\leq\overline{G}\left(x\right)\leq
1$, which leads to
$\int_{0}^{+\infty}\left[\overline{G}\left(x\right)\right]^{d}dx\leq\int_{0}^{+\infty}\overline{G}\left(x\right)dx=1/\mu.$
It is easy to see that $0<\theta<\mu^{d-1}$, and thus if $0<\mu<+\infty$, then
$0<\theta<+\infty$.
In what follows we analyze five simple and useful examples. In first two
examples, the service time distribution is light-tailed; while in the last
three examples, the service time distribution is heavy-tailed. Specifically,
the examples with heavy-tailed service times illustrate two important
observations: the first one indicates that the fixed point for the supermarket
model is different from the tail of stationary queue length distribution for
the ordinary M/G/1 queue, and the second one is to show that the doubly
exponential solution to the fixed point can exist extensively if the service
time mean is non-zero and finite.
Example one: Exponential distribution. Let $\overline{G}\left(x\right)=e^{-\mu
x}$. Then $\theta=\mu^{d-1}/d$. It is easy to see that when
$\mu>\sqrt[d-1]{d}$, $\theta>1$; when $\mu=\sqrt[d-1]{d}$, $\theta=1$; and
when $\mu<\sqrt[d-1]{d}$, $0<\theta<1$. If $d=2$, then $\theta$ is a linear
function of $\mu$, and if $d=3$, then $\theta$ is a nonlinear function of
$\mu$. Figures 3 and 4 show the functions $\theta=\mu/2$ and
$\theta=\mu^{2}/3$, respectively.
Figure 3: $\theta$ is a linear function of $\mu$ for $d=2$ Figure 4: $\theta$
is a nonlinear function of $\mu$ for $d=3$
Example two: Erlang distribution $E\left(m,\mu\right)$. Let
$\overline{G}\left(x\right)=e^{-\mu x}\sum_{k=0}^{m}\frac{\left(\mu
x\right)^{k}}{k!}$. Then $\theta$ is given by
$\theta=\left(\frac{\mu}{m}\right)^{d}\int_{0}^{+\infty}e^{-\mu
dx}\left[\sum_{k=0}^{m}\frac{\left(\mu x\right)^{k}}{k!}\right]^{d}\text{d}x.$
Let $\mu=1$. Table 1 lists how $\theta$ depends on the parameter pair
$\left(m,d\right)$. As seen from Table 1, $\theta$ is decreasing for each of
the two parameters $m$ and $d$.
Table 1: $\theta$ depends on the Erlang parameter pair $(m,d)$ $(m,d)$ | (2, 2) | (2, 5) | (2, 10) | (5, 2) | (10, 2) | (5, 5) | (10, 10)
---|---|---|---|---|---|---|---
$\theta$ | 0.52 | 0.19 | $9.15\times 10^{-2}$ | $4.13\times 10^{-2}$ | $9.48\times 10^{-4}$ | $1.11\times 10^{-3}$ | $6.51\times 10^{-10}$
Example three: Weibull distribution $W\left(\tau,\mu\right)$. Let
$\overline{G}\left(x\right)=\exp\left\\{-\left(\mu t\right)^{\tau}\right\\}$.
It is easy to check that the mean of the Weibull distribution is given by
$\frac{1}{\mu}\Gamma\left(1+\frac{1}{\tau}\right),$
which follows that $\theta$ is given by
$\theta=\frac{\mu^{d-1}}{d^{\frac{1}{\tau}}\left[\Gamma\left(1+\frac{1}{\tau}\right)\right]^{d-1}},$
where $\Gamma(\alpha)=\int_{0}^{+\infty}x^{\alpha-1}e^{-x}$d$x$. Obviously,
the Weibull distribution $W\left(\tau,\mu\right)$ is heavy-tailed if
$0<\tau<1$; and the Weibull distribution $W\left(\tau,\mu\right)$ is light-
tailed if $\tau>1$. To indicate the role played by the heavy-tailed parameter
$\tau$ for $0<\tau<1$, taking $\mu=5$ and $d=2$ we have
$\theta=\frac{5}{2^{\frac{1}{\tau}}\Gamma\left(1+\frac{1}{\tau}\right)}.$
Table 2 indicates how $\theta$ depends on the heavy-tailed parameter $\tau$,
such as, $0<\theta<1$ if $\tau=0.2$; $\theta>1$ if $\tau=0.9$. This example,
together with Theorem 1, illustrates an important observation that the fixed
point
$\pi=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$
is doubly exponential (clearly, it is light-tailed) even if the service time
distribution is heavy-tailed. Based on this, the the fixed point is different
from the tail of stationary queue length distribution of the ordinary M/G/1
queue, since for the ordinary M/G/1 queue, the stationary queue length
distribution is heavy-tailed if the service time distribution is heavy-tailed,
e.g., see Adler, Feldman and Taqqu [1].
Table 2: $\theta$ depends on the heavy-tailed parameter $\tau$ for $0<\tau<1$ $\tau$ | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9
---|---|---|---|---|---|---|---|---
$\theta$ | $1.3\times 10^{-3}$ | $5.3\times 10^{-2}$ | 0.27 | 0.63 | 1.05 | 1.47 | 1.86 | 2.19
Example four: Power law distribution. Let
$\overline{G}\left(x\right)=\left(\mu+x\right)^{-\alpha}$. If $0<\alpha\leq
1$, then the power law distribution does not exist the finite mean. In this
case, we can not setup the system of integral-differential equations for the
fraction density vector
$S\left(t,x\right)=\left(S_{0}\left(t\right),S_{1}\left(t,x\right),S_{2}\left(t,x\right),\ldots\right)$
which leads to the analysis for the fixed point. Thus we only deal with the
case with $\alpha>1$. Note that for each $\alpha>1$
$\int_{0}^{+\infty}\overline{G}\left(x\right)\text{d}x=\frac{1}{\mu}$
and
$\int_{0}^{+\infty}\left[\overline{G}\left(x\right)\right]^{d}\text{d}x=\frac{1}{\mu},$
thus we obtain $\theta=\mu^{d-1}$. It is easy to see that when $\mu>1$,
$\theta>1$; when $\mu=1$, $\theta=1$; and when $0<\mu<1$, $0<\theta<1$. It
follows from Theorem 1 that for $k\geq 1$
$\pi_{k}\left(x\right)=\mu^{d^{k-1}-1}\rho^{\frac{d^{k}-1}{d-1}}\cdot\mu\overline{G}\left(x\right).$
This indicates that the fixed point
$\pi=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$
is doubly exponential (of course, it is light-tailed) if the service time
distribution is power law.
Example five: Almost exponential distribution. Let
$\overline{G}\left(x\right)=\exp\left\\{-x\left(\ln
x\right)^{-\alpha}\right\\}$. Then it is easy to see that the almost
exponential distribution is heavy-tailed if $\alpha>0$
$\theta=\frac{\int_{0}^{+\infty}\exp\left\\{-dx\left(\ln
x\right)^{-\alpha}\right\\}\text{d}x}{\left[\int_{0}^{+\infty}\exp\left\\{-x\left(\ln
x\right)^{-\alpha}\right\\}\text{d}x\right]^{d}}$
Table 3 lists how $\theta$ depends on the parameter pair $(d,\alpha)$. As seen
from Table 3, $\theta$ is decreasing for each of the two parameters $d$ and
$\alpha$.
Table 3: $\theta$ depends on the parameter pair $(d,\alpha)$ $(d,\alpha)$ | (2, 2) | (4, 2) | (2, 4) | (4, 4)
---|---|---|---|---
$\theta$ | $2.24\times 10^{-2}$ | $2.01\times 10^{-4}$ | $3.44\times 10^{-5}$ | $1.18\times 10^{-13}$
## 5 The key parameter $\theta$ for PH Service Times
In this section, as an important example we provide three methods to analyze a
supermarket model with Poisson arrivals and PH service times. Our purpose is
to provide three different ways to determine the key parameter $\theta$ and
compute the doubly exponential solution to the fixed point. Also, we indicate
that the doubly exponential solution to the fixed point is not unique for a
more general supermarket model.
The supermarket model with Poisson arrivals and PH service times is described
as follows. Customers arrive at a queueing system of $n>1$ servers as a
Poisson process with arrival rate $n\lambda$ for $\lambda>0$. The service
times of these customers are of phase type with irreducible representation
$\left(\alpha,T\right)$ of order $m$. Each arriving customer chooses $d\geq 1$
servers independently and uniformly at random from these $n$ servers, and
waits for service at the server which currently contains the fewest number of
customers. If there is a tie, servers with the fewest number of customers will
be chosen randomly. All customers in any service center will be served in the
FCFS manner. For the PH service time distribution, we use the irreducible
representation $\left(\alpha,T\right)$ of order $m$, where the row vector
$\alpha$ is a probability vector whose $j$th entry is the probability that a
service begins in phase $j$ for $1\leq j\leq m$; and $T$ is a matrix of order
$m$ whose $\left(i,j\right)^{\text{th}}$ entry is denoted by $t_{i,j}$ with
$t_{i,i}<0$ for $1\leq i\leq m$, and $t_{i,j}\geq 0$ for $1\leq i,j\leq m$ and
$i\neq j$. Let $T^{0}=-Te\gvertneqq 0$, where $e$ is a column vector of ones
with a suitable dimension in the context. When a PH service time is in phase
$i$, the transition rate from phase $i$ to phase $j$ is $t_{i,j}$, the service
completion rate is $t_{i}^{0}$. At the same time, the mean service rate is
given by
$\mu=-\frac{1}{\alpha T^{-1}e}.$
Unless we state otherwise, we assume that all the random variables defined
above are independent, and that the system is operating at the stable region:
$\rho=\lambda/\mu<1$.
We introduce some useful notation. Let $n_{k}^{\left(i\right)}\left(t\right)$
be the number of queues with at least $k$ customers and the service time in
phase $i$ at time $t\geq 0$. Clearly, $0\leq
n_{k}^{\left(i\right)}\left(t\right)\leq n$ for $1\leq i\leq m$ and $0\leq
k\leq n$. We define
$s_{k}^{\left(i\right)}\left(t\right)=\frac{n_{k}^{\left(i\right)}\left(t\right)}{n},$
which is the fraction of queues with at least $k$ customers and the service
time in phase $i$. We write
$S_{0}\left(t\right)=\left(s_{0}\left(t\right)\right)$
and for $k\geq 1$,
$S_{k}\left(t\right)=\left(s_{k}^{\left(1\right)}\left(t\right),s_{k}^{\left(2\right)}\left(t\right),\ldots,s_{k}^{\left(m\right)}\left(t\right)\right),$
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right).$
We now introduce Hadamard Product of two matrices $A=\left(a_{i,j}\right)$ and
$B=\left(b_{i,j}\right)$ as follows:
$A\odot B=\left(a_{i,j}b_{i,j}\right).$
Specifically, for $k\geq 2$ we have
$A^{\odot k}=\underset{k\text{ matrix }A}{\underbrace{A\odot A\odot\cdots\odot
A}}.$
Let $a=\left(a_{1},a_{2},a_{3},\ldots\right)$. We write
$a^{\odot\frac{1}{d}}=\left(a_{1}^{\frac{1}{d}},a_{2}^{\frac{1}{d}},a_{3}^{\frac{1}{d}},\ldots\right).$
Using a similar analysis to that in Equations (4) to (7), we can obtain the
following systems of differential vector equations for the fraction density
vector
$S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$.
$S_{0}\left(t\right)=1,\text{ \ for }t\geq 0,$ (21)
$\frac{\mathtt{d}}{\text{d}t}S_{0}\left(t\right)=-\lambda S_{0}^{\odot
d}\left(t\right)+S_{1}\left(t\right)T^{0},$ (22)
$\frac{\mathtt{d}}{\text{d}t}S_{1}\left(t\right)=\lambda\alpha S_{0}^{\odot
d}\left(t\right)-\lambda S_{1}^{\odot
d}\left(t\right)+S_{1}\left(t\right)T+S_{2}\left(t\right)T^{0}\alpha,$ (23)
and for $k\geq 2$,
$\frac{\mathtt{d}}{\text{d}t}S_{k}\left(t\right)=\lambda S_{k-1}^{\odot
d}\left(t\right)-\lambda S_{k}^{\odot
d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha.$ (24)
If $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is a fixed point of the
fraction density vector $S\left(t\right)$, then the system of differential
vector equations (21) to (24) can be simplified as
$\pi_{0}=1$ (25) $-\lambda\pi_{0}^{\odot d}+\pi_{1}T^{0}=0,$ (26)
$\lambda\alpha\pi_{0}^{\odot d}-\lambda\pi_{1}^{\odot
d}+\pi_{1}T+\pi_{2}T^{0}\alpha=0,$ (27)
and for $k\geq 2$,
$\lambda\pi_{k-1}^{\odot d}-\lambda\pi_{k}^{\odot
d}+\pi_{k}T+\pi_{k+1}T^{0}\alpha=0.$ (28)
In what follows we provide three methods to solve the system of nonlinear
equations (25) to (28), and give three different doubly exponential solutions
to the fixed point.
### 5.1 The first method
The first method is based on Theorem 1 given in this paper. For the PH service
time distribution
$\overline{G}\left(x\right)=\alpha\exp\left\\{Tx\right\\}e$
Let
$\theta=\int_{0}^{+\infty}\left[\mu\overline{G}\left(x\right)\right]^{d}$d$x$,
and we assume that $0<\theta<+\infty$. Then the fixed point
$\pi=\left(\pi_{0},\pi_{1}\left(x\right),\pi_{2}\left(x\right),\ldots\right)$
is given by
$\pi_{0}=1,$
and for $k\geq 1$
$\pi_{k}\left(x\right)=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\mu\overline{G}\left(x\right).$
(29)
### 5.2 The second method
The second method is proposed in Li, Wang and Liu [11], and the key parameter
$\theta$ is based on the stationary probability vector $\omega$ of the
irreducible Markov chain $T+T^{0}\alpha$, that is, $\theta=\omega^{\odot d}e$.
It follows from Equation (26) that
$\pi_{1}T^{0}=\lambda.$
Note that
$\omega T^{0}=\mu,$ $\frac{\lambda}{\mu}\omega T^{0}=\lambda.$ (30)
Thus, we obtain
$\pi_{1}=\frac{\lambda}{\mu}\omega=\rho\cdot\,\omega.$
Based on the fact that $\pi_{0}=1$ and $\pi_{1}=\rho\cdot\omega$, it follows
from Equation (27) that
$\lambda\alpha-\lambda\rho^{d}\cdot\omega^{\odot d}+\rho\cdot\omega
T+\pi_{2}T^{0}\alpha=0,$
which leads to
$\lambda-\lambda\rho^{d}\cdot\omega^{\odot d}e+\rho\cdot\omega
Te+\pi_{2}T^{0}=0.$
Note that $\omega Te=-\mu$ and $\rho=\lambda/\mu$, we obtain
$\pi_{2}T^{0}=\lambda\rho^{d}\omega^{\odot d}e.$
Let $\theta=\omega^{\odot d}e$. Then it is easy to see that
$\theta\in\left(0,1\right)$, and
$\pi_{2}T^{0}=\lambda\theta\rho^{d}.$
Using a similar analysis on Equation (30), we have
$\pi_{2}=\frac{\lambda\theta\rho^{d}}{\mu}\omega=\theta\rho^{d+1}\cdot\omega.$
Based on $\pi_{1}=\rho\,\omega$ and $\pi_{2}=\theta\rho^{d+1}\cdot\omega$, it
follows from Equation (28) that for $k=2$,
$\lambda\rho^{d}\cdot\omega^{\odot
d}-\lambda\theta^{d}\rho^{d^{2}+d}\cdot\omega^{\odot
d}+\theta\rho^{d+1}\cdot\omega T+\pi_{3}T^{0}\alpha=0,$
which leads to
$\lambda\theta\rho^{d}-\lambda\theta^{d+1}\rho^{d^{2}+d}+\theta\rho^{d+1}\cdot\omega
Te+\pi_{3}T^{0}=0,$
thus we obtain
$\pi_{3}T^{0}=\lambda\theta^{d+1}\rho^{d^{2}+d}.$
Using a similar analysis on Equation (30), we have
$\pi_{3}=\frac{\lambda\theta^{d+1}\rho^{d^{2}+d}}{\mu}\omega=\theta^{d+1}\rho^{d^{2}+d+1}\cdot\omega.$
Now, we assume that
$\pi_{k}=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega$
is correct for the cases with $l=k$. Then it follows from Equation (28) that
for $l=k+1$, we have
$\displaystyle\lambda$
$\displaystyle\theta^{d^{k-2}+d^{k-3}+\cdots+d}\rho^{d^{k-1}+d^{k-2}+\cdots+d}\cdot\omega^{\odot
d}-\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d}\rho^{d^{k}+d^{k-1}+\cdots+d}\cdot\omega^{\odot
d}$
$\displaystyle+\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega
T+\pi_{k+1}T^{0}\alpha=0,$
which leads to
$\displaystyle\lambda$
$\displaystyle\theta^{d^{k-2}+d^{k-3}+\cdots+d+1}\rho^{d^{k-1}+d^{k-2}+\cdots+d}-\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}$
$\displaystyle+\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega
Te+\pi_{k+1}T^{0}=0,$
thus we obtain
$\pi_{k+1}T^{0}=\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}.$
By a similar analysis to (30), we have
$\displaystyle\pi_{k+1}$
$\displaystyle=\frac{\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}}{\mu}\omega$
$\displaystyle=\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d+1}\cdot\omega.$
Therefore, by induction the fixed point
$\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is given by
$\pi_{0}=1,$
and for $k\geq 1$
$\pi_{k}=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega.$
(31)
### 5.3 The third method
The third method is based on the matrix computation for the system of
nonlinear equations (25) to (28), and shows that the key parameter $\theta$ is
based on the initial probability vector $\alpha$ in the PH service time
distribution, that is, $\theta=1/\alpha^{\odot\frac{1}{d}}e.$
It follows from (25) to (28) that
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}-\lambda&\lambda&&&\\\
&-\lambda&\lambda&&\\\ &&-\lambda&\lambda&\\\
&&&\ddots&\ddots\end{array}\right)+\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)\left(\begin{array}[c]{cccc}T&&&\\\
T^{0}\alpha&T&&\\\ &T^{0}\alpha&T&\\\ &&\ddots&\ddots\end{array}\right)$
$\displaystyle=-\left(\lambda\alpha,0,0,\ldots\right),$
which leads to
$\displaystyle\left(\pi_{1},\pi_{2},\pi_{3},\ldots\right)=$
$\displaystyle\left(\pi_{1}^{\odot d},\pi_{2}^{\odot d},\pi_{3}^{\odot
d},\ldots\right)\left(\begin{array}[c]{ccccc}R&V&&&\\\ &R&V&&\\\ &&R&V&\\\
&&&\ddots&\ddots\end{array}\right)$
$\displaystyle+\left(\lambda\alpha\left(-T\right)^{-1},0,0,\ldots\right),$
where
$V=\lambda\left(-T\right)^{-1}$
and
$R=\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}.$
Thus we obtain
$\pi_{1}=\lambda\alpha\left(-T\right)^{-1}+\pi_{1}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]$ (32)
and for $k\geq 2$
$\pi_{k}=\pi_{k-1}^{\odot
d}\left[\lambda\left(-T\right)^{-1}\right]+\pi_{k}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right].$ (33)
To omit the term $\pi_{k}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]$ for $k\geq
1$, we assume that $\left\\{\pi_{k},k\geq 1\right\\}$ has the following
expression
$\pi_{k}=r\left(k\right)\alpha^{\odot\frac{1}{d}}.$
In this case, we have
$\pi_{k}^{\odot
d}\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]=r^{d}\left(k\right)\alpha\left[\lambda\left(-I+e\alpha\right)\left(-T\right)^{-1}\right]=0,$
thus it follows from (32) and (33) that
$\pi_{1}=\lambda\alpha\left(-T\right)^{-1}$ (34)
and for $k\geq 2$
$\pi_{k}=\pi_{k-1}^{\odot d}\left[\lambda\left(-T\right)^{-1}\right].$ (35)
It follows from (34) that
$r\left(1\right)\alpha^{\odot\frac{1}{d}}=\lambda\alpha\left(-T\right)^{-1},$
which follows that
$r\left(1\right)=\theta\rho,$
where
$\theta=\frac{1}{\alpha^{\odot\frac{1}{d}}e}.$
It follows from (35) that
$r\left(k\right)\alpha^{\odot\frac{1}{d}}=r^{d}\left(k-1\right)\alpha\left[\lambda\left(-T\right)^{-1}\right],$
which follows that
$r\left(k\right)=r^{d}\left(k-1\right)\theta\rho=\left(\theta\rho\right)^{\frac{d^{k}-1}{d-1}}.$
Therefore, we can obtain
$\pi_{0}=1$
and for $k\geq 1$
$\pi_{k}=\left(\theta\rho\right)^{\frac{d^{k}-1}{d-1}}\cdot\alpha^{\odot\frac{1}{d}}.$
(36)
### 5.4 Non-uniqueness
Based on the above three methods, we can summarize the key parameter and the
doubly exponential solution to the fixed point in the following table.
Table 4: Comparison for the three methods | Key Parameter | Fixed point
---|---|---
Method 1 | $\theta=\frac{\int_{0}^{+\infty}\left[\alpha\exp\left\\{Tx\right\\}e\right]^{d}\text{d}x}{\left[-\alpha T^{-1}e\right]^{d}}$ | $\pi_{k}=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\mu\overline{G}\left(x\right)$
Method 2 | $\theta=\omega^{\odot d}e$ | $\pi_{k}=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega$
Method 3 | $\theta=1/\alpha^{\odot\frac{1}{d}}e$ | $\pi_{k}=\left(\theta\rho\right)^{\frac{d^{k}-1}{d-1}}\cdot\alpha^{\odot\frac{1}{d}}$
When the PH service time is an $m$-order Erlang distribution with the
irreducible representation $(\alpha,T)$, where
$\alpha=\left(1,0,\ldots,0\right)$
and
$T=\left(\begin{array}[c]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\
&&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ &&&&-\eta\end{array}\right),\text{ \ \
}T^{0}=\left(\begin{array}[c]{c}0\\\ 0\\\ \vdots\\\ 0\\\
\eta\end{array}\right).$
We have
$\alpha^{\odot\frac{1}{d}}=\left(1,0,\ldots,0\right)$
and
$\theta=\frac{1}{\alpha^{\odot\frac{1}{d}}e}=1.$
Thus the doubly exponential solution by the third method is given by
$\pi_{k}=\rho^{\frac{d^{k}-1}{d-1}}\cdot\left(1,0,\ldots,0\right),\text{ \
}k\geq 1.$ (37)
It is clear that
$T+T^{0}\alpha=\left(\begin{array}[c]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\
&&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ \eta&&&&-\eta\end{array}\right),$
which leads to the stationary probability vector of the Markov chain
$T+T^{0}\alpha$ as follows:
$\omega=\left(\frac{1}{m},\frac{1}{m},\ldots,\frac{1}{m}\right),$ $\mu=\omega
T^{0}=\frac{\eta}{m},$ $\rho=\frac{\lambda}{\mu}=\frac{m\lambda}{\eta}$
and
$\theta=\omega^{\odot d}e=m\left(\frac{1}{m}\right)^{d}=m^{1-d}.$
Thus the doubly exponential solution by the second method is given by
$\displaystyle\pi_{k}$
$\displaystyle=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\left(\frac{1}{m},\frac{1}{m},\ldots,\frac{1}{m}\right)$
$\displaystyle=\rho^{\frac{d^{k}-1}{d-1}}\cdot\left(m^{-d^{k-1}},m^{-d^{k-1}},\ldots,m^{-d^{k-1}}\right),\text{
\ }k\geq 1.$ (38)
It is clear that the three doubly exponential solutions (29), (37) and (38)
are different for $m\geq 2$.
###### Remark 4
For the supermarket model with Poisson arrivals and PH service times, we have
obtained three different doubly exponential solutions to the fixed point. It
is interesting but difficult how to be able to find another new doubly
exponential solution. We believe that it is an open problem how to give all
the doubly exponential solutions to the fixed point for a more general
supermarket model including the case with MAP arrivals, PH service times or
general service times.
## 6 Exponential convergence to the fixed point
In this section, we study the exponential convergence of the current location
$S\left(t,x\right)$ of the supermarket model to its fixed point
$\pi\left(x\right)$ for $t\geq 0$ and $x\geq 0$. Not only does the exponential
convergence indicates the existence of the fixed point, but it also explains
such a convergent process is very fast.
For the supermarket model, the initial point $S\left(0,x\right)$ can affect
the current location $S\left(t,x\right)$ for each $t>0$, since the service
process in the supermarket model is under a unified structure. Here, we
provide notation for comparison of two vectors. Let
$a=\left(a_{1},a_{2},a_{3},\ldots\right)$ and
$b=\left(b_{1},b_{2},b_{3},\ldots\right)$. We write $a\prec b$ if
$a_{k}<b_{k}$ for all $k\geq 1$; $a\preceq b$ if $a_{k}\leq b_{k}$ for all
$k\geq 1$.
Now, we can obtain the following useful proposition whose proof is clear from
a sample path analysis and is omitted here.
###### Proposition 2
If $S\left(0,x\right)\preceq\widetilde{S}\left(0,x\right)$, then
$S\left(t,x\right)\preceq\widetilde{S}\left(t,x\right)$.
Based on Proposition 2, the following theorem shows that the fixed point
$\pi\left(x\right)$ is an upper bound of the current location
$S\left(t,x\right)$ for all $t\geq 0$ and $x\geq 0$.
###### Theorem 4
For the supermarket model, if there exists some $k$ such that
$S_{k}\left(0,x\right)=0$, then the sequence
$\left\\{S_{k}\left(t,x\right)\right\\}$ has a upper bound sequence which
decreases doubly exponentially for all $t\geq 0$ and $x\geq 0$, that is,
$S\left(t,x\right)\preceq\pi\left(x\right)$ for all $t\geq 0$ and $x\geq 0$.
Proof Let
$\widetilde{S}_{0}\left(0\right)=\pi_{0}$
$\widetilde{S}_{k}\left(0,x\right)=\pi_{k}\left(x\right),\text{ \ }k\geq 1.$
Then
$\widetilde{S}_{0}\left(t\right)=\widetilde{S}_{0}\left(0\right)=\pi_{0}$, and
for each $k\geq 1$,
$\widetilde{S}_{k}\left(t,x\right)=\widetilde{S}_{k}\left(0,x\right)=\pi_{k}\left(x\right)$
for all $t\geq 0$ and $x\geq 0$, since $\widetilde{S}\left(0\right)$ is a
fixed point in the supermarket model. If $S_{k}\left(0,x\right)=0$ for some
$k$, then $S_{k}\left(0,x\right)<\widetilde{S}_{k}\left(0,x\right)$ and
$S_{j}\left(0,x\right)\leq\widetilde{S}_{j}\left(0,x\right)$ for all $j\geq 1$
and $j\neq k$, thus $S\left(0,x\right)\preceq\widetilde{S}\left(0,x\right)$.
It is easy to see from Proposition 2 that
$S_{k}\left(t,x\right)\leq\widetilde{S}_{k}\left(t,x\right)=\pi_{k}\left(x\right)$
for all $k\geq 1$, $t\geq 0$ and $x\geq 0$. Thus we obtain that for all $k\geq
1$, $t\geq 0$ and $x\geq 0$
$S_{k}\left(t,x\right)\leq\pi_{k}\left(x\right)=\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\left[\mu\overline{G}\left(x\right)\right].$
This completes the proof.
To show the exponential convergence, we use Theorem 4 to define a potential
function (or Lyapunov function) $\Phi\left(t\right)$ as follows:
$\Phi\left(t\right)=\sum_{k=1}^{\infty}w_{k}\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x,$
where $\left\\{w_{k}\right\\}$ is a positive scalar sequence with
$w_{k}>w_{k-1}\geq w_{1}=1$ for $k\geq 2$. Note that
$\pi_{0}=S_{0}\left(t\right)=1$. It is easy to see from Proposition 2 that
$\Phi\left(t\right)\geq 0$ for all $t\geq 0$.
When
$\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]$d$x>0$
for $k\geq 1$, we write
$\frac{\int_{0}^{+\infty}S_{k}^{d}\left(t,x\right)\text{d}x}{\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x}=c_{k}\left(t\right)$
and
$\frac{\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(t,x\right)\text{d}x}{\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x}=d_{k}\left(t\right).$
The following lemma provide a method to determine the positive scalar sequence
$\left\\{w_{k}\right\\}$ with $w_{k}>w_{k-1}\geq w_{1}=1$ for $k\geq 2$. This
proof is easy by means of some simple computation.
###### Lemma 2
If $\delta$ is a positive constant,
$w_{1}=1,$ $\lambda\left(w_{1}-w_{2}\right)c_{1}\left(t\right)=-\delta w_{1}$
and for $k\geq 2$
$\lambda\left(w_{k}-w_{k+1}\right)c_{k}\left(t\right)+\left(w_{k}-w_{k-1}\right)d_{k}\left(t\right)=-\delta
w_{k},$
then
$w_{2}=1+\frac{\delta}{\lambda c_{1}\left(t\right)},$
and for $k\geq 3$
$w_{k}=w_{k-1}+\frac{\delta
w_{k-1}+\left(w_{k-1}-w_{k-2}\right)d_{k-1}\left(t\right)}{\lambda
c_{k-1}\left(t\right)}.$
The following theorem measures the distance $\Phi\left(t\right)$ of the
current location $S\left(t,x\right)$ for $t\geq 0$ and the fixed point
$\pi\left(x\right)$ for $x\geq 0$, and illustrates that the distance
$\Phi\left(t\right)$ to the fixed point from the current location is very go
to zero with exponential convergence. Hence, it shows that from any suitable
starting point, the supermarket model can be quickly close to the fixed point,
that is, there always exists a fixed point in the supermarket model.
###### Theorem 5
For $t\geq 0$,
$\Phi\left(t\right)\leq c_{0}e^{-\delta t},$
where $c_{0}$ and $\delta$ are two positive constants, and they possibly
depend on time $t\geq 0$. In this case, the potential function
$\Phi\left(t\right)$ is exponentially convergent.
Proof Note that
$\Phi\left(t\right)=\sum_{k=1}^{\infty}w_{k}\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x,$
we have
$\displaystyle\frac{\text{d}}{\text{d}t}\Phi\left(t\right)$
$\displaystyle=\frac{\text{d}}{\text{d}t}\sum_{k=1}^{\infty}w_{k}\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x$
$\displaystyle=-\sum_{k=1}^{\infty}w_{k}\frac{\text{d}}{\text{d}t}\int_{0}^{+\infty}S_{k}\left(t,x\right)\text{d}x$
by means of the Dominated Convergence Theorem. It follows from (4) to (7) that
$\int_{0}^{+\infty}\mu\left(x\right)S_{1}\left(t,x\right)\text{d}x=\lambda,$
(39)
and using (39) we obtain
$\displaystyle\frac{d}{dt}\Phi\left(t\right)=$
$\displaystyle-\sum_{k=1}^{\infty}w_{k}\frac{\mathtt{d}}{\text{d}t}\int_{0}^{+\infty}S_{k}\left(t,x\right)\text{d}x$
$\displaystyle=$ $\displaystyle-
w_{1}[\lambda-\lambda\int_{0}^{+\infty}S_{1}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)S_{1}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{2}\left(t,x\right)\text{d}x]$
$\displaystyle-\sum_{k=2}^{\infty}w_{k}[\lambda\int_{0}^{+\infty}S_{k-1}^{d}\left(t,x\right)\text{d}x-\lambda\int_{0}^{+\infty}S_{k}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{k+1}\left(t,x\right)\text{d}x]$
$\displaystyle=$ $\displaystyle-
w_{1}[-\lambda\int_{0}^{+\infty}S_{1}^{d}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{2}\left(t,x\right)\text{d}x]$
$\displaystyle-\sum_{k=2}^{\infty}w_{k}[\lambda\int_{0}^{+\infty}S_{k-1}^{d}\left(t,x\right)\text{d}x-\lambda\int_{0}^{+\infty}S_{k}^{d}\left(t,x\right)\text{d}x$
$\displaystyle-\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(t,x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{k+1}\left(t,x\right)\text{d}x],$
which follows
$\displaystyle\frac{d}{dt}\Phi\left(t\right)=$
$\displaystyle\lambda\left(w_{1}-w_{2}\right)c_{1}\left(t\right)\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x$
$\displaystyle+\sum_{k=2}^{\infty}\left[\lambda\left(w_{k}-w_{k+1}\right)c_{k}\left(t\right)+\left(w_{k}-w_{k-1}\right)d_{k}\left(t\right)\right]$
$\displaystyle\cdot\int_{0}^{+\infty}\left[\pi_{k}\left(x\right)-S_{k}\left(t,x\right)\right]\text{d}x.$
Using Lemma 2 we can easily choose a parameter $\delta>0$ and a suitable
positive scalar sequence $\left\\{w_{k}\right\\}$ with $w_{k}>w_{k-1}\geq
w_{1}=1$ for $k\geq 2$ such that
$\lambda\left(w_{1}-w_{2}\right)c_{1}\left(t\right)\leq-\delta w_{1}$
and for $k\geq 2$
$\left(w_{k}-w_{k-1}\right)d_{k}\left(t\right)-\lambda\left(w_{k+1}-w_{k}\right)c_{k}\left(t\right)\leq-\delta
w_{k},$
thus we can obtain
$\frac{d}{dt}\Phi\left(t\right)\leq-\delta\Phi\left(t\right),$
which leads to
$\Phi\left(t\right)\leq c_{0}e^{-\delta t}.$
This completes the proof.
###### Remark 5
We have provided an algorithm for computing the positive scalar sequence
$\left\\{w_{k}\right\\}$ with $1=w_{1}\leq w_{k-1}<w_{k}$ for $k\geq 2$ as
follows:
Step one:
$w_{1}=1.$
Step two:
$w_{2}=1+\frac{\delta}{\lambda c_{1}\left(t\right)}.$
Step three: for $k\geq 2$
$w_{k}=w_{k-1}+\frac{\delta
w_{k-1}+\left(w_{k-1}-w_{k-2}\right)d_{k-1}\left(t\right)}{\lambda
c_{k-1}\left(t\right)}.$
This illustrates that $w_{k}$ is a function of time $t$. Note that
$\lambda,\delta,c_{k}\left(t\right),d_{l}\left(t\right)>0$, it is clear that
for $k\geq 2$
$1=w_{1}\leq w_{k-1}<w_{k}.$
## 7 Lipschitz Condition
In this section, we apply the Kurtz Theorem to study the supermarket model
with general service times, and analyze the Lipschitz condition with respect
to general service times.
The supermarket model can be analyzed by a density dependent jump Markov
process, where the density dependent jump Markov process is a Markov process
with a single parameter $n$ which corresponds to the population size. Kurtz’s
work provides a basis for the density dependent jump Markov processes in order
to relate the infinite-size system of differential equations to the
corresponding finite-size system of differential equations. Readers may refer
to Kurtz [9] for more details.
In the supermarket model, the states of density dependent jump Markov process
can be normalized and interpreted as measuring population densities, so that
the transition rates depend only on these densities. Hence, the infinite-size
system of differential equations can be regarded as the limiting model of the
corresponding finite-size system of differential equations as the population
size grows arbitrarily large. When the population size is $n$, we write
$E_{n}=\left\\{k:k=0,1,\ldots,n\right\\}.$
For $k\geq 1$ and $x\geq 0$, we write
$s_{k}^{\left(n\right)}\left(x\right)=\left(\frac{k}{n},x\right),$
where $x$ is the residual service time of each server, and
$s_{k}^{\left(n\right)}=\int_{0}^{+\infty}s_{k}^{\left(n\right)}\left(x\right)\text{d}x.$
Let
$S_{0}=\lim_{n\rightarrow\infty}s_{0}^{\left(n\right)}$
and for $k\geq 1$
$S_{k}=\lim_{n\rightarrow\infty}\int_{0}^{+\infty}s_{k}^{\left(n\right)}\left(x\right)\text{d}x.$
Let $\left\\{\widehat{X}_{n}\left(t\right):t\geq 0\right\\}$ be a density
dependent jump Markov process on the state space $E_{n}$ whose transition
rates are given by
$q_{k,k+l}^{\left(n\right)}=n\beta_{l}\left(\frac{k}{n}\right)=n\beta_{l}\left(s_{k}^{\left(n\right)}\right).$
In this supermarket model, $\widehat{X}_{n}\left(t\right)$ is the unscaled
process which records the number of servers with at least $k$ customers for
$0\leq k\leq n$.
Let $a$ and $b$ denote an arrival and a service completion, respectively.
Hence taking $l=a$ or $b$ for $a>b>0$, we write
$\beta_{a}\left(s_{0}^{\left(n\right)}\right)=-\lambda,$
$\beta_{b}\left(s_{0}^{\left(n\right)}\right)=\int_{0}^{+\infty}\mu\left(x\right)s_{1}^{\left(n\right)}\left(x\right)\text{d}x;$
$\beta_{a}\left(s_{1}^{\left(n\right)}\right)=\lambda-\lambda\int_{0}^{+\infty}\left[s_{1}^{\left(n\right)}\left(x\right)\right]^{d}\text{d}x,$
$\beta_{b}\left(s_{1}^{\left(n\right)}\right)=-\int_{0}^{+\infty}\mu\left(x\right)s_{1}^{\left(n\right)}\left(x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)s_{2}^{\left(n\right)}\left(x\right)\text{d}x;$
and for $n\geq k\geq 2,$
$\beta_{a}\left(s_{k}^{\left(n\right)}\right)=\lambda\int_{0}^{+\infty}\left[s_{k-1}^{\left(n\right)}\left(x\right)\right]^{d}\text{d}x-\lambda\int_{0}^{+\infty}\left[s_{k}^{\left(n\right)}\left(x\right)\right]^{d}\text{d}x,$
$\beta_{b}\left(s_{k}^{\left(n\right)}\right)=-\int_{0}^{+\infty}\mu\left(x\right)s_{k}^{\left(n\right)}\left(x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)s_{k+1}^{\left(n\right)}\left(x\right)\text{d}x.$
Using Chapter 7 in Kurtz [9] or Subsection 3.4.1 in Mitzenmacher [20], the
Markov process $\left\\{\widehat{X}_{n}\left(t\right):t\geq 0\right\\}$ with
transition rates $q_{k,k+l}^{\left(n\right)}$ is given by
$\widehat{X}_{n}\left(t\right)=\widehat{X}_{n}\left(0\right)+\sum_{l=a,b}lY_{l}\left(n\int_{0}^{t}\beta_{l}\left(\frac{\widehat{X}_{n}\left(u\right)}{n}\right)\text{d}u\right),$
(40)
where $Y_{l}\left(x\right)$ for $l=a$ and $b$ are two independent standard
Poisson processes. Clearly, the jump Markov process by Equation (40) at time
$t$ is determined by the starting point and the transition rates which are
integrated over its history.
Let
$F\left(y\right)=a\beta_{a}\left(y\right)+b\beta_{b}\left(y\right).$ (41)
Taking $X_{n}\left(t\right)=n^{-1}\widehat{X}_{n}\left(t\right)$ which is an
appropriate scaled process, we have
$X_{n}\left(t\right)=X_{n}\left(0\right)+\sum_{l=a,b}ln^{-1}\widehat{Y}_{l}\left(n\int_{0}^{t}\beta_{l}\left(X_{n}\left(u\right)\right)\text{d}u\right)+\int_{0}^{t}F\left(X_{n}\left(u\right)\right)\text{d}u,$
(42)
where $\widehat{Y}_{l}\left(y\right)=Y_{l}\left(y\right)-y$ is a Poisson
process centered at its expectation. Note that in (42), the function
$F\left(y\right)$ given in (41) is for $y=s_{k}^{\left(n\right)},0\leq k\leq
n$.
Taking $X\left(t\right)=\lim_{n\rightarrow\infty}X_{n}\left(t\right)$ and
$x_{0}=\lim_{n\rightarrow\infty}X_{n}\left(0\right)$, we obtain
$X\left(t\right)=x_{0}+\int_{0}^{t}F\left(X\left(u\right)\right)\text{d}u,\text{
\ }t\geq 0,$ (43)
due to the fact that
$\lim_{n\rightarrow\infty}\frac{1}{n}\widehat{Y}_{l}\left(n\int_{0}^{t}\beta_{l}\left(X_{n}\left(u\right)\right)\text{d}u\right)=0$
by means of the law of large numbers. Note that in (43), the function
$F\left(y\right)$ given in (41) is for $y=S_{k},k\geq 1$. In the supermarket
model, the deterministic and continuous process $\left\\{X\left(t\right),t\geq
0\right\\}$ is described by the infinite-size system of integral-differential
equations (4) to (7), or simply in the below
$\frac{d}{dt}X\left(t\right)=F\left(X\left(t\right)\right)$ (44)
with the initial condition
$X\left(0\right)=x_{0}.$ (45)
Now, we consider the uniqueness of the limiting deterministic process
$\left\\{X\left(t\right),t\geq 0\right\\}$ with (44) to (45), or the
uniqueness of solution to the infinite-size system of integral-differential
equations (4) to (7). To that end, a sufficient condition is Lipschitz, that
is, for some constant $M>0,$
$|F\left(y\right)-F\left(z\right)|\leq M|y-z|.$
In general, the Lipschitz condition is standard and sufficient for the
uniqueness of solution to the finite-size system of differential equations;
while for the countable infinite-size case, readers may refer to Theorem 3.2
in Deimling [4] and Subsection 3.4.1 in Mitzenmacher [20] for some
generalization.
To check the Lipschitz condition, as $n\rightarrow\infty$ we have
$\beta_{a}\left(S_{0}\right)=-\lambda,$
$\beta_{b}\left(S_{0}\right)=\int_{0}^{+\infty}\mu\left(x\right)S_{1}\left(x\right)\text{d}x;$
$\beta_{a}\left(S_{1}\right)=\lambda-\lambda\int_{0}^{+\infty}\left[S_{1}\left(x\right)\right]^{d}\text{d}x,$
$\beta_{b}\left(S_{1}\right)=-\int_{0}^{+\infty}\mu\left(x\right)S_{1}\left(x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{2}\left(x\right)\text{d}x;$
and for $k\geq 2,$
$\beta_{a}\left(S_{k}\right)=\lambda\int_{0}^{+\infty}\left[S_{k-1}\left(x\right)\right]^{d}\text{d}x-\lambda\int_{0}^{+\infty}\left[S_{k}\left(x\right)\right]^{d}\text{d}x,$
$\beta_{b}\left(S_{k}\right)=-\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(x\right)\text{d}x+\int_{0}^{+\infty}\mu\left(x\right)S_{k+1}\left(x\right)\text{d}x.$
Let
$\zeta_{k}=\frac{\int_{0}^{+\infty}\left[S_{k}\left(x\right)\right]^{d}\text{d}x}{\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x}$
and
$\eta_{k}=\frac{\int_{0}^{+\infty}\mu\left(x\right)S_{k}\left(x\right)\text{d}x}{\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x}.$
Then $\zeta_{k},\eta_{k}>0$ for $k\geq 1$.
The following theorem shows that the supermarket model with general service
times satisfies the Lipschitz condition for the infinite-size system of
integral-differential equations (4) to (7).
###### Theorem 6
The supermarket model with general service times satisfies the Lipschitz
condition.
Proof Let
$\Omega=\left\\{S_{k}:k\geq 0\right\\}.$
For two arbitrary entries $y,z\in\Omega$, we have
$|F\left(y\right)-F\left(z\right)|\leq
a|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|+b|\beta_{b}\left(y\right)-\beta_{b}\left(z\right)|.$
Now, we analyze the following four cases for the function
$\beta_{a}\left(y\right)$, while the function $\beta_{b}\left(y\right)$ can be
analyzed similarly.
Case one: $y=S_{0},z=S_{1}$. In this case, we have
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle=|-\lambda-\lambda+\lambda\int_{0}^{+\infty}\left[S_{1}\left(x\right)\right]^{d}\text{d}x|$
$\displaystyle=\lambda|2-\zeta_{1}\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x|$
$\displaystyle=\lambda\left[2-\zeta_{1}\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x\right],$
since $0<\zeta_{1},\int_{0}^{+\infty}S_{1}\left(x\right)$d$x<1$. Taking
$M_{a}\left(0,1\right)\geq\frac{2-\zeta_{1}\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x}{2-\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x},$
it is clear that
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle\leq
M_{a}\left(0,1\right)\lambda\left[2-\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x\right]$
$\displaystyle=M_{a}\left(0,1\right)|y-z|.$
Case two: $y=S_{0},z=S_{k}$ for $k\geq 2$. In this case, we have
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle=|-\lambda-\lambda\int_{0}^{+\infty}\left[S_{k-1}\left(x\right)\right]^{d}\text{d}x+\lambda\int_{0}^{+\infty}\left[S_{k}\left(x\right)\right]^{d}\text{d}x|$
$\displaystyle=\lambda|-1-\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|$
$\displaystyle=\lambda\left[1+\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x-\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x\right]$
due to that $0<\zeta_{k},\int_{0}^{+\infty}S_{k}\left(x\right)$d$x<1$. Let
$M_{a}\left(0,k\right)\geq\frac{1+\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x-\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x}{1+\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x-\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x.}$
Then
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle\leq
M_{a}\left(0,k\right)\lambda\left[1+\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x-\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x\right]$
$\displaystyle=M_{a}\left(0,k\right)|y-z|.$
Case three: $y=S_{1},z=S_{k}$ for $k\geq 2$. In this case, we have
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle=|\lambda-\lambda\int_{0}^{+\infty}\left[S_{1}\left(x\right)\right]^{d}\text{d}x-\lambda\int_{0}^{+\infty}\left[S_{k-1}\left(x\right)\right]^{d}\text{d}x+\lambda\int_{0}^{+\infty}\left[S_{k}\left(x\right)\right]^{d}\text{d}x|$
$\displaystyle=\lambda|1-\zeta_{1}\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x-\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|$
Let
$M_{a}\left(1,k\right)\geq\frac{|1-\zeta_{1}\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x-\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|}{|1-\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x-\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|}.$
Then
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|$
$\displaystyle\leq
M_{a}\left(1,k\right)\lambda|1-\int_{0}^{+\infty}S_{1}\left(x\right)\text{d}x-\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|$
$\displaystyle=M_{a}\left(1,k\right)|y-z|.$
Case four: $y=S_{l},z=S_{k}$ for $k>l\geq 2$. In this case, we have
$\displaystyle|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|=$
$\displaystyle|\lambda\int_{0}^{+\infty}\left[S_{l-1}\left(x\right)\right]^{d}\text{d}x-\lambda\int_{0}^{+\infty}\left[S_{l}\left(x\right)\right]^{d}\text{d}x$
$\displaystyle-\lambda\int_{0}^{+\infty}\left[S_{k-1}\left(x\right)\right]^{d}\text{d}x+\lambda\int_{0}^{+\infty}\left[S_{k}\left(x\right)\right]^{d}\text{d}x|$
$\displaystyle=$
$\displaystyle\lambda|\zeta_{l-1}\int_{0}^{+\infty}S_{l-1}\left(x\right)\text{d}x+\zeta_{l}\int_{0}^{+\infty}S_{l}\left(x\right)\text{d}x$
$\displaystyle-\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|.$
Let
$M_{a}\left(l,k\right)\geq\frac{|\zeta_{l-1}\int_{0}^{+\infty}S_{l-1}\left(x\right)\text{d}x+\zeta_{l}\int_{0}^{+\infty}S_{l}\left(x\right)\text{d}x-\zeta_{k-1}\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\zeta_{k}\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|}{|\int_{0}^{+\infty}S_{l-1}\left(x\right)\text{d}x+\int_{0}^{+\infty}S_{l}\left(x\right)\text{d}x-\int_{0}^{+\infty}S_{k-1}\left(x\right)\text{d}x+\int_{0}^{+\infty}S_{k}\left(x\right)\text{d}x|}.$
Then
$|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq
M_{a}\left(l,k\right)|y-z|.$
Based on the above four cases, taking
$M_{a}=\max\left\\{M_{a}\left(l,k\right):k>l\geq 0\right\\}$
we obtain that for two arbitrary entries $y,z\in\Omega,$
$|\beta_{a}\left(y\right)-\beta_{a}\left(z\right)|\leq M_{a}|y-z|.$
Similarly, we can choose a positive number $M_{b}$ such that for two arbitrary
entries $y,z\in\Omega,$
$|\beta_{b}\left(y\right)-\beta_{b}\left(z\right)|\leq M_{b}|y-z|.$
Let $M=\max\left\\{aM_{a},bM_{b}\right\\}$. Then for two arbitrary entries
$y,z\in\Omega,$
$|F\left(y\right)-F\left(z\right)|\leq M|y-z|.$
This completes the proof.
Based on Theorem 6, the following theorem easily follows from Theorem 3.13 in
Mitzenmacher [20].
###### Theorem 7
In the supermarket model with general service times,
$\left\\{X_{n}\left(t\right)\right\\}$ and $\left\\{X\left(t\right)\right\\}$
are respectively given by (42) and (43), we have
$\lim_{n\rightarrow\infty}\sup_{u\leq
t}|X_{n}\left(u\right)-X\left(u\right)|=0,\text{ \ }a.s.$
Proof It is seen from that in the supermarket model with general service
times, the function $F\left(y\right)$ for $y\in\Omega$ satisfies the Lipschitz
condition. At the same time, it is easy to take a subset
$\Omega^{\ast}\subset\Omega$ such that
$\left\\{X\left(u\right):u\leq t\right\\}\subset\Omega^{\ast}$
and
$a\sup_{y\in\Omega^{\ast}}\beta_{a}\left(y\right)+a\sup_{y\in\Omega^{\ast}}\beta_{a}\left(y\right)<+\infty.$
Thus, this proof can easily be completed by means of Theorem 3.13 in
Mitzenmacher [20]. This completes the proof.
Using Theorem 3.11 in Mitzenmacher [20] and Theorem 7, we can obtain the
following theorem for the expected sojourn time that a customer spends in an
initially empty supermarket model with general service times over the time
interval $\left[0,T\right]$.
###### Theorem 8
In the supermarket model with general service times, the expected sojourn time
that a customer spends in an initially empty system over the time interval
$\left[0,T\right]$ is bounded above by
$\theta\rho^{d}\left\\{E\left[X_{R}\right]-E\left[X\right]\right\\}+E\left[X\right]\left[\sum_{k=1}^{\infty}\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k}-d}{d-1}}\right]+o\left(1\right),$
where $o\left(1\right)$ is understood as $n\rightarrow\infty$.
## 8 Concluding remarks
In this paper, we provide a novel and simple approach to study the randomized
load balancing model with general service times, which is described as an
infinite-size system of integral-differential equations. This approach is
based on the supplementary variable method, which is always applied in dealing
with stochastic models of M/G/1 type, e.g., see Li and Zhao [13, 14] and Li
[10]. We organize an infinite-size system of integral-differential equations
by means of the density dependent jump Markov process, and obtain a close-form
solution: doubly exponential structure, for the fixed point satisfying the
system of nonlinear equations, which is always a key in the study of
supermarket models. Since the fixed point is decomposited into two groups of
information under a product form, we indicate three important observations:
1. 1.
the fixed point for the supermarket model is different from the tail of
stationary queue length distribution for the ordinary M/G/1 queue;
2. 2.
the doubly exponential solution to the fixed point can exist extensively for
$0<\mu<+\infty$ even if the service time distribution is heavy-tailed; and
3. 3.
the doubly exponential solution to the fixed point is not unique for a more
general supermarket model.
Furthermore, we analyze the exponential convergence of the current location of
the supermarket model to its fixed point, and study the Lipschitz condition in
the Kurtz Theorem under general service times. Finally, we present numerical
examples to illustrate the effectiveness of our approach in analyzing the
randomized load balancing schemes with the non-exponential service
requirements. Based on this analysis, one can gain a new and important
understanding how workload probing can help in load balancing jobs with
general service times such as heavy-tailed service.
The approach of this paper is useful in analyzing the randomized load
balancing schemes in resource allocation in computer networks. We expect that
this approach will be applicable to the study other randomized load balancing
schemes with general service times, for example, generalizing the arrival
process to non-Poisson: the renewal arrival process or the Markovian arrival
process.
## Acknowledgements
The work of Q.L. Li was supported by the National Science Foundation of China
under grant No. 10871114 and the National Grand Fundamental Research 973
Program of China under grant No. 2006CB805901.
## References
* [1] R. Adler, R. Feldman and M.S. Taqqu (1998). A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions. Birkhäuser: Boston.
* [2] Y. Azar, A.Z. Broder, A.R. Karlin and E. Upfal (1999). Balanced allocations. SIAM Journal on Computing 29, 180–200. A preliminary version of this paper appeared in Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, 1994.
* [3] M. Dahlin (1999). Interpreting stale load information. In Proceedings of the Nineteenth Annual IEEE International Conference on Distributed Computing Systems.
* [4] K. Deimling (1977). Ordinary Differential Equations in Banach Spaces. Springer-Verlag, Lecture Notes in Math., Vol. 96.
* [5] D.L. Eager, E.D. Lazokwska and J. Zahorjan (1986). Adaptive load sharing in homogeneous distributed systems. IEEE Transactions on Software Engineering 12, 662–675.
* [6] D.L. Eager, E. D. Lazokwska and J. Zahorjan (1986). A comparison of receiver-initiated and sender-initiated adaptive load sharing. Performance Evaluation Review 6, 53–68.
* [7] D.L. Eager, E.D. Lazokwska and J. Zahorjan (1988). The limited performance benefits of migrating active processes for load sharing. Performance Evaluation Review 16, 63–72.
* [8] M. harchol-Balter and A.B. Downey. Exploiting process lifetime distributions for dynamic load balancing. ACM Transactions on Computer Systems 15, 253–285, 1997.
* [9] T.G. Kurtz (1981). Approximation of Population Processes. SIAM.
* [10] Q.L. Li (2010). Constructive Computation in Stochastic Models with Applications: The RG-Factorizations. Tsinghua Press and Springer.
* [11] Q.L. Li, John C.S. Lui and Y. Wang (2010). A matrix-analytic solution for randomized load balancing models with PH service times. Submited for publication.
* [12] Q.L. Li, John C.S. Lui and Y. Wang (2010). Doubly exponential asymptotics for randomized load balancing with Markovian arrival processes. Submited for publication.
* [13] Q.L. Li and Y.Q. Zhao (2004). A MAP/G/1 queue with negative customers. Queueing Systems 47, 5-43.
* [14] Q.L. Li, Y. Ying and Y.Q. Zhao (2006). A BMAP/G/1 retrial queue with a server subject to breakdowns and repairs. Annals of Operations Research 141, 233-270.
* [15] M. Luczak and C. McDiarmid (2005). On the power of two choices: Balls and bins in continuous time. The Annals of Applied Probability 15, 1733–1764.
* [16] M. Luczak and C. McDiarmid (2006). On the maximum queue length in the supermarket model. The Annals of Probability 34, 493–527.
* [17] J.B. Martin (2001). Point processes in fast Jackson networks. Annals of Applied Probability 11, 650-663.
* [18] J.B. Martin and Y.M Suhov (1999). Fast Jackson networks. Annals of Applied Probability 9, 854–870.
* [19] M.D. Mitzenmacher (1996). Load balancing and density dependent jump Markov processes. In Proceedings of the Thirty-Seventh Annual Symposium on Foundations of Computer Science, pages 213–222.
* [20] M.D. Mitzenmacher (1996). The power of two choices in randomized load balancing. PhD thesis, University of California at Berkeley, Department of Computer Science, Berkeley, CA, 1996.
* [21] M. Mitzenmacher (1998). Analyses of load stealing models using differential equations. In Proceedings of the Tenth ACM Symposium on Parallel Algorithms and Architectures, pages 212–221.
* [22] M. Mitzenmacher (1999). On the analysis of randomized load balancing schemes. Theory of Computing Systems 32, 361–386.
* [23] M. Mitzenmacher (1999). Studying balanced allocations with differential equations. Combinatorics, Probability, and Computing 8, 473–482.
* [24] M. Mitzenmacher (2000). How useful is old information? IEEE Transactions on Parallel and Distributed Systems 11, 6–20.
* [25] M. Mitzenmacher (2001). The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Computing 12, 1094-1104.
* [26] M. Mitzenmacher, A. Richa, and R. Sitaraman (2001). The power of two random choices: a survey of techniques and results. In Handbook of Randomized Computing: volume 1, edited by P. Pardalos, S. Rajasekaran and J. Rolim, pp. 255-312.
* [27] M. Mitzenmacher and B. Vöcking (1998). The asymptotics of selecting the shortest of two, improved. In Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, pages 326–327. A full version is available as Harvard Computer Science TR-08-99.
* [28] R. Mirchandaney, D. Towsley, and J.A. Stankovic (1989). Analysis of the e ects of delays on load sharing. IEEE Transactions on Computers 38, 1513–1525.
* [29] Y.M. Suhov and N.D. Vvedenskaya (2002). Fast Jackson Networks with Dynamic Routing. Problems of Information Transmission 38, 136{153.
* [30] B. Vöcking (1999). How asymmetry helps load balancing. In Proceedings of the Fortieth Annual Symposium on Foundations of Computer Science, pages 131–140.
* [31] N.D. Vvedenskaya, R.L. Dobrushin and F.I. Karpelevich (1996). Queueing system with selection of the shortest of two queues: An asymptotic approach. Problems of Information Transmissions 32, 20–34.
* [32] N.D. Vvedenskaya and Y.M. Suhov (1997). Dobrushin’s mean-field approximation for a queue with dynamic routing. Markov Processes and Related Fields 3, 493–526.
* [33] S. Zhou (1988). A trace-driven simulation study of dynamic load balancing. IEEE Transactions on Software Engineering, 1327–1341.
|
arxiv-papers
| 2010-08-23T10:10:38 |
2024-09-04T02:49:12.365070
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan-Lin Li",
"submitter": "Quan-Lin Li",
"url": "https://arxiv.org/abs/1008.3788"
}
|
1008.3808
|
A New Operator Theory Similar to Pseudo-Differential Operators
††footnotetext: † Corresponding author. Yan’an Second School, Yan’an 71600,
China
E-mail: guangqingbi@yahoo.cn
b) School of Electronic and Information Engineering, BUAA, Beijing 100191,
China
E-mail: yuekaifly@163.com
Guangqing Bi a)†, Yuekai Bi b)
###### Abstract
We summarize and extend the correlative definitions and principles of abstract
operators, discuss the relation between abstract operators and pseudo-
differential operators, add several new algorithms, furthermore, develop the
theory of partial differential equations with abstract operators, and then
systematically expound the basic methods. By combining abstract operators with
the Laplace transform, we can easily derive the explicit solution of initial
value problem of linear higher-order partial differential equations for
n-dimensional space, and establish the general theory of linear higher-order
partial differential equations.
Keywords: Abstract operators, Symbol, Partial differential equations, Laplace
transform, Pseudo-differential operators
MSC(2010) Subject Classification: 35G10, 44A10, 47A56, 35S05, 35G05
## 1 Introduction
In 1960s, the general theory of linear partial differential equations made
important progress by using the generalized function and its Fourier
transform, further, the theory of pseudo-differential operators. The
significance of the latter theory lies in introducing the concept of symbol,
but due to the lack of awareness of the analytic continuous fundamental
theorem (See [1]), the concept of abstract operators remained unclear. Without
the theories of abstract operators, the operator method of differential
equations is just a technique, and the solution of partial differential
equations is normally complex. In 1997, the author delivered the analytic
continuous fundamental theorem, based on which the author introduced the
concept of abstract operators, and derived the algorithms of five types of
abstract operators (See [1]) such as
$\exp(h\partial_{x}),\;\sin(h\partial_{x}),\;\cos(h\partial_{x}),\;\sinh(h\partial_{x})\;\mbox{and}\;\cosh(h\partial_{x}).$
Where $\cos(ih\partial_{x})=\cosh(h\partial_{x})$,
$\sin(ih\partial_{x})=i\sinh(h\partial_{x})$. Obviously, the abstract operator
uses the same notation as infinite order differential operator, as it is the
extension of the latter. However, within the frame of infinite order
differential operators, complex operators are difficult to define, such as
$\exp(tP(\partial_{x})),\;\cos(tP(\partial_{x})^{1/2}),\;\frac{\sin(tP(\partial_{x})^{1/2})}{P(\partial_{x})^{1/2}},\;\cosh(tP(\partial_{x})^{1/2}),\;\frac{\sinh(tP(\partial_{x})^{1/2})}{P(\partial_{x})^{1/2}}.$
But as abstract operators, we can easily establish their algorithms (See [1],
[2]), therefore the general solving procedure of initial value problem of
linear partial differential equations with constant coefficients is derived
[1], [2], [3], this approach can also be applied in solving the partial
differential equations with variable coefficients containing $t$.
According to [4], pseudo-differential operators are defined as:
If $a(x,\xi)\in{C}^{\infty}(\mathbb{R}_{x}^{n}\times\mathbb{R}_{\xi}^{n})$,
and for arbitrary $\alpha,\beta\in\mathbb{N}^{n}$ and an real number $m$,
$|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}{a}(x,\xi)|\leq{C}_{\alpha,\beta}(1+|\xi|)^{m-|\alpha|}$
is tenable, where $C_{\alpha,\beta}$ is a constant, then the linear continuous
mapping $A$ of
$\mathscr{S}(\mathbb{R}^{n})\rightarrow\mathscr{S}(\mathbb{R}^{n})$ can be
defined as:
$Au(x)=(2\pi)^{-n}\int{e}^{i\xi{x}}a(x,\xi)\hat{u}(\xi)d\xi,$ (1)
which are called the pseudo-differential operators, denoted by $a(x,D)$, where
$a(x,\xi)$ are the symbols of $a(x,D)$, $D$ denotes
$D^{\alpha}=\left(\frac{1}{i}\partial_{x_{1}}\right)^{\alpha_{1}}\left(\frac{1}{i}\partial_{x_{2}}\right)^{\alpha_{2}}\cdots\left(\frac{1}{i}\partial_{x_{n}}\right)^{\alpha_{n}}.$
Obviously, this definition means the following equation is tenable, namely
$a(x,D)e^{i\xi{x}}=a(x,\xi)e^{i\xi{x}}.$ (2)
Therefore pseudo-differential operators are similar to abstract operators. As
the definition formula of pseudo-differential operators is a Fourier integral
expression, their symbols are largely restricted to ensure the convergence of
Fourier integrals. But for many specific problems, the symbols of pseudo-
differential operators should not be restricted that much. Thus, a better
definition of symbols can be expressed as:
$\sigma_{A}(x,\xi)=e^{-i\xi{x}}Ae^{i\xi{x}}.$ (3)
Where $A$ are properly extended pseudo-differential operators. Certainly, this
extension still has its limitations as it involves another type of Fourier
integrals.
Then can we get rid of Fourier integrals when defining pseudo-differential
operators? Let us study the following expression:
$Ae^{i\xi{x}}=\sigma_{A}(x,\xi)e^{i\xi{x}}.$ (4)
Obviously, the properties of $A$ depend on $\sigma_{A}(x,\xi)$. (4) indicates
the mapping relation between $A$ and $\sigma_{A}(x,\xi)$. For a given function
$\sigma_{A}(x,\xi)$, if the corresponding algorithms of $A$ can be derived
from this type of mapping relations, to determine the domain and range of $A$,
then this type of mapping relations can be the best definition of pseudo-
differential operators. Luckily, this idea totally works by using the analytic
continuous fundamental theorem or the abstract operator fundamental theorem.
The pseudo-differential operators established by using this method are the
abstract operators.
Now let us carry out a systematic review of the basic methods of abstract
operators, and prove that if we combine abstract operators with the Laplace
transform on $C^{\infty}(\mathbb{R}_{x}^{n}\times\mathbb{R}_{t}^{1})$, then no
matter how complex the initial value problem of linear partial differential
equations is, the solving process is simply easy. In addition, in order to
achieve self consistency theoretically, it is necessary to derive rules of
differentiation from the view of abstract operators.
## 2 Basic theory of abstract operators
### 2.1 Basic concepts, research method and basic operations
Definition 1. (See [1]) Within each convergence circle of analytic functions,
if the effects on the series term by term from the linear operator converge
uniformly to the effects on the sum function, then the operator is called
having the analytic continuity.
Definition 2. The linear continuous mapping $A$ of
$\mathscr{S}(\mathbb{R}^{n})\rightarrow\mathscr{S}(\mathbb{R}^{n})$ has the
Fourier continuity, if $A$ acts on an arbitrary function
$f\in\mathscr{S}(\mathbb{R}^{n})$, we have
$Af(x)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)\,Ae^{i\xi{x}}\,d\xi.$
Where the Fourier transform $\hat{f}=\mathscr{F}f$ is defined as
$\hat{f}(\xi)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}f(x)e^{-i\xi{x}}\,dx,\quad\forall{f}\in\mathscr{S}(\mathbb{R}^{n}).$
The Fourier transform $\mathscr{F}$ is an isomorphic mapping in the space
$\mathscr{S}(\mathbb{R}^{n})$, and its inverse transform is
$\mathscr{F}^{-1}\hat{f}(\xi)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)e^{i\xi{x}}\,d\xi=f(x).$
Definition 3. (See [1]) $x^{\alpha}$, $x\in{\mathbb{R}^{n}}$ is called the
base function, its exponential form is $e^{\xi{x}}$, where
$\alpha\in\mathbb{N}^{n},\;\xi\in{\mathbb{R}_{n}}$ or $\mathbb{C}^{n}$ are
respectively called the character of $x^{\alpha}$ and $e^{\xi{x}}$.
Definition 4. (See [1]) A class of linear operators is called the abstract
operators, denoted by
$f(t,\partial_{x}),\,x\in\mathbb{R}^{n},\,t\in\mathbb{R}^{1}$, and if it acts
on the exponential base function $e^{\xi{x}}$, we have
$f(t,\partial_{x})e^{\xi{x}}=f(t,\xi)e^{\xi{x}},\qquad\forall{f(t,\xi)}\in{C^{\infty}}(\mathbb{R}_{t}^{1}\times\mathbb{R}_{\xi}^{n}).$
Which is called the abstract operators taking $\partial_{x}$ as the operator
element, where $f(t,\xi)$ is called the symbol of the abstract operators
$f(t,\partial_{x})$.
If abstract operators $f(t,\partial_{x})$ is a linear operator having the
Fourier continuity, then it similar to a pseudo-differential operators.
Definition 5. A class of linear operators is called the abstract operators,
denoted by $f\left(t,x\frac{\partial}{\partial{x}}\right)$,
$x\in\mathbb{R}^{n},\,t\in\mathbb{R}^{1}$, where
$x\frac{\partial}{\partial{x}}=\left(x_{1}\frac{\partial}{\partial{x}_{1}},\;x_{2}\frac{\partial}{\partial{x}_{2}},\;\ldots,\;x_{n}\frac{\partial}{\partial{x}_{n}}\right),$
and if it acts on the base function $x^{\alpha}$, we have
$f\left(t,\,x\frac{\partial}{\partial{x}}\right)x^{\alpha}=f(t,\alpha)x^{\alpha},\qquad\forall{f(t,\alpha)}\in{C^{\infty}}(\mathbb{R}_{t}^{1}\times\mathbb{N}^{n}).$
Which is called the abstract operators taking $x\frac{\partial}{\partial{x}}$
as the operator element, where $f(t,\alpha)$ is called the symbol of the
abstract operators $f\left(t,x\frac{\partial}{\partial{x}}\right)$.
Definition 6. We call any operator element and common variables the abstract
element, denoted by $X,Y,\ldots$ and $X=(X_{1},X_{2},\ldots,X_{n})$,
$Y=(Y_{1},Y_{2},\ldots,Y_{n})$, etc.; Accordingly, $f(X)$, $f(Y)$, etc.,
represent the abstract operators or common functions, which is called the
operator function.
Abstract operators fundamental theorem. Let $A,B,\;A^{\prime},B^{\prime}$ be
the abstract operators, if there are functions
$v(t)\in\mathscr{D}(A^{\prime}),\;u(t)\in\mathscr{D}(B^{\prime}),\;t\in\mathbb{R}^{1}$
and abstract elements $X,Y$, making one of the following two expressions
tenable
$AX^{\alpha}{A}^{\prime}v(t)=BY^{\alpha}{B}^{\prime}u(t)\quad\mbox{or}\quad
Ae^{\xi{X}}{A}^{\prime}v(t)=Be^{\xi{Y}}{B}^{\prime}u(t),$
and the expressions of $A,B,\;A^{\prime},B^{\prime}$ do not explicitly contain
the character $\alpha\in\mathbb{N}^{n}$,
$\xi\in\mathbb{R}_{n}\,\mbox{or}\,\mathbb{C}_{n}$, then
$Af(X)A^{\prime}v(t)=Bf(Y)B^{\prime}u(t),\qquad\forall{f(X),f(Y)}\in\mathscr{D}(A)\cap\mathscr{D}(B).$
Where $\mathscr{D}(\cdot),\,\mathscr{R}(\cdot)$ are domains and ranges of
operators.
$\mathscr{D}(f(X))=\mathscr{R}(A^{\prime}),\;\mathscr{D}(f(Y))=\mathscr{R}(B^{\prime})$,
$\mathscr{D}(A)=\mathscr{R}(f(X)),\;\mathscr{D}(B)=\mathscr{R}(f(Y))$.
When $A^{\prime}v(t)=B^{\prime}u(t)=I$, and
$X=x\in\mathbb{R}^{n},\;Y=y\in\mathbb{R}^{n}$, this theorem becomes the
following Analytic continuous fundamental theorem:
If there are functions $x(t),y(t)\in\mathbb{R}^{n},\;t\in\mathbb{R}^{1}$,
making one of the following two expressions tenable
$Ax^{\alpha}=By^{\alpha}\quad\mbox{or}\quad Ae^{\xi{x}}=Be^{\xi{y}},$
and the expressions of abstract operators $A,B$ do not explicitly contain the
character $\alpha\in\mathbb{N}^{n}$,
$\xi\in\mathbb{R}_{n}\,\mbox{or}\,\mathbb{C}_{n}$, then
$Af(x)=Bf(y),\qquad\forall f(x),f(y)\in\mathscr{D}(A)\cap\mathscr{D}(B).$
Especially, when $A,B$ are the linear operators having the analytic
continuity, the analytic continuous fundamental theorem has been proved by G.Q
Bi in 1997 (See [1]).
The abstract operators fundamental theorem is the fundamental property of
abstract operators, which directly extends the domains of abstract operators
from power functions and exponential functions to a wider range of functions.
Definition 7. A linear operator equation, if it’s true for arbitrary functions
or operator functions in a certain range, then it is called an operator
formula. Particularly, the operator formulas that determine the domain and
range of abstract operators are called the algorithm of the abstract
operators.
Definition 8. The relational expression between each component of the
character $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ or
$\xi=(\xi_{1},\xi_{2},\ldots,\xi_{n})$ is called the characteristic equation.
In general, an operator formula becomes the corresponding characteristic
equation, when where arbitrary function of which is the base function.
Corollary 1. As the base function can be expressed by both power function and
exponential function, a single operator formula can be corresponding to two
different characteristic equations, conversely, a single characteristic
equation can also be corresponding to two different operator formulas.
Constructing the operator equation tenable for the base function by the
characteristic equation through the definition of abstract operators, and then
deducing that it is also tenable for arbitrary functions or operator functions
in a certain range by using the analytic continuous fundamental theorem or the
abstract operators fundamental theorem, thus we derive new operator formulas.
Finding or establishing a new operator formula requires the knowledge of the
corresponding characteristic equation in advance, without knowing the specific
form of the new operator formula. Therefore, it all boils down to seek or
construct appropriate characteristic equations. The key to transform the
characteristic equation to the corresponding operator formula, is constructing
the operator equation true for the base function by using the specific form of
the characteristic equation and the definition of abstract operators, and
also, only when the operator constructed is linear and the expression of the
linear operator doesn’t explicitly contain the character of the base function
can we derive that the operator equation is not only true for the base
function, but also for arbitrary functions in a certain range, according to
the analytic continuous fundamental theorem or the abstract operators
fundamental theorem. This is the research method of abstract operators.
Example 1. Let $x\in\mathbb{R}^{n},\;\xi\in\Omega\subseteq\mathbb{R}_{n}$,
then $\forall{f}(\xi)\in{C}^{\infty}(\Omega)$ we have
$\displaystyle f(\partial_{x})\sin{bx}$ $\displaystyle=$
$\displaystyle\left.\sin\left(bx+\frac{\pi}{2}\,\xi_{1}\frac{\partial}{\partial\xi_{1}}+\frac{\pi}{2}\,\xi_{2}\frac{\partial}{\partial\xi_{2}}+\cdots+\frac{\pi}{2}\,\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)f(\xi)\right|_{\xi=b},$
$\displaystyle f(\partial_{x})\cos{bx}$ $\displaystyle=$
$\displaystyle\left.\cos\left(bx+\frac{\pi}{2}\,\xi_{1}\frac{\partial}{\partial\xi_{1}}+\frac{\pi}{2}\,\xi_{2}\frac{\partial}{\partial\xi_{2}}+\cdots+\frac{\pi}{2}\,\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)f(\xi)\right|_{\xi=b}.$
Where $bx=b_{1}x_{1}+b_{2}x_{2}+\cdots+b_{n}x_{n}$.
Proof. Selecting the following two expressions from the known formulas as the
characteristic equations, namely
$\partial_{x}^{\alpha}\sin{bx}=b^{\alpha}\sin\left(bx+\frac{\pi}{2}|\alpha|\right)\quad\mbox{and}\quad\partial_{x}^{\alpha}\cos{bx}=b^{\alpha}\cos\left(bx+\frac{\pi}{2}|\alpha|\right).$
Taking $\alpha\in\mathbb{N}^{n}$ as the character of the base function,
$\partial_{x}$ and $\xi\frac{\partial}{\partial\xi}$ as the operator elements,
we have the characteristic equation with the following two operator equations
by Definition 5:
$\partial_{x}^{\alpha}\sin{bx}=\left.\sin\left(bx+\frac{\pi}{2}\,\xi_{1}\frac{\partial}{\partial\xi_{1}}+\frac{\pi}{2}\,\xi_{2}\frac{\partial}{\partial\xi_{2}}+\cdots+\frac{\pi}{2}\,\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)\xi^{\alpha}\right|_{\xi=b},$
$\partial_{x}^{\alpha}\cos{bx}=\left.\cos\left(bx+\frac{\pi}{2}\,\xi_{1}\frac{\partial}{\partial\xi_{1}}+\frac{\pi}{2}\,\xi_{2}\frac{\partial}{\partial\xi_{2}}+\cdots+\frac{\pi}{2}\,\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)\xi^{\alpha}\right|_{\xi=b}.$
According to the abstract operators fundamental theorem, we get the Example 1.
Definition 9. Let $A$ be a linear operator having the analytic continuity or
Fourier continuity, if there is another linear operator, denoted by $A^{-1}$,
making $AA^{-1}=A^{-1}A=I$, then $A^{-1}$ is called the inverse operator of
$A$.
By Definition 4 we have the following Corollary:
Corollary 2. (See [1]) The operator algebras formed by all the abstract
operators $f(t,\partial_{x})$, are isomorphic to the algebras formed by their
symbols $f(t,\xi)$. This isomorphism is determined by
$f(t,\partial_{x})\leftrightarrow{f}(t,\xi)$, and
$f(t,\partial_{x})\pm{g}(t,\partial_{x})\leftrightarrow{f}(t,\xi)\pm{g}(t,\xi),\quad{f}(t,\partial_{x})\circ{g}(t,\partial_{x})\leftrightarrow{f}(t,\xi)g(t,\xi).$
Especially, the abstract operators $f(\partial_{x})$ and
$g(\partial_{x}),\;x\in\mathbb{R}^{n}$ are each other’s inverse operator, if
and only if their symbols $f(\xi)$ and $g(\xi)$ satisfy
$f(\xi)g(\xi)=1,\;\xi\in\mathbb{R}_{n}$.
For instance,
$\partial_{x_{i}}\sin(x\partial_{y})=\partial_{y_{i}}\cos(x\partial_{y})$,
$\partial_{x_{i}}\cos(x\partial_{y})=-\partial_{y_{i}}\sin(x\partial_{y})$,
where $x,y\in\mathbb{R}^{n}$.
Corollary 3. (See [1]) Let $g(x)\in{C^{\infty}}(\mathbb{R}^{n})$, if
$g(\partial_{\xi})(e^{\xi{x}}f(\xi))$ is continuous at $\xi=a$, then
$f(\partial_{x})(e^{ax}g(x))=g(\partial_{\xi})(e^{\xi{x}}f(\xi))|_{\xi=a}.$
(6)
Obviously, by Definition 5 we can also get similar Corollary.
Example 2. If function $f(x),x\in\mathbb{R}^{n}$ satisfies
$\int_{\mathbb{R}^{n}}f(\eta)\exp\left(-\frac{|\eta-2\alpha{x}|^{2}}{4\alpha}\right)d\eta<+\infty,\quad\eta\in\mathbb{R}^{n},\;\alpha>0,$
then
$f(\partial_{x})e^{\alpha|x|^{2}}=e^{\alpha|x|^{2}}\frac{1}{2^{n}(\alpha\pi)^{n/2}}\int_{\mathbb{R}^{n}}f(\eta)\exp\left(-\frac{|\eta-2\alpha{x}|^{2}}{4\alpha}\right)d\eta.$
(7)
Proof. In the Corollary 3, let
$g(x)=e^{\alpha|x|^{2}}\in{C^{\infty}}(\mathbb{R}^{n})$ and $a=0$, we have
$\displaystyle f(\partial_{x})e^{\alpha|x|^{2}}$ $\displaystyle=$
$\displaystyle\left.\exp(\alpha|\partial_{\xi}|^{2})(e^{\xi{x}}f(\xi))\right|_{\xi=0}=\left.\exp(\alpha|x+\partial_{\xi}|^{2})f(\xi)\right|_{\xi=0}$
$\displaystyle=$ $\displaystyle
e^{\alpha|x|^{2}}\left.\exp(2\alpha{x}\partial_{\xi})\exp(\alpha|\partial_{\xi}|^{2})f(\xi)\right|_{\xi=0}$
$\displaystyle=$ $\displaystyle
e^{\alpha|x|^{2}}\frac{1}{2^{n}(\alpha\pi)^{n/2}}\left.\exp(2\alpha{x}\partial_{\xi})\int_{\mathbb{R}^{n}}f(\eta)\exp\left(-\frac{|\eta-\xi|^{2}}{4\alpha}\right)d\eta\right|_{\xi=0}$
$\displaystyle=$ $\displaystyle
e^{\alpha|x|^{2}}\frac{1}{2^{n}(\alpha\pi)^{n/2}}\left.\int_{\mathbb{R}^{n}}f(\eta)\exp\left(-\frac{|\eta-\xi-2\alpha{x}|^{2}}{4\alpha}\right)d\eta\right|_{\xi=0},\;\alpha>0.$
Where
$x\partial_{\xi}=x_{1}\partial_{\xi_{1}}+x_{2}\partial_{\xi_{2}}+\cdots+x_{n}\partial_{\xi_{n}}$,
$|x|^{2}=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{n}$.
Clearly, by (7) the $f(\eta)$ can also be a bounded measurable function.
Therefore, Corollary 3 is an important way to extend the symbols and domains
of abstract operators. In fact, limitations on the symbol $f(t,\xi)$ and the
operator element $X$ will be broadened, and more general abstract operators
will be introduced to deal with problems involved in our further discussion.
There are two major research methods of the operator theory. One determines
the domain and range of an operator first, then derives its operator algebra.
The other abstractly determines an operator algebra, then derives the domain
and range of the operator. The former is applied by pseudo-differential
operators, while the latter is applied by abstract operators. The definition
of abstract operators actually determines a kind of excellent operator
algebras, then establishes different algorithms for different operators using
the abstract operator fundamental theorem, further, derives different domains
and ranges for abstract operators with different symbols. In other words, the
domain and range of an abstract operator depend on the algorithm it applies.
Therefore, a significant feature of abstract operators is focusing on the
research of algorithms, without depending on the Fourier integral transform.
The basic theory of abstract operators is a theory about algorithms, which has
already become an important tool to seek and calculate explicit solutions of
linear higher-order partial differential equations.
### 2.2 Algorithms of Abstract operators on differentiable function
In order to achieve theoretical rigor, firstly let us derive the rules of
differentiation from the view of abstract operators, making the concept of
ordinary or partial differential operator on the basis of the abstract
operators itself.
Differential operators can be defined as the following abstract operators,
namely
$x\frac{d}{dx}\,x^{\alpha}=\alpha{x}^{\alpha},\quad\frac{d}{dx}\,e^{\xi{x}}=\xi{e}^{\xi{x}},\qquad\forall{x}\in\mathbb{R}^{1}.$
Where $\alpha\in\mathbb{N}^{1},\;\xi\in\mathbb{R}_{1}$ are the symbols of
abstract operators $x\frac{d}{dx}$ and $\frac{d}{dx}$ respectively.
Let $a$ and $b$ be the characters of base functions, as the characteristic
equation, the binomial formula of integer power can be expressed as
$(a+b)^{n}=\sum^{n}_{j=0}{n\choose{j}}a^{j}b^{n-j},\quad\forall{n}\in\mathbb{N}.$
By base functions $e^{ax}$ and $e^{bx}$ we can combine the binomial formula
with the following operator equations:
$\frac{d^{n}}{dx^{n}}\left(e^{ax}\cdot{e}^{bx}\right)=\sum^{n}_{j=0}{n\choose{j}}\frac{d^{j}}{dx^{j}}e^{ax}\cdot\frac{d^{n-j}}{dx^{n-j}}e^{bx}.$
According to the analytic continuous fundamental theorem, it is tenable,
namely
$\frac{d^{n}}{dx^{n}}(vu)=\sum^{n}_{j=0}{n\choose{j}}\frac{d^{j}v}{dx^{j}}\frac{d^{n-j}u}{dx^{n-j}},\quad\forall{v,u}\in{C^{n}}(\mathbb{R}^{1}).$
(8)
This is the Leibniz rule. Similarly we have
$v\frac{d^{n}u}{dx^{n}}=\sum^{n}_{j=0}(-1)^{j}{n\choose{j}}\frac{d^{n-j}}{dx^{n-j}}\left(u\frac{d^{j}v}{dx^{j}}\right),\quad\forall{v,u}\in{C^{n}}(\mathbb{R}^{1}).$
(9)
$\frac{d^{k}}{dx^{k}}f(x^{2})=\sum^{[k/2]}_{j=0}\frac{k!}{j!\,(k-2j)!}(2x)^{k-2j}\frac{d^{k-j}}{dy^{k-j}}f(y),\quad\forall{f(x^{2})}\in{C^{k}}(\mathbb{R}^{1}),\;y=x^{2}.$
(10)
By (10) we easily obtain the Hermite polynomials
$H_{k}(x)=(-1)^{k}e^{x^{2}}\frac{d^{k}}{dx^{k}}e^{-x^{2}}=\sum^{[k/2]}_{j=0}(-1)^{j}\frac{k!}{j!\,(k-2j)!}(2x)^{k-2j}.$
The major parts of rules of differentiation are the derivative principle of
function product, the derivative principle of compound function and the chain
rule of multivariate function. Firstly let us establish the derivative
principle of function product. Without losing the universality, we only
consider the situation of the product of two functions. Let $n=1$ in (8), then
for $v=f(x)$ and $u=g(x)$ we have
$\frac{d}{dx}(f(x)g(x))=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x),\quad\forall{f,g}\in{C}^{1}(\mathbb{R}^{1}).$
(11)
Let $\varphi(x)=f_{1}(x)f_{2}(x)\cdots{f}_{n}(x)$, generally we have
$\frac{d}{dx}\varphi(x)=\sum^{n}_{j=1}\prod^{n}_{i=1\atop
i\neq{j}}f_{i}(x)\frac{d}{dx}f_{j}(x),\quad\forall{f_{j}(x)}\in{C}^{1}(\mathbb{R}^{1}).$
Let $f_{1}(x)=f_{2}(x)=\cdots=f_{n}(x)=g(x)=y$, then for differentiable
function $y=g(x)$ we have
$\frac{d}{dx}y^{n}=ny^{n-1}\frac{dy}{dx}\quad\mbox{or}\quad\frac{d}{dx}y^{n}=\frac{dy}{dx}\frac{d}{dy}y^{n},\qquad\forall{n}\in\mathbb{N}^{1}.$
According to the analytic continuous fundamental theorem, we have the
derivative principle of compound function $f(g(x))\in{C^{1}}(\mathbb{R}^{1})$:
$\frac{d}{dx}f(g(x))=\frac{dy}{dx}\frac{d}{dy}f(y),\qquad{y}=g(x),\quad{x}\in\mathbb{R}^{1}.$
(12)
By using (11) and (12), for differentiable functions $x_{1}(t)$ and
$x_{2}(t)$, $t\in\mathbb{R}^{1}$ we have
$\frac{d}{dt}(x^{\alpha_{1}}_{1}x^{\alpha_{2}}_{2})=\frac{dx^{\alpha_{1}}_{1}}{dx_{1}}\frac{dx_{1}}{dt}x^{\alpha_{2}}_{2}+x^{\alpha_{1}}_{1}\frac{dx^{\alpha_{2}}_{2}}{dx_{2}}\frac{dx_{2}}{dt},\qquad\alpha_{1},\alpha_{2}\in\mathbb{N}.$
Taking this one as the characteristic equation, by base functions
$x^{\alpha}$, $x\in\mathbb{R}^{2},\;\alpha\in\mathbb{N}^{2}$, combining it
with the following operator equation:
$\frac{d}{dt}x^{\alpha}=\frac{dx_{1}}{dt}\frac{\partial}{\partial{x}_{1}}x^{\alpha}+\frac{dx_{2}}{dt}\frac{\partial}{\partial{x}_{2}}x^{\alpha},\qquad{x}(t)\in\mathbb{R}^{2}\quad{t}\in\mathbb{R}^{1}\quad\alpha\in\mathbb{N}^{2}.$
According to the analytic continuous fundamental theorem, it is tenable
$\forall{f(x)}\in{C^{1}}(\mathbb{R}^{2})$, namely
$\frac{d}{dt}f(x)=\frac{dx_{1}}{dt}\frac{\partial}{\partial{x}_{1}}f(x)+\frac{dx_{2}}{dt}\frac{\partial}{\partial{x}_{2}}f(x),\qquad{x}(t)\in\mathbb{R}^{2}\quad{t}\in\mathbb{R}^{1}.$
Clearly, for $x(t)\in\mathbb{R}^{n}$, $t\in\mathbb{R}^{1}$, we can generally
derive $\forall{f(x)}\in{C^{1}}(\mathbb{R}^{n})$
$\frac{d}{dt}f(x)=\frac{dx_{1}}{dt}\frac{\partial}{\partial{x}_{1}}f(x)+\frac{dx_{2}}{dt}\frac{\partial}{\partial{x}_{2}}f(x)+\cdots+\frac{dx_{n}}{dt}\frac{\partial}{\partial{x}_{n}}f(x).$
(13)
If taking $n\in\mathbb{N}$ as the character of base function, the binomial
formula can be expressed as the following characteristic equation:
$(x+h)^{n}=\sum^{\infty}_{j=0}\frac{h^{j}}{j!}\frac{d^{j}}{dx^{j}}x^{n}.$
According to the analytic continuous fundamental theorem, it is tenable for
any analytic function $f(x)\in{C^{\infty}}(\mathbb{C}^{1})$, namely
$f(x+h)=\sum^{\infty}_{j=0}\frac{h^{j}}{j!}\frac{d^{j}}{dx^{j}}f(x),\quad{x}\in\mathbb{C}^{1}.$
(14)
This is the Taylor formula. If taking (14) as a characteristic equation, then
similarly we have
Theorem 1. If $\exists{v(s),u(s)}\in{C^{\infty}}(\mathbb{C}^{1})$ can make the
infinite series on the right side of (15) uniform convergent, then it will
uniform converges to the left side of the formula, namely
$f\\!\left(\lambda\frac{\partial}{\partial{s}}\right)(vu)=\sum_{k=0}^{\infty}\frac{\lambda^{k}}{k!}\,\frac{\partial^{k}v}{\partial{s^{k}}}\,f^{(k)}\\!\left(\lambda\frac{\partial}{\partial{s}}\right)u,\quad{s}\in\mathbb{C}^{1},\;|\lambda|\leq
1.$ (15)
Corollary 4. In (15), when $\lambda=-1,\;v=s,\;u=1/s,\,s\neq 0$, we have
$f^{\prime}\left(-\frac{\partial}{\partial{s}}\right)\frac{1}{s}=sf\left(-\frac{\partial}{\partial{s}}\right)\frac{1}{s}-f(0)\quad\mbox{or}\quad\mathscr{L}f^{\prime}(t)=s\mathscr{L}f(t)-f(0).$
(16)
Where $\mathscr{L}f(t)$ denotes
$\mathscr{L}f(t)=f\left(-\frac{\partial}{\partial{s}}\right)\frac{1}{s},\quad\;\Re(s)>0,\;t\in\overline{\mathbb{R}_{+}^{1}}.$
(17)
This is the internal connection between abstract operators and Laplace
transform, thus the abstract operator becomes a significant tool to compute
Laplace transform. For instance, by (17) we have
$\mathscr{L}[g(t)f(t)]=g\left(-\frac{\partial}{\partial{s}}\right)F(s),\quad(F(s)=\mathscr{L}f(t)).$
(18)
Similar to (11)-(13), Guangqing Bi has already established a series of
algorithms of abstract operators in reference [1]:
$\cos(h\partial_{x})f(x)=\Re[f(x+ih)],\quad\sin(h\partial_{x})f(x)=\Im[f(x+ih)],\quad\forall{f(z)}\in{C}^{\infty}(\mathbb{C}^{n}).$
(19)
Where
$bx=b_{1}x_{1}+b_{2}x_{2}+\cdots+b_{n}x_{n},\;bh=b_{1}h_{1}+b_{2}h_{2}+\cdots+b_{n}h_{n}$.
$\displaystyle\sin(h\partial_{x})(uv)$ $\displaystyle=$
$\displaystyle\cos(h\partial_{x})v\cdot\sin(h\partial_{x})u+\sin(h\partial_{x})v\cdot\cos(h\partial_{x})u,$
$\displaystyle\cos(h\partial_{x})(uv)$ $\displaystyle=$
$\displaystyle\cos(h\partial_{x})v\cdot\cos(h\partial_{x})u-\sin(h\partial_{x})v\cdot\sin(h\partial_{x})u.$
$\displaystyle\sin(h\partial_{x})\frac{u}{v}$ $\displaystyle=$
$\displaystyle\frac{\cos(h\partial_{x})v\cdot\sin(h\partial_{x})u-\sin(h\partial_{x})v\cdot\cos(h\partial_{x})u}{(\cos(h\partial_{x})v)^{2}+(\sin(h\partial_{x})v)^{2}},$
$\displaystyle\cos(h\partial_{x})\frac{u}{v}$ $\displaystyle=$
$\displaystyle\frac{\cos(h\partial_{x})v\cdot\cos(h\partial_{x})u+\sin(h\partial_{x})v\cdot\sin(h\partial_{x})u}{(\cos(h\partial_{x})v)^{2}+(\sin(h\partial_{x})v)^{2}}.$
Where $x\in\mathbb{R}^{n},\;h\in\mathbb{R}_{n}\,\mbox{or}\,\mathbb{C}_{n}$.
$\displaystyle\sin(h\partial_{t})f(x(t))$ $\displaystyle=$
$\displaystyle\sin(Y\partial_{X})f(X),$
$\displaystyle\cos(h\partial_{t})f(x(t))$ $\displaystyle=$
$\displaystyle\cos(Y\partial_{X})f(X).$
$\displaystyle
x(t)\in\mathbb{R}^{n},\;t\in\mathbb{R}^{1},\;h\in\mathbb{R}_{1}\,\mbox{or}\,\mathbb{C}_{1}.$
Where
$X=(X_{1},X_{2},\cdots,X_{n}),\;X_{j}=\cos(h\partial_{t})x_{j}(t),\;Y\in\mathbb{R}_{n},\;Y_{j}=\sin(h\partial_{t})x_{j}(t)$.
For example, when $n=1$, the (2.2) can easily be restated as
$\displaystyle\sin\left(h\frac{d}{dt}\right)f(x(t))$ $\displaystyle=$
$\displaystyle\sin\left(Y\frac{\partial}{\partial{X}}\right)f(X),$
$\displaystyle\cos\left(h\frac{d}{dt}\right)f(x(t))$ $\displaystyle=$
$\displaystyle\cos\left(Y\frac{\partial}{\partial{X}}\right)f(X).$
Where
$Y=\sin(h\partial_{t})x(t),\;X=\cos(h\partial_{t})x(t),\;t\in\mathbb{R}^{1},\;h\in\mathbb{R}_{1}\,\mbox{or}\,\mathbb{C}_{1}$.
If $n=2$, then (2.2) can easily be restated as
$\displaystyle\sin\left(h\frac{d}{dt}\right)f(x(t),y(t))$ $\displaystyle=$
$\displaystyle\sin\left(Y_{x}\frac{\partial}{\partial{X}_{x}}+Y_{y}\frac{\partial}{\partial{X}_{y}}\right)f(X_{x},X_{y}),$
$\displaystyle\cos\left(h\frac{d}{dt}\right)f(x(t),y(t))$ $\displaystyle=$
$\displaystyle\cos\left(Y_{x}\frac{\partial}{\partial{X}_{x}}+Y_{y}\frac{\partial}{\partial{X}_{y}}\right)f(X_{x},X_{y}).$
Where
$Y_{x}=\sin(h\partial_{t})x(t),\;X_{x}=\cos(h\partial_{t})x(t),\;Y_{y}=\sin(h\partial_{t})y(t),\;X_{y}=\cos(h\partial_{t})y(t)$.
Theorem 7 in reference [1] can be more generally expressed as:
Theorem 2. If $y=f(bx)$ in set of analytic functions is the inverse function
of $bx=g(y)$, namely $g(f(bx))=bx$, then $\sin(h\partial_{x})f(bx)$(denoted by
$Y$) and $\cos(h\partial_{x})f(bx)$(denoted by $X$) can be determined by the
following set of equations:
$\displaystyle\cos\left(Y\frac{\partial}{\partial{X}}\right)g(X)$
$\displaystyle=$ $\displaystyle bx,$
$\displaystyle\sin\left(Y\frac{\partial}{\partial{X}}\right)g(X)$
$\displaystyle=$ $\displaystyle bh.$
$\displaystyle
x\in\mathbb{R}^{n},\quad{b}\in\mathbb{R}_{n},\quad{h}\in\mathbb{R}_{n}.$
By using (2.2)-(2.2), we can constructively extend the domains and ranges of
the following five abstract operators
$\exp(h\partial_{x}),\;\sin(h\partial_{x}),\;\cos(h\partial_{x}),\;\sinh(h\partial_{x})\;\mbox{and}\;\cosh(h\partial_{x})$
from power functions and exponential functions to all the $C^{\infty}$
functions. For instance (See [5])
$\displaystyle\sin(h\partial_{x})\arctan{bx}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\textrm{tanh}^{-1}\frac{2bh}{1+(bx)^{2}+(bh)^{2}},$
$\displaystyle\cos(h\partial_{x})\arctan{bx}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\arctan\frac{2bx}{1-(bx)^{2}-(bh)^{2}}.$
Therefore, establishing such a set of algorithms as (2.2)-(2.2) is the second
important way to extend the domains and ranges of abstract operators.
Because of the introduction of Definition 5, it is necessary to add new
algorithms.
Theorem 3. Let $a\in\mathbb{C}^{1}$, $\rho\in\mathbb{R}^{1}$, then
$\forall{f}(z)\in{C}^{\infty}(\mathbb{C}^{1})$, we have
$a^{\rho\frac{\partial}{\partial\rho}}f(\rho)=f(a\rho).$ (27)
Proof. By Definition 5, for $a\in\mathbb{C}^{1}$, $\rho\in\mathbb{R}^{1}$, we
have
$a^{\rho\frac{\partial}{\partial\rho}}\rho^{\alpha}=a^{\alpha}\rho^{\alpha}\quad\mbox{or}\quad
a^{\rho\frac{\partial}{\partial\rho}}\rho^{\alpha}=(a\rho)^{\alpha},\qquad\forall{\alpha\in\mathbb{N}^{1}}.$
According to the analytic continuous fundamental theorem, we have Theorem 3.
Theorem 4. Let
$X=(\rho_{1}\cos\theta_{1},\cdots,\rho_{n}\cos\theta_{n}),\;Y=(\rho_{1}\sin\theta_{1},\cdots,\rho_{n}\sin\theta_{n})$,
introducing the notation
$\theta\,\rho\frac{\partial}{\partial\rho}=\theta_{1}\,\rho_{1}\frac{\partial}{\partial\rho_{1}}+\cdots+\theta_{n}\,\rho_{n}\frac{\partial}{\partial\rho_{n}},\;\;Y\partial_{X}=Y_{1}\frac{\partial}{\partial{X}_{1}}+\cdots+Y_{n}\frac{\partial}{\partial{X}_{n}},$
then
$\forall{f}(z)\in{C}^{\infty}(\mathbb{C}^{n}),\;\rho\in\mathbb{R}^{n},\;\theta\in\mathbb{R}_{n}$,
we have
$\displaystyle\cos\left(\theta\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)$
$\displaystyle=$ $\displaystyle\cos(Y\partial_{X})f(X),$
$\displaystyle\sin\left(\theta\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)$
$\displaystyle=$ $\displaystyle\sin(Y\partial_{X})f(X).$
Proof. Let $a_{1}=e^{i\theta_{1}},\,\ldots,\,a_{n}=e^{i\theta_{n}}$, according
to Theorem 3, then
$\exp\left(i\theta\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)=f(\rho{e}^{i\theta})=f(X+iY)=\exp(iY\partial_{X})f(X),\qquad
X\in\mathbb{R}^{n},\;\;Y\in\mathbb{R}_{n}.$
Considering
$\displaystyle\exp\left(i\theta\,\rho\frac{\partial}{\partial\rho}\right)$
$\displaystyle=$
$\displaystyle\cos\left(\theta\,\rho\frac{\partial}{\partial\rho}\right)+i\sin\left(\theta\,\rho\frac{\partial}{\partial\rho}\right),$
$\displaystyle\exp(iY\partial_{X})$ $\displaystyle=$
$\displaystyle\cos(Y\partial_{X})+i\sin(Y\partial_{X}),$
thus we have Theorem 4, which transforms the abstract operators taking
$\rho\frac{\partial}{\partial\rho}$ as the operator element into those taking
$\partial_{X}$ as the operator element.
Corollary 5. Let $f(z),\,z=x+iy\in\mathbb{C}^{1}$ be an arbitrary analytic
function in a complex plane, then on the line $y=kx$ we have
$\displaystyle\left.\cos\left(y\frac{\partial}{\partial{x}}\right)f(x)\right|_{y=kx}$
$\displaystyle=$
$\displaystyle\beta^{x\frac{\partial}{\partial{x}}}\cos\left(\alpha{x}\frac{\partial}{\partial{x}}\right)f(x),$
$\displaystyle\left.\sin\left(y\frac{\partial}{\partial{x}}\right)f(x)\right|_{y=kx}$
$\displaystyle=$
$\displaystyle\beta^{x\frac{\partial}{\partial{x}}}\sin\left(\alpha{x}\frac{\partial}{\partial{x}}\right)f(x).$
Where $\beta=(1+k^{2})^{1/2},\;\alpha=\arctan{k}$, which can be used to solve
boundary value problems of 2-dimensional Laplace equation on polygonal
domains.
Corollary 6. Making use of the analytic continuous fundamental theorem and the
Pochhammer symbol $(\lambda)_{m}$ defined by
$(\lambda)_{m}=\frac{\Gamma(\lambda+m)}{\Gamma(\lambda)}=\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle
1,&(m=0),\\\
\displaystyle\lambda(\lambda+1)\cdots(\lambda+m-1)&(m\in\mathbb{N}_{+}).\end{array}\right.$
(30)
It is easily seen
$(-X)_{m}u(x)=(-1)^{m}x^{m}\frac{d^{m}u}{dx^{m}},\quad\forall{u}(x)\in{C^{m}}(\mathbb{R}^{1}),\,X=x\frac{d}{dx}=\frac{d}{dz}=D_{z}.$
(31)
Where $x=e^{z}$ or $z=\ln{x}$. Using this set of formulas, we can easily
transform the Euler equation into an ordinary differential equation with
constant coefficients.
Therefore, for an arbitrary abstract operators, its symbol are $C^{\infty}$
functions. Further descriptions require concrete analysis under specific
circumstances. For instance, the symbol of the abstract operators on the right
side of (17) can be further described by using conditions of the Laplace
transform. The domain of the abstract operators $P(\partial_{x})$,
$\sin(h\partial_{x})\;\mbox{and}\;\cos(h\partial_{x})$ are the $C^{\infty}$
whole space.
### 2.3 Abstract operators and bounded function
The abstract operators can also acting on the differentiable functions on
bounded domain.
For instance, let $a\leq{b}$, without losing the universality, assuming $a\geq
0,\;b\geq 0$ we have
$na^{n-1}\leq(a^{n-1}+a^{n-2}b+a^{n-3}b^{2}+\cdots+ab^{n-2}+b^{n-1})\leq{n}b^{n-1}.$
$na^{n-1}\leq\frac{b^{n}-a^{n}}{b-a}\leq{n}b^{n-1}\quad\mbox{or}\quad{a}\leq\left(\frac{1}{n}\frac{b^{n}-a^{n}}{b-a}\right)^{\frac{1}{n-1}}\leq{b}.$
Therefore, if $a\leq{b}$, then we have $a\leq{c}\leq{b}$, making
$nc^{n-1}=\frac{b^{n}-a^{n}}{b-a}\quad\mbox{or}\quad\frac{d}{dc}c^{n}=\frac{b^{n}-a^{n}}{b-a},\qquad\forall{n}\in\mathbb{N}^{1}.$
Taking this one as the characteristic equation, according to the analytic
continuous fundamental theorem we get the Lagrange mean value theorem, namely
If $a\leq{b}$, and $f(x)\in{C^{1}}[a,b]$, then $\exists{c}\in[a,b]$ making
$\frac{d}{dc}f(c)=\frac{f(b)-f(a)}{b-a},\quad\forall{f(x)}\in{C^{1}}[a,b].$
(32)
In reference [5], Guangqing Bi and Yuekai Bi discussed the following
functions:
$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^{z})\right|_{z=0},$
$\displaystyle g(x)$ $\displaystyle=$
$\displaystyle\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^{z})\right|_{z=0},\quad{z}\in\mathbb{R}^{1},\;a<x<b.$
Where $f(x),g(x)\in{C}_{0}^{\infty}(\mathbb{R}^{1})$,
$S(t)\in{C}^{\infty}(\Omega_{0})$, the $\Omega_{0}$ is a certain neighborhood
at $t=0$.
If $S(t)$ is expanded in power series on $\Omega_{0}$, then (2.3) are
equivalent to Fourier series, and their domain $(a,b)$ can be extended to all
$(-\infty,+\infty)$ periodically. So $f(x),g(x)\in{L}^{2}([-c,c])$ and
$f(x+2l)=f(x),\;g(x+2l)=g(x)$.
For instance, by (2.2) and (2.2), we have
$\displaystyle\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\arctan{e^{z}}\right|_{z=0}\,=\,\left.\cos\left(Y\frac{\partial}{\partial{X}}\right)\arctan{X}\right|_{z=0}$
$\displaystyle=$
$\displaystyle\left.\frac{1}{2}\arctan\frac{2X}{1-(X^{2}+Y^{2})}\right|_{z=0}\,=\,\frac{1}{2}\arctan\frac{2\cos(\pi{x}/c)}{1-\left(\cos^{2}(\pi{x}/c)+\sin^{2}(\pi{x}/c)\right)}.$
When $\cos(\pi{x}/c)\neq 0$ or $|x|\neq{kc}+c/2,\;k=0,1,2,\cdots$, the above
expression turns into
$\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\arctan{e^{z}}\right|_{z=0}=\left\\{\begin{array}[]{r@{\qquad}l}(1/2)\arctan(+\infty)=+\pi/4,&2Kc-c/2<x<2Kc+c/2\\\
(1/2)\arctan(-\infty)=-\pi/4,&2Kc+c/2<x<2Kc+3c/2.\end{array}\right.$
Where $K=0,\pm 1,\pm 2,\cdots$. This function represents a square wave.
By algorithms (19)-(2.2), and some basic formulas similar to (2.2) (See [5]),
we obtained
$\displaystyle\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)(-\ln(1-e^{z}))\right|_{z=0}$
$\displaystyle=$ $\displaystyle\frac{\pi}{2}-\frac{\pi{x}}{2c},\quad 0<x<2c.$
$\displaystyle\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)(-\ln(1-e^{z}))\right|_{z=0}$
$\displaystyle=$ $\displaystyle-\ln\left(2\sin\frac{\pi{x}}{2c}\right),\quad
0<x<2c.$
$\displaystyle\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^{z})\right|_{z=0}$
$\displaystyle=$ $\displaystyle\frac{\pi{x}}{2c},\quad|x|<c.$
$\displaystyle\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^{z})\right|_{z=0}$
$\displaystyle=$ $\displaystyle\ln\left(2\cos\frac{\pi
x}{2c}\right),\quad|x|<c.$
$\displaystyle\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\arctan{e^{z}}\right|_{z=0}$
$\displaystyle=$ $\displaystyle\frac{\pi}{4},\quad|x|<c/2.$
In other words, the ranges of the abstract operators $\sin(h\partial_{x})$ and
$\cos(h\partial_{x})$ may be extended from the
$C_{0}^{\infty}(\mathbb{R}^{1})$ to the $L^{2}([-c,c])$.
In order to solve an initial-boundary value problems of partial differential
equations, we need to express a function with boundary conditions. For
instance, (2.3) are infinitely differentiable within a finite domain, such as
$\frac{d^{k}}{dx^{k}}\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)(-\ln(1-e^{z}))\right|_{z=0}=\frac{d^{k}}{dx^{k}}\left(\frac{\pi}{2}-\frac{\pi{x}}{2c}\right),\quad\forall{k}\in\mathbb{N},\;0<x<2c.$
Example 3. $\forall{f(z)}\in{C}^{\infty}(\mathbb{C}^{1})$, if $f(z)$ is
continuous at $z=1/2$, then
$\frac{2}{\pi}\int^{\pi/2}_{0}\\!\left.\cos\left(x\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)\right|_{\rho=\cos{x}}dx=f\left(\frac{1}{2}\right).$
(35)
Proof. Taking the following integral formula as the characteristic equation
$\int^{\pi/2}_{0}\\!\cos^{n}x\,\cos{nx}\,dx=\frac{\pi}{2^{n+1}},\quad\forall{n}\in\mathbb{N}.$
Combining the characteristic equation with the following operator equation by
using Definition 5 and the base function $x^{\alpha}$:
$\frac{2}{\pi}\int^{\pi/2}_{0}\\!\cos\left(x\,\rho\frac{\partial}{\partial\rho}\right)\rho^{n}\,dx=\left(\frac{1}{2}\right)^{n},\qquad\forall{n}\in\mathbb{N}^{1},\quad
x\in[0,\pi/2].$
Where $\rho=\cos{x}$, according to the analytic continuous fundamental
theorem, we get (35).
Example 4. $\forall{f(z)}\in{C}^{\infty}(\mathbb{C}^{1})$, if $f(z)$ is
differentiable at $z=1/2$, then
$\int^{\pi/2}_{0}\\!\left.\sin\left(x\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)\right|_{\rho=\cos{x}}dx=\frac{1}{2}\int^{1}_{0}\\!\frac{f(\xi)-f(1/2)}{\xi-1/2}\,d\xi.$
(36)
Proof. Considering another integral formula
$\int^{\pi/2}_{0}\\!\cos^{n}x\,\sin{nx}\,dx=\frac{1}{2^{n+1}}\sum^{n}_{k=1}\frac{2^{k}}{k},\quad\forall{n}\in\mathbb{N}.$
Obviously it cannot be taken as the characteristic equation directly, but
considering $\sum^{n}_{k=1}2^{k}/k=\int^{2}_{0}\frac{q^{n}-1}{q-1}dq$, and
according to the analytic continuous fundamental theorem, we get (36).
Example 5. Let $k\in\mathbb{N}_{+}$,
$\forall{f(z)}\in{C}^{\infty}(\mathbb{C}^{1})$, if $f(z)$ is $k$ times
continuously differentiable at $z=1/2$, then
$\frac{2}{\pi}\int^{\pi/2}_{0}\\!\cos(2kx)\left.\cos\left(x\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)\right|_{\rho=\cos{x}}dx=\frac{1}{2^{k+1}k!}f^{(k)}\left(\frac{1}{2}\right).$
(37)
$\frac{2}{\pi}\int^{\pi/2}_{0}\\!\sin(2kx)\left.\sin\left(x\,\rho\frac{\partial}{\partial\rho}\right)f(\rho)\right|_{\rho=\cos{x}}dx=\frac{1}{2^{k+1}k!}f^{(k)}\left(\frac{1}{2}\right).$
(38)
Proof. In [6], Connon has given the following useful identical
equations$(\forall{n}\in\mathbb{N})$:
$2^{n}\cos^{n}x\cos(nx)=\sum^{n}_{k=0}{n\choose{k}}\cos(2kx).$ (39)
$2^{n}\cos^{n}x\sin(nx)=\sum^{n}_{k=0}{n\choose{k}}\sin(2kx).$ (40)
Using the orthogonality of trigonometric functions, by (39) and (40) we have
$\frac{1}{\pi}\int_{0}^{\pi}\cos(2kx)\cos^{n}x\cos(nx)dx={n\choose{k}}\left(\frac{1}{2}\right)^{n+1},\quad\forall{k}\in\mathbb{N}_{+}.$
(41)
$\frac{1}{\pi}\int_{0}^{\pi}\sin(2kx)\cos^{n}x\sin(nx)dx={n\choose{k}}\left(\frac{1}{2}\right)^{n+1},\quad\forall{k}\in\mathbb{N}_{+}.$
(42)
Which can also be written as
$\frac{2}{\pi}\int_{0}^{\pi/2}\cos(2kx)\cos^{n}x\cos(nx)dx={n\choose{k}}\left(\frac{1}{2}\right)^{n+1}.$
(43)
$\frac{2}{\pi}\int_{0}^{\pi/2}\sin(2kx)\cos^{n}x\sin(nx)dx={n\choose{k}}\left(\frac{1}{2}\right)^{n+1}.$
(44)
Clearly, (43) is the characteristic equation of (37), and 44) is the
characteristic equation of (38). So by Definition 5 we have
$\frac{2}{\pi}\int^{\pi/2}_{0}\\!\cos(2kx)\left.\cos\left(x\,\rho\frac{\partial}{\partial\rho}\right)\rho^{n}\right|_{\rho=\cos{x}}dx=\frac{1}{2^{k+1}k!}\left.\frac{d^{k}}{d\xi^{k}}\xi^{n}\right|_{\xi=1/2},\quad\forall{n}\in\mathbb{N}^{1}.$
(45)
$\frac{2}{\pi}\int^{\pi/2}_{0}\\!\sin(2kx)\left.\sin\left(x\,\rho\frac{\partial}{\partial\rho}\right)\rho^{n}\right|_{\rho=\cos{x}}dx=\frac{1}{2^{k+1}k!}\left.\frac{d^{k}}{d\xi^{k}}\xi^{n}\right|_{\xi=1/2},\quad\forall{n}\in\mathbb{N}^{1}.$
(46)
Then according to the analytic continuous fundamental theorem, we get (37) and
(38).
In reference [5] we get
Let
$S(t)=\sum^{\infty}_{n=0}a_{n}t^{n},\;t\in\mathbb{R}^{1},\;0\leq{t}\leq{r},\;0<r<+\infty$,
if
$\sum^{\infty}_{n=0}a_{n}\cos\frac{n\pi{x}}{c}=\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^{z})\right|_{z=0},$
$\sum^{\infty}_{n=0}a_{n}\sin\frac{n\pi{x}}{c}=\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^{z})\right|_{z=0}$
are tenable in $a<x<b,\;x\in\mathbb{R}^{1}$, then
$\forall{x_{0}}\in[0,\frac{b-a}{2})$,
$\displaystyle\sum^{\infty}_{n=0}a_{n}\cos\frac{n\pi{x_{0}}}{c}\cos\frac{n\pi{x}}{c}$
$\displaystyle=$
$\displaystyle\cosh\left(x_{0}\frac{\partial}{\partial{x}}\right)\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^{z})\right|_{z=0},$
$\displaystyle\sum^{\infty}_{n=0}a_{n}\cos\frac{n\pi{x_{0}}}{c}\sin\frac{n\pi{x}}{c}$
$\displaystyle=$
$\displaystyle\cosh\left(x_{0}\frac{\partial}{\partial{x}}\right)\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^{z})\right|_{z=0}$
are tenable in $a+x_{0}<x<b-x_{0}$. Accordingly, for any definite value
$x\in(a,b)$, $x_{0}$ in (2.3) takes values in the following interval:
$0<x_{0}<\left\\{\begin{array}[]{lr}x-a,&a<x\leq(a+b)/2,\\\
b-x,&b>x\geq(a+b)/2.\end{array}\right.$
Where the left side of (2.3) is:
$\cos\frac{n\pi{x_{0}}}{c}\cos\frac{n\pi{x}}{c}=\cosh\left(x_{0}\frac{\partial}{\partial{x}}\right)\cos\frac{n\pi{x}}{c},$
$\cos\frac{n\pi{x_{0}}}{c}\sin\frac{n\pi{x}}{c}=\cosh\left(x_{0}\frac{\partial}{\partial{x}}\right)\sin\frac{n\pi{x}}{c}.$
Therefore, $\sinh(h\partial_{x})\;\mbox{and}\;\cosh(h\partial_{x})$ are the
continuous operators on $L^{2}([-c,c])$.
### 2.4 Abstract operators and Fourier analysis
The third way to extend the domains and ranges of abstract operators is using
Fourier analysis.
Theorem 5. Let $t\in\overline{\mathbb{R}_{+}^{1}}$,
$\forall{f(x)}\in{S}(\mathbb{R}^{1})$, if $F_{+}(s),\,F_{-}(s)$ are
respectively
$F_{+}(s)=\frac{1}{\pi}\mathscr{L}\int^{\infty}_{-\infty}f(\xi)\cos(t\xi)d\xi,\qquad
F_{-}(s)=\frac{1}{\pi}\mathscr{L}\int^{\infty}_{-\infty}f(\xi)\sin(t\xi)d\xi,$
(48)
where $\mathscr{L}$ is the Laplace transform operator, then we have
$f(x)=\left.\cos\left(x\frac{\partial}{\partial{s}}\right)F_{+}(s)\right|_{s=0}+\left.\sin\left(-x\frac{\partial}{\partial{s}}\right)F_{-}(s)\right|_{s=0}.$
(49)
Or
$\left.\cos\left(x\frac{\partial}{\partial{s}}\right)F_{+}(s)\right|_{s=0}=\frac{f(x)+f(-x)}{2},\qquad\frac{1}{\pi}\int^{\infty}_{-\infty}f(x)\cos(tx)dx=\mathscr{L}^{-1}F_{+}(s).$
(50)
$\left.\sin\left(-x\frac{\partial}{\partial{s}}\right)F_{-}(s)\right|_{s=0}=\frac{f(x)-f(-x)}{2},\qquad\frac{1}{\pi}\int^{\infty}_{-\infty}f(x)\sin(tx)dx=\mathscr{L}^{-1}F_{-}(s).$
(51)
Proof. By (17) we have
$\mathscr{L}\cos(t\xi)=\cos\left(\xi\frac{\partial}{\partial{s}}\right)\frac{1}{s},\quad\mathscr{L}\sin(t\xi)=\sin\left(-\xi\frac{\partial}{\partial{s}}\right)\frac{1}{s},\;\;t\in\overline{\mathbb{R}_{+}^{1}},\;\Re(s)>0.$
So by (48) and (49),
$\forall{f(x)}\in{S}(\mathbb{R}^{1}),\,\exists{f_{s}(x)}\in{C^{\infty}}(\mathbb{R}^{1})$
can be expressed as
$\displaystyle f_{s}(x)$ $\displaystyle=$
$\displaystyle\cos\left(x\frac{\partial}{\partial{s}}\right)F_{+}(s)+\sin\left(-x\frac{\partial}{\partial{s}}\right)F_{-}(s)$
$\displaystyle=$
$\displaystyle\cos\left(x\frac{\partial}{\partial{s}}\right)\left[\frac{1}{\pi}\int^{\infty}_{-\infty}f(\xi)\mathscr{L}\cos(t\xi)d\xi\right]+\sin\left(-x\frac{\partial}{\partial{s}}\right)\left[\frac{1}{\pi}\int^{\infty}_{-\infty}f(\xi)\mathscr{L}\sin(t\xi)d\xi\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\cos\left(x\frac{\partial}{\partial{s}}\right)\int^{\infty}_{-\infty}f(\xi)\cos\left(\xi\frac{\partial}{\partial{s}}\right)\frac{1}{s}\,d\xi+\frac{1}{\pi}\sin\left(-x\frac{\partial}{\partial{s}}\right)\int^{\infty}_{-\infty}f(\xi)\sin\left(-\xi\frac{\partial}{\partial{s}}\right)\frac{1}{s}\,d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}f(\xi)\left[\cos\left(x\frac{\partial}{\partial{s}}\right)\cos\left(\xi\frac{\partial}{\partial{s}}\right)+\sin\left(-x\frac{\partial}{\partial{s}}\right)\sin\left(-\xi\frac{\partial}{\partial{s}}\right)\right]\frac{1}{s}\,d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{-\infty}f(\xi)\cos\left((x-\xi)\frac{\partial}{\partial{s}}\right)\frac{1}{s}\,d\xi=\frac{1}{\pi}\int^{\infty}_{-\infty}f(\xi)\frac{s}{(x-\xi)^{2}+s^{2}}d\xi,\;\Re(s)>0.$
Therefore, we get
$\displaystyle\lim_{s\rightarrow 0^{+}}f_{s}(x)$ $\displaystyle=$
$\displaystyle\lim_{s\rightarrow
0^{+}}\frac{1}{\pi}\int^{\infty}_{-\infty}f(\xi)\frac{s}{(x-\xi)^{2}+s^{2}}d\xi$
$\displaystyle=$ $\displaystyle\int^{\infty}_{-\infty}f(\xi)\lim_{s\rightarrow
0^{+}}\frac{1}{\pi}\frac{s}{(x-\xi)^{2}+s^{2}}d\xi\rightharpoonup\int^{\infty}_{-\infty}f(\xi)\delta(x-\xi)d\xi=f(x).$
In the same manner we find
Theorem 6. Let ${f(x)}\in{L}^{2}([-l,l]),\;f(x+2l)=f(x)$. If $S_{+}(t)$ and
$S_{-}(t)$ are respectively
$S_{+}(t)=\frac{1}{2l}\int^{l}_{-l}f(\xi)\frac{1-t^{2}}{1-2t\cos(\pi\xi/l)+t^{2}}d\xi,$
(52)
$S_{-}(t)=\frac{1}{l}\int^{l}_{-l}f(\xi)\frac{t\sin(\pi\xi/l)}{1-2t\cos(\pi\xi/l)+t^{2}}d\xi,$
(53)
then $\exists{f_{z}(x)}\in{C^{\infty}}(\mathbb{R}^{1})$ with the following
form
$f_{z}(x)=\cos\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)S_{+}(e^{z})+\sin\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)S_{-}(e^{z}),\quad-\infty<z<0$
(54)
making
$\lim_{z\rightarrow
0^{-}}f_{z}(x)\rightharpoonup{f(x)},\quad\forall{f(x)}\in{C}(-l,l)\subset{L}^{2}([-l,l]).$
$\lim_{z\rightarrow
0^{-}}\|f(x)-f_{z}(x)\|_{L^{2}([-l,l])}=0,\quad\forall{f(x)}\in{L}^{2}([-l,l]).$
Proof. By the algorithms (2.2) we have
$\cos\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\frac{1+e^{z}}{1-e^{z}}=\frac{1-e^{2z}}{1-2e^{z}\cos(\pi\xi/l)+e^{2z}}.$
$\sin\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\frac{e^{z}}{1-e^{z}}=\frac{e^{z}\sin(\pi\xi/l)}{1-2e^{z}\cos(\pi\xi/l)+e^{2z}}.$
So $f_{z}(x)$ can be expressed as
$\displaystyle f_{z}(x)$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)\left[\frac{1}{2l}\int^{l}_{-l}f(\xi)\cos\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\frac{1+e^{z}}{1-e^{z}}d\xi\right]$
$\displaystyle+\,\sin\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)\left[\frac{1}{l}\int^{l}_{-l}f(\xi)\sin\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\frac{e^{z}}{1-e^{z}}d\xi\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2l}\int^{l}_{-l}f(\xi)\cos\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)\cos\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\left[1+\frac{2e^{z}}{1-e^{z}}\right]d\xi$
$\displaystyle+\,\frac{1}{l}\int^{l}_{-l}f(\xi)\sin\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)\sin\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\frac{e^{z}}{1-e^{z}}d\xi=\frac{1}{2l}\int^{l}_{-l}f(\xi)d\xi$
$\displaystyle+\,\frac{1}{l}\int^{l}_{-l}f(\xi)\left[\cos\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)\cos\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)+\sin\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)\sin\left(\frac{\pi\xi}{l}\frac{\partial}{\partial{z}}\right)\right]\frac{e^{z}}{1-e^{z}}d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{2l}\int^{l}_{-l}f(\xi)d\xi+\frac{1}{l}\int^{l}_{-l}f(\xi)\cos\left(\frac{\pi(x-\xi)}{l}\frac{\partial}{\partial{z}}\right)\frac{e^{z}}{1-e^{z}}d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{2l}\int^{l}_{-l}f(\xi)\cos\left(\frac{\pi(x-\xi)}{l}\frac{\partial}{\partial{z}}\right)\frac{1+e^{z}}{1-e^{z}}d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{2l}\int^{l}_{-l}f(\xi)\frac{1-e^{2z}}{1-2e^{z}\cos(\pi(x-\xi)/l)+e^{2z}}d\xi,\quad-\infty<z<0.$
Therefore, if $f(x)\in{C}(-l,l)\subset{L}^{2}([-l,l])$, we get
$\displaystyle\lim_{z\rightarrow 0^{-}}f_{z}(x)$ $\displaystyle=$
$\displaystyle\lim_{z\rightarrow
0^{-}}\frac{1}{2l}\int^{l}_{-l}f(\xi)\frac{1-e^{2z}}{1-2e^{z}\cos(\pi(x-\xi)/l)+e^{2z}}d\xi$
$\displaystyle=$ $\displaystyle\int^{l}_{-l}f(\xi)\lim_{z\rightarrow
0^{-}}\frac{1}{2l}\frac{1-e^{2z}}{1-2e^{z}\cos(\pi(x-\xi)/l)+e^{2z}}d\xi\rightharpoonup\int^{l}_{-l}f(\xi)\delta(x-\xi)d\xi=f(x).$
As we all know, $\forall{f(x)}\in{L}^{2}([-l,l]),\;f(x+2l)=f(x)$ we have
$f(x)=a_{0}+\sum^{\infty}_{k=1}\left(a_{k}\cos\frac{k\pi{x}}{l}+b_{k}\sin\frac{k\pi{x}}{l}\right).$
Where $\forall{k}\in\mathbb{N}_{+}$
$a_{0}=\frac{1}{2l}\int^{l}_{-l}f(\xi)d\xi,\quad{a_{k}=\frac{1}{l}\int^{l}_{-l}f(\xi)\cos\frac{k\pi{x}}{l}d\xi},\quad{b_{k}=\frac{1}{l}\int^{l}_{-l}f(\xi)\sin\frac{k\pi{x}}{l}d\xi}.$
Accordingly $f_{z}(x)\in{C}^{\infty}(\mathbb{R}^{1})$ can be expressed as
$f_{z}(x)=a_{0}+\sum^{\infty}_{k=1}\left(a_{k}e^{kz}\cos\frac{k\pi{x}}{l}+b_{k}e^{kz}\sin\frac{k\pi{x}}{l}\right),\quad-\infty<z<0.$
(55)
So we have
$\frac{1}{l}\int^{l}_{-l}|f(x)-f_{z}(x)|^{2}dx=\sum^{\infty}_{k=1}[(a^{2}_{k}+b^{2}_{k})(1-e^{kz})^{2}],\quad-\infty<z<0.$
(56) $\lim_{z\rightarrow
0^{-}}\|f(x)-f_{z}(x)\|_{L^{2}([-l,l])}=0,\quad\forall{f(x)}\in{L}^{2}([-l,l]),\;f(x+2l)=f(x).$
(57)
If $f(x)$ is an even function, then we have the integral equation:
$\frac{1}{l}\int^{l}_{0}f(x)\frac{1-t^{2}}{1-2t\cos(\pi{x}/l)+t^{2}}dx=S_{+}(t),\quad\forall{S_{+}(t)}\in{C}^{\infty}(\Omega_{0}).$
(58)
Its solution $f(x)\in{L}^{2}([-l,l]),\;f(x+2l)=f(x)$, and we have
$f(x)=\left.\cos\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)S_{+}(e^{z})\right|_{z=0}.$
(59)
If $f(x)$ is an odd function, then we have the integral equation:
$\frac{2}{l}\int^{l}_{0}f(x)\frac{t\sin(\pi{x}/l)}{1-2t\cos(\pi{x}/l)+t^{2}}dx=S_{-}(t),\quad\forall{S_{-}(t)}\in{C}^{\infty}(\Omega_{0}).$
(60)
Its solution $f(x)\in{L}^{2}([-l,l]),\;f(x+2l)=f(x)$, and we have
$f(x)=\left.\sin\left(\frac{\pi{x}}{l}\frac{\partial}{\partial{z}}\right)S_{-}(e^{z})\right|_{z=0}.$
(61)
For instance, in the integral equation (58), if $S_{+}(t)=\arctan{t},\;0<t<1$,
then its solution is a square wave, namely
$f(x)=\left\\{\begin{array}[]{r@{\qquad}l}+\pi/4,&2Kl-l/2<x<2Kl+l/2\\\
-\pi/4,&2Kl+l/2<x<2Kl+3l/2.\end{array}\right.$
Where $K=0,\pm 1,\pm 2,\cdots$.
Let $\rho_{z}(x)\in{C}_{0}^{\infty}(\mathbb{R}^{n})$,
supp$\rho_{z}(x)\subset\\{x\in\mathbb{R}^{n}|\,|x|\leq{l}\\}$ be a polished
kernel, taken as
$\rho_{z}(x)=\rho_{z_{1}}(x_{1})\rho_{z_{2}}(x_{2})\cdots\rho_{z_{i}}(x_{i})\cdots\rho_{z_{n}}(x_{n})$
and
$\rho_{z_{i}}(x_{i})=\frac{1}{2l_{i}}\frac{1-e^{2z_{i}}}{1-2e^{z_{i}}\cos(\pi{x_{i}}/l_{i})+e^{2{z_{i}}}},\quad-\infty<z_{i}<0,\;|x_{i}|\leq{l_{i}}\;(i=1,2,\cdots,n).$
(62)
Then we easily prove the following results:
$\lim_{z\rightarrow
0^{-}}g(\partial_{x})\int^{l}_{-l}f(\xi)\rho_{z}(x-\xi)d\xi\rightharpoonup{g}(\partial_{x})f(x),\quad\forall{f(x)}\in{L}^{2}([-l,l]),\;g(\xi)\in{C}^{\infty}(\mathbb{R}_{n}).$
(63)
Where
$f(x+2l)=f(x),\;l\in\mathbb{R}^{n},\;[-l,l]=\\{x\in\mathbb{R}^{n}|-l\leq{x}\leq{l}\\}$.
If $g(\partial_{x})=\partial_{x}^{\alpha},\;\forall\alpha\in\mathbb{N}^{n}$,
then (63) gives a definition of generalized derivative on $L^{2}([-l,l])$.
### 2.5 Integral representation of abstract operators
Clearly, such algorithms as (2.2)-(2.2) do not exist for most abstract
operators. Therefore, for more complex abstract operators, establishing the
integral expression is the fourth important way to extend the domain of
abstract operators. In reference [2] Guangqing Bi has obtained the following
results:
Theorem BI1. Let $P(\partial_{x})$ be an $m$-order partial differential
operator of any kind and there exist $a_{1},\,a_{2},\,\cdots,\,a_{k}$ of real
and partial differential operators $A_{1},\,A_{2},\,\cdots,\,A_{k}$ of the
order less than $[(m+1)/2]$ such that
$P(\partial_{x})\equiv{a}_{1}A_{1}^{2}+a_{2}A_{2}^{2}+\cdots+a_{k}A_{k}^{2}$.
If $k=2\nu+3,\;\nu=0,\,1,\,2,\cdots$, then
$\forall{f(x)}\in{C^{m\nu}}(\mathbb{R}^{n})$ we have
$\displaystyle\frac{\sinh\left(\alpha{t}P(\partial_{x})^{1/2}\right)}{\alpha
P(\partial_{x})^{1/2}}f(x)$ $\displaystyle=$ $\displaystyle
t\underbrace{\int^{t}_{0}tdt\cdots}_{\nu}\int^{t}_{0}\,tdt\frac{(\alpha^{2}P(\partial_{x}))^{\nu}}{2^{\nu+2}\pi^{\nu+1}}$
(64)
$\displaystyle\times\,\int^{\pi}_{-\pi}\underbrace{\int^{\pi}_{0}\cdots}_{k-2}\int^{\pi}_{0}e^{\eta_{1}a_{1}^{1/2}A_{1}+\cdots+\eta_{k}a_{k}^{1/2}A_{k}}f(x)\,d\sigma_{k}$
$\displaystyle+\,\sum^{\nu-1}_{i=0}\frac{t^{2i+1}}{(2i+1)!}(\alpha^{2}P(\partial_{x}))^{i}f(x).$
Where $\alpha$ is a real numbers,
$x\in\mathbb{R}^{n},\;t\in\mathbb{R}^{1}.\;\;\eta\in\mathbb{R}_{k}$ is the
integral variable and
$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle\alpha{t}\cos\theta_{1},$ $\displaystyle\eta_{2}$
$\displaystyle=$ $\displaystyle\alpha{t}\sin\theta_{1}\cos\theta_{2},$
$\displaystyle\cdots$ $\displaystyle\eta_{p}$ $\displaystyle=$
$\displaystyle\alpha{t}\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{p-1}\cos\theta_{p},$
$\displaystyle\eta_{p+1}$ $\displaystyle=$
$\displaystyle\alpha{t}\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{p}\cos\phi,$
$\displaystyle\eta_{p+2}$ $\displaystyle=$
$\displaystyle\eta_{k}\;=\;\alpha{t}\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{p}\sin\phi;$
$d\sigma_{k}=\sin^{k-2}\theta_{1}\sin^{k-3}\theta_{2}\cdots\sin\theta_{k-2}d\theta_{1}d\theta_{2}\cdots{d}\theta_{k-2}d\phi.$
When $A_{1},\,\cdots,\,A_{k}$ in the right-hand of (64) are one order partial
differential operators, then the abstract operator
$e^{\eta_{1}a_{1}^{1/2}A_{1}+\cdots+\eta_{k}a_{k}^{1/2}A_{k}}$ is one of the
following five simplest operators:
$\exp(h\partial_{x}),\;\sin(h\partial_{x}),\;\cos(h\partial_{x}),\;\sinh(h\partial_{x})\;\mbox{and}\;\cosh(h\partial_{x}).$
Proof. In (64), let
$f(x)=e^{\xi{x}},\,x\in\mathbb{R}^{n},\xi\in\mathbb{R}_{n}$, and the symbol of
the partial differential operator $A_{j},j=1,2,\cdots,k$ is denoted by
$\chi_{j}(\xi),\;\beta_{j}=a_{j}^{1/2}\chi_{j}(\xi)$. By Definition 4, the
formula degenerates to its characteristic equation:
$\displaystyle\frac{\sinh\left(\alpha{t}P(\xi)^{1/2}\right)}{\alpha
P(\xi)^{1/2}}$ $\displaystyle=$ $\displaystyle
t\underbrace{\int^{t}_{0}tdt\cdots}_{\nu}\int^{t}_{0}\,tdt\frac{(\alpha^{2}P(\xi))^{\nu}}{2^{\nu+2}\pi^{\nu+1}}\int^{\pi}_{-\pi}\underbrace{\int^{\pi}_{0}\cdots}_{k-2}\int^{\pi}_{0}e^{\eta_{1}\beta_{1}+\cdots+\eta_{k}\beta_{k}}d\sigma_{k}$
(65)
$\displaystyle+\,\sum^{\nu-1}_{i=0}\frac{t^{2i+1}}{(2i+1)!}(\alpha^{2}P(\xi))^{i},\quad\nu=\frac{k-3}{2}.$
According to the analytic continuous fundamental theorem, we only need to
prove (65). Solving the integral on a hypersphere on the right side of (65),
we have
$\displaystyle\frac{\sinh\left(\alpha{t}P(\xi)^{1/2}\right)}{\alpha
P(\xi)^{1/2}}=t\underbrace{\int^{t}_{0}tdt\cdots}_{\nu}\int^{t}_{0}tdt\sum^{\infty}_{j=0}\frac{(\alpha^{2}P(\xi))^{\nu+j}t^{2j}}{(2j)!!(2j+2\nu+1)!!}+\sum^{\nu-1}_{i=0}\frac{t^{2i+1}(\alpha^{2}P(\xi))^{i}}{(2i+1)!}.$
Then it is proved by the termwise integration of the infinite series on the
right side of the equation.
Similarly, we have (See [1])
Theorem BI1’. Let $P(\partial_{x})$ be an $m$-order partial differential
operator of any kind and there exist $a_{1},\,a_{2},\,\cdots,\,a_{k}$ of real
and partial differential operators $A_{1},\,A_{2},\,\cdots,\,A_{k}$ of the
order less than $[(m+1)/2]$ such that
$P(\partial_{x})\equiv{a}_{1}A_{1}^{2}+a_{2}A_{2}^{2}+\cdots+a_{k}A_{k}^{2}$.
If $k=2\nu+3,\;\nu=0,\,1,\,2,\cdots$, then
$\forall{f(z)}\in{C^{\infty}}(\mathbb{C}^{n})$
$\displaystyle\frac{\sin\left({t}P(\partial_{x})^{1/2}\right)}{P(\partial_{x})^{1/2}}f(x)$
$\displaystyle=$ $\displaystyle
t\underbrace{\int^{t}_{0}tdt\cdots}_{\nu}\int^{t}_{0}tdt[-P(\partial_{x})]^{\nu}$
(66)
$\displaystyle\times\,\left(\frac{2}{\pi}\right)^{\nu+1}\underbrace{\int^{\pi/2}_{0}\cdots}_{k-1}\int^{\pi/2}_{0}\cos(\eta_{1}a_{1}^{1/2}A_{1})\cdots\cos(\eta_{k}a_{k}^{1/2}A_{k})f(x)\,d\sigma_{k}$
$\displaystyle+\,\sum^{\nu-1}_{i=0}\frac{t^{2i+1}}{(2i+1)!}[-P(\partial_{x})]^{i}f(x).$
Where $x\in\mathbb{R}^{n},\;t\in\mathbb{R}^{1}.\;\;\eta\in\mathbb{R}_{k}$ is
the integral variable and
$\displaystyle\eta_{1}$ $\displaystyle=$ $\displaystyle t\cos\theta_{1},$
$\displaystyle\eta_{2}$ $\displaystyle=$ $\displaystyle
t\sin\theta_{1}\cos\theta_{2},$ $\displaystyle\cdots$ $\displaystyle\eta_{p}$
$\displaystyle=$ $\displaystyle
t\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{p-1}\cos\theta_{p},$
$\displaystyle\eta_{p+1}$ $\displaystyle=$ $\displaystyle
t\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{p}\cos\phi,$
$\displaystyle\eta_{p+2}$ $\displaystyle=$
$\displaystyle\eta_{k}\;=\;t\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{p}\sin\phi;$
$d\sigma_{k}=\sin^{k-2}\theta_{1}\sin^{k-3}\theta_{2}\cdots\sin\theta_{k-2}d\theta_{1}d\theta_{2}\cdots{d}\theta_{k-2}d\phi.$
If $A_{1},\,\cdots,\,A_{k}$ in (64) and (66) are partial differential
operators of the order great than 1, then the order can be lowered by taking
the following Theorem:
Theorem BI2. Let $P(\partial_{x}),\;x\in\mathbb{R}^{n}$ be a partial
differential operator of any order and $f(x)\in{C}(\mathbb{R}^{n})$ which make
the integral in the follow formula meaningful, then
$e^{\lambda
P(\partial_{x})}f(x)=\frac{1}{2\sqrt{\pi}}\int^{\infty}_{-\infty}e^{-\zeta^{2}/4}e^{\lambda^{1/2}\zeta
P(\partial_{x})^{1/2}}f(x)d\zeta,\quad\forall\lambda\in\mathbb{C}^{1}.$ (67)
Continuing the processing one can arrive at order one.
Example 6. By using the Theorem BI2 and Corollary 2 we have
$\exp\left(-a^{2}t\frac{\partial^{2}}{\partial{x}_{j}^{2}}\right)g(x)=\frac{1}{2\sqrt{\pi}}\int^{\infty}_{-\infty}e^{-\zeta^{2}/4}\cos\left(a\sqrt{t}\,\zeta\frac{\partial}{\partial{x_{j}}}\right)g(x)d\zeta,\quad{t}\in\overline{\mathbb{R}_{+}^{1}}.$
$\cos\left(\lambda\frac{\partial^{2}}{\partial{x}_{j}^{2}}\right)g(x)=\frac{1}{2\sqrt{\pi}}\int^{\infty}_{-\infty}e^{-\zeta^{2}/4}e^{\sqrt{\lambda/2}\,\zeta\frac{\partial}{\partial{x}_{j}}}\cos\left(\sqrt{\frac{\lambda}{2}}\,\zeta\frac{\partial}{\partial{x}_{j}}\right)g(x)d\zeta,$
$\sin\left(a\lambda\frac{\partial}{\partial{x}_{j}}+\lambda\frac{\partial^{2}}{\partial{x}_{j}^{2}}\right)g(x)=\frac{1}{2\sqrt{\pi}}\int^{\infty}_{-\infty}e^{-\zeta^{2}/4}e^{\sqrt{\lambda/2}\,\zeta\frac{\partial}{\partial{x}_{j}}}\sin\left(h_{\lambda,a}(\zeta)\frac{\partial}{\partial{x}_{j}}\right)g(x)d\zeta.$
Where
$h_{\lambda,a}(\zeta)=a\lambda+\sqrt{\lambda/2}\zeta,\;\forall\lambda,a\in\mathbb{C}^{1}$.
With the expansion of definitions of abstract operators, it is necessary to
add new algorithms continuously.
## 3 Abstract operators and linear higher-order partial differential
equations
### 3.1 Applications of Laplace transform for n-dimensional space
Solving the ordinary or partial differential equations, is constructing the
algorithms of the inverse operators of ordinary or partial differential
operators. In terms of abstract operators, solving the initial value problem
of partial differential equations for n-dimensional space is similar to
solving the ordinary differential equations with respect to the variable $t$,
thus we can introduce the Laplace transform to further simplify the solving
process.
Corollary 7. Let $g(\int^{t}_{0}\\!\cdot\,{t}dt)$ be the abstract operators
taking $\int^{t}_{0}\\!\cdot\,{t}dt$ as the operator element, and
$g(-\frac{1}{s}\frac{\partial}{\partial{s}})$ be the abstract operator taking
$-\frac{1}{s}\frac{\partial}{\partial{s}}$ as the operator element, and the
symbols of these two abstract operators are the same, then for any Laplace
transformable function $f(t)$ we have
$g\left(\int^{t}_{0}\\!\cdot\,{t}dt\right)f(t)=\mathscr{L}^{-1}\,g\left(-\frac{1}{s}\frac{\partial}{\partial{s}}\right)\mathscr{L}f(t),\quad{t}\in\overline{\mathbb{R}_{+}^{1}}.$
(68)
Where $\mathscr{L}$ and $\mathscr{L}^{-1}$ are the Laplace transform operator
and its inverse operator respectively, and $\mathscr{L}f(t)=F(s)$.
Proof. According to the properties of Laplace transform, we generally have
$\left(\int^{t}_{0}\\!\cdot\,{t}dt\right)^{m}f(t)=\mathscr{L}^{-1}\,\left(-\frac{1}{s}\frac{\partial}{\partial{s}}\right)^{m}F(s),\quad\forall{m}\in\mathbb{N}.$
According to the abstract operator fundamental theorem, we get the Corollary
7.
Similarly we have
$g\left(\int_{0}^{t}\\!\cdot\,dt\right)f(t)=\mathscr{L}^{-1}g\left(\frac{1}{s}\right)\mathscr{L}f(t),\quad{t}\in\overline{\mathbb{R}_{+}^{1}}.$
(69)
Therefore, using properties of the Laplace transform and the abstract operator
fundamental theorem, we respectively obtain the abstract operators taking
$\int^{t}_{0}\\!\cdot\,{t}dt$ and $\int^{t}_{0}\\!\cdot\,dt$ as the operator
elements. This type of formulas can also be applied in solving certain
integral equations.
Theorem 7. Let $m\in\mathbb{N}_{+}$, $P(\partial_{x})$ be arbitrary order
partial differential equations for n-dimensional space. Then we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle\left(\frac{\partial^{2}}{\partial{t^{2}}}-P(\partial_{x})\right)^{m}u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{j}u}{\partial{t^{j}}}\right|_{t=0}=\varphi_{j}(x),&j=0,1,2,\ldots,2m-1.\end{array}\right.$
(70) $\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}\int^{t-\tau}_{0}\frac{\left[(t-\tau)^{2}-\tau^{\prime
2}\right]^{m-2}}{(2m-2)!!\,(2m-4)!!}\,\frac{\sinh\left(\tau^{\prime}P(\partial_{x})^{1/2}\right)}{P(\partial_{x})^{1/2}}\,f(x,\tau)\,\tau^{\prime}d\tau^{\prime}\,d\tau$
(71)
$\displaystyle+\,\sum^{m-1}_{k=0}(-1)^{k}{m\choose{k}}P(\partial_{x})^{k}\sum^{2m-1-2k}_{j=0}\frac{\partial^{2m-1-2k-j}}{\partial
t^{2m-1-2k-j}}\int^{t}_{0}\frac{(t^{2}-\tau^{2})^{m-2}\tau}{(2m-2)!!\,(2m-4)!!}$
$\displaystyle\times\,\frac{\sinh\left(\tau
P(\partial_{x})^{1/2}\right)}{P(\partial_{x})^{1/2}}\,\varphi_{j}(x)\,d\tau.$
Proof. The Laplace transform of the partial differential equation, with
respect to $t$ and considering the initial condition, is
$\sum^{m}_{k=0}(-1)^{k}{m\choose{k}}P(\partial_{x})^{k}\left(s^{2m-2k}U(x,s)-\sum^{2m-1-2k}_{j=0}s^{2m-1-2k-j}\varphi_{j}(x)\right)=F(x,s).$
Where $U(x,s)=\mathscr{L}u(x,t),\;F(x,s)=\mathscr{L}f(x,t)$. Let
$G_{m}(\partial_{x},t)=\mathscr{L}^{-1}[1/(s^{2}-P(\partial_{x}))^{m}]$,
solving $U(x,s)$ and its inverse Laplace transform is
$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\mathscr{L}^{-1}U(x,s)=\mathscr{L}^{-1}\frac{1}{(s^{2}-P(\partial_{x}))^{m}}F(x,s)$
$\displaystyle+\,\mathscr{L}^{-1}\sum^{m-1}_{k=0}(-1)^{k}{m\choose{k}}P(\partial_{x})^{k}\sum^{2m-1-2k}_{j=0}\frac{s^{2m-1-2k-j}}{(s^{2}-P(\partial_{x}))^{m}}\varphi_{j}(x)$
$\displaystyle=$ $\displaystyle G_{m}(\partial_{x},t)\ast f(x,t)$
$\displaystyle+\,\sum^{m-1}_{k=0}(-1)^{k}{m\choose{k}}P(\partial_{x})^{k}\sum^{2m-1-2k}_{j=0}\frac{\partial^{2m-1-2k-j}}{\partial
t^{2m-1-2k-j}}G_{m}(\partial_{x},t)\varphi_{j}(x).$
Now let us solve $G_{m}(\partial_{x},t)$. In Corollary 7, taking
$g(\xi)=\xi^{m-1},\;\xi\in\mathbb{R}_{1}$ as the symbol of the abstract
operator, and let $f(t)=\sin{bt},\;t\in\mathbb{R}^{1}$, we have
$\left(\int^{t}_{0}\cdot\,tdt\right)^{m-1}\\!\\!\sin{bt}=\mathscr{L}^{-1}\left(-\frac{1}{s}\frac{\partial}{\partial{s}}\right)^{m-1}\\!\\!\frac{b}{s^{2}+b^{2}}=\mathscr{L}^{-1}\frac{2^{m-1}(m-1)!}{(s^{2}+b^{2})^{m}}b.$
Let $b=iP(\xi)^{1/2},\;\xi\in\mathbb{R}_{n}$, we have
$\mathscr{L}^{-1}\frac{1}{(s^{2}-P(\xi))^{m}}=\frac{1}{(2m-2)!!}\left(\int^{t}_{0}\cdot\,tdt\right)^{m-1}\frac{\sinh\left(tP(\xi)^{1/2}\right)}{P(\xi)^{1/2}}.$
Taking this one as the characteristic equation, according to the analytic
continuous fundamental theorem, we have
$G_{m}(\partial_{x},t)=\mathscr{L}^{-1}\frac{1}{(s^{2}-P(\partial_{x}))^{m}}=\frac{1}{(2m-2)!!}\left(\int^{t}_{0}\cdot\,tdt\right)^{m-1}\frac{\sinh\left(tP(\partial_{x})^{1/2}\right)}{P(\partial_{x})^{1/2}}.$
(72)
By using (8) in reference [3], we can easily derive the following integral
formula
$\left(\int^{x}_{a}\cdot\,xdx\right)^{m}f(x)=\underbrace{\int^{x}_{a}xdx\cdots}_{m}\int^{x}_{a}f(x)\,xdx=\int^{x}_{a}\frac{(x^{2}-\xi^{2})^{m-1}}{(2m-2)!!}f(\xi)\,\xi
d\xi.$ (73)
Applying (73) to (72), we have the expression of abstract operator
$G_{m}(\partial_{x},t)$:
$G_{m}(\partial_{x},t)g(x)=\int^{t}_{0}\frac{(t^{2}-\tau^{\prime
2})^{m-2}}{(2m-2)!!\,(2m-4)!!}\,\frac{\sinh\left(\tau^{\prime}P(\partial_{x})^{1/2}\right)}{P(\partial_{x})^{1/2}}g(x)\,\tau^{\prime}d\tau^{\prime}.$
(74)
Thus Theorem 7 is proved.
In reference [2] the Guangqing Bi has obtained the following results:
Theorem BI3. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real or complex
numbers different from each other, $P(\partial_{x})$ be a partial differential
operator of any order, then we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle\prod^{m}_{i=1}(\frac{\partial}{\partial{t}}-a_{i}P(\partial_{x}))u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{j}u}{\partial{t^{j}}}\right|_{t=0}=0,&j=0,1,2,\ldots,m-1.\end{array}\right.$
(75)
$u(x,t)=\int^{t}_{0}\int^{t-\tau}_{0}\frac{(t-\tau-\tau^{\prime})^{m-2}}{(m-2)!}\sum^{m}_{j=1}\frac{a_{j}^{m-1}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}-a_{i})}e^{\tau^{\prime}a_{j}P(\partial_{x})}f(x,\tau)\,d\tau^{\prime}d\tau.$
(76)
Theorem BI4. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real or complex
numbers different from each other, $P(\partial_{x})$ be a partial differential
operator of any order, then we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle\prod^{m}_{i=1}(\frac{\partial^{2}}{\partial{t^{2}}}-a_{i}^{2}P(\partial_{x}))u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{j}u}{\partial{t^{j}}}\right|_{t=0}=0,&j=0,1,2,\ldots,2m-1.\end{array}\right.$
(77)
$u(x,t)=\int^{t}_{0}\int^{t-\tau}_{0}\frac{(t-\tau-\tau^{\prime})^{2m-3}}{(2m-3)!}\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}\frac{\sinh(\tau^{\prime}a_{j}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}f(x,\tau)\,d\tau^{\prime}d\tau.$
(78)
On this basis, by using the abstract operators and Laplace transform we have
obtained the following results:
Theorem 8. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real or complex roots
different from each other for
$b_{0}+b_{1}\chi+b_{2}\chi^{2}+\cdots+b_{m}\chi^{m}=0$, and
$P(\partial_{x},\partial_{t})$ be a partial differential operator defined by
$P(\partial_{x},\partial_{t})=\sum^{m}_{k=0}b_{k}P(\partial_{x})^{m-k}\frac{\partial^{k}}{\partial{t^{k}}},\quad{x}\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+}.$
Where $P(\partial_{x})$ is a partial differential operator of any order. Then
we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle
P(\partial_{x},\partial_{t})u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{r}u}{\partial{t^{r}}}\right|_{t=0}=\varphi_{r}(x),&r=0,1,2,\ldots,m-1.\end{array}\right.$
(79) $\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}\int^{t-\tau}_{0}\frac{(t-\tau-\tau^{\prime})^{m-2}}{(m-2)!}\sum^{m}_{j=1}\frac{a_{j}^{m-1}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}-a_{i})}\,e^{\tau^{\prime}a_{j}P(\partial_{x})}f(x,\tau)\,d\tau^{\prime}d\tau$
(80)
$\displaystyle+\,\sum^{m}_{k=1}b_{k}P(\partial_{x})^{m-k}\sum^{k-1}_{r=0}\frac{\partial^{k-1-r}}{\partial
t^{k-1-r}}\int^{t}_{0}\frac{(t-\tau)^{m-2}}{(m-2)!}$
$\displaystyle\times\,\sum^{m}_{j=1}\frac{a_{j}^{m-1}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}-a_{i})}\,e^{\tau{a_{j}}P(\partial_{x})}\,\varphi_{r}(x)\,d\tau.$
Theorem 9. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real or complex roots
different from each other, satisfy
$\sum^{m}_{k=0}b_{2k}\chi^{2k}=\prod^{m}_{i=1}(\chi^{2}-a_{i}^{2})$, and
$P(\partial_{x},\partial_{t})$ be a partial differential operators defined by
$P(\partial_{x},\partial_{t})=\sum^{m}_{k=0}b_{2k}P(\partial_{x})^{m-k}\frac{\partial^{2k}}{\partial{t^{2k}}},\quad{x}\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+}.$
Where $P(\partial_{x})$ be a partial differential operator of any order, then
we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle
P(\partial_{x},\partial_{t})u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{r}u}{\partial{t^{r}}}\right|_{t=0}=\varphi_{r}(x),&r=0,1,2,\ldots,2m-1.\end{array}\right.$
(81) $\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}\int^{t-\tau}_{0}\frac{(t-\tau-\tau^{\prime})^{2m-3}}{(2m-3)!}\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}\frac{\sinh(\tau^{\prime}a_{j}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}f(x,\tau)\,d\tau^{\prime}d\tau$
(82)
$\displaystyle+\,\sum^{m}_{k=1}b_{2k}P(\partial_{x})^{m-k}\sum^{2k-1}_{r=0}\frac{\partial^{2k-1-r}}{\partial
t^{2k-1-r}}\int^{t}_{0}\frac{(t-\tau)^{2m-3}}{(2m-3)!}$
$\displaystyle\times\,\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}\frac{\sinh(\tau{a_{j}}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}\,\varphi_{r}(x)\,d\tau.$
Let us prove these two theorems. According to the Theorem BI3 and Theorem BI4,
we just need to prove the following Corollary of Theorem 8 and Theorem 9:
Corollary 8. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real or complex roots
different from each other for
$b_{0}+b_{1}\chi+b_{2}\chi^{2}+\cdots+b_{m}\chi^{m}=0$, and
$P(\partial_{x},\partial_{t})$ be a partial differential operators defined by
$P(\partial_{x},\partial_{t})=\sum^{m}_{k=0}b_{k}P(\partial_{x})^{m-k}\frac{\partial^{k}}{\partial{t^{k}}},\quad{x}\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+}.$
Where $P(\partial_{x})$ be a partial differential operator of any order, then
we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle
P(\partial_{x},\partial_{t})u=0,&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{r}u}{\partial{t^{r}}}\right|_{t=0}=\varphi_{r}(x),&r=0,1,2,\ldots,m-1.\end{array}\right.$
(83) $\displaystyle
u(x,t)=\sum^{m}_{k=1}b_{k}P(\partial_{x})^{m-k}\sum^{k-1}_{r=0}\frac{\partial^{k-1-r}}{\partial
t^{k-1-r}}\int^{t}_{0}\frac{(t-\tau)^{m-2}}{(m-2)!}\sum^{m}_{j=1}\frac{a_{j}^{m-1}e^{\tau{a_{j}}P(\partial_{x})}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}-a_{i})}\,\varphi_{r}(x)\,d\tau.$ (84)
Proof. Considering initial conditions, the Laplace transform of the Eq (83)
with respect to $t$ is
$\sum^{m}_{k=0}b_{k}P(\partial_{x})^{m-k}\left(s^{k}U(x,s)-\sum^{k-1}_{r=0}s^{k-1-r}\varphi_{r}(x)\right)=0.$
Where $U(x,s)=\mathscr{L}u(x,t)$, considering
$\prod^{m}_{i=1}(s-a_{i}P(\partial_{x}))=\sum^{m}_{k=0}b_{k}s^{k}P(\partial_{x})^{m-k}$
we have
$\prod^{m}_{i=1}(s-a_{i}P(\partial_{x}))U(x,s)-\sum^{m}_{k=1}b_{k}P(\partial_{x})^{m-k}\sum^{k-1}_{r=0}s^{k-1-r}\varphi_{r}(x)=0.$
We need to introduce an abstract operators $G_{m}(\partial_{x},t)$, defined as
$G_{m}(\partial_{x},t)=\mathscr{L}^{-1}\frac{1}{\prod^{m}_{i=1}(s-a_{i}P(\partial_{x}))}.$
By solving $U(x,s)$, we have its inverse Laplace transform:
$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\mathscr{L}^{-1}U(x,s)=\sum^{m}_{k=1}b_{k}P(\partial_{x})^{m-k}\sum^{k-1}_{r=0}\mathscr{L}^{-1}\frac{s^{k-1-r}}{\prod^{m}_{i=1}(s-a_{i}P(\partial_{x}))}\,\varphi_{r}(x)$
(85) $\displaystyle=$
$\displaystyle\sum^{m}_{k=1}b_{k}P(\partial_{x})^{m-k}\sum^{k-1}_{r=0}\frac{\partial^{k-1-r}}{\partial{t^{k-1-r}}}\,G_{m}(\partial_{x},t)\varphi_{r}(x).$
Now let us solve $G_{m}(\partial_{x},t)$. Considering initial conditions, the
Laplace transform of the Eq (75) with respect to $t$ is
$\prod^{m}_{i=1}(s-a_{i}P(\partial_{x}))U(x,s)=F(x,s),$
where $F(x,s)=\mathscr{L}f(x,t)$. By solving $U(x,s)$ and using the
convolution theorem, we have its inverse Laplace transform:
$u(x,t)=\mathscr{L}^{-1}U(x,s)=\mathscr{L}^{-1}\frac{1}{\prod^{m}_{i=1}(s-a_{i}P(\partial_{x}))}F(x,s)=G_{m}(\partial_{x},t)*f(x,t).$
By comparing (76) with $u(x,t)=G_{m}(\partial_{x},t)*f(x,t)$, we have the
expression of the abstract operators $G_{m}(\partial_{x},t)$:
$G_{m}(\partial_{x},t)=\int^{t}_{0}\frac{(t-\tau)^{m-2}}{(m-2)!}\sum^{m}_{j=1}\frac{a_{j}^{m-1}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}-a_{i})}e^{\tau{a_{j}}P(\partial_{x})}d\tau.$ (86)
Applying (86) to (85), thus the Corollary 8 is proved.
Corollary 9. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real or complex roots
different from each other, which satisfy
$\sum^{m}_{k=0}b_{2k}\chi^{2k}=\prod^{m}_{i=1}(\chi^{2}-a_{i}^{2})$, and
$P(\partial_{x},\partial_{t})$ be a partial differential operator defined by
$P(\partial_{x},\partial_{t})=\sum^{m}_{k=0}b_{2k}P(\partial_{x})^{m-k}\frac{\partial^{2k}}{\partial{t^{2k}}},\quad{x}\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+}.$
Where $P(\partial_{x})$ is a partial differential operator of any order, then
we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle
P(\partial_{x},\partial_{t})u=0,&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{r}u}{\partial{t^{r}}}\right|_{t=0}=\varphi_{r}(x),&r=0,1,2,\ldots,2m-1.\end{array}\right.$
(87) $\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\sum^{m}_{k=1}b_{2k}P(\partial_{x})^{m-k}\sum^{2k-1}_{r=0}\frac{\partial^{2k-1-r}}{\partial
t^{2k-1-r}}\int^{t}_{0}\frac{(t-\tau)^{2m-3}}{(2m-3)!}$ (88)
$\displaystyle\times\,\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}\frac{\sinh(\tau{a_{j}}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}\,\varphi_{r}(x)\,d\tau.$
Proof. Considering initial conditions, the Laplace transform of the Eq (87)
with respect to $t$ is
$\sum^{m}_{k=0}b_{2k}P(\partial_{x})^{m-k}\left(s^{2k}U(x,s)-\sum^{2k-1}_{r=0}s^{2k-1-r}\varphi_{r}(x)\right)=0.$
Where $U(x,s)=\mathscr{L}u(x,t)$, considering
$\prod^{m}_{i=1}(s^{2}-a_{i}^{2}P(\partial_{x}))=\sum^{m}_{k=0}b_{2k}s^{2k}P(\partial_{x})^{m-k}$
we have
$\prod^{m}_{i=1}(s^{2}-a_{i}^{2}P(\partial_{x}))U(x,s)-\sum^{m}_{k=1}b_{2k}P(\partial_{x})^{m-k}\sum^{2k-1}_{r=0}s^{2k-1-r}\varphi_{r}(x)=0.$
We need to introduce an abstract operators $G_{m}(\partial_{x},t)$, defined as
$G_{m}(\partial_{x},t)=\mathscr{L}^{-1}\frac{1}{\prod^{m}_{i=1}(s^{2}-a_{i}^{2}P(\partial_{x}))}.$
By solving $U(x,s)$, we have its inverse Laplace transform:
$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\mathscr{L}^{-1}U(x,s)=\sum^{m}_{k=1}b_{2k}P(\partial_{x})^{m-k}\sum^{2k-1}_{r=0}\mathscr{L}^{-1}\frac{s^{2k-1-r}}{\prod^{m}_{i=1}(s^{2}-a_{i}^{2}P(\partial_{x}))}\,\varphi_{r}(x)$
(89) $\displaystyle=$
$\displaystyle\sum^{m}_{k=1}b_{2k}P(\partial_{x})^{m-k}\sum^{2k-1}_{r=0}\frac{\partial^{2k-1-r}}{\partial{t^{2k-1-r}}}\,G_{m}(\partial_{x},t)\varphi_{r}(x).$
Now let us solve the $G_{m}(\partial_{x},t)$. Considering initial conditions,
the Laplace transform of the Eq (77) with respect to $t$ is
$\prod^{m}_{i=1}(s^{2}-a_{i}^{2}P(\partial_{x}))U(x,s)=F(x,s),$
where $F(x,s)=\mathscr{L}f(x,t)$. By solving $U(x,s)$ and using the
convolution theorem, we have its inverse Laplace transform:
$u(x,t)=\mathscr{L}^{-1}U(x,s)=\mathscr{L}^{-1}\frac{1}{\prod^{m}_{i=1}(s^{2}-a_{i}^{2}P(\partial_{x}))}F(x,s)=G_{m}(\partial_{x},t)*f(x,t).$
By comparing (78) with $u(x,t)=G_{m}(\partial_{x},t)*f(x,t)$, we have the
expression of the abstract operators $G_{m}(\partial_{x},t)$:
$G_{m}(\partial_{x},t)=\int^{t}_{0}\frac{(t-\tau)^{2m-3}}{(2m-3)!}\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}\frac{\sinh(\tau{a_{j}}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}\,d\tau.$
(90)
Applying (90) to (89), thus the Corollary 9 is proved.
### 3.2 Analytic solution of Cauchy problem
Theorem 10. Let
$\Delta_{n}=\frac{\partial^{2}}{\partial{x_{1}}^{2}}+\frac{\partial^{2}}{\partial{x_{2}}^{2}}+\cdots+\frac{\partial^{2}}{\partial{x_{n}}^{2}}$
be an n-dimensional Laplacian. If $n-2=2\nu+1,\;\nu\in\mathbb{N}$, then
$\forall{f(x,\tau)}\in{C^{2\nu}}(\mathbb{R}^{n})$, and
$\forall\varphi_{j}(x)\in{C}^{2m+2\nu-2}(\mathbb{R}^{n})$, we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle\left(\frac{\partial^{2}}{\partial{t^{2}}}-a^{2}\Delta_{n}\right)^{m}u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{j}u}{\partial{t^{j}}}\right|_{t=0}=\varphi_{j}(x),&j=0,1,2,\ldots,2m-1.\end{array}\right.$
(91) $\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}d\tau\int^{t-\tau}_{0}d\tau^{\prime}\frac{\left[(t-\tau)^{2}-\tau^{\prime
2}\right]^{m-2}\tau^{\prime 2}}{(2m-2)!!\,(2m-4)!!}$ (92)
$\displaystyle\times\underbrace{\int^{\tau^{\prime}}_{0}\tau^{\prime}d\tau^{\prime}\cdots}_{\nu}\int^{\tau^{\prime}}_{0}\frac{(a^{2}\Delta_{n})^{\nu}}{S^{\prime}_{n}}\int_{S^{\prime}_{n}}f(\xi^{\prime},\tau)\,dS^{\prime}_{n}\,\tau^{\prime}d\tau^{\prime}$
$\displaystyle+\,\frac{1}{(2m-2)!!}\sum^{\nu-1}_{r=0}\int^{t}_{0}\frac{(t-\tau)^{2m+2r-1}}{(2m+2r-1)!!\,(2r)!!}(a^{2}\Delta_{n})^{r}f(x,\tau)\,d\tau$
$\displaystyle+\,\sum^{m-1}_{k=0}(-1)^{k}{m\choose{k}}(a^{2}\Delta_{n})^{k+\nu}\sum^{2m-1-2k}_{j=0}\frac{\partial^{2m-1-2k-j}}{\partial{t}^{2m-1-2k-j}}\int^{t}_{0}d\tau\frac{(t^{2}-\tau^{2})^{m-2}\tau^{2}}{(2m-2)!!\,(2m-4)!!}$
$\displaystyle\times\underbrace{\int^{\tau}_{0}\tau{d}\tau\cdots}_{\nu}\int^{\tau}_{0}\frac{1}{S_{n}}\int_{S_{n}}\varphi_{j}(\xi)\,dS_{n}\,\tau{d}\tau+\sum^{m-1}_{k=0}(-1)^{k}{m\choose{k}}\sum^{2m-1-2k}_{j=0}$
$\displaystyle\times\sum^{\nu-1}_{i=0}{m-1+i\choose{i}}\frac{t^{2k+2i+j}}{(2k+2i+j)!}(a^{2}\Delta_{n})^{k+i}\varphi_{j}(x).$
Where $n-2=2\nu+1$, $S^{\prime}_{n}=2(2\pi)^{\nu+1}(a\tau^{\prime})^{n-1}$,
$S_{n}=2(2\pi)^{\nu+1}(a\tau)^{n-1}$, and $\xi^{\prime}\in\mathbb{R}_{n}$ is
the integral variable. The integral is on the hypersphere
$(\xi^{\prime}_{1}-x_{1})^{2}+(\xi^{\prime}_{2}-x_{2})^{2}+\cdots+(\xi^{\prime}_{n}-x_{n})^{2}=(a\tau^{\prime})^{2}$,
and $dS^{\prime}_{n}$ is its surface element. $\xi\in\mathbb{R}_{n}$ is the
integral variable on the hypersphere
$(\xi_{1}-x_{1})^{2}+(\xi_{2}-x_{2})^{2}+\cdots+(\xi_{n}-x_{n})^{2}=(a\tau)^{2}$,
and $dS_{n}$ is its surface element.
Proof. According to (64), we can easily derive:
$\displaystyle\frac{\sinh\left(at\Delta_{n}{}^{1/2}\right)}{a\Delta_{n}{}^{1/2}}f(x)$
$\displaystyle=$ $\displaystyle
t\underbrace{\int^{t}_{0}tdt\cdots}_{\nu}\int^{t}_{0}\frac{(a^{2}\Delta_{n})^{\nu}}{S_{n}}\int_{S_{n}}f(\xi)\,dS_{n}\,tdt$
(93)
$\displaystyle+\,\sum^{\nu-1}_{i=0}\frac{t^{2i+1}}{(2i+1)!}(a^{2}\Delta_{n})^{i}f(x),\quad\forall{f(x)}\in{C}^{2\nu}(\mathbb{R}^{n}).$
Where $\Delta_{n}$ is an n-dimensional Laplacian, $n-2=2\nu+1$,
$S_{n}=2(2\pi)^{\nu+1}(at)^{n-1}$. $\xi\in\mathbb{R}_{n}$ is the integral
variable on the hypersphere
$(\xi_{1}-x_{1})^{2}+(\xi_{2}-x_{2})^{2}+\cdots+(\xi_{n}-x_{n})^{2}=(at)^{2}$,
and $dS_{n}$ is its surface element.
In Theorem 7, let $P(\partial_{x})=a^{2}\Delta_{n}$, then Theorem 10 is proved
by the substitution of (93).
Theorem 11. Let $a_{1},a_{2},\ldots,a_{m}$ be arbitrary real roots different
from each other, which satisfy
$\sum^{m}_{k=0}b_{2k}\chi^{2k}=\prod^{m}_{i=1}(\chi^{2}-a_{i}^{2})$, and
$P(\partial_{x},\partial_{t})$ be a partial differential operator defined by
$P(\partial_{x},\partial_{t})=\sum^{m}_{k=0}b_{2k}\Delta_{n}^{m-k}\frac{\partial^{2k}}{\partial{t^{2k}}},\quad{x}\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+}.$
Where $\Delta_{n}$ is an n-dimensional Laplacian. If
$n-2=2\nu+1,\;\nu=0,1,2,\cdots$, then
$\forall\varphi_{r}(x)\in{C^{2m+2\nu-2}}(\mathbb{R}^{n})$, and
$\forall{f(x,\tau)}\in{C^{2\nu}}(\mathbb{R}^{n})$, we have
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle
P(\partial_{x},\partial_{t})u=f(x,t),&x\in\mathbb{R}^{n},\;t\in\overline{\mathbb{R}_{+}^{1}},\;\forall{m}\in\mathbb{N}_{+},\\\
\displaystyle\left.\frac{\partial^{r}u}{\partial{t^{r}}}\right|_{t=0}=\varphi_{r}(x),&r=0,1,2,\ldots,2m-1.\end{array}\right.$
(94) $\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}d\tau\int^{t-\tau}_{0}d\tau^{\prime}\frac{(t-\tau-\tau^{\prime})^{2m-3}}{(2m-3)!}\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}$ (95)
$\displaystyle\times\left(\tau^{\prime}\underbrace{\int^{\tau^{\prime}}_{0}\tau^{\prime}d\tau^{\prime}\cdots}_{\nu}\int^{\tau^{\prime}}_{0}\frac{(a_{j}^{2}\Delta_{n})^{\nu}}{S^{\prime}_{n,j}}\int_{S^{\prime}_{n,j}}f(\xi^{\prime},\tau)\,dS^{\prime}_{n,j}\,\tau^{\prime}d\tau^{\prime}+\sum^{\nu-1}_{l=0}\frac{a_{j}^{2l}\tau^{\prime
2l+1}}{(2l+1)!}\Delta_{n}^{l}f(x,\tau)\right)$
$\displaystyle+\,\sum^{m}_{k=1}b_{2k}\Delta_{n}^{m-k}\sum^{2k-1}_{r=0}\frac{\partial^{2k-1-r}}{\partial
t^{2k-1-r}}\int^{t}_{0}d\tau\frac{(t-\tau)^{2m-3}}{(2m-3)!}\sum^{m}_{j=1}\frac{a_{j}^{2m-2}}{\prod^{m}_{i=1\atop
i\neq{j}}(a_{j}^{2}-a_{i}^{2})}$
$\displaystyle\times\left(\tau\underbrace{\int^{\tau}_{0}\tau{d\tau}\cdots}_{\nu}\int^{\tau}_{0}\frac{(a_{j}^{2}\Delta_{n})^{\nu}}{S_{n,j}}\int_{S_{n,j}}\varphi_{r}(\xi)\,dS_{n,j}\,\tau{d\tau}+\sum^{\nu-1}_{l=0}\frac{a_{j}^{2l}\tau^{2l+1}}{(2l+1)!}\Delta_{n}^{l}\varphi_{r}(x)\right).$
Where $S^{\prime}_{n,j}=2(2\pi)^{\nu+1}(a_{j}\tau^{\prime})^{n-1}$,
$S_{n,j}=2(2\pi)^{\nu+1}(a_{j}\tau)^{n-1}$, and
$\xi^{\prime}\in\mathbb{R}_{n}$ is the integral variable. The integral is on
the hypersphere
$(\xi^{\prime}_{1}-x_{1})^{2}+(\xi^{\prime}_{2}-x_{2})^{2}+\cdots+(\xi^{\prime}_{n}-x_{n})^{2}=(a_{j}\tau^{\prime})^{2}$,
and $dS^{\prime}_{n,j}$ is its surface element. $\xi\in\mathbb{R}_{n}$ is the
integral variable on the hypersphere
$(\xi_{1}-x_{1})^{2}+(\xi_{2}-x_{2})^{2}+\cdots+(\xi_{n}-x_{n})^{2}=(a_{j}\tau)^{2}$,
and $dS_{n,j}$ is its surface element.
Proof. In Theorem 9, let $P(\partial_{x})=\Delta_{n}$, then Theorem 11 is
proved by the substitution of (93).
Similarly, we can easily obtain explicit solutions of the Cauchy problem of
more complex partial differential equations. Therefore, using the method of
abstract operators, we have established a general theory of initial value
problems for linear higher-order partial differential equations.
### 3.3 Initial-boundary value problem and Hilbert space
Clearly, similar to $\sinh(h\partial_{x})\;\mbox{and}\;\cosh(h\partial_{x})$,
the abstract operators
$\frac{\sinh\left(at\Delta_{n}{}^{1/2}\right)}{a\Delta_{n}{}^{1/2}}\quad\mbox{and}\quad\cos\left(at\Delta_{n}{}^{1/2}\right)=\frac{\partial}{\partial{t}}\,\frac{\sinh\left(at\Delta_{n}{}^{1/2}\right)}{a\Delta_{n}{}^{1/2}}$
on the left side of (93) are bounded operators in a Hilbert space, and can act
on the whole Hilbert space. Thus we can attach proper boundary conditions to
the initial value problem introduced by Theorem 7, which makes the given
functions $f(x,t),\;\varphi_{j}(x)$ in Eq (91) and (94) become functions with
boundary conditions. Then solving this initial-boundary value problem comes
down to expressing $f(x,t),\;\varphi_{j}(x)$ in a Hilbert space $H$ within the
given domain $\Omega$.
For the initial-boundary value problem, the operator $P(\partial_{x})$ must
have the characteristic function related to boundary conditions, in order to
expand the known function $f(x,\tau),\;\varphi_{r}(x)$ in (71), (80) and (82)
by using the characteristic function of $P(\partial_{x})$. $P(\partial_{x})$
in Theorem 7, Theorem 8 and Theorem 9 can be variable-coefficient partial
differential operators. For instance, if $P(\partial_{x})$ is a self-adjoint
operator defined on a Hilbert space $H$, then the abstract operators
$\frac{\sinh(tP(\partial_{x})^{1/2})}{P(\partial_{x})^{1/2}}\quad\mbox{and}\quad\cosh(tP(\partial_{x})^{1/2})=\frac{\partial}{\partial{t}}\,\frac{\sinh(tP(\partial_{x})^{1/2})}{P(\partial_{x})^{1/2}}$
are the continuous operators on the Hilbert space. In this case, we can attach
proper boundary conditions to the initial value problems in (70), (79) and
(81). Therefore, the given function $f(x,t),\varphi_{r}(x)$ becomes a function
with boundary conditions, and can be expressed in the Hilbert space $H$ within
the given domain $\Omega$. In order to solve the corresponding initial-
boundary value problem, we need to solve the characteristic value problem of
$P(\partial_{x})$ under given boundary conditions to determine a set of
orthogonal functions, which generates a linear manifold of the Hilbert space,
thus $f(x,t),\varphi_{r}(x)$ can be expressed in the Hilbert space.
In (71), (80) and (82), if $f(x,\tau),\varphi_{r}(x)\in{L^{2}}(\Omega)$, and
$P(\partial_{x})$ is a second-order linear self-adjoint elliptic operator,
namely
$P(\partial_{x})u=\sum^{n}_{i,j=1}\frac{\partial}{\partial{x}_{j}}\left(a_{ij}(x)\frac{\partial{u}}{\partial{x}_{i}}\right)+c(x)u,\quad{x}\in\Omega\subset\mathbb{R}^{n},$
(96)
then boundary conditions can be added for the definite solution problems (70),
(79) and (81): $\overline{B}u|_{\partial\Omega}=0$ representing
$u|_{\partial\Omega}=0$ or
$\left[\sum^{n}_{i,j=1}a_{ij}(x)\frac{\partial{u}}{\partial{x}_{j}}\cos\langle\mathbf{a},x_{i}\rangle+b(x)u\right]_{\partial\Omega}=0.$
Where $\mathbf{a}$ is the unit outward normal of $\partial\Omega$. Thus this
kind of initial boundary value problems boils down to solving the
characteristic value problem of the first boundary value problem of second-
order linear self-adjoint elliptic operator:
$\left\\{\begin{array}[]{l@{\qquad}l}\displaystyle\sum^{n}_{i,j=1}\frac{\partial}{\partial{x}_{j}}\left(a_{ij}(x)\frac{\partial{u}}{\partial{x}_{i}}\right)+c(x)u=-\lambda{u},\quad{x}\in\Omega\subset\mathbb{R}^{n},\\\
\displaystyle\overline{B}u|_{\partial\Omega}=0\end{array}\right.$ (97)
Reference [7] quotes the following theorem:
Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open domain, and
$\partial\Omega$ be smooth. Let $a_{ij}=a_{ji}$, and there exists $\theta>0$
such that
$\sum^{n}_{i,j=1}a_{ij}(x)\xi_{i}\xi_{j}\geq\theta|\xi|^{2},\quad{x\in\Omega}.$
And let
$a_{ij}\in{C}^{1}(\overline{\Omega}),\;c(x)\in{C}(\overline{\Omega}),\;b(x)\in{C}(\partial\Omega)$,
then (97) has the following countable characteristic values:
$0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{\nu}\leq\cdots,\quad\lim_{\nu\rightarrow\infty}\lambda_{\nu}=\infty$
(If $(a_{ij})=I$ is a unit matrix, then $\lambda_{1}=0$ when $b(x)=c(x)=0$.
When $b(x)\geq 0,\,c(x)\geq 0$ and one of them does not identically equal to
zero, $\lambda_{1}>0$) and the corresponding characteristic functions
$e_{1}(x),e_{2}(x),\cdots,e_{\nu}(x),\cdots,$ satisfy
$\sum^{n}_{i,j=1}\frac{\partial}{\partial{x}_{j}}\left(a_{ij}(x)\frac{\partial{e_{i}}}{\partial{x}_{i}}\right)+c(x)e_{i}=-\lambda_{i}{e_{i}},\quad(e_{i},\,e_{j})=\delta_{ij}$
(98)
and $\\{e_{j}(x)\\}^{\infty}_{j=1}$ is complete in $L^{2}(\Omega)$, that is
for an arbitrary $f(x)\in{L^{2}}(\Omega)$, there exists $c_{j}$ such that
$\lim_{m\rightarrow\infty}\|f-\sum^{m}_{j=1}c_{j}e_{j}\|_{L^{2}(\Omega)}=0.$
Therefore, for (71), (80) and (82), by
$f(x,\tau),\varphi_{r}(x)\in{L^{2}}(\Omega)$ we also have
$\lim_{m\rightarrow\infty}\|f(x,\tau)-\sum^{m}_{j=1}c_{j}(\tau)e_{j}(x)\|_{L^{2}(\Omega)}=0.$
$\lim_{m\rightarrow\infty}\|\varphi_{r}(x)-\sum^{m}_{j=1}c_{j}e_{j}(x)\|_{L^{2}(\Omega)}=0.$
Clearly, based on (96) and (98), we have the extension of abstract operators
in Hilbert space:
$e^{\tau{a_{j}}P(\partial_{x})}e_{i}(x)=e^{-\tau{a_{j}}\lambda_{i}}e_{i}(x),\quad\frac{\sinh(\tau{a_{j}}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}e_{i}(x)=\frac{\sin(a_{j}\sqrt{\lambda_{i}}\,\tau)}{a_{j}\sqrt{\lambda_{i}}}e_{i}(x).$
(99)
Where $x\in\Omega\subset\mathbb{R}^{n},\;j=1,2,\cdots,m$. Then we have
$\lim_{m\rightarrow\infty}\|e^{\tau^{\prime}{a_{j}}P(\partial_{x})}f(x,\tau)-\sum^{m}_{j=1}(f,e_{j})e^{-\tau^{\prime}{a_{j}}\lambda_{i}}e_{j}(x)\|_{L^{2}(\Omega)}=0.$
$\lim_{m\rightarrow\infty}\|e^{\tau{a_{j}}P(\partial_{x})}\varphi_{r}(x)-\sum^{m}_{j=1}(\varphi_{r},e_{j})e^{-\tau{a_{j}}\lambda_{i}}e_{j}(x)\|_{L^{2}(\Omega)}=0.$
$\lim_{m\rightarrow\infty}\left\|\frac{\sinh(\tau^{\prime}a_{j}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}f(x,\tau)-\sum^{m}_{j=1}(f,e_{j})\frac{\sin(a_{j}\sqrt{\lambda_{i}}\,\tau^{\prime})}{a_{j}\sqrt{\lambda_{i}}}e_{j}(x)\right\|_{L^{2}(\Omega)}=0.$
$\lim_{m\rightarrow\infty}\left\|\frac{\sinh(\tau{a_{j}}P(\partial_{x})^{1/2})}{a_{j}P(\partial_{x})^{1/2}}\varphi_{r}(x)-\sum^{m}_{j=1}(\varphi_{r},e_{j})\frac{\sin(a_{j}\sqrt{\lambda_{i}}\,\tau)}{a_{j}\sqrt{\lambda_{i}}}e_{j}(x)\right\|_{L^{2}(\Omega)}=0.$
Thus when (70), (79) and (81) satisfy the boundary condition
$\overline{B}u|_{\partial\Omega}=0$, if the orthogonal function system
$\\{e_{j}(x)\\}^{\infty}_{j=1}$ is known, we have the solution of the
corresponding initial boundary value problem.
In conclusion, based on the method of abstract operators and comprehensively
applying the Laplace transform and Hilbert space theory, we establish a
general theory of initial value and initial boundary value problems of linear
partial differential equations, and illustrate broad application prospects of
the abstract operator theory, which will definitely have a wide and profound
influence on various scientific fields.
## References
* [1] G.Q. Bi, Applications of abstract operator in partial differential equation(i), _Pure and Applied Mathematics_. 13(1997), 7-14 (in Chinese)
* [2] G.Q. Bi, Applications of abstract operator in partial differential equation(ii), _Chinese Quarterly Journal of Mathematic_. 14(1999), 80-87
* [3] G.Q. Bi, Operator methods in high order partial differential equation, _Chinese Quarterly Journal of Mathematics_. 16(2001), 88-101
* [4] S.X. Chen, _Pseudodifferential Operators_ , (Second Edition), Higher Education Press, Beijing, China, 2006, pp. 8-25 (in Chinese)
* [5] G.Q. Bi and Y.K. Bi, New properties of Fourier series and Riemann Zeta function, 2010. arXiv:1008.5046
* [6] D.F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers, Volume I, 2007\. arXiv:0710.4022
* [7] S. Wang, _Sobolev Space and Introduction Partial Differential Equations_ , Science Press, Beijing, China, 2009, pp. 156-157 (in Chinese)
|
arxiv-papers
| 2010-08-23T12:37:07 |
2024-09-04T02:49:12.376101
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Guang-Qing Bi, Yue-Kai Bi",
"submitter": "Yuekai Bi",
"url": "https://arxiv.org/abs/1008.3808"
}
|
1008.3856
|
# Electric field-dependent dynamic polarizability and state-insensitive
conditions for optical trapping of diatomic polar molecules
Svetlana Kotochigova1∗ and David DeMille2 1Department of Physics, Temple
University, Philadelphia, PA 19122-6082, USA
2Department of Physics, Yale University, New Haven, CT 06520, USA
###### Abstract
Selection of state-insensitive or “magic” trapping conditions with ultracold
atoms or molecules, where pairs of internal states experience identical
trapping potentials, brings substantial benefits to precision measurements and
quantum computing schemes. Working at such conditions could ensure that
detrimental effects of inevitable inhomogeneities across an ultracold sample
are significantly reduced. However, this aspect of confinement remains
unexplored for ultracold polar molecules. Here, we present means to control
the AC Stark shift of rotational states of ultracold diatomic polar molecules,
when subjected to both trapping laser light and an external electric field. We
show that both the strength and relative orientation of the two fields
influence the trapping potential. In particular, we predict “magic electric
field strengths” and a “magic angle”, where the Stark shift is independent of
the DC external field for certain rotational states of the molecule.
The advantage of using state-independent light traps for precision frequency
measurements with ultracold atoms has been demonstrated in several experiments
Katori ; TrapScience08 . Applications of this approach were analyzed in the
context of optical atomic clocks and coherent control of atoms and photons
within an optical cavity. For these applications the frequency of the laser
beam that creates a far-off resonant optical dipole trap is chosen such that
the AC Stark shift of the ground and one excited electronic atomic level are
the same. In this way any optical atomic transition between these levels is
unaffected by the trapping light. In Refs. Flambaum ; Beloy ; Derevianko ;
Lundblad state-insensitive trapping conditions have also been found for
microwave transitions in the atomic ground state by using a combination of the
vector and tensor components of the AC Stark shift and an external magnetic
field.
Ultracold molecular systems possess unique proporties that are considered to
make them potentially useful as tools for precision measurements special_issue
; NJP_review and quantum computing DeMille02 ; Zhao . Therefore it is
desirable to extend the zero-differential AC Stark shift technique to these
more complex systems. The idea of “magic” frequencies for vibrational Raman
transitions in homo-nuclear Sr2 molecules was first explored in Refs.
Zelevinsky ; Kotochigova09 . This molecule is proposed for a search of
possible time variation of the electron-to-proton mass ratio. The AC Stark
shift of a molecule is determined by the dynamic molecular polarizability
$\alpha(\nu)$, which is a function of radiation frequency $\nu$ and its
polarization.
Polar molecules have a permanent dipole moment and their levels can be shifted
and mixed with one another by applying an external electric field. This opens
up a new way to create “magic” trapping conditions for two rotational levels
of the molecule. In the presence of an external electric field, $J$ is not a
good quantum number and all states, even the “rotationless” ground state, have
an anisotropic polarizability Ospelkaus . The anisotropy of the dynamic
polarizability of these levels manifests itself as a dependence on the
relative orientation of the polarization of the trapping laser and the DC
electric field. The combined action of these two fields can be a powerful tool
to manipulate and control ultracold molecules trapped in an optical potential.
The behavior described here has potential applications to several experiments
that have been envisioned for diatomic polar molecules held in optical traps.
The implications are particularly striking for the use of polar molecules in
an optical lattice as quantum bits. As was described in Ref. DeMille02 , a
pair of rotational states forms a suitable quantum bit. However, it was
pointed out that this system is susceptible to decoherence due to intensity
fluctuations in the optical trapping lasers, if the dynamic polarizabilities
of these levels differ as is generally the case. This in turn leads to very
stringent requirements on the laser intensity stability for such a system to
be practical. As we will show, this limitation can be removed by adjusting the
experimental parameters to guarantee that the dynamic polarizabilities of
these states are equal. In particular, for this proposed system (where a
spatial gradient of the electric field $\vec{\mathcal{E}}$ is required, so
working at a magic electric field value is impossible) it should be possible
to use light polarized at the “magic angle” relative to the static field in
order to eliminate this potentially dangerous source of decoherence.
The existence of magic electric field values is also of possible use for
envisioned applications where polar molecules in optical lattices are employed
in novel types of many-body systems. This includes, for example, cases where
the properties of long-range molecular interactions are tailored by a
combination of a static electric field and resonant microwave fields coupling
different rotational states Micheli06 ; Buchler06 ; Buchler07 . Working at a
magic electric field value in such systems could ensure that the inevitable
inhomogeneities in the intensity of the trapping light across a large sample
would not change the resonant condition for the microwave drive fields. Hence,
working under such “magic” conditions might be necessary to implement
proposals of this type.
Motivated by these ideas, we calculate the near-infrared dynamic
polarizability of various rotational levels of the $v=0$ vibrational state of
the X${}^{1}\Sigma^{+}$ potential of the KRb and RbCs molecules, under the
simultaneous influence of trapping electromagnetic and static electric fields.
We calculate dynamic polarizability using the computational techniques
developed in previous publications Kotochigova1 ; Kotochigova2 ; Kotochigova3
. Our calculations with an external electric field are predominantly performed
at a laser frequency of 9174 cm-1 (or wavelength of 1090 nm), which
corresponds to an often-used frequency to trap atoms and molecules in
ultracold experiments. In the near infrared this laser frequency is
sufficiently far away from molecular resonances of the excited electronic
states that heating due to photon scattering is negligible. We focus on
external electric field strengths up to 15 kV/cm, a value which should be
experimentally accessible.
Figure 1: Dynamic polarizability in the absence of an electric field and in
atomic units for the $v$=0, $J$=0 and 1 levels of the X${}^{1}\Sigma^{+}$
ground state of KRb and RbCs as a function of trapping frequency in the near
infrared domain. The dashed lines show the polarizability for the $J$=0 and
$M$ = 0 level, which is independent of the light polarization. The solid lines
correspond to the $J$= 1 state with magnetic sublevels $M$ = 0 and $\pm$ 1
illuminated by linear polarized light along the $\hat{z}$ direction. One
atomic unit of polarizability corresponds to 4.68645$\times 10^{-8}$
MHz/(W/cm2).
We begin by studying the dynamic polarizability or AC Stark shift for the
$J=0$ and $J=1$ rotational levels of the $v=0$ X${}^{1}\Sigma^{+}$ state of
KRb and RbCs in the near infrared, without an external electric field. The
procedure used to determine the complex molecular polarizability has already
been described in our previous paper Kotochigova1 . Figure 1 illustrates the
results of this study in the absence of an electric field by showing the
dynamic polarizability $\alpha$ as a function of laser frequency $h\nu$. The
range of laser frequencies spans a technologically relevant near-infrared
optical domain and includes the lasing frequency near 9174 cm-1 used in Ref.
Ospelkaus . The figure also shows that the dynamic polarizability of the
molecular state depends on the rotational quantum number $J$ and its
projection $M$. The curves for the different states do not cross as a function
of the laser frequency. In other words, at zero electric field we can not find
a “magic” frequency for J=0 and J=1 states using trapping light in the near
infrared.
Field-dependent dynamic polarizability
Here we extend the idea of the AC Stark shift for the rotational levels of the
X${}^{1}\Sigma^{+}$ ground state to the mixing of these levels in a static
electric field $\vec{\mathcal{E}}=\mathcal{E}\hat{z}$ along the space-fixed
$\hat{z}$ direction. Our analyses show that the experimentally accessible
electric fields mix only a few low-lying rotational states. The ultimate goal
is to introduce the dynamic polarizability of the mixed rotational levels as a
function of an external DC electric field and the polarization of the AC
trapping field. In a heteronuclear molecule this Stark mixing is primarily due
to the permanent electronic dipole moment $\vec{d}$; we neglect the much
smaller effects due to Stark mixing with other electronic states. For the
Hund’s case (a) X${}^{1}\Sigma^{+}$ state of an alkali-metal dimer, the total
electron spin and orbital angular momentum are not coupled to the molecular
rotation. The molecular wavefunction in the lab frame, $|vJM\rangle_{z}$, is
then given by
$|vJM\rangle_{z}\equiv|vJ\rangle\times
Y_{JM}(\hat{R})=\left\\{\frac{\psi_{vJ}(R)}{R}|{\rm
X}^{1}\Sigma^{+}\rangle\right\\}\times Y_{JM}(\hat{R})\,\,,$ (1)
where $v$ is a vibrational quantum number, $J$ and $M$ are the molecular
rotational angular momentum and its projection along the $z$ axis,
$\psi_{vJ}(R)$ is the radial rovibrational wavefunction, $Y_{JM}(\hat{R})$ is
a spherical harmonic, $\hat{R}$ is the orientation of the molecule relative to
the electric field direction $z$, $|{\rm X}^{1}\Sigma^{+}\rangle$ is the
electronic wavefunction with projections defined along the internuclear axis.
For the X${}^{1}\Sigma^{+}$ state, in even modest electric fields the nuclear
spins are well-decoupled from the other spins and angular momenta; hence we
ignore the nuclear spins here. Then the Hamiltonian for such a system becomes
$H=\sum_{vJM}E_{vJ}|vJM\rangle_{z}\times{}_{z}\langle
vJM|-\vec{d}\cdot\vec{\mathcal{E}}\,,$ (2)
where $E_{vJ}=G_{v}+B_{v}J(J+1)$ and $G_{v}$ and $B_{v}$ are the vibrational
energy and rotational constant of vibrational level $v$, respectively. Higher
order rotational corrections are negligible. For the alkali-metal dimers KRb
and RbCs we have that $\Delta G_{v}=G_{v+1}-G_{v}$ is on the order of 50-100
cm-1 for small $v$, while $B_{v}$ is on the order of 0.017-0.037 cm-1.
We evaluate the matrix elements of the operator
$-\vec{d}\cdot\vec{\mathcal{E}}$ by noting that after averaging over the
electronic wavefunction $|{\rm X}^{1}\Sigma^{+}\rangle$ it reduces to
$-d(R)C_{10}(\hat{R})\mathcal{E}_{z}$, where $d(R)=\langle{\rm
X}^{1}\Sigma^{+}|d|{\rm X}^{1}\Sigma^{+}\rangle$ is the $R$-dependent
permanent electric dipole moment,
$C_{lm}(\hat{R})=\sqrt{4\pi/(2l+1)}Y_{lm}(\hat{R})$ are tensors of rank $l$,
and $\mathcal{E}_{z}$ is the electric field strength. Consequently, this
operator conserves the projection quantum number $M$. The matrix element
between two rovibrational states is
${}_{z}\langle
vJM|-\vec{d}\cdot\vec{\mathcal{E}}|v^{\prime}J^{\prime}M^{\prime}\rangle_{z}=-\delta_{MM^{\prime}}\,d_{vJ,v^{\prime}J^{\prime}}\,\mathcal{E}_{z}\int
d\hat{R}\,Y_{JM}^{*}(\hat{R})C_{10}(\hat{R})Y_{J^{\prime}M^{\prime}}(\hat{R})$
(8)
$\displaystyle\quad\quad=-\delta_{MM^{\prime}}d_{vJ,v^{\prime}J^{\prime}}\mathcal{E}_{z}(-1)^{M}\sqrt{(2J+1)(2J^{\prime}+1)}\left(\begin{array}[]{ccc}J&1&J^{\prime}\\\
-M&0&M^{\prime}\end{array}\right)\left(\begin{array}[]{ccc}J&1&J^{\prime}\\\
0&0&0\end{array}\right)\,,$
where $\delta_{MM^{\prime}}$ is the Kronecker delta function,
$d_{vJ,v^{\prime}J^{\prime}}=\int_{0}^{\infty}dR\,\psi_{vJ}(R)d(R)\psi_{v^{\prime}J^{\prime}}(R)$,
and $(\cdots)$ are 3-$j$ symbols (see e.g. Brink ). This matrix element is
nonzero when $J+1+J^{\prime}$ is even, according to the parity selection
rules, and is independent of the sign of $M$ and $M^{\prime}$. For the small
$J$ values of interest here we can assume that the $J$ dependence of
$\psi_{vJ}(R)$ is negligible. Moreover, coupling between vibrational levels
can also be ignored, as the dipole moment $d(R)$ is a slowly varying function
with $R$ and the spacing between vibrational levels is large compared to the
rotational splitting. For tensor operators $C_{lm}(\hat{R})$ of rank 0, 1, and
2 the relationship between Cartesian $x$, $y$, and $z$ components and
spherical $m=-1$, $0$, and $+1$ components can be found in Refs. Bonin ;
Varsholovich .
For each projection $M$ and vibrational level $v$, eigenvalues and
eigenvectors of the Hamiltonian, Eq. (2), are obtained by the direct
diagonalization of the Hamiltonian matrix including rotational states $J$ from
$|M|$ up to some value $J_{\rm max}$. For the external electric field strength
and $J$ values of interest, $J_{\rm max}=10$ is sufficient for convergence. We
label the eigenenergies by $E_{v\tilde{J}M}$ with corresponding eigenvectors
$|v\tilde{J}M\rangle=\sum_{J}U^{vM}_{\tilde{J},J}|vJM\rangle_{z}$. Here
$\tilde{J}$ is an integer index with values $\tilde{J}=|M|,|M|+1,...$, such
that the eigenstate $|v\tilde{J}M\rangle$ adiabatically connects to the
electric field-free eigenstate $|vJM\rangle_{z}$ with $J=\tilde{J}$. For
$|M|>0$ the levels with projection quantum number $-M$ and $M$ remain
degenerate. The dipole matrix element between states $|v\tilde{J}M\rangle$ and
$|v^{\prime}\tilde{J}^{\prime}M^{\prime}\rangle$ of the X potential is given
by $\langle
v\tilde{J}M|d_{\sigma}|v^{\prime}\tilde{J}^{\prime}M^{\prime}\rangle=\sum_{J,J^{\prime}}U^{vM}_{\tilde{J},J}\,U^{v^{\prime}M^{\prime}}_{\tilde{J}^{\prime},J^{\prime}}\times{}_{z}\langle
vJM|d_{\sigma}|v^{\prime}J^{\prime}M^{\prime}\rangle_{z}$, where $\sigma$ is a
spatial index that can be expressed either in Cartesian coordinates $x,y,z$,
or spherical coodinates $q=0,\pm 1$.
Now we are able to calculate the dynamic polarizability of the mixed
rotational eigenstates at laser frequency $\nu$. It is determined by the
properties of the operator $\alpha_{\sigma\sigma^{\prime}}(\nu)$ defined by
Bonin ; Stone
$\displaystyle\alpha_{\sigma\sigma^{\prime}}(\nu)$ $\displaystyle=$
$\displaystyle\sum_{\gamma}\left\\{\frac{1}{E_{\gamma}-E_{v\tilde{J}M}+h\nu}+\frac{1}{E_{\gamma}-E_{v\tilde{J}M}-h\nu}\right\\}d_{\sigma}|\gamma\rangle\langle\gamma|d_{\sigma^{\prime}}$
(9)
with $\gamma$ enumerating eigenstates of the ground as well as excited
electronic potentials in the presence of the electric field and
$\sigma,\sigma^{\prime}=x,y,$ or $z$. We will focus on the $J=0$ and $J=1$
levels and without loss of generality assume that the $x$ axis of our
coordinate system lies in the plane spanned by the electric-field direction
and the orientation of the linearly-polarized laser light. Moreover, we are
interested in the situation where the level shifts due to the laser are small
compared to those induced by the electric field. For the isolated levels with
M=0, the polarizability is determined by the diagonal matrix element
$\alpha^{v\tilde{J}M=0}_{\sigma\sigma^{\prime}}(\nu)=\langle
v\tilde{J}M=0|\alpha_{\sigma\sigma^{\prime}}(\nu)|v\tilde{J}M=0\rangle$. The
$|M|>0$ dynamic polarizability needs to be treated by degenerate perturbation
theory within the two-dimensional subspace $|v\tilde{J}M\rangle$ and
$|v\tilde{J}-\\!M\rangle$. In fact, our choice of the $x$ direction and the
symmetry properties of the dynamic polarizability ensure that the linear
combinations
$|v\tilde{J}M,\pm\rangle=\\{|v\tilde{J}M\rangle\pm|v\tilde{J}-\\!M\rangle\\}/\sqrt{2}$
with $M>0$ are the correct eigenstates. For these states the dynamic
polarizability $\alpha_{\sigma\sigma^{\prime}}^{v\tilde{J}M,\pm}(\nu)=\langle
v\tilde{J}M,\pm|\alpha_{\sigma\sigma^{\prime}}(\nu)|v\tilde{J}M,\pm\rangle$.
For diatomic species, only the diagonal elements
$\alpha^{v\tilde{J}M,\pm}_{xx}$, $\alpha^{v\tilde{J}M,\pm}_{yy}$, and
$\alpha^{v\tilde{J}M,\pm}_{zz}$ are nonzero. Hence in an oscillating electric
field
$\vec{\mathcal{E}}_{o}(t)=\mathcal{E}_{o}(0)\textrm{Re}\\{\vec{\epsilon}e^{i2\pi\nu
t}\\}$ (where $\vec{\epsilon}$ is the complex unit vector indicating the
polarization), the state $|v\tilde{J}M,\pm\rangle$ shifts in energy by an
amount $\Delta E$ given by $\Delta
E=-\sum_{\sigma,\sigma^{\prime}}|\mathcal{E}_{o}(0)|^{2}\alpha^{v\tilde{J}M,\pm}_{\sigma\sigma^{\prime}}(\nu)\epsilon_{\sigma}\epsilon^{*}_{\sigma^{\prime}}/4$.
The sum is over the spatial indices $\sigma,\sigma^{\prime}=x,y,z$.
We are interested in near infrared laser frequencies, which are detuned away
from resonances with rovibrational levels of the electronically excited
potentials. In particular, we focus on wavelengths between 1000 nm and 1100
nm. Starting from the $v=0$ vibrational level of the X${}^{1}\Sigma^{+}$
states of KRb and RbCs, photons of this wavelength do not possess sufficient
energy to reach rovibrational levels of the electronically excited singlet
${}^{1}\Lambda$ potentials. The photons do have enough energy to reach
vibrational level of the triplet b${}^{3}\Pi$ potential considering only
single-photon excitations. However, such transitions require relativistic
spin-orbit coupling to the A${}^{1}\Sigma^{+}$ potential to acquire a non-zero
dipole matrix element; such spin-orbit induced couplings are small enough to
neglect under the conditions of interest here.
This allows us to make several approximations. Firstly, we can use non-
relativistic potentials and transition dipole moments and in the calculation
of the dynamic polarizability only consider singlet ${}^{1}\Sigma^{+}$ and
${}^{1}\Pi$ potentials. Secondly, assuming a large detuning (such that
$|E_{\gamma}-E_{v\tilde{J}M}-h\nu|\gg E_{vJM}$ for all states of interest) we
can neglect the electric field and rotational dependence of the energy
denominators in Eq. (9). Finally, we find that in the near infrared the
contribution to the polarizability from intermediate states $\gamma$ in the
ground X${}^{1}\Sigma^{+}$ state is small.
With this in mind the polarizability becomes
$\displaystyle\alpha^{v\tilde{J}M,\pm}_{\sigma\sigma^{\prime}}(\nu)$
$\displaystyle\cong$
$\displaystyle\sum_{J,J^{\prime}}U^{v|M|}_{\tilde{J},J}U^{v|M|}_{\tilde{J},J^{\prime}}\sum_{ev_{e}\Lambda}\left\\{\frac{1}{E_{ev_{e}\Lambda}-E_{v}+h\nu}+\frac{1}{E_{ev_{e}\Lambda}-E_{v}-h\nu}\right\\}$
$\displaystyle\quad\quad\quad\quad\times\sum_{J_{e}M_{e}}{}_{z}\langle
vJM,\pm|d_{\sigma}|ev_{e}J_{e}M_{e}\Lambda\rangle_{z}\,{}_{z}\langle
ev_{e}J_{e}M_{e}\Lambda|d_{\sigma^{\prime}}|vJ^{\prime}M,\pm\rangle_{z}\,,$
where the energy $E_{v}$ is a typical vibrational energy in the ground state
potential, and the rovibrational wavefunctions
$|ev_{e}J_{e}M_{e}\Lambda\rangle$ of the electronically excited states with
approximate energy $E_{ev_{e}\Lambda}$ are given by
$|ev_{e}J_{e}M_{e}\Lambda\rangle_{z}\equiv|ev_{e}\Lambda\rangle\times|J_{e}M_{e}\Lambda\rangle_{z}\equiv\left\\{\frac{\phi_{v_{e}}(R)}{R}|e~{}^{1}\Lambda\rangle\right\\}\,\times\left\\{\sqrt{\frac{2J+1}{4\pi}}D^{J_{e}*}_{M_{e}\Lambda}(\hat{R})\right\\}\,,$
where the vibrational and electronic dependence has been isolated in the ket
$|ev_{e}\Lambda\rangle$ and the rotational dependence in
$|J_{e}M_{e}\Lambda\rangle_{z}$, respectively. The wavefunction
$\phi_{v_{e}}(R)$ is the radial rovibrational wavefunction, the ket
$|e~{}^{1}\Lambda\rangle$ is the electronic state with projection quantum
number $\Lambda$ defined along the internuclear axis, and
$D^{J}_{MM^{\prime}}(\hat{R})$ is a Wigner rotation matrix that describes a
symmetric top rotational wavefunction. The sum over $\Lambda$ includes both
positive and negative values, where $\Lambda=0$ corresponds to excited
${}^{1}\Sigma^{+}$ electronic states and $\Lambda=\pm 1$ to ${}^{1}\Pi$
states.
As stated before the energy of the excited state and $\phi_{v_{e}}(R)$ depend
on the electronic state $|e~{}^{1}\Lambda\rangle$ and vibrational level
$v_{e}$, but not $J_{e}$ and $M_{e}$. Consequently, the transition dipole
moments separate into ${}_{z}\langle
vJM,\pm|d_{\sigma}|ev_{e}J_{e}M_{e}\Lambda\rangle_{z}=\langle
vJ|d(R)|ev_{e}\Lambda\rangle F^{J_{e}M_{e}\Lambda}_{JM\pm,\sigma}$, where
$F^{J_{e}M_{e}\Lambda}_{JM\pm,\sigma}$ is an integral of the product of three
Wigner rotation matrices $D^{J}_{MM^{\prime}}(\hat{R})$ over the orientation
of the molecule $\hat{R}$ that can be evaluated using angular momentum algebra
Brink .
Moreover, we have verified that $\langle vJ|d(R)|ev_{e}\Lambda\rangle$ is
nearly independent of $J$. The sums over $J_{e}$ and $M_{e}$ in Eq. (Electric
field-dependent dynamic polarizability and state-insensitive conditions for
optical trapping of diatomic polar molecules) can now be performed and we
finally find
$\displaystyle\langle
v\tilde{J}M,\pm|\alpha_{\sigma\sigma^{\prime}}(\nu)|v\tilde{J}M,\pm\rangle=$
$\displaystyle\sum_{\Lambda}\alpha^{\Lambda}(\nu)\sum_{J,J^{\prime}}U^{v|M|}_{\tilde{J},J}U^{v|M|}_{\tilde{J},J^{\prime}}\times\sum_{J_{e}M_{e}}F^{J_{e}M_{e}\Lambda}_{JM\pm,\sigma}F^{J_{e}M_{e}\Lambda}_{J^{\prime}M\pm,\sigma^{\prime}}$
with the $\Lambda$-dependent
$\displaystyle\alpha^{\Lambda}(\nu)=\sum_{ev_{e}}\langle
vJ|d|ev_{e}\Lambda\rangle\langle
ev_{e}\Lambda|d|vJ\rangle\left\\{\frac{1}{E_{ev_{e}\Lambda}-E_{v}+h\nu}+\frac{1}{E_{ev_{e}\Lambda}-E_{v}-h\nu}\right\\}\,.$
(12)
The parallel $\alpha^{0}(\nu)$ and perpendicular $\alpha^{1}(\nu)$
contributions to the polarizability are due to transitions to the $\Sigma$ and
$\Pi$ states, respectively.
Magic DC Electric Field
Figure 2: Dynamic polarizability at a wavenumber of 9174 cm-1 of the $v$=0,
$\tilde{J}$=0 and 1 levels of the X${}^{1}\Sigma^{+}$ ground state of KRb (top
row) and RbCs (bottom row) as a function of the field strength of an external
electric field. The polarization of the trapping field is parallel (left
panel) and perpendicular (right panel) to the direction of the electric field.
The circles indicate a crossing point between $\tilde{J}$=0 and 1
polarizabilities. The degeneracy of the states $\tilde{J}=1$ and $M=\pm 1$ is
lifted by perpendicularly polarized laser light. The new states are labeled
$|+1\rangle\pm|-1\rangle$, corresponding to $|\tilde{J}=1,M\pm\rangle$
respectively. The $M=\pm 1$ states remain degenerate for parallel
polarization.
The polarizability depends on the stength of an external electric field
through the use of the unitary matrices $U^{vM}_{\tilde{J},J}$ of Eq.
(Electric field-dependent dynamic polarizability and state-insensitive
conditions for optical trapping of diatomic polar molecules). Figure 2 shows
the dynamic polarizability of the ground states of the KRb and RbCs molecules
as a function of the external electric field strength. The left panels of
these figures correspond to the parallel $(\vec{\epsilon}=\hat{z})$ and the
right panels to the perpendicular $(\vec{\epsilon}=\hat{x})$ polarization of
the trapping light relative to the electic field direction
$\vec{\mathcal{E}}$. In all cases the polarizability depends on both the state
index $\tilde{J}$ and the angular momentum projection $M$. For KRb (Fig. 2,
top row) a “magic” electric field strength exists at $\mathcal{E}=10$ kV/cm,
where the polarizability of $\tilde{J}$=0, $M$=0 and $\tilde{J}$=1, $M$=0
states coincide. This is possible due to the polar character of the molecules.
For RbCs (Fig. 2, bottom row) two “magic” electric field strengths exist. For
the pair of states $\tilde{J}=0,M=0$ and $\tilde{J}=1,M=0$ a crossing occurs
near 2 kV/cm. Another crossing appears for the states $\tilde{J}=1,M=0$ and
$\tilde{J}=1,M=\pm 1$ at 4.7 kV/cm. They both are at a much smaller field
strength than for KRb, since RbCs has a larger permanent dipole moment and
smaller rotational splittings in the ground state. Note that the $M=0$ “magic”
electric field occurs at the same field strength for both parallel and
perpendicular polarization of the trapping light. In fact, they are the same
for any polarization.
Magic Angle
Figure 3: Dynamic polarizability of the $v$=0, $\tilde{J}$=0, $M$= 0 level of
the X${}^{1}\Sigma^{+}$ ground state of RbCs as a function of external
electric field strength and the angle $\theta$ between the direction of the
electric field $\vec{\mathcal{E}}$ and the polarization $\vec{\epsilon}$ of
the trapping light at a wavenumber of 9174 cm-1. The contour lines are marked
by the polarizability value in atomic units.
Figure 4: Dependence of the dynamic polarizability on the angle between the
polarization $\vec{\epsilon}$ of the trapping light at a wavenumber of 9174
cm-1 and the direction of the external electric field $\vec{\mathcal{E}}$, for
different values of $\mathcal{E}$. The polarizability of states with
projections $M=0$ and $M=\pm 1$ of the X${}^{1}\Sigma^{+}$ ground state of KRb
and RbCs are shown in the left and right panels, respectively. For clarity
$\mathcal{E}$ ranges from 0.6 kV/cm to 6 kV/cm in steps of 0.6 kV/cm for KRb
and from 0.3 kV/cm to 3 kV/cm in steps of 0.3 kV/cm for RbCs. At the “magic
angle” $\theta=\theta_{0}=54^{o}$, the polarizability of the states
$\tilde{J}=0,M=0$ and $\tilde{J}=1,M=0$ are identical and independent of the
strength of the static electric field. For the $\tilde{J}=1,M=\pm 1$ states no
magic angle exists.
In future experiments one would expect to be able to change the angle between
the static and dynamic electric fields. Figure 3 shows a surface plot of the
dynamic polarizability as a function of $\mathcal{E}$ and the angle $\theta$
between $\vec{\mathcal{E}}$ and polarization of the trapping light
$\vec{\epsilon}$, for linearly polarized light. The polarizability depends
smoothly on both $\theta$ and $\mathcal{E}$.
Figure 4 compares the dynamic polarizability of the states
$|\tilde{J}=0\rangle$ and $|\tilde{J}=1,M=0,\pm 1\rangle$ as a function of the
angle $\theta$ of the linear polarization of the optical field relative to the
static field (such that
$\hat{\epsilon}=\cos{\theta}\hat{z}+\sin{\theta}\hat{x}$), for several static
electric field strengths within the range from 0 to 6 kV/cm for KRb and 0 to 3
kV/cm for RbCs. All curves with $M=0$ cross at the angle $\theta=\theta_{0}$
such that cos${}^{2}\theta_{0}$ = 1/3, or $\theta_{0}\approx 54$ degrees. We
refer to this as the ”magic angle” since here the AC Stark shift is
independent of the internal state of the molecule. This behavior occurs in
many contexts and is a simple consequence of the rank-2 tensor structure of
the polarizability BKDbook . The angular dependence of the $\tilde{J}=0$ level
is smaller than that of the $\tilde{J}=1$ levels as expected. It inherited the
zero-electric field properties of the scalar $J=0$ state. As seen in Fig. 2,
the RbCs polarizability of the $\tilde{J}=0,M=0$ and $\tilde{J}=1,M=0$ levels
cross at a magic electric field strength of 2 kV/cm. For Fig. 4 this implies
that the curves start to “overlap” when $\cal E$ is equal or larger than this
magic field value.
The polarizability operator $\alpha_{\sigma\sigma^{\prime}}(\nu)$, defined by
Eq. (9), is a reducible rank-two tensor operator. Hence it can be expressed as
a sum of irreducible tensor operators $\alpha^{(k)}(\nu)$ of rank $k=0,1,2$.
In terms of these irreducible tensor operators, the AC Stark shift $\Delta E$
is proportional to the diagonal matrix element of the operator
$\sum_{\sigma\sigma^{\prime}}\alpha_{\sigma\sigma^{\prime}}(\nu)\epsilon_{\sigma}\epsilon^{*}_{\sigma^{\prime}}$,
which can be written in the general form
$\sum_{k=0}^{2}\sum_{q=-k}^{k}(-1)^{q}\alpha^{k}_{q}(\nu)\epsilon\epsilon^{k}_{-q},$
(13)
where $q$ is a spherical tensor projection index and the explicit forms of the
irreducible spherical tensors $\alpha^{k}_{q}(\nu)$ and
$\epsilon\epsilon^{k}_{q}$ are given in Refs Bonin ; Varsholovich .
In order to derive the “magic angle” condition for $M=0$ states, we consider
the effect of each term in the expansion (13). The term with $k=0$ corresponds
to the scalar polarizability; the operator
$\alpha^{0}_{0}(\nu)=\sum_{\sigma\sigma^{\prime}}\alpha_{\sigma\sigma^{\prime}}(\nu)\delta_{\sigma\sigma^{\prime}}$
has, under our approximations, diagonal matrix elements that are independent
of $J$ (or $\tilde{J}$) and $M$ for all states of interest, and similarly the
quantity
$\epsilon\epsilon^{0}_{0}\propto\vec{\epsilon}\cdot\vec{\epsilon}\,{}^{*}=1$
is independent of the polarization of the optical field. The term with $k=1$,
corresponding to the vector polarizability, in general is significant.
However, for the special case of linearly polarized light where
$\vec{\epsilon}$ is real, the quantities
$\epsilon\epsilon^{1}_{q}=(\vec{\epsilon}\times\vec{\epsilon}\,{}^{*})_{q}$
vanish and hence the effect of the vector polarizability is zero. For the
tensor polarizability terms (with $k=2$), from the Wigner-Eckhart theorem only
the operator component with $q=0$ gives rise to a non-zero diagonal matrix
element for $M=0$ states. Hence the contribution of this term is proportional
to $\epsilon\epsilon^{2}_{0}$. Without loss of generality we can define the
linear polarization as
$\vec{\epsilon}=\epsilon_{x}\hat{x}+\epsilon_{z}\hat{z}=\cos{\theta}\hat{z}+\sin{\theta}\hat{x}$.
In this case
$\epsilon\epsilon^{2}_{0}\propto\epsilon_{z}\epsilon_{z}^{*}-\vec{\epsilon}\cdot\vec{\epsilon}\,{}^{*}/3=\cos^{2}{\theta}-1/3$.
Hence the contribution to $\Delta E$ due to the tensor polarizability also
vanishes for all states with $M=0$, when the optical field is linearly
polarized at the “magic angle” $\theta=\theta_{0}$. Under this condition the
only contribution to the dynamic polarizability is from the $k=0$ scalar term,
which is the same for all states of interest.
The property of the “magic” electric field discussed in Fig. 2 can also be
understood in terms of the tensor structure of the polarizability. At certain
values of the applied DC electric field $\mathcal{E}$, the rank-2 components
of the polarizability, $\alpha^{2}_{0}$, of the $\tilde{J}=0,M=0$ and
$\tilde{J}=1,M=0$ levels becomes the same. When this condition is met, the AC
Stark shift becomes independent of the direction of the trapping light’s
linear polarization.
Here we finish with an example of a specific implementation of a “magic angle”
3-D lattice. Let the lattice be formed by three orthogonal retroreflected
laser beams $a,b$, and $c$, with initial propagation directions $\hat{k}$
given by $\hat{k}_{a}=\hat{y}$, $\hat{k}_{b}=(\hat{x}+\hat{z})/\sqrt{2}$, and
$\hat{k}_{c}=(\hat{x}-\hat{z})/\sqrt{2}$. These three beams each have a
different frequency $\nu$, such that
$\nu_{a}=\nu_{b}-\delta_{b}=\nu_{c}-\delta_{c}$; here the offset frequencies
$\delta_{b,c}$ must satisfy $\nu_{a,b,c}\gg\delta_{b,c}\gg f_{mot}$, where
$f_{mot}$ is the motional frequency of the molecules in the optical trapping
potential. The use of different frequencies (which can be generated from a
single laser by using e.g. acousto-optic modulators) in this manner eliminates
the effect of interference terms between the different laser beams: such terms
average to zero rapidly over the time of motion of the atom, and hence can be
neglected. The resulting average trap potential is then simply the sum of the
potentials due to each individual laser beam. Finally, the polarizations of
the three beams can be chosen as
$\hat{\epsilon}_{a}=\sqrt{2/3}\hat{x}+\sqrt{1/3}\hat{z}$;
$\hat{\epsilon}_{b}=\sqrt{2/3}\hat{k}_{c}+\sqrt{1/3}\hat{y}$; and
$\hat{\epsilon}_{c}=\sqrt{2/3}\hat{k}_{b}+\sqrt{1/3}\hat{y}$. In each case,
$|\hat{\epsilon}\cdot\hat{z}|=\cos{\theta_{0}}$.
## References
* (1) J. Ye, H. J. Kimble, and H. Katori, Science 320, 1734 (2008).
* (2) H. Katori, M. Takamoto, V. G. Pal’chikov, V. D. Ovsiannikov, Phys. Rev. Lett. 91, 173005 (2003).
* (3) V. V. Flambaum, V. A. Dzuba, and A. Derevianko, Phys. Rev. Lett. 101, 220801 (2008).
* (4) K. Beloy, A. Derevianko, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. Lett. 102, 120801 (2009).
* (5) N. Lundblad, M. Schlosser, and J. V. Porto, Phys. Rev. A 81, 031611(R) (2010).
* (6) A. Derevianko, Phys. Rev. A 81, 051606(R) (2010).
* (7) J. Doyle, B. Friedrich, R. V. Krems, and F. Masnow-Seeuws, Special issue on ulatracold polar molecules, Eur. Phys. J. D 31, 149 (2004).
* (8) L.D. Carr, D. DeMille, R.V. Krems, Jun Ye, New J. Phys. 11, 055049 (2009).
* (9) D. DeMille, Phys. Rev. Lett. 88, 067901 (2002).
* (10) R. Zhao, el al. Nat. Phys. 5, 100 (2009).
* (11) T. Zelevinsky, S. Kotochigova, and Jun Ye, Phys. Rev. Lett. 100, 043201 (2008).
* (12) S. Kotochigova, T. Zelevinsky, and Jun Ye, Phys. Rev. A 79, 012504 (2009).
* (13) S. Ospelkaus, K.-K. Ni, M. H. G. de Miranda, B. Neyenhuis, D. Wang, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Faraday Discuss. 142, 351 (2009).
* (14) A. Micheli, G.K. Brennen, and P. Zoller, Nature Phys. 2, 341 (2006).
* (15) H. P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo, and P. Zoller, Phys. Rev. Lett. 98, 060404 (2007).
* (16) H. P. Büchler, A. Micheli, and P. Zoller. Nature Phys. 3, 726 (2007).
* (17) S. Kotochigova and E. Tiesinga, Phys. Rev. A 73, 041405(R) (2006).
* (18) S. Kotochigova, E. Tiesinga, P. S. Julienne, New J. Phys. 11, 055043 (2009).
* (19) S. Kotochigova, New J. Phys. 12, 073041 (2010).
* (20) D. M. Brink and G. R. Satchler, Angular momentum, (Clarendon Press, London, 1993).
* (21) K.D. Bonin and V.V. Kresin, Electric-dipole polarizabilities of atoms, molecules, and clusters, (World Scientific, Singapore, 1997).
* (22) D.A. Varsholovich, A.N. Moskalev, and V.K. Khersonskii, Quantum theory of angular momentum: irreducible tensors, spherical harmonics, vectors coupling coefficients, 3nj symbols, (World Scientific, Singapore, 1988).
* (23) A. J. Stone, The theory of intermolecular forces, (Clarendon Press, London, 1996).
* (24) D. Budker, D.F. Kimball, and D.P. DeMille, Atomic Physics: an exploration through problems and solutions, 2nd ed., (Oxford Univ. Press, Oxford, 2008).
## I Acknowledgments
This work is supported by a MURI grant of the Air Force Office of Scientific
Research; by NSF; and (for DD) by DOE. SK acknowledges helpful discussions
with J. Ye, D. Jin, and B. Neyenhuis.
|
arxiv-papers
| 2010-08-23T16:53:29 |
2024-09-04T02:49:12.386520
|
{
"license": "Public Domain",
"authors": "Svetlana Kotochigova and David DeMille",
"submitter": "Svetlana Kotochigova",
"url": "https://arxiv.org/abs/1008.3856"
}
|
1008.3871
|
# Symmetry and uniqueness of minimizers of Hartree type equations with
external Coulomb potential
Vladimir Georgiev Dipartimento di Matematica, Università di Pisa
Largo Bruno Pintecorvo 5, 56127 Pisa, Italy
e-mail: georgiev@dm.unipi.it George Venkov Faculty of Applied Mathematics
and Informatics, Technical University of Sofia
Kliment Ohridski 8, 1756 Sofia, Bulgaria
e-mail: gvenkov@tu-sofia.bg
###### Abstract
In the present article we study the radial symmetry of minimizers of the
energy functional, corresponding to the repulsive Hartree equation in external
Coulomb potential. To overcome the difficulties, resulting from the ”bad” sign
of the nonlocal term, we modify the reflection method and then, by using
Pohozaev integral identities we get the symmetry result.
###### keywords:
Hartree equations , minimizers , symmetry , variational methods , nonlinear
solitary waves
###### MSC:
35J50 , 35J60 , 35Q55
††journal: Journal of Differential Equationslabel2label2footnotetext: The
first author was supported by the Italian National Council of Scientific
Research (project PRIN No. 2008BLM8BB ) entitled: ”Analisi nello spazio delle
fasi per E.D.P.”
## 1 Introduction
Solitary waves associated with the Hartree type equation in external Coulomb
potential are solutions of type
$\chi(x)e^{-i\omega t},\quad x\in\mathbb{R}^{3},t\in\mathbb{R},$
where $\omega>0$ and $\chi$ satisfies the nonlinear elliptic equation
$\displaystyle\ \ \ \ \
-\Delta\chi(x)+\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}\
\chi(x)-\frac{\chi(x)}{|x|}+\omega\chi(x)=0.$ (1)
The natural energy functional associated with this problem is (see [4])
$\displaystyle\mathcal{E}(\chi)=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{1}{4}A(|\chi|^{2})-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx,$ (2)
where we shall denote
$A(f)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{f(x)f(y)}{|x-y|}dydx.$
(3)
The corresponding minimization problem is associated with the quantity
$I_{N}=\min\\{\mathcal{E}(\chi);\chi\in H^{1},\|\chi\|^{2}_{L^{2}}=N\\}.$ (4)
The existence of positive minimizers $\chi_{0}(x)$, such that
$\mathcal{E}(\chi_{0})=I_{N},\ \ \|\chi_{0}\|^{2}_{L^{2}}=N,$
is established by Cazenave and Lions in [4] by the aid of the concentration
compactness method.
For a given $\omega>0$, the constrained minimization problem (4) can be
compared with the unconstrained minimization problem
$S^{min}_{\omega}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\},$
where $S_{\omega}(\chi)$ is the corresponding action functional, defined by
$S_{\omega}(\chi)=\mathcal{E}(\chi)+\frac{\omega}{2}\|\chi\|^{2}_{L^{2}}.$ (5)
There are different results on the symmetry (and uniqueness) of the
minimizers. The basic result due to Gidas, Ni and Nirenberg [9] implies the
radial symmetry of the minimizers associated with the semilinear elliptic
equation
$\Delta u+f(u)=0,$
provided suitable assumptions on the function $f(u)$ are satisfied and the
scalar function $u$ is positive. As in the previous result due to Serrin [19],
the proof is based on the maximum principle and the Hopf’s lemma.
Therefore, the first natural question is to ask if the linear operator
$P_{\omega}=-\Delta-\frac{1}{|x|}+\omega$
in (1), satisfies the weak maximum principle in the sense that
$u\in H^{2},\ P_{\omega}(u)=g\geq 0,\Longrightarrow u\geq 0.$ (6)
The above maximum principle is incomplete, since additional behavior of $u$
and $g$ at infinity has to be imposed, namely, we shall suppose that
$(1+|x|)^{-M}e^{\sqrt{\omega}|x|}u\in
H^{2},\quad(1+|x|)^{-M}e^{\sqrt{\omega}|x|}g\in H^{2},$ (7)
for some real number $M>0.$
Note, that the energy levels of the hydrogen atom are described by the
eigenvalues $\omega_{k}>0$ of the eigenvalue problem
$\Delta e_{k}(x)+\frac{e_{k}(x)}{|x|}=\omega_{k}e_{k}(x),\ \ e_{k}(x)\in
H^{2}.$
One has
$\omega_{k}=\frac{1}{4(k+1)^{2}},\quad k=0,1,...$
and $e_{0}(x)=ce^{-|x|/2},c>0.$ The first observation is that all
eigenfunctions $e_{k}(x)$, $k\geq 1$, are expressed in terms of Laguerre
polynomials of $|x|$, having exactly $k$ roots. This fact guarantees that the
maximum principle is not valid for $\omega=\omega_{k}.$ More precisely, we can
show the following.
###### Lemma 1.
The weak maximum principle (6) is valid if an only if
$\omega\geq\frac{1}{4}.$
This result can be compared with the existence of action minimizers for the
corresponding functional $S_{\omega}$, obtained by Lions for $0<\omega<1/4$
(see for details [14]).
###### Theorem 2.
We have the properties:
a) for any $\omega>0,$ the inequality
$\min_{\chi\in H^{1}}S_{\omega}(\chi)=S^{min}_{\omega}>-\infty$
holds;
b) if $0<\omega<1/4$, then $S^{min}_{\omega}<0;$
c) if $0<\omega<1/4$, then there exists a positive function $\chi(x)\in
H^{1},$ such that
$S_{\omega}(\chi)=S^{min}_{\omega}.$
Our main goal of this paper is to clarify if the positive minimizers of
$S_{\omega}$ are radially symmetric and unique. The above results show that we
have to consider the domain $0<\omega<1/4,$ where the key tool of Gidas, Ni
and Nirenberg (i.e. the maximum principle for the corresponding linear
operator) is not applicable.
The symmetry of the energy functional (even with constraint conditions) can
not imply, in general, the radial symmetry of the minimizers. This phenomena
was discovered and studied in the works [6], [7] and [8] in the scalar case.
Some sufficient conditions that guarantee the symmetry of minimizers have been
studied by Lopes in [15], by means of the reflection method that (for the case
of plane $x_{1}=0$) uses the functions
$u_{1}(x)=\left\\{\begin{array}[]{ll}u(\hat{x}),\
\hat{x}=(-x_{1},x_{2},\cdots,x_{n}),&\hbox{if $x_{1}>0$;}\\\ u(x),&\hbox{if
$x_{1}<0$}\end{array}\right.$
and
$u_{2}(x)=\left\\{\begin{array}[]{ll}u(\hat{x}),\
\hat{x}=(-x_{1},x_{2},\cdots,x_{n}),&\hbox{if $x_{1}<0$;}\\\ u(x),&\hbox{if
$x_{1}>0$.}\end{array}\right.$
If the functional to be minimized has the form
$E(u)=\frac{1}{2}\|\nabla u\|^{2}_{L^{2}}+\int_{\mathbb{R}^{n}}F(u(x))\ dx,$
then we have the relation
$E(u_{1})+E(u_{2})=2E(u)$
and this enables one to obtain the symmetry of minimizer, when $F(u)$ is a
combination of functions of type $|u|^{p},p\geq 2$.
The reflection method works effectively when $u(x)$ is a vector-valued
function and constraint conditions (as in the problem (4)) are involved too.
Recently, the reflection method was generalized in [16] and [17] for very
general situations and one example of possible application is the functional
of type
$E(u)=\frac{1}{2}\|\nabla u\|^{2}_{L^{2}}+\int_{\mathbb{R}^{n}}F(u(x))\
dx-A(|u|^{2}),$
involving nonlocal term as in (2). This Choquard type functional has the
specific property
$E(u_{1})+E(u_{2})\leq 2E(u),$
exploiting the negative sign of the nonlocal term $A(|u|^{2})$.
An analogous result for the scalar case can be obtained by means of the
Schwarz symmetrization (or spherical decreasing rearrangement [12])
$u^{*}(|x|)$ of the non-negative $u\in H^{1}.$ Indeed, we have the equality
$\int_{\mathbb{R}^{n}}F(u(x))\ dx=\int_{\mathbb{R}^{n}}F(u^{*}(x))\ dx,$
as well as the inequalities
$\|\nabla u\|^{2}_{L^{2}}\geq\|\nabla u^{*}\|^{2}_{L^{2}},\ \ \ A(|u|^{2})\leq
A(|u^{*}|^{2}),$
so, we get
$E(u^{*})\leq E(u)$
and one can use the property that $u$ is minimizer.
The functional in (2) is a typical example, when reflection method and Schwarz
symmetrization meet essential difficulty to be applied directly.
The main goal of this work is to find an approach to establish the symmetry of
the minimizer for functionals of Hartree type (2), involving nonlocal terms
with ”bad” sign.
To state this main result, we shall try first to connect the minimizers of the
constraint problem (4) (associated with the energy functional
$\mathcal{E}(\chi)$) with the minimization of the action functional
$S_{\omega}(\chi)$. Similar relation for local type interactions is discussed
in chapter IX of [3]. Then, we shall establish that the minimizer of Theorem 2
is a radially symmetric function.
###### Theorem 3.
The solution $\chi(x)$ from Theorem 2 is a radially symmetric function for
$\frac{1}{16}<\omega<\frac{1}{4}.$
###### Remark 1.
The result of Theorem III.1 in [4] treats more general case of potentials of
type
$V(x)=-\sum_{j=1}^{K}\frac{Z}{|x-x_{j}|},$
while in our case we have
$V(x)=-\frac{Z}{|x|}.$
Therefore, the energy functional $\mathcal{E}(\chi)$ is rotationally invariant
in our case. From Theorem 2 and Theorem 3 one can see that the solution
$\chi_{0}(x)$ of (4) is radially symmetric and unique (up to a multiplication
with complex number $z$, with $|z|=1$).
As it was mentioned above the energy (and therefore the action) is a
functional involving the nonlocal term with ”bad” sign. To explain the main
idea to treat this case, we recall the rotational symmetry of the energy (and
action) functional. Therefore, if $\chi$ is the action minimizer from Theorem
2, it is sufficient to show that the solution is symmetric with respect to
$x_{1}$-plane, for any choice of the $x_{1}$-direction. In other words, we
consider $\hat{\chi}(x)=\chi(\hat{x})$, with $\hat{x}=(-x_{1},x_{2},x_{3})$
and we aim to prove that $\chi=\hat{\chi}.$
To show this, we shall consider the two terms
$\chi_{\pm}=\frac{\chi\pm\hat{\chi}}{2}.$
So, our goal is to verify the inequality
$S_{\omega}(\chi_{+})+S_{\omega}(\chi_{-})\leq S_{\omega}(\chi)$ (8)
and see that the condition $\chi\neq\hat{\chi}$ implies
$S_{\omega}(\chi_{-})>0.$
The form of the functional $S_{\omega}$ suggests one, in order to verify (8),
to use an appropriate version of the Clarkson inequality for the quadratic
form $A(f)$. Namely, we can prove that the following inequality
$A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)\leq\frac{A(f^{2})+A(g^{2})}{2}$
holds true. Unfortunately, the usual Clarkson inequality in the form given
above, is too rough to serve as a tool for proving (8). Therefore, we shall
use a refined version of Clarkson inequality (see Lemma 5 below) in the form
$\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)\leq\frac{A(f^{2})+A(g^{2})}{8}$
$\displaystyle+\frac{3\sqrt{A(f^{2})A(g^{2})}}{4}.$
The final step is to treat the uniqueness of positive minimizers. of the
problem
$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$ (9)
Our proof can not follow the Lieb’s uniqueness proof for the ground state
solution of the Choquard equation [11]. In general, the Lieb’s proof strongly
depends on the specific features of the nonlocal nonlinear equation (1) and
differs from the corresponding results for semilinear elliptic equation given
by Kwong in [10]. Indeed, once the radial symmetry is established, one can use
Pohozaev identities and reduce the nonlocal nonlinear elliptic problem (1) to
an ordinary differential equation of the type
$u^{\prime\prime}(r)+W(r)u(r)+4\pi\int^{r}_{0}\left(\frac{1}{s}-\frac{1}{r}\right)u^{2}(s)dsu(r)=\omega
u(r),$
where
$W_{\chi}(r)=\frac{1}{r}-4\pi\int^{\infty}_{0}\chi^{2}(s)sds.$
The positive sign in front of the nonlinear term is the main obstacle to apply
Sturm type argument and derive the uniqueness of positive solutions to this
ordinary differential equation. However, for $\frac{1}{16}<\omega<\frac{1}{4}$
we can apply the approach based on the refined Clarkson inequality and using
the orthogonal projection on the eigenspace of the first eigenvalue of the
operator $\Delta+1/|x|$, we can establish the following result.
###### Theorem 4.
Let $\frac{1}{16}<\omega<\frac{1}{4}$. Then, the solution $\chi$ of
minimization problems (9) is unique.
Let’s mention that the results in Theorems 3 and 4 can be compared with the
results in [1], where the uniqueness of minimizers for the constrained
variational problem (4) is studied. To show the relations between action
minimization and (4) one has to apply the uniqueness of action minimizers or
alternatively the uniqueness of minimizers of constrained variational problem.
The plan of the work is the following. In Section 2 we consider the maximum
principle for the linear Schrödinger equation with Coulomb potential and prove
Lemma 1. The proof of Theorem 3, stating that the minimizers are radially
symmetric is presented in Section 3 by the aid of a refined version of
Clarkson inequality. In Section 4 we establish the Pohozaev integral
relations, corresponding to equation (1), and in Section 5 we prove uniqueness
Theorem 4. Finally, in A we prove for completeness the existence of positive
action minimizers, stated in Theorem 2, while in B the connection between
energy and action minimizers is discussed.
The authors are grateful to Louis Jeanjean for important discussions and
remarks on symmetry of minimizers as well as to the referee for pointing out a
gap in the proof of the Theorem 3.
## 2 Maximum principle for Schrödinger equation with Coulomb potential
The maximum principle, stated in (6) will be verified by the aid of the
substitution
$u=\varphi w,\varphi(x)=\varphi(|x|),$
where $\varphi$ is a radial function, satisfying the property
$-\Delta\varphi-\frac{\varphi}{|x|}+\omega\varphi=h(|x|)\geq 0.$ (10)
Our goal is to construct $\varphi$, so that $\varphi(|x|)>0.$ We have several
possibilities, depending on $\omega.$ If $\omega>1/4$, we shall show that such
a function exists and it is of type
$\varphi(r)=e^{-\beta r}Q(r),\ \ \beta=\sqrt{\omega},\ Q(r)=Ar^{2}+Br+C.$ (11)
If $\omega=1/4$, then we can take simply $\varphi(r)=e^{-r/2}.$ If
$0<\omega<1/4$, we shall see that a function $\varphi$ of type (11) exists,
but $\varphi(r)$ changes the sign for $r>0.$ Hence, this function gives a
counterexample, showing that the weak maximum principle (6) is not fulfilled
in this case.
Therefore, to complete the proof of Lemma 1, we have to explain how the
existence of positive $\varphi(r)$, satisfying (10) will imply the weak
maximum principle and then to construct in different cases the function $Q(r)$
in (11), so that (10) is satisfied.
###### Proof of Lemma 1 1.
After the substitution $u=\varphi w$, we have
$P_{\omega}(u)=-\varphi\Delta w-2\nabla\varphi\nabla
w+P_{\omega}(\varphi)w=\varphi\Delta w+2\nabla\varphi\nabla w+hw.$
If $\varphi(|x|)>0$, then we can write
$-\Delta w-\frac{2}{\varphi}\nabla\varphi\nabla
w+\frac{h}{\varphi}w=\frac{g}{\varphi}.$
Choosing $M=1,$ we see that
$\frac{g}{\varphi}\in H^{2},$
so we can apply the classical maximum principle (since $h\geq 0$) and obtain
$w\geq 0.$ This argument shows that the maximum principle is fulfilled if the
function $\varphi(r)$ satisfies inequality (10) and its polynomial term
$Q(r)>0$ for $r\geq 0$.
To construct $Q$, we substitute $\varphi(r)=e^{-\beta r}Q(r)$ into (10) and
find that
$e^{\beta r}rh(r)=-(2B+C(-2\beta+1))-(6A+B(-4\beta+1))r+(6\beta-1)Ar^{2}.$
We take for simplicity $A=1$ and
$B=C(\beta-1/2),\,C=\frac{12}{(2\beta-1)(4\beta-1)}.$
Then the condition $\beta>1/2$ implies that
$B=\frac{6}{(4\beta-1)}>0,\,C=\frac{12}{(2\beta-1)(4\beta-1)}>0$
so $Q(r)>0$ and
$e^{\beta r}rh(r)=(6\beta-1)r^{2}\geq 0.$
This argument completes the proof of the weak maximum principle for
$\omega>1/4.$
If $1/16<\omega<1/4$, then we can take the same $A,B,C$ and see that
$e^{\beta r}rh(r)=(6\beta-1)r^{2}\geq 0.$
Since $A=1$ and $C<0$ in this case, the function $Q(r)$ changes the sign.
Finally, if $0<\omega<1/16$, then we choose
$A=0,B=-1,C=\frac{1}{1/2-\beta}$
and then
$Q(r)=\frac{2}{1-2\beta}-r,\ \ e^{\beta r}rh(r)=(1-4\beta)r\geq 0.$
Again, it is clear that $Q(r)$ changes the sign, and the proof of the Lemma is
completed.
## 3 Radial symmetry of action minimizers
Even in the non-local case, the problem that action and energy minimizers are
nonnegative functions, is easy to be proved. Indeed, if $\chi(x)\in H^{1}$ is
a real-valued minimizer of the functional
$S_{\omega}(\chi)=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{1}{4}A(\chi^{2})-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{\chi(x)^{2}}{|x|}\
dx+\frac{\omega}{2}\|\chi\|^{2}_{L^{2}},$ (12)
then $|\chi(x)|$ satisfies the inequality
$\|\nabla|\chi|\|^{2}_{L^{2}}\leq\|\nabla\chi\|^{2}_{L^{2}},$
as well as the identities
$A(|\chi|^{2})=A(\chi^{2}),\ \ \int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx=\int_{\mathbb{R}^{3}}\frac{\chi(x)^{2}}{|x|}\ dx,$
so $|\chi(x)|\geq 0$ is also a minimizer of $S_{\omega}.$
Let us define the bilinear form
$L_{\omega}(\chi,\psi)=\langle(-\Delta-\frac{1}{|x|}+\omega)\chi,\psi\rangle_{L^{2}},\quad\omega>0$
(13)
and the corresponding quadratic form
$L_{\omega}(\chi)=\langle(-\Delta-\frac{1}{|x|}+\omega)\chi,\chi\rangle_{L^{2}}.$
(14)
The quadratic form $A(\chi)$ defined in (3) generates the corresponding
bilinear form
$A(\chi,\psi)=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{\chi(x)\psi(y)}{|x-y|}dydx.$
(15)
Then, the action functional $S_{\omega}$ can be written as
$S_{\omega}(\chi)=\frac{1}{2}L_{\omega}(\chi)+\frac{1}{4}A(\chi^{2}).$ (16)
Also, for any function $\chi$ we shall denote $\hat{\chi}(x)=\chi(\hat{x})$,
where $\hat{x}=(-x_{1},x_{2},x_{3})$ for any choice of our $x_{1}$-axis. It is
easy to check that
$S_{\omega}(\chi)=S_{\omega}(\hat{\chi}),\quad
L_{\omega}(\chi)=L_{\omega}(\hat{\chi}).$ (17)
With our next result, we shall establish Clarkson type inequalities for the
forms $A$ and $L_{\omega}$. In fact, we shall prove the Lemma.
###### Lemma 5.
The following inequalities hold
$L_{\omega}\left(\frac{f+g}{2}\right)+L_{\omega}\left(\frac{f-g}{2}\right)=\frac{L_{\omega}(f)+L_{\omega}(g)}{2},$
(18) $\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)\leq\frac{A(f^{2})+A(g^{2})}{8}$
$\displaystyle+\frac{3\sqrt{A(f^{2})A(g^{2})}}{4}.$ (19)
###### Proof 1.
It is easy to verify the relation
$\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)$
$\displaystyle=\frac{1}{16}A(f^{2}+g^{2}+2fg)+\frac{1}{16}A(f^{2}+g^{2}-2fg).$
(20)
Note that from
$A(a+b)+A(a-b)=2A(a)+2A(b),$
equality (1) becomes
$\displaystyle
A\left(\left(\frac{f+g}{2}\right)^{2}\right)+A\left(\left(\frac{f-g}{2}\right)^{2}\right)=\frac{1}{8}\left[A(f^{2}+g^{2})+4A((fg)^{2})\right]$
$\displaystyle=\frac{1}{8}\left[A(f^{2})+A(g^{2})+2A(f^{2},g^{2})+4A((fg)^{2})\right]$
$\displaystyle\leq\frac{A(f^{2})+A(g^{2})}{8}+\frac{3\sqrt{A(f^{2})A(g^{2})}}{4},$
(21)
which proves (5). The first relation (18) in the Lemma, follows directly.
The next result will play the crucial role in the present study. We shall
prove the following Lemma.
###### Lemma 6.
If $L_{\omega}(f)=L_{\omega}(g)$ and $\mu,\nu\geq 0$ satisfy
$2(\mu^{2}+\nu^{2})=1,$ then
$L_{\omega}\left(\mu f+\nu g\right)+L_{\omega}\left(\mu f-\nu
g\right)=L_{\omega}(f).$ (22)
If $A(f^{2})=A(g^{2})$ and $\mu,\nu\geq 0$ satisfy $2(\mu^{2}+\nu^{2})=1,$
then we have
$A\left(\left(\mu f+\nu g\right)^{2}\right)+A\left(\left(\mu f-\nu
g\right)^{2}\right)\leq A(f^{2}).$ (23)
###### Proof 2.
Setting $\mu_{1}=2\mu,$ $\nu_{1}=2\nu$, we apply (18) with $f,g$ replaced by
$\mu_{1}f$ and $\nu_{1}g$ respectively. Thus, we get
$L_{\omega}\left(\frac{\mu_{1}f+\nu_{1}g}{2}\right)+L_{\omega}\left(\frac{\mu_{1}f-\nu_{1}g}{2}\right)=\frac{\mu_{1}^{2}L_{\omega}(f)+\nu_{1}^{2}L_{\omega}(g)}{2}.$
(24)
From $L_{\omega}(f)=L_{\omega}(g)$ and $\mu_{1}^{2}+\nu_{1}^{2}=2$, we
complete the proof of (22).
Similarly, applying (5) and the assumption $A(f^{2})=A(g^{2})$, we find
$\displaystyle
A\left(\left(\frac{\mu_{1}f+\nu_{1}g}{2}\right)^{2}\right)+A\left(\left(\frac{\mu_{1}f-\nu_{1}g}{2}\right)^{2}\right)\leq\frac{\mu_{1}^{4}+\nu_{1}^{4}+6\mu_{1}^{2}\nu_{1}^{2}}{8}A(f^{2})$
or, equivalently
$\displaystyle A\left(\left(\frac{\mu f+\nu
g}{2}\right)^{2}\right)+A\left(\left(\frac{\mu f-\nu
g}{2}\right)^{2}\right)\leq 2(\mu^{4}+\nu^{4}+6\mu^{2}\nu^{2})A(f^{2})$ (25)
Consider now the homogeneous quartic polynomial
$2(\mu^{4}+\nu^{4}+6\mu^{2}\nu^{2})$ (26)
on the circle $\mu^{2}+\nu^{2}=\frac{1}{2}$. Substituting
$\nu^{2}=\frac{1}{2}-\mu^{2}$, we obtain the following estimate
$\displaystyle
2(\mu^{4}+\nu^{4}+6\mu^{2}\nu^{2})=2((\mu^{2}+\nu^{2})^{2}+4\mu^{2}\nu^{2})$
$\displaystyle=\frac{1}{2}+4\mu^{2}-8\mu^{4}=1-\frac{(1-4\mu^{2})^{2}}{2}\leq
1.$ (27)
Then, from (25) and (2) follows the proof of the Lemma.
Turning back to the minimization problem of the action functional
$S_{\omega}$, we observe the following fact. If $\chi(x)$ is a minimizer of
the problem
$\min_{\chi\in H^{1}}S_{\omega}(\chi),$ (28)
then $\hat{\chi}(x)$ and $-\hat{\chi}(x)$ are also minimizers of
$S_{\omega}(\chi)$. Moreover, we have the property.
###### Lemma 7.
Assume that $\chi(x)$ is a minimizer of the problem (28) and one of the
following alternatives:
1. 1.
$L_{\omega}(\chi-\hat{\chi})\geq 0$;
2. 2.
$L_{\omega}(\chi+\hat{\chi})\geq 0$
holds. Then $\chi=\hat{\chi}$.
###### Proof 3.
For simplicity, we shall consider the first case only. Suppose
$\chi\neq\hat{\chi}$, then from (18) we have
$L_{\omega}\left(\frac{\chi+\hat{\chi}}{2}\right)+L_{\omega}\left(\frac{\chi-\hat{\chi}}{2}\right)=L_{\omega}(\chi),$
(29)
implying
$L_{\omega}\left(\frac{\chi+\hat{\chi}}{2}\right)\leq L_{\omega}(\chi).$ (30)
On the other hand, it is easy to check that the following Cauchy inequalities
$A(f^{2},g^{2})\leq\sqrt{A(f^{2})A(g^{2})},\quad
A(fg)\leq\sqrt{A(f^{2})A(g^{2})}$ (31)
hold true. Applying now (5), we obtain
$\displaystyle
A\left(\left(\frac{\chi+\hat{\chi}}{2}\right)^{2}\right)+A\left(\left(\frac{\chi-\hat{\chi}}{2}\right)^{2}\right)\leq\frac{A(\chi^{2})+A(\hat{\chi}^{2})}{8}$
$\displaystyle+\frac{3\sqrt{A(\chi^{2})A(\hat{\chi}^{2})}}{4}\leq\frac{A(\chi^{2})+A(\hat{\chi}^{2})}{2}=A(\chi^{2}),$
(32)
which, together with the assumption $\chi\neq\hat{\chi}$ gives that
$A\left(\left(\frac{\chi+\hat{\chi}}{2}\right)^{2}\right)<A(\chi^{2}).$ (33)
Thus, from (30), (33) and the definition (16) it follows
$S_{\omega}\left(\frac{\chi+\hat{\chi}}{2}\right)\leq S_{\omega}(\chi),$ (34)
which contradicts to the assumption that $\chi$ is a minimizer. This proves
the Lemma.
Now, we are ready to prove the radial symmetry of the action minimizer, stated
in Theorem 3.
###### Proof of Theorem 3 1.
Taking into account Lemma 7, we shall take a minimizer $\chi(x)\geq 0$ of
$S_{\omega}$ and shall show that the condition
$\frac{1}{16}<\omega<\frac{1}{4},$
implies that $\chi=\hat{\chi}$ or
$L_{\omega}\left(\chi-\hat{\chi}\right)>0.$ (35)
Let
$\chi(x)=e_{0}(x)+f(x),$
where $e_{0}(x)=ce^{-|x|/2},c>0$ is the eigenvector corresponding to the first
eigenvalue of the operator $\Delta+1/|x|,$ while $\langle
f,e_{0}\rangle_{L^{2}}=0.$ Since $e_{0}$ is a radial function, we have
$\hat{e_{0}}=e_{0},$ so
$\chi-\hat{\chi}=f-\hat{f}=g,\ \ \langle g,e_{0}\rangle_{L^{2}}=0,\ g\neq 0.$
###### Lemma 8.
Let us assume that $g\perp e_{0}$ in $L^{2}$. Then
$L_{\omega}\left(g\right)\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}.$
###### Proof 4.
Note that $g\perp e_{0}$ in $L^{2}$ implies
$g=\sum_{k\geq 1}c_{k}e_{k}+h,$
where $h$ is in the absolutely continuous space of the self-adjoint operator
$\Delta+\frac{1}{|x|}$ in $L^{2}$, while $e_{k}$ are eigenvectors of the same
operator in $\\{g\in L^{2};g\perp e_{0}\\}$ with eigenvalues $\omega_{k}\leq
1/16.$ On the absolutely continuous space the operator has spectrum on
$(-\infty,0)$ and it is non positive, so
$\left\langle\left(\Delta+\frac{1}{|x|}\right)h,h\right\rangle\leq 0.$
Hence, we have
$\left\langle\left(\Delta+\frac{1}{|x|}\right)g,g\right\rangle\leq\sum|c_{k}|^{2}\omega_{k}\leq\frac{1}{16}\left(\sum|c_{k}|^{2}\right)=\frac{1}{16}\|g\|^{2}_{L^{2}}$
and
$L_{\omega}\left(g\right)=-\left\langle\left(\Delta+\frac{1}{|x|}\right)g,g\right\rangle+\omega\|g\|^{2}_{L^{2}}\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}.$
This completes the proof of the Lemma.
Applying the above Lemma, we find
$L_{\omega}\left(\chi-\hat{\chi}\right)=L_{\omega}\left(g\right)\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}>0,$
since $\omega>1/16$ and $g\neq 0.$ Hence, (35) is fulfilled and the proof of
the Theorem is complete.
## 4 Pohozaev identities
In this part we shall establish the so-called Pohozaev identities for (1).
More precisely, we shall prove the following
###### Lemma 9.
If $\chi\in H^{1}(\mathbb{R}^{3})$ and satisfies (1) in
$H^{-1}(\mathbb{R}^{3})$, then the following identities hold
$\|\nabla\chi\|^{2}_{L^{2}}+\omega\|\chi\|^{2}_{L^{2}}=\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx-A(|\chi|^{2}),$ (36)
$\|\nabla\chi\|^{2}_{L^{2}}+3\omega\|\chi\|^{2}_{L^{2}}=2\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx-\frac{5}{2}A(|\chi|^{2}).$ (37)
###### Proof 5.
To prove (36) we multiply equation (1) by $\bar{\chi}$, take the real part and
integrate over $\mathbb{R}^{3}$. To prove (37) we shall use the following
relations
$\nabla\cdot(x|\chi|^{2})=3|\chi|^{2}+2\;\mathrm{Re}\;\chi(x\cdot\nabla\bar{\chi}),$
(38)
$\nabla\cdot\left(x|\nabla\chi|^{2}-2\;\mathrm{Re}\;\nabla\chi(x\cdot\nabla\bar{\chi})\right)=|\nabla\chi|^{2}-2\;\mathrm{Re}\;\Delta\chi(x\cdot\nabla\bar{\chi}),$
(39)
$\nabla\cdot(x\frac{|\chi|^{2}}{|x|})=2\frac{|\chi|^{2}}{|x|}+2\;\mathrm{Re}\;\frac{\chi(x\cdot\nabla\bar{\chi})}{|x|},$
(40)
and
$\displaystyle\nabla\cdot\left(x\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}|\chi|^{2}\right)=3\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}|\chi|^{2}$
$\displaystyle\qquad-\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}dy}{|x-y|^{3}}|\chi|^{2}+2\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}dy}{|x-y|}\;\mathrm{Re}\;\chi(x\cdot\nabla\bar{\chi}).$
(41)
Integrating (38)–(5) over $\mathbb{R}^{3}$ implies the equalities
$\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\chi(x\cdot\nabla\bar{\chi})\
dx=-\frac{3}{2}\|\chi\|^{2}_{L^{2}},$ (42)
$\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\Delta\chi(x\cdot\nabla\bar{\chi})\
dx=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}},$ (43)
$\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\frac{1}{|x|}\chi(x\cdot\nabla\bar{\chi})\
dx=-\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\ dx,$ (44)
$\displaystyle\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}\chi(x)(x\cdot\nabla\bar{\chi}(x))}{|x-y|}dydx=-\frac{3}{2}A(|\chi|^{2})$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}dydx.$
(45)
On the other hand, observing the symmetry
$\displaystyle\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx$
$\displaystyle=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{y(y-x)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx,$ (46)
we calculate
$\displaystyle\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{x(x-y)|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{(x-y)^{2}|\chi(y)|^{2}|\chi(x)|^{2}}{|x-y|^{3}}\
dydx=\frac{1}{2}A(|\chi|^{2}).$ (47)
Substituting (5) into (5) we get
$\displaystyle\;\mathrm{Re}\;\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{|\chi(y)|^{2}\chi(x)(x\cdot\nabla\bar{\chi}(x))}{|x-y|}dydx=-\frac{5}{4}A(|\chi|^{2}).$
(48)
Finally, multiplying equation (1) by $x\cdot\nabla\bar{\chi}$, taking the real
part, integrating over $\mathbb{R}^{3}$ and using (42), (43), (44) and (48) we
complete the proof of the Lemma.
The Pohozaev identities are useful to treat the uniqueness of the minimizers
(modulo multiplication by complex constant $z$ with $|z|=1$). Indeed, let
$\chi_{1}$ and $\chi_{2}$ are minimizers of the problem
$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$ (49)
Since
$S_{\omega}(\chi)=\frac{1}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi\|^{2}_{L^{2}}-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx+\frac{1}{4}A(|\chi|^{2}),$
we can apply the Pohozaev identities of Lemma 9. In this way we find
$S_{\omega}(\chi)=-\frac{1}{4}A(|\chi|^{2})$ (50)
and
$A(|\chi_{1}|^{2})=A(|\chi_{2}|^{2}),\
L_{\omega}(\chi_{1})=L_{\omega}(\chi_{2}),$ (51)
where $L_{\omega}(\chi)$ is defined according to (14).
## 5 Uniqueness of minimizers
In this section we shall prove the uniqueness result of Theorem 4. The
classical approach for proving the uniqueness of minimizers is to reduce the
initial nonlinear equation to an ordinary differential equation, using the
radial symmetry. Uniqueness of positive ground state solutions for nonlinear
Schrödinger equation on ${\mathbb{R}}^{n}$ with local nonlinearities of the
form $|u|^{p}u$ for $0<p<\frac{4}{n-2}$, is a well-known fact, due to Kwong
[10]. The proof in this case relies on Sturm comparison theorems, but it
cannot be applied directly to nonlocal equations, such as (1). For the
attractive Choquard equation, Lieb in [11] prove uniqueness of energy
minimizer by using Newton’s theorem for radial function $f(x)=f(|x|)$, that is
$\displaystyle\int\frac{f(|y|)}{|x-y|^{n-2}}dy=\int\frac{f(|y|)}{\max\\{|x|,|y|\\}}dy.$
(52)
.
The repulsive sign of the Hartree term in (1) is again the main obstacle for
applying directly the standard technique.
###### Proof of Theorem 4 1.
Let $\chi_{1}$ and $\chi_{2}$ are non negative minimizers of the problem
$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$
Since they are radial functions, one can rewrite the elliptic equation (1),
using Newton’s theorem (52), as an ordinary differential equation of the form
$-\chi^{\prime\prime}(r)-\frac{2}{r}\chi^{\prime}(r)-\frac{\chi(r)}{r}+4\pi\int^{\infty}_{0}\frac{\chi^{2}(s)s^{2}ds}{\max\\{r,s\\}}\chi(r)+\omega\chi(r)=0.$
(53)
The above equation can be rewritten in the form
$-\chi^{\prime\prime}(r)-\frac{2}{r}\chi^{\prime}(r)-W(r)\chi(r)+4\pi\int^{r}_{0}\chi^{2}(s)\left(\frac{1}{r}-\frac{1}{s}\right)s^{2}ds\chi(r)+\omega\chi(r)=0,$
where
$W(r)=\frac{1}{r}-4\pi\int^{\infty}_{0}\chi^{2}(s)sds.$
If we set $u(r)=r\chi(r)$, then from the identity
$\chi^{\prime\prime}(r)+\frac{2}{r}\chi^{\prime}(r)=\frac{u^{\prime\prime}(r)}{r},$
the last equation becomes
$u^{\prime\prime}(r)+W(r)u(r)-4\pi\int^{r}_{0}\left(\frac{1}{r}-\frac{1}{s}\right)u^{2}(s)dsu(r)=\omega
u(r).$ (54)
This observation shows that the assumption $\chi(x)$ is a non negative
minimizer implies $u(r)>0$ for $r>0.$ Hence $\chi_{1}(x)$ and $\chi_{2}(x)$
are positive functions.
Our goal is to use the projection of $\chi_{1}$ and $\chi_{2}$ on the one
dimensional eigenspace
$E_{0}=\\{\alpha e^{-|x|/2},\alpha\in(-\infty,\infty)\\}$
is the eigenvector corresponding to the first eigenvalue $\omega_{0}=1/4$ of
the operator $\Delta+1/|x|.$ First, we have to observe that $\chi_{1}$ is not
orthogonal to $E_{0}.$ Indeed, if $\chi_{1}\perp E_{0}$, then Lemma 8 implies
$L_{\omega}(\chi_{1})\geq\left(\omega-\frac{1}{16}\right)\|\chi_{1}\|^{2}_{L^{2}}>0.$
The relation (16) guarantees now $S_{\omega}(\chi_{1})>0$ and this contradicts
the relation (50). The contradiction shows that $\chi_{1}$ (and also
$\chi_{2}$) is not orthogonal to $E_{0}.$
Let
$\chi_{1}=\mu_{1}\alpha e^{-|x|/2}+f_{1},\ \ \chi_{2}=\mu_{2}\alpha
e^{-|x|/2}+f_{2},$
where $\alpha e^{-|x|/2}\in E_{0},$ with $\alpha>0$ and $f_{1},f_{2}\perp
E_{0}.$ Note that $\mu_{1},\mu_{2}>0$, since $\chi_{1},\chi_{2}$ and $e_{0}$
are positive functions. We can choose $\alpha>0$, such that
$2(\mu_{1}^{2}+\mu_{2}^{2})=1,$ (55)
used as assumption in Lemma 6. The other assumption
$A(|\chi_{1}|^{2})=A(|\chi_{2}|^{2}),\
L_{\omega}(\chi_{1})=L_{\omega}(\chi_{2}),$
is already established in (51).
Applying Lemma 6, we find the identity
$L_{\omega}\left(\mu_{2}\chi_{1}+\mu_{1}\chi_{2}\right)+L_{\omega}\left(\mu_{2}\chi_{1}-\mu_{1}\chi_{2}\right)=L_{\omega}(\chi_{1}),$
as well as the inequality
$A\left(\left(\mu_{2}\chi_{1}+\mu_{1}\chi_{2}\right)^{2}\right)+A\left(\left(\mu_{2}\chi_{1}-\mu_{1}\chi_{2}\right)^{2}\right)\leq
A(\chi_{1}^{2}).$
Then, we have the relation
$\mu_{2}\chi_{1}-\mu_{1}\chi_{2}=\mu_{2}f_{1}-\mu_{1}f_{2}=g\perp E_{0}.$
If $g=0$, then $\chi_{1}=\mu_{1}\chi_{2}/\mu_{2}$ and one can use the ODE (54)
and the corresponding integral identities (36) and (37), to show that
$\chi_{1}=\chi_{2}.$ If $g\neq 0,$ then one can apply Lemma 8 and find
$L_{\omega}\left(\mu_{2}\chi_{1}-\mu_{1}\chi_{2}\right)\geq\left(\omega-\frac{1}{16}\right)\|g\|^{2}_{L^{2}}>0.$
Hence,
$S(\mu_{2}\chi_{1}+\mu_{1}\chi_{2})=\frac{1}{2}L_{\omega}(\mu_{2}\chi_{1}+\mu_{1}\chi_{2})+\frac{1}{4}A((\mu_{2}\chi_{1}+\mu_{1}\chi_{2})^{2})<S_{\omega}(\chi_{1})$
and this is a contradiction. The contradiction shows that $\chi_{1}=\chi_{2}$
and this completes the proof of Theorem 4.
## Appendix A Existence of action minimizers
The existence of action minimizers for Hartree type equation is already
established in [14]. For completeness, we shall sketch the proof.
To show the boundedness from below of $S_{\omega}$, we shall prove the
following inequalities involving homogeneous Sobolev norms
$\|f\|_{\dot{H}^{s}(\mathbb{R}^{3})}=\|(-\Delta)^{s/2}f\|_{L^{2}(\mathbb{R}^{3})},\
\ s>-3/2.$
###### Lemma 10.
For any $p_{1}\in[3,6]$ and $p_{2}\in[2,3]$ we have the estimates
$\left(\int_{|x|\leq 1}|\chi(x)|^{p_{1}}dx\right)^{1/p_{1}}\leq
C\|\chi\|^{\theta_{1}}_{\dot{H}^{1}}\|\chi^{2}\|^{(1-\theta_{1})/2}_{\dot{H}^{-1}}$
(56) $\left(\int_{|x|\geq 1}|\chi(x)|^{p_{2}}dx\right)^{1/p_{2}}\leq
C\|\chi\|^{\theta_{2}}_{L^{2}}\|\chi\|^{\theta_{3}}_{\dot{H}^{1}}\|\chi^{2}\|^{(1-\theta_{2}-\theta_{3})/2}_{\dot{H}^{-1}},$
(57)
where
$\theta_{1}=\frac{5}{3}-\frac{4}{p_{1}},\ \
\theta_{2}=\frac{4(3-p_{2})}{p_{2}},\ \ \theta_{3}=\frac{p_{2}-2}{p_{2}}.$
###### Remark 2.
The assumptions $p_{1}\in[3,6]$ and $p_{2}\in[2,3]$ guarantee that all
parameters $\theta_{1},\theta_{2},\theta_{3},\theta_{2}+\theta_{3}$ are in the
interval $[0,1].$
###### Remark 3.
The relation
$\|f\|^{2}_{\dot{H}^{-1}}=\langle(-\Delta)^{-1}f,f\rangle_{L^{2}}=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{f(x)f(y)}{|x-y|}dydx$
implies
$\|\chi^{2}\|^{2}_{\dot{H}^{-1}}=\frac{1}{4\pi}A(\chi^{2}).$
###### Proof 6.
For $p_{1}=6$ the inequality (56) becomes
$\left(\int_{|x|\leq 1}|\chi(x)|^{6}dx\right)^{1/6}\leq
C\|\chi\|_{\dot{H}^{1}}$
and this is the standard Sobolev embedding. For $p_{1}=3$ we have to verify
the following estimate
$\left(\int_{\mathbb{R}^{3}}|\chi(x)|^{3}dx\right)^{1/3}\leq
C\|\chi\|^{1/3}_{\dot{H}^{1}}\|\chi^{2}\|^{1/3}_{\dot{H}^{-1}}.$ (58)
This inequality follows from
$\left|\int f(x)g(x)dx\right|\leq\|f\|_{\dot{H}^{1}}\|g\|_{\dot{H}^{-1}}$
with $f(x)=|\chi(x)|,$ $g(x)=|\chi(x)|^{2}=\chi^{2}(x)$ and the observation
that
$\||\chi|\|_{\dot{H}^{1}}=\|\chi\|_{\dot{H}^{1}}.$
Interpolation between $p_{1}=6$ and $p_{1}=3$ proves (56).
The inequality (57) for $p_{2}=3$ follows from (58).
For $p_{2}=2$ (57) reduces to the simple inequality
$\left(\int_{|x|\geq 1}|\chi(x)|^{2}dx\right)^{1/2}\leq C\|\chi\|_{L^{2}}.$
An interpolation argument implies (57) and completes the proof of the Lemma.
After this Lemma we can show that the action functional is bounded from below.
###### Lemma 11.
For any $\omega>0$ the inequality
$\min_{\chi\in H^{1}}S_{\omega}(\chi)=S^{min}_{\omega}>-\infty$
holds. For $0<\omega<1/4$ we have $S^{min}_{\omega}<0.$
###### Proof 7.
The only negative term in $S_{\omega}$ is
$-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\ dx.$
Decomposing the integration domain into $|x|\leq 1$ and $|x|>1$ we apply
Hölder inequality and obtain
$\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\ dx\leq C\left(\int_{|x|\leq
1}|\chi(x)|^{p_{1}}dx\right)^{2/p_{1}}+C\left(\int_{|x|>1}|\chi(x)|^{p_{2}}dx\right)^{2/p_{2}},$
where $p_{1}>3>p_{2}$. Applying Lemma 10 as well as the Young inequality
$X^{\theta_{1}}Y^{\theta_{2}}Z^{\theta_{3}}\leq\varepsilon X+\varepsilon
Y+C_{\varepsilon}Z,$
with
$\theta_{j}\in(0,1),\theta_{1}+\theta_{2}+\theta_{3}=1,$
we get
$\int_{\mathbb{R}^{3}}\frac{|\chi(x)|^{2}}{|x|}\
dx\leq\varepsilon\|\chi\|^{2}_{L^{2}}+\varepsilon\|\nabla\chi\|^{2}_{L^{2}}+C_{\varepsilon}\sqrt{A(\chi^{2})}.$
This estimate implies
$S_{\omega}(\chi)\geq\frac{1-\varepsilon}{2}\|\nabla\chi\|^{2}_{L^{2}}+\frac{\omega-\varepsilon}{2}\|\chi\|^{2}_{L^{2}}+\frac{1}{4}A(\chi^{2})-C_{\varepsilon}\sqrt{A(\chi^{2})}.$
Choosing $\varepsilon>0$ so small that $\varepsilon<\min(1,\omega),$ we find
$S_{\omega}(\chi)\geq\frac{1}{4}A(\chi^{2})-C_{\varepsilon}\sqrt{A(\chi^{2})}\geq-2C_{\varepsilon}^{2}.$
To finish the proof we take $\chi_{\delta}(x)=\delta e^{-|x|/2},$ such that
$\left(\Delta+\frac{1}{|x|}\right)\chi_{\delta}=\frac{1}{4}\chi_{\delta}.$
Then
$2S_{\omega}(\chi_{\delta})=(\omega-1/4)\|\chi_{\delta}\|^{2}_{L^{2}}+A(\chi_{\delta}^{2})/2.$
Since
$\|\chi_{\delta}\|^{2}_{L^{2}}=C_{0}\delta^{2},\ \
A(\chi_{\delta}^{2})/2=O(\delta^{4}),$
the condition $\omega\in(0,1/4)$ implies $2S_{\omega}(\chi_{\delta})<0$ and
this completes the proof.
###### Proof of Theorem 2 1.
Take a minimizing sequence $\chi_{k}\in H^{1},$ so that
$\lim_{k\rightarrow\infty}S_{\omega}(\chi_{k})=S_{\omega}^{min}<0.$ (59)
The argument of the proof of Lemma 11 guarantees that there exists a constant
$C>0,$ so that
$\|\chi_{k}\|_{H^{1}}\leq C.$ (60)
One can find $\chi_{*}(x)\in H^{1}$ so that (after taking a subsequence)
$\chi_{k}$ tends weakly in $H^{1}$ to $\chi_{*}.$ Using the inequality
$\int_{|x|>R}\frac{|\chi(x)|^{2}}{|x|}\ dx\leq\frac{C}{R},$
and the compactness of the embedding $L^{p}(|x|<R)\hookrightarrow
H^{1}(|x|<R),$ when $2\leq p<6,$ we see that (choosing a suitable subsequence)
$\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx=\int_{\mathbb{R}^{3}}\frac{|\chi_{*}(x)|^{2}}{|x|}\ dx.$ (61)
Then we introduce $\varphi_{k},\ \varphi_{*}$ so that
$\Delta\varphi_{k}=-4\pi\chi_{k}^{2}(x),\ \
\Delta\varphi_{*}=-4\pi\chi_{*}^{2}(x).$
One can show that $\varphi_{k}$ tends weakly to $\varphi_{*}$ in
$\dot{H}^{1}.$ We have also the identities
$A(\chi_{k}^{2})=\int\varphi_{k}(x)\chi^{2}_{k}(x)dx=\frac{1}{4\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}$
and
$A(\chi_{*}^{2})=\int\varphi_{*}(x)\chi^{2}_{*}(x)dx=\frac{1}{4\pi}\|\nabla\varphi_{*}\|^{2}_{L^{2}}$
so we obtain
$S_{\omega}(\chi_{k})=\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{4}A(\chi_{k}^{2})-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx$
$=\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}-\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx.$
Using (59) and (61), we get
$\lim_{k\rightarrow\infty}S_{\omega}(\chi_{k})+\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{k}(x)|^{2}}{|x|}\
dx$
$=\lim_{k\rightarrow\infty}\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}=S_{\omega}^{min}+\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{*}(x)|^{2}}{|x|}\
dx.$
It is well - known that for any sequence $f_{k}$ in a Hilbert space $H$
tending weakly (in $H$) to $f_{*}\in H$, one has
$\liminf_{k\rightarrow\infty}\|f_{k}\|_{H}\geq\|f_{*}\|_{H}$ (62)
and
$\lim_{k\rightarrow\infty}\|f_{k}-f_{*}\|_{H}=0\ \ \Longleftrightarrow\ \
\lim_{k\rightarrow\infty}\|f_{k}\|_{H}=\|f_{*}\|_{H}.$ (63)
From (62) we have
$S_{\omega}^{min}+\frac{1}{2}\int_{\mathbb{R}^{3}}\frac{|\chi_{*}(x)|^{2}}{|x|}\
dx\geq\|\nabla\chi_{*}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{*}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{*}\|^{2}_{L^{2}}$
and a strict inequality is impossible since this will contradicts the
definition of $S_{\omega}^{min}.$ Hence
$\lim_{k\rightarrow\infty}\frac{1}{2}\|\nabla\chi_{k}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{k}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{k}\|^{2}_{L^{2}}=$
$=\frac{1}{2}\|\nabla\chi_{*}\|^{2}_{L^{2}}+\frac{\omega}{2}\|\chi_{*}\|^{2}_{L^{2}}+\frac{1}{16\pi}\|\nabla\varphi_{*}\|^{2}_{L^{2}}$
and applying (63) we conclude that
$\lim_{k\rightarrow\infty}\|\chi_{k}-\chi_{*}\|_{H^{1}}=0.$
This completes the proof of the Theorem.
## Appendix B Connection between the action and energy minimization problems
Consider the minimization problem
$S_{\omega}^{min}=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\},$ (64)
associated with the action functional $S_{\omega}(\chi)$ and the
Lions–Cazenave minimization problem
$I_{N}=\min\\{\mathcal{E}(\chi);\chi\in H^{1},\|\chi\|^{2}_{L^{2}}=N\\}.$ (65)
As we have seen before, for every $\omega\in(1/16,1/4)$, there exists (at most
one) solution $\chi_{\omega}\in H^{1}(\mathbb{R}^{3})$, which is positive and
radially symmetric, and such that
$S_{\omega}(\chi_{\omega})=S_{\omega}^{min}.$ (66)
Let us denote
$N(\omega)=\|\chi_{\omega}\|^{2}_{L^{2}}.$ (67)
The above definition of the function $N(\omega)$ poses the question if
$S_{\omega}^{min}=I_{N(\omega)}+\frac{\omega}{2}N(\omega).$
For completeness, in this section we shall prove the following Lemma.
###### Lemma 12.
If $\chi_{1}$ is a solution of (65) with $N=N(\omega)$, then $\chi_{1}$
satisfies the equation
$\displaystyle-\Delta\chi_{1}(x)+\int_{\mathbb{R}^{3}}\frac{\chi_{1}^{2}(y)dy}{|x-y|}\
\chi_{1}(x)-\frac{\chi_{1}(x)}{|x|}+\omega\chi_{1}(x)=0.$ (68)
and
$S_{\omega}(\chi_{1})=\min\\{S_{\omega}(\chi);\chi\in H^{1}\\}.$
###### Proof 8.
To prove the Lemma we shall follow the idea of the proof of Corollary 8.3.8 in
[3]. It is obvious, that the relation
$S_{\omega}(\chi_{1})=\mathcal{E}(\chi_{1})+\frac{\omega}{2}N(\omega),$
guarantees that $\chi_{1}$ is a minimizer of the problem
$\min_{\|\chi\|^{2}_{L^{2}}=N(\omega)}S_{\omega}(\chi).$
Since,
$S_{\omega}(\chi_{1})=\min_{\|\chi\|^{2}_{L^{2}}=N(\omega)}S_{\omega}(\chi)\geq\min
S_{\omega}(\chi)=S_{\omega}(\chi_{\omega}),$
we can use (67) and see that this inequality becomes equality, so
$S_{\omega}(\chi_{1})=\min_{\|\chi\|^{2}_{L^{2}}=N(\omega)}S_{\omega}(\chi)=\min
S_{\omega}(\chi)=S_{\omega}(\chi_{\omega}).$
Now, the uniqueness result of Theorem 4 implies $\chi_{1}=\chi_{\omega}$ and
completes the proof.
## References
* [1] N. Bazley and R. Seydel, Existence and bounds for critical energies of the Hartree equation, Chem. Phys. Lett., 24(1), 128-132, (1974).
* [2] J. Byeon, L. Jeanjean, M. Mariş Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations 36 (2009) 481 – 492.
* [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes 10, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, 2003.
* [4] T. Cazenave, P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations , Comm. Math. Physics 85 (1982) 549–561.
* [5] G. Coclite, V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Diff. Equations 94 (2004) 1–31.
* [6] V. Coti-Zelati, M. Esteban, Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain, Proc. Royal Soc. Edinburgh 116A (1990) 327 – 339.
* [7] M. Esteban, Nonsymmetric ground states of symmetric variational problems, Comm. Pure. Appl. Math. 44 (1991) 259–274.
* [8] M. Esteban, Rupture de symétrie pour des problémes de Neumann sur- linéaires dans les ouverts extérieur, Note C.R.A.S. 308 Série I (1989) 281 – 286.
* [9] B. Gidas, W. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^{N}$, Math. Anal. Appl. 7A (1981) 369 – 402.
* [10] M. Kwong, Uniqueness of positive solutions of $\triangle u-u+u^{p}=0$ in ${\mathbb{R}}^{n}$, Arch. Rational Mech. Anal. 105 (1989) 243–266.
* [11] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math., 57 2, (1977), p. 93 -105.
* [12] E. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, 1996.
* [13] E. Lieb, B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977) 185 – 194.
* [14] P.L.Lions, Some remarks on Hartree equation, Nonlinear Analysis: Theory, Methods $\&$ Applications, vol. 5, No 11, pg 1245-1256, 1981.
* [15] O. Lopes, Radial and non radial minimizers for some radially symmetric functionals, Electron. J. Diff. Equations 3 (1996) 1–14.
* [16] O. Lopes, M. Mariş, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Analysis 254 (2008) 535 – 592.
* [17] M. Mariş, On symmetry of minimizers , Arch. Ration. Mech. Anal. 192 (2009) 311 – 330
* [18] M. Reeken, A general theorem on bifurcation and its application to the Hartree equation of the Helium atom, J. Math. Phys. ll, (1970) 2505 – 2512.
* [19] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 (1971) 304 – 318.
|
arxiv-papers
| 2010-08-23T17:47:48 |
2024-09-04T02:49:12.392955
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir Georgiev and George Venkov",
"submitter": "Vladimir Georgiev",
"url": "https://arxiv.org/abs/1008.3871"
}
|
1008.3977
|
11institutetext:
%the␣affiliations␣are␣given␣next;␣don’t␣give␣your␣e-mail␣address%unless␣you␣accept␣that␣it␣will␣be␣publishedhttp://www.citizendium.org/User:Tom_Morris
22institutetext: http://www.citizendium.org/User:Daniel_Mietchen
Correspondence: Daniel.Mietchen (at) uni-jena (dot) de
Tom Morris and Daniel Mietchen
# Collaborative Structuring of Knowledge
by Experts and the Public
Tom Morris 11 Daniel Mietchen 22
###### Abstract
There is much debate on how public participation and expertise can be brought
together in collaborative knowledge environments. One of the experiments
addressing the issue directly is Citizendium. In seeking to harvest the
strengths (and avoiding the major pitfalls) of both user-generated wiki
projects and traditional expert-approved reference works, it is a wiki to
which anybody can contribute using their real names, while those with specific
expertise are given a special role in assessing the quality of content. Upon
fulfillment of a set of criteria like factual and linguistic accuracy, lack of
bias, and readability by non-specialists, these entries are forked into two
versions: a stable (and thus citable) approved ”cluster” (an article with
subpages providing supplementary information) and a draft version, the latter
to allow for further development and updates. We provide an overview of how
Citizendium is structured and what it offers to the open knowledge
communities, particularly to those engaged in education and research. Special
attention will be paid to the structures and processes put in place to provide
for transparent governance, to encourage collaboration, to resolve disputes in
a civil manner and by taking into account expert opinions, and to facilitate
navigation of the site and contextualization of its contents.
###### Keywords:
open knowledge, open education, open science, open
governance, wikis, expertise, Citizendium, Semantic Web
## 1 Introduction
> Science is already a wiki if you look at it a certain way. It’s just a
> highly inefficient one – the incremental edits are made in papers instead of
> wikispace, and significant effort is expended to recapitulate existing
> knowledge in a paper in order to support the one to three new assertions
> made in any one paper.
>
> John Wilbanks [21]
There are many ways to structure knowledge, including collaborative
arrangements of digital documents. Only a limited number of the latter ones
have so far been employed on a larger scale. Amongst them are wikis – online
platforms which allow the aggregation, interlinking and updation of diverse
sets of knowledge in an Open Access manner, i.e. with no costs to the reader.
### 1.1 Wikis as an example of public knowledge environments online
As implied by the introductory quote, it is probably fair to say that turning
science (or any system of knowledge production, for that matter) into a wiki
(or a set of interlinked collaborative platforms) would make research,
teaching and outreach much more transparent, less prone to hype, and more
efficient. Just imagine you had a time slider and could watch the history of
research on general relativity, plate tectonics, self-replication, or cell
division unfold from the earliest ideas of their earliest proponents (and
opponents) onwards up to you, your colleagues, and those with whom you compete
for grants. So why don’t we do it?
Traditionally, given the scope of a particular journal, knowledge about
specialist terms (which may describe completely non-congruent concepts in
different fields), methodologies, notations, mainstream opinions, trends, or
major controversies could reasonably be expected to be widespread amongst the
audience, which reduced the need to redundantly say and then repeat the same
things all over again and again (in cross-disciplinary environments, there is
a higher demand for proper disambiguation of the various meanings of a term).
Nonetheless, redundancy is still quite visible in journal articles, especially
in the introduction, methods, and discussion sections and the abstracts, often
in a way characteristic of the authors (such that services like eTBLAST and
JANE can make qualified guesses on authors of a particular piece of text, with
good results if some of the authors have a lot of papers in the respective
database, mainly PubMed, and if they have not changed their individual
research scope too often in between).
A manuscript well-adapted to the scope of one particular journal is often not
very intelligible to someone outside its intended audience, which hampers
cross-fertilization with other research fields (we will get back to this
below). When using paper as the sole medium of communication there is not much
to be done about this limitation. Indeed, we have become so used to it that
some do not perceive it as a limitation at all. Similar thoughts apply to
manuscript formatting. However, the times when paper alone reigned over
scholarly communication have certainly passed, and wiki-like platforms provide
for simple and efficient means of storing information, updating it and
embedding it into a wider context.
Cross-field fertilization, for example, is crucial with respect to
interdisciplinary research projects, digital libraries and multi-journal (or
indeed cross-disciplinary) bibliographic search engines (e.g. Google Scholar),
since these dramatically increase the likelihood of, say, a biologist
stumbling upon a not primarily biological source relevant to her research
(think shape quantification or growth curves, for instance). What options do
we have to systematically integrate such cross-disciplinary hidden treasures
with the traditional intra-disciplinary background knowledge and with new
insights resulting from research?
The by now classical example of a wiki environment are the Wikipedias, a set
of interlinked wikis in multiple languages where basically anyone can edit any
page, regardless of subject matter expertise or command of the respective
language. As a consequence of this openness, the larger Wikipedias have a
serious problem with vandalism: take an article of your choice and look at its
history page for reverts - most of them will be about neutralizing subtle or
blunt forms of destructive edits that do nothing to improve the quality of the
articles, but may reduce it considerably. Few of these malicious edits persist
for long [14], but finding and fixing them takes time that could better be
spent on improving articles. This is less of an issue with more popular topics
for which large numbers of volunteers may be available to correct ”spammy”
entries but it is probably fair to assume that most researchers value their
time too much to spend it on repeatedly correcting information that had
already been correctly entered. Other problems with covering scientific topics
at the Wikipedias include the nebulous notability criteria which have to be
fulfilled to avoid an article being deleted, and the rejection of ”original
research” in the sense of not having been peer reviewed before publication.
Despite these problems, one scientific journal – RNA Biology – already
requires an introductory Wikipedia article for a subset of papers it is to
publish [16].
Peer review is indeed a central aspect of scholarly communication, as it paves
the way towards the reproducibility that forms one of the foundations of
modern science. Yet we know of no compelling reason to believe that it works
better before than after the content concerned has been made public (doing it
beforehand was just a practical decision in times when journal space was
measured in paper pages), while emerging movements like Open Notebook Science
– where claims are linked directly to the underlying data that are being made
public as they arise – represent an experiment in this direction whose initial
results look promising and call into question Wikipedia’s ”no original
research” as a valid principle for generating encyclopaedic content.
Although quite prominent at the moment, the Wikipedias are not the only wikis
around, and amongst the more scholarly inclined alternatives, there are even a
number of wiki-based journals, though usually with a very narrow scope and/or
a low number of articles. On the other hand, Scholarpedia (which has classical
peer review and an ISSN and may thus be counted as a wiki journal, too [17]),
OpenWetWare [12], Citizendium [2] and the Wikiversities [20] are cross-
disciplinary and structured (and of a size, for the moment) such that
vandalism and notability are not really a problem. With minor exceptions, real
names are required at the first three, and anybody can contribute to entries
about anything, particularly in their fields of expertise. None of these is
even close to providing the vast amount of context existing in the English
Wikipedia but the difference is much less dramatic if the latter were broken
down to scholarly useful content. Out of these four wikis, only OpenWetWare is
explicitly designed to harbour original research, while the others allow
different amounts thereof. Furthermore, a growing number of yet more
specialized scholarly wikis exist (e.g. WikiGenes [18], the Encyclopedia of
Earth [8], the Encyclopedia of Cosmos [7], the Dispersive PDE Wiki [6], or the
Polymath Wiki [13]), which can teach us about the usefulness of wikis within
specific academic fields.
## 2 The Citizendium model of wiki-based collaboration
Despite the above-mentioned tensions between public participation and
expertise in the collaborative structuring of knowledge, it is not
unreasonable to expect that these can be overcome by suitably designed public
knowledge environments, much like Citizen Science projects involve the public
in the generation of scientific data. One approach at such a design is
represented by Citizendium. The founder of Citizendium – Larry Sanger – is the
co-founder of Wikipedia. The two projects share the common goal of providing
free knowledge to the public, they are based on variants of the same software
platform, and they use the same Creative Commons-Attribution-Share Alike
license [4]. Yet they differ in a number of important ways, such that
Citizendium can be seen as composed of a Wikipedia core (stripped down in
terms of content, templates, categories and policies), with elements added
that are characteristic of the other wiki environments introduced above: A
review process leading to stable versions (as at Scholarpedia), an open
education environment (as at Wikiversity) and an open research environment (as
at OpenWetWare). Nonetheless, assuming that the reader is less familiar with
these three latter environments, we will follow previous commenters and frame
the discussion of Citizendium in terms of properties differentiating it from
Wikipedia, and specifically the latter’s English language branch [19].
### 2.1 Real names
The first of these is simply an insistence on real names. While unusual from a
Wikipedia perspective, this is custom in professional environments, including
traditional academic publishing and some of the above-mentioned wikis, e.g.
Scholarpedia and Encyclopedia of Earth. It certainly excludes a number of
legitimate contributors who prefer to remain anonymous but otherwise gives
participants accountability and allows to bring in external reputation to the
project.
### 2.2 Expert guidance
Figure 1: Screenshot of the main page of the [[Crystal Palace]] cluster while
logged in using a monobook skin that is the default at Wikipedia. It shows the
green cluster bar that indicates that the page has been approved and links to
all the supages. Also visible is the status indicator (green dots on the left
topped by green tick), mention of ”an editor” to distinguish the number of
editors involved (some pages can be approved by one rather than three
editors), links to the workgroups which have approval rights for the article
(in this case: the History and Architecture Workgroups), a prominent
disclaimer (unapproved articles have a much strong disclaimer), and links to
the ’unstable’ draft version of the article which any registered contributor
can update. Like traditional encyclopaedic environments, Citizendium does not
require every statement to be referenced, in the interest of narrative flow.
To compose and develop articles and to embed them in the multimedial context
of a digital knowledge environment, expert guidance is important. Of course,
many experts contribute to Wikipedia, and the Wikipedias in turn have long
started to actively seek out expert involvement, yet the possibility to see
their edits overturned by anonymous users that may lack even the most basic
education in that field keeps professionals away from spending their precious
time on such a project. The Citizendium approach of verifying expertise takes
a different approach – sometimes termed ”credentialism” – that rests on a
common sense belief that some people do know more than others: it is sometimes
the case that the thirteen-year-old kid in Nebraska does know more than the
physics professor. But most of the time, at least when matters of physics are
concerned, this is not the case. The role the experts have at Citizendium is
not, as frequently stated in external comments, that of a supreme leader who
is allowed to exercise his will on the populace. On the contrary, it is much
more about guiding. We use the analogy of a village elder wandering around the
busy marketplace [15] who can resolve disputes and whom people respect for
their mature judgement, expertise and sage advice. Wikipedia rejects
”credentialism” in much the same way that the Internet Engineering Task Force
(IETF) does. David Clark summarised the IETF process thusly [3]: ”We reject
kings, presidents and voting. We believe in rough consensus and running code.”
In an open source project, or an IETF standardisation project, one can decide
a great many of the disputes with reference to technical reality: the
compiler, the existing network protocols etc. If the code doesn’t compile,
think again. For rough consensus to happen under such circumstances, one needs
to get the people together who have some clear aim in mind: getting two
different servers to communicate with one another. The rough consensus
required for producing an encyclopaedia article is different – it should
attempt to put forward what is known, and people disagree on this to a higher
degree than computers do on whether a proper connection has been established.
It is difficult to get ”rough consensus, running code” when two parties are
working on completely different epistemological standards. At this point, one
needs the advice of the village elderly who will vet existing content and
provide feedback on how it can be expanded or otherwise improved. Upon
fulfillment of a set of criteria like factual and linguistic accuracy, lack of
bias, and readability by non-specialists, these vetted entries are forked into
two versions: a stable (and thus citable) approved ”cluster” (an article with
subpages providing supplementary information) and a draft version, the latter
to allow for further development and updates (cf. Fig. 1).
The respect for experts because of their knowledge of facts is only part of
the reasoning: the experts point out and correct factual mistakes, but they
also help to guide the structuring of content within an article and by means
of the subpages. The experts bring with them the experience and knowledge of
years of in-depth involvement with their subject matter, and the project is
designed to make best use of this precious resource, while still allowing
everyone to participate in the process. Of course, experts are likewise free
to bring in content, be it within their specialty or in other areas, where
others take over the guiding role. The Citizendium can also host ’Signed
Articles’, which are placed in a subpage alongside the main article. A Signed
Article is an article on the topic described by a recognised expert in the
field, but can express opinions and biases in a way that the main article
ought not to.
Figure 2: Top: Screenshot of the Related Articles subpage from the [[Biology]]
cluster (which is approved) while logged in. It shows the Parent topics and
the first section of the Subtopics – subdisciplines. For each related article,
there is a short definition or description of the topic, and a link to its
Related Articles subpage (hidden behind the [r]), as well as instructions on
mouseover and a Table of Content. Bottom: Related Articles subpage from the
[[Open Knowledge Conference]] cluster (while logged out) which has not yet
been converted to subpage style but can already be used for structuring
information related to the topic. In principle, on could also think of adding
[[Open Knowledge Conference 2010]] as a subtopic and using this article for
conference blogging. However, the current MediaWiki software cannot handle
parallel editing by multiple users, though tools like Etherpad [9] have shown
that it is feasible.
### 2.3 Contextualization
Citizendium attempts to structure knowledge in a different way. Each article
on Citizendium can make comprehensive use of Subpages, i.e. pages providing
additional information that are subordinate to an article’s page. Some of
these – e.g. the Sculptures subpage in Fig. 1 – are similar to but more
flexible than the supplementary online materials now being published routinely
along scholarly articles. Two subpages types are different, with keywords and
running title being the closest analogues from academic papers: All pages are
encouraged to have a short Definition subpage (around 30 words or 150
characters) which defines or describes the subject of the page. They are also
encouraged to have a comprehensive Related Articles subpage, which uses
templates to pull in the definitions from the pages that it links to (a
feature that relies on the absence of vandalism). If one looks at the Related
Articles subpage of [[Biology]] (cf. Fig. 2, top), one can see the parent
topics of biology (science), the subtopics - subdisciplines of biology like
zoology, genetics and biochemistry, articles on the history of biology and
techniques used by biologists - and finally other related topics, including
material on the life cycle, the various biochemical substances like DNA and
proteins, the components of the cell, and other specialised language. This
Related Articles page gives a pretty comprehensive contextual introduction to
what biology is all about, and is structured by the authors of the article in
a way that is consistent across the site (cf. Fig. 2, bottom). This goes
beyond Wikipedias categories, ”See also” sections and ad-hoc infoboxes.
Citizendium’s approach can be considered as an exploratory next step towards
linking encyclopaedic content with the Semantic Web.
Subpages (a further usage example is in (cf. Fig. 3) are one way in which
Citizendium is attempting to go beyond what is provided in either traditional
paper-based encyclopaedias or by Wikipedia: to engage with context, with
related forms of knowledge, and to emancipate knowledge from the page format
to which it was confined in the print era. Marx wrote that ”Philosophers have
hitherto only interpreted the world in various ways; the point is to change
it” [11]. Traditional encyclopaedias attempt to reflect the world, but we are
attempting to go further. The open science movement - which has formed around
the combination of providing open access to journal articles, making
scientific data more openly available in raw forms, using and sharing open
source software and experimenting with some of the new techniques appearing
from the community that is formed under the ’Web 2.0’ banner - is exploring
the edge of what is now possible for scientists to do to create new knowledge.
Some of the electronic engagements by academics has been for actual research
benefit, some has just been PR for universities - doing podcasts to sound
’relevant’. The Citizendium model, while a little bit more traditional than
some of the open science platforms, is willing to try a variety of new things.
Wikipedia has produced a pretty good first version of a collaboratively
written encyclopedia – the challenge is to see if we can go further and
produce a citizens’ compendium of structured and comprehensive knowledge and
update it as new evidence or insights arise.
Figure 3: Top: Screenshot of the main page of the [[English spellings]]
cluster while logged out. It shows the blue cluster bar that indicates that
the page has not been approved and links to all the supages. Also visible is
the status indicator (red dots on the left topped by grey dots, indicating the
status of a ’developing’ article), and a stronger disclaimer than on approved
pages. Below this standard header, a set of templates links to ’Catalog’
subpages that collect links, for each letter of the English alphabet, to
Alphabetical and Retroalphabetical lists of spellings, to lists of Common
misspellings as well as to an article on the specific Letter. Bottom: Close-up
of the Catalogs subpage hosting the retroalphabetical list of English
spellings for the letter T, again cross-linked with all the other subpages in
that cluster.
### 2.4 Open governance
Citizendium has an evolving, but hopefully soon-to-be clearly defined
governance process - a Charter is in the process of being drafted by an
elected group of writers that will allow for democratic governance and
oversight. The broad outline is this: we will have a democratically elected
Editorial Council which will deal with content policy and resolving disputes
regarding content, and we will also have a Management Committee, responsible
for anything not related to content. The Management Committee appoint
Constables who uphold community policy regarding behaviour. Disputes with the
Constables can be brought to an Ombudsman selected by the Editorial Council
and Management Committee. At the time of writing, the charter is still to be
ratified by the community. One of the reasons we have this is that although
there is a cost to having bureaucracy and democracy, the benefits of having an
open governance process outweigh the costs. We have a real problem when
governments of real-life communities are controlled by shadowy cabals who
invoke byzantine legal codes - all the same problems would seem to apply to
online communities. With a Wikipedia article, the debate seems to shift very
quickly from the truth value or relevance of the content itself into often
ritualized arguments about acronyms (AfDs, NPOV, CSD, ArbCom, OR etc.). There
is always a challenge in any knowledge-based community in attempting to
reconcile a fair and democratic process with a meritocratic respect for
expertise. There are no easy answers - if we go too far towards bureaucracy,
we risk creating a system where management is separated from the actual day-
to-day writing of the site, while if we attempt to let the site ’manage
itself’, we risk creating a rather conservative mob rule that doesn’t afford
due process to interested outsiders. A more traditional management structure,
combined with real names and civility, should help those outside of the online
community - the many experts in real life who work in universities, in
business and in public life - to participate on an equal footing. Hopefully,
if we get the governance decisions right, we can also not get in the way of
the people who engage on hobbyist terms with Citizendium.
### 2.5 Open education
An important part of the governance process is collaboration with partners
external to the Citizendium. One of our initiatives – called Eduzendium –
provides for educators in higher education to assign work on wiki articles as
part of a course. We have most recently had politics students from the
Illinois State University work on articles on pressure groups in American
public life, as well as medical students from Edinburgh, biologists from City
University of New York and the University of Colorado at Boulder, finance
students from Temple University and others. These courses reserve a batch of
articles for the duration of the course, and assign each article to one or
more students. The course instructor can reserve the articles for just the
group of students enrolled in the course, or invite the wider Citizendium
community to participate. Much of the formatting is achieved via course-
specific templates that can be generated semi-automatically by the instructor
and applied throughout the course pages, so that course participants can
concentrate on content.
## 3 Open questions
The project is still young, resulting in a number of challenges and
opportunities. In many fields, Citizendium does not meet our own standards –
we do not have a full range of expert editors. Larry Sanger once envisioned
that Citizendium could reach 100,000 articles by 2012. This would, on average,
require about 150 new articles a day to reach; the current level is around 15.
It is not obvious how the necessary shift from linear to exponential growth
can be achieved.
Motivating both editors and authors to take part in both writing and approving
of content remains a difficult challenge – most experts have very little time
to offer for projects that do not contribute to the metrics according to which
their performance is evaluated, and others shy away from contributing under
their real name and in the presence of experts. Another problem is that the
initial structure of the community, and the nature of its interaction with
Wikipedia, has led to a few articles on popular pseudoscientific topics which
are hard to handle from an editorial perspective because those willing to
invest their time on the topics are usually heavily biased in their approach,
and most of those capable of evidence-based comment prefer not to contribute
to these topics.
The project also needs to allow for more feedback by non-registered readers,
without harming the currently very collegial atmosphere that is to a large
extent due to the real-name policy and the respect for expertise. We may need
to explore how to codify our core policies and collaboration model as a
possible MediaWiki extension, from which other wikis could possibly benefit –
online, ”code is law” [10], as is currently being highlighted by sites like
Stack Overflow which have changed the social interactions of participants by
changing formal features of user experience and social structure. We need to
find financial backing and support. So far, the project has been run on a
basically volunteer-only basis, yet the envisioned growth and improvement of
English-language content and the possible start of branches in other languages
require a higher degree of professionalisation, for which the upcoming Charter
is meant as a basis.
## 4 Open perspectives
Citizendium is open for partnerships with other open science and online
knowledge communities and projects. Possible candidate projects would include,
for instance, AcaWiki [1] for references, OpenWetWare for primary research,
and Open Access journals [5] as possible content providers, and of course the
Wikipedias and other public wikis for exchange on matters of content
management, community development and user experience. The key strength we
think the Citizendium model brings is a greater focus on knowledge
contextualization: it will be interesting to see whether we can evolve the
social model for knowledge production to keep up with changes in the
technological possibilities. Many in the Citizendium community are looking
forward to working alongside both academics and those working in the Semantic
Web community to tie Citizendium into data projects. We feel that despite the
commoditization of Web 2.0 technologies, there is still plenty of
opportunities for reinventing and experimenting with new ways to render and
collaborate on knowledge production and to see if we can build a more stable,
sustainable and collegial atmosphere – with democratic and meritocratic
elements – for experts and the public to work together.
#### Acknowledgments.
The authors wish to thank Russell D. Jones, Howard C. Berkowitz, Steven
Mansour and Peter Schmitt for critical comments on earlier versions of this
draft as well as Claudia Koltzenburg, François Dongier and Charles van den
Heuvel for helpful discussions.
## References
* [1] AcaWiki,
http://acawiki.org/
All URLs referenced in this article were functional as of March 31, 2010.
* [2] Citizendium,
http://www.citizendium.org/
* [3] Clark, D.: Plenary lecture, ”A Cloudy Crystal Ball – Visions of the Future”, Proc. 24th IETF: 539 (1992),
http://www.ietf.org/proceedings/prior29/IETF24.pdf
* [4] Creative Commons-Attribution-Share Alike license 3.0,
http://creativecommons.org/licenses/by-sa/3.0/
* [5] Directory of Open Access Journals,
http://www.doaj.org/
* [6] Dispersive PDE Wiki,
http://tosio.math.utoronto.ca/wiki/
* [7] Encyclopedia of Cosmos,
http://www.cosmosportal.org/
* [8] Encyclopedia of Earth,
http://www.eoearth.org
* [9] Etherpad source code,
http://code.google.com/p/etherpad/
* [10] Lessig, Lawrence: Code and Other Laws of Cyberspace,
http://codev2.cc/
* [11] Marx, Karl: Theses on Feuerbach,
http://www.marxists.org/archive/marx/works/1845/theses/theses.htm
* [12] OpenWetWare,
http://www.openwetware.org/
* [13] Polymath WIki,
http://michaelnielsen.org/polymath1/
* [14] Priedhorsky R, Chen J, Lam STK, Panciera K, Terveen L, et al. (2007) Creating, destroying, and restoring value in wikipedia. In: GROUP ’07: Proceedings of the 2007 international ACM conference on Supporting group work. New York, NY, USA: ACM, pp. 259–268. http://doi.acm.org/10.1145/1316624.1316663.
* [15] Raymond, Eric S.: The Cathedral and the Bazaar,
http://www.catb.org/~esr/writings/homesteading/
* [16] RNA Biology, Guidelines for the RNA Families Track,
http://www.landesbioscience.com/journals/rnabiology/guidelines/
* [17] Scholarpedia,
http://www.scholarpedia.org/
* [18] WikiGenes,
http://www.wikigenes.org/
* [19] English Wikipedia,
http://en.wikipedia.org
* [20] Wikiversity,
http://www.wikiversity.org
* [21] Wilbanks, J.: Publishing science on the web,
http://scienceblogs.com/commonknowledge/2009/07/publishing_science_on_the_web.php
|
arxiv-papers
| 2010-08-24T07:50:14 |
2024-09-04T02:49:12.401679
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tom Morris and Daniel Mietchen",
"submitter": "Daniel Mietchen",
"url": "https://arxiv.org/abs/1008.3977"
}
|
1008.4034
|
# Testing Lorentz Invariance with Neutrinos from Ultrahigh Energy Cosmic Ray
Interactions
Sean T. Scully scullyst@jmu.edu Department of Physics and Astronomy, James
Madison University,Harrisonburg, VA 22807 Floyd W. Stecker
Floyd.W.Stecker@nasa.gov NASA/Goddard Space Flight Center, Greenbelt, MD 20771
###### Abstract
We have previously shown that a very small amount of Lorentz invariance
violation (LIV), which suppresses photomeson interactions of ultrahigh energy
cosmic rays (UHECRs) with cosmic background radiation (CBR) photons, can
produce a spectrum of cosmic rays that is consistent with that currently
observed by the Pierre Auger Observatory (PAO) and HiRes experiments. Here, we
calculate the corresponding flux of high energy neutrinos generated by the
propagation of UHECR protons through the CBR in the presence of LIV. We find
that LIV produces a reduction in the flux of the highest energy neutrinos and
a reduction in the energy of the peak of the neutrino energy flux spectrum,
both depending on the strength of the LIV. Thus, observations of the UHE
neutrino spectrum provide a clear test for the existence and amount of LIV at
the highest energies. We further discuss the ability of current and future
proposed detectors make such observations.
###### keywords:
cosmic rays; neutrinos; Lorentz invariance; quantum gravity
††journal: Astroparticle Physics
## 1 Introduction
Ultrahigh energy cosmic rays and neutrinos are of interest as possible probes
of new physics [1]. In particular, some quantum gravity models predict that
Lorentz invariance may be weakly broken at the very high energies, leading to
potentially observable consequences. The possibility of using ultrahigh energy
cosmic rays (UHECRs) to probe for a small violation of Lorentz invariance was
suggested over a decade ago [2]. Indeed, a detailed analysis of the effects of
LIV on the UHECR spectrum has yielded the tightest constraint on LIV to date
[3]. Shortly after the discovery of the CBR it was pointed out that photomeson
interactions of UHECRs with photons of the cosmic background radiation (CBR)
would result in a sharp steepening of their spectrum above E $\sim$ 50 EeV now
known as the ”GZK effect” [4, 5]. However, even a very small amount of LIV
will kinematically inhibit some of these interactions. It has been previously
shown that a possible signature of LIV in the UHECR spectrum would be a
recovery of the cosmic ray spectrum at energies greater than $\sim$ 200 EeV
([3, 6]).
Given the current state of the UHECR observational data as reported by HiRes
[7, 8] and Auger [9], it is possible to constrain LIV, but not to rule it out.
However, unlike cosmic-ray baryons and photons, ultrahigh energy neutrinos do
not suffer significant energy losses over cosmological distances. Studies of
UHE neutrinos with both ground-based and space-based detectors could provide a
new and less ambiguous test of LIV. This is because they are a guaranteed
byproduct of the photomeson interactions of UHECRs with the CBR followed by
subsequent pion decay [10, 11, 12, 13].
In this paper, we further consider the observational implications of the
effect of a very small amount of LIV, viz. the suppression of photomeson
production on the resulting UHE neutrino spectrum. In our previous work [3, 6]
we undertook a detailed calculation of the modification of the UHECR spectrum
caused by LIV using the formalism of reference [2] and the kinematical
approach originally developed in reference [14]. We employ the same techniques
used in references [3] and [6] to determine the resulting photomeson neutrino
spectrum. We again consider here the case where the primary UHECRs are
protons.
ANITA II [15] has placed an upper limit on the neutrino flux for energies
greater than $>10^{18}$ eV. IceCube is nearing completion and will be
sensitive to neutrinos of energies up to $\sim 10^{17}$eV. Proposed future
ground-based and space-based neutrino detectors could be capable of detecting
and studying photomeson neutrinos. We will discuss the ability of such
detectors to constrain LIV or observe its effect.
## 2 LIV and the Spectrum of UHECRs
We now extend the calculation of reference [3] to determine the photomeson
neutrino fluxes.111We use the usual convention $c=1$. A full description of
the LIV formalism we use is given in references [2] and [3]. We summarize the
salient points here. The free particle Lagrangian is modified by the inclusion
of a leading order perturbative, Lorentz violating term. This term leads to
the modified free particle dispersion relations
$E^{2}~{}=~{}\vec{p}\ ^{2}+m^{2}+2\delta\vec{p}\ ^{2}.$ (1)
These relations can be put in the standard form
$E^{2}~{}=~{}\vec{p\ }{{}^{2}}c_{MAV}^{2}+m^{2}c_{MAV}^{4},$ (2)
by shifting the renormalized mass by the small amount $m\rightarrow
m/(1+2\delta)$ and shifting the velocity by the amount
$c_{MAV}=\sqrt{(1+2\delta)}\simeq 1+\delta$ (3)
where $c_{MAV}$ is identified as the maximum attainable velocity of the free
particle in the reference frame of the CBR. Using this formalism, different
particles can have different maximum attainable velocities (MAVs) that can all
be different from $1$ as well as different from one another. Hereafter, we
denote the MAV of a particle of type $i$ by $c_{i}$ and the difference
$c_{i}-c_{j}~{}~{}\equiv~{}\delta_{ij}$ (4)
These modified dispersion relations are then applied to the kinematical
relations governing the dominant single meson photomeson interaction:
$p+\gamma\rightarrow N+\pi.$ (5)
From equations (1) and (4), a dispersion relation can be constructed for a
particle $a$
$E^{2}=p^{2}+2\delta_{a}p^{2}+{m_{a}}^{2}$ (6)
where $\delta_{a}$ is the difference between the MAV for the particle a and
the speed of light in the low momentum limit, $c=1$.
In order to modify the effect of photomeson production on the UHECR spectrum
above the GZK energy, $\delta_{\pi p}>0$ as shown in reference [2]. The
condition for photomeson interactions to take place is
$\delta_{\pi p}\leq 3.23\times 10^{-24}(\omega/\omega_{0})^{2}.$ (7)
where $\omega$ is the energy of the CBR photon and $\omega_{0}\equiv
kT_{CBR}=2.35\times 10^{-4}$ eV with $T_{CBR}=2.725\pm 0.02$ K [2, 3].
If LIV occurs and $\delta_{\pi p}>0$, photomeson production can only take
place for interactions of CBR photons with energies large enough to satisfy
equation (7). This condition implies that while photomeson interactions
leading to GZK suppression can occur for “lower energy” UHE protons
interacting with higher energy CBR photons on the Wien tail of the spectrum,
other interactions involving higher energy protons and photons with smaller
values of $\omega$ will be forbidden [3]. Thus, the observed UHECR spectrum
may exhibit the characteristics of GZK suppression near the normal GZK
threshold, but the UHECR spectrum can “recover” at higher energies owing to
the possibility that photomeson interactions at higher proton energies may be
forbidden [3, 6, 16, 17]. Even a small violation of Lorentz invariance changes
the inelasticity of the interaction, (i.e., the amount of energy transferred
from the incident proton to the created pion). This is the key to
understanding the effect of LIV on photomeson production. With an increase in
proton energy, the range of kinematically allowed angles of interaction
between it and the photon becomes more restricted, thus reducing the phase
space and, in turn, the total inelasticity. Figure 1, reproduced from
reference [3], shows the calculated inelasticity modified by LIV for a value
of $\delta_{\pi p}=3\times 10^{-23}$ as a function of both CBR photon energy
and incident proton energy. Other choices for $\delta_{\pi p}$ yield similar
plots but change the energy at which LIV effects become significant. The
inelasticity precipitously drops above a certain energy because the LIV term
in the pion rest energy from equation (6) becomes comparable to $m_{\pi}$.
Figure 1: The calculated proton inelasticity modified by LIV for $\delta_{\pi
p}=3\times 10^{-23}$ as a function of CBR photon energy and proton energy
(from reference [3]).
The proton energy loss rate by photomeson production is given by
${{1}\over{E}}{{dE}\over{dt}}=-{{\omega_{0}c}\over{2\pi^{2}\gamma^{2}}\hbar^{3}c^{3}}\int\limits_{\eta}^{\infty}d\epsilon~{}\epsilon~{}\sigma(\epsilon)K(\epsilon)\ln[1-e^{-\epsilon/2\gamma\omega_{0}}]$
(8)
where $\epsilon$ is the photon energy in the center of mass system,
$K(\epsilon)$ is the modified inelasticity calculated from the kinematics, and
$\sigma(\epsilon)$ is the total $\gamma$-p cross section. The lower limit of
the integration,$\eta$, is the photon threshold energy for the interaction in
the center of mass frame.
As in reference [3], we assume that the source spectrum of UHE protons can be
approximated over a limited energy range by a power-law that fits the UHECR
data below 60 EeV. This spectrum is then of the form $A(z)E_{i}^{-\Gamma}$
where $E_{i}$ is the energy of the proton. The UHECRs suffer energy losses
from pair and pion production through interactions with the CBR and also
cosmological redshifting. The energy losses from pion production are
determined according to equation (8). The pair-production loss rate comes from
[18]. In order to determine redshift loses, a flat $\Lambda$CDM universe with
a Hubble constant of H0 = 70 km s-1 Mpc-1 is assumed, taking
$\Omega_{\Lambda}$ = 0.7 and $\Omega_{m}$ = 0.3. The source evolution is
additionally assumed $\propto(1+z)^{\zeta}$ with $\zeta$ = 3.6, out to a
maximum redshift of 2.5 which tracks the star formation rate. This value is a
mean between the fast evolution and baseline models used in reference [19].
(See also references [20, 21].) The spectrum of UHECRs on Earth can then be
determined from
$\displaystyle J(E)={{3cA(0)}\over{8\pi
H_{0}}}E^{-\Gamma}\int_{0}^{z_{max}}{{(1+z)^{(\zeta-1)}}\over{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}}\left({E_{i}\over{E}}\right)^{-\Gamma}{{dE_{i}}\over{dE}}dz.$
(9)
where A(0) is determined by fitting our final calculated spectrum to the
observational UHECR data assuming $\Gamma=2.55$, which is consistent with the
spectrum derived by the Pierre Auger Observatory (PAO) collaboration below 60
EeV.222We have chosen a maximum UHECR energy of $3\times 10^{21}$ eV. However,
our results are insensitive to this value because they are determined by the
LIV kinematics. The results of this calculation for various choices of the
parameter $\delta_{\pi p}$ are shown in Figure 2 plotted along with the most
recent results from PAO.
Figure 2: Comparison of the PAO data [9] with calculated spectra for various
values of $\delta_{\pi p}.$ From top to bottom, the curves give the predicted
spectra for $\delta_{\pi p}=1\times 10^{-22},6\times 10^{-23},3\times
10^{-23},1\times 10^{-23},0$ (no Lorentz violation).
It can be seen from Figure 2 that a small amount of LIV can still preserve the
GZK suppression effect, but produces a ”recovery” of the UHECR spectrum at
higher energies. Since this recovery is due to the virtual elimination of
photomeson interactions at higher UHECR energies, it will also suppress the
production of higher energy photomeson neutrinos.
## 3 The Photomeson Neutrino Spectrum
We now turn our attention to calculating the photomeson neutrino spectrum that
would result from the UHECR calculation detailed in the previous section. We
use the data on the cross section for pion production compiled in reference
[22] and summarized in reference [23]. Near threshold the the total photomeson
cross section is dominated by the emission of single pions. The most
significant channel to consider involves the intermediate production of the
$\Delta$ resonance [24]:
$p+\gamma\rightarrow\Delta\rightarrow N+\pi$ (10)
This channel strongly dominates the photomeson production process near
threshold. Since the UHECR flux falls steeply with energy, it follows that the
bulk of the pions leading to the production of neutrinos will be produced
close to the threshold.
For a proton interacting with the CBR, a pion and a nucleon are produced. The
outgoing nucleon has probability of 2/3 to be a proton and 1/3 probability to
be a neutron from isospin considerations. Should the resulting nucleon be a
neutron, then the resulting pion is a $\pi^{+}$. Thus approximately twice the
number of neutral pions are produced relative to charged pions from resonant
pion production. However direct pion production, which accounts for about 20%
of the total cross section, produces charged pions almost exclusively meaning
that all told, approximately equal numbers of neutral and charged pions are
produced around threshold. Neutral pions decay into photons so we need only
consider the charged pions for neutrino production. Three neutrinos of roughly
equal energy result from the decay chain of the
$\pi^{+}\rightarrow\mu^{+}\nu_{\mu}\rightarrow
e^{+}\bar{\nu{{}_{\mu}}}\nu_{e}$.
It is straightforward to determine the neutrinos produced and their energies
from the UHECRs. We follow closely the calculation of the neutrino flux as
described in reference [12] and references therein. The key is to determine
the amount of energy that is carried away by the pion. This follows directly
from the inelasticity and the incident proton energy calculated using equation
(8). We assume that all of the sources have the same primary injection
spectrum and distribution as detailed in section 2. We calculate the total
neutrino flux by integrating over proton energy, photon energy, and redshift,
assuming the standard $\Lambda$CDM cosmology. We find that the shape of the
neutrino spectrum we obtain when assuming Lorentz invariance is very similar
to that obtained from the more detailed Monte Carlo calculations that include
all the relevant baryonic resonances and possible meson and multi-pion
production channels (See, e.g., reference [13]).
The effect of LIV on the photomeson neutrino production is again manifested
through the modification of the inelasticity of the interaction, since this
determines the amount of energy that is carried away by the pion and therefore
the resultant neutrino energy. The biggest impact of including LIV is to
suppress the production of the higher energy photopions and therefore the
resulting higher energy neutrinos.
Figure 3 shows the corresponding total neutrino flux (all species) for the
same choices of $\delta_{\pi p}$ as the UHECR spectra presented in figure 2.
As expected, increasing $\delta_{\pi p}$ leads to a decreased flux of higher
energy photomeson neutrinos as the interactions involving higher energy UHECRs
are suppressed [3]. It is also evident that the peak energy of the neutrino
energy flux spectrum (EFS), $E\Phi(E)$, shifts to lower energies with
increasing $\delta_{\pi p}$.
Figure 3: Neutrino fluxes (of all species) corresponding to the UHECR models
considered in Figure 2. From left to right, the curves give the predicted
fluxes for $\delta_{\pi p}=1\times 10^{-22},6\times 10^{-23},3\times
10^{-23},1\times 10^{-23},0$.
We note that a similar effect can be produced on the photomeson neutrino
spectrum from another possible mechanism involving LIV, namely ”neutrino
splitting” [25]. With LIV, the decay of one neutrino into three neutrinos can
be kinematically allowed. This effect may also produce a decreased flux at the
high energy end of the neutrino spectrum. The feature that distinguishes
between the two possible LIV effects is that neutrino splitting results in an
increase in the flux of lower energy photomeson neutrinos.
## 4 Considerations of UHECR Composition
Throughout this paper, we have made the assumption that the highest energy
cosmic rays, i.e., those above 100 EeV, are protons. The composition of these
primary particles is presently unknown. The highest energy events for which
composition measurements have been attempted are in the range between 40 and
50 EeV, and the composition of these events is uncertain [26],[27],[28].
We note that in the case where the UHECRs with total energy above $\sim$100
EeV are not protons, both the photomeson threshold and the LIV effects are
moved to higher energies because (i) the threshold is dependent on
$\gamma\propto E/A$, where $A$ is the atomic weight of the UHECR [29], and
(ii) it follows from equation (1) that the LIV effect depends on the
individual nucleon momentum. We also note that the neutrino spectrum at the
high energy end is the same for the mixed composition case as in the pure
proton case [30, 31, 32].
## 5 Observational Prospects
Several experiments currently place upper limits on the photomeson neutrino
flux. The ANITA long duration balloon experiment launched in December of 2008
searched for electromagnetic cascades initiated by UHE neutrinos within the
Antarctic ice shelf via the Askaryan effect. Their analysis yielded a model-
independent 90% CL limit on neutrino fluxes in the range of 1018 – 1023 eV
with a sensitivity capable of excluding several optimistic photomeson neutrino
flux models [15]. However, ANITA does not have sufficient sensitivity or
energy range to distinguish a possible LIV effect on the neutrino spectrum.
ANITA has an effective threshold energy $\sim 10^{18}$ eV and its sensitivity
about an order of magnitude too weak. IceCube is capable of detecting
neutrinos $<10^{17}$ eV, but its sensitivity is two orders of magnitude too
weak to detect photomeson neutrinos [33]. Proposed future space-borne missions
such as the Extreme-Universe Space Observatory (EUSO) [34], super EUSO [35],
and Orbiting Wide-angle Light Detectors (OWL) [36] would have much larger
effective aperatures than presently available detectors. They would be capable
of making accurate determinations of the energy, arrival direction, and
composition of the cosmic-rays and the associated photomeson neutrinos using a
target volume far greater than is possible from ground-based experiments
presently in operation. While such experiments could potentially provide the
statistics necessary to observe the recovery of the UHECR spectrum at high
energies, both EUSO and OWL as proposed would not achieve either the
sensitivity or energies necessary to distinguish LIV effects in the neutrino
spectrum.
However, the proposed full Antarctic Ross Ice shelf ANtenna Neutrino Array
(ARIANNA) would be capable of detecting photomeson neutrinos with a
sensitivity an order of magnitude better than ANITA and other existing
detectors. Like ANITA, ARIANNA exploits the Askaryan effect, i.e. the
detection of coherent Čerenkov emission at radio wavelengths produced in the
ice shelf by neutrino induced cascades. Because the power of coherent radio
emission grows as the square of the shower energy and therefore neutrino
energy, ARIANNA is capable of detecting lower energy neutrinos than ANITA
since the distances to balloon-borne detectors can be quite large while
ARIANNA utilizes detectors situated directly on the ice shelf surface. ARIANNA
is expected to observe $\sim$40 events per 6 months in the energy range of
photomeson neutrinos with energies in excess of $\sim 10^{17}$ eV [37]. This
lower energy threshold is crucial for searching for LIV effects. As such, we
shall restrict ourselves to the discussing the potential of ARIANNA for
distinguishing the effect of LIV on the photomeson neutrino spectrum. We note
that another proposed detector called IceRay would also make use of the
Askaryan effect. IceRay would be placed at the location of IceCube and would
also be capable of detecting photomeson neutrinos [38].
ARIANNA’s high event rate, combined with its low energy threshold can
distinguish LIV effects if $\delta_{\pi p}\leq 3.0\times 10^{-23}$ with a 5
year exposure. We note that this limit is close to the upper limit indicated
by the current PAO data [3]. Figure 4 shows our calculated fluxes along with
the sensitivity of ARIANNA for exposure times of 6 months [37] and 5 years.
Here we have plotted $E\Phi(E)$ for all neutrino flavors and $\nu$-$\bar{\nu}$
combinations compared with the proposed ARIANNA sensitivities to more clearly
illustrate the threshold effect. We note that ARIANNA is sensitive to all
neutrino flavors since it would primarily detect the hadronic shower and not
the outgoing lepton. In Figure 4, we present our neutrino fluxes and
experimental sensitivities as the total of all flavors and $\nu$-$\bar{\nu}$
combinations.
Figure 4: All-flavor neutrino flux spectra (EFS) that correspond to the UHECR
models considered in Figure 2. The six month sensitivity from the proposed
ARIANNA array is shown as a dashed curve. The dot-dashed curve shows the
sensitivity scaled to a 5 year exposure. From bottom to top, the solid curves
give the predicted spectra for $\delta_{\pi p}=1\times 10^{-22},6\times
10^{-23},3\times 10^{-23},1\times 10^{-23},0$.
It is clear from Figure 4 that at the lower energy threshold of $\sim 10^{17}$
eV for ARIANNA is very close to the peak energy in $E\Phi(E)$ for $\delta_{\pi
p}=3\times 10^{-23}$ and would have sufficient sensitivity at $10^{17}$ eV to
detect the expected neutrino flux provided it runs long enough to produce the
desired sensitivity. ARIANNA is therefore very promising in terms of its
ability to distinguish the effect of an amount of LIV that is compatible with
the current PAO results.
### 5.1 A Caveat
We note that suggestions have been made to use the derivation of the
extragalactic $\gamma$-ray background flux up to 100 GeV using data from the
Fermi Gamma-ray Space Telescope [39] to place limits on the absolute value of
the photomeson neutrino flux, assuming that the 100 GeV $\gamma$\- rays are
produced by an electromagnetic cascade off the CBR initiated by UHE pion decay
$\gamma$-rays produced along with the neutrinos [40],[41]. Such a constraint
would lower the neutrino flux to almost an order of magnitude below the
expected value and would make it much more difficult to test Lorentz
invariance with photomeson neutrinos. These results follow earlier neutrino
constraints [42] obtained using various analyses of the EGRET-GRO results on
the extragalactic $\gamma$-ray background [43],[44]. However, the argument
here is contingent on the assumption that the extragalactic magnetic field is
so small that a UHE electromagnetic pair-production-Compton cascade leading to
the production of $\gamma$-rays in the GeV energy range will not be cut off by
synchrotron losses of the UHE electrons dominating over Compton losses [45,
11]. At this point in time, the strength of the extragalactic B-field is only
constrained to be within the range $\sim 3\times 10^{-16}$ – $\sim 3\times
10^{-9}$ G. [46],[47].
In addition, we note that the Fermi spectrum is not the result of direct
observation, but of analysis. It critically involves the subtraction of both
galactic foreground $\gamma$-rays and, in the case of Fermi instrumental
calibration, by Monte Carlo modeling [48]. These are non-negligible
uncertainties. As an example of the uncertainties involved, we note the
significant differences between the EGRET-GRO results [43],[44] and the Fermi
results [39] on the extragalactic $\gamma$-ray background.
We also note that since both the photomeson production cross section and the
CBR photon spectrum are very well determined, and since the GZK cutoff effect
is well documented [8],[9], a significant decrease in the predicted photomeson
neutrino flux could either imply an unexpectedly small production of UHECRs or
the existence of new physics.
## 6 Conclusion
With future improved data from PAO, tighter constraints can be placed on the
amount of LIV allowed by the UHECR spectrum. However even after a decade more
of operation it seems unlikely that PAO would be able to determine the UHECR
spectrum with adequate statistics at energies greater that 300 EeV, where the
effect of LIV would be manifested. While space-borne missions such as JEM-EUSO
and OWL could provide the statistics necessary to observe the effect, these
missions are currently only in the planning stage and are many years from
being realized.
We have shown here that additional information on LIV or its constraints can
be obtained from studying the spectrum of photomeson neutrinos. By calculating
the flux of high energy neutrinos generated by the propagation of UHECR
protons through the CBR in the presence of LIV, we find that LIV produces a
reduction in the flux of the highest energy photomeson neutrinos and a
reduction in the energy of the peak of the neutrino energy flux spectrum with
both effects increasing with the strength of the LIV. Thus, observations of
the UHE neutrino spectrum could provide a clear test for the existence and
amount of LIV that would be exhibited in the highest energy cosmic-ray
interactions.
ARIANNA would have a sufficiently low threshold energy and the sensitivity
necessary to determine the location of the energy peak in the photomeson
neutrino EFS to further test LIV in the range $\delta_{\pi p}\leq 3\times
10^{-23}$, consistent with the current limit indicated by the PAO data. The
amount of LIV or its nondetection has important consequences for Planck scale
physics and quantum gravity theories.
## Acknowledgments
We would like to thank Steve Barwick, Peter Gorham, and John Krizmanic for
enlightening discussions and comments about the various present and potential
future neutrino detectors.
## References
## References
* [1] F. W. Stecker, High Energy Astrophysics Tests of Lorentz Invariance Violation, Intl. J. of Mod. Phys. A 20 (2005) 3139–3142. arXiv:astro-ph/0409731.
* [2] S. Coleman, S. L. Glashow, High-energy tests of Lorentz invariance, Phys. Rev. D 59 (1999) 116008.
* [3] F. W. Stecker, S. T. Scully, Searching for new physics with ultrahigh energy cosmic rays, New J. Phys. 11 (2009) 085003.
* [4] K. Greisen, End to the Cosmic Ray Spectrum?, Phys. Rev. Lett. 16 (1966) 748–750.
* [5] G. T. Zatsepin, V. A. Kuzmin, Upper limit of the Spectrum of Cosmic Rays, JETP Lett. 4 (1966) 78–80.
* [6] S. T. Scully, F. W. Stecker, Lorentz Invariance Violation and the Observed Spectrum of Ultrahigh Energy Cosmic Rays, Astropart. Phys. 31 (2009) 220–225.
* [7] J. Belz, Overview of recent HiRes results, Nucl. Phys. B (Proc. Suppl.) 190 (2009) 5–11.
* [8] P. Sokolsky, Observation of the GZK cutoff by the HiRes experiment, Nucl. Phys. B (Proc. Suppl.) 196 (2009) 67–73.
* [9] J. Abraham, for the Pierre Auger Collaboration, Measurement of the energy spectrum of cosmic rays above 1018 eV using the Pierre Auger Observatory, Phys. Lett. B 685 (2010) 239–246.
* [10] V. S. Berezinsky, G. T. Zatsepin, Cosmic neutrinos of superhigh energy, Yad. Fiz. 11 (1970) 200–205.
* [11] F. W. Stecker, Ultrahigh energy protons electrons and neutrinos the microwave background and the universal cosmic ray hypothesis, Astrophys. and Space Sci. 20 (1973) 47–57.
* [12] F. W. Stecker, Diffuse fluxes of cosmic high energy neutrinos, Astrophysical Journal 228 (1979) 919–927.
* [13] R. Engel, D. Seckel, T. Stanev, Neutrinos from propagation of ultrahigh energy protons, Phys. Rev. D 64 (2001) 093010.
* [14] J. Alfaro, G. Palma, Loop quantum gravity and ultrahigh energy cosmic rays, Phys. Rev. D 67 (2003) 083003.
* [15] P. W. Gorham, et al., Observational constraints on the ultrahigh energy cosmic neutrino flux from the second flight of the ANITA experiment, Phys. Rev. D 82 (2010) 022004.
* [16] L. Maccione, et al., Planck-scale Lorentz violation constrained by Ultra-High-Energy Cosmic Rays, J. Cosmology and Astropart. Phys. 04 (2009) 022\.
* [17] X. Bi, et al., Testing Lorentz invariance with the ultrahigh energy cosmic ray spectrum, Phys. Rev. D 79 (2009) 083015.
* [18] G. R. Blumenthal, Energy loss of high-energy cosmic rays in pair-producing collisions with ambient photons, Phys. Rev. D 1 (1970) 1596–1602.
* [19] F. W. Stecker, M. A. Malkan, S. T. Scully, Intergalactic Photon Spectra from the Far-IR to the UV Lyman Limit for z between 0 and 6 and the Optical Depth of the Universe to High-Energy Gamma Rays, Astrophys. J. 648 (2006) 774–783.
* [20] E. Le Floc’h, et al., Infrared Luminosity Functions from the Chandra Deep Field-South: The Spitzer View on the History of Dusty Star Formation at z between 0 and 1, Astrophys. J. 632 (2005) 169–190.
* [21] A. M. Hopkins, J. F. Beacom, On the Normalization of the Cosmic Star Formation History, Astrophys. J. 651 (2006) 142–154.
* [22] R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, Analysis of pion photoproduction data, Phys. Rev. C 66 (2002) 055213.
* [23] A. M. Bernstein, S. Stave, Energy dependence of the delta resonance: Chiral dynamics in action, Few-Body Systems 41. arXiv:0710.3387.
* [24] F. W. Stecker, Effect of Photomeson Production by the Universal Radiation Field on High-Energy Cosmic Rays, Phys. Rev. Letters 21 (1968) 1016–1018.
* [25] D. Mattingly, et al., Possible neutrino constraints on Planck scale Lorentz violation, J. Cosmology and Astropart. Phys. 02(2010)007.
* [26] J. Belz, Composition Measurements near the Second Knee with the Telescope Array Low-Energy Extension (TALE), in: Intl. Cosmic Ray Conf., Vol. 5, 2008, pp. 1163–1166.
* [27] M. Unger, for the Pierre Auger Collaboration, Composition Studies with the Pierre Auger Observatory, ArXiv: 0902.3787.
* [28] R. Ulrich, for the Pierre Auger Collaboration, Investigation of Hadronic Interactions at Ultra-High Energies with the Pierre Auger Observatory, ArXiv: 0906.4691.
* [29] F. W. Stecker, Photodisintegration of Ultrahigh-Energy Cosmic Rays by the Universal Radiation Field, Physical Review 180 (1969) 1264–1266.
* [30] D. Hooper, A. Taylor, S. Sarkar, The impact of heavy nuclei on the cosmogenic neutrino flux, Astroparticle Physics 23 (2005) 11–17.
* [31] M. Ave, et al., Cosmogenic neutrinos from ultra-high energy nuclei, Astroparticle Physics 23 (2005) 19–29. arXiv:arXiv:astro-ph/0409316.
* [32] Allard, et al., Cosmogenic neutrinos from the propagation of ultrahigh energy nuclei, J. Cosmology and Astropart. Phys. 9 (2006) 005.
* [33] J. Ahrens, et al., Sensitivity of the IceCube detector to astrophysical sources of high energy muon neutrinos, Astropart. Phys. 20 (2004) 507–532.
* [34] A. Santangelo, et al., The Science Case of the JEM-EUSO Mission-Unveiling the Universe at ultra-high energies, in: H. Susa and others (Ed.), AIP Conf. Ser., Vol. 1238, 2010, pp. 380–387.
* [35] A. Petrolini, Ultra-High Energy Cosmic Particles studies from space: super-EUSO, a possible next-generation experiment, arXiv: 0909.5220.
* [36] F. W. Stecker, et al., Observing the Ultrahigh Energy Universe with OWL Eyes, Nucl. Phys. B Proc. Suppl. 136 (2004) 433–438. arXiv:astro-ph/0408162.
* [37] S. W. Barwick, ARIANNA: A New Concept for UHE Neutrino Detection, in: Intl. Cosmic Ray Conf., Vol. 5, 2008, pp. 1601–1604.
* [38] P. Allison, et al., IceRay: An IceCube-centered radio-Cherenkov GZK neutrino detector, Nucl. Instr. Meth. A 604 (2009) 64–69. arXiv:0904.1309.
* [39] A. A. Abdo, et al., Spectrum of the Isotropic Diffuse Gamma-Ray Emission Derived from First-Year Fermi Large Area Telescope Data, Phys. Rev. Letters 104 (2010) 101101.
* [40] V. Berezinsky, et al., Fermi-LAT restrictions on UHECRs and cosmogenic neutrinos, arXiv: 1003.1496.
* [41] M. Ahlers, et al., GZK neutrinos after the Fermi-LAT diffuse photon flux measurement, Astropart. Phys. 34 (2010) 106–115.
* [42] D. V. Semikoz, G. Sigl, Ultra-high energy neutrino fluxes: new constraints and implications, J. Cosm. Astropart. Phys. 04(2004)003.
* [43] P. Sreekumar, et al., EGRET Observations of the Extragalactic Gamma-Ray Emission, Astrophys. J. 494 (1998) 523–534.
* [44] A. W. Strong, I. V. Moskalenko, O. Reimer, Diffuse Galactic Continuum Gamma Rays: A Model Compatible with EGRET Data and Cosmic-Ray Measurements, Astrophys. J. 613 (2004) 962–976.
* [45] J. Wdowczyk, W. Tkaczyk, A. W. Wolfendale, Primary cosmic $\gamma$-rays above $10^{12}$ eV, J. Phys. A 5 (1972) 1419–1432.
* [46] P. P. Kronberg, et al., A Global Probe of Cosmic Magnetic Fields to High Redshifts, Astrophys. J. 676 (2008) 70–79.
* [47] A. Neronov, I. Vovk, Evidence for Strong Extragalactic Magnetic Fields from Fermi Observations of TeV Blazars, Science 328 (2010) 73–75.
* [48] W. Atwood, et al., The Large Area Telescope on the Fermi Gamma-Ray Space Telescope Mission, Astrophys. J. 697 (2009) 1071–1102.
|
arxiv-papers
| 2010-08-24T13:06:14 |
2024-09-04T02:49:12.410566
|
{
"license": "Public Domain",
"authors": "Sean T. Scully and Floyd W. Stecker",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/1008.4034"
}
|
1008.4074
|
Uniform Theory
of Geometric Spaces
Alexander Popa
email: alpopa@gmail.com
To my wife Raisa and my mother Tamara.
###### Contents
1. Introduction
2. 1 Geometric Space Model Construction
1. 1.1 Three Kinds of Plane Rotations. Rotation Characteristic
2. 1.2 Functions $C(x)$, $S(x)$ and $T(x)$
3. 1.3 Representation of Translation as Rotation. Its Characteristics
4. 1.4 Kinds of Space Rotations. Bundles of Unconnectable Points
5. 1.5 Main Space Rotations
6. 1.6 Vector Product. Invariant Quadric Form
7. 1.7 Space Definition by its Specification
8. 1.8 Definition of Measure Using Motions
3. 2 Measure Calculus
1. 2.1 General Triangle Equations
2. 2.2 Right (Quasi)–Triangle Equations
3. 2.3 More rotations
4. 2.4 Generalized Orthogonal Matrix
5. 2.5 Orthogonal Matrix Decomposition
6. 2.6 Coordinate and State Matrix
7. 2.7 Plane definition and Specification
8. 2.8 Projection of Vector on Lineal
9. 2.9 Basis Change in Lineal
10. 2.10 Measure Calculus Between Lineals
11. 2.11 Volume Calculation
4. 3 Theory Application
1. 3.1 Space and Lineal Specification Search Algorithm
2. 3.2 Some Special Spaces
3. 3.3 Spaces as Product of their Subspaces
## Introduction
The first documented attempt to construct the geometry theory in an axiomatic
way was made, as we know, by Euclid (III cent BC) in his Elements. And while
the word ‘geometry’ literally means ‘earth measuring’, Euclidean geometry
doesn’t describe elliptic space, as more proper for measuring of our planet.
New axiomatic approach was revolutionary one, however the axiomatic has
limitations. Euclid study what can be constructed calculated or demonstrated
starting with compass and straightedge. It was sufficient for that time.
However today, despite the fact Euclidean geometry is studied in the school,
many people, including geometriests, can’t remember its axioms. Exception
makes famous Euclid’s V-th postulate, which many of us remember in the form:
“At most one line can be drawn through any point not on a given line parallel
to the given line in a plane”. Euclid decided to formulate it so: “If a line
segment intersects two straight lines forming two interior angles on the same
side that sum to less than two right angles, then the two lines, if extended
indefinitely, meet on that side on which the angles sum to less than two right
angles”.
For modern geometry Euclid’s axiomatic has several limitations:
* •
Euclid’s axiomatic theory covered the only geometry system and only
two–dimensional case. The axiomatic of Euclidian geometry used today was
developed by David Hilbert (1862 — 1943), has 20 axioms and covers two and
three dimensions.
* •
The four–dimensional case uses much more axioms. Development of axiomatic for
spaces of further dimensions is non–trivial.
* •
Except hyperbolic geometry, construction of good axiomatic for other
geometries is also non–trivial. Usually, an axiomatic is constructed after the
geometry is well studied with aim of some model (for example, [3] describes
the space–time axiomatic).
* •
Undefined notions in geometry (point, line, between) differ very much from
undefined notions in other mathematic disciplines (number, function, space).
Undefined notions of different geometries differ from each other.
* •
Mathematicians successful study Euclidean space of any dimension using
analytic geometry and forget Euclid’s axiomatic.
Euclid’s axiomatic played one important role. Its V-th postulate is so hard
expressed and creates so artificial feeling that urged mathematicians to
create the hyperbolic geometry. Sad, when Nikolai Lobachevsky (1792 — 1856)
and János Bolyai (1802 — 1860) published their results, the new geometry was
slow in acceptance. Only after decades it was demonstrated that hyperbolic
geometry is interior geometry of surfaces with constant negative curvature.
After next several years some models of hyperbolic geometry were elaborated.
Due to that fact the new geometry became accessible.
Author of a model, Felix Klein (1849 — 1925) proposed “Erlangen Program” [2] —
the unified view over different geometries as complex of different
transformation groups of space. The invariants of these groups are figures of
the geometries. In such way, Klein presented 9 two–dimensional spaces.
However, 6 of them he considered practic unaplicable [1]. Till now speaking
about “non–euclidean geometry”, elliptic or hyperbolic geometry is primarily
understood. Obviously, in order to make all geometries to be taken seriously,
an accessible model is required. One of such model for two–dimensional case
proposed [4] Isaak Moiseevich Yaglom (1921 — 1988), using the notion of
generalized complex number. Among more recent results you can refer to [5, 6,
7].
In this work, supposed to your attention an uniform model of geometric spaces
and based on it general analytic geometry are described. Among its advantages
there are its universality and linearity, hence easyness to use. It isn’t
limited to specific dimension.
The first chapter describes different types of distance and angular measure
and their models. Different variants of axioms valid for different geometries
are analyzed, as well as one variant of them, depending on some parameter and
universally valid. A analytic model depending on some parameters is
constructed. Lengths and angles are defined as parameters of corresponding
motions.
In the second chapter you can find triangle equations valid for all
geometries. The chapter describes generalized orthogonal matrix as general
form of motion matrix. A vector approach will be shown for description of
points, lines and planes, and for linear calculus of lengths and angles. At
the end of chapter, the reader will find a linear way to calculate volumes.
The third chapter has more philosophical character then practical one. Your
attention will be set on proper terminology and several well known spaces will
be described in terms of constructed theory.
Uniform model of geometric spaces becomes the background of the GeomSpace
project111http://sourceforge.net/projects/geomspace/. The last version of this
book can also be downloaded form here..
## Chapter 1 Geometric Space Model Construction
### 1.1 Three Kinds of Plane Rotations. Rotation Characteristic
Consider real plane $\mathbb{R}^{2}$. Consider three different transformations
of $\mathbb{R}^{2}$: rotation $\mathfrak{R}^{\prime}(\phi)$, Galilean
transformation $\mathfrak{R}^{\prime\prime}(\phi)$ and Lorintz transformation
$\mathfrak{R}^{\prime\prime\prime}(\phi)$ defined by matrices:
$\mathfrak{R}^{\prime}(\phi)=\begin{pmatrix}\cos\phi&-\sin\phi\\\
\sin\phi&\cos\phi\end{pmatrix},$
$\mathfrak{R}^{\prime\prime}(\phi)=\begin{pmatrix}1&0\\\ \phi&1\end{pmatrix},$
and
$\mathfrak{R}^{\prime\prime\prime}(\phi)=\begin{pmatrix}\cosh\phi&\sinh\phi\\\
\sinh\phi&\cosh\phi\end{pmatrix},$
where $\phi\in\mathbb{R}$.
Transformations $\mathfrak{R}^{\prime}(\phi)$,
$\mathfrak{R}^{\prime\prime}(\phi)$ and
$\mathfrak{R}^{\prime\prime\prime}(\phi)$ have several common properties. The
determinant of all their matrices is 1, all them have the only fixed point —
origin $O=(0,0)$, $\mathfrak{R}(0)=I$ — unit matrix,
$\mathfrak{R}(x)\mathfrak{R}(y)=\mathfrak{R}(x+y)=\mathfrak{R}(y)\mathfrak{R}(x)$
and the trajectory of point $P=(1,0)$ verifies equations:
$x_{0}^{2}+x_{1}^{2}=1,\text{ for }\mathfrak{R}^{\prime},$ $x_{0}=1,\text{ for
}\mathfrak{R}^{\prime\prime},$ $x_{0}^{2}-x_{1}^{2}=1,\text{ for
}\mathfrak{R}^{\prime\prime\prime}.$
More general, the trajectory equation can be written as (Figure 1.1):
$x_{0}^{2}+k\,x_{1}^{2}=1,k=-1,0,1.$
Figure 1.1: Trajectory of point $P$ on transformations
$\mathfrak{R}^{\prime}$, $\mathfrak{R}^{\prime\prime}$ and
$\mathfrak{R}^{\prime\prime\prime}$.
We will name $\mathfrak{R}^{\prime}$ elliptic rotation,
$\mathfrak{R}^{\prime\prime}$ parabolic rotation,
$\mathfrak{R}^{\prime\prime\prime}$ hyperbolic rotation and $\phi$ respective
angle. We will name the coefficient $k$ characteristic of a rotation. $k=1$
corresponds to elliptic, $k=0$ to parabolic and $k=-1$ to hyperbolic rotation.
### 1.2 Functions $C(x)$, $S(x)$ and $T(x)$
We can see, that the matrices $\mathfrak{R}^{\prime}$,
$\mathfrak{R}^{\prime\prime}$ and $\mathfrak{R}^{\prime\prime\prime}$ have
elements $r_{11}=r_{22}$ and $r_{12}=-k\,r_{21}$. We can write:
$\mathfrak{R}(\phi)=\begin{pmatrix}C(\phi)&-kS(\phi)\\\
S(\phi)&C(\phi)\end{pmatrix},$ (1.1)
where
$C(x)=\begin{cases}\cos x,&k=1\\\ 1,&k=0\\\ \cosh x,&k=-1\end{cases}$
and
$S(x)=\begin{cases}\sin x,&k=1\\\ x,&k=0\\\ \sinh x,&k=-1\end{cases}$
Finally, we can define formally the functions $C(x)$ and $S(x)$ as111Here and
further we will consider for simplicity that $k^{0}=1$ for $k=0$ too. We will
say $x$ divide $k^{i}$, $k=0$ if in expression $x/k^{i}$ the exponent of $k$
in numerator is not less then $i$.:
$C(x)=C(x,k)=\sum_{n=0}^{\infty}(-k)^{n}\frac{x^{2n}}{(2n)!},$ (1.2)
$S(x)=S(x,k)=\sum_{n=0}^{\infty}(-k)^{n}\frac{x^{2n+1}}{(2n+1)!}.$ (1.3)
We will introduce one more function:
$T(x)=\frac{S(x)}{C(x)}.$ (1.4)
Note, that always has place the equality:
$C^{2}(x)+kS^{2}(x)=1,\,\forall x\in\mathbb{R}.$ (1.5)
We will name transformation (1.1) generalized rotation. Note, that along the
angle it has one more parameter — its characteristic.
### 1.3 Representation of Translation as Rotation. Its Characteristics
Generalized rotation has a fixed point. However, the translation usually
doesn’t have a fixed point222A fixed point is present for example in
translation on elliptic plane.. We will define the translation through
generalized rotation using an extra-dimension, as it is done in projective
geometry.
For $n$-dimensional space consider vector space $\mathbb{R}^{n+1}$. Rotation
matrices have two different rows compared to unit matrix. When one of them is
the first row, we will consider them translations. When none of them is the
first one, we will consider them rotations. Therefore, the first coordinate
(we will count it 0) will be additional.
Let $n=1$. We will name the vector $o=\\{1,0\\}$ origin. If
$a=\mathfrak{R}(\phi)o$, then the angle $\phi$ between $o$ and $a$ we can name
distance $oa$. Different values of $k$ correspond to different translations:
when $k=1$ translations are elliptic, when $k=0$ they are parabolic, and when
$k=-1$ they are hyperbolic.
There is a kind of distance measure for each kind of translation: elliptic,
parabolic and hyperbolic. The difference between them can be seen in variants
of V postulate of Euclid for elliptic, linear and hyperbolic geometry (Figure
1.2).
Figure 1.2: Variants of Euclid’s V postulate — elliptic a), linear b) and
hyperbolic c).
Elliptic postulate (Figure 1.2 a) is333For this case it is necessary to modify
another two postulates, namely that from any three points on a line exactly
one lies between two others, and that any line can be extended infinitely in
any direction.: For a given line $l$ and a point $P\notin l$, exists no line
$p\ni P\,|\,l\cap p=\varnothing$. It is identical to the following: For a
given line $l$ and a point $P\notin l$, all lines $p\ni P$ intersect $l$.
The linear postulate (Figure 1.2 b) is: For a given line $l$ and a point
$P\notin l$, exists one line $p\ni P\,|\,l\cap p=\varnothing$.
The hyperbolic postulate (Figure 1.2 c) is: For a given line $l$ and a point
$P\notin l$, exist at least two lines $p^{\prime},p^{\prime\prime}\ni
P\,|\,l\cap p^{\prime}=\varnothing,l\cap p^{\prime\prime}=\varnothing$.
Generally, V postulate of Euclid can be formulated as: For a given line $l$
and a point $P\notin l$, exist $0^{k_{1}}$ lines $p\ni P\,|\,l\cap
p=\varnothing$. It should be mentioned that $0^{k_{1}}$ is a symbol, not a
number used in calculus. Its value equals to 0 for $k_{1}=1$, 1 for $k_{1}=0$
and $\infty$ for $k_{1}=-1$.
### 1.4 Kinds of Space Rotations. Bundles of Unconnectable Points
It’s easy to see that classic rotations in Euclidean geometry, as well as in
the elliptic (Riemannian) geometry and the hyperbolic (Bolyai–Lobachevsky)
geometry has the characteristic $k=1$. We can extend the notion of space
rotation to generalized space rotation with some characteristic. The best way
to illustrate difference between them is to formulate angular equivalent of V
Postulate of Euclid — axiom of points connectability (Figure 1.3). In order to
do this we will change the following phrases between them:
$\displaystyle\text{line }l$ $\displaystyle\longleftrightarrow$
$\displaystyle\text{ point }L,$ $\displaystyle P\in l$
$\displaystyle\longleftrightarrow$ $\displaystyle p\ni L,$ $\displaystyle
P\notin l$ $\displaystyle\longleftrightarrow$ $\displaystyle p\not\ni L,$
$\displaystyle AB=\phi$ $\displaystyle\longleftrightarrow$
$\displaystyle\angle ab=\phi,$ $\displaystyle a\cap b=C$
$\displaystyle\longleftrightarrow$ $\displaystyle c=AB,$ $\displaystyle a\cap
b=\varnothing$ $\displaystyle\longleftrightarrow$ $\displaystyle A\text{ is
unconnectable with }B.$
The last statement is unusual for the above three geometries444It conflicts
with axiom which states that through any two points goes a line. This axiom
should be changed by one of the following in order to consider the geometries
with non-elliptic rotations.. It makes sense in geometries with angular
characteristic 0 or $-1$. The unconnectable property of points is similar to
parallel property of lines.
Figure 1.3: Different variants of points unconnectability axiom — elliptic a),
linear b) and hyperbolic c).
The angle equivalent of V Postulate for elliptic characteristic (Figure 1.3 a)
is: On a line $l\not\ni P$ exist no points $L$ unconnectable with $P$.
For parabolic characteristic (Figure 1.3 b) it is: On a line $l\not\ni P$
exists the only point $L$ unconnectable with $P$.
For the hyperbolic characteristic (Figure 1.3) it is: On a line $l\not\ni P$
exist at least two points $L^{\prime}$ and $L^{\prime\prime}$ unconnectable
with $P$.
Generally this axiom can be formulated as: On a line $l\not\ni P$ exist
$0^{k_{2}}$ points $L$ unconnectable with $P$. As in case of parallel lines,
symbol $0^{k_{2}}$ isn’t used in calculus.
Similar to bundles of lines — intersected, parallel or divergent we can speak
about bundles of points. More exactly, let $X,Y\in\mathbb{R}^{n+1}$. All
linear combinations $Z=\alpha X+\beta Y,\alpha,\beta\in\mathbb{R}$ form a set
we will name bundle of points. As we will see, this set has one constraint.
Therefore, it has one free parameter. As every two lines define a bundle of
lines, every two points ($X$ and $Y$) define bundle of points. If $X$ is
connectable with $Y$ this bundle is a line (similar to intersection point of
bundle of intersected lines). Lines has blue color on figure 1.3. If $X$ and
$Y$ are unconnectable, this bundle of points isn’t a line (similar to bundle
of parallel or divergent lines). Bundles of unconnectable points are green and
red on figure 1.3.
For any angle characteristic there are infinity of bundles of connectable
points. For angle characteristic 1 all point bundles are lines. For angle
characteristic 0 for any point there is the only bundle of unconnectable
points (green). For angle characteristic $-1$ there are infinity bundles of
unconnectable points (red). In thes case the bundles of connectable points and
the bundles of unconnectable points for some point form two categories of
bundles. The limit (marginal) bundles of unconnectable points (green) can be
viewed as the third category (similar to differencee between parallel and
divergent lines). There are exactly two limit bundles. Note that bundles of
connectable points intersect all circles with centre in the centre of bundle,
all bundles of unconnectable points don’t intersect these circles and limit
bundles are asymptotic to circles (Figure 1.4).
Figure 1.4: Mutual position of different bundles and circles a) elliptic
angular characteristic, b) linear angular characteristic and c) hyperbolic
angular characteristic.
Emphasize that the angle between two lines and the angle between two
two–dimensional planes are the different measures. The angle between two
threedimensional planes is different from them both and so on. Thus, the angle
between lines can have the different characteristic then the angle between
two–dimensional planes and so on.
### 1.5 Main Space Rotations
Consider $\mathbb{R}^{n+1}$ and $k_{1},k_{2},...k_{n}\in\\{-1,0,1\\}$. We will
note $C_{i}(x)=C(x,k_{i})$, $S_{i}(x)=S(x,k_{i})$ and
$T_{i}(x)=S_{i}(x)/C_{i}(x)$. Let
$\mathfrak{R}_{1}(\phi)=\begin{pmatrix}C_{1}(\phi)&-k_{1}S_{1}(\phi)&0&\ldots&0\\\
S_{1}(\phi)&C_{1}(\phi)&0&\ldots&0\\\ 0&0&1&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\ldots&1\end{pmatrix},$
$\mathfrak{R}_{2}(\phi)=\begin{pmatrix}1&0&0&\ldots&0\\\
0&C_{2}(\phi)&-k_{2}S_{2}(\phi)&\ldots&0\\\
0&S_{2}(\phi)&C_{2}(\phi)&\ldots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\ldots&1\end{pmatrix},$ $\vdots$
$\mathfrak{R}_{n}(\phi)=\begin{pmatrix}1&0&\ldots&0&0\\\ 0&1&\ldots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\ldots&C_{n}(\phi)&-k_{n}S_{n}(\phi)\\\
0&0&\ldots&S_{n}(\phi)&C_{n}(\phi)\end{pmatrix}.$
We will name $\mathfrak{R}_{1},...\mathfrak{R}_{n}$ main space rotations.
### 1.6 Vector Product. Invariant Quadric Form
Let
$K_{m}=\prod_{i=1}^{m}k_{i},\,\forall m=\overline{0,n}$ (1.6)
We can see that $K_{m}\in\\{-1,0,1\\},\,\forall m=\overline{0,n}$ as well as
$k_{m}$. Let define vector product $\odot$ as
$x\odot y=\sum_{i=0}^{n}K_{i}x_{i}y_{i}$ (1.7)
For some vectors $x=\\{x_{0},x_{1},...x_{n}\\}$ and
$y=\\{y_{0},y_{1},...y_{n}\\}$,
$x^{\prime}=\mathfrak{R}_{m}(\phi)x=\\{x_{0},...x_{m-2},x_{m-1}$
$C_{m}(\phi)-k_{m}x_{m}S_{m}(\phi),x_{m-1}S_{m}(\phi)+x_{m}C_{m}(\phi),x_{m+1},...x_{n}\\}$
and $y^{\prime}=\mathfrak{R}_{m}(\phi)y$. We can see that
$\displaystyle x^{\prime}\odot y^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}K_{i}x^{\prime}_{i}y^{\prime}_{i}$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{m-2}K_{i}x_{i}y_{i}$
$\displaystyle+$
$\displaystyle((x_{m-1}C_{m}(\phi)-k_{m}x_{m}S_{m}(\phi))(y_{m-1}C_{m}(\phi)-k_{m}y_{m}S_{m}(\phi))$
$\displaystyle+$ $\displaystyle
k_{m}(x_{m-1}S_{m}(\phi)+x_{m}C_{m}(\phi))(y_{m-1}S_{m}(\phi)+y_{m}C_{m}(\phi)))K_{m-1}$
$\displaystyle+$ $\displaystyle\sum_{i=m+1}^{n}K_{i}x_{i}y_{i}$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{m-2}K_{i}x_{i}y_{i}$
$\displaystyle+$
$\displaystyle((x_{m-1}y_{m-1}C_{m}^{2}(\phi)-k_{m}(x_{m-1}y_{m}+x_{m}y_{m-1})S_{m}(\phi)C_{m}(\phi)+k_{m}^{2}x_{m}y_{m}S_{m}^{2}(\phi)$
$\displaystyle+$ $\displaystyle
k_{m}x_{m-1}y_{m-1}S_{m}^{2}(\phi)+k_{m}(x_{m-1}y_{m}+x_{m}y_{m-1})S_{m}(\phi)C_{m}(\phi)+k_{m}x_{m}y_{m}C_{m}^{2}(\phi)))K_{m-1}$
$\displaystyle+$ $\displaystyle\sum_{i=m+1}^{n}K_{i}x_{i}y_{i}$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{m-2}K_{i}x_{i}y_{i}$
$\displaystyle+$
$\displaystyle(x_{m-1}y_{m-1}(C_{m}^{2}(\phi)+k_{m}S_{m}^{2}(\phi))+k_{m}x_{m}y_{m}(C_{m}^{2}(\phi)+k_{m}S_{m}^{2}(\phi)))K_{m-1}$
$\displaystyle+$ $\displaystyle\sum_{i=m+1}^{n}x_{i}y_{i}K_{i}$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{n}K_{i}x_{i}y_{i}=x\odot y$
This is true for all $m=\overline{1,n}$. So the quadric form $x\odot y$ is
invariant in respect to main rotations of $\mathfrak{R}_{m}$.
### 1.7 Space Definition by its Specification
Consider $\mathbb{RP}^{n}$ projective space and
$k_{i}\in\\{-1,0,1\\},\,\forall i=\overline{1,n}$. We can now introduce a
geometric space ‘unit sphere’
$\mathbb{B}^{n}=\\{x\in\mathbb{RP}^{n}\,|\,x\odot x=1\\}$ (Figure 1.5). As all
main rotations preserves the quadric form defined by product $\odot$, they
also preserves $\mathbb{B}^{n}$. We will name $k_{i},i=\overline{1,n}$ space
specification. We will name ‘point’ $X\in\mathbb{B}^{n}$ the corresponding
vector $x\in\mathbb{RP}^{n}$ and will use homogeneous coordinates normalized
in order to $x\odot x=1$.
Figure 1.5: Sphere of space with specification $\\{-1,-1\\}$.
We will name ‘origin’ of $\mathbb{B}^{n}$ the point
$O=\left[1:0:...:0\right]\in\mathbb{B}^{n}$. It isn’t origin of
$\mathbb{R}^{n+1}$, $(0,0,...0)\notin\mathbb{B}^{n}$ and we will refer to $O$
as origin if isn’t specified otherwise.
It’s easy to see that for any $k_{1},k_{2},...k_{n}$,
$O=\mathbb{B}^{0}\subset\mathbb{B}^{1}\subset...\subset\mathbb{B}^{n}$.
We will define motions of $\mathbb{B}^{n}$ all transformations that result on
finite product of main rotations.
We will define ‘lines’ all images of $\mathbb{B}^{1}$ on any motion of
$\mathbb{B}^{n}$. Similarly, we define ‘$m$-dimensional’ planes all images of
$\mathbb{B}^{m}$ on any motions of $\mathbb{B}^{n}$ for any
$m\in\overline{0,n-1}$.
For each characteristic parameter $k_{i}$ we can introduce a scale parameter
$r_{i}\in\mathbb{R}_{+}$, $i=\overline{1,n}$. The $k_{1}/r_{1}^{2}$ is exactly
the gaussian curvature of space. Others have no representation since finite
angle measure doesn’t require scaling. In this case the radian measure is
native. An example of angle scale is degree measure which has scale $180/\pi$.
However when the angle is not bounded a scale introduction has sense. All
scales can be easy embedded in functions $C_{i}(x)$, $S_{i}(x)$ and $T_{i}(x)$
by using instead $C_{i}\left(\frac{x}{r_{i}}\right)$,
$S_{i}\left(\frac{x}{r_{i}}\right)$ and $T_{i}\left(\frac{x}{r_{i}}\right)$
respectively, $i=\overline{1,n}$.
### 1.8 Definition of Measure Using Motions
A traditional way of definition the measures and motions is to provide a way
to calculate the distances as is and then to define motions in such way that
all maps $M:\mathbb{R}^{n}\to\mathbb{R}^{n}$ preserve the distance. We go
another way. We provide motions as is and then search for a way to define
measures in such way that motions preserve them.
We will say point $A\in\mathbb{B}^{1}\subset\mathbb{B}^{n}$ has the distance
$\phi$ from origin $O$ if $A=\mathfrak{R}_{1}(\phi)O$. Having
$O=\left[1:0:...:0\right]$, $A=\left[C_{1}(\phi):S_{1}(\phi):0:...:0\right]$,
$O\odot A=C_{1}(\phi)$. We will say one–dimensional (planar) angle between
$\mathbb{B}^{1}$ and some one–dimensional line $\mathbb{B}^{\prime
1}\subset\mathbb{B}^{2}$ equals $\phi$ if $\mathbb{B}^{\prime
1}=\mathfrak{R}_{2}(\phi)\mathbb{B}^{1}$. Similarly, we will define the
$m$-dimensional angle $\phi$ between $\mathbb{B}^{m}$ and $m$-dimensional
plane $\mathbb{B}^{\prime m}\subset\mathbb{B}^{m+1}$ if $\mathbb{B}^{\prime
m}=\mathfrak{R}_{m+1}(\phi)\mathbb{B}^{m},\,\forall m=\overline{0,m-1}$. Note,
that $n$-dimensional angle between any planes is 0 since all them are subset
of $B^{n}$.
Let $X,Y\in\mathbb{B}^{n}$. If there exists a motion that maps
$\mathbb{B}^{1}$ to $XY$ we will name points $X$ and $Y$ connectable and
distance $XY$ measurable. If not, we will name points $X$ and $Y$
unconnectable (just as lines can be parallel) and strictly speaking the
distance $XY$ doesn’t exists555In this case there exist a measure $XY$, but it
may have different characteristic then distance. We will name this measure
also distance, keeping in mind that it is generalized distance..
We can find a motion $\mathfrak{M}$ of space $\mathbb{B}^{n}$ that maps origin
$O$ to $X$ and some point $A\in\mathbb{B}^{1}\subset\mathbb{B}^{n}$ to $Y$. As
motion $\mathfrak{M}$ preserves the quadric form $\odot$, we can see that
$X\odot Y=O\odot A$. We can define the distance $\phi$ between $X$ and $Y$ as
$C_{1}(\phi)=X\odot Y.$ (1.8)
It’s easy to see that all motions preserve the distance. In case of elliptic,
Euclidian and hyperbolic space it is sufficient, because all other measures
can be calculated from distances. However, in some spaces angles can be scaled
in a manner distances are scaled in Euclidean space. So we should find the way
to measure all the measures in general case.
## Chapter 2 Measure Calculus
### 2.1 General Triangle Equations
Consider triangle $ABC\in\mathbb{B}^{2}$ with the edges $a$, $b$, $c$,
interior angles $\alpha$, $\gamma$ and exterior angle $\beta^{\prime}$ (Figure
2.1). Let $A=\left[1:0:0\right]$ the origin,
$C=\mathfrak{R}_{1}(b)A=\left[C_{1}(b):S_{1}(b):0\right]$ and
$B=\mathfrak{R}_{2}(\alpha)\mathfrak{R}_{1}(c)A=\left[C_{1}(c):S_{1}(c)C_{2}(\alpha):S_{1}(c)S_{2}(\alpha)\right]$.
Note, that the interior angle $\beta$ does not exist in case of $k_{2}=0$ or
$k_{2}=-1$. The exterior angle $\beta^{\prime}$ always exists.
Figure 2.1: General triangle equations deduction.
Now, let $A^{\prime}B^{\prime}C^{\prime}=\mathfrak{R}(-b)(ABC)$ (Figure 2.1,
cyan). The point we are interested in is
$B^{\prime}=\left[C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha):-S_{1}(b)C_{1}(c)+C_{1}(b)S_{1}(c)C_{2}(\alpha):S_{1}(c)S_{2}(\alpha)\right]$.
At the other hand, now
$B^{\prime}=\left[C_{1}(a):-S_{1}(a)C_{2}(\gamma):S_{1}(a)S_{2}(\gamma)\right]$.
It means:
$\displaystyle C_{1}(a)$ $\displaystyle=$ $\displaystyle
C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha),$ $\displaystyle-
S_{1}(a)C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle-
S_{1}(b)C_{1}(c)+C_{1}(b)S_{1}(c)C_{2}(\alpha),$ $\displaystyle
S_{1}(a)S_{2}(\gamma)$ $\displaystyle=$ $\displaystyle S_{1}(c)S_{2}(\alpha).$
The first equation is the form of the Cosine I law. Similarly we have
$C_{1}(c)=C_{1}(a)C_{1}(b)+k_{1}S_{1}(a)S_{1}(b)C_{2}(\gamma).$
The third equation is equivalent to
$\frac{S_{1}(a)}{S_{2}(\alpha)}=\frac{S_{1}(c)}{S_{2}(\gamma)},$
which is the form of the Sine law.
Let now
$A^{\prime\prime}B^{\prime\prime}C^{\prime\prime}=\mathfrak{R}_{1}(-c)\mathfrak{R}_{2}(-\alpha)(ABC)$
(Figure 2.1, brown). Now we are interested in vertex
$C^{\prime\prime}=\left[C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha):-C_{1}(b)S_{1}(c)+S_{1}(b)C_{1}(c)C_{2}(\alpha):-S_{1}(b)S_{2}(\alpha)\right]$.
At the other hand,
$C^{\prime\prime}=\left[C_{1}(a):S_{1}(a)C_{2}(\beta^{\prime}):-S_{1}(a)S_{2}(\beta^{\prime})\right]$.
From here we have:
$\displaystyle C_{1}(a)$ $\displaystyle=$ $\displaystyle
C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha),$ $\displaystyle
S_{1}(a)C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle-
C_{1}(b)S_{1}(c)+S_{1}(b)C_{1}(c)C_{2}(\alpha),$ $\displaystyle-
S_{1}(a)S_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle-
S_{1}(b)S_{2}(\alpha).$
The first one is the Cosine I law, the third one is equivalent to:
$\frac{S_{1}(a)}{S_{2}(\alpha)}=\frac{S_{1}(b)}{S_{2}(\beta^{\prime})}=\frac{S_{1}(c)}{S_{2}(\gamma)}$
(2.1)
which is the Sine law. Note that in case $k_{2}=1$ we have
$\beta=\pi-\beta^{\prime}$, $S_{2}(\beta)=S_{2}(\beta^{\prime})$. Let
calculate the value of $C_{2}(\alpha)$ from the first equation and put it to
the second one:
$\displaystyle S_{1}(a)C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle-
C_{1}(b)S_{1}(c)+S_{1}(b)C_{1}(c)\frac{C_{1}(a)-C_{1}(b)C_{1}(c)}{k_{1}S_{1}(b)S_{1}(c)}$
$\displaystyle=$ $\displaystyle-
C_{1}(b)S_{1}(c)+C_{1}(c)\frac{C_{1}(a)-C_{1}(b)C_{1}(c)}{k_{1}S_{1}(c)},$
$\displaystyle k_{1}S_{1}(a)S_{1}(c)C_{2}(\beta^{\prime})$ $\displaystyle=$
$\displaystyle-
k_{1}S_{1}(c)^{2}C_{1}(b)+C_{1}(a)C_{1}(c)-C_{1}(b)C_{1}(c)^{2}$
$\displaystyle=$ $\displaystyle
C_{1}(a)C_{1}(c)-C_{1}(b)(C_{1}(c)^{2}+k_{1}S_{1}(c)^{2})$ $\displaystyle=$
$\displaystyle C_{1}(a)C_{1}(c)-C_{1}(b),$ $\displaystyle C_{1}(b)$
$\displaystyle=$ $\displaystyle
C_{1}(a)C_{1}(c)-k_{1}S_{1}(a)S_{1}(c)C_{2}(\beta^{\prime}).$
Note the ‘$-$’ sign in the right part of the equation. It is so because the
$\beta^{\prime}$ angle is external. For the case $k_{2}=1$, the internal angle
$\beta=\pi-\beta^{\prime}$, $C_{2}(\beta)=-C_{2}(\beta^{\prime})$.
What about the Cosine II law? We will use these two equations:
$\displaystyle-S_{1}(a)C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle-
S_{1}(b)C_{1}(c)+C_{1}(b)S_{1}(c)C_{2}(\alpha),$ $\displaystyle
S_{1}(a)C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle-
C_{1}(b)S_{1}(c)+S_{1}(b)C_{1}(c)C_{2}(\alpha)$
First, replace $S_{1}(b)$ with $S_{1}(a)S_{2}(\beta^{\prime})/S_{2}(\alpha)$
and $S_{1}(c)$ with $S_{1}(a)S_{2}(\gamma)/S_{2}(\alpha)$:
$\displaystyle-S_{1}(a)C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle-
S_{1}(a)\frac{S_{2}(\beta)}{S_{2}(\alpha)}C_{1}(c)+C_{1}(b)S_{1}(a)\frac{S_{2}(\gamma)}{S_{2}(\alpha)}C_{2}(\alpha),$
$\displaystyle-S_{2}(\alpha)C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle-
C_{1}(c)S_{2}(\beta^{\prime})+C_{1}(b)S_{2}(\gamma)C_{2}(\alpha),$
$\displaystyle S_{2}(\beta^{\prime})C_{1}(c)$ $\displaystyle=$ $\displaystyle
S_{2}(\alpha)C_{2}(\gamma)+C_{2}(\alpha)S_{2}(\gamma)C_{1}(b),$
and
$\displaystyle S_{1}(a)C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle-
C_{1}(b)S_{1}(a)\frac{S_{2}(\gamma)}{S_{2}(\alpha)}+S_{1}(a)\frac{S_{2}(\beta^{\prime})}{S_{2}(\alpha)}C_{1}(c)C_{2}(\alpha),$
$\displaystyle S_{2}(\alpha)C_{2}(\beta^{\prime})$ $\displaystyle=$
$\displaystyle-
C_{1}(b)S_{2}(\gamma)+C_{1}(c)S_{2}(\beta^{\prime})C_{2}(\alpha),$
$\displaystyle S_{2}(\gamma)C_{1}(b)$ $\displaystyle=$ $\displaystyle-
S_{2}(\alpha)C_{2}(\beta^{\prime})+C_{2}(\alpha)S_{2}(\beta^{\prime})C_{1}(c).$
Now from the first equation let calculate $C_{1}(c)$ and put it in the second
one:
$\displaystyle S_{2}(\gamma)C_{1}(b)$ $\displaystyle=$ $\displaystyle-
S_{2}(\alpha)C_{2}(\beta^{\prime})+C_{2}(\alpha)S_{2}(\beta^{\prime})\frac{S_{2}(\alpha)C_{2}(\gamma)+C_{2}(\alpha)S_{2}(\gamma)C_{1}(b)}{S_{2}(\beta^{\prime})}$
$\displaystyle=$ $\displaystyle-
S_{2}(\alpha)C_{2}(\beta^{\prime})+C_{2}(\alpha)S_{2}(\alpha)C_{2}(\gamma)+C_{2}(\alpha)^{2}S_{2}(\gamma)C_{1}(b),$
$\displaystyle S_{2}(\gamma)C_{1}(b)(1-C_{2}(\alpha)^{2})$ $\displaystyle=$
$\displaystyle
S_{2}(\alpha)(C_{2}(\alpha)C_{2}(\gamma)-C_{2}(\beta^{\prime})),$
$\displaystyle k_{2}S_{2}(\gamma)C_{1}(b)S_{2}(\alpha)^{2}$ $\displaystyle=$
$\displaystyle S_{2}(\alpha)(C_{2}(\alpha)C_{2}(\gamma)-C_{2}(\beta)),$
$\displaystyle k_{2}S_{2}(\alpha)S_{2}(\gamma)C_{1}(b)$ $\displaystyle=$
$\displaystyle C_{2}(\alpha)C_{2}(\gamma)-C_{2}(\beta^{\prime}),$
$\displaystyle C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle
C_{2}(\alpha)C_{2}(\gamma)-k_{2}S_{2}(\alpha)S_{2}(\gamma)C_{1}(b).$
When $k_{2}=1$ we have
$\displaystyle-\cos\beta$ $\displaystyle=$ $\displaystyle\cos\alpha\cos\gamma-
k_{2}\sin\alpha\sin\gamma C_{1}(b),$ $\displaystyle\cos\beta$ $\displaystyle=$
$\displaystyle-\cos\alpha\cos\gamma+k_{2}\sin\alpha\sin\gamma C_{1}(b).$
Similarly, calculating $C_{1}(b)$ form the second equation and putting it in
the first one, obtain:
$\displaystyle S_{2}(\beta^{\prime})C_{1}(c)$ $\displaystyle=$ $\displaystyle
S_{2}(\alpha)C_{2}(\gamma)+C_{2}(\alpha)S_{2}(\gamma)\frac{C_{2}(\alpha)S_{2}(\beta^{\prime})C_{1}(c)-S_{2}(\alpha)C_{2}(\beta^{\prime})}{S_{2}(\gamma)}$
$\displaystyle=$ $\displaystyle
S_{2}(\alpha)C_{2}(\gamma)+C_{2}(\alpha)^{2}S_{2}(\beta^{\prime})C_{1}(c)-C_{2}(\alpha)S_{2}(\alpha)C_{2}(\beta^{\prime}),$
$\displaystyle S_{2}(\beta^{\prime})C_{1}(c)(1-C_{2}(\alpha)^{2})$
$\displaystyle=$ $\displaystyle
S_{2}(\alpha)(C_{2}(\gamma)-C_{2}(\alpha)C_{2}(\beta^{\prime})),$
$\displaystyle k_{2}S_{2}(\beta^{\prime})C_{1}(c)S_{2}(\alpha)^{2}$
$\displaystyle=$ $\displaystyle
S_{2}(\alpha)(C_{2}(\gamma)-C_{2}(\alpha)C_{2}(\beta^{\prime})),$
$\displaystyle k_{2}S_{2}(\alpha)S_{2}(\beta^{\prime})C_{1}(c)$
$\displaystyle=$ $\displaystyle
C_{2}(\gamma)-C_{2}(\alpha)C_{2}(\beta^{\prime}),$ $\displaystyle
C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle
C_{2}(\alpha)C_{2}(\beta^{\prime})+k_{2}S_{2}(\alpha)S_{2}(\beta^{\prime})C_{1}(c).$
When $k_{2}=1$ we have as above
$\cos\gamma=-\cos\alpha\cos\beta+k_{2}\sin\alpha\sin\beta C_{1}(c).$
Similarly, we have
$C_{2}(\alpha)=C_{2}(\beta^{\prime})C_{2}(\gamma)+k_{2}S_{2}(\beta^{\prime})S_{2}(\gamma)C_{1}(a).$
We will find the form of the Cosine I and II law that does not contain $C_{1}$
or $C_{2}$ functions in the left part. However, it contains these functions in
the right part. It makes sense since in the case $k_{1}\neq 0$ (for the Cosine
I law) and $k_{2}\neq 0$ (for the Cosine II law) when we can calculate their
respective $C^{-1}$ functions, but when $k_{1}=0$, the space admit distance
scaling and the angle values does not determine the distances (Cosine II law
is a equality which doesn’t contain $C_{1}$ function), while when $k_{2}=0$,
the space admit the angular scaling and distances does not determine angles
(Cosine I law is a equality which doesn’t contain $C_{2}$ function).
Note also that we can deduce one form of Cosine I and one form of Cosine II
law if we introduce a (may be virtual) angle $\beta$ so as:
$\displaystyle S_{2}(\beta)$ $\displaystyle=$ $\displaystyle
S_{2}(\beta^{\prime}),$ $\displaystyle C_{2}(\beta)$ $\displaystyle=$
$\displaystyle-C_{2}(\beta^{\prime}),$ $\displaystyle T_{2}(\beta)$
$\displaystyle=$ $\displaystyle-T_{2}(\beta^{\prime}).$
Then both Cosine I and II law have identical form. Now let calculate
$\displaystyle k_{1}S_{1}^{2}(a)$ $\displaystyle=$ $\displaystyle
1-C_{1}^{2}(a)$ $\displaystyle=$
$\displaystyle(C_{1}^{2}(b)+k_{1}S_{1}^{2}(b))(C_{1}^{2}(c)+k_{1}S_{1}^{2}(c))$
$\displaystyle-$
$\displaystyle(C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha))^{2}$
$\displaystyle=$ $\displaystyle
C_{1}^{2}(b)C_{1}^{2}(c)+k_{1}C_{1}^{2}(b)S_{1}^{2}(c)+k_{1}S_{1}^{2}(b)C_{1}^{2}(c)+k_{1}^{2}S_{1}^{2}(b)S_{1}^{2}(c)$
$\displaystyle-$ $\displaystyle
C_{1}^{1}(b)C_{1}^{2}(c)-2k_{1}C_{1}(b)C_{1}(c)S_{1}(b)S_{1}(c)C_{2}(\alpha)-k_{1}^{2}S_{1}^{2}(b)S_{1}^{2}(c)C_{2}^{2}(\alpha)$
$\displaystyle=$ $\displaystyle
k_{1}(C_{1}^{2}(b)S_{1}^{2}(c)+S_{1}^{2}(b)C_{1}^{2}(c)-2C_{1}(b)C_{1}(c)S_{1}(b)S_{1}(c)C_{2}(\alpha))$
$\displaystyle+$ $\displaystyle
k_{1}^{2}S_{1}^{2}(b)S_{1}^{2}(c)(1-C_{2}^{2}(\alpha))$ $\displaystyle=$
$\displaystyle
k_{1}(C_{1}^{2}(b)S_{1}^{2}(c)+S_{1}^{2}(b)C_{1}^{2}(c)-2C_{1}(b)C_{1}(c)S_{1}(b)S_{1}(c)C_{2}(\alpha))$
$\displaystyle+$ $\displaystyle
k_{1}^{2}k_{2}S_{1}^{2}(b)S_{1}^{2}(c)S_{2}^{2}(\alpha),$ $\displaystyle
S_{1}^{2}(a)$ $\displaystyle=$ $\displaystyle
C_{1}^{2}(b)S_{1}^{2}(c)+S_{1}^{2}(b)C_{1}^{2}(c)-2C_{1}(b)C_{1}(c)S_{1}(b)S_{1}(c)C_{2}(\alpha)$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}S_{1}^{2}(b)S_{1}^{2}(c)S_{2}^{2}(\alpha),$
or, having:
$\displaystyle C_{1}(a)$ $\displaystyle=$ $\displaystyle
C_{1}(b)C_{1}(c)(1+k_{1}T_{1}(b)T_{1}(c)C_{2}(\alpha)),$ $\displaystyle
T_{1}^{2}(a)$ $\displaystyle=$
$\displaystyle\frac{T_{1}^{2}(b)+T_{1}^{2}(c)-2T_{1}(b)T_{1}(c)C_{2}(\alpha)+k_{1}k_{2}T_{1}^{2}(b)T_{1}^{2}(c)S_{1}^{2}(\alpha)}{(1+k_{1}T_{1}(b)T_{1}(c)C_{2}(\alpha))^{2}}.$
(2.2)
Similarly,
$\displaystyle S_{1}^{2}(b)$ $\displaystyle=$ $\displaystyle
C_{1}^{2}(a)S_{1}^{2}(c)+S_{1}^{2}(a)C_{1}^{2}(c)+2C_{1}(a)C_{1}(c)S_{1}(a)S_{1}(c)C_{2}(\beta^{\prime})$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}S_{1}^{2}(a)S_{1}^{2}(c)S_{2}^{2}(\beta^{\prime}),$ $\displaystyle
T_{1}^{2}(b)$ $\displaystyle=$
$\displaystyle\frac{T_{1}^{2}(a)+T_{1}^{2}(c)+2T_{1}(a)T_{1}(c)C_{2}(\beta^{\prime})+k_{1}k_{2}T_{1}^{2}(a)T_{1}^{2}(c)S_{1}^{2}(\beta^{\prime})}{(1-k_{1}T_{1}(a)T_{1}(c)C_{2}(\beta^{\prime}))^{2}},$
(2.3)
and
$\displaystyle S_{1}^{2}(c)$ $\displaystyle=$ $\displaystyle
C_{1}^{2}(a)S_{1}^{2}(b)+S_{1}^{2}(a)C_{1}^{2}(b)-2C_{1}(a)C_{1}(b)S_{1}(a)S_{1}(b)C_{2}(\gamma)$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}S_{1}^{2}(a)S_{1}^{2}(b)S_{2}^{2}(\gamma),$ $\displaystyle
T_{1}^{2}(c)$ $\displaystyle=$
$\displaystyle\frac{T_{1}^{2}(a)+T_{1}^{2}(b)-2T_{1}(a)T_{1}(b)C_{2}(\gamma)+k_{1}k_{2}T_{1}^{2}(a)T_{1}^{2}(b)S_{1}^{2}(\gamma)}{(1+k_{1}T_{1}(a)T_{1}(b)C_{2}(\gamma))^{2}}.$
(2.4)
Now, let calculate
$\displaystyle k_{2}S_{2}^{2}(\alpha)$ $\displaystyle=$ $\displaystyle
1-C_{2}^{2}(\alpha)$ $\displaystyle=$
$\displaystyle(C_{2}^{2}(\beta^{\prime})+k_{2}S_{2}^{2}(\beta^{\prime}))(C_{2}^{2}(\gamma)+k_{2}S_{2}^{2}(\gamma))$
$\displaystyle-$
$\displaystyle(C_{2}(\beta^{\prime})C_{2}(\gamma)+k_{2}S_{2}(\beta^{\prime})S_{2}(\gamma)C_{1}(a))^{2}$
$\displaystyle=$ $\displaystyle
C_{2}^{2}(\beta^{\prime})C_{2}^{2}(\gamma)+k_{2}C_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)+k_{2}S_{2}^{2}(\beta^{\prime})C_{2}^{2}(\gamma)+k_{2}^{2}S_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)$
$\displaystyle-$ $\displaystyle
C_{2}^{2}(\beta^{\prime})C_{2}^{2}(\gamma)-2k_{2}C_{2}(\beta^{\prime})S_{2}(\beta^{\prime})C_{2}(\gamma)S_{2}(\gamma)C_{1}(a)-k_{2}^{2}S_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)C_{1}^{2}(a)$
$\displaystyle=$ $\displaystyle
k_{2}(C_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)+S_{2}^{2}(\beta^{\prime})C_{2}^{2}(\gamma)-2C_{2}(\beta^{\prime})S_{2}(\beta^{\prime})C_{2}(\gamma)S_{2}(\gamma)C_{1}(a))$
$\displaystyle+$ $\displaystyle
k_{2}^{2}S_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)(1-C_{1}^{2}(a))$
$\displaystyle=$ $\displaystyle
k_{2}(C_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)+S_{2}^{2}(\beta^{\prime})C_{2}^{2}(\gamma)-2C_{2}(\beta^{\prime})S_{2}(\beta^{\prime})C_{2}(\gamma)S_{2}(\gamma)C_{1}(a))$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}^{2}S_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)S_{1}^{2}(a),$
$\displaystyle S_{2}^{2}(\alpha)$ $\displaystyle=$ $\displaystyle
C_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)+S_{2}^{2}(\beta^{\prime})C_{2}^{2}(\gamma)-2C_{2}(\beta^{\prime})S_{2}(\beta^{\prime})C_{2}(\gamma)S_{2}(\gamma)C_{1}(a)$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}S_{2}^{2}(\beta^{\prime})S_{2}^{2}(\gamma)S_{1}^{2}(a),$
or
$T_{2}^{2}(\alpha)=\frac{T_{2}^{2}(\beta^{\prime})+T_{2}^{2}(\gamma)-2T_{2}(\beta^{\prime})T_{2}(\gamma)C_{1}(a)+k_{1}k_{2}T_{2}^{2}(\beta^{\prime})T_{2}^{2}(\gamma)S_{1}^{2}(a)}{(1+k_{2}T_{2}(\beta^{\prime})T_{2}(\gamma)C_{1}(a))^{2}}.$
(2.5)
Similarly,
$\displaystyle S_{2}^{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle
C_{2}^{2}(\alpha)S_{2}^{2}(\gamma)+S_{2}^{2}(\alpha)C_{2}^{2}(\gamma)+2C_{2}(\alpha)S_{2}(\alpha)C_{2}(\gamma)S_{2}(\gamma)C_{1}(b)$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}S_{2}^{2}(\alpha)S_{2}^{2}(\gamma)S_{1}^{2}(b),$ $\displaystyle
T_{2}^{2}(\beta^{\prime})$ $\displaystyle=$
$\displaystyle\frac{T_{2}^{2}(\alpha)+T_{2}^{2}(\gamma)+2T_{2}(\alpha)T_{2}(\gamma)C_{1}(b)+k_{1}k_{2}T_{2}^{2}(\alpha)T_{2}^{2}(\gamma)S_{1}^{2}(b)}{(1-k_{2}T_{2}(\alpha)T_{2}(\gamma)C_{1}(b))^{2}},$
(2.6)
and
$\displaystyle S_{2}^{2}(\gamma)$ $\displaystyle=$ $\displaystyle
C_{2}^{2}(\alpha)S_{2}^{2}(\beta^{\prime})+S_{2}^{2}(\alpha)C_{2}^{2}(\beta^{\prime})-2C_{2}(\alpha)S_{2}(\alpha)C_{2}(\beta^{\prime})S_{2}(\beta^{\prime})C_{1}(c)$
$\displaystyle+$ $\displaystyle
k_{1}k_{2}S_{2}^{2}(\alpha)S_{2}^{2}(\beta^{\prime})S_{1}^{2}(c),$
$\displaystyle T_{2}^{2}(\gamma)$ $\displaystyle=$
$\displaystyle\frac{T_{2}^{2}(\alpha)+T_{2}^{2}(\beta^{\prime})-2T_{2}(\alpha)T_{2}(\beta^{\prime})C_{1}(c)+k_{1}k_{2}T_{2}^{2}(\alpha)T_{2}^{2}(\beta^{\prime})S_{1}^{2}(c)}{(1+k_{2}T_{2}(\alpha)T_{2}(\beta^{\prime})C_{1}(c))^{2}}.$
(2.7)
What does it mean for triangle? From the Sine law (2.1), having function
$S(x)$ monotonically increasing result that the longest side of any triangle
is opposite to the largest angle and the shortest side is opposed to the
smallest angle. From the Cosine I law in its form that uses $C_{1}(x)$
function, having
$C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)=C_{1}(b-c),$
$C_{1}(a)C_{1}(b)-k_{1}S_{1}(a)S_{1}(b)=C_{1}(a+b),$
and $C_{i}(x)le1$ and is decreasing when $k_{i}=1$, $C_{i}(x)=1$ and is
constant when $k_{i}=0$, $C_{i}(x)ge1$ is increasing when $k_{i}=-1$, we can
see:
$C_{1}(a)=C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha)$
is equivalent to
$a\begin{cases}>b-c,\,k_{2}=1\\\ =b-c,\,k_{2}=0\\\ <b-c,\,k_{2}=-1\end{cases}$
Similarly,
$C_{1}(b)=C_{1}(a)C_{1}(c)-k_{1}S_{1}(a)S_{1}(c)C_{2}(\beta^{\prime})$
is equivalent to
$b\begin{cases}<a+c,\,k_{2}=1\\\ =a+c,\,k_{2}=0\\\ >a+c,\,k_{2}=-1\end{cases}$
From the Cosine II law we can see:
$C_{2}(\alpha)=C_{2}(\beta^{\prime})C_{2}(\gamma)+k_{2}S_{2}(\beta^{\prime})S_{2}(\gamma)C_{1}(a)$
is equivalent to
$\alpha\begin{cases}>\beta^{\prime}-\gamma,\,k_{1}=1\\\
=\beta^{\prime}-\gamma,\,k_{1}=0\\\
<\beta^{\prime}-\gamma,\,k_{1}=-1\end{cases}$
Similarly,
$C_{2}(\beta^{\prime})=C_{2}(\alpha)C_{2}(\gamma)-k_{2}S_{2}(\alpha)S_{2}(\gamma)C_{1}(b)$
is equivalent to
$\beta^{\prime}\begin{cases}<\alpha+\gamma,\,k_{1}=1\\\
=\alpha+\gamma,\,k_{1}=0\\\ >\alpha+\gamma,\,k_{1}=-1\end{cases}$
### 2.2 Right (Quasi)–Triangle Equations
We can define orthogonality in $\mathbb{B}^{n}$ using the orthogonality in
$\mathbb{RP}^{n}$. Namely, two vectors $v_{1}$ and $v_{2}$ of space
$\mathbb{RP}^{n}$ are orthogonal, if $v_{1}\odot v_{2}=0$.
For $\mathbb{B}^{2}$ plane and a line, the orthogonal bundle is line only if
$k_{2}=1$. In this case when line rotates count–clockwise, its orthogonal line
rotates count–clockwise and vice–versa (Figure 1.4 a). When $k_{2}=0$ there is
the only orthogonal bundle, which doesn’t rotate (Figure 1.4 b). When
$k_{2}=-1$ the orthogonal bundle rotates clockwise when the line rotates
count–clockwise toward to the same limit bundle and vice–versa (Figure 1.4 c).
Generally we can’t speak about right triangle as one of its catheti is line
and another isn’t (when $k_{2}\neq 1$). However, as we will see, this figure
is important. We will name it right (quasi)–triangle, which means right
triangle, when $k_{2}=1$ and right quasi–triangle when $k_{2}\neq 1$.
Figure 2.2: Right (quasi)–triangle equations deduction.
We will construct a (quasi)–triangle as half of isosceles one (Figure 2.2).
Consider a triangle $A_{0}B_{0}A^{\prime}_{0}$ with
$A_{0}B_{0}=A^{\prime}_{0}B_{0}=c$, $A_{0}A^{\prime}_{0}=2b$, $\angle
A^{\prime}_{0}A_{0}B_{0}=\angle A_{0}A^{\prime}_{0}B_{0}=\alpha$ and external
angle $\angle A_{0}B_{0}A^{\prime}_{0}=2\beta^{\prime}$.
Let $A^{\prime}_{0}=O=\left[1:0:0\right]$ be origin,
$A_{0}=\mathfrak{R}_{1}(2b)A^{\prime}_{0}=\left[C_{1}(2b):S_{1}(2b):0\right]$,
$B_{0}=\mathfrak{R}_{2}(\alpha)\mathfrak{R}_{1}(c)$
$A_{0}=\left[C_{1}(c):S_{1}(c)C_{2}(\alpha):S_{1}(c)S_{2}(\alpha)\right]$.
Let $ABA^{\prime}=\mathfrak{R}_{1}(-b)(A_{0}B_{0}A^{\prime}_{0})$ (Figure 2.2,
black). Now,
$A^{\prime}=\mathfrak{R}_{1}(-b)A^{\prime}_{0}=\left[C_{1}(b):-S_{1}(b):\right.$
$\left.0\right]$,
$A=\mathfrak{R}_{1}(-b)A_{0}=\left[C_{1}(b):S_{1}(b):0\right]$ and
$B=\mathfrak{R}_{1}(-b)B_{0}=$
$\begin{pmatrix}C_{1}(b)&k_{1}S_{1}(b)&0\\\ -S_{1}(b)&C_{1}(b)&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}C_{1}(c)\\\ S_{1}(c)C_{2}(\alpha)\\\
S_{1}(c)S_{2}(\alpha)\end{pmatrix}$
$=\left[C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha):-S_{1}(b)C_{1}(c)+C_{1}(b)S_{1}(c)C_{2}(\alpha):S_{1}(c)\right.$
$\left.S_{2}(\alpha)\right]$.
Finally, let $C\in AA^{\prime},AC=A^{\prime}C=b$. Then $C=\left[1:0:0\right]$
is origin. From figure equality $A^{\prime}BC=ABC$ result $BC\perp
A^{\prime}A$. Therefore we can consider $ABC$ right (quasi)–triangle.
Having figures $A^{\prime}BC=ABC$ and $C$ is origin, result $B$ have form
$B=(x,0,y)$, where
$\displaystyle C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha)$
$\displaystyle=$ $\displaystyle x,$ $\displaystyle-
S_{1}(b)C_{1}(c)+C_{1}(b)S_{1}(c)C_{2}(\alpha)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle S_{1}(c)S_{2}(\alpha)$ $\displaystyle=$
$\displaystyle y.$
From the second equality, have
$T_{1}(b)=T_{1}(c)C_{2}(\alpha).$ (2.8)
Using the value of $C_{2}(\alpha)$ from this equality and putting it in the
first one, have
$\displaystyle x$ $\displaystyle=$ $\displaystyle
C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)\frac{T_{1}(b)}{T_{1}(c)}$
$\displaystyle=$
$\displaystyle\frac{C_{1}(c)}{C_{1}(b)}(C_{1}^{2}(b)+k_{1}S_{1}^{2}(b))=\frac{C_{1}(c)}{C_{1}(b)}.$
Let now calculate the value of
$\displaystyle x^{2}+k_{1}k_{2}y^{2}$ $\displaystyle=$
$\displaystyle\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}+k_{1}k_{2}S_{1}^{2}(c)S_{2}^{2}(\alpha)$
$\displaystyle=$
$\displaystyle\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}+k_{1}S_{1}^{2}(c)(1-C_{2}^{2}(\alpha))$
$\displaystyle=$
$\displaystyle\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}+k_{1}S_{1}^{2}(c)-k_{1}S_{1}^{2}(c)\frac{T_{1}^{2}(b)}{T_{1}^{2}(c)}$
$\displaystyle=$
$\displaystyle\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}+k_{1}S_{1}^{2}(c)-k_{1}S_{1}^{2}(b)\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}$
$\displaystyle=$
$\displaystyle\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}(1-k_{1}S_{1}^{2}(b))+k_{1}S_{1}^{2}(c)$
$\displaystyle=$
$\displaystyle\frac{C_{1}^{2}(c)}{C_{1}^{2}(b)}C_{1}^{2}(b)+k_{1}S_{1}^{2}(c)$
$\displaystyle=$ $\displaystyle C_{1}^{2}(c)+k_{1}S_{1}^{2}(c)=1.$
It means that exists
$a\in\mathbb{R},C_{12}(a)=\frac{C_{1}(c)}{C_{1}(b)},S_{12}(a)=S_{1}(c)S_{2}(\alpha)$
that has characteristic $k=k_{1}k_{2}$. It is a ‘distance’ parameter $BC$. We
have two more equations:
$\displaystyle C_{1}(c)$ $\displaystyle=$ $\displaystyle C_{12}(a)C_{1}(b)$
(2.9) $\displaystyle S_{12}(a)$ $\displaystyle=$ $\displaystyle
S_{1}(c)S_{2}(\alpha)$ (2.10)
From (2.8) and (2.10) have:
$\frac{S_{12}(a)}{T_{1}(b)}=\frac{S_{1}(c)S_{2}(\alpha)}{T_{1}(c)C_{2}(\alpha)}=C_{1}(c)T_{2}(\alpha),$
using the value of $C_{1}(c)$ from (2.9) have
$\displaystyle\frac{S_{12}(a)}{T_{1}(b)}$ $\displaystyle=$ $\displaystyle
C_{12}(a)C_{1}(b)T_{2}(\alpha),$ $\displaystyle T_{12}(a)$ $\displaystyle=$
$\displaystyle S_{1}(b)T_{2}(\alpha).$ (2.11)
The last 6 equations will include $\beta^{\prime}$. In order to be able to
deduce them we will introduce translation $\mathfrak{T}(-a)$, so as
$\mathfrak{T}(-a)B=C$. Having characteristic $a$ is $k_{1}k_{2}=K_{2}$,
$\mathfrak{T}(-a)=\begin{pmatrix}C_{12}(a)&0&K_{2}S_{12}(a)\\\ 0&1&0\\\
-S_{12}(a)&0&C_{12}(a)\end{pmatrix}.$
We can check this map preserves vector product.
Applying $\mathfrak{T}(-a)$, obtain
$B_{1}=\mathfrak{T}(-a)B=\left[1:0:0\right]$,
$C_{1}=\mathfrak{T}(-a)C=\left[C_{12}(a):0:-S_{12}(a)\right]$ and
$A_{1}=\mathfrak{T}(-a)A=$
$\begin{pmatrix}C_{12}(a)&0&K_{2}S_{12}(a)\\\ 0&1&0\\\
-S_{12}(a)&0&C_{12}(a)\end{pmatrix}\begin{pmatrix}C_{1}(b)\\\ S_{1}(b)\\\
0\end{pmatrix}$
$=\left[C_{12}(a)C_{1}(b):S_{1}(b):-S_{12}(a)C_{1}(b)\right]=\left[C_{1}(c):S_{1}(c)C_{2}(\beta^{\prime}):-S_{1}(c)S_{2}(\beta^{\prime})\right]$
(Figure 2.2, cyan). From here we have
$S_{1}(b)=S_{1}(c)C_{2}(\beta^{\prime}).$ (2.12)
Moreover, having (2.9), obtain
$\displaystyle S_{12}(a)C_{1}(b)$ $\displaystyle=$ $\displaystyle
S_{1}(c)S_{2}(\beta^{\prime}),$ $\displaystyle
S_{12}(a)\frac{C_{1}(c)}{C_{12}(a)}$ $\displaystyle=$ $\displaystyle
S_{1}(c)S_{2}(\beta^{\prime}),$ $\displaystyle T_{12}(a)$ $\displaystyle=$
$\displaystyle T_{1}(c)S_{2}(\beta^{\prime}).$ (2.13)
Combining the last 2 equalities (2.12), (2.13) with (2.9), we have
$\displaystyle\frac{T_{12}(a)}{S_{1}(b)}$ $\displaystyle=$
$\displaystyle\frac{T_{1}(c)S_{2}(\beta^{\prime})}{S_{1}(c)C_{2}(\beta^{\prime})}$
$\displaystyle=\frac{T_{2}(\beta^{\prime})}{C_{1}(c)}$ $\displaystyle=$
$\displaystyle\frac{T_{2}(\beta^{\prime})}{C_{12}(a)C_{1}(b)},$ $\displaystyle
S_{12}(a)$ $\displaystyle=$ $\displaystyle T_{1}(b)T_{2}(\beta^{\prime}).$
(2.14)
Now, having (2.8) and (2.12):
$T_{1}(c)C_{2}(\alpha)=T_{1}(b)=\frac{S_{1}(b)}{C_{1}(b)}=\frac{S_{1}(c)C_{2}(\beta^{\prime})}{C_{1}(b)},$
calculate with (2.9):
$\displaystyle C_{1}(b)$ $\displaystyle=$
$\displaystyle\frac{S_{1}(c)C_{2}(\beta^{\prime})}{T_{1}(c)C_{2}(\alpha)}$
$\displaystyle=$ $\displaystyle
C_{1}(c)\frac{C_{2}(\beta^{\prime})}{C_{2}(\alpha)}$ $\displaystyle=$
$\displaystyle C_{12}(a)C_{1}(b)\frac{C_{2}(\beta^{\prime})}{C_{2}(\alpha)},$
$\displaystyle C_{12}(a)\frac{C_{2}(\beta^{\prime})}{C_{2}(\alpha)}$
$\displaystyle=$ $\displaystyle 1,$ $\displaystyle C_{2}(\alpha)$
$\displaystyle=$ $\displaystyle C_{12}(a)C_{2}(\beta^{\prime}).$ (2.15)
Now from (2.11), (2.12) and (2.13),
$\displaystyle T_{12}(a)=S_{1}(b)T_{2}(\alpha)$ $\displaystyle=$
$\displaystyle T_{1}(c)S_{2}(\beta^{\prime}),$ $\displaystyle
S_{1}(c)C_{2}(\beta^{\prime})T_{2}(\alpha)$ $\displaystyle=$ $\displaystyle
T_{1}(c)S_{2}(\beta^{\prime}),$ $\displaystyle T_{2}(\beta^{\prime})$
$\displaystyle=$ $\displaystyle C_{1}(c)T_{2}(\alpha).$ (2.16)
Finally, by multiplying the last equations (2.15) and (2.16), have using
(2.9):
$\displaystyle C_{1}(c)T_{2}(\alpha)C_{2}(\alpha)$ $\displaystyle=$
$\displaystyle T_{2}(\beta^{\prime})C_{12}(a)C_{2}(\beta^{\prime}),$
$\displaystyle C_{1}(c)S_{2}(\alpha)$ $\displaystyle=$ $\displaystyle
C_{12}(a)C_{2}(\beta^{\prime}),$ $\displaystyle
C_{12}(a)C_{1}(b)S_{2}(\alpha)$ $\displaystyle=$ $\displaystyle
C_{12}(a)S_{2}(\beta^{\prime}),$ $\displaystyle S_{2}(\beta^{\prime})$
$\displaystyle=$ $\displaystyle C_{1}(b)S_{2}(\alpha).$ (2.17)
It is necessary to modify equations (2.9) and (2.15) in order to not contain
the $C(x)$ function.
$\displaystyle k_{1}S_{1}^{2}(c)$ $\displaystyle=$ $\displaystyle
1-C_{1}^{2}(c)=(C^{2}_{12}(a)+k_{1}k_{2}S^{2}_{12}(a))(C_{1}^{2}(b)+k_{1}S_{1}^{2}(b))-C^{2}_{12}(a)C_{1}^{2}(b)$
$\displaystyle=$ $\displaystyle
k_{1}k_{2}S^{2}_{12}(a)C_{1}^{2}(b)+k_{1}C^{2}_{12}(a)S_{1}^{2}(b)+k_{1}^{2}k_{2}S^{2}_{12}(a)S_{1}^{2}(b),$
$\displaystyle S_{1}^{2}(c)$ $\displaystyle=$ $\displaystyle
k_{2}S^{2}_{12}(a)C_{1}^{2}(b)+C^{2}_{12}(a)S_{1}^{2}(b)+k_{1}k_{2}S^{2}_{12}(a)S_{1}^{2}(b)$
By dividing the last equality by its $C(x)$ form, obtain:
$T_{1}^{2}(c)=k_{2}T^{2}_{12}(a)+T_{1}^{2}(b)+k_{1}k_{2}T^{2}_{12}(a)T_{1}^{2}(b).$
(2.18)
Similarly,
$\displaystyle k_{2}S_{2}^{2}(\alpha)$ $\displaystyle=$ $\displaystyle
1-C_{2}^{2}(\alpha)=(C^{2}_{12}(a)+k_{1}k_{2}S^{2}_{12}(a))(C_{2}^{2}(\beta^{\prime})+k_{2}S_{2}^{2}(\beta^{\prime}))-C^{2}_{12}(a)C_{2}^{2}(\beta^{\prime})$
$\displaystyle=$ $\displaystyle
k_{2}C^{2}_{12}(a)S_{2}^{2}(\beta^{\prime})+k_{1}k_{2}S^{2}_{12}(a)C_{2}^{2}(\beta^{\prime})+k_{1}k_{2}^{2}S^{2}_{12}(a)S_{2}^{2}(\beta^{\prime}),$
$\displaystyle S_{2}^{2}(\alpha)$ $\displaystyle=$ $\displaystyle
C^{2}_{12}(a)S_{2}^{2}(\beta^{\prime})+k_{1}S^{2}_{12}(a)C_{2}^{2}(\beta^{\prime})+k_{1}k_{2}S^{2}_{12}(a)S_{2}^{2}(\beta^{\prime})$
By dividing the last equality by its $C(x)$ form, obtain:
$T_{2}^{2}(\alpha)=k_{1}T^{2}_{12}(a)+T_{2}^{2}(\beta^{\prime})+k_{1}k_{2}T^{2}_{12}(a)T_{2}^{2}(\beta^{\prime})$
(2.19)
Note that for $k_{2}=1$ equations (2.8) — (2.19) can be used if external angle
$\beta^{\prime}$ change to internal $\beta$ with the following changes:
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}-\beta^{\prime}$ $\displaystyle\cos\beta$
$\displaystyle=$ $\displaystyle\sin\beta^{\prime}$ $\displaystyle\sin\beta$
$\displaystyle=$ $\displaystyle\cos\beta^{\prime}$ $\displaystyle\tan\beta$
$\displaystyle=$ $\displaystyle\cot\beta^{\prime}$ $\displaystyle\cot\beta$
$\displaystyle=$ $\displaystyle\tan\beta^{\prime}$
### 2.3 More rotations
As we can see, transformation $\mathfrak{T}(-a)$ preserves vector product. In
order to be a motion it needs to be presented as finite product of main
rotations. If $a$, $b$, $c$, $\alpha$ and $\beta^{\prime}$ are real numbers
for which have place equalities (2.8) — (2.17) then it can be checked that
$\mathfrak{T}(a)=\mathfrak{R}_{2}(\beta^{\prime})\mathfrak{R}_{1}(c)\mathfrak{R}_{2}(-\alpha)\mathfrak{R}_{1}(-b)$.
We will introduce new transformations as following:
$\mathfrak{R}_{ij}(\phi)=\begin{pmatrix}1&\ldots&0&\ldots&0&\ldots&0\\\
\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&\ldots&C_{i+1,...j}(\phi)&\ldots&-\frac{K_{j}}{K_{i}}S_{i+1,...j}(\phi)&\ldots&0\\\
\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&\ldots&S_{i+1,...j}(\phi)&\ldots&C_{i+1,...j}(\phi)&\ldots&0\\\
\vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&\ldots&0&\ldots&0&\ldots&1\end{pmatrix}.$
It’s easy to see that all $\mathfrak{R}_{ij}(\phi)$ are motions. All they can
be presented as finit product of main rotations. We will name them rotations
of the space $\mathbb{B}^{n}$. In special case, $\mathfrak{R}_{0i}(\phi)$ we
will name them translations $\mathfrak{T}_{i}(\phi)$ of the space.
### 2.4 Generalized Orthogonal Matrix
For $\mathbb{RP}^{n}$ with given specification $k_{p},\,p=\overline{1,n}$ we
will name the vector $x\in\mathbb{RP}^{n}$ upper $i$-normalized,
$i\in\overline{0,n}$ if $\frac{1}{K_{i}}x\odot x=1$. For $0\leq i<j\leq n$ we
will name two vectors $x$ and $y$ upper $ij$-orthogonal if
$\frac{1}{K_{min(i,j)}}x\odot y=0$. We will name the matrix
$M_{(n+1)\times(n+1)}$ composed of columns $c_{i}$ upper orthogonal if all
columns $c_{i}$ are upper $i$-normalized and any two columns $c_{i}$ and
$c_{j}$ are upper $ij$-orthogonal.
It’s easy to see that all main rotation matrixes are upper orthogonal.
Moreover product of two upper orthogonal matrix is upper orthogonal. Really,
let $X,Y$ are two upper orthogonal matrices. It means that $X$ is composed of
$(x_{0},...,x_{n})$ columns and $Y$ — from $(y_{0},...,y_{n})$ columns, and
$\frac{1}{K_{min(i,j)}}x_{i}\odot x_{j}=\frac{1}{K_{min(i,j)}}y_{i}\odot
y_{j}=\delta_{ij}$ for all $i,j=\overline{0,n}$, where $\delta_{ij}=1,i=j$ and
$\delta_{ij}=0,i\neq j$. Let $Z=XY$ with elements
$z_{ij}=\sum_{p=0}^{n}x_{ip}y_{pj}$. Let $z_{i}$ and $z_{j}$ be 2 columns of
$Z$. Let calculate
$\displaystyle\frac{1}{K_{min(i,j)}}z_{i}\odot z_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{K_{min(i,j)}}\sum_{p=0}^{n}K_{p}z_{pi}z_{pj}$
$\displaystyle=$
$\displaystyle\frac{1}{K_{min(i,j)}}\sum_{p=0}^{n}K_{p}\left(\sum_{m_{1}=0}^{n}x_{pm_{1}}y_{m_{1}i}\right)\left(\sum_{m_{2}=0}^{n}x_{pm_{2}}y_{m_{2}j}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{K_{min(i,j)}}\sum_{p=0}^{n}K_{p}\sum_{m_{1}=0}^{n}\sum_{m_{2}=0}^{n}x_{pm_{1}}x_{pm_{2}}y_{m_{1}i}y_{m_{2}j}$
$\displaystyle=$
$\displaystyle\frac{1}{K_{min(i,j)}}\sum_{m_{1}=0}^{n}\sum_{m_{2}=0}^{n}y_{m_{1}i}y_{m_{2}j}\sum_{p=0}^{n}K_{p}x_{pm_{1}}x_{pm_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{K_{min(i,j)}}\sum_{m_{1}=0}^{n}\sum_{m_{2}=0}^{n}y_{m_{1}i}y_{m_{2}j}K_{min(m_{1},m_{2})}\delta_{m_{1}m_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{K_{min(i,j)}}\sum_{m=0}^{n}y_{mi}y_{mj}K_{m}$
$\displaystyle=$ $\displaystyle\delta_{ij}$
We will name the vector $x\in\mathbb{RP}^{n}$ lower $i$-normalized,
$i\in\overline{0,n}$ if $K_{i}\sum_{j=0}^{n}\frac{x_{j}^{2}}{K_{j}}=1$. For
$0\leq i<j\leq n$ we will name two vectors $x$ and $y$ lower $ij$-orthogonal
if $K_{max(i,j)}\sum_{p=0}^{n}\frac{x_{p}y_{p}}{K_{p}}=0$. We will name the
matrix $M_{(n+1)\times(n+1)}$ composed of rows $r_{i}$ lower orthogonal if all
rows $r_{i}$ are lower $i$-normalized and any two rows $r_{i}$ and $r_{j}$ are
lower $ij$-orthogonal.
It’s easy to see that all main rotation matrixes are also lower orthogonal.
Moreover product of two lower orthogonal matrices is lower orthogonal. Really,
let $X,Y$ are two lower orthogonal matrices. It means that $X$ is composed of
$(x_{0},...,x_{n})$ rows and $Y$ — from $(y_{0},...,y_{n})$ rows, where
$K_{max(i,j)}\sum_{p=0}^{n}\frac{x_{ip}x_{jp}}{K_{p}}=K_{max(i,j)}\sum_{p=0}^{n}\frac{y_{ip}y_{jp}}{K_{p}}=\delta_{ij}$
for all $i,j=\overline{0,n}$. Let $Z=XY$. Let calculate
$\displaystyle K_{max(i,j)}\sum_{p=0}^{n}\frac{z_{ip}z_{jp}}{K_{p}}$
$\displaystyle=$ $\displaystyle
K_{max(i,j)}\sum_{p=0}^{n}\frac{1}{{K_{p}}}\left(\sum_{m_{1}=0}^{n}x_{im_{1}}y_{m_{1}p}\right)\left(\sum_{m_{2}=0}^{n}x_{jm_{2}}y_{m_{2}p}\right)$
$\displaystyle=$ $\displaystyle
K_{max(i,j)}\sum_{p=0}^{n}\frac{1}{K_{p}}\sum_{m_{1}=0}^{n}\sum_{m_{2}=0}^{n}x_{im_{1}}x_{jm_{2}}y_{m_{1}p}y_{m_{2}p}$
$\displaystyle=$ $\displaystyle
K_{max(i,j)}\sum_{m_{1}=0}^{n}\sum_{m_{2}=0}^{n}x_{im_{1}}x_{jm_{2}}\sum_{p=0}^{n}\frac{1}{K_{p}}y_{m_{1}p}y_{m_{2}p}$
$\displaystyle=$ $\displaystyle
K_{max(i,j)}\sum_{m_{1}=0}^{n}\sum_{m_{2}=0}^{n}x_{im_{1}}x_{jm_{2}}\frac{\delta_{m_{1}m_{2}}}{K_{max(m_{1},m_{2})}}$
$\displaystyle=$ $\displaystyle
K_{max(i,j)}\sum_{m=0}^{n}\frac{x_{im}x_{jm}}{K_{m}}$ $\displaystyle=$
$\displaystyle\delta_{ij}$
For some upper orthogonal matrix $X$ has place the equality
$\displaystyle\frac{1}{K_{j}}\sum_{i=0}^{n}K_{i}x^{2}_{ij}$ $\displaystyle=$
$\displaystyle 1,$ $\displaystyle\sum_{i=0}^{n}K_{i}x^{2}_{ij}$
$\displaystyle=$ $\displaystyle K_{j},$
$\displaystyle\sum_{i=0}^{n}x^{2}_{ij}\prod_{p=1}^{i}k_{p}$ $\displaystyle=$
$\displaystyle\prod_{p=1}^{j}k_{p}.$
Let divide it to $K_{q}$, $q\leq j$:
$\sum_{i=0}^{q-1}\frac{K_{i}}{K_{q}}x^{2}_{ij}+\sum_{i=q}^{n}\frac{K_{i}}{K_{q}}x^{2}_{ij}=\frac{K_{j}}{K_{q}}.$
As $K_{q}$ divides $K_{j}$ and $K_{i},i\geq q$, but $K_{q}$ doesn’t divide
$K_{i},i<q$, result that for $0\leq i<j\leq n$, $x_{ij}$ divide $K_{q}/K_{i}$
for all $i<q\leq j$, or $x_{ij}$ divide $K_{j}/K_{i}$.
Having for some upper orthogonal matrix $X$, elements $x_{ij}$ divide
$K_{j}/K_{i}$, construct the matrix $Y$ of the same size with elements
$y_{ij}=\sqrt{\frac{K_{i}}{K_{j}}}x_{ij}$. The matrix $Y$ is orthogonal
one111consider $\sqrt{0}=0$. (if may be complex, in this case it isn’t unitar,
but orthogonal). Really, for some $i=\overline{0,n}$,
$\sum_{i=0}^{n}y^{2}_{ij}=\sum_{i=0}^{n}\frac{K_{i}}{K_{j}}x^{2}_{ij}=\frac{1}{K_{j}}\sum_{i=0}^{n}K_{i}x^{2}_{ij}=1$
and for some $j_{1}\neq j_{2}=\overline{0,n}$,
$\displaystyle\sum_{i=0}^{n}y_{ij_{1}}y_{ij_{2}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\frac{K_{i}}{\sqrt{K_{j_{1}}K_{j_{2}}}}x_{ij_{1}}x_{ij_{2}}$
$\displaystyle=$
$\displaystyle\frac{K_{min(j_{1},j_{2})}}{\sqrt{K_{j_{1}}K_{j_{2}}}}\frac{1}{K_{min(j_{1},j_{2})}}\sum_{i=0}^{n}K_{i}x_{ij_{1}}x_{ij_{2}}$
$\displaystyle=$
$\displaystyle\frac{K_{min(j_{1},j_{2})}}{\sqrt{K_{j_{1}}K_{j_{2}}}}0=0,$
because $x_{ij_{1}}x_{ij_{2}}$ divides
$\frac{\sqrt{K_{j_{1}}K_{j_{2}}}}{K_{i}}$.
For some orthogonal matrix always has place also the following equalities for
$i_{1}\neq i_{2}=\overline{0,n}$:
$\sum_{j=0}^{n}y^{2}_{ij}=\sum_{j=0}^{n}\frac{K_{i}}{K_{j}}x^{2}_{ij}=K_{i}\sum_{j=0}^{n}\frac{1}{K_{j}}x^{2}_{ij}=1,$
$\displaystyle\sum_{j=0}^{n}y_{i_{1}j}y_{i_{2}j}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\frac{\sqrt{K_{i_{1}}K_{i_{2}}}}{K_{j}}x_{i_{1}j}x_{i_{2}j}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{K_{i_{1}}K_{i_{2}}}}{K_{max(i_{1},i_{2})}}K_{max(i_{1},i_{2})}\sum_{j=0}^{n}\frac{1}{K_{j}}x_{i_{1}j}x_{i_{2}j}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{K_{i_{1}}K_{i_{2}}}}{K_{max(i_{1},i_{2})}}0=0$
So, $x_{i_{1}j}x_{i_{2}j}$ divides $\frac{K_{j}}{\sqrt{K_{i_{1}}K_{i_{2}}}}$.
It means that $X$ is also lower orthogonal matrix.
Inverse orthogonal matrix $Y^{-1}$ is easy constructed as
$y^{\prime}_{ji}=y_{ij}$. Then
$\displaystyle\sqrt{\frac{K_{j}}{K_{i}}}x^{\prime}_{ji}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{K_{i}}{K_{j}}}x_{ij},$ $\displaystyle
x^{\prime}_{ji}$ $\displaystyle=$ $\displaystyle\frac{K_{i}}{K_{j}}x_{ij}.$
The last equality isn’t applicable if some characteristic $k_{i}=0$. Although
is true, it isn’t determinable having the form of $0/0$. If some
characteristic $k_{m}=0,m<n$, the matrix has the form:
$M=\begin{pmatrix}A&O\\\ B&C\end{pmatrix}$
Really, for the first $m$ columns the upper orthogonality condition is
equivalent to:
$\displaystyle\sum_{i=0}^{n}\frac{K_{i}}{K_{j}}x_{ij}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{m-1}\frac{K_{i}}{K_{j}}x_{ij}^{2},\forall
j=\overline{0,m-1},$
$\displaystyle\sum_{i=0}^{n}\frac{K_{i}}{K_{j_{1}}}x_{ij_{1}}x_{ij_{2}}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{m-1}\frac{K_{i}}{K_{j_{1}}}x_{ij_{1}}x_{ij_{2}},\forall
j_{1}=\overline{0,m-1},j_{2}=\overline{0,n},j_{1}<j_{2},$
Having $K_{j}\neq 0$ and $K_{i}=0$, all terms, starting with $i=m$ equals to
zero. So, matrix $A$ is upper orthogonal of size $m\times m$ and matrix $B$ is
free of size $(n-m+1)\times m$. For the last $n-m+1$ columns upper
orthogonality has form:
$\displaystyle\sum_{i=0}^{n}\frac{K_{i}}{K_{j}}x_{ij}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=m}^{n}\frac{\prod_{p=m}^{i}k_{p}}{\prod_{p=m}^{j}k_{p}}x_{ij}^{2},\forall
j=\overline{m,n},$
$\displaystyle\sum_{i=0}^{n}\frac{K_{i}}{K_{j_{1}}}x_{ij_{1}}x_{ij_{2}}$
$\displaystyle=$
$\displaystyle\sum_{i=m}^{n}\frac{\prod_{p=m}^{i}k_{p}}{\prod_{p=m}^{j_{1}}k_{p}}x_{ij_{1}}x_{ij_{2}},\forall
j_{1}<j_{2}=\overline{m,n},$
because $K_{j}=0$. It means the matrix $C$ is upper orthogonal of size
$(n-m+1)\times(n-m+1)$ and the matrix $O$ is obligatory zero one of size
$m\times(n-m+1)$ (otherwise elements of $M$ aren’t finite).
It’s easy to verify that inverse matrix has form:
$M^{-1}=\begin{pmatrix}A^{-1}&O\\\ -C^{-1}BA^{-1}&C^{-1}\end{pmatrix}$
This way of calculating the inverse matrix can easy be generalized to either
number of null characteristics.
We will name upper orthogonal matrixes (which also are lower orthogonal)
generalized orthogonal. We will use the term orthogonal matrix meaning
generalized orthogonal matrix if isn’t stated otherwise. As we can see, the
orthogonal matrix set is closed in respect of multiplying, it contains the
unit element and for any element it contains its inverse. So the orthogonal
matrix set form isomentry group of space. All motion matrices are generalized
orthogonal.
### 2.5 Orthogonal Matrix as Product of Rotations
Let $X$ will be orthogonal matrix. We will search the rotation matrices, the
product of which gives $X$. Note that The matrix $X\mathfrak{R}_{ij}(\phi)$
have all columns $x_{p}$ of $X$ except $i$ and $j$ ones. These columns are
$x^{\prime}_{i}=x_{i}C_{i+1...j}(\phi)+x_{j}S_{i+1...j}(\phi)$ and
$x^{\prime}_{j}=-\frac{K_{j}}{K_{i}}x_{i}S_{i+1...j}(\phi)+x_{j}C_{i+1...j}$.
For the last row let separate elements $x_{i},i=\overline{0,n}$ in three
categories: having characteristics $K_{n}/K_{i}$ equals to 1, 0 and $-1$. Note
that for $i$ row the $i$ element is always of the category 1, because its
characteristic is $K_{i}/K_{i}=1$. We will multiply $X$ on the right by
$\mathfrak{R}_{in}(\phi),i=\overline{0,n}$ in order to have in the $n$-th row
a single element of category 1 and single element of category $-1$, different
from 0. All these rotations are elliptic ones. For elements of the same
characteristic $x_{ni}$ and $x_{nj}$ we can use
$\cos\phi=\frac{x_{ni}}{\sqrt{x^{2}_{ni}+x^{2}_{nj}}}$ and
$\sin\phi=-\frac{x_{nj}}{\sqrt{x^{2}_{ni}+x^{2}_{nj}}}$. Moreover, always
$x_{nn}\neq 0$.
Now, we have one element of category 1 and one of category $-1$, different
from zero (the $n$-th one and, for example, the $p$-th one) and element of
category 1 has absolute value greater then the element of category $-1$
because for this lower $n$-normalized row have place equality
$x^{2}_{nn}-x^{2}_{np}=1$. It means that exists $\phi\in\mathbb{R}$ so that
$\cosh\phi=\frac{x_{nn}}{\sqrt{x^{2}_{nn}-x^{2}_{np}}}$ and
$\sinh\phi=-\frac{x_{np}}{\sqrt{x^{2}_{nn}-x^{2}_{np}}}$ and hyperbolic
rotation that transforms the element of category $-1$, $x_{np}$ in
$x^{\prime}_{np}=0$ and the element of category 1, $x_{nn}$ in
$x^{\prime}_{nn}\neq 0$.
For category 0 there exist parabolic rotations, that preserves the element of
category 1 ($x_{nn}$) and elements of category 0 transform in 0. For this
case, if one this element is on $q$-th column, $\phi=-x_{nq}$. The last
non–zero element $x_{nn}$ equals to 1 or $-1$, because the last row is lower
$n$-normalized.
We can consider the first $n$ columns as having $n$ elements (the last one
equals to zero). They form orthogonal matrix of size $n$. The last, $(n+1)$-th
column (without the last element) is upper $in$-orthogonal to first $n$
columns, $i=\overline{0,n-1}$. As these columns have $n$ elements each,
$(n+1)$-th column is obligatory null (excluding the last its element).
In this stage we can consider the resulting matrix as having size $n$ instead
of $n+1$ and repeat the process for it. Finally obtain the matrix $E$ which
has elements on main diagonal 1 or $-1$ and all rest elements 0. It is the
reflection matrix on a point or line or plane or hyperplane. Obtain the
equality: $X\prod_{j=1}^{q}\mathfrak{M}_{j}=E$. It’s easy to see that
$X=E\prod_{j=q}^{1}\mathfrak{M}^{-1}_{j}$ ($q=(n+1)n/2$). Strictly speaking,
the matrix $E$ can’t be presented as product of rotations. In order to
identify motions of $\mathbb{B}^{n}$ with orthogonal matrices, we should name
$E$ (which preserve vector product) motion. However, these motions are
improper (there is no continuous parameterization of motion
$\mathfrak{M}(\alpha)$ on a segment $[0,1]$ such that
$\mathfrak{M}(0)=I,\mathfrak{M}(1)=E$ and all $\mathfrak{M}(\alpha)$ are
motions on all $\alpha\in[0,1]$). Having in expression for $X$ determinant of
matrices $\mathfrak{M}_{i}$ equals to 1 and determinant of $E$ is $\pm 1$,
determinant of $X$ equals $\pm 1$.
### 2.6 Coordinate and State Matrix
Consider in some space $\mathbb{B}^{n}$ $n+1$ vectors $v_{0},...v_{n}$. Let
coordinates of $v_{i}$ are $\left[v_{0i}:...:v_{ni}\right],i=\overline{0,n}$.
Let vectors $v_{i}$ are ordered and form basis of $\mathbb{B}^{n}$ (not
obligatory orthonormal). Compose the matrix $V$ with elements
$v_{ij},i,j=\overline{0,n}$. Will name it coordinate matrix for vectors
$v_{i}$. Construct also the matrix $M$ of size $(n+1)\times(n+1)$ with
elements $m_{i,j}=\frac{v_{i}\odot v_{j}}{K_{i}}$. We will name the matrix $M$
state matrix of $v_{i}$. Having $v_{i}$ is space basis, elements $m_{ij}$ are
all finite. State matrix shows how orthonormal is some vector family. It tends
to unite one when vectors are more normalized and orthogonal to each
other222If space specification contains null characteristics, some elements of
state matrix can have any value, even for orthonormal vector family.. We will
demonstrate that volume of parallelepiled constructed on vectors $v_{i}$
equals to $|\det V|$ and
$\det M=\left(\det V\right)^{2}.$ (2.20)
First, let $v_{i},i=\overline{0,n}$ are orthonormal. Then the parallelepiped
volume is 1, the matrix $V$ is orthogonal one and $\det V=\pm 1$. So, $|\det
V|$ equals to parallelepiped volume. All elements on main diagonal $m_{ii}=1$,
because all vectors $v_{i}$ are upper $i$-normalized. All elements above main
diagonal are $m_{ij}=0,i<j$, because all vectors $v_{i}$ and $v_{j}$ are upper
$ij$-orthogonal (elements under the main diagonal may differ from 0). It means
that the matrix $M$ is lower triangular with all elements on main diagonal
equls to 1 and $\det M=1=\left(\det V\right)^{2}$.
Further, note that $x\odot(y+z)=x\odot y+x\odot z,(x+y)\odot z=x\odot z+y\odot
z,(\alpha x)\odot y=\alpha(x\odot y)=x\odot(\alpha y),\forall
x,y,z\in\mathbb{B}^{n},\alpha\in\mathbb{R}$. Matrix determinant equals to zero
if it contain proportional columns or rows. When some row or column of a
matrix is multiplied by $\alpha$, the resulting matrix determinant is $\alpha$
times original matrix determinant. When some row or column is sum of two rows
/ columns, then the matrix determinant equals to sum of determinants of
matrices containing the first and the second row / column.
Second, let instead of some $v_{i}$ use $v^{\prime}_{i}=\alpha v_{i}$. In this
case parallelepiped volume grows in $\alpha$ times and $|\det
V^{\prime}|=|\alpha||\det V|$. Moreover, $\det M^{\prime}=\alpha^{2}\det
M=\left(\alpha\det V\right)^{2}=\left(\det V^{\prime}\right)^{2}$.
Third, let instead of some $v_{i}$ use $v^{\prime}_{i}=v_{i}+\alpha v_{j}$. In
this case parallelepiped volume remains unchanged, as well as determinant of
$V$, and $\det M^{\prime}=\det M=\left(\det V\right)^{2}=\left(\det
V^{\prime}\right)^{2}$.
Finally, observe, that all matrices $V$ result from orthogonal matrices using
operations form the second and the third step. It means the equation (2.20) is
true for all matrices and volume of parallelepiped constructed on vectors
$v_{i}$ equals to $|\det V|$.
When somebody calculates the parallelepiped volume it’s usefull to use the
state matrix. Its elements don’t change on motions and it is always square,
even when the number of vectors is less then the space dimension (the matrix
$V$ isn’t square in this case).
### 2.7 Plane definition and Specification. Lineals and their Specification
Having $\mathbb{B}^{m}\subset\mathbb{RP}^{m}$, all $m$-dimensional planes
$\mathbb{B}^{n}$ ($m<n$) lie in $\mathbb{RP}^{n}$ with one global condition
for vectors $X\in\mathbb{RP}^{m}\subset\mathbb{PR}^{n}$: $X\odot X=1$. Leaving
this condition (it doesn’t change on any motion), we can consider
$m$-dimensional planes of $\mathbb{B}^{n}$ as $m$-dimensional planes of
$\mathbb{RP}^{n}$.
By definition, $m$-dimensional plane $L^{m}$ results from subspace
$\mathbb{B}^{m}$ on some motion. Subspace $\mathbb{B}^{m}$ has the first $m+1$
columns of unite matrix with dimension $n+1$ as its basis. Multiplying the
basis matrix of $\mathbb{B}^{m}$ by some orthogonal matrix result basis matrix
of $L^{m}$ as first $m+1$ columns of orthogonal matrix. Being a subspace,
specification of $\mathbb{B}^{m}$ contain the first $m$ characteristics of
specification $\mathbb{B}^{n}$.
What happens if we take any $m+1$ columns of some orthogonal matrix as basis?
Let column indices $i_{0},i_{1},...i_{m}$ and $i_{p}=\overline{0,n}$. It’s
easy to see that motions that preserve this figure only change these columns
(interior figure motions) or change no these columns (motions of
$\mathbb{B}^{n}$ that preserve all its points). Thus, figure characteristics
$K_{p}^{\prime}=K_{i_{p}},p=\overline{0,m}$, or
$k_{p}^{\prime}=K_{p}^{\prime}/K_{p-1}^{\prime}=K_{i_{p}}/K_{i_{p-1}}=\prod_{j=i_{p-1}+1}^{i_{p}}k_{j},p=\overline{1,m}$
is its specification. These figures generally speaking are not planes. We will
name them lineals. We will name planes also lineals.
It may happen, that some lineal has $K_{0}^{\prime}\neq 1$ (space and planes
have it equals to 1). In this case lineal may not intersect the space sphere
and may not have image. We will name lineals that have no image improper.
Although they have no image, their properties help studying the space
geometry.
One more interesting case is when the space specification has characteristic
$-1$ and some lineal is constructed on limit vectors for this characteristic.
Such lineals can’t be constructed from matrices get as finite product of
motions. They can be constructed as limit of infinite products. These lineal
specifications can’t be deduced from space specification.
For example, let space $\mathbb{B}^{2}$ has specification $\\{1,-1\\}$.
Vectors $\left[0:1:1\right]$ and $\left[0:-1:1\right]$ can’t result from
coordinate vectors on finite product of motions. However, there exist
translations along these vectors (interior lineal translations):
$\mathfrak{M}=\begin{pmatrix}1&-2S_{0}(\frac{\phi}{2})&-2S_{0}(\frac{\phi}{2})\\\
2S_{0}(\frac{\phi}{2})&1-2S_{0}^{2}(\frac{\phi}{2})&-2S_{0}^{2}(\frac{\phi}{2})\\\
-2S_{0}(\frac{\phi}{2})&2S_{0}^{2}(\frac{\phi}{2})&1+2S_{0}^{2}(\frac{\phi}{2})\end{pmatrix},$
$\mathfrak{W}=\begin{pmatrix}1&2S_{0}(\frac{\psi}{2})&-2S_{0}(\frac{\psi}{2})\\\
-2S_{0}(\frac{\psi}{2})&1-2S_{0}^{2}(\frac{\psi}{2})&2S_{0}^{2}(\frac{\psi}{2})\\\
-2S_{0}(\frac{\psi}{2})&-2S_{0}^{2}(\frac{\psi}{2})&1+2S_{0}^{2}(\frac{\psi}{2})\end{pmatrix}.$
These motions matrices use functions $C_{0}(x)$ and $S_{0}(x)$ have
characteristic 0, despite the fact the space specification doesn’t contain
zero. These translations are border space motions between elliptic and
hyperbolic ones.
### 2.8 Projection of Vector on Lineal and on its Orthogonal Completion
We will name some vector $v^{\prime}$ projection of vector $v$ on lineal
$L^{m}$, if $v^{\prime}\in L^{m},v-v^{\prime}\perp L^{m}$. Let lineal $L^{m}$
is constructed on vectors $l_{0},...l_{m}$. Then
$v^{\prime}=\sum_{i=0}^{m}\frac{v\odot l_{i}}{K_{i}}l_{i}$. Evident,
$v^{\prime}\in L_{m}$. Let’s see:
$\displaystyle(v-v^{\prime})\odot l_{j}$ $\displaystyle=$
$\displaystyle\left(v-\sum_{i=0}^{m}\frac{v\odot
l_{i}}{K_{i}}l_{i}\right)\odot l_{j}$ $\displaystyle=$ $\displaystyle v\odot
l_{j}-\sum_{i=0}^{m}\frac{v\odot l_{i}}{K_{i}}(l_{i}\odot l_{j})$
$\displaystyle=$ $\displaystyle v\odot l_{j}-\frac{v\odot
l_{j}}{K_{j}}(l_{j}\odot l_{j})$ $\displaystyle=$ $\displaystyle v\odot
l_{j}-\frac{v\odot l_{j}}{K_{j}}K_{j}=0,$
for all $j=\overline{0,m}$, in other words,
$v^{\prime\prime}=v-v^{\prime}\perp L^{m}$. When some $K_{i}=0$, expression
$\frac{v\odot l_{i}}{K_{i}}$ has undefined value. It happens when some vector
direction is orthogonal to all others. In this case there is impossible to
determine unique orthogonal vector. However, any value of this expression, for
example 0, is valid, as it corresponds to some orthogonal vector.
### 2.9 Basis Change in Lineal. Unique Form of Lineal
Let $L^{m}\subset\mathbb{B}^{n}$ is some space lineal, defined by matrix of
size $(n+1)\times(m+1)$. The matrix columns $l_{i},i=\overline{0,m}$ form
basis of lineal. Consider vector $a=\left[a_{0}:...:a_{m}\right]\in L^{m}$.
Let vector coordinates in $\mathbb{B}^{m}$ are
$v=\left[v_{0}:...:v_{n}\right]$. Then $v=L^{m}a$. Let $\mathfrak{M}$ be
interior motion of lineal $L^{m}$, defined by matrix of size
$(m+1)\times(m+1)$. And let coordinates of $a$ in new basis $L^{\prime m}$ are
$b=\left[b_{0}:...:b_{m}\right]$. Then $b=\mathfrak{M}a$. Now $v=L^{\prime
m}b$. Having the fact the coordinates of vector $v$ in $\mathbb{B}^{n}$ don’t
change, result matrix equality:
$L^{m}a=v=L^{\prime m}b=L^{\prime m}(\mathfrak{M}a)=(L^{\prime
m}\mathfrak{M})a.$
This equality doesn’t depend on vector $a$, then
$L^{m}=L^{\prime m}\mathfrak{M}$ (2.21)
is equation of basis change in lineal.
It is necessary to find the unique form of lineal definition. Consider the
following algorithm for the unique basis search:
1. 1.
Let $i_{p},p=\overline{0,n}$ is basis of $\mathbb{B}^{n}$. Start with empty
basis of $L^{m}$.
2. 2.
Until new basis has less then $m+1$ elements, search for $i^{\prime}_{p}$ as
projection of next $i_{p}$ on $L^{m}$.
1. (a)
If projection isn’t null, find new vector $i^{\prime\prime}_{p}$ as projection
of $i^{\prime}_{p}$ on orthogonal completion of existing basis $l_{i}$.
2. (b)
If $i^{\prime\prime}_{p}$ isn’t null, find its position as free index $0\leq
q\leq m$ so that $r^{2}=\frac{1}{K_{q}}i^{\prime\prime}_{p}\odot
i^{\prime\prime}_{p}>0$.
3. (c)
Norm it and add to existing basis $l_{q}=\frac{i^{\prime\prime}_{p}}{r}$.
### 2.10 Measure Calculus Between Lineals
Figure 2.3: Measure calculus between lineals $X^{2}$ and $Y^{2}$.
Let $X^{p},Y^{q},p\leq q\leq n$ are two lineals. Let $x_{i},i=\overline{0,p}$
is the basis of $X^{p}$. Let $X^{\prime p}$ be projection of $X^{p}$ on
$Y^{p}$ (Figure 2.3) and let $x^{\prime}_{i}$ be projection of $x_{i}$ on
$Y^{p}$ (they are not orthonormal). If the volumes of parallelepipeds
constructed on vectors $x_{i}$ and $x^{\prime}_{i}$ are equals to $V_{x}$ and
$V^{\prime}_{x}$ respectively and the angle between $X^{p}$ and $Y^{q}$ is
measurable and equals to $\phi$, then has place the equality:
$V^{\prime}_{x}=V_{x}C(\phi).$
This equality is a particular case of (2.8), when $k_{1}=0,T(x)=x$. In our
case always $k_{1}=0$, because the space model is linear. As were discussed
earlier, $V_{x}=1$ and $V^{\prime}_{x}=\sqrt{\det M^{\prime}_{x}}$, where
$M^{\prime}_{x}$ is state matrix of vectors $x^{\prime}_{i}$:
$C(\phi)=\sqrt{\det M^{\prime}_{x}}.$ (2.22)
It may happen that characteristic of $\phi$ equals to zero and we can’t
calculate $C^{-1}(\phi)$. If we project vectors $x_{i}$ on $Y^{q}_{\perp}$
orthogonal completion of $Y^{q}$ (suppose it has dimension at least $p$), get
$X^{\prime\prime p}$ constructed on vectors $x^{\prime\prime}_{i}$ with the
volume $V^{\prime\prime}_{x}$, then by (2.13) get:
$V^{\prime\prime}_{x}=V_{x}S(\phi).$
Or, having $M^{\prime\prime}_{x}$ state matrix for vectors
$x^{\prime\prime}_{i}$,
$S(\phi)=\sqrt{\det M^{\prime\prime}_{x}}.$ (2.23)
If the dimension of $Y^{q}_{\perp}$ is less then $p$, then we can get
$S(\phi)$ by projecting of $Y^{q}_{\perp}$ on $X^{p}$. If $\phi$ isn’t
measurable, then the angle $\psi$ between $X^{p}$ and $Y^{q}_{\perp}$ is
measurable and:
$\displaystyle S(\psi)=\sqrt{\det M^{\prime}_{x}},$ (2.24) $\displaystyle
C(\psi)=\sqrt{\det M^{\prime\prime}_{x}}.$ (2.25)
The angles $\phi$ and $\psi$ present measure between lineals $X^{p}$ and
$Y^{q}$. Having values $C(\phi),S(\phi)$ and $C(\psi),S(\psi)$, it is possible
to determine $\phi$ and $\psi$. The measure characteristic of $\phi$ and
$\psi$ equals to measure characteristic between $X^{\prime p}$ and
$X^{\prime\prime p}$. Depending on this characteristic, situation can be one
of the following:
* •
If characteristic equals to 1, then $\det M^{\prime}_{x}+\det
M^{\prime\prime}_{x}=1$ and $\phi=\tan^{-1}\sqrt{\frac{\det
M^{\prime\prime}_{x}}{\det M^{\prime}_{x}}}$, $\psi=\tan^{-1}\sqrt{\frac{\det
M^{\prime}_{x}}{\det M^{\prime\prime}_{x}}}$.
* •
If characteristic equals to 0, then either $\det M^{\prime}_{x}=1$ and
$\phi=\sqrt{\det M^{\prime\prime}_{x}}$, $\psi=\infty$, or $\det
M^{\prime\prime}_{x}=1$, and $\phi=\infty$, $\psi=\sqrt{\det M^{\prime}_{x}}$.
* •
If characteristic equals to $-1$, then either $\det M^{\prime}_{x}-\det
M^{\prime\prime}_{x}=1$ and $\phi=\tanh^{-1}\sqrt{\frac{\det
M^{\prime\prime}_{x}}{\det M^{\prime}_{x}}}$, $\psi$ isn’t measurable, or
$\det M^{\prime\prime}_{x}-\det M^{\prime}_{x}=1$ and $\phi$ isn’t measurable,
$\psi=\tanh^{-1}\sqrt{\frac{\det M^{\prime}_{x}}{\det M^{\prime\prime}_{x}}}$,
or $\det M^{\prime\prime}_{x}=\det M^{\prime}_{x}$ and $\phi=\psi=\infty$.
### 2.11 Volume Calculation
We can see that for any $\mathbb{B}^{n}$ seen as a unit sphere in
$\mathbb{R}^{n+1}$, the surface is orthogonal to radius. Let
$X,Y\in\mathbb{B}^{n}$ and the distance between $X$ and $Y$ is small. Let
$O=(0,0,...,0)$ is origin of $\mathbb{R}^{n+1}$. We will see that
$(O-X)\odot(Y-X)=0$ when $Y\rightarrow X$ in sense of distance between them.
$O-X=-X$, $(O-X)\odot(Y-X)=-X\odot(Y-X)=X\odot X-X\odot Y=1-C_{1}(d(X,Y))$,
where $d(X,Y)$ is distance between $X$ and $Y$. When $Y\rightarrow X$,
$d(X,Y)\rightarrow 0$ and $1-C_{1}(d(X,Y))\rightarrow 0$.
Let $A,B\in\mathbb{B}^{1}$. $A=\left[C_{1}(\alpha):S_{1}(\alpha)\right]$,
$B=\left[C_{1}(\beta):S_{1}(\beta)\right]$, where $C_{1}(x)$, $S_{1}(x)$ and
$T_{1}(x)$ are defined as (1.2), (1.3) and (1.4). Let calculate in
$\mathbb{R}^{2}$ the area of $\mathbb{B}^{1}$ sector between $A$ and $B$. In
Euclidean polar system the argument
$\tan\phi_{e}=y/x=S_{1}(\phi)/C_{1}(\phi)=T_{1}(\phi)$, where $\phi$ is native
argument in $\mathbb{B}^{1}$. The Euclidean radius
$\rho=\sqrt{x^{2}+y^{2}}=\sqrt{C_{1}^{2}(\phi)+S_{1}^{2}(\phi)}$. Having
$d\phi_{e}=\frac{d\phi}{(1+T_{1}^{2}(\phi))C_{1}^{2}(\phi)}$, the area is:
$S=\frac{1}{2}\int_{A}^{B}\rho(\phi_{e})^{2}d\phi_{e}=\frac{1}{2}\int_{A}^{B}(C_{1}^{2}(\phi)+S_{1}^{2}(\phi))\frac{d\phi}{C_{1}^{2}(\phi)(1+T_{1}^{2}(\phi))}$
$=\frac{1}{2}\int_{A}^{B}\frac{C_{1}^{2}(\phi)+S_{1}^{2}(\phi)}{C_{1}^{2}(\phi)+S_{1}^{2}(\phi)}d\phi=\frac{1}{2}\int_{A}^{B}d\phi=\left.\frac{1}{2}\phi\right|_{\alpha}^{\beta}=\frac{\beta-\alpha}{2}.$
That is, $2S$ equals to length $AB$.
Figure 2.4: Figure $F\subset\mathbb{B}^{n}$ volume calculation with aid of
cone in $\mathbb{R}^{n+1}$.
Let $F\subset\mathbb{B}^{n}$ be some figure with volume (in sense of
$\mathbb{B}^{n}$) $V_{\mathbb{B}}$. We will name $V_{\mathbb{R}}$ the volume
(in sense of $\mathbb{R}^{n+1}$) of cone with base $F\subset\mathbb{B}^{n}$
and vertex $O\notin\mathbb{B}^{n}$ origin of $\mathbb{R}^{n+1}$ (Figure 2.4).
As $\mathbb{B}^{n}$ is orthogonal to radius, $F$ also is orthogonal. The
radius equals 1, because $\forall X\in\mathbb{B}^{n},X\odot X=1$. Then for
each figure $F\subset\mathbb{B}^{n}$ have place equality:
$V_{\mathbb{B}}=(n+1)V_{\mathbb{R}}$
As motions preserve $\mathbb{B}^{n}$ and the absolute value of their matrices’
determinant is 1, all motions preserve $V_{\mathbb{R}}$ and thus, they
preserves also $V_{\mathbb{B}}$.
## Chapter 3 Theory Application
### 3.1 Space and Lineal Specification Search Algorithm
As we can see, the theory described in this book is universal and easy
applicable. However, one issue stops somebody from using it. Geometric spaces
are classified and defined different from the way adopted here. Therefore, in
order to not loose the feeling of reality, we will describe an algorithm aimed
to find specification for some geometric space. The algorithm can be applied
to any space where have sense notions of points, lines, planes, subspaces,
distances, angles and / or motions.
1. 1.
Let $m$ equals to the greatest number of general situated points, or same, the
lowest number of vertices in a polyhedron of positive volume.
2. 2.
Count space dimension as $n=m-1$.
3. 3.
Name points 0-dimensional planes and lines 1-dimensional planes.
4. 4.
For $i=\overline{1,n}$ do:
1. (a)
If among $(i-1)$-dimensional planes there are non-congruent ones, then the
space definition or space terminology is inconsistent. Theory still can be
used, however, in order to understand it correctly, it is necessary to modify
terminology or to define otherwise some space elements (about it later).
2. (b)
If the measure between $(i-1)$-dimensional planes is bounded, then $k_{i}=1$.
3. (c)
If the measure between $(i-1)$-dimensional planes is scalable, then $k_{i}=0$.
4. (d)
Otherwise, $k_{i}=-1$.
5. 5.
Having space dimension $n$ and specification $\\{k_{1},...,k_{n}\\}$, use
theory.
The necessity of proper terminology, uniform among all spaces is required by
wish to have such a theory, that isn’t misleading and helps us to study the
space structure and to compare it with other spaces. Still, under inconsistent
theory / terminology we should understand it has a contradiction, but failing
it to match to theory / terminology that is common today. We assume the
following here:
* •
All the planes of any dimension are congruent, including points and lines.
* •
Theory allows the duality principle of $(m-1)$-dimensional planes and
$(n-m)$-dimensional ones.
We should mention that ‘common terminology’ may change over the time. In order
to understand what it is consider an example of inconsistent terminology. The
Minkowskii space is successfully used in physics to describe the theory of
relativity. Unfortunately, from geometry point of view, it have no proper
terminology. The notions of ‘space–like lines’, ‘time–like lines’ and
‘light–like lines’ have sense in physics, but not in geometry. Corresponding
geometric notions are: ‘I-st category lines’, ‘II-nd category lines’ and ‘III-
rd category lines’. No space motion maps some line of a category into some
line of another category. There is no contradiction here, but there is an
inconsistence. What happens if somebody wants to define a space with five
categories of lines111Depending on concrete space, there can exist more
categories of two–dimensional planes. For further dimensions of planes the
number of their categories grows.? Nobody defines several kinds of points. All
points are congruent222The notion of points on infinity is used in projective
geometry. These points are non-congruent with others. The terminology is not
common in a scope of analytic geometry.. Why shouldn’t be lines all congruent?
At the other hand, relative position of points may differ. If we name lines
only the I-st category of lines, then we should exclude II-nd and III-rd
category of lines from lines. At the first look, it conflicts with the axiom
that claims any two points can be connected with a line. But this axiom may
have no place in other spaces. In contrast, just Euclidean geometry, where all
points are connectable, gives us an example of parallel lines (that have no
common point). Using the duality principle, it should exist the notion of non-
connectable points (that have no common line).
It should be mention, that even for somebody feels comfortable using this
theory, the algorithm described earlier may help to determinate the
specification of some exotic lineals (for example, of ones defined as limit
lineals, which aren’t deductable from the space specification).
### 3.2 Some Special Spaces
Many linear spaces are defined using the quadric form of distance
$d^{2}(X,Y)=(Y-X)\odot(Y-X)$. As for these spaces $k_{1}=0$,
$K_{0}=1,K_{i}=0,i>0$. In this case, the equality $1=C(d(X,Y))=X\odot Y=1$ is
trivial and can’t be used for distance calculation. Consider one more vector
product — $\otimes$ such as $(X\odot Y)^{2}+k_{1}(X\otimes Y)^{2}=1,\forall
X,Y\in\mathbb{B}^{n},k=\\{-1,0,1\\}$. This product is similar to exterior
vector product. Change $1=(X\odot X)(Y\odot Y)$:
$\displaystyle(X\otimes Y)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{1}}((X\odot X)(Y\odot Y)-(X\odot Y)^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{k_{1}}\left(\left(\sum_{i=0}^{n}K_{i}x_{i}^{2}\right)\left(\sum_{j=0}^{n}K_{j}y_{j}^{2}\right)-\left(\sum_{i=0}^{n}K_{i}x_{i}y_{i}\right)\left(\sum_{j=0}^{n}K_{j}x_{j}y_{j}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{k_{1}}\sum_{i=0}^{n}\sum_{j=0}^{n}K_{i}K_{j}(x_{i}^{2}y_{j}^{2}-x_{i}x_{j}y_{i}y_{j})$
$\displaystyle=$
$\displaystyle\frac{1}{k_{1}}\sum_{i<j=0}^{n}K_{i}K_{j}(x_{i}^{2}y_{j}^{2}+x_{j}^{2}y_{i}^{2}-2x_{i}x_{j}y_{i}y_{j})$
$\displaystyle=$
$\displaystyle\frac{1}{k_{1}}\sum_{i<j=0}^{n}K_{i}K_{j}(x_{i}y_{j}-x_{j}y_{i})^{2}$
So,
$X\otimes
Y=\sqrt{\frac{1}{k_{1}}\sum_{i<j=0}^{n}K_{i}K_{j}(x_{i}y_{j}-x_{j}y_{i})^{2}}.$
(3.1)
Note that from $\mathbb{B}^{n}=\\{x\in\mathbb{RP}^{n}\,|\,x\odot
x=x_{0}^{2}=1\\}$ result $x_{0}=1$ or $x_{0}=-1$. As $x\in\mathbb{B}^{n}$
implies $-x\in\mathbb{B}^{n}$ we can consider $x_{0}=1$. We will use $\otimes$
operator for distance $d(X,Y)$ between points $X$ and $Y$. Note that having
$C^{2}(x)+k_{1}S^{2}(X)=1,\forall x\in\mathbb{R},k_{1}=\\{-1,0,1\\}$ and
$(X\odot X)^{2}+k_{1}(X\otimes Y)^{2}=1,\forall
X,Y\in\mathbb{B}^{n},k_{1}=\\{-1,0,1\\}$ result $S(d(X,Y))=X\otimes Y,\forall
X,Y\in\mathbb{B}^{n}$:
$d^{2}(X,Y)=S^{2}(d(X,Y))=(X\otimes
Y)^{2}=\frac{1}{k_{1}}\sum_{i<j=0}^{n}K_{i}K_{j}(x_{i}y_{j}-x_{j}y_{i})^{2}$
In this sum all non–zero terms are those for which $i=0$:
$\displaystyle d^{2}(X,Y)$ $\displaystyle=$
$\displaystyle\frac{1}{k_{1}}\sum_{j=1}^{n}K_{j}(x_{0}y_{j}-x_{j}y_{0})^{2}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\frac{K_{j}}{k_{1}}(y_{j}-x_{j})^{2}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}(y_{j}-x_{j})^{2}\prod_{p=2}^{n}k_{p}$
In this equality don’t appear $x_{0}$ or $y_{0}$. We can consider
$\mathbb{B}^{n}$ a hyperplane of $\mathbb{R}^{n+1}$ with equation $x_{0}=1$
and specification $\\{k_{2},...k_{n}\\}$. We can identify it with
$\mathbb{R}^{n}$. Then the equality above is equivalent to $(Y-X)\odot(Y-X)$.
It means firstly, that scalar product of vestors in lnear spaces ($k_{1}=0$)
induces the same metrics that is used in the model, and secondly, that
non–linear spaces with specification $\\{k_{1},k_{2},...k_{n}\\}$ ($k_{1}\neq
0$) are best approximated by linear spaces with specification
$\\{0,k_{2},...k_{n}\\}$. Note also that from here deduce that non–linear
space with specification $\\{k_{1},...k_{n}\\}$ is enclosed in model
meta–space of greater by one dimension, of which specification is
$\\{0,k_{1},...k_{n}\\}$.
We can use this quadric form to search for all characteristics except
$k_{1}=0$. We will use this method in order to describe some special spaces by
specifying their specifications.
#### Case 1. Elliptic, Euclidean and Hyperbolic Spaces.
Elliptic, linear (Euclidean) and hyperbolic (Bolyai-Lobachevsky) spaces have
characteristic $k_{1}$ equals to sign of space curvature $k_{1}=1$ for
elliptic space, $k_{1}=0$ for linear space and $k_{0}=-1$ for hyperbolic
space.
All these spaces are usually approximated by Euclidean one. We can calculate
the rest of characteristics using euclidean quadric form. Let dimension is 3:
$\displaystyle d(X,Y)^{2}$ $\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}+(y_{3}-x_{3})^{2}$
$\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+k_{2}(y_{2}-x_{2})^{2}+k_{2}k_{3}(y_{3}-x_{3})^{2}$
so $k_{2}=1$ and $k_{2}k_{3}=1$, $k_{3}=1$.
#### Case 2. Minkowskii Space.
The distance between $X$ and $Y$ is calculated (for time–like vectors) as
$\displaystyle d^{2}(X,Y)$ $\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}-(y_{2}-x_{2})^{2}-(y_{3}-x_{3})^{2}-(y_{4}-x_{4})^{2}$
$\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+k_{2}(y_{2}-x_{2})^{2}+k_{2}k_{3}(y_{3}-x_{3})^{2}+k_{2}k_{3}k_{4}(y_{4}-x_{4})^{2}$
where coordinate 1 is time–like and coordinates 2, 3 and 4 are space–like. So
$k_{2}=-1$, $k_{2}k_{3}=-1$, $k_{3}=1$ and $k_{2}k_{3}k_{4}=-1$, $k_{4}=1$. As
Minkowskii space is linear, $k_{1}=0$.
If we introduce curvature in space, its structure changes. For example, let
$k_{1}=1$. Then
$\displaystyle X\odot Y$ $\displaystyle=$ $\displaystyle
x_{0}y_{0}+k_{1}x_{1}y_{1}+k_{1}k_{2}x_{2}y_{2}+k_{1}k_{2}k_{3}x_{3}y_{3}+k_{1}k_{2}k_{3}k_{4}x_{4}y_{4}$
$\displaystyle=$ $\displaystyle
x_{0}y_{0}+x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}$
so time characteristic becomes elliptic and space characteristic becomes
hyperbolic. If $k_{1}=1$, then
$\displaystyle X\odot Y$ $\displaystyle=$ $\displaystyle
x_{0}y_{0}+k_{1}x_{1}y_{1}+k_{1}k_{2}x_{2}y_{2}+k_{1}k_{2}k_{3}x_{3}y_{3}+k_{1}k_{2}k_{3}k_{4}x_{4}y_{4}$
$\displaystyle=$ $\displaystyle
x_{0}y_{0}-x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}$
and time characteristic becomes hyperbolic and space characteristic becomes
elliptic.
#### Case 3. Minkowskii Space with 2-dimensional Time.
Consider a 4-dimensional space with distance quadric form that has 2 positive
signs and 2 negative. This space is sometimes named Minkowskii space with
2-dimensional time:
$\displaystyle d^{2}(X,Y)$ $\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}-(y_{3}-x_{3})^{2}-(y_{4}-x_{4})^{2}$
$\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+k_{2}(y_{2}-x_{2})^{2}+k_{2}k_{3}(y_{3}-x_{3})^{2}+k_{2}k_{3}k_{4}(y_{4}-x_{4})^{2}$
So $k_{2}=1$, $k_{2}k_{3}=-1$, $k_{3}=-1$ and $k_{2}k_{3}k_{4}=-1$, $k_{4}=1$.
As for all lineal spaces, $k_{1}=0$ for it.
#### Case 4. Spaces with Degenerate Distance Quadric Form.
Consider linear 4-dimensional space ($k_{1}=0$) with degenerate distance
quadric form:
$\displaystyle d^{2}(X,Y)$ $\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}+(y_{3}-x_{3})^{2}$
$\displaystyle=$
$\displaystyle(y_{1}-x_{1})^{2}+k_{2}(y_{2}-x_{2})^{2}+k_{2}k_{3}(y_{3}-x_{3})^{2}+k_{2}k_{3}k_{4}(y_{4}-x_{4})^{2}$
so $k_{2}=k_{3}=1$ and $k_{4}=0$.
It means that motions:
$\displaystyle x^{\prime}_{1}$ $\displaystyle=$ $\displaystyle x_{1}$
$\displaystyle x^{\prime}_{2}$ $\displaystyle=$ $\displaystyle x_{2}$
$\displaystyle x^{\prime}_{3}$ $\displaystyle=$ $\displaystyle x_{3}$
$\displaystyle x^{\prime}_{4}$ $\displaystyle=$
$\displaystyle\phi_{1}x_{1}+\phi_{2}x_{2}+\phi_{3}x_{3}+x_{4}$
are all valid.
However, transformation:
$\displaystyle x^{\prime}_{1}$ $\displaystyle=$ $\displaystyle x_{1}$
$\displaystyle x^{\prime}_{2}$ $\displaystyle=$ $\displaystyle x_{2}$
$\displaystyle x^{\prime}_{3}$ $\displaystyle=$ $\displaystyle x_{3}$
$\displaystyle x^{\prime}_{4}$ $\displaystyle=$ $\displaystyle\phi_{4}x_{4}$
is not a motion. Although it preserved distance, it doesn’t preserve volume
except $\phi_{4}=1$ or $-1$. It is an example of angle scaling.
### 3.3 Spaces as Product of their Subspaces
Another way to define spaces is by product of their subspaces. It is necessary
to be accurate here. The geometric space isn’t only a structure of points. It
is also the structure of all its subspaces. It is mistake to think that having
$\mathbb{R}^{1}$ is isomorphic to one–dimensional Euclidean space
$\mathbb{E}^{1}$, from $\mathbb{R}^{1}\times\mathbb{R}^{1}=\mathbb{R}^{2}$
results $\mathbb{E}^{1}\times\mathbb{E}^{1}=\mathbb{E}^{2}$ (using
specification notation, $\\{0\\}\times${0} = {0, 1}). The problem is the
product doesn’t define way to measure the angle between multiplied subspaces.
It can be defined in several ways, for example, $\\{0,0\\}$ or $\\{0,-1\\}$.
The situation is even worse when multiplied subspaces with different
specification $X^{m}$ and $Y^{n}$. One–dimensional images can be constructed
in two ways: $X^{1}\times Y^{0}$ (isomorphic to $X^{1}$) and $X^{0}\times
Y^{1}$ (isomorphic to $Y^{1}$). And if $X^{1}$ and $Y^{1}$ have different
specifications, these two one–dimensional lines aren’t congruent. For example,
if somebody wants to construct geometry on a cylinder, first thing he or she
thinks of is $\mathbb{S}^{1}\times\mathbb{E}^{1}$ ($\\{1\\}\times\\{0\\}$),
where $\mathbb{S}^{1}$ is one–dimensional elliptic space. In this case some
lines are circles, some are lines and others are right and left helices, that
may not intersect, intersect in one point or intersect in infinity of points.
As an example of complete geometry on a cylinder you can take the space with
specification $\\{1,0\\}$.
Additionally, one should not consider that if from algebraic point of view
$\mathbb{E}^{1}$ is isomorphic to $\mathbb{H}^{1}$ (one–dimensional hyperbolic
space), then constructions like $\mathbb{H}^{2}\times\mathbb{E}^{1}$
($\\{-1,1\\}\times\\{0\\}$) and $\mathbb{H}^{2}\times\mathbb{H}^{1}$
($\\{-1,1\\}\times\\{-1\\}$) are also isomorphic. From geometric point of
view, $\mathbb{E}^{1}$ is scalable, while $\mathbb{H}^{1}$ is not (the mutual
departure of points is possible, however it can’t be linear). In contrast, it
is possible to construct spaces with specifications $\\{-1,1,0\\}$ and
$\\{-1,1,-1\\}$, which differ one from another by the fact that in first one
on a twodimensional plane doesn’t containing some point there is the only
point that isn’t connectable with it, and for the second space the number of
such a points is infinity.
## Bibliography
* [1] Felix Klein. Vorlesungen Nicht-Euklidische Geometrie. B.G.Teubner, Leipzig 1890.
* [2] Felix Klein. _A comparative review of recent researches in geometry_. Bull. New York Math. Soc. 2, (1892-1893), 215-249, 1893.
* [3] Edwin B. Wilson & Gilbert N. Lewis. _The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics_. Proceedings of the American Academy of Arts and Sciences 48:387-507, 1912.
* [4] Isaak Yaglom. A simple non-euclidean geometry and its physical basis. Springer (New York) 1979.
* [5] A. V. Khachaturean. Galilean Geometry. (in russian) Moskow, MCCME, 2005.
* [6] L. N. Romakina. Geometries of coeuclidean and copseudoeuclidean planes. (in russian) http://window.edu.ru/window_catalog/redir?id=66242&file=geom.pdf
* [7] A. Artykbaev, D. D. Sokolov. Geometry as whole in planar space–time. (in russian) Taskent, Fan, 1991.
|
arxiv-papers
| 2010-08-24T15:54:43 |
2024-09-04T02:49:12.419581
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexander Popa",
"submitter": "Alexandru Popa",
"url": "https://arxiv.org/abs/1008.4074"
}
|
1008.4171
|
# Measuring the masses of the charged hadrons using a RICH as a precision
velocity spectrometer
Peter. S. Cooper Fermi National Accelerator Laboratory, Batavia, IL 60510,
U.S.A. Jürgen Engelfried Universidad Autónoma de San Luis Potosí, San Luis
Potosí, Mexico
###### Abstract
The Selex experiment measured several billion charged hadron tracks with a
high precision magnetic momentum spectrometer and high precision RICH velocity
spectrometer. We have analyzed these data to simultaneously measure the masses
of all the long lived charged hadrons and anti-hadrons from the $\pi$ to the
$\Omega$ using the same detector and technique. The statistical precision
achievable with this data sample is more than adequate for $0.1\%$ mass
measurements.
We have used these measurements to develop and understand the systematic
effects in using a RICH as a precision velocity spectrometer with the goal of
measuring 10 masses with precision ranging from 100 KeV for the lightest to
1000 KeV for the heaviest. This requires controlling the radius measurement of
RICH rings to the $\sim 10^{-4}$ level. Progress in the mass measurements and
the required RICH analysis techniques developed are discussed.
††journal: NIM-A
The Selex RICH was orginally conceived as a particle identifier for a fixed
target multi-particle spectrometer [1]. Once we saw real data from this
detector we learned that this technique had serious potential as a precision
velocity spectrometer [2, 3, 4, 5]. Plots like Figure 1 [2] clearly
demonstrated that precision mass measurements of many of the hadrons, at the
same time with the same detectors, was possible. This paper is an exploration
of the systematic limits of this technique.
Recently the MIPP experiment proposed further data taking to resolve the $\sim
100$ ppm discrepancy in the latest charged kaon x-ray mass measurements [6].
They now have the Selex RICH [7] (with $CO_{2}$ instead of $Ne$ as a radiator)
but with insufficient data to reach the $100$ ppm level of statistical
precision. Selex has large amounts of data already recorded, and analysed, in
a well understood apparatus. Selex events are multi-hadronic interactions, not
single tracks. Nonetheless we thought it would be good to see how far we could
take the idea of making precision particle mass measurements using a precision
momentum (magnetic) and velocity (RICH) spectrometers. We have more than
enough data to make mass measurements with statistical precsion better than
the $<1000$ppm$=0.1\%$ level, even with very tight cuts to select clean
individual track measurements.
The goals of this study are to identify the important systematic uncertainties
of RICHs used a precision velocity spectrometers. Only studies with real data
can fully illuminate the the systematic limitations of such precision
spectrometry. It worth noting that $m=p/(\beta\gamma)$, where $p$ is the
momentum, and $\beta\gamma$ the relativisitic velocity, so that systematics in
the magnetic momentum spectrometry are equally important: both must be in
control to achieve a precision mass measurement to the $<100$ ppm level.
Selex was a Fermilab fixed target experiment designed to study the production
and decay of charmed baryons. It took data in $1996-7$ in a
$600~{}{\mbox{$\mathrm{GeV/c}$}}$ $\Sigma^{-}$ beam with excellent vertex and
momentum spectrometers. The Selex RICH, one of the first large multi-pixel PMT
RICHes, provided $\sim 1\%$ velocity resolution for all particles above
threshold ($\beta\gamma>86$). This is well matched to the $0.5-1\%$ momentum
resolution for tracks which reach the RICH.
The same detectors and analysis provide common systematics for all particles
species. Figure 1 displays 18 particles species from a 12.5M track sample with
RICH Ring Radius plotted as a function of measured track momentum. We have no
sensitivity to the electron and muon masses and we can’t resolve the
$\Sigma^{+}$ from the anti-$\Sigma^{-}$. We can measure the masses of the
other 10 particles.
Figure 1: RICH Ring radius vs measured momentum for $12.5$M tracks.
The usual small angle and ultra-relativistic approximations
[$\Theta_{c}=\sqrt{2\delta-1/\gamma^{2}}$] to the Cherenkov equation
[$cos(\Theta_{c})=1/\beta n$ $\delta=n-1=67x10^{-6}$] is good to $0.05\%$. The
measured ring radius is related through the spherical mirror focal length
($F$) to the momentum, mass and maximum Cherenkov angle by
$R(p)=F\sqrt{(\Theta_{c}^{max})^{2}-(m/p)^{2}}$.
Exploiting this relationship to measure mass requires three calibrations; the
momentum scale, determined by reconstructing $K^{0}_{s}$ decays, the mirror
focal length, which varies from $989-992cm$ across the 16 mirrors in the Selex
RICH, and the maximum Cherenkov angle $\Theta_{c}^{max}$.
Rings are fit for the radius and center coordinates [8] from a PMT list
generated around the measured track angles in the RICH cut around the
predicted radius for a given mass hypothesis. At least 5 hits are required. As
a clean place to start the Selex data used are low statistics, low rate, low
multiplicity data with one of the two spectrometer magnets off. A mass
spectrum generated after an initial calibration for tracks with ring radii in
the interval $5-8cm$ ($R^{max}=11.4cm$) is shown in Figure 2(top). Gaussian
mass fits in $2mm$ R bins give poor $\chi^{2}$ mass averages shown in Figure
2(bottom, open black points). The difference with the PDG [9] mass values are
plotted. The statistical mass uncertainties are ($\pi=70~{}KeV/c^{2}$,
$K=160~{}KeV/c^{2}$ and $p=350~{}KeV/c^{2}$). Systematic uncertainties clearly
dominate these mass measurements.
Figure 2: Mass from measured ring radius and momentum [top]. Fits to masses in
2mm radius bins; open (before), and solid (after) acceptance corrections
[bottom].
The momentum resolution is shown in Figure 3. For the data set analysed here
the momentum cutoff for a track to reach the RICH is $12~{}GeV/c$. For the
Selex charm data the resolution is much better but the cutoff is $20~{}GeV/c$,
putting the pions out of reach but greatly improving the momentum resolution
for the hyperons. The momentum scale was calibrated using
$K_{s}^{0}\rightarrow\pi^{+}\pi^{-}$ decays and the $K_{s}^{0}$ mass from the
PDG [9]. With this calibration the masses reconstructed with charged particle
decays of 12 other hadrons from the $\phi$ through the $\Omega_{c}^{0}$ are
correctly reproduced.
We’ve identified several systematic effects. The first is just geometry; as
shown in Figure 4(top), the intersection of a circular ring with a circular
tube isn’t symmetric. The acceptance correction (proportional the the arc
length shown) is biased towards radii larger than the center of ring to center
of tube distance. This acceptance is a just geometric. It is plotted as a
function of $z=R^{\prime}-R$ in Figure 4(bottom) for different ring radii.
We’ve build a second ring radius fitter based on maximizing the joint
likelihood for all tubes on a ring where the likelihood for each tube is from
the acceptance curves shown. The average shift in ring radius is $\sim 0.1mm$
or $0.1-0.2\%$ for rings in the radius interval ($5-8$ cm) we use to fit the
mass.
This likelihood fitter has an interesting application in pattern recognition.
The acceptance plotted goes exactly to zero at the radius difference where the
ring no longer crosses the hit tube. To deal with these cases we assign a
noise probability of $0.002$ for the minimum likelihood of any tube in the
ring fit. The likelihood as a function of fit radius can have one or more
maxima. Consider the case of two tubes along a ring radius: only one can fit
the ring, the other being “noise” in this model. This likelihood has two
maxima (if all the other tubes in the fit are well behaved). Rejecting these
ambiguous fits is an objective way to remove non-Gaussian tails due to pattern
recognition mistakes from the ring radius resolution function.
Figure 3: Momentum resolution for the data set analysed here (solid) and for
the Selex charm data (dashed). The momentum region corresponding to
$5<R(cm)<8$ for each particle species is also shown.
Figure 4: The intersection of a ring with a tube isn’t symmetric. The
acceptance is biased to R beyond the tube center [top]. The acceptance
function as a function of ring radius [bottom].
Repeating the mass analysis after applying cuts based on the criteria outlined
above yield the mas values shown in Figure 2. The $\chi^{2}$ for the mass
averages as a function of radius are improved but systematics in the mass as a
function of ring radius remain for the pions and kaons when the calibration
constants are determined using the protons.
We have made some progress. The proton and anti-proton mass now agree with
each other. We used the protons for calibration so the proton mass has to
agree with the PDG value. The kaon masses are close to each other and to the
PDG values but still several 100 KeV and several $\sigma$ off. The pion masses
are still quite far from the PDG values for reasons which are unclear. We can
and will apply these methods to the charged hyperons when the systematics are
better understood.
This study is beginning to illuminate some of the systematics of the
resolution of RICHs as precision velocity spectrometers. More work will be
required to reach and understand the resolution limits of this technique.
We are indebted to Selex for the RICH data, our home institutions, Consejo
Nacional de Ciencia y Tecnología (CONACyT), Fondo de Apoyo a la Investigación
(UASLP), and the U.S. Department of Energy (contract DE-AC02-76CHO3000), for
support.
## References
* [1] J. Engelfried et al., “The E781 (SELEX) RICH detector,” Nucl. Instrum. Meth. A409 (1998) 439-442.
* [2] J. Engelfried, I. S. Filimonov, J. Kilmer et al., “SELEX RICH performance and physics results,” Nucl. Instrum. Meth. A502 (2003) 285-288. [hep-ex/0208046].
* [3] J. Engelfried, P. S. Cooper, A. Morelos Pineda et al., “Two RICH detectors as velocity spectrometers in the CKM Experiment,” Nucl. Instrum. Meth. A502 (2003) 62-66. [hep-ex/0209020].
* [4] P. S. Cooper et al., “Redesign of the CKM RICH velocity spectrometers for use in a 1/4 GHz unseparated beam,” Nucl. Instrum. Meth. A553 (2005) 220-224.
* [5] A. Morelos Pineda, J. Mata, P. S. Cooper et al., “Radial tail resolution in the SELEX RICH,” Nucl. Instrum. Meth. A553 (2005) 237-241.
* [6] N. Graf et al., “Charged Kaon Mass Measurement using the Cherenkov Effect,” Nucl. Instrum. Meth. A 615, 27 (2010) [arXiv:0909.0971 [hep-ex]].
* [7] J. Engelfried et al., “The SELEX phototube RICH detector,” Nucl. Instrum. Meth. A 431, 53 (1999) [arXiv:hep-ex/9811001].
* [8] J.F. Crawford, “A Non-Iterative Method for Fitting Circualr Arcs to Measured Points,” Nucl. Instrum. Meth. A 211, 223 (1983).
* [9] K. Nakamura et al. (Particle Data Group), Journal of Physics G 37, 075021 (2010).
|
arxiv-papers
| 2010-08-24T22:43:52 |
2024-09-04T02:49:12.431802
|
{
"license": "Public Domain",
"authors": "Peter S. Cooper, Jurgen Engelfried",
"submitter": "Jurgen Engelfried",
"url": "https://arxiv.org/abs/1008.4171"
}
|
1008.4194
|
Binary Bell polynomials approach to the integrability of nonisospectral and
variable-coefficient nonlinear equations
Engui Fan111 E-mail address: faneg@fudan.edu.cn
School of Mathematical Sciences and Key Laboratory of Mathematics for
Nonlinear Science, Fudan University, Shanghai, 200433, P.R. China
Abstract. Recently, Lembert, Gilson et al proposed a lucid and systematic
approach to obtain bilinear Bäcklund transformations and Lax pairs for
constant-coefficient soliton equations based on the use of binary Bell
polynomials. In this paper, we would like to further develop this method with
new applications. We extend this method to systematically investigate complete
integrability of nonisospectral and variable-coefficient equations. In
addiction, a method is described for deriving infinite conservation laws of
nonlinear evolution equations based on the use of binary Bell polynomials. All
conserved density and flux are given by explicit recursion formulas. By taking
variable-coefficient KdV and KP equations as illustrative examples, their
bilinear formulism, bilinear Bäcklund transformations, Lax pairs, Darboux
covariant Lax pairs and conservation laws are obtained in a quick and natural
manner. In conclusion, though the coefficient functions have influences on a
variable-coefficient nonlinear equation, under certain constrains the equation
turn out to be also completely integrable, which leads us to a canonical
interpretation of their $N$-soliton solutions in theory.
Keywords: binary Bell polynomial; variable-coefficient equation; bilinear
Bäcklund transformation; Lax pair; Darboux covariance; conservation law.
PACS numbers: 11\. 30. Pb; 05. 45. Yv; 02. 30. Gp; 02.30.Ik.
1\. Introduction
In many physical situations, it is often preferable to have an equation with
variable-coefficients, which may allow us to describe real phenomena in
physical and engineering fields. For example, variable-coefficient nonlinear
Schr odinger-typed ones, which describe such situations more realistically
than their constant-coefficient counterparts, in plasma physics, arterial
mechanics and long-distance optical communications [2]-[6]. Many physical and
mechanical situations governed by variable-coefficient KdV (vc-KdV) equation,
e.g., the pulse wave propagation in blood vessels and dynamics in the
circulatory system, matter waves and nonlinear atom optics enhanced by the
observations of Bose-Einstein condensation in the weakly interacting atomic
gases, the nonlinear excitations of a Bose gas of impenetrable bosons with
longitudinal confinement, the nonlinear waves in types of rods [7]-[12]. In
recent years, there has been considerable interest in the study of variable-
coefficient nonlinear equations, such as vc-KdV, vc-KP, vc-Schrödinger, vc-
Boussinesq and cylindrical KdV equations. Recent progress in the investigation
of the complete integrability and exact solutions for such equations via
Painleve analysis, inverse scattering transformation, Hirota bilinear method
and Darboux transformation has been reported, the details can be seen in
reference and references therein [15]-[22]. It is obvious that variable-
coefficient equations are often more complicated and difficult to be solved
than constant-coefficient ones. As well-known, investigation of integrability
for a nonlinear equation can be regarded as a pre-test and the first step of
its exact solvability. There are many significant properties, such as Lax
pairs, infinite conservation laws, infinite symmetries, Hamiltonian structure,
Painlevé test that can characterize integrability of nonlinear equations. This
may pave the way for constructing their exact solutions explicitly in a
future. But in contrast to constant-coefficient cases, very little of detail
is known about complete integrability of variable-coefficient nonlinear
equations.
Among the direct algebraic methods applicable to nonlinear partial
differential equations in soliton theory, there is on which has proved
particularly powerful: the bilinear method developed by Hirota [28, 29]. Once
a nonlinear equation is written in bilinear forms by a dependent variable
transformation, then multi-soliton solutions are usually obtained [30]-[35].
The search for a Hirota representation of a given nonlinear equation is
generally recognized as an important first step in the construction of multi-
soliton solutions. Yet, the construction of such bilinear Bäcklund
transformation is not as one would wish. It relies on a particular skill in
using appropriate exchange formulas which are connected with the linear
representation of the system. Recently, Lembert, Gilson et al proposed an
alternative procedure based on the use of Bell polynomials which enable one to
obtain parameter families of bilinear Bäcklund transformation for soliton
equations in a lucid and systematic way [37]-[39]. The Bell polynomials are
found to play an important role in the characterization of bilinearizable
equations. As a consequence bilinear Bäcklund transformation with single field
can be linearize into corresponding Lax pairs. Their method provides a
shortest way to bilinear Bäcklund transformation and Lax pairs of nonlinear
equations, which establishes a deep relation between integrability of a
nonlinear equation and the Bell polynomials.
The problem that we consider in this paper is to further develop this method
with new applications. We extend the binary Bell polynomials approach to a
large class of nonisospectral and variable-coefficient equations, such as
nonisospectral and variable-coefficient KdV and KP equations etc. One of the
many remarkable properties that deemed to characterize soliton equations is
existence of an infinite sequence of conservation laws. Here we propose a
approach to construct infinite conservation laws of nonlinear evolution
equations through decoupling binary Bell polynomials into a Riccati type
equation and a divergence type equation. As illustrative examples, the
bilinear representations, bilinear Bäcklund transformations, Lax pais and
infinite conservation laws of the vc-KdV and vc-KP equations are obtained in a
quick and natural manner. The integrable constraint conditions on the
variable-coefficient functions can be naturally found in the procedure of
applying binary Bell polynomials. We can also find that though the coefficient
functions have influences on a variable-coefficient equation, under certain
constrains the equation still can admit many integrability properties which
are similar to those of its standard constant-coefficient equation. The
organization of this paper is as follows. In section 2, we briefly present
necessary notations on multi-dimensional binary Bell polynomial that will be
used in this paper. In the sections 3 and 4, we deal with integrability of
nonisospectral vc-KdV equation and vc-KdV equation, respectively. We aim at
integrability of nonisospectral vc-KP equation and vc-KP equation in the
sections 5 and 6, respectively.
2\. Multi-dimensional binary Bell polynomials
The main tool used in this paper is a class of the Bell polynomials, named
after E. T. Bell [36]. To make our presentation easy understanding and self-
contained, we simply recall some necessary notations on the Bell polynomials,
the details refer, for instance, to Lembert and Gilson’s work [37]-[39].
Let $f=f(x_{1},\cdots,x_{n})$ be a $C^{\infty}$ function with multi-variables,
the following polynomials
$Y_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(f)\equiv
Y_{n_{1},\cdots,n_{\ell}}(f_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}})=e^{-f}\partial_{x_{1}}^{n_{1}}\cdots\partial_{x_{\ell}}^{n_{\ell}}e^{f}$
is called multi-dimensional Bell polynomials, in which we denote that
$f_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}=\partial_{x_{1}}^{r_{1}}\cdots\partial_{x_{\ell}}^{r_{\ell}}f,\
\ r_{1}=0,\cdots,n_{1};\cdots;\ r_{\ell}=0,\cdots,n_{\ell}$.
For example, for the simplest case $f=f(x)$, the associated one-dimensional
Bell polynomials read
$\displaystyle{Y}_{1}(f)=f_{x},\ {Y}_{2}(f)=f_{2x}+f_{x}^{2},\ \ \
{Y}_{3}(f)=f_{3x}+3f_{x}f_{2x}+f_{x}^{3},\cdots.$
For $f=f(x,t)$, the of associated two-dimensional Bell polynomials are
$\displaystyle{Y}_{x,t}(f)=f_{x,t}+f_{x}f_{t},\ \
{Y}_{2x,t}(f)=f_{2x,t}+f_{2x}f_{t}+2f_{x,t}f_{x}+f_{x}^{2}f_{t},\cdots.$
Based on the use of above Bell polynomials, the multi-dimensional binary Bell
polynomials can be defined as follows
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v,w)=Y_{n_{1},\cdots,n_{\ell}}(f)\mid_{f_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}=\left\\{\begin{matrix}v_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}},&r_{1}+\cdots+r_{\ell}\
\ {\rm is\ \ odd},\cr\cr
w_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}},&r_{1}+\cdots+r_{\ell}\ \ {\rm is\ \
even},\end{matrix}\right.}$
which inherit the easily recognizable partial structure of the Bell
polynomials.
The first few lowest order binary Bell Polynomials are
$None$ $\displaystyle\mathcal{Y}_{x}(v)=v_{x},\
\mathcal{Y}_{2x}(v,w)=w_{2x}+v_{x}^{2},\ \
\mathcal{Y}_{x,t}(v,w)=w_{xt}+v_{x}v_{t}.$
$\displaystyle\mathcal{Y}_{3x}=v_{3x}+3v_{x}w_{2x}+v_{x}^{3},\cdots.$
The link between $\mathcal{Y}$-polynomials and the standard Hirota bilinear
equation $D_{x_{1}}^{n_{1}}\cdots D_{x_{\ell}}^{n_{\ell}}F\cdot G$ can be
given by an identity
$None$ $\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v=\ln F/G,w=\ln
FG)=(FG)^{-1}D_{x_{1}}^{n_{1}}\cdots D_{x_{\ell}}^{n_{\ell}}F\cdot G,$
in which ${n_{1}}+n_{2}+\cdots+n_{\ell}\geq 1$. In the particular case when
$G=F$, the formula (2.2) becomes
$None$ $\displaystyle F^{-2}D_{x_{1}}^{n_{1}}\cdots
D_{x_{\ell}}^{n_{\ell}}F\cdot
F=\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(0,q=2\ln F)$
$\displaystyle=\left\\{\begin{matrix}0,&n_{1}+\cdots+n_{\ell}\ \ {\rm is\ \
odd},\cr\cr P_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(q),&n_{1}+\cdots+n_{\ell}\
\ {\rm is\ \ even},\end{matrix}\right.$
in which the $P$-polynomials can be characterized by an equally recognizable
even part partitional structure
$None$ $\displaystyle P_{2x}(q)=q_{2x},\ P_{x,t}(q)=q_{xt},\
P_{4x}(q)=q_{4x}+3q_{2x}^{2},$ $\displaystyle
P_{6x}(q)=q_{6x}+15q_{2x}q_{4x}+15q_{2x}^{2},\cdots.$
The formulae (2.2) and (2.3) will prove particularly useful in connecting
nonlinear equations with their corresponding bilinear equations. This means
that once a nonlinear equation is expressible as a linear combination of
$P$-polynomials, then it can be transformed into a linear equation.
It follows that the binary Bell polynomials
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v,w)$ can be separated into
$P$-polynomials and $Y$-polynomials
$None$ $\displaystyle(FG)^{-1}D_{x_{1}}^{n_{1}}\cdots
D_{x_{\ell}}^{n_{\ell}}F\cdot
G=\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v,w)|_{v=\ln F/G,w=\ln FG}$
$\displaystyle=\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v,v+q,)|_{v=\ln
F/G,q=2\ln G}$
$\displaystyle=\sum_{n_{1}+\cdots+n_{\ell}=even}\sum_{r_{1}=0}^{n_{1}}\cdots\sum_{r_{\ell}=0}^{n_{\ell}}\prod_{i=1}^{\ell}\left(\begin{matrix}n_{i}\cr
r_{i}\end{matrix}\right)P_{r_{1}x_{1},\cdots,r_{\ell}x_{\ell}}(q)Y_{(n_{1}-r_{1})x_{1},\cdots,(n_{\ell}-r_{\ell})x_{\ell}}(v).$
The key property of the multi-dimensional Bell polynomials
$None$
$Y_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v)|_{v=\ln\psi}=\frac{\psi_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}}{\psi},$
implies that the binary Bell polynomials
$\mathcal{Y}_{n_{1}x_{1},\cdots,n_{\ell}x_{\ell}}(v,w)$ can still be
linearized by means of the Hopf-Cole transformation $v=\ln\psi$, that is,
$\psi=F/G$. The formulae (2.5) and (2.6) will then provide the shortest way to
the associated Lax system of nonlinear equations.
We start with construction of infinite conservation laws by virtue of binary
Bell polynomials. We define a new auxiliary field variable
$\eta=(q^{\prime}_{x_{k}}-q_{x_{k}})/2,$
where $q^{\prime}$ and $q$ are given by $q^{\prime}=w+v,\ q=w-v$ and $x_{k}$
is a appropriate variable chosen from $x_{1},\cdots,x_{\ell}$. The two-filed
condition
$None$ $C(q^{\prime},q)=E(q^{\prime})-E(q)=0$
can be regarded as the natural ansatz for a bilinear Bäckbend transformation.
By expressing the two-filed condition (2.7) in terms of binary Bell
$\mathcal{Y}$-polynomials and their derivatives, we expect that the resulting
condition is then decoupled into a pair of constrains, i.e. often a Riccati
type equation with respect to $x_{k}$,
$None$ $\eta_{x_{k}}+f(\eta)=0,$
and a divergence-type equation
$None$
$\partial_{x_{1}}F_{1}(\eta)+\cdots+\partial_{x_{\ell}}F_{\ell}(\eta)=0.$
The recursion formulas of conversed density come from the equation (2.8), the
formulas of associated flux are obtained by using the equation (2.9). It is
often the case that the first few conservation laws of a nonlinear equation
have a physical interpretation.
3\. Nonisospectral variable coefficient KdV equation
Consider nonisospectral vc-KdV equation [15]
$None$ $u_{t}+h_{1}(u_{3x}+6uu_{x})+4h_{2}u_{x}-h_{3}(2u+xu_{x})=0,$
where $h_{1}=h_{1}(t),\ h_{2}=h_{2}(t)$ and $\ h_{3}=h_{3}(t)$ are all
arbitrary functions with respect to time variable $t$. The equation (3.1)
includes some governing physical equations as special reduction, such as
celebrated constant-coefficient KdV equation
$u_{t}+6uu_{x}+u_{3x}=0,$
cylindrical KdV equation ($h_{1}=1,\ h_{2}=1/8t,\ h_{3}=0$) [14]
$None$ $u_{t}+6uu_{x}+u_{3x}+\frac{1}{2t}u_{x}=0,$
and the vc-KdV equation ($h_{1}=1,\ h_{2}=c_{0}/4,\ h_{3}=-\gamma$)
$u_{t}+6uu_{x}+u_{3x}+\gamma u+[(c_{0}+\gamma x)u]_{x}=0,$
which describes the effect of relaxation inhomogeneous medium [16]. It can be
observed that the equation (3.1) is invariant under Galiean transformation
$u\rightarrow u+\lambda,\ x\rightarrow x+6\lambda t,\ t\rightarrow t.$
The inverse scattering transformation of the equation (3.1) was considered by
Chan and Li [15]. Lou and Ruan obtained infinite conservation laws [18]. Here
we shall investigate the integrability of the equation (3.1) from bilinear
representation, Bäcklund transformation, Lax pair, Darboux covariant Lax pair
and infinite conservation laws.
3.1. Bilinear representation
In order to detect its existence of linearizable representation, we introduce
a potential field $q$ by setting
$None$ $u=c(t)q_{2x},$
with $c=c(t)$ being free function to be the appropriate choice such that the
equation (3.1) connect with $P$-polynomials. Substituting (3.3) into (3.1) and
integrating with respect to $x$ yields
$None$ $E(q)\equiv
q_{xt}+h_{1}(q_{4x}+3cq_{2x}^{2})+4h_{2}q_{2x}-h_{3}(q_{x}+xq_{2x})+q_{x}\partial_{t}\ln
c=0.$
Comparing the second term of this equation together with the formula (2.4), we
require $c(t)=1$. The result equation is then cast into a combination form of
$P$-polynomials
$None$
$E(q)=P_{xt}(q)+h_{1}P_{4x}(q)+4h_{2}P_{2x}(q)-h_{3}(xP_{2x}(q)+q_{x})=0.$
Making a change of dependent variable
$q=2\ln F\ \ \Longleftrightarrow\ \ u=cq_{2x}=2(\ln F)_{2x}$
and noting the property (2.3), the equation (3.5) gives the bilinear
representation as follows
$(D_{x}D_{t}+h_{1}D_{x}^{4}+4h_{2}D_{x}^{2}-xh_{3}D_{x}^{2}-h_{3}\partial_{x})F\cdot
F=0,$
in which we have used the notation $\partial_{x}F\cdot
F\equiv\partial_{x}F^{2}=2FF_{x}$. This equation is easy to be solved for
multi-soliton solutions by using Hirota’s bilinear method. For example, the
regular one-soliton like solution reads
$\displaystyle u=\frac{k^{2}}{2}{\rm sech}^{2}\frac{kx+\omega}{2},$
where $k=k(t)$ and $\omega=\omega(t)$ are two functions about $t$, given by
$k(t)=\alpha e^{\int h_{3}dt},\ \ \omega(t)=-\int(h_{1}k^{3}+4h_{2}k)dt.$
The multi-soliton solution are omitted here since exactly solving the equation
(3.1) is not our main purpose in this paper.
3.2. Bäcklund transformation and Lax pair
Next, we search for the bilinear Bäcklund transformation and Lax pair of the
vc-KdV equation (3.1). Let $q$ and $q^{\prime}$ be two different solutions of
the equation (3.4), respectively, we associate the two-field condition
$None$ $\displaystyle
E(q^{\prime})-E(q)=(q^{\prime}-q)_{xt}+h_{1}[(q^{\prime}-q)_{4x}+3(q^{\prime}+q)_{2x}(q^{\prime}-q)_{2x}]$
$\displaystyle+4h_{2}(q^{\prime}-q)_{2x}-h_{3}[(q^{\prime}-q)_{x}+x(q^{\prime}-q)_{2x}]=0.$
This two-field condition can be regarded as the natural ansatz for a bilinear
Bäcklund transformation and may produce the required transformation under
appropriate additional constraints.
To find such constraints, we introduce two new variables
$None$ $v=(q^{\prime}-q)/2,\ \ w=(q^{\prime}+q)/2,$
and rewrite the condition (3.5) into the form
$None$ $\displaystyle
E(q^{\prime})-E(q)=v_{xt}+h_{1}(v_{4x}+6v_{2x}w_{2x})+4h_{2}v_{2x}-h_{3}(v_{x}+xv_{2x})$
$\displaystyle=\partial_{x}[\mathcal{Y}_{t}(v)+h_{1}\mathcal{Y}_{3x}(v,w)]+R(v,w)=0,$
with
$R(v,w)=3h_{1}{\rm
Wronskian}[\mathcal{Y}_{2x}(v,w),\mathcal{Y}_{x}(v)]+\partial_{x}[4h_{2}\mathcal{Y}_{x}(v)-xh_{3}\mathcal{Y}_{x}(v)].$
In order to decouple the two-field condition (3.7) into a pair of constraints,
we impose such a constraint which enable us to express $R(v,w)$ as the
$x$-derivative of a combination of $\mathcal{Y}$-polynomials. The simplest
possible choice of such constraint may be
$None$ $\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{x}(v)=\lambda,$
where $\alpha$ and $\lambda$ are arbitrary parameters. On account of the
equation (3.8), then $R(v,w)$ can be rewritten in the form
$None$
$R(v,w)=\partial_{x}[3h_{1}\lambda\mathcal{Y}_{x}(v)+4h_{2}\mathcal{Y}_{x}(v)-xh_{3}\mathcal{Y}_{x}(v)].$
Then from (3.7)-(3.9), we deduce a coupled system of $\mathcal{Y}$-polynomials
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{x}(v)-\lambda=0,$
$\displaystyle\partial_{x}\mathcal{Y}_{t}(v)+\partial_{x}[h_{1}\mathcal{Y}_{3x}(v,w)+(3h_{1}\lambda+4h_{2}-xh_{3})\mathcal{Y}_{x}(v)]=0.$
where prefer the second equation in the conserved form without integration
with respect to $x$, which is useful to construct conservation laws later. By
application of the identity (2.2), the system (3.10) immediately leads to the
bilinear Bäcklund transformation
$\displaystyle(D_{x}^{2}+\alpha D_{x}-\lambda)F\cdot G=0,$
$\displaystyle[D_{t}+h_{1}D_{x}^{3}+(3h_{1}\lambda+4h_{2}-xh_{3})D_{x}+\beta]F\cdot
G=0,$
where $\beta$ is a arbitrary parameter.
By transformation $v=\ln\psi$, it follows from the formulae (2.5) and (2.6)
that
$\displaystyle\mathcal{Y}_{x}(v)=\psi_{x}/\psi,\ \
\mathcal{Y}_{2x}(v,w)=q_{2x}+\psi_{2x}/\psi,$
$\displaystyle\mathcal{Y}_{3x}(v,w)=3q_{2x}\psi_{x}/\psi+\psi_{3x}/\psi,\ \
\mathcal{Y}_{t}(v)=\psi_{t}/\psi,$
on account of which, the system (3.10) is then linearized into a Lax pair with
double parameters about $\lambda$ and $\beta$
$None$ $\displaystyle
L_{1}\psi\equiv(\partial_{x}^{2}+\alpha\partial_{x}+q_{2x})\psi=\lambda\psi,\
\ \lambda_{t}=2h_{3}\lambda,$ $None$
$\displaystyle(\partial_{t}+L_{2})\psi\equiv[\partial_{t}+h_{1}\partial_{x}^{3}+3h_{1}(q_{2x}+\lambda)\partial_{x}+(4h_{2}-xh_{3})\partial_{x}]\psi$
or equivalently replacing $q_{2x}$ by $u$,
$\displaystyle\psi_{2x}+\alpha\psi_{x}+(u-\lambda)\psi=0,\ \
\lambda_{t}=2h_{3}\lambda,$
$\displaystyle\psi_{t}+[h_{1}(2u+4\lambda+\alpha^{2})+4h_{2}-xh_{3}]\psi_{x}+[\beta-
h_{1}u_{x}-\alpha h_{1}(u-\lambda)]\psi=0.$
Starting from this Lax pair, the Darboux transformation and soliton-like
solutions of the vc-KdV equation (3.1) can be established [15]. It is easy to
check that the integrability condition
$[L_{1}-\lambda,\partial_{t}+L_{2}]\psi=0$
is satisfied if $u$ is a solution of the vc-KdV equation (3.1) and
nonisospectral condition $\lambda_{t}=2h_{3}\lambda$ holds.
3.4. Infinite conservation laws
Finally, we show how to derive the infinite conservation laws for vc-KdV
equation (3.1) based on the use of the binary Bell polynomials. The
conservation laws actually have been hinted in the two-filed constraint system
(3.10), which can be rewritten in the conserved form
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{x}(v)-\lambda=0,$
$\displaystyle\partial_{t}\mathcal{Y}_{x}(v)+\partial_{x}[h_{1}\mathcal{Y}_{3x}(v,w)+(3h_{1}\lambda+4h_{2}-xh_{3})\mathcal{Y}_{x}(v)]=0.$
by applying the relation
$\partial_{x}\mathcal{Y}_{t}(v)=\partial_{t}\mathcal{Y}_{x}(v)=v_{xt}.$
By introducing a new potential function
$\eta=(q^{\prime}_{x}-q_{x})/2,$
it follows from the relation (3.6) that
$None$ $v_{x}=\eta,\ \ w_{x}=q_{x}+\eta.$
Substituting (3.14) into (3.13), we get a Riccati-type equation
$None$ $\displaystyle\eta_{x}+\eta^{2}+q_{2x}=\lambda=\varepsilon^{2},$
and a divergence-type equation
$None$
$\displaystyle\eta_{t}+\partial_{x}[h_{1}\eta_{2x}+6h_{1}(\eta+\varepsilon)\varepsilon^{2}-2h_{1}(\eta+\varepsilon)^{3}+(4h_{2}-xh_{3})(\eta+\varepsilon)]=0,$
where we have used the equation (3.15) to get the equation (3.16) and set
$\lambda=\varepsilon^{2}$.
To proceed, inserting the expansion
$None$
$\eta=\varepsilon+\sum_{n=1}^{\infty}I_{n}(q,q_{x},\cdots)\varepsilon^{-n},$
into the equation (3.15) and equating the coefficients for power of
$\varepsilon$, we then obtain the recursion relations for $I_{n}$
$None$ $\displaystyle I_{1}=-p_{x}=-\frac{1}{2}u,\ \ \
I_{2}=\frac{1}{4}p_{2x}=\frac{1}{4}u_{x},$ $\displaystyle
I_{n+1}=-\frac{1}{2}(I_{n,x}+\sum_{k=1}^{n}I_{k}I_{n-k}),\ \ n=2,3,\cdots,$
By applying the nonisospectral condition
$\lambda_{t}=2h_{3}\lambda\ \
\Longrightarrow\varepsilon_{t}=h_{3}\varepsilon,$
then substituting (3.17) into (3.16) yields
$\displaystyle\sum_{n=1}^{\infty}I_{n,t}\varepsilon^{-n}+\partial_{x}\left[h_{1}\sum_{n=1}^{\infty}I_{n,2x}\varepsilon^{-n}-6h_{1}\varepsilon(\sum_{n=1}^{\infty}I_{n}\varepsilon^{-n})^{2}-2h_{1}(\sum_{n=1}^{\infty}I_{n}\varepsilon^{-n})^{3}\right.$
$\displaystyle\left.+(4h_{2}-xh_{3})\sum_{n=1}^{\infty}I_{n}\varepsilon^{-n}-h_{3}\sum_{n=1}^{\infty}n\partial_{x}^{-1}I_{n}\varepsilon^{-n}\right]=0,$
which provides us infinite consequence of conservation laws
$None$ $I_{n,t}+F_{n,x}=0,\ n=1,2,\cdots.$
In the equation (3.19), the conversed densities $I_{n}^{\prime}s$ are given by
formula (3.15) and the fluxes $F_{n}^{\prime}s$ are given by recursion
formulas explicitly
$None$ $\displaystyle
F_{1}=-\frac{1}{2}\left[h_{1}(u_{2x}+3u^{2})+4h_{2}u-h_{3}(xu+\partial^{-1}_{x}u)\right],$
$\displaystyle
F_{2}=\frac{1}{4}\left[h_{1}(u_{3x}+6uu_{x})+4h_{2}u_{x}-h_{3}(xu_{x}+2u)\right],$
$\displaystyle
F_{n}=h_{1}I_{n,2x}-6h_{1}\sum_{k=1}^{n}I_{k}I_{n+1-k}-2h_{1}\sum_{i+j+k=n}I_{i}I_{j}I_{k}+(4h_{2}-xh_{3})I_{n}$
$\displaystyle\ \ \ \ \ \ \ \ \ +nh_{3}\partial_{x}^{-1}I_{n},\ \
n=3,4,\cdots.$
We present recursion formulas for generating an infinite sequence of
conservation laws for each equation, the first few conserved density and
associated flux are explicit. The first equation of conservation law equation
(3.19) is exactly the vc-KdV equation (3.1). The expressions (3.20) indicate
that the fluxes $F_{n}^{\prime}s$ of the vc-KdV equation are not local, which
are different from those of standard constant-coefficient KdV equation. In
conclusion, the vc-KdV equation (3.1) is complete integrable in the sense that
it admits bilinear Bäcklund transformation, Lax pair and infinite conservation
laws.
4\. Generalized variable-coefficient KdV equation
A more general example, we consider vc-KdV equation [21]
$None$ $u_{t}+h_{1}u_{3x}+h_{2}uu_{x}+h_{3}u_{x}+h_{4}u=0,$
where $h_{j}=h_{j}(t),\ j=1,2,3,4$ are all arbitrary functions with respect to
time variable $t$. Special cases of the equation (4.1) include cylindrical
equation [23]-[25]
$u_{t}+f(t)uu_{x}+g(t)u_{3x}=0$
and other special variable-coefficient equation [26, 27]
$u_{t}+at^{n}uu_{x}+bt^{m}u_{3x}=0.$
Recently, Zhang et al obtained the bilinear form, Bäcklund transformation and
exact solutions for the equation (4.1) under the constrain [21]
$None$ $h_{1}=c_{0}h_{2}e^{-\int h_{4}dt}.$
Here we construct bilinear representation, Bäcklund transformation, Lax pair
and conservation laws of the equation (4.1) based on the sue of binary Bell
polynomials technique, which will be seen to be a natural way to find such a
constraint (4.2). We find that the bilinear representation of the equation
(4.1) existence without need of any constraint. The constraint only need it
for construction of the Bäcklund transformation, Lax pair and conservation
laws.
4.1. Bilinear representation
As before, we introduce a field $q$ by setting
$None$ $u=c(t)q_{2x},$
in which $c=c(t)$ is free function to be determined. Substituting (4.3) into
(4.1) and integrating with respect to $x$ yields
$None$ $E(q)\equiv
q_{xt}+h_{1}q_{4x}+\frac{1}{2}h_{2}cq_{2x}^{2}+h_{3}q_{2x}+(h_{4}+\partial_{t}\ln
c)q_{x}=0.$
which can be cast into a combination form of $P$-polynomials by using the
formula (2.4)
$None$ $E(q)=P_{xt}(q)+h_{1}P_{4x}(q)+h_{3}P_{2x}(q)+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})q_{x}=0,$
if one chooses the function $c(t)=6h_{1}h_{2}^{-1}.$
By transformation
$q=2\ln F\ \ \Longleftrightarrow\ \ u=c(t)q_{2x}=12h_{1}h_{2}^{-1}(\ln
F)_{2x}$
and using the property (2.3), then the equation (4.5) implies the bilinear
form for the vc-KdV equation (4.1) as follows
$None$ $[D_{x}D_{t}+h_{1}D_{x}^{4}+h_{3}D_{x}^{2}+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})\partial_{x}]F\cdot F=0,$
which is obviously more general than that obtained in [21], since we have no
any constraint on the $h_{1},h_{2},h_{3}$ and $h_{4}$. Starting the bilinear
equation (4.6), we can get multi-soliton solutions to the vc-KdV equation
(4.1). For example, one-soliton solution takes the form
$\displaystyle u=6h_{1}h_{2}^{-1}k^{2}{\rm sech}^{2}\frac{kx+\omega(t)}{2},$
where $k$ is a constant and $\omega(t)$ given by
$\omega(t)=-\int(k^{3}h_{1}+kh_{3}+h_{4}+\partial_{t}\ln h_{1}h_{2}^{-1})dt.$
4.2. Bäcklund transformation and Lax pair
In order to obtain the bilinear Bäcklund transformation and Lax pairs of the
equation (4.1), let $q$, $q^{\prime}$ be two solutions of the equation (4.4)
and consider the associated two-field condition
$None$ $\displaystyle
E(q^{\prime})-E(q)=(q^{\prime}-q)_{xt}+h_{1}[(q^{\prime}-q)_{4x}+3(q^{\prime}+q)_{2x}(q^{\prime}-q)_{2x})$
$\displaystyle+h_{3}(q^{\prime}-q)_{2x}+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})(q^{\prime}-q)_{x}]=0,$
which may produce the required bilinear Bäcklund transformation under an
appropriate additional constraint. By introducing variables
$None$ $v=(q^{\prime}-q)/2,\ \ w=(q^{\prime}+q)/2$
we can rewrite the condition (4.7) as the form
$None$ $\displaystyle
E(q^{\prime})-E(q)=v_{xt}+h_{1}(v_{4x}+6v_{2x}w_{2x})+h_{3}v_{2x}+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})v_{x}$
$\displaystyle=\partial_{x}[\mathcal{Y}_{t}(v)+h_{1}\mathcal{Y}_{3x}(v,w)]+R(v,w)=0,$
with
$R(v,w)=3h_{1}{\rm
Wronskian}[\mathcal{Y}_{2x}(v,w),\mathcal{Y}_{x}(v)]+h_{3}v_{2x}+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})v_{x}.$
In order to express $R(v,w)$ as the $x$-derivative of a linear combination of
$\mathcal{Y}$-polynomials, we choose a constraint
$None$ $\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{x}(v)=\lambda,$
where $\alpha$ and $\lambda$ are arbitrary parameters. Direct calculation
gives
$R(v,w)=3h_{1}\lambda v_{2x}+h_{3}v_{2x}+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})v_{x},$
which can be written as $x$-derivative of $\mathcal{Y}$-polynomials
$None$
$R(v,w)=\partial_{x}[3h_{1}\lambda\mathcal{Y}_{x}(v)+h_{3}\mathcal{Y}_{x}(v)+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})v].$
From (4.10)-(4.11), we infer that
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{x}(v)-\lambda=0,$
$\displaystyle\partial_{x}\mathcal{Y}_{t}(v)+\partial_{x}[h_{1}\mathcal{Y}_{3x}(v,w)+(3h_{1}\lambda+h_{3})\mathcal{Y}_{x}(v)+(h_{4}+\partial_{t}\ln
h_{1}h_{2}^{-1})v]=0,$
which can be cast into a bilinear Bäcklund transformation by using the
property (2.2)
$None$ $\displaystyle(D_{x}^{2}+\alpha D_{x}-\lambda)F\cdot G=0,$
$\displaystyle[D_{t}+h_{1}D_{x}^{3}+(3h_{1}\lambda+h_{3})D_{x}+\beta]F\cdot
G=0,$
if we set the constraint
$\displaystyle h_{4}+\partial_{t}\ln h_{1}h_{2}^{-1}=0\ \ \Longrightarrow
h_{1}=c_{0}e^{-\int h_{4}dt}h_{2},$
with $c_{0}$ being a arbitrary parameter. Without loss of generality, taking
$c_{0}=1/6$, then
$c(t)=6h_{1}/h_{2}=e^{-\int h_{4}dt}.$
As $\alpha=\beta=0$, the Bäcklund transformation (4.13) reduces to the one
obtained in [21].
Making use of the Hopf-Cole transformation $v=\ln\psi$ and the formula
(2.5)-(2.6), then the system (4.10) can be linearized into a Lax pair
$None$ $\displaystyle
L_{1}\psi=(\partial_{x}^{2}+\alpha\partial_{x}+q_{2x})\psi=\lambda\psi,$
$None$
$\displaystyle(\partial_{t}+L_{2})\psi=[\partial_{t}+h_{1}\partial_{x}^{3}+(3h_{1}q_{2x}+3\lambda
h_{1}+h_{3})\partial_{x}]\psi=0,$
or equivalently,
$\displaystyle\psi_{2x}+\alpha\psi_{x}+(ue^{\int h_{4}dt}-\lambda)\psi=0,$
$\displaystyle\psi_{t}+[h_{1}(2ue^{\int
h_{4}dt}+4\lambda+\alpha^{2})+h_{3}]\psi_{x}+[\beta+\alpha\lambda-
h_{1}(u_{x}e^{\int h_{4}dt}-\alpha ue^{\int h_{4}dt})]\psi=0.$
This Lax pair can be used to construct Darboux transformation, inverse
scattering transformation for soliton solutions. It is easy to check that the
integrability condition
$[L_{1}-\lambda,\partial_{t}+L_{2}]\psi=0$
is satisfied if $u$ is a solution of the vc-KdV equation (4.1).
4.3. Darboux covariant Lax pair
Let us go back to the vc-KdV equation (4.1) and the associated Lax pair
(4.14)-(4.15). Assume that $\phi$ is a solution eigenvalue equation (4.14)
(taking $\alpha=0$ for simplicity). It is well-known that the gauge
transformation
$None$ $T=\phi\partial_{x}\phi^{-1}=\partial_{x}-\sigma,\ \
\sigma=\partial_{x}\ln\phi$
map the operator $L_{1}=\partial_{x}^{2}+q_{2x}$ onto a similar operator:
$T(L_{1}(q)-\lambda)T^{-1}=\tilde{{L}}_{1}(\tilde{q})-\lambda,$
which satisfies the covariance condition
$\tilde{{L}}_{1}(\tilde{q})=L_{1}(\tilde{q}=q+\Delta q),\ \ {\rm with}\ \
\Delta q=2\ln\phi.$
But it can verified that similar property does not hold for the evolution
equation (4.15). Next step is to find another third order operator
${L}_{2,{\rm cov}}(q)$ with appropriate coefficients, such that
$\partial_{t}+L_{2,{\rm cov}}(q)$ be mapped, by gauge transformation (4.16),
onto a similar operator $\partial_{t}+\tilde{L}_{2,{\rm{\rm cov}}}(\tilde{q})$
which satisfies the covariance condition
$\tilde{L}_{2,{\rm cov}}(\tilde{q})={L}_{2,{\rm cov}}(\tilde{q}=q+\Delta q).$
Suppose that $\phi$ is a solution of the following Lax pair
$None$ $\displaystyle L_{1}\phi=\lambda\phi,$
$\displaystyle(\partial_{t}+{L}_{2,{\rm cov}})\phi=0,\ \ {L}_{2,{\rm
cov}}=4h_{1}\partial_{x}^{3}+b_{1}\partial_{x}+b_{2},$
where $b_{1}$ and $b_{2}$ are functions to be determined. It suffice that we
require that the transformation $T$ map the operator $\partial_{t}+L_{2,{\rm
cov}}$ onto the similar one
$None$ $T(\partial_{t}+L_{2,{\rm cov}})T^{-1}=\partial_{t}+\tilde{L}_{2,{\rm
cov}},\ \ \tilde{L}_{2,{\rm
cov}}=4h_{1}\partial_{x}^{3}+\tilde{b}_{1}\partial_{x}+\tilde{b}_{2},$
where $\tilde{b}_{1}$ and $\tilde{b}_{2}$ satisfy the covariant condition
$None$ $\tilde{b}_{j}={b}_{j}(q)+\Delta b_{j}={b}_{j}(q+\Delta q),\ \ j=1,2.$
It follows from (4.16) and (4,18) that
$None$ $\Delta b_{1}=\tilde{b}_{1}-b_{1}=12h_{1}\sigma_{x},\ \ \Delta
b_{2}=\tilde{b}_{2}-b_{2}=b_{1,x}+\sigma\Delta b_{1}+12h_{1}\sigma_{2x},$
and $\sigma$ satisfies
$None$ $\sigma_{t}+4h_{1}\sigma_{1x}+b_{2,x}+\Delta
b_{2}\sigma+\tilde{b}_{1}\sigma_{x}=0.$
According to (4.19), it remains to determine $b_{1}$ and $b_{2}$ in the form
of polynomial expressions in terms of derivatives of $q$
$b_{j}=F_{j}(q,q_{x},q_{2x},q_{3x},\cdots),\ \ j=1,2$
such that
$None$ $\Delta F_{j}=F_{j}(q+\Delta q,q_{x}+\Delta q_{x},q_{2x}+\Delta
q_{2x},\cdots)-F_{j}(q,q_{x},q_{2x},\cdots)=\Delta b_{j},$
with $\Delta q_{rx}=2(\ln q)_{rx},\ r=1,2,\cdots$, the $\Delta b_{j}$ being
determined by the relations (4.19).
Expanding the left hand of the equation (4.22) , we obtain
$\Delta b_{1}=\Delta F_{1}=F_{1,q}\Delta q+F_{1,q_{x}}\Delta
q_{x}+F_{1,q_{2x}}\Delta q_{2x}+\cdots=12h_{1}\sigma_{x}=6h_{1}\Delta q_{2x},$
which implies that we can choose
$None$ $b_{1}=F_{1}(q_{2x})=6h_{1}q_{2x}+c_{1}(t),$
with $c_{1}(t)$ being arbitrary function about $t$.
From the eigenvalue equation in (4.17), we can find the following relation
$None$ $q_{3x}=-\sigma_{2x}-2\sigma\sigma_{x}.$
Substituting (4.23) and (4.24) into (4.19) leads to
$\Delta
b_{2}=12h_{1}q_{3x}+12h_{1}\sigma\sigma_{x}+12h_{1}\sigma_{2x}=6h_{1}\sigma_{2x}=3h_{1}\Delta
q_{3x}.$
The second condition
$\Delta F_{2}=F_{2,q}\Delta q+F_{2,q_{x}}\Delta q_{x}+F_{2,q_{2x}}\Delta
q_{2x}+F_{2,q_{3x}}\Delta q_{3x}+\cdots=\Delta b_{2},$
can be satisfied, if one chooses
$b_{2}=F_{2}(q,q_{x}q_{2x},q_{3x})=3h_{1}q_{3x}+c_{2}(t),$
in which $c_{2}(t)$ is arbitrary constant.
Setting $c_{1}(t)=h_{3},\ c_{2}(t)=0$, we find the following Darboux covariant
evolution equation
$(\partial_{t}+L_{2,{\rm cov}})\phi=0,\ \ L_{2,{\rm
cov}}=4h_{1}\partial_{x}^{3}+(6h_{1}q_{2x}+h_{3})\partial_{x}+3h_{1}q_{3x},$
which is in agreement with the equation (4.21). Moreover, the relation between
the operator $L_{2,{\rm cov}}$ and the operator $L_{2}$ is given by
$L_{2,{\rm cov}}=L_{2}+3h_{1}\partial_{x}(L_{1}-\lambda).$
The integrability condition of the Darboux covariant Lax pair (4.15) precisely
give rise to the equation (4.1) in Lax representation
$[\partial_{t}+L_{2,{\rm
cov}},L_{1}]=-[q_{xt}+h_{1}q_{4x}+3h_{1}q_{2x}^{2}+h_{3}q_{2x}+h_{4}q_{x}]_{x}.$
The higher operators can be obtained in a similar way step by step
$L_{k,{\rm
cov}}(q)=4h_{1}\partial_{x}^{k}+b_{1}\partial_{x}^{k-2}+\cdots+b_{p},\ \
k=3,4,\cdots$
which are Darboux covariant with respect to $L_{1}$, so as to produce higher
order members of the vc-KdV hierarchy.
4.4. Infinite conservation laws
Finally, we construct the conservation laws of vc-KdV equation. The second
equation of (4.12) has been conserved form due to the relation
$\partial_{x}\mathcal{Y}_{t}(v)=\partial_{t}\mathcal{Y}_{x}(v)=v_{xt}.$ By
introducing a new potential function
$\eta=(q^{\prime}_{x}-q_{x})/2,$
it follows from the relation (4.8) that
$None$ $v_{x}=\eta,\ \ w_{x}=q_{x}+\eta.$
Substituting (4.25) into (4.12), we get a Riccati type equation
$None$ $\displaystyle\eta_{x}+\eta^{2}+q_{2x}=\lambda=\varepsilon^{2},$
and a divergence type equation
$None$
$\displaystyle\eta_{t}+\partial_{x}[h_{1}\eta_{2x}+6h_{1}(\eta+\varepsilon)\varepsilon^{2}-2h_{1}(\eta+\varepsilon)^{3}+h_{3}(\eta+\varepsilon)]=0,$
where we have used the equation (4.26) to get the equation (4.27) and set
$\lambda=\varepsilon^{2}$.
Substituting the expansion
$None$
$\eta=\varepsilon+\sum_{n=1}^{\infty}I_{n}(q,q_{x},\cdots)\varepsilon^{-n}.$
into the equation (4.26) and equating the coefficients of $\varepsilon^{-1}$,
the conserved densities are explicitly obtained by recursion relations
$None$ $\displaystyle I_{1}=-p_{x}=-\frac{1}{2}e^{\int h_{4}dt}u,\ \
I_{2}=\frac{1}{4}e^{\int h_{4}dt}u_{x},$ $\displaystyle
I_{n+1}=-\frac{1}{2}(I_{n,x}+\sum_{k=1}^{n}I_{k}I_{n-k}),\ \ n=2,3,\cdots,$
In addition, substituting (4.28) into (4.27) leads to
$\displaystyle\sum_{n=1}^{\infty}I_{n,t}\varepsilon^{-n}+\partial_{x}\left[h_{1}\sum_{n=1}^{\infty}I_{n,2x}\varepsilon^{-n}-6h_{1}\varepsilon(\sum_{n=1}^{\infty}I_{n}\varepsilon^{-n})^{2}-2h_{1}(\sum_{n=1}^{\infty}I_{n}\varepsilon^{-n})^{3}+h_{3}\sum_{n=1}^{\infty}I_{n}\varepsilon^{-n}\right]=0,$
which provides us infinite conservation laws
$None$ $I_{n,t}+F_{n,x}=0,\ n=1,2,\cdots.$
In the equation (4.30), the conversed densities $I_{n}^{\prime}s$ are given by
recursion formulas (4.29), and the fluxes $F_{n}^{\prime}s$ are given by
$\displaystyle F_{1}=-\frac{1}{2}e^{\int
h_{4}dt}(h_{1}u_{2x}+3h_{2}u^{2}+h_{3}u),$ $\displaystyle
F_{2}=\frac{1}{4}e^{\int h_{4}dt}(h_{1}u_{3x}+6h_{2}e^{\int
h_{4}dt}uu_{x}+h_{3}u_{x}),$ $\displaystyle
F_{n}=h_{1}I_{n,2x}-6h_{1}\sum_{k=1}^{n}I_{k}I_{n+1-k}-2h_{1}\sum_{i+j+k=n}I_{i}I_{j}I_{k}+h_{3}I_{n},\
\ n=3,4,\cdots.$
The first equation of (4.30) is exactly the vc-KdV equation (4.1). We see that
above fluxes $F_{n}^{\prime}s$ can be obtained from solution $u$ by algebraic
and differential manipulation, thus they are local. Taking the boundary
condition of $u$ into account, the conservation equation (4.30) implies that
$\\{I_{n},\ n=1,2,\cdots,\\}$ constitute infinite conserved densities of the
vc-KdV equation (4.1). In conclusion, the vc-KdV equation (4.1) is complete
integrable under the constraint $h_{2}=6h_{1}e^{\int h_{7}dt}$ in the sense
that it admits bilinear Bäcklund transformation, Lax pair and infinite
conservation laws.
5\. Nonisospectral variable-coefficient KP equation
Consider nonisospectral and vc-KP equation [17]
$None$ $\displaystyle
u_{t}+h_{1}(u_{3x}+6uu_{x}+3\alpha^{2}\partial_{x}^{-1}u_{yy})+h_{2}(u_{x}-\alpha
xu_{y}-2\alpha\partial_{x}^{-1}u_{y})$ $\displaystyle-
h_{3}(xu_{x}+2u+2yu_{y})=0,$
where $h_{1}=h_{1}(t),\ h_{2}=h_{2}(t)$ and $\ h_{3}=h_{3}(t)$ are all
arbitrary functions with respect to time variable $t$. The equation (5.1)
reduces to the vc-KdV equation (3.1) when $u=u(x,t)$ is independent of the
variable $y$. In the case $h_{1}=1,\ h_{2}=h_{3}=0$, it reduce to standard KP
equation.
5.1. Bilinear representation
By introducing a potential field $q$
$None$ $u=c(t)q_{2x},$
with $c=c(t)$ is a free function about $t$ to be determined, the resulting
equation (5.1) for $q$ (integrating with respect to $x$) reads
$None$ $\displaystyle E(q)\equiv
q_{xt}+h_{1}(q_{4x}+3cq_{2x}^{2}+3\alpha^{2}q_{2y})+h_{2}(q_{2x}-x\alpha
q_{xy}-\alpha q_{y})$ $\displaystyle-
h_{3}(q_{x}+xq_{2x}+2yq_{xy})+q_{x}\partial_{t}\ln c=0,$
which can be expressed in the form of $P$-polynomials
$None$ $\displaystyle
E(q)=P_{xt}(q)+h_{1}[P_{4x}(q)+3\alpha^{2}P_{2y}(q)]+h_{2}[P_{2x}(q)-\alpha
xP_{xy}(q)-\alpha q_{y}]$ $\displaystyle-
h_{3}[xP_{2x}(q)+2yP_{xy}(q)+q_{x}]=0,$
if one chooses $c(t)=1$ and use the formula (2.4).
By application of the variable transformation
$q=2\ln F\ \ \Longleftrightarrow\ \ u=2(\ln F)_{2x}$
and using the property (2.3), then the equation (5.4) give the bilinear form
for the vc-KP equation (5.1) as follows
$[D_{x}D_{t}+h_{1}(D_{x}^{4}+3\alpha^{2}D_{y}^{2})+h_{2}(D_{x}^{2}-\alpha
xD_{x}D_{y}-\alpha\partial_{y})-h_{3}(xD_{x}^{2}+2yD_{x}D_{y}+\partial_{x})]F\cdot
F=0,$
starting from which, we can get multi-soliton solutions to the equation (5.1).
For example, one-soliton solution takes the form
$\displaystyle u=\frac{(k+s)^{2}}{2}{\rm
sech}^{2}\frac{\xi+\zeta+\ln\omega}{2},$
in which $\xi=kx-k^{2}y/\alpha,\ \zeta=sx+s^{2}y/\alpha$, while
$k=k(t),s=s(t)$ and $\omega=\omega(t)$ are all functions about $t$, satisfying
$\displaystyle k_{t}=h_{3}k-bk^{2},\ \ s_{t}=h_{3}s+bs^{2},$
$\displaystyle\omega(t)=\exp(\int[h_{3}(k^{3}+s^{3})-h_{2}(k+s)]dt).$
5.2. Bäcklund transformation and Lax pair
In following we consider bilinear Bäcklund transformation and Lax pairs of the
equation (5.1). Assume $q$ and $q^{\prime}$ are two solutions of the equation
(5.3), we consider the corresponding two-field condition
$None$ $\displaystyle
E(q^{\prime})-E(q)=(q^{\prime}-q)_{xt}+h_{1}[(q^{\prime}-q)_{4x}+3(q^{\prime}+q)_{2x}(q^{\prime}-q)_{2x}+3\alpha^{2}(q^{\prime}-q)_{2y}]$
$\displaystyle+h_{2}[(q^{\prime}-q)_{2x}-\alpha
x(q^{\prime}-q)_{xy}-\alpha(q^{\prime}-q)_{y}]-h_{3}[x(q^{\prime}-q)_{2x}$
$\displaystyle+2y(q^{\prime}-q)_{xy}+(q^{\prime}-q)_{x}]=0,$
which may produce the required bilinear Bäcklund transformation under an
appropriate additional constraint.
By change of the variables
$None$ $v=(q^{\prime}-q)/2,\ \ w=(q^{\prime}+q)/2$
we rewrite the condition (5.5) in the form
$None$ $\displaystyle
E(q^{\prime})-E(q)=v_{xt}+h_{1}(v_{4x}+6v_{2x}w_{2x}+3\alpha^{2}v_{2y})+h_{2}(v_{2x}-\alpha
xv_{xy}-\alpha v_{y})$ $\displaystyle-
h_{3}(xv_{2x}+2yv_{xy}+v_{x})=\partial_{x}[\mathcal{Y}_{t}(v)+h_{1}\mathcal{Y}_{3x}(v,w)]+R(v,w)=0,$
with
$\displaystyle R(v,w)=3h_{1}{\rm
Wronskian}[\mathcal{Y}_{2x}(v,w),\mathcal{Y}_{x}(v)]+3h_{1}\alpha^{2}v_{2y}+h_{2}(v_{2x}-\alpha
xv_{xy}-\alpha v_{y})$ $\displaystyle-h_{3}(xv_{2x}+2yv_{xy}+v_{x}).$
In order to express $R(v,w)$ as the $x$-derivative of a linear combination of
$\mathcal{Y}$-polynomials, we choose the constraint
$None$ $\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{y}(v)=\lambda,$
on account of which
$None$ $\displaystyle
R(v,w)=\partial_{x}\\{h_{1}[3\lambda\mathcal{Y}_{x}(v)-3\alpha\mathcal{Y}_{xy}(v,w)]+h_{2}[\mathcal{Y}_{x}(v)-\alpha
x\mathcal{Y}_{y}(v)]$ $\displaystyle-
h_{3}[x\mathcal{Y}_{x}(v)+2y\mathcal{Y}_{y}(v)]\\}.$
Combining relations (5.7)-(5.9), we then obtain a pair of constraints in Bell
polynomial form
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{y}(v)-\lambda=0,$
$\displaystyle\partial_{t}\mathcal{Y}_{x}(v)+\partial_{x}\\{h_{1}[\mathcal{Y}_{3x}(v,w)-3\alpha\mathcal{Y}_{xy}(v,w)+3\lambda\mathcal{Y}_{x}(v)]+h_{2}[\mathcal{Y}_{x}(v)-\alpha
x\mathcal{Y}_{y}(v)]$ $\displaystyle-
h_{3}[x\mathcal{Y}_{x}(v)+2y\mathcal{Y}_{y}(v)]\\}=0,$
which result in bilinear Bäcklund transformation
$None$ $\displaystyle(D_{x}^{2}+\alpha D_{y}-\lambda)F\cdot G=0,$
$\displaystyle[D_{t}+h_{1}(D_{x}^{3}-3\alpha D_{x}D_{y}+3\lambda
D_{x})+h_{2}(D_{x}-\alpha xD_{y})$ $\displaystyle-
h_{3}(xD_{x}+2yD_{y})+\beta]F\cdot G=0$
by applying the property (2.2).
It only remains to linearize the expression (5.10) with the formulae (2.5) and
(2.6). By using the Hopf-Cole transformation $v=\ln\psi$, the bilinear
Bäcklund transformation (5.10) can be linearized into a Lax pair
$\displaystyle(\partial_{y}+L_{1})\psi\equiv\psi_{y}+\alpha^{-1}\psi_{2x}+\alpha^{-1}(q_{2x}-\lambda)\psi=0,\
\ \lambda_{t}=2h_{3}\lambda,$
$\displaystyle(\partial_{t}+L_{2})\psi\equiv\psi_{t}+4h_{1}\psi_{3x}+(xh_{2}+2y\alpha^{-1}h_{3})\psi_{2x}+(6h_{1}q_{2x}+h_{2}-xh_{3})\psi_{x}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +[3h_{1}q_{3x}-3\alpha
h_{1}q_{xy}+(q_{2x}-\lambda)(xh_{2}+2y\alpha^{-1}h_{3})]\psi=0,$
which can be used to construct Darboux transformation, inverse scattering
transformation for getting soliton solutions. It is easy to check that the
integrability condition
$[\partial_{y}+L_{1},\partial_{t}+L_{2}]\psi=0$
is satisfied if $u$ is a solution of the vc-KP equation (5.1) and
nonisospectral condition $\lambda_{t}=2h_{3}\lambda$ holds.
5.3. Infinite conservation laws
Finally, we precede to construct the conservation laws of vc-KP equation. For
this purpose, we decompose the two-filed condition (5.7) into $x$\- and
$y$-derivative of a linear combination of $\mathcal{Y}$-polynomials. Let us go
back to the two-field condition (5.7) and consider the following decomposition
$None$ $\displaystyle R(v,w)=\partial_{x}[3h_{1}\lambda v_{x}-3\alpha
h_{1}v_{x}v_{y}+h_{2}(v_{x}-\alpha xv_{y})-h_{3}(xv_{x}+2yv_{y})]$
$\displaystyle+\partial_{y}(3h_{1}\alpha^{2}v_{y}+3h_{1}\alpha v_{x}^{2}).$
It follows from (5.7), (5.8) and (5.11) that
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{y}(v)-\lambda=0,$
$\displaystyle(\mathcal{Y}_{x}(v))_{t}+\partial_{x}\\{h_{1}[\mathcal{Y}_{3x}(v,w)-3\alpha\mathcal{Y}_{y}(v,w)+3\lambda\mathcal{Y}_{x}(v)]+h_{2}[\mathcal{Y}_{x}(v)-\alpha
x\mathcal{Y}_{y}(v)]$ $\displaystyle-
h_{3}[x\mathcal{Y}_{x}(v)+2y\mathcal{Y}_{y}(v)]\\}+\partial_{y}[3h_{1}\alpha^{2}\mathcal{Y}_{y}(v)+3h_{1}\alpha\mathcal{Y}_{x}^{2}(v)]=0,$
which is slightly different from (5.10) and can produce desired conservation
laws. By introducing a new potential function
$\eta=(q^{\prime}_{x}-q_{x})/2,$
and it follows from the relation (5.6) that
$None$ $v_{x}=\eta,\ \ w_{x}=q_{x}+\eta.$
Substituting (5.13) into (5.12), we decompose the two-field condition (5.7)
into a Riccati type equation
$None$
$\displaystyle\eta_{x}+\eta^{2}+\alpha\partial^{-1}_{x}\eta_{y}+q_{2x}-\varepsilon^{2}=0,$
and a divergence type equation
$None$
$\displaystyle\eta_{t}+\partial_{x}[h_{1}(\eta_{2x}+6\eta\varepsilon^{2}-2\eta^{3}-6\alpha\eta\partial_{x}^{-1}\eta_{y})+h_{2}(\eta-x\alpha\partial_{x}^{-1}\eta_{y})$
$\displaystyle-
h_{3}(x\eta+2y\partial_{x}^{-1}\eta_{y})]+\partial_{y}(3h_{1}\alpha^{2}\partial_{x}^{-1}\eta_{y}+3h_{1}\alpha\eta^{2})=0,$
where we have used the equation (5.14) to get the equation (5.15) and set
$\lambda=\varepsilon^{2}$.
Substituting the expansion
$None$
$\eta=\varepsilon+\sum_{n=1}^{\infty}I_{n}(q,q_{x},\cdots)\varepsilon^{-n}.$
Inserting the equation (5.16) into the equation (5.14), equation the
coefficients for power of $\varepsilon$, then we have the recursion relations
for $I_{n}$
$None$ $\displaystyle I_{1}=-\frac{1}{2}q_{2x}=-\frac{1}{2}u,\ \
I_{2}=\frac{1}{4}(u_{2x}+\alpha\partial^{-1}_{x}u_{y}),\ \
I_{3}=-\frac{1}{8}(u_{3x}+u^{2}+\alpha u_{y}),$ $\displaystyle
I_{n+1}=-\frac{1}{2}(I_{n,x}+\sum_{k=1}^{n}I_{k}I_{n-k}+\alpha\partial^{-1}_{x}I_{n,y}),\
\ n=3,4,\cdots,$
Again substituting (5.16) into (5.15) and noting the nonisospectral condition
$\lambda_{t}=2h_{3}\lambda\ \
\Longrightarrow\varepsilon_{t}=h_{3}\varepsilon,$
we then obtain the following infinite conservation laws
$None$ $I_{n,t}+F_{n,x}+G_{n,y}=0,\ n=1,2,\cdots.$
In the equation (5.18), the conversed densities $I_{n}^{\prime}s$ are given by
formula (5.17), the first fluxes $F_{n}^{\prime}s$ are given by
$\displaystyle F_{1}=h_{1}I_{1,2x}+(h_{2}-xh_{3})I_{1}-(x\alpha
h_{1}+2yh_{3})\partial^{-1}_{x}I_{1,y}-6\alpha
h_{1}\partial^{-1}_{x}I_{2,y}+h_{3}\partial^{-1}_{x}I_{1},$ $\displaystyle
F_{2}=h_{1}I_{2,2x}+(h_{2}-xh_{3})I_{2}-(x\alpha
h_{1}+2yh_{3})\partial^{-1}_{x}I_{2,y}-12h_{1}I_{1}I_{2}-12\alpha
h_{1}I_{1}\partial^{-1}_{x}I_{2,y}$ $\displaystyle-6\alpha
h_{1}\partial^{-1}_{x}I_{3,y}+2h_{3}\partial^{-1}_{x}I_{2},$ $\displaystyle
F_{n}=-6h_{1}\sum_{k=1}^{n}I_{k}(I_{n+1-k}+\alpha\partial_{x}^{-1}I_{n-k,y})-2h_{1}\sum_{i+j+k=n}I_{i}I_{j}I_{k}+(h_{2}-xh_{3})I_{n}+h_{1}I_{n,2x}$
$\displaystyle\ \ \ \ \ \ \ \ \ -(x\alpha
h_{2}+2yh_{3})\partial_{x}^{-1}I_{n,y}+nh_{3}\partial_{x}^{-1}I_{n}-6\alpha
h_{1}I_{n+1},\ \ n=3,4,\cdots.$
and the second fluxes $G_{n}^{\prime}s$ are given by
$\displaystyle G_{1}=3h_{1}\alpha^{2}\partial^{-1}_{x}I_{1,y},\ \
G_{2}=3h_{1}\alpha I_{1}^{2}+3h_{1}\alpha^{2}\partial^{-1}_{x}I_{2,y},$
$\displaystyle
G_{n}=3h_{1}\alpha\sum_{k=1}^{n}I_{k}I_{n-k}+3h_{1}\alpha^{2}\partial_{x}^{-1}I_{n,y},\
\ n=2,3,\cdots.$
The first equation of (5.18) is exactly the vc-KP equation (5.1). The
expressions $F_{n}^{\prime}s$ and $G_{n}^{\prime}s$ indicate that the fluxes
of the vc-KP equation are not local. To summarize, the vc-KP equation (5.1) is
complete integrable, since it admits bilinear Bäcklund transformation, Lax
pair and infinite conservation laws.
6\. General variable-coefficient KP equation
Consider a general vc-KP equation [40]
$None$
$(u_{t}+h_{1}u_{3x}+h_{2}uu_{x})_{x}+h_{3}u_{2y}+h_{4}u_{xy}+(h_{5}+h_{6}y)u_{2x}+h_{7}u_{x}=0,$
where $h_{i}=h_{i}(t),\ i=1,2,\cdots,7$ are arbitrary functions with respect
to time variable $t$. The equation (6.1) include many special variable-
coefficient equations in physics, such as cylindrical KdV equation [23]
$None$ $u_{t}+uu_{x}+u_{3x}+\frac{1}{2t}u_{x}=0,$
cylindrical KP equation [41, 42]
$None$
$(u_{t}+h_{1}u_{3x}+h_{2}uu_{x})_{x}+\frac{1}{2t}u_{x}+\frac{3\sigma^{2}}{t^{2}}u_{2y}=0,$
generalized cylindrical KP equation [43, 44]
$None$
$(u_{t}+h_{1}u_{3x}+h_{2}uu_{x})_{x}+\frac{1}{2t}u_{x}+\frac{3\sigma^{2}}{t^{2}}u_{2y}+r(t)u_{xy}+[f(t)+g(t)y]u_{2x}=0.$
Here we attempt to find the integrability condition that the equation (6.1)
possesses bilinear representation, Bäckbend transformation, Lax pair, Darboux
covariant Lax pair and infinite conservation laws.
6.1. Bilinear representation
By introducing a potential field $q$
$u=c(t)q_{2x},$
with $c=c(t)$ is free function to be determined, the resulting equation (6.1)
for $q$ (integrating with respect to $x$ twice) reads
$None$ $\displaystyle E(q)\equiv
q_{xt}+h_{1}q_{4x}+\frac{c}{2}h_{2}q_{2x}^{2}+h_{3}q_{2y}+h_{4}q_{xy}+(h_{5}+h_{6}y)q_{2x}$
$\displaystyle+(h_{7}+\partial_{t}\ln c)q_{x}=0,$
which can be expressible as $P$-polynomials
$None$ $\displaystyle
E(q)=P_{xt}(q)+h_{1}P_{4x}(q)+h_{3}P_{2y}(q)+h_{4}P_{xy}(q)+(h_{5}+h_{6})P_{2x}(q)$
$\displaystyle+(h_{7}+\partial_{t}\ln h_{1}h_{2}^{-1})q_{x}=0,$
if one chooses $c=6h_{1}h_{2}^{-1}$ and use the formula (2.4). By application
of the transformation
$q=2\ln F\ \ \Longleftrightarrow\ \ u=cq_{2x}=12h_{1}h_{2}^{-1}(\ln F)_{2x}$
and using the property (2.3), then the equation (6.6) gives the bilinear
representation for the vc-KP equation (6.1) as follows
$[D_{x}D_{t}+h_{1}D_{x}^{4}+h_{3}D_{y}^{2}+h_{4}D_{x}D_{y}+(h_{5}+h_{6}y)D_{x}^{2}+(h_{7}+\partial_{t}\ln
h_{1}h_{2}^{-1})\partial_{x}]F\cdot F=0.$
Starting form this bilinear equation, we can get multi-solutions, for example,
the regular one-soliton like solution is
$\displaystyle u=6h_{1}h_{2}^{-1}k^{2}{\rm sech}^{2}\frac{kx+sy+\omega}{2},$
in which $k$ is a arbitrary constant, while $s=s(t)$ and $\omega=\omega(t)$
are two function with respect to $t$, given by
$s(t)=k\int h_{6}dt,\ \
\omega(t)=-\int(k^{3}h_{1}+k^{-1}s^{2}h_{3}+sh_{4}+h_{7}+\partial_{t}\ln
h_{1}h_{2}^{-1})dt.$
6.2. Bäcklund transformation and Lax pair
We now consider bilinear Bäcklund transformation and Lax pairs of the equation
(6.1). Let $q^{\prime}$ and $q$ be two solutions of the equation (6.5), we
consider the following two-field condition
$None$ $\displaystyle
E(q^{\prime})-E(q)=(q^{\prime}-q)_{xt}+h_{1}(q^{\prime}-q)_{4x}+3h_{1}(q^{\prime}+q)_{2x}(q^{\prime}-q)_{2x}+h_{3}(q^{\prime}-q)_{2y}$
$\displaystyle+h_{4}(q^{\prime}-q)_{xy}+(h_{5}+h_{6}y)(q^{\prime}-q)_{2x}+(h_{7}+\partial_{t}\ln
h_{1}h_{2}^{-1})(q^{\prime}-q)_{x}=0,$
which may produce the required bilinear Bäcklund transformation under an
appropriate additional constraint.
On introducing two new variables
$None$ $v=(q^{\prime}-q)/2,\ \ w=(q^{\prime}+q)/2$
we rewrite the condition (6.7) as the form
$None$ $\displaystyle
E(q^{\prime})-E(q)=v_{xt}+h_{1}v_{4x}+6h_{1}v_{2x}w_{2x}+h_{3}v_{2y}+h_{4}v_{xy}+(h_{5}+h_{6}y)v_{2x}$
$\displaystyle+(h_{7}+\partial_{t}\ln
h_{1}h_{2}^{-1})v_{x}=\partial_{x}[\mathcal{Y}_{t}(v)+h_{1}\mathcal{Y}_{3x}(v,w)]+R(v,w)=0,$
with
$\displaystyle R(v,w)=3h_{1}{\rm
Wronskian}[\mathcal{Y}_{2x}(v,w),\mathcal{Y}_{x}(v)]+h_{3}v_{2y}+h_{4}v_{xy}+(h_{5}+h_{6}y)v_{2x}$
$\displaystyle+(h_{7}+\partial_{t}\ln h_{1}h_{2}^{-1})v_{x}.$
In order to express $R(v,w)$ as the $x$\- and $y$-derivative of
$\mathcal{Y}$-polynomials, we choose the constraint
$None$ $\mathcal{Y}_{y}(v)+\alpha(t)\mathcal{Y}_{2x}(v,w)=\lambda,$
where $\alpha=\alpha(t)$ is to be determined. Direct calculation show that
$None$ $\displaystyle R(v,w)=3h_{1}\lambda
v_{2x}-\alpha^{-1}[h_{3}w_{2x,y}+(2h_{3}-3h_{1}\alpha^{2})v_{x}v_{xy}+3h_{1}\alpha^{2}v_{2x}v_{y}]$
$\displaystyle+h_{4}v_{xy}+(h_{5}+h_{6}y)v_{2x}+(h_{7}+\partial_{t}\ln
h_{1}h_{2}^{-1})v_{x},$
which can be expressible $R(v,w)$ as the $x$-derivative of a linear
combination of $\mathcal{Y}$-polynomials
$\displaystyle R(v,w)=\partial_{x}[3h_{1}\lambda\mathcal{Y}_{x}-3\alpha
h_{1}\mathcal{Y}_{3x}(v,w)+h_{4}\mathcal{Y}_{y}+(h_{5}+h_{6}y)\mathcal{Y}_{x}],$
if we take a simple constraints
$h_{7}+\partial_{t}\ln h_{1}h_{2}^{-1}=0,\ \
h_{3}=2h_{3}-3h_{1}\alpha^{2}=3h_{1}\alpha^{2},$
namely,
$h_{2}=6h_{1}e^{\int h_{7}dt},\ \ \ 3\alpha^{2}=h_{3}h_{1}^{-1}.$
From (6.9)-(6.11), one infers that
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{y}(v)-\lambda=0,$
$\displaystyle\partial_{x}\mathcal{Y}_{t}(v)+\partial_{x}\\{h_{1}[\mathcal{Y}_{3x}(v,w)-3\alpha\mathcal{Y}_{xy}(v,w)+3\lambda\mathcal{Y}_{x}(v)]+h_{4}\mathcal{Y}_{y}$
$\displaystyle+(h_{5}+h_{6}y)\mathcal{Y}_{x}\\}=0,$
which leads to bilinear Bäcklund transformation of variable coefficient KP
equation
$\displaystyle(D_{x}^{2}+\alpha D_{y}-\lambda)F\cdot G=0,$
$\displaystyle[D_{t}+h_{1}(D_{x}^{3}-3\alpha D_{x}D_{y}+3\lambda
D_{x})+h_{4}D_{y}+(h_{5}+h_{6}y)D_{x}+\beta]F\cdot G=0$
by using the property (2.2).
By using the Hopf-Cole transformation $v=\ln\psi$ and the formulae (2.5) and
(2.6), the system (6.12) can be linearized into a Lax pair
$None$
$\displaystyle(\alpha\partial_{y}+L_{1})\psi\equiv(\alpha\partial_{y}+\partial_{x}^{2}+q_{2x}-\lambda)\psi=0,$
$\displaystyle(\partial_{t}+L_{2})\psi\equiv[\partial_{t}+4h_{1}\partial_{x}^{3}-h_{4}\alpha^{-1}\partial_{x}^{2}+(6h_{1}q_{2x}+h_{5}+h_{6}y)\partial_{x}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +3h_{1}q_{3x}-3h_{1}\alpha
q_{xy}-h_{4}\alpha^{-1}q_{2x}+h_{4}\alpha^{-1}\lambda]\psi=0,$
whose integrability condition
$[\alpha\partial_{y}+L_{1},\partial_{t}+L_{2}]\psi=0$
is satisfied if $u$ is a solution of the vc-KP equation (6.1) and
$\alpha_{t}=0$, or equivalently, $h_{3}h_{1}^{-1}={\rm constant}$. This Lax
pair can be used to construct Darboux transformation, inverse scattering
transformation for soliton solutions to the vc-KP equation (6.1).
6.3. Darboux covariant Lax pair
Let us go back to the vc-KP equation (6.1) and the associated Lax pair (6.13).
Assume that $\phi$ is a solution of the following Lax pair
$None$ $\displaystyle(\alpha\partial_{y}+L_{1})\phi=\lambda\phi,\ \
L_{1}=\partial_{x}^{2}+q_{2x},$ $\displaystyle(\partial_{t}+{L}_{2,{\rm
cov}})\phi=0,\ \ {L}_{2,{\rm
cov}}=h_{1}\partial_{x}^{3}+b_{1}\partial_{x}^{2}+b_{2}\partial_{x}+b_{3},$
where $b_{1}$, $b_{2}$ and $b_{3}$ are functions to be determined. It is shown
that the gauge transformation
$None$ $T=\phi\partial_{x}\phi^{-1}=\partial_{x}-\sigma,\ \
\sigma=\partial_{x}\ln\phi$
map the operator $\alpha\partial_{y}+L_{1}(q)$ onto a similar operator:
$T(\alpha\partial_{y}+L_{1}(q))T^{-1}=\alpha\partial_{y}+\tilde{{L}}_{1}(\tilde{q}=q+\Delta
q)\ \ {\rm with}\ \ \Delta q=2\ln\phi.$
It is suffices to verify that such transformation (6.15) map the
$\partial_{t}+{L}_{2,{\rm cov}}$ into similar one
$None$ $T(\partial_{t}+L_{2,{\rm cov}})T^{-1}=\partial_{t}+\tilde{L}_{2,{\rm
cov}},\ \ \tilde{L}_{2,{\rm
cov}}=h_{1}\partial_{x}^{3}+\tilde{b}_{1}\partial_{x}^{2}+\tilde{b}_{2}\partial_{x}+\tilde{b}_{3},$
where $\tilde{b}_{j},j=1,2,3$ and $\tilde{L}_{2,{\rm cov}}$ satisfy the
covariant conditions
$\tilde{b}_{j}={b}_{j}(q)+\Delta b_{j}={b}_{j}(q+\Delta q),\ \ j=1,2,3.$
$\tilde{L}_{2,{\rm cov}}(\tilde{q})={L}_{2,{\rm cov}}(\tilde{q}=q+\Delta q),\
\ \Delta q=2\ln\phi.$
It follows from (6.15) and (3,16) that
$None$ $\displaystyle\Delta b_{1}=\tilde{b}_{1}-b_{1}=0,\ \ \Delta
b_{2}=12h_{1}\sigma_{x}+b_{1,x}+\Delta b_{1}\sigma,$ $\displaystyle\Delta
b_{3}=12h_{1}\sigma_{2x}+2\sigma_{x}\tilde{b}_{1}+\Delta b_{2}\sigma+b_{2,x},$
We require $b_{1},b_{2}$ and $b_{3}$ in the differential polynomial form of
potential filed $q$
$b_{j}=F_{j}(q,q_{x},q_{2x},q_{3x},\cdots),\ \ j=1,2$
such that
$\Delta F_{j}=F_{j}(q+\Delta q,q_{x}+\Delta q_{x},q_{2x}+\Delta
q_{2x},\cdots)-F_{j}(q,q_{x},q_{2x},\cdots)=\Delta b_{j},$
with $\Delta q_{rx}=2(\ln q)_{rx},\ r=1,2,\cdots$.
From eigenvalue equation in (6.14), we get the relation
$q_{3x}=-\alpha\sigma_{xy}-(\sigma_{x}+\sigma^{2})_{x},$
on account of which, solving the system (6.17) yields
$None$ $\displaystyle b_{1}=c_{1}(y,t),\ \ b_{2}=6h_{1}q_{2x}+c_{2}(y,t),$
$\displaystyle b_{3}=3h_{1}(q_{3x}-\alpha q_{xy})+c_{1}q_{2x}+c_{3}(y,t),$
with $c_{1}(y,t),c_{2}(y,t)$ and $c_{3}(y,t)$ being arbitrary functions with
respect to $y$ and $t$. From (6.14) and (6.18), we then find a Darboux
covariant evolution equation
$\displaystyle(\partial_{t}+L_{2,{\rm cov}})\phi=0,$ $\displaystyle L_{2,{\rm
cov}}=4h_{1}\partial_{x}^{3}+c_{1}\partial_{x}^{2}+(6h_{1}q_{2x}+c_{2})\partial_{x}+3h_{1}q_{3x}-3h_{1}\alpha
q_{2x}+c_{3}.$
In particular, if setting
$c_{1}=-h_{4}\alpha^{-1},\ c_{2}=h_{5}+h_{6}y,\ c_{3}=h_{4}\lambda,$
the Darboux covariant operator $L_{2,{\rm cov}}$ reduce to operator $L_{2}$,
namely
$L_{2,{\rm cov}}=L_{2}.$
Therefore the Lax pair (6.13) is Darboux covariant under constraint
$h_{3}h_{1}^{-1}={\rm constant}$.
In a similar way, we can get higher operators
$L_{p,{\rm
cov}}(q)=h_{1}\partial_{x}^{p}+b_{1}\partial_{x}^{p-2}+\cdots+b_{p},\ \
p=3,4,\cdots$
which are Darboux covariant with respect to $L_{1}$ step by step, so as to
produce higher order members of the vc-KP hierarchy.
6.4. Infinite conservation laws
Finally, we turn to construct the conservation laws of vc-KP equation. For
this purpose, we expect re-decompose the two-filed condition (6.7) into $x$\-
and $y$-derivative of $\mathcal{Y}$-polynomials. We return to revisit $R(v,w)$
in two-field condition (6.9) and write it as another form
$None$ $\displaystyle R(v,w)=[3h_{1}\lambda v_{x}-3\alpha
h_{1}v_{x}v_{y}+(h_{5}+h_{6}y)v_{x}]_{x}+(h_{3}v_{y}+h_{4}v_{x}+3h_{1}\alpha
v_{x}^{2})_{y}.$
It follows from the relations (6.9), (6.10) and (6.19) that
$None$
$\displaystyle\mathcal{Y}_{2x}(v,w)+\alpha\mathcal{Y}_{y}(v)-\lambda=0,$
$\displaystyle\partial_{t}\mathcal{Y}_{x}(v)+\partial_{x}[h_{1}\mathcal{Y}_{3x}(v,w)-3h_{1}\alpha\mathcal{Y}_{y}(v,w)+3h_{1}\lambda\mathcal{Y}_{x}(v)+(h_{5}+h_{6}y)\mathcal{Y}_{x}(v)]$
$\displaystyle+\partial_{y}[h_{3}\mathcal{Y}_{y}(v)+3h_{1}\alpha\mathcal{Y}_{x}(v)^{2}+h_{4}\mathcal{Y}_{x}(v)]=0,$
which is slightly different from (6.10) and can produce desired conservation
laws. Especially we don’t need the constraint $3\alpha^{2}=h_{3}h_{1}^{-1}$.
By introducing a new potential function
$\eta=(q^{\prime}_{x}-q_{x})/2,$
and it follows from the relation (6.8) that
$None$ $v_{x}=\eta,\ \ w_{x}=q_{x}+\eta.$
Substituting (6.21) into (6.20) yields a coupled system
$None$
$\displaystyle\eta_{x}+\eta^{2}+\alpha\partial^{-1}_{x}\eta_{y}+q_{2x}-\varepsilon^{2}=0,$
$None$
$\displaystyle\eta_{t}+\partial_{x}[h_{1}(\eta_{2x}+6\eta\varepsilon^{2}-2\eta^{3}-6\alpha\eta\partial_{x}^{-1}\eta_{y})+(h_{5}+h_{6}y)\eta]$
$\displaystyle+\partial_{y}(h_{3}\partial_{x}^{-1}\eta_{y}+3h_{1}\alpha\eta^{2}+h_{4}\eta)=0,$
where we have used the equation (6.22) to get the equation (6.23) and set
$\lambda=\varepsilon^{2}$.
Substituting the expansion
$None$
$\eta=\varepsilon+\sum_{n=1}^{\infty}I_{n}(p,p_{x},\cdots)\varepsilon^{-n}$
into the equation (6.22), equating the coefficients for power of
$\varepsilon$, then we obtain the recursion relations for $I_{n}$ as follows
$None$ $\displaystyle I_{1}=-\frac{1}{2}q_{2x}=-\frac{1}{2}ue^{\int h_{4}dt},\
\ I_{2}=\frac{1}{4}e^{\int h_{4}dt}(u_{2x}+\alpha\partial^{-1}_{x}u_{y}),$
$\displaystyle
I_{n}=-\frac{1}{2}(I_{n,x}+\sum_{k=1}^{n}I_{k}I_{n-k}+\alpha\partial^{-1}_{x}I_{n,y}),\
\ n=2,3,\cdots,$
Again substituting (6.24) into (6.23) and comparing the coefficients for power
of $\varepsilon$ provide us infinite conservation laws
$None$ $I_{n,t}+F_{n,x}+G_{n,y}=0,\ n=1,2,\cdots.$
In the equation (6.26), the conversed densities $I_{n}^{\prime}s$ obtained by
recursion formulas (6.25), and the first fluxes $F_{n}^{\prime}s$ are
expressible in $I_{n}^{\prime}s$
$\displaystyle F_{1}=h_{1}I_{1,2x}-6\alpha h_{1}I_{1}^{2}-6\alpha
h_{1}\partial^{-1}_{x}I_{2,y}+(h_{5}+h_{6}y)I_{1},$ $\displaystyle
F_{2}=h_{1}I_{2,2x}-6h_{1}I_{1}(2I_{2}+\alpha
I_{1}+\alpha\partial^{-1}_{x}I_{1,y})-6h_{1}\alpha\partial^{-1}_{x}I_{3,y}+(h_{5}+h_{6}y)I_{2},$
$\displaystyle
F_{n}=-6h_{1}\sum_{k=1}^{n}I_{k}(I_{n+1-k}+\alpha\partial_{x}^{-1}I_{n-k,y})-2h_{1}\sum_{i+j+k=n}I_{i}I_{j}I_{k}+h_{1}I_{n,2x}$
$\displaystyle\ \ \ \ \ \ \ \ \ -6\alpha
h_{1}\partial^{-1}_{x}I_{n+1,y}+(h_{5}+h_{6}y)I_{n},\ \ n=3,4,\cdots.$
and the second fluxes $G_{n}^{\prime}s$ are given by
$\displaystyle G_{1}=h_{3}\partial^{-1}_{x}I_{1,y}+6\alpha
h_{1}I_{2}+h_{4}I_{1},$ $\displaystyle
G_{n}=3h_{1}\alpha\sum_{k=1}^{n}I_{k}I_{n-k}+3h_{1}\alpha^{2}\partial_{x}^{-1}I_{n,y},\
\ n=2,3,\cdots.$
The first equation of the conservation law equation (6.26) is exactly the vc-
KP equation (6.1). Taking the boundary condition of $p$ into account, the
equation (6.26) implies that $I_{n}^{\prime}s,\ n=1,2,\cdots$ constitute
infinite conserved densities of the vc-KP equation (6.1). To this end, we
remark that as application of these results, all equations (6.2)-(6.4) are
complete integrable under the constraint $h_{2}=6h_{1}e^{\int h_{7}dt}$, since
they possess bilinear Bäcklund transformation, Lax pair and infinite
conservation laws.
Acknowledgment
The work described in this paper was supported by grants from the National
Science Foundation of China (No. 10971031), Shanghai Shuguang Tracking Project
(No. 08GG01).
## References
* [1]
* [2] V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85(2000) 4502
* [3] V. I. Kruglov and A.C. Peacock, J.D. Harvey, Phys. Rev. E 71(2005) 056619
* [4] J. K. Xue, Phys. Plasmas 11(2004) 1860
* [5] V. Serkin, M. Matsumoto, T. Belyaeva, Opt. Comm. 196 (2001) 159
* [6] A. Biswas, J. Nonl. Opt. Phys. Mat. 12(2003) 17
* [7] M. Zamir, The Physics of Pulsatile Flow, Springer-Verlag, New York, 2000\.
* [8] C. C. Bradley, C.A. Sackett, J.J. Tollett and R.G. Hulet, Phys. Rev. Lett. 75(1995) 1687
* [9] F. Dalfovo, S. Giorgini, L.P. Pitaevskii and S. Stringari, Rev. Mod. Phys. 71(1999) 463
* [10] W. M. Liu, B.Wu and Q. Niu, Phys. Rev. Lett. 84(2000) 2294
* [11] G. X. Huang, J. Szeftel and S.H. Zhu, Phys. Rev. A 65(2002) 053605
* [12] H. H. Dai and Y. Huo, Wave Motion 35(2002) 55
* [13] W. L. Chan and K. S. Li, J. Math. Phys. 30 (1989) 2521
* [14] S. Maxon and J. Viecelli, Phys. Fluids. 17 (1989) 2521
* [15] W. L. Chan and K. S. Li, J. Math. Phys. 30 (1989) 1614
* [16] R. Hirota, J. Phys. Soc. Jpn. 46 (1979) 1681
* [17] W. L. Chan, K. S. Li and Y. S. Li, J. Math. Phys. 33 (1992) 3759
* [18] S. Y. Lou and H. Y. Ruan, Acta Phys. Sin. 41 (1992) 182
* [19] B. Tian, W. R. San, C. Y. Zhang, G. M. Wei, Y. T. Gao, Eur. Phys. J. B 47 (2005) 329
* [20] G. M. Wei, Y. T. Gao, W. Hu. C. Y. Zhang, Eur. Phys. J. B 53 (2006) 343
* [21] C. Y. Zhang, Y. T. Gao. X. H. Meng J. Li, T. Xu, G. M. Wei, H. W. Zhu, J. Phys. A 39 (2006) 1453
* [22] Y. T. Gao, B. Tian, Phys. Plasmas 10(2003) 4306
* [23] V. Hlavaty, J. Phys. Soc. Jpn. 55(1986) 1405
* [24] P. Clarkson, IMA J. Appl. Math. 44(1990) 27
* [25] E. G. Fan, Phys. Lett. A 294 (2002) 26
* [26] R. Grimshaw, Proc. R. Soc. Lond. A 368(1979) 359
* [27] N. Joshi, Phys. Lett. A 125 (1987) 456
* [28] R. Hirota and J. Satsuma: Prog. Theor. Phys. 57 (1977) 797
* [29] R. Hirota: Direct methods in soliton theory (Springer-verlag, Berlin, 2004)
* [30] X. B. Hu and P. A. Clarkson, J. Phys. A 28 (1995) 5009
* [31] X. B. Hu, C X Li, J. J. C. Nimmo and G. F. Yu, J. Phys. A 38 (2005) 195
* [32] Q. P. Liu, X. B. Hu and M. X. Zhang, Nonlinearity 18 (2005) 1597
* [33] D. J. Zhang, J. Phys. Soc. Jpn. 71 (2002) 2649
* [34] A. Nakamura, J. Phys. Soc. Jpn. 48(1980) 1365
* [35] E. G. Fan and Y. C. Hon, Phys Rev E 78 (2008) 036607
* [36] E. T. Bell, Ann. Math. 35 (1934) 258
* [37] C. Gilson, F. Lambert, J. Nimmo and R. Willox, Proc. R. Soc. Lond. A 452 (1996) 223
* [38] F. Lambert, I. Loris and J.Springael, Inverse Probl. 17 (2001) 1067
* [39] F. Lambert and J.Springael, Acta Appl. Math. 102 (2008) 147
* [40] P. A. Clarkson, J. Appl. Math. 44 (1990) 27
* [41] A. Nakamura, Prog. Theor. Phys. Suppl. 94 (1988) 195
* [42] R. Johnson, J. Fluid. Mech. 97 (1980) 701
* [43] D. David, D. Levi and P. Winternitz, Stud. Appl. Math. 76 (1987) 133
* [44] D. Levi and P. Winternitz, Phys. Lett. A 102 (2008) 147
|
arxiv-papers
| 2010-08-25T03:20:05 |
2024-09-04T02:49:12.437394
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1008.4194"
}
|
1008.4198
|
Generalized Super Bell Polynomials with Applications to Superymmetric
Equations
Engui Fan111 Corresponding author and E-mail address: faneg@fudan.edu.cn
School of Mathematical Sciences, Institute of Mathematics and Key Laboratory
of Mathematics for Nonlinear Science, Fudan University, Shanghai, 200433, P.R.
China
Y. C. Hon
Department of Mathematics, City University of Hong Kong, Hong Kong, P.R. China
Abstract. In this paper, we introduce a class of new generalized super Bell
polynomials on a superspace, explore their properties, and show that they are
a natural and effective tool to systematically investigate integrability of
supersymmetric equations. The connections between the super Bell polynomials
and super bilinear representation, bilinear Bäcklund transformation, Lax pair
and infinite conservation laws of supersymmetric equations are established. We
take supersymmetric KdV equation and supersymmetric sine-Gordon equation to
illustrate this procedure.
Keywords: super Bell polynomial; supersymmetric equation; bilinear Bäcklund
transformation; Lax pair; conservation law.
1\. Introduction
The supersymmetry represents a kind of symmetrical characteristic between
boson and fermion in physics. The concept of supersymmetry was originally
introduced and developed for applications in elementary particle physics
thirty years ago [2]–[4]. It is found that supersymmetry can be applied to a
variety of problems such as relativistic, non-relativistic physics and nuclear
physics. In recent years, supersymmetry has been a subject of considerable
interest both in physics and mathematics. The mathematical formulation of the
supersymmetry is based on the introduction of Grassmann variables along with
the standard ones [37]. In a such way, a number of well known mathematical
physical equations have been generalized into the supersymmetric analogues,
such as supersymmetric versions of sine-Gordon, KdV, KP hierarchy, Boussinesq,
MKdV etc. It has been shown that these supersymmetric integrable systems
possess bi-Hamiltonian structure, Painlevé property, infinite many symmetries,
Darboux transformation, Bäcklund transformation, bilinear form, super soliton
solutions and super quasi-periodic solutions [5]–[19]. In our present paper,
we investigate the integrability of supersymmetric equations by using a class
of super Bell polynomials which are a multidimensional and super
generalization of ordinary Bell polynomials.
The ordinary Bell polynomials introduced by Bell during the early 1930s are a
class of exponential polynomials, which are specified by a generating function
and exhibit important properties [23]. The Bell polynomials have been
exploited in combinatorics, statistics and other fields [24]-[26]. Some
generalized forms of Bell polynomials already appeared in literature
[27]-[31]. More recently Lambert, Gilson et al found that the Bell polynomials
also play important role in the characterization of bilinearizable equations.
They presented an alternative procedure based on the use of the properties of
Bell polynomials to obtain parameter families of bilinear Bäcklund
transformation for soliton equations. As a consequence bilinear Bäcklund
transformation with single field can be linearize into corresponding Lax pairs
[32]-[34].
Our paper is a further contribution to the theory of Bell polynomials and
supersymmetric equations. We reconsider Bell polynomials in a more extended
context– superspace. We define a kind of new generalized super Bell
polynomials and discuss their relations with super bilinear equations, which
actually provides an approach to systematically investigate complete
integrability of supersymmetric systems. As illustrative examples, the
bilinear representations, bilinear Bäcklund transformations, Lax pais and
infinite conservation laws of the supersymmetric KdV equation and
supersymmetric sine-Gordon equation are obtained in a quick and natural
manner.
The layout of this paper is as follows. In Section 2, we briefly recall
elementary notations about superdifferential, integrals and super bilinear
operators on superspace. As In Section 3, we propose theory of super Bell
polynomials and establish their connections with supersymmetric equations. As
consequence a approach to investigate integrability of supersymmetric
equations is presented. In the Sections 4 and 5, as applications of super Bell
polynomials, we study integrability of supersymmetric KdV equation
supersymmetric sine-Gordon equation, respectively. At last, we briefly discuss
further possible generalization and applications of Bell polynomials and
future work in Section 6.
2\. Derivatives and bilinear operators on superspace
To make our presentation easily understanding and self-contained, in this
section we first briefly review some notations about superanalysis [35]-[38]
and super-Hirota bilinear operators [20, 21].
A superalgebra is a $Z_{2}$-graded space
$\Lambda=\Lambda_{0}\oplus\Lambda_{1}$ in which, $\Lambda_{0}$ is a subspace
consisting of even elements and $\Lambda_{1}$ is a subspace consisting of odd
elements. A parity function is introduced for homogeneous elements on the
$\Lambda$, namely, $|a|=0$ if $a\in\Lambda_{0}$ and $|a|=1$ if
$a\in\Lambda_{1}$.
The superalgebra is said to be commutative if the supercommutator
$[a,b]=ab-(-1)^{|a||b|}ba=0$, for arbitrary homogeneous elements
$a,b\in\Lambda$.
A commutative superalgebra $\Lambda$ with unit $e=1$ is called a finite-
dimensional Grassmann algebra if it contains a system of anticommuting
generators $\theta_{j},j=1,\cdots,n$ with the anticommutative property:
$[\theta_{j},\theta_{k}]=\theta_{j}\theta_{k}+\theta_{k}\theta_{j}=0,\
\theta_{j}^{2}=0,\ j,k=1,2,\cdots,n$.
Let $\Lambda=\Lambda_{0}\oplus\Lambda_{1}$ be a finite-dimensional Grassmann
algebra, then the Banach space
$\mathbb{R}_{\Lambda}^{m,n}=\Lambda_{0}^{m}\times\Lambda_{1}^{n}$ is called a
superspace of dimension $(m,n)$ over $\Lambda$. In particular, if
$\Lambda_{0}=\mathbb{C}$ and $\Lambda_{1}=0$, then
$\mathbb{R}_{\Lambda}^{m,n}=\mathbb{C}^{m}.$ We may take even-valued complex
space $\Lambda_{0}=\mathbb{C}$ in our context.
A function
$f(\boldsymbol{x},\boldsymbol{\theta}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda$
is said to be superdifferentiable at the point
$(\boldsymbol{x},\boldsymbol{\theta})\in\mathbb{R}_{\Lambda}^{m,n}$ with even
coordinates $\boldsymbol{x}=(x_{1},\cdots,x_{m})$ and odd coordinates
$\boldsymbol{\theta}=(\theta_{1},\cdots,\theta_{n})$, if there exist elements
$F_{j}(\boldsymbol{x},\boldsymbol{\theta}),\widetilde{F}_{k}(\boldsymbol{x},\boldsymbol{\theta})\in\Lambda,\
j=1,\cdots,m;k=1,\cdots,n$, such that
$f(\boldsymbol{x}+\boldsymbol{h},\boldsymbol{\theta}+\boldsymbol{\widetilde{h}})=f(\boldsymbol{x},\boldsymbol{\theta})+\sum_{j=1}^{m}\langle
F_{j}(\boldsymbol{x},\boldsymbol{\theta}),h_{j}\rangle+\sum_{k=1}^{n}\langle\widetilde{F}_{k}(\boldsymbol{x},\boldsymbol{\theta}),\widetilde{h}_{k}\rangle+o(||(\boldsymbol{h},\boldsymbol{\widetilde{h}})||),$
where the vectors $\boldsymbol{h}=(h_{1},\cdots,h_{m})\in\Lambda_{0}^{m}$ and
$\boldsymbol{\widetilde{h}}=(\widetilde{h}_{1},\cdots,\widetilde{h}_{n})\in\Lambda_{1}^{n}$.
The
$F_{j}(\boldsymbol{x},\boldsymbol{\theta}),\widetilde{F}_{k}(\boldsymbol{x},\boldsymbol{\theta})$
are called the super partial derivative of $f$ with respect to
$x_{j},\theta_{k}$ at the point $(\boldsymbol{x},\boldsymbol{\theta})$ and are
denoted, respectively, by
$\frac{\partial f(\boldsymbol{x},\boldsymbol{\theta})}{\partial
x_{j}}=F_{j}(\boldsymbol{x},\boldsymbol{\theta}),\ \ \frac{\partial
f(\boldsymbol{x},\boldsymbol{\theta})}{\partial\theta_{k}}=F_{k}(\boldsymbol{x},\boldsymbol{\theta}),\
j=1,\cdots,m;k=1,\cdots,n.$
The derivatives $\frac{\partial
f(\boldsymbol{x},\boldsymbol{\theta})}{\partial x_{j}}$ with respect to even
variables $x_{j},\ j=1,2,\cdots m$ are uniquely defined. While the derivatives
$\frac{\partial f(\boldsymbol{x},\boldsymbol{\theta})}{\partial\theta_{k}}$ to
odd variables $\theta_{k},\ k=1,2,\cdots n$ are not uniquely defined, but with
an accuracy to within an addition constant
$c\theta_{1}\cdots\theta_{n},c\in\Lambda_{0}$ from the annihilator
${}^{\perp}L_{n}$,
$L_{n}=\\{\theta_{1}\cdots\theta_{n},\boldsymbol{\theta}\in\Lambda_{1}^{n}\\}$.
Let
$f=f(\boldsymbol{x},\boldsymbol{\theta}),g=g(\boldsymbol{x},\boldsymbol{\theta}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda$
be a superdifferentiable function, then super derivative also satisfies
Leibnitz formula
$\displaystyle\partial_{x_{j}}(fg)=(\partial_{x_{j}}f)g+f(\partial_{x_{j}}g),\
\ j=1,\cdots,m,$
$\displaystyle\partial_{\theta_{k}}(fg)=(\partial_{\theta_{k}}f)g+(-1)^{|f|}f(\partial_{\theta_{k}}g),\
\ k=1,\cdots,n.$
Let differential operators
$\mathcal{D}_{k}=\partial_{\theta_{k}}+\theta_{k}\partial_{x_{r}}$ (
$k=1,\cdots,n;r\in\\{1,\cdots,m\\}$ ) be supersymmetric covariant derivatives,
we can show that they satisfy
$None$
$\displaystyle\mathcal{D}_{k}(fg)=(\mathcal{D}_{k}f)g+(-1)^{|f|}f(\mathcal{D}_{k}g),$
$\displaystyle[\mathcal{D}_{j},\mathcal{D}_{k}]=0,\ \
\mathcal{D}_{k}^{2}=\partial_{x_{r}}.$
Denote by $\mathcal{P}(\Lambda_{1}^{n},\Lambda)$ the set of polynomials
defined on $\Lambda_{1}^{n}$ with value in $\Lambda$. We say that a super
integral is a map $I:\mathcal{P}(\Lambda_{1}^{n},\Lambda)\rightarrow\Lambda$
satisfying the following condition is an super Berezin integral about
Grassmann variables
(1) A linearity: $I(\mu f+\nu g)=\mu I(f)+\nu I(g),\ \mu,\nu\in\Lambda,\
f,g\in\mathcal{P}(\Lambda_{1}^{n},\Lambda);$
(2) translation invariance: $I(f_{\xi})=I(f)$, where
$f_{\xi}=f(\boldsymbol{\theta}+\boldsymbol{\xi})$ for all
$\boldsymbol{\xi}\in\Lambda_{1}^{n}$,
$f\in\mathcal{P}(\Lambda_{1}^{n},\Lambda).$
We denote $I(\theta^{\varepsilon})=I_{\varepsilon}$, where $\varepsilon$
belongs to the set of multiindices
$N_{n}=\\{\boldsymbol{\epsilon}=(\varepsilon_{1},\cdots,\varepsilon_{n}),\varepsilon_{j}=0,1,\boldsymbol{\theta}^{\varepsilon}=\theta_{1}^{\varepsilon_{1}}\cdots\theta_{n}^{\varepsilon_{n}}\not\equiv
0\\}$. In the case when $I_{\varepsilon}=0,\varepsilon\in
N_{n},|\varepsilon|\leq n=n-1$, such kind of integral has the form
$I(f)=J(f)I(1,\cdots,1)\equiv\frac{\partial^{n}f(0)}{\partial\theta_{1}\cdots\partial\theta_{n}}I(1,\cdots,1),$
Since the derivative is defined with an accuracy to with an additive constant
form the annihilator ${}^{\perp}L_{n}$, it follows that
$J:\mathcal{P}\rightarrow\Lambda/^{\perp}L_{n}$ is single-valued mapping. This
mapping also satisfies the conditions 1 and 2, and therefore we shall call it
an integral and denote
$J(f)=\int
f(\boldsymbol{\theta})d\boldsymbol{\theta}=\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots
d\theta_{n},$
which has properties:
$None$ $\displaystyle\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots
d\theta_{n}=1,$ $\displaystyle\int\frac{\partial
f}{\partial\theta_{j}}d\theta_{1}\cdots d\theta_{n}=0,\ j=1,\cdots,n.$
$\displaystyle\int f(\boldsymbol{\theta})\frac{\partial
g(\boldsymbol{\theta})}{\partial\theta_{j}}d\boldsymbol{\theta}=(-1)^{1+|g|}\int\frac{\partial
f(\boldsymbol{\theta})}{\partial\theta_{j}}g(\boldsymbol{\theta})d\boldsymbol{\theta}.$
For a pair of Grassmann-valued functions
$f(\boldsymbol{x},\boldsymbol{\theta}),\
g(\boldsymbol{x},\boldsymbol{\theta}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda$,
the ordinary Hirota bilinear operator is defined by
$\displaystyle D_{x_{j}}f(\boldsymbol{x},\boldsymbol{\theta})\cdot
g(\boldsymbol{x},\boldsymbol{\theta})=(\partial_{x_{j}}-\partial_{x_{j}^{\prime}})f(\boldsymbol{x},\boldsymbol{\theta})g(\boldsymbol{x}^{\prime},\boldsymbol{\theta}^{\prime})|_{\boldsymbol{x^{\prime}=x,\theta^{\prime}=\theta}},\
j=1,\cdots,m,$
and super-Hirota bilinear operators are defined as
$\displaystyle S_{k}D_{x_{j}}f(\boldsymbol{x},\boldsymbol{\theta})\cdot
g(\boldsymbol{x},\boldsymbol{\theta})=(\mathcal{D}_{k}-\mathcal{D}_{k}^{\prime})f(\boldsymbol{x},\boldsymbol{\theta})g(\boldsymbol{x}^{\prime},\boldsymbol{\theta}^{\prime})|_{\boldsymbol{x^{\prime}=x,\theta^{\prime}=\theta}},\
\ k=1,\cdots,n,$
here we have denoted
$\mathcal{D}_{k}^{\prime}=\partial_{\theta^{\prime}}+\theta^{\prime}\partial_{x}$.
It can be shown that these super-Hirota bilinear operators have properties
$\displaystyle S_{k}^{2N}f\cdot g=D_{x_{r}}^{N}f\cdot g,\ \ N\in\mathbb{Z}$
$\displaystyle S_{k}f\cdot
g=(\mathcal{D}_{k}f)g-(-1)^{|f|}f(\mathcal{D}_{k}g).$
In our context, we are interested in bosonic, also called even, superfield
function
$f(\boldsymbol{x},\boldsymbol{\theta}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\mathbb{R}_{\Lambda}^{1,0}=\Lambda_{0}$.
It can be expanded in powers of odd coordinates $\theta_{k},\ k=1,\cdots,n$,
that is,
$f=f_{0}(\boldsymbol{x})+\sum_{k\geq 0}\sum_{j_{1}<\cdots<j_{k}}f_{j_{1}\cdots
j_{k}}(\boldsymbol{x})\theta_{j_{1}}\cdots\theta_{j_{k}},$
where the coefficients $f_{j_{1}\cdots j_{k}}(\boldsymbol{x})\in\Lambda_{0}$
are even functions with respect to $x_{1},\cdots,x_{m}$.
3\. Generalized super Bell polynomials on superspace
Based on the above fundamental notations, in this section we develop theory of
generalized super Bell polynomials, which are a main tool to study the
integrability of supersymmetric equations.
2.1. Generalized super Bell polynomials
To well compare our super Bell polynomials with ordinary ones, let’s first
simply recall the rdinary Bell polynomials. During the early 1930s, Bell
introduced three kinds of exponential polynomials [23].
The first Bell polynomials are defined as
$None$ $\xi_{n}(x,t,r)=e^{-tx^{r}}\partial_{x}^{n}e^{tx^{r}},$
where $r>0$ is a constant integer, $n\geq 0$ an arbitrary integer, and
$x,t\in\mathbb{R}$ independent variables. For $r=2$, the Bell polynomials
$\xi_{n}(x,t)$ are exactly Hermite polynomials.
The Bell polynomials are algebraic polynomials in two elements $x$ and $t$.
The first few lowest order Bell Polynomials are
$\displaystyle\xi_{0}(x,t,r)=1,\ \ \xi_{1}(x,t,r)=rtx^{r-1},\ \
\xi_{2}(x,t,r)=r^{2}t^{2}x^{2r-2}+r(r-1)tx^{r-2},$
$\displaystyle\xi_{3}(x,t,r)=r^{3}t^{3}x^{3r-3}+3r^{2}(r-1)t^{2}x^{2r-3}+r(r-1)(r-2)tx^{r-3}.$
The second Bell polynomials are a generalization of the Bell polynomials (3.1)
and defined by
$\phi_{n}=\phi(\alpha_{1},\cdots,\alpha_{n}),\ \ \phi_{0}=1,\ \
\phi_{n+1}=\sum_{s=1}^{n}\left(\begin{matrix}n\cr
s\end{matrix}\right)\alpha_{s+1}\phi_{n-s},$
where $(\alpha_{1},\cdots,\alpha_{n},\cdots)$ is an infinite sequence of
independent variables. For the particular sequence
$\alpha_{j}=j!\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)xt^{r-j},\ j=1,\cdots,r;\ \alpha_{j}=0,\ j>r$, we have
$\phi_{n}=\xi_{n}(x,t,r).$
The Bell polynomial $\phi_{n}$ is a polynomial about variables
$\alpha_{1},\cdots,\alpha_{n}$. For instance, the first three of second Bell
polynomials read
$\displaystyle\phi_{0}=1,\ \ \phi_{1}=\alpha_{1}^{2}+\alpha_{2},\ \
\phi_{3}=\alpha_{1}^{3}+3\alpha_{1}\alpha_{2}+\alpha_{3}.$
The third Bell polynomials, further generalization of the $\xi_{n}$ and
$\phi_{n}$, are defined by
$None$ $\displaystyle
Y_{n}=Y_{n}(y_{t},\cdots,y_{nt})=e^{-y}\partial_{t}^{n}e^{y},$
where $y=e^{\alpha t}-\alpha_{0}\equiv\alpha_{1}t+\alpha_{2}t^{2}/2!+\cdots$,
and we have denoted derivative notation $y_{kt}=\partial_{t}^{k}y$. For the
spacial case when $\alpha_{r}=r!x,\ \alpha_{k}=0,\ k\not=r$, then we have
$Y_{n}=\xi_{n}(x,t,r).$
The polynomials (3.3) are polynomials about the derivatives of function $y$,
for example, the first three are
$\displaystyle Y_{0}=1,\ \ Y_{1}=y_{t},\ Y_{2}=y_{t}^{2}+y_{2t},\ \
Y_{3}=y_{t}^{3}+3y_{t}y_{2t}+y_{3t},$
More recently Lambert et al generalized the third Bell polynomials as
$None$ $\displaystyle
Y_{n}=e^{-y}\partial_{x_{1}}^{n_{1}}\cdots\partial_{x_{m}}^{n_{m}}e^{y},$
where $y=y(x_{1},\cdots,x_{m}):\mathbb{R}^{m}\rightarrow\mathbb{R}$ [32]-[34].
We now propose the following multi-dimensional and super extension to the
ordinary Bell polynomials (3.1)-(3.3).
Definition 1. Let
$f=f(\boldsymbol{x},\boldsymbol{\theta}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda_{0}$
be a superdifferential bosonic function, the generalized super Bell
polynomials (super $Y$-polynomials) is defined as follows
$None$
$Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(f)=Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}]\equiv
e^{-f}\mathcal{D}_{1}\cdots\mathcal{D}_{n}\partial_{x_{1}}^{\ell_{1}}\cdots\partial_{x_{m}}^{\ell_{m}}e^{f},$
where $\ell_{j}\geq 0,\ j=1,\cdots,m$ denote arbitrary integers. To make
subscript in expressions simple, we use some abbreviation notations in our
context, for example,
$\displaystyle\boldsymbol{\ell}=(\ell_{1},\cdots,\ell_{m}),\ \
\boldsymbol{\theta}=(\theta_{1},\cdots,\theta_{n}),\ \
\boldsymbol{\ell}\cdot\boldsymbol{x}=(\ell_{1}x_{1},\cdots,\ell_{m}x_{m}),$
$\displaystyle\boldsymbol{r}=(r_{1},\cdots,r_{m}),\ \
\boldsymbol{r}\cdot\boldsymbol{x}=(r_{1}x_{1},\cdots,r_{m}x_{m}),$
$\displaystyle\boldsymbol{\mu}=(\mu_{1},\cdots,\mu_{n}),\ \
\boldsymbol{\mu}\cdot\boldsymbol{\theta}=(\mu_{1}\theta_{1},\cdots,\mu_{n}\theta_{n}).$
Remark 1. The first notation
$Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(f)$ in (3.4)
denotes the $\ell_{j}$-order derivatives of $f$ with respect to the variable
$x_{j},j=1,\cdots,m$ and covariant derivatives with respect to ${\theta_{k}},\
k=1,\cdots,n$. The second notation
$Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}]$
implies that the super Bell polynomials (3.4) should be understood as such a
multivariable differential polynomial with respect to partial derivatives
$f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}$
( $r_{j}=0,\cdots,\ell_{j},\ j=1,\cdots,m,\mu_{k}=0,1,k=1,\cdots,n$), but not
variable elements $x_{j},\theta_{k}\ (j=1,\cdots,m;k=1,\cdots,n)$ as ordinary
Bell polynomials. For instance, the ${Y}_{3x}(f)$ in the next example is a
polynomial $Y_{3x}(f_{x},f_{2x},f_{3x})$ with respect to three variable
elements $f_{x},f_{2x},f_{3x}$.
To better understanding our generalized super Bell polynomials, let us see an
illustrative example. For the special case $f=f(x,\theta_{1},\theta_{2})$, the
associated super Bell polynomials defined by (3.4) read
$\displaystyle{Y}_{x}(f)=f_{x},\ \ \ {Y}_{2x}(f)=f_{2x}+f_{x}^{2},$
$\displaystyle{Y}_{3x}(f)=f_{3x}+3f_{x}f_{2x}+f_{x}^{3},\ \
Y_{\theta_{1}}(f)=\mathcal{D}_{1}f,$ $\displaystyle
Y_{\theta_{1}\theta_{2}}(f)=\mathcal{D}_{1}\mathcal{D}_{2}f+(\mathcal{D}_{1}f)\mathcal{D}_{2}f,\
\ {Y}_{x,\theta_{1}}(f)=\mathcal{D}_{1}f_{x}+f_{x}\mathcal{D}_{1}f,$
$\displaystyle{Y}_{2x,\theta_{1}}(f)=f_{2x}\mathcal{D}_{1}f+\mathcal{D}_{1}f_{2x}+f_{x}^{2}\mathcal{D}_{1}f+2f_{x}\mathcal{D}_{1}f_{x},$
$\displaystyle{Y}_{3x,\theta_{1}}(f)=f_{3x}\mathcal{D}_{1}f+3f_{x}f_{2x}\mathcal{D}_{1}f+3f_{2x}\mathcal{D}_{1}f_{x}+3f_{x}\mathcal{D}_{1}f_{2x}+3f_{x}^{2}\mathcal{D}_{1}f_{x}+\mathcal{D}_{1}f_{3x}.$
Let see the relations between our generalized super polynomials and ordinary
Bell polynomials, as well as ordinary generalized Bell polynomials.
For the special case $\mathbb{R}_{\Lambda}^{m,n}=\mathbb{R}^{2},\ell_{2}=0$,
$f=f(x_{1},x_{2})=x_{2}x_{1}^{r}$ with the constant integer $r>0$, then (3.4)
reduces to the first Bell polynomials (3.1)
$Y_{\ell_{1}x_{1}}(f)=e^{{-x_{2}x_{1}^{r}}}\partial_{x_{1}}^{\ell_{1}}e^{{x_{2}x_{1}^{r}}}=\xi_{\ell_{1}}(x_{1},x_{2}).$
For the case $\mathbb{R}_{\Lambda}^{m,n}=\mathbb{R}^{m}$, the corresponding
generalized super Bell polynomials (3.4) degenerates to generalized Bell
polynomials (3.3) given by Lambert et al. The Bell polynomials admit
partitional representation [32]
$None$
$Y_{\boldsymbol{\ell}\cdot\boldsymbol{x}}(f)=\sum\frac{\ell_{1}!\cdots\ell_{m}!}{c_{1}!\cdots
c_{k}!}\prod_{j=1}^{k}\left(\frac{f_{r_{1j},\cdots,r_{mj}}}{r_{1j}!\cdots
r_{mj}!}\right)^{c_{j}},$
where the sum is to taken over all partitions
$[(r_{j1},\cdots,r_{m1})^{c_{1}},\cdots,(r_{1k},\cdots,r_{mk})^{c_{k}}]$ the
$m$-tuple $(\ell_{1},\cdots,\ell_{m})$.
In following, we investigate properties of super Bell polynomials which are
key results to establish connects with supersymmetric equations.
Theorem 1. Under the Hopf-Cole transformation $f=\ln\psi$, the generalized
super Bell polynomials
$Y_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}(f)$
can be “linearized” into the form
$None$
$Y_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}(f)|_{f=\ln\psi}=\psi_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}/\psi.$
Proof. According to the definition (3.4), we have
$\displaystyle
Y_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}(f)|_{f=\ln\psi}=e^{-\ln\psi}\mathcal{D}_{1}^{\mu_{1}}\cdots\mathcal{D}_{n}^{\mu_{n}}\partial_{x_{1}}^{r_{1}}\cdots\partial_{x_{m}}^{r_{m}}e^{\ln\psi}=\psi_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}/{\psi},$
which finishes the proof of Theorem 1. $\square$
Remark 2. According to the theorem, under the Hopf-Cole transformation
$f=\ln\psi$, a equation in term of linear combination of super Bell
polynomials, i.e.
$\sum_{\boldsymbol{r},\boldsymbol{\mu}}C_{\boldsymbol{r},\boldsymbol{\mu}}(\boldsymbol{x},\boldsymbol{\theta})Y_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}(f)=0$
can be linearized into the form
$\sum_{\boldsymbol{r},\boldsymbol{\mu}}C_{\boldsymbol{r},\boldsymbol{\mu}}(\boldsymbol{x},\boldsymbol{\theta})\psi_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}=0,$
where
$C_{\boldsymbol{r},\boldsymbol{\mu}}(\boldsymbol{x},\boldsymbol{\theta})$ are
functions independent of the function $f$. This is a key property to construct
the Lax pair of supersymmetric equations.
Theorem 2. The super Bell polynomials (3.4) admit recursion formula
$None$
$Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(f)=\prod_{k=1}^{n}(\mathcal{D}_{k}+\mathcal{D}_{k}f)Y_{\boldsymbol{\ell}\cdot\boldsymbol{x}}(f).$
Proof. By the definition (3.4), direct computation leads to
$\displaystyle
Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(f)=\mathcal{D}_{1}Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\theta_{2},\cdots,\theta_{n}}(f)+(\mathcal{D}_{1}f)Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\theta_{2},\cdots,\theta_{n}}(f)$
$\displaystyle=(\mathcal{D}_{1}+\mathcal{D}_{1}f)Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\theta_{2},\cdots,\theta_{n}}(f).$
Similarly,
$\displaystyle
Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\theta_{2},\cdots,\theta_{n}}(f)=(\mathcal{D}_{2}+\mathcal{D}_{2}f)Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\theta_{3},\cdots,\theta_{n}}(f).$
Repeating the above arguments then proves the formula (3.7). $\square$
Theorem 3. The super Bell polynomials (3.4) possess parity property
$None$ $\displaystyle
Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[(-1)^{\sum
r_{j}+\sum\mu_{k}}f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}]=(-1)^{n+\sum\ell_{j}}Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}].$
Proof. From the recursion relation (3.7), it follows that
$None$ $\displaystyle
Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[(-1)^{\footnotesize\sum
r_{j}+\sum\mu_{k}}f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}]\stackrel{{\scriptstyle}}{{=}}\prod_{k=1}^{n}(-\mathcal{D}_{k}-\mathcal{D}_{k}f)Y_{\boldsymbol{\ell}\cdot\boldsymbol{x}}[(-1)^{\sum
r_{j}}f_{\boldsymbol{r}\cdot\boldsymbol{x}}]$
$\displaystyle=(-1)^{n}\prod_{k=1}^{n}(\mathcal{D}_{k}+\mathcal{D}_{k}f)Y_{\boldsymbol{\ell}\cdot\boldsymbol{x}}[(-1)^{\sum
r_{j}}f_{\boldsymbol{r}\cdot\boldsymbol{x}}].$
While applying the partitional representation (3.5), we have
$None$ $\displaystyle Y_{\boldsymbol{\ell}\cdot\boldsymbol{x}}[(-1)^{\sum
r_{j}}f_{\boldsymbol{r}\cdot\boldsymbol{x}}]\stackrel{{\scriptstyle}}{{=}}(-1)^{\sum\ell_{j}}Y_{\boldsymbol{\ell}\cdot\boldsymbol{x}}[f_{\boldsymbol{r}\cdot\boldsymbol{x}}].$
Hence, combing (3.7), (3.9) and (3.10) proves the formula (3.8). $\square$
Theorem 4. The super Bell polynomials (3.4) obey addition property
$None$ $\displaystyle
Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(f+g)=\sum_{\mu_{1},\cdots,\mu_{n}=0}^{1}(-1)^{\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot
n}\\}\setminus\\{0\\}]}\sum_{r_{1}=0}^{\ell_{1}}\cdots\sum_{r_{m}=0}^{\ell_{m}}\prod_{i=1}^{m}\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)$ $\displaystyle\times Y_{\boldsymbol{(\ell-r)\cdot
x},\boldsymbol{(1-\mu)\cdot\theta}}(f)Y_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}(g),$
where $\boldsymbol{n}=(1,2,\cdots,n)$; $\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot n}\\}\setminus\\{0\\}]$ denotes the reverse order
numbers of the $n$-order permutation $\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot n}\\}\setminus\\{0\\}$, which is generated from
anticommutation of covariant derivatives $\mathcal{D}_{j},\ j=1,\cdots,n$, and
obtained from a $2n$-order permutation
$\\{(1-\mu_{1})1,\cdots,(1-\mu_{n})n,\mu_{1}1,\cdots,\mu_{n}n\\}$ ($\mu_{j}=0$
or $1$) by taking off all zero terms. The $2n$-order permutation is the
subscript of corresponding covariant derivatives of term
$Y_{\boldsymbol{(\ell-r)\cdot
x},\boldsymbol{(1-\mu)\cdot\theta}}(f)Y_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}(g)$ kept in original order.
Proof. According to the commutative properties of covariant derivatives (3.1),
a minus sign in the Leibnitz rule exactly corresponds to an inverse order of
the $n$-order permutation $\\{1,2,\cdots,n\\}$. So direct computation shows
that
$\displaystyle(FG)^{-1}\mathcal{D}_{1}\cdots\mathcal{D}_{n}\partial_{x_{1}}^{\ell_{1}}\cdots\partial_{x_{m}}^{\ell_{m}}(FG)=\sum_{\mu_{1},\cdots,\mu_{n}=0}^{1}(-1)^{\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot
n}\\}\setminus\\{0\\}]}\sum_{r_{1}=0}^{\ell_{1}}\cdots\sum_{r_{m}=0}^{\ell_{m}}\prod_{i=1}^{m}\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)$
$\displaystyle\times\left(F\mathcal{D}_{1}^{1-\mu_{1}}\cdots\mathcal{D}_{n}^{1-\mu_{n}}\partial_{x_{1}}^{\ell_{1}-r_{1}}\cdots\partial_{x_{m}}^{\ell_{m}-r_{m}}F\right)\left(G\mathcal{D}_{1}^{\mu_{1}}\cdots\mathcal{D}_{n}^{\mu_{n}}\partial_{x_{1}}^{r_{1}}\cdots\partial_{x_{m}}^{r_{m}}G\right),$
which implies (3.11) by replacing $F=e^{f}$ and $G=e^{g}$. $\square$
Let $F,G:\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda_{0}$ be two bosonic
superdifferential functions, then direct computation yields
$\displaystyle\mathcal{D}_{1}\mathcal{D}_{2}\mathcal{D}_{3}(FG)=(\mathcal{D}_{1}\mathcal{D}_{2}\mathcal{D}_{3}F)G+(\mathcal{D}_{2}\mathcal{D}_{3}F)\mathcal{D}_{1}G-(\mathcal{D}_{1}\mathcal{D}_{3}F)\mathcal{D}_{2}G$
$\displaystyle+(\mathcal{D}_{2}F)(\mathcal{D}_{1}F)\mathcal{D}_{2}G+(\mathcal{D}_{1}\mathcal{D}_{2}F)\mathcal{D}_{3}G-(\mathcal{D}_{2}F)(\mathcal{D}_{1}F)\mathcal{D}_{3}G$
$\displaystyle+(\mathcal{D}_{1}F)(\mathcal{D}_{2}F)\mathcal{D}_{3}G+F(\mathcal{D}_{1}\mathcal{D}_{2}\mathcal{D}_{3}G),$
in which corresponding six even permutations are
$\\{1,2,3\\},\\{2,3,1\\},\\{3,1,2\\},\\{1,2,3\\}$, $\\{1,2,3\\},\\{1,2,3\\}$,
and two odd permutations are $\\{1,3,2\\},\\{2,1,3\\}$.
2.2. Generalized super binary Bell polynomials
We further define a class of super binary Bell polynomials which play an
important role in the study of integrability for supersymmetric equations.
Definition 2. Based on the use of above super Bell polynomials (3.4), the
super binary Bell polynomials ( $\mathcal{Y}$-polynomials) can be defined as
follows
$None$
$\mathcal{Y}_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(v,w)=Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}],$
in which we replace the function $f$ and its derivatives by corresponding
terms of functions $w$ and $v$ respectively, according the following rule
${f_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}=\left\\{\begin{matrix}v_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}},&{\rm
if}\ \sum_{j=1}^{m}r_{j}+\sum_{k=1}^{n}\mu_{k}\ {\rm is\ \ odd},\cr\cr
w_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}},&{\rm
if}\ \sum_{j=1}^{m}r_{j}+\sum_{k=1}^{n}\mu_{k}\ \ {\rm is\ \
even},\end{matrix}\right.}$
The super binary Bell polynomials (3.6) is multi-variable polynomials with
respect to various partial derivatives
$v_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}$
and
$w_{\boldsymbol{r}\cdot\boldsymbol{x},\boldsymbol{\mu}\cdot\boldsymbol{\theta}}$,
$r_{j}=0,\cdots,\ell_{j},\ j=0,\cdots,m,\mu_{k}=0,1,k=1,\cdots,n$.
The super binary Bell polynomials also inherit the easily recognizable partial
structure of the super Bell polynomials. The first few are explicitly
calculated as
$None$ $\displaystyle\mathcal{Y}_{x}(v)=v_{x},\ \
\mathcal{Y}_{2x}(v,w)=w_{2x}+v_{x}^{2},\ \
\mathcal{Y}_{3x}(v,w)=v_{3x}+3v_{x}w_{2x}+v_{x}^{3},$
$\displaystyle\mathcal{Y}_{\theta_{1}}(v)=\mathcal{D}_{1}v,\ \
\mathcal{Y}_{\theta_{1}\theta_{2}}(w,v)=\mathcal{D}_{1}\mathcal{D}_{2}w+(\mathcal{D}_{1}v)\mathcal{D}_{2}v,$
$\displaystyle\mathcal{Y}_{x,\theta_{1}}(v,w)=\mathcal{D}_{1}w_{x}+v_{x}\mathcal{D}_{1}v,$
$\displaystyle\mathcal{Y}_{2x,\theta_{1}}(v,w)=w_{2x}\mathcal{D}_{1}v+\mathcal{D}_{1}v_{2x}+v_{x}^{2}\mathcal{D}_{1}v+2v_{x}\mathcal{D}_{1}w_{x},$
$\displaystyle\mathcal{Y}_{3x,\theta_{1}}(v,w)=v_{3x}\mathcal{D}_{1}v+3v_{x}w_{2x}\mathcal{D}_{1}v+3w_{2x}\mathcal{D}_{1}w_{x}+3v_{x}\mathcal{D}_{1}v_{2x}$
$\displaystyle+3v_{x}^{2}\mathcal{D}_{1}w_{x}+\mathcal{D}_{1}w_{3x}.$
We denote the special case of super Bell polynomials by
$\mathcal{Y}_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(v=0,w)=P_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(w)$,
then it follows from (3.13) that
$None$ $\displaystyle P_{2x}(w)=w_{2x},\ \ P_{4x}(w)=w_{4x}+3w_{2x}^{2},\ \
P_{\theta_{1},\theta_{2}}(w)=\mathcal{D}_{1}\mathcal{D}_{2}w,$ $\displaystyle
P_{x,\theta_{1}}(w)=\mathcal{D}_{1}w_{x},\ \
P_{3x,\theta_{1}}(w)=\mathcal{D}_{1}w_{3x}+3w_{2x}\mathcal{D}_{1}w_{x},\cdots.$
Theorem 5. The link between super binary Bell polynomials
$\mathcal{Y}_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(v,w)$
and the super Hirota bilinear equation $S_{1}\cdots
S_{n}D_{x_{1}}^{\ell_{1}}\cdots D_{x_{m}}^{\ell_{m}}F\cdot G$ can be
established by an identity
$None$
$\displaystyle\mathcal{Y}_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(v=\ln
F/G,w=\ln FG)=(FG)^{-1}S_{1}\cdots S_{n}D_{x_{1}}^{\ell_{1}}\cdots
D_{x_{m}}^{\ell_{m}}F\cdot G.$
This formula will be sued to obtain bilinear Bäcklund transformations of
supersymmetric equations.
Proof. Let $f=\ln F,\ g=\ln G$, then we have $v=f-g,w=f+g$. It follows from
the definition 2 that
$\displaystyle\mathcal{Y}_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(v=\ln
F/G,w=\ln
FG)=Y_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}[f_{\boldsymbol{s}\cdot\boldsymbol{x},\boldsymbol{\tilde{\mu}}\boldsymbol{\theta}}+(-1)^{\sum
s_{i}+\sum\tilde{\mu}_{j}}g_{\boldsymbol{s}\cdot\boldsymbol{x},\boldsymbol{\tilde{\mu}}\boldsymbol{\theta}}]$
$\displaystyle\stackrel{{\scriptstyle(3.11)}}{{=}}\sum_{\mu_{1},\cdots,\mu_{n}=0}^{1}(-1)^{\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot
n}\\}/0]}\sum_{r_{1}=0}^{\ell_{1}}\cdots\sum_{r_{m}=0}^{\ell_{m}}\prod_{i=1}^{m}\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)$ $\displaystyle\ \ \ \ \times
Y_{\boldsymbol{(\ell-r)\cdot
x},\boldsymbol{(1-\mu)\cdot\theta}}[f_{\boldsymbol{s}\cdot\boldsymbol{x},\boldsymbol{\tilde{\mu}}\boldsymbol{\theta}}]Y_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}[(-1)^{\sum
s_{i}+\sum\tilde{\mu}_{j}}g_{\boldsymbol{s}\cdot\boldsymbol{x},\boldsymbol{\tilde{\mu}}\boldsymbol{\theta}}]$
$\displaystyle\stackrel{{\scriptstyle(3.8)}}{{=}}\sum_{\mu_{1},\cdots,\mu_{n}=0}^{1}(-1)^{\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot
n}\\}/0]+\sum_{j=1}^{m}r_{j}+\sum_{k=1}^{n}{\mu}_{k}}\sum_{r_{1}=0}^{\ell_{1}}\cdots\sum_{r_{m}=0}^{\ell_{m}}\prod_{i=1}^{m}\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)$ $\displaystyle\ \ \ \ \times
Y_{\boldsymbol{(\ell-r)\cdot
x},\boldsymbol{(1-\mu)\cdot\theta}}(f)Y_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}(g)$ $\displaystyle=(FG)^{-1}S_{1}\cdots
S_{n}D_{x_{1}}^{\ell_{1}}\cdots D_{x_{m}}^{\ell_{m}}F\cdot G.$
$\square$
For the particular case when $F=G$, the formula (3.12) reduces to
$None$ $\displaystyle G^{-2}S_{1}\cdots S_{n}D_{x_{1}}^{\ell_{1}}\cdots
D_{x_{m}}^{\ell_{m}}G\cdot
G=\mathcal{Y}_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(0,w=2\ln
G)$ $\displaystyle=\left\\{\begin{matrix}0,&n+\sum_{j=1}^{m}\ell_{j}\ \ {\rm
is\ \ odd},\cr\cr
P_{\boldsymbol{\ell}\cdot\boldsymbol{x},\boldsymbol{\theta}}(w),&n+\sum_{j=1}^{m}\ell_{j}\
\ {\rm is\ \ even},\end{matrix}\right.$
which implies that the $P$-polynomials can be characterized by an equally
recognizable even part partitional structure. The formulae (3.15) and (3.16)
will prove particularly useful in connecting supersymmetric equations with
their corresponding super bilinear equations. Once a nonlinear equation is
expressible as a linear combination of super Bell $\mathcal{Y}$-polynomials or
$P$-polynomials, then it can be transformed into a super linear equation.
Theorem 6. The super binary Bell polynomials
$\mathcal{Y}_{\boldsymbol{\ell\cdot x},\boldsymbol{\theta}}(v,w)$ can be
separated into super $P$-polynomials and super Bell $Y$-polynomials
$None$ $\displaystyle\mathcal{Y}_{\boldsymbol{\ell\cdot
x},\boldsymbol{\theta}}(v,w)=\sum_{\mu_{1},\cdots,\mu_{n}=0}^{1}(-1)^{\large\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot
n}\\}/0]}\sum_{r_{1}=0}^{\ell_{1}}\cdots\sum_{r_{m}=0}^{\ell_{m}}\prod_{i=1}^{m}\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)$ $\displaystyle\ \ \ \times P_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}(w-v)Y_{\boldsymbol{(\ell-r)\cdot
x},\boldsymbol{(1-\mu)\cdot\theta}}(v),$
where only non-vanishing contributions being those for which $\sum
r_{j}+\sum\mu_{k}$ is even integer.
Proof. According Definition 2 of the super Bell polynomials, we have
$\mathcal{Y}_{\boldsymbol{p\cdot x},\boldsymbol{\nu\cdot\theta}}(v=0,w)=0,\ \
{\rm as}\ \ \sum p_{j}+\sum\nu_{k}\ {\rm is\ odd},$
so that by using Theorem 4,
$None$ $\displaystyle\mathcal{Y}_{\boldsymbol{\ell\cdot
x},\boldsymbol{\theta}}(v,w)=\mathcal{Y}_{\boldsymbol{\ell\cdot
x},\boldsymbol{\theta}}(v,v+q)={Y}_{\boldsymbol{\ell\cdot
x},\boldsymbol{\theta}}(v+q)|_{\large q_{\boldsymbol{p\cdot
x},\boldsymbol{\nu\cdot\theta}}=0}$
$\displaystyle=\sum_{\mu_{1},\cdots,\mu_{n}=0}^{1}(-1)^{\large\tau[\\{\boldsymbol{(1-\mu)\cdot
n},\boldsymbol{\mu\cdot
n}\\}/0]}\sum_{r_{1}=0}^{\ell_{1}}\cdots\sum_{r_{m}=0}^{\ell_{m}}\prod_{i=1}^{m}\left(\begin{matrix}\ell_{i}\cr
r_{i}\end{matrix}\right)$ $\displaystyle\times\ \ \ Y_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}(q)Y_{\boldsymbol{(\ell-r)\cdot
x},\boldsymbol{(1-\mu)\cdot\theta}}(v)|_{\large q_{\boldsymbol{p\cdot
x},\boldsymbol{\nu\cdot\theta}}=0},$
where $q=w-v$, the sum $\sum p_{j}+\sum\nu_{k}$ is odd integer.
Substituting the relation
$\displaystyle Y_{\boldsymbol{r\cdot
x},\boldsymbol{\mu\cdot\theta}}(q)|_{q_{\boldsymbol{p\cdot
x},\boldsymbol{\nu\cdot\theta}}=0,\ \ \sum p_{j}+\sum\nu_{k}\ {\rm is\
odd}}=P_{\boldsymbol{r\cdot x},\boldsymbol{\mu\cdot\theta}}(q)|_{\sum
p_{j}+\sum\nu_{k}\ {\rm is\ even}},$
into (3.18) then leads to the formula (3.17). $\square$
This theorem implies that the super binary Bell polynomials (3.12) can still
be “linearized” by means of the Hopf-Cole transformation $v=\ln\psi$,
$\psi=F/G$. The formulae (3.6) and (3.17) will then provide a way to find the
associated Lax system of supersymmetric equations.
Finally, let’s through a graph describe general procedure how to use the
theory of super Bell polynomials that we have developed above.
$\begin{array}[]{ccccc}&&{\rm SUSY\ equation:}\ {\footnotesize
F(\Phi)=0}&&\\\\[4.0pt] &&\vbox{ \hbox to0.0pt{\hss }
}{\Big{\downarrow}}\vbox{ \hbox to0.0pt{ {{\footnotesize\rm dimensionless\
field:\ }q} \hss} }&&\\\\[4.0pt] {}\hfil&&{\rm Bell}\ P(q)\ {\rm\
system:}{\footnotesize E(q)=0}&\xrightarrow[]{q=2\ln G}&{{\bf\small Bilinear\
form}}\\\\[4.0pt] &&\vbox{ \hbox to0.0pt{\hss } }{\Big{\downarrow}}\vbox{
\hbox to0.0pt{ {{\footnotesize\rm Two-fold condition }}\hss} }&&\\\\[8.0pt]
&&{}\ E(\widetilde{q})-E(q)=0&&\\\\[4.0pt] &&\vbox{ \hbox to0.0pt{\hss }
}{\Big{\downarrow}}\vbox{ \hbox to0.0pt{ {{\footnotesize constaint}} \hss}
}&&\\\\[4.0pt] {{\bf\small Binlear\ BT}}&\xleftarrow[w=\ln FG]{v=\ln F/G}&{\rm
Binary\ Bell}\ \ {\footnotesize\mathcal{Y}(v,w)}\ {\rm\
system}&\xrightarrow[]{{{v=\ln\psi}}}&{{\bf\small Lax\ pair}}\\\\[4.0pt]
&&\vbox{ \hbox to0.0pt{\hss{\footnotesize} } }{\Big{\downarrow}}\vbox{ \hbox
to0.0pt{ {{\footnotesize$w=h_{1}(\eta,q),\ \ v=h_{2}(\eta,q)$}}\hss}
}&&\\\\[4.0pt] &&{{\bf\small Infinite\ conservation\ laws}}&&\\\\[4.0pt]
\end{array}$
It is clear from this graph to see the close connections among Bell
polynomials with bilinear equation, bilinear Bäcklund transformation, Lax pair
and conservation laws.
4\. The supersymmetric KdV equation
Consider the supersymmetric KdV equation of Manin-Radul-Mathieu [5, 7]
$None$
$\displaystyle\Phi_{t}+3\left(\Phi\mathcal{D}\Phi\right)_{x}+\Phi_{3x}=0,$
where $\Phi=\Phi(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda_{1}$
is a fermionic super field function with independent variables $x$, $t$ and
Grassmann variable $\theta$. The symbol
$\mathcal{D}=\partial_{\theta}+\theta\partial_{x}$ denotes the super
derivative differential operator, which satisfies
$\mathcal{D}^{2}=\partial_{x},\ \ \theta^{2}=0$. The supersymmetric version of
the KdV equation (4.1) describe the time evolution of a Grassmann-valued
superfield $\Phi(x,t,\theta)=\tilde{u}(x,t)+\theta u(x,t)$, where $u(x,t)$ is
an ordinary function and $\tilde{u}(x,t)$ is a Grassmann valued function. The
variable $x,t$ acquire a Grassmann partner $\theta$, so $(x,t,\theta)$ are
coordinates in a one dimensional superspace $\mathbb{R}_{\Lambda}^{2,1}$.
Since the introduction of the supersymmetric KdV equation (1.1) by Manin,
Radul and Mathieu [5, 7], much attention has been given to its mathematical
structure and integrable properties. For instances, bi-Hamiltonian structure,
Painlevé property, infinite many symmetries, Darboux transformation, Bäcklund
transformation, bilinear form, super soliton solutions and super quasi-
periodic solutions had been investigated in [11]–[22]. Here we see how to
apply the super polynomials to investigate complete integrability of the
supersymmetric KdV equation (4.1).
Theorem 7. Under the transformation $\Phi=2\mathcal{D}(\ln G)_{x}$, the
supersymmetric KdV equation (4.1) can be bilinearized into
$None$ $(SD_{t}+SD_{x}^{3})G\cdot G=0.$
Proof. The invariance of the equation (4.1) under the scale transformation
$x\rightarrow\lambda x,\ \ t\rightarrow\lambda^{3}t,\ \
\theta\rightarrow\lambda^{{1}/{2}}\theta,\ \
\Phi\rightarrow\lambda^{-3/2}\Phi$
shows that the dimension of the fermionic field $\Phi$ is $-3/2$, and it can
be related to a dimensionless bosonic field
$q:\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda_{0}$, by setting
$None$ $\Phi=c\mathcal{D}q_{x},$
with $c\in\Lambda_{0}$ being free function to be the appropriate choice such
that the equation (4.1) connects with $P$-polynomials. Substituting (4.3) into
(4.1) and integrating with respect to $x$ yields
$None$
$E(q)\equiv\mathcal{D}q_{t}+\mathcal{D}q_{3x}+3cq_{2x}\mathcal{D}_{\theta}q_{x}=0.$
Comparing the last two terms of this equation with the formula (3.16) implies
that we should require $c=1$. The equation (4.4) is then cast into a
combination form of $P$-polynomials
$None$ $E(q)=P_{t,\theta}(q)+P_{3x,\theta}(q)=0.$
Making a change of dependent variable
$q=2\ln G,\ \ \Longleftrightarrow\ \ \Phi=2\mathcal{D}(\ln G)_{x}$
with $G:\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda_{0}$, then the property
(3.16) shows that the equation (4.5) is equivalent to the bilinear equation
(4.2). $\square$
Starting from the bilinear equation (4.2), it is easy to get super soliton
solutions. For example, the regular one-soliton like solution reads
$\displaystyle\Phi=\mathcal{D}[\ln(1+\exp(kx-k^{3}t+\theta\zeta)]_{x},$
where $k\in\Lambda_{0},\zeta\in\Lambda_{1}$. Since solving the equation (4.1)
is not our main purpose in this paper, the super soliton solutions can be
found in details [21].
Next, we search for the bilinear Bäcklund transformation and Lax pair of the
supersymmetric KdV equation (4.1).
Theorem 8. Let $F$ be a solution of the equation (4.2), then $G$ satisfying
$None$ $\displaystyle(SD_{x}-\lambda S)F\cdot G=0,$
$\displaystyle(D_{t}+D_{x}^{3}+3\lambda^{2}D_{x}-3\lambda D_{x})F\cdot G=0$
is another solution of the equation (4.2). This kind of Bäcklund
transformation is exactly the same with that given by Liu [18]
Proof. Let $q=2\ln G,\ \widetilde{q}=2\ln
F:\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda_{0}$ be two different solutions
of the equation (4.4), respectively, we associate the two-field condition
$None$ $\displaystyle
E(\widetilde{q})-E(q)=\mathcal{D}(\widetilde{q}-q)_{t}+\mathcal{D}(\widetilde{q}-q)_{3x}+3\widetilde{q}_{2x}\mathcal{D}\widetilde{q}_{x}-3{q}_{2x}\mathcal{D}{q}_{x}=0.$
This two-field condition can be regarded as a ansatz for a bilinear Bäcklund
transformation and may produce the required transformation under appropriate
additional constraints.
To find such constraints, we introduce two new variables
$None$ $v=(\widetilde{q}-q)/2=\ln F/G,\ \ w=(\widetilde{q}+q)/2=\ln FG,$
and rewrite the condition (4.6) into the form
$None$ $\displaystyle
E(\widetilde{q})-E(q)=2\mathcal{D}v_{t}+2\mathcal{D}v_{3x}+6v_{2x}\mathcal{D}w_{x}+6w_{2x}\mathcal{D}v_{x}$
$\displaystyle=2\mathcal{D}[\mathcal{Y}_{t}(v)+\mathcal{Y}_{3x}(v,w)]+6R(v,w)=0,$
with
$R(v,w)=v_{2x}\mathcal{D}w_{x}-v_{x}\mathcal{D}w_{2x}-v_{x}^{2}\mathcal{D}v_{x}={\rm
Wronskian}[\mathcal{Y}_{x,\theta}(v,w),\mathcal{Y}_{x}(v)].$
In order to decouple the two-field condition (4.8) into a pair of constraints,
we impose such a constraint which enable us to express $R(v,w)$ as the
$\mathcal{D}$-derivative of a combination of $\mathcal{Y}$-polynomials. A
possible choice of such constraint may be
$None$ $\mathcal{Y}_{x,\theta}(v,w)=\lambda\mathcal{Y}_{\theta}(v),$
where $\lambda\in\Lambda_{0}$ is an arbitrary parameter. It follows from the
identity (4.9) that
$(v_{x}\mathcal{D}v)_{x}=\lambda\mathcal{D_{\theta}}v_{x}-\mathcal{D}w_{2x},$
on account which, then $R(v,w)$ can be rewritten in the form
$None$ $\displaystyle R(v,w)=\lambda(v_{x}\mathcal{D}v)_{x}-2\lambda
v_{x}\mathcal{D}v_{x}=\lambda^{2}\mathcal{D}v_{x}-\lambda\mathcal{D}w_{2x}-2\lambda
v_{x}\mathcal{D}v_{x}$
$\displaystyle=\mathcal{D}[\lambda^{2}\mathcal{Y}_{x}(v)-\lambda\mathcal{Y}_{2x}(v,w)].$
Then from (4.7)-(4.10), we deduce a coupled system of super binary Bell
$\mathcal{Y}$-polynomials
$None$
$\displaystyle\mathcal{Y}_{x,\theta}(v,w)-\lambda\mathcal{Y}_{\theta}(v)=0,$
$\displaystyle\mathcal{Y}_{t}(v)+\mathcal{Y}_{3x}(v,w)+3\lambda^{2}\mathcal{Y}_{x}(v)-3\lambda\mathcal{Y}_{2x}(v,w)=0.$
By application of the identity (3.15), under transformation $v=\ln F/G,w=\ln
FG$, the system (4.11) then leads to the bilinear Bäcklund transformation
(4.5). $\square$
Theorem 9. The supersymmetric KdV equation (4.1) admits a Lax pair
$None$
$\displaystyle(\partial_{x}^{2}+\Phi\mathcal{D}-\lambda\partial_{x})\varphi=0,$
$\displaystyle[\mathcal{D}\partial_{t}+\mathcal{D}\partial_{x}^{3}-3\lambda\mathcal{D}\partial_{x}^{2}+3(\mathcal{D}\Phi+\lambda)\mathcal{D}+(\mathcal{D}\Phi)\mathcal{D}]\varphi=0,$
where $\varphi:\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda_{1}$ is a
fermionic eigenfunction.
Proof. By transformation $v=\ln\psi$, it follows from the formulae (3.6) and
(3.17) that
$\displaystyle\mathcal{Y}_{\theta}(v)=\mathcal{D}\psi/\psi,\ \
\mathcal{Y}_{x,\theta}(v,w)=\mathcal{D}q_{x}+\mathcal{D}\psi_{x}/\psi,$
$\displaystyle\mathcal{Y}_{t}(v)=\psi_{t}/\psi,\ \
\mathcal{Y}_{2x}(v,w)=q_{2x}+\psi_{2x}/\psi,\ \
\mathcal{Y}_{3x}(v,w)=3q_{2x}\psi_{x}/\psi+\psi_{3x}/\psi,$
on account of which, the system (4.12) is then linearized into a Lax pair with
a parameter $\lambda$
$\displaystyle
L_{1}\psi\equiv(\mathcal{D}\partial_{x}-\lambda\mathcal{D}+\mathcal{D}q_{x})\psi=0,$
$\displaystyle
L_{2}\psi\equiv(\partial_{t}+\partial_{x}^{3}+3q_{2x}\partial_{x}+3\lambda^{2}\partial_{x}-3\lambda\partial^{2}+q_{2x})\psi=0,$
which is equivalent to the formula (4.12) b by replacing $\mathcal{D}q_{x}$
with $\Phi$, and $\psi$ with $\mathcal{D}\varphi$. It is easy to check that
the integrability condition of the Lax pair
$[L_{1},L_{2}]\psi=0$
is satisfied if $\Phi$ is a solution of the supersymmetric KdV equation (4.1).
$\square$
Finally, we show how to derive the infinite conservation laws for super KdV
equation (4.1) based on the use of the binary Bell polynomials.
Theorem 10. The supersymmetric KdV equation (4.1) possesses the following
infinite conservation laws
$None$ $I_{n,t}+\mathcal{D}F_{n}=0,\ n=1,2,\cdots.$
where the fermionic conserved densities $I_{n}^{\prime}s$ are explicitly given
by recursion relations
$None$ $\displaystyle I_{1}=\mathcal{D}q_{x}=\Phi,\ \ \
I_{2}=I_{1,x}=\Phi_{x},$ $\displaystyle
I_{n+1}=I_{n,x}+\sum_{k=1}^{n}I_{k}\mathcal{D}I_{n-k},\ \ n=2,3,\cdots,$
and the bosonic fluxes $F_{n}^{\prime}s$ are given by recursion formulas
$None$ $\displaystyle
F_{1}=\mathcal{D}\Phi_{2x}+3\Phi\Phi_{x}+3(\mathcal{D}\Phi)^{2},$
$\displaystyle
F_{2}=\mathcal{D}\Phi_{3x}+3(\Phi\Phi_{2x}+\Phi_{x}^{2})+6\mathcal{D}\Phi\mathcal{D}\Phi_{x},$
$\displaystyle
F_{n}=\mathcal{D}I_{n,2x}+3\sum_{k=1}^{n}(I_{k}I_{n+1-k}+\mathcal{D}I_{k}\mathcal{D}I_{n+1-k,x})+3\mathcal{D}\Phi\mathcal{D}I_{n}$
$\displaystyle+\sum_{i+j+k=n}\mathcal{D}I_{i}\mathcal{D}I_{j}\mathcal{D}I_{k},\
\ n=3,4,\cdots.$
Proof. The conservation laws actually have been hinted in the two-filed
constraint system (4.9)-(4.11), which can be rewritten in the conserved form
$None$
$\displaystyle\mathcal{Y}_{x,\theta}(v,w)-\lambda\mathcal{Y}_{\theta}(v)=0,$
$\displaystyle\partial_{t}\mathcal{Y}_{\theta}(v)+\mathcal{D}[\mathcal{Y}_{3x}(v,w)+3\lambda^{2}\mathcal{Y}_{x}(v)-3\lambda\mathcal{Y}_{2x}(v,w)]=0.$
by applying the relation
$\mathcal{D}\mathcal{Y}_{t}(v)=\partial_{t}\mathcal{Y}_{\theta}(v)=\mathcal{D}v_{t}.$
By introducing a new fermionic potential function
$\eta=(\mathcal{D}\widetilde{q}-\mathcal{D}q)/2:\
\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda_{1},$
it follows from the relation (4.8) that
$None$ $\mathcal{D}v=\eta,\ \ \mathcal{D}w=\eta+\mathcal{D}q.$
Substituting (4.17) into (4.16), we get a super Riccati-type equation
$None$
$\displaystyle\eta_{x}+\eta\mathcal{D}\eta+\mathcal{D}q_{x}-\lambda\eta=0,$
and a divergence-type equation
$None$
$\displaystyle\eta_{t}+\mathcal{D}[\mathcal{D}\eta_{2x}+3\lambda\eta\eta_{x}+3q_{2x}\mathcal{D}\eta+3\mathcal{D}\eta\mathcal{D}\eta_{x}+(\mathcal{D}\eta)^{3}]=0,$
where we have used the equation (4.18) to get the equation (4.19).
To proceed, inserting the expansion
$None$ $\eta=\sum_{n=1}^{\infty}I_{n}(\mathcal{D}q,q_{x},\cdots)\lambda^{-n},$
into the equation (4.18) and equating the coefficients for power of $\lambda$,
we then obtain the formulas (4.13).
Finally, substituting (4.20) into (4.19) yields
$\displaystyle\sum_{n=1}^{\infty}I_{n,t}\lambda^{-n}+\mathcal{D}\left[\sum_{n=1}^{\infty}\mathcal{D}I_{n,2x}\varepsilon^{-n}+3\lambda\sum_{n=1}^{\infty}I_{n}\lambda^{-n}\sum_{n=1}^{\infty}I_{n,x}\lambda^{-n}+3q_{2x}\sum_{n=1}^{\infty}\mathcal{D}I_{n}\lambda^{-n}\right.$
$\displaystyle\left.+3\sum_{n=1}^{\infty}\mathcal{D}I_{n}\lambda^{-n}\sum_{n=1}^{\infty}\mathcal{D}I_{n,x}\lambda^{-n}+(\sum_{n=1}^{\infty}\mathcal{D}I_{n}\lambda^{-n})^{3}\right]=0,$
which leads to infinite consequence of conservation law equation (4.13) by
equating the coefficients for power of $\lambda$. $\square$
It follows from the conservation equation (4.13) by using (2.2) that
$\left(\iint I_{n}dxd\theta\right)_{t}=-\iint(\mathcal{D}F_{n})dxd\theta=0,$
which implies that $I_{n}^{\prime}s$ are fermionic conserved densities. We
present recursion formulas for generating an infinite sequence of conservation
laws for each equation, the first few conserved density and associated flux
are explicit. The first equation of conservation law equation (4.13) is
exactly the supersymmetric KdV equation (4.1). In conclusion, the
supersymmetric KdV (4.1) is complete integrable in the sense that it admits
bilinear Bäcklund transformation, Lax pair and infinite conservation laws.
5\. The supersymmetric sine-Gordon equation
The classical sine-Gordon equation
$None$ $\phi_{xt}=\sin\phi$
has applications in various areas of physics including nonlinear field theory,
solid-state physics, nonlinear optics, elementary particle theory and fluid
dynamics, see [39]-[43] and references therein. The supersymmetric extension
of the equation (5.1), i.e. the supersymmetric sine-Gordon equation [44]-[51]
$None$ $\displaystyle\mathcal{D}_{1}\mathcal{D}_{2}\Phi=\sin\Phi$
is constructed on the four dimensional superspace
$(x,t,\theta_{1},\theta_{2})\in\mathbb{R}_{\Lambda}^{2,2}$ . Here,
$\Phi=\Phi(x,t,\theta_{1},\theta_{2}):\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{0}$
is a scalar bosonic superfield; The variables $x$ and $t$ represent the even
coordinates on the two-dimensional super-Minkowski space, while the quantities
$\theta_{1}$ and $\theta_{2}$ are anticommuting odd coordinates which satisfy
the anticommutation relations
$\theta_{1}^{2}=\theta_{2}^{2}=0,\ \ [\theta_{1},\theta_{2}]=0.$
The $\mathcal{D}_{1}=\partial_{\theta_{1}}+\theta_{1}\partial_{x}$ and
$\mathcal{D}_{2}=\partial_{\theta_{2}}+\theta_{2}\partial_{t}$ are two
covariant derivatives which satisfy the anticommutation relations
$\mathcal{D}_{1}^{2}=\partial_{x},\ \ \mathcal{D}_{2}^{2}=\partial_{t},\ \
[\mathcal{D}_{1},\mathcal{D}_{2}]=0.$
The supersymmetric version of the sine-Gordon equation was introduced from
purely physical motivations [44]. It is becoming increasingly interesting to
investigate the supersymmetric sine-Gordon equation because of its close
relation to string theories and statistical physics [45]-[47]. In recent
publications, a superspace extension of the Lagrangian formulation has been
established for the supersymmetric sine-Gordon equation [19]. The bilinear
method is used to construct multi-super soliton solutions [48]. The
supersymmetric sine-Gordon equation admits a Lax pair, and a connection was
established between its super-Backlund and super-Darboux transformations [49,
50]. The method of symmetry reduction is systematically applied in order to
derive invariant solutions of the supersymmetric sine-Gordon equation [51].
The prolongation method of Wahlquist and Estabrook was used to find an
infinite-dimensional superalgebra and the associated super Lax pairs [52].
Here we study the integrable properties of the supersymmetric sine-Gordon
based on the use of generalized super Bell polynomials. The bilinear form,
bilinear Backlund transformation, Lax pair and infinite conservation laws
systematically are obtained with our method.
Theorem 11. Under the transformation
$\Phi=2i\ln(F/G),$
the supersymmetric sine-Gordon equation (5.2) admits the bilinear form
$None$ $\displaystyle 2S_{1}S_{2}F\cdot F+G^{2}=0,\ \ 2S_{1}S_{2}G\cdot
G+F^{2}=0,$
where $F,G:\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{0}$ are two bosonic
functions.
Proof. As before, the invariance of the supersymmetric sine-Gordon equation
(5.2) under the scale transformation
$x\rightarrow\lambda x,\ \ t\rightarrow\lambda^{-1}t,\ \
\theta_{1}\rightarrow\lambda^{1/2}\theta_{1},\ \ \ \
\theta_{2}\rightarrow\lambda^{-1/2}\theta_{2},\ \ \Phi\rightarrow\Phi$
shows that the dimension of the bosonic superfield $\Phi$ is zero, and so we
may introduce a dimensionless bosonic field $q$ by setting
$None$ $\Phi=cq,$
in which $c\in\Lambda_{0}$ is free constant to be determined. Substituting
(5.4) into (5.2) yields
$None$
$2\mathcal{D}_{1}\mathcal{D}_{2}q=P_{\theta_{1}\theta_{2}}(p+q)-P_{\theta_{1}\theta_{2}}(p-q)=i(e^{-icq}-e^{icq})/c$
where $p:\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{0}$ is an auxiliary
function. If one chooses the constant $c=2i$, the equation (5.5) is then cast
into a linear combination form of $P$-polynomials
$2P_{\theta_{1}\theta_{2}}(p+q)-2P_{\theta_{1}\theta_{2}}(p-q)+\exp(-2q)-\exp(2q)=0,$
which can be decoupled into a system
$None$ $\displaystyle E_{1}(p,q)=2P_{\theta_{1}\theta_{2}}(p+q)+\exp(-2q)=0,$
$\displaystyle E_{2}(p,q)=2P_{\theta_{1}\theta_{2}}(p-q)+\exp(2q)=0.$
Multiplying the first equation by $\exp(p+q)$, the second equation by
$\exp(p-q)$ in the equation (5.6) yields
$None$ $\displaystyle 2\exp(p+q)P_{\theta_{1}\theta_{2}}(p+q)+\exp(p-q)=0,$
$\displaystyle 2\exp(p-q)P_{\theta_{1}\theta_{2}}(p-q)+\exp(p+q)=0.$
By transformation
$q=\ln(F/G),\ \ p=\ln(FG)\Longleftrightarrow\ \ \Phi=2iq=2i\ln(F/G),\ \
p=\ln(FG)$
and using the property (3.16), then the equation (5.7) gives the bilinear form
(5.3) for the supersymmetric sine-Gordon equation (5.2). $\square$
Theorem 12. Let $(F,G)$ be a solution of the equation (5.3), then
$(\widetilde{F},\widetilde{G})$ satisfying
$None$ $\displaystyle S_{1}\widetilde{G}\cdot G=\lambda g\widetilde{F}F,\ \
S_{1}\widetilde{F}\cdot F=-\lambda g\widetilde{G}G,$ $\displaystyle
S_{2}\widetilde{F}\cdot G=\frac{1}{4\lambda}g\widetilde{G}F,\ \
S_{2}\widetilde{G}\cdot F=-\frac{1}{4\lambda}g\widetilde{F}G,$
$\displaystyle\mathcal{D}_{1}g=\lambda\left(\frac{F\widetilde{F}}{G\widetilde{G}}-\frac{G\widetilde{G}}{F\widetilde{F}}\right),\
\
\mathcal{D}_{2}g=\frac{1}{4\lambda}\left(\frac{G\widetilde{F}}{F\widetilde{G}}-\frac{F\widetilde{G}}{G\widetilde{F}}\right).$
is another solution of the equation (5.3), where
$g:\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{1}$ fermionic auxiliary
superfield and $\lambda\in\Lambda_{0}$ is even parameter.
Proof. In order to obtain the bilinear Bäcklund transformation and Lax pairs
of the equation (5.2), let $p,q$ and $\widetilde{p},\widetilde{q}$ be two
solutions of the equation (5.6) and consider the associated two-field
condition
$None$ $\displaystyle
E_{1}(\widetilde{p},\widetilde{q})-E_{1}(p,q)=2\mathcal{D}_{1}\mathcal{D}_{2}(\widetilde{p}-p)-2\mathcal{D}_{1}\mathcal{D}_{2}(\widetilde{q}-q)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+e^{\widetilde{q}+q}(e^{\widetilde{q}-q}-e^{q-\widetilde{q}})=0,$
$\displaystyle
E_{2}(\widetilde{p},\widetilde{q})-E_{2}(p,q)=2\mathcal{D}_{1}\mathcal{D}_{2}(\widetilde{p}-p)+2\mathcal{D}_{1}\mathcal{D}_{2}(\widetilde{q}-q)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+e^{-(\widetilde{q}+q)}(e^{\widetilde{q}-q}-e^{q-\widetilde{q}})=0,$
where
$\displaystyle\widetilde{p}=\ln(\widetilde{F}\widetilde{G}),\ \
\widetilde{q}=\ln(\widetilde{F}/\widetilde{G}),$
We introduce variables
$\displaystyle v_{1}=\ln(\widetilde{G}/G),\ v_{2}=\ln(\widetilde{F}/F),\
v_{3}=\ln(\widetilde{F}/G),\ v_{4}=\ln(\widetilde{G}/F),$ $\displaystyle
w_{1}=\ln(\widetilde{G}G),\ w_{2}=\ln(\widetilde{F}F),\
w_{3}=\ln(\widetilde{F}G),\ w_{4}=\ln(\widetilde{G}F),$
from which, we have relations
$None$ $\displaystyle\widetilde{q}-q=v_{2}-v_{1}=w_{3}-w_{4},\ \
\widetilde{q}+q=v_{3}-v_{4}=w_{2}-w_{1},$
$\displaystyle\widetilde{p}-p=v_{1}+v_{2}=v_{3}+v_{4},\ \
\widetilde{p}+p=w_{1}+w_{2}=w_{3}+w_{4}$
and
$None$ $\displaystyle v_{1}=v_{4}+q,\ \ v_{2}=v_{3}-q,\ \ w_{1}=w_{4}-q,\ \
w_{2}=w_{3}+q.$
By using the mixed variables (5.10), it follows that from (5.9)
$None$ $\displaystyle
4\mathcal{D}_{1}\mathcal{D}_{2}v_{1}+e^{v_{3}-v_{4}}(e^{v_{2}-v_{1}}-e^{v_{1}-v_{2}})=0,$
$\displaystyle
4\mathcal{D}_{1}\mathcal{D}_{2}v_{2}+e^{v_{4}-v_{3}}(e^{v_{1}-v_{2}}-e^{v_{2}-v_{1}})=0,$
which may produce the required bilinear Bäcklund transformation under an
appropriate additional constraint. We choose a constraint
$None$ $\mathcal{D}_{1}v_{1}=\mathcal{Y}_{\theta_{1}}(v_{1})=\lambda
ge^{v_{3}-v_{4}},$
where $g:\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{1}$ fermionic auxiliary
superfield and $\lambda\in\Lambda_{0}$ is even parameter. The fermionic
function $g$ is introduced because of supersymmetry and the oddness of the
superspace derivatives $\mathcal{D}_{1},\mathcal{D}_{2}$. The constraint
(5.13) reduces the first equation in (5.12) into
$None$ $-4\lambda\mathcal{D}_{2}g-4\lambda
g\mathcal{D}_{2}(v_{3}-v_{4})+e^{v_{2}-v_{1}}-e^{v_{1}-v_{2}}=0.$
Since the term $\mathcal{D}_{2}(v_{3}-v_{4})$ should be fermionic function, we
make a constraint
$None$ $\mathcal{D}_{2}(v_{3}-v_{4})=\frac{1}{4\lambda}gh,$
where $h$ is a bosonic function to be determined. On account of this
constraint, it follows from (5.14) that
$None$ $\mathcal{D}_{2}g=\frac{1}{4\lambda}(e^{v_{2}-v_{1}}-e^{v_{1}-v_{2}}),$
which holds because $g^{2}=0$, $g$ being fermionic.
By means of the system (5.15) and (5.16), the second equation in (5.12) reads
$\displaystyle\mathcal{D}_{2}(\mathcal{D}_{1}v_{2}+\lambda
ge^{v_{4}-v_{3}})=0,$
which is satisfied if we choose
$None$
$\displaystyle\mathcal{Y}_{\theta_{1}}(v_{2})=\mathcal{D}_{1}v_{2}=-\lambda
ge^{v_{4}-v_{3}}.$
On the one hand, using the relation (5.11), we have
$None$ $\displaystyle
4\mathcal{D}_{1}\mathcal{D}_{2}(v_{3}-v_{4})+4\mathcal{D}_{2}\mathcal{D}_{1}(v_{1}-v_{2})=8\mathcal{D}_{1}\mathcal{D}_{2}q$
$\displaystyle=(e^{v_{2}-v_{1}}-e^{v_{1}-v_{2}})(e^{v_{3}-v_{4}}+e^{v_{4}-v_{3}})+(e^{v_{2}-v_{1}}+e^{v_{1}-v_{2}})(e^{v_{3}-v_{4}}-e^{v_{4}-v_{3}}).$
On the other hand, it follows from (5.13), (5.16) and (5.17) that
$None$ $\displaystyle
4\mathcal{D}_{2}\mathcal{D}_{1}(v_{1}-v_{2})=4\lambda(\mathcal{D}_{2}g)(e^{v_{3}-v_{4}}+e^{v_{4}-v_{3}})$
$\displaystyle=(e^{v_{2}-v_{1}}-e^{v_{1}-v_{2}})(e^{v_{3}-v_{4}}+e^{v_{4}-v_{3}}).$
Combining (5.15), (5.18) and (5.19) yields
$\displaystyle
4\mathcal{D}_{2}\mathcal{D}_{1}(v_{3}-v_{4})=\frac{1}{\lambda}(\mathcal{D}_{1}g)h=(e^{v_{3}-v_{4}}-e^{v_{4}-v_{3}})(e^{v_{2}-v_{1}}+e^{v_{1}-v_{2}}),$
which implies that we may choose
$None$
$\displaystyle\mathcal{D}_{1}g=\lambda(e^{v_{3}-v_{4}}-e^{v_{4}-v_{3}})$
and
$\displaystyle h=(e^{v_{2}-v_{1}}+e^{v_{1}-v_{2}}).$
Thus, we have
$\displaystyle\mathcal{D}_{2}(v_{3}-v_{4})=\frac{1}{4\lambda}g(e^{v_{2}-v_{1}}+e^{v_{1}-v_{2}}),$
which can be written as a pair of $\mathcal{Y}$-polynomials
$None$
$\mathcal{Y}_{\theta_{2}}(v_{3})=\mathcal{D}_{2}v_{3}=\frac{1}{4\lambda}ge^{v_{1}-v_{2}},\
\
\mathcal{Y}_{\theta_{2}}(v_{4})=\mathcal{D}_{2}v_{4}=-\frac{1}{4\lambda}ge^{v_{2}-v_{1}},$
Combining (5.13), (5.16), (5.17), (5.20) and (5.21) gives bilinear Bäcklund
transformation (5.8) of the supersymmetric sine-Gordon equation. $\square$
Finally we derive Lax pair of the supersymmetric sine-Gordon equation.
Theorem 13. The supersymmetric sine-Gordon equation (5.2) admits a Lax pair
$None$
$\displaystyle\mathcal{D}_{1}\Psi=M\Psi=\left(\begin{matrix}\displaystyle{-\frac{1}{2}}i\mathcal{D}_{1}\Phi&\lambda
g\cr\displaystyle{\lambda
g}&\displaystyle{\frac{1}{2}}i\mathcal{D}_{1}\Phi\end{matrix}\right)\Psi,$
$\displaystyle\mathcal{D}_{2}\Psi=N\Psi=\left(\begin{matrix}0&\displaystyle{-\frac{1}{4\lambda}}ge^{-i\Phi}\cr\displaystyle{-\frac{1}{4\lambda}ge^{i\Phi}}&0\end{matrix}\right)\Psi,$
together with
$\displaystyle\mathcal{D}_{1}g=\lambda\left(\frac{\psi_{3}}{\psi_{4}}-\frac{\psi_{4}}{\psi_{3}}\right),\
\
\mathcal{D}_{2}g=\frac{1}{\lambda}\left(e^{i\Phi}\frac{\psi_{3}}{\psi_{4}}-e^{-i\Phi}\frac{\psi_{4}}{\psi_{3}}\right),$
where $\Psi=(\psi_{3},\psi_{4})^{T}$.
Making use of the Hopf-Cole transformation
$v_{3}=\ln\psi_{3},\ \ v_{4}=\ln\psi_{4},$
then the system (5.13), (5.16), (5.17), (5.20) and (5.21) can be linearized
into a Lax pair (5.22). It is easy to check that the integrability condition
$\mathcal{D}_{2}M+\mathcal{D}_{1}N-[M,N]=0$
is satisfied if $\Phi$ is a solution of the sine-Gordon equation (5.2).
If we choose a transformation
$\phi_{1}=\psi_{4}^{2},\ \ \phi_{2}=\psi_{3}^{3},\ \
g=\frac{\phi_{3}}{2i\psi_{3}\psi_{4}}$
then the Lax pair (5.22) is also equivalent to a linear system in $3\times 3$
matrix form
$None$
$\displaystyle\mathcal{D}_{1}\Omega=\frac{1}{4}\left(\begin{matrix}4\mathcal{D}_{1}\Phi&0&\lambda\cr
0&-4\mathcal{D}_{1}\Phi&-\lambda\cr-4\lambda&4\lambda&0\end{matrix}\right)\Omega,$
$\displaystyle\mathcal{D}_{2}\Omega=\frac{1}{16\lambda}\left(\begin{matrix}0&0&e^{i\Phi}\cr
0&0&-e^{-i\Phi}\cr 4e^{-i\Phi}&-4e^{-i\Phi}&0\end{matrix}\right)\Omega,$
where $\Omega=(\phi_{1},\phi_{2},\phi_{3})^{T}$,
$\phi_{1},\phi_{2}:\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{0}$ are
bosonic functions and
$\phi_{3}:\mathbb{R}_{\Lambda}^{2,2}\rightarrow\Lambda_{1}$ is a fermionic
function. The system (5.23) also can be obtained from (5.13), (5.16), (5.17),
(5.20) and (5.21) by setting
$2(v_{4}-v_{3})=\ln\frac{\phi_{1}}{\phi_{2}},\ \
2(v_{2}-v_{1})=-2i\Phi+\ln\frac{\phi_{1}}{\phi_{2}},\ \
g=\frac{\phi_{3}}{2i\sqrt{\phi_{1}\phi_{2}}}.$
The compatibility of the linear system (5.23) in superspace is equivalent to
the equation (5.2). The system (5.23) is the same as obtained in [42], but
here it is derived systematically from the super Bell polynomials and Lax
pairs. $\square$
Noting the transformation relation
$v_{1}-v_{2}=i(\Phi-\widetilde{\Phi})/2,\ \
v_{3}-v_{4}=-i(\Phi+\widetilde{\Phi})/2,$
then it follows from equations (5.13), (5.17), (5.18), (5.21) and (5.22) that
$None$ $\displaystyle\mathcal{D}_{1}(\Phi-\widetilde{\Phi})=\lambda
g\cos\left(\frac{\Phi+\widetilde{\Phi}}{2}\right),$
$\displaystyle\mathcal{D}_{2}(\Phi+\widetilde{\Phi})=\frac{1}{4\lambda}g\cos\left(\frac{\Phi-\widetilde{\Phi}}{2}\right),$
$\displaystyle\mathcal{D}_{1}g=\lambda\sin\left(\frac{\Phi+\widetilde{\Phi}}{2}\right),\
\
\mathcal{D}_{2}g=\frac{1}{4\lambda}\sin\left(\frac{\Phi-\widetilde{\Phi}}{2}\right),$
which is the Bäcklund transformation of the supersymmetric sine-Gordon
equation. The compatibility of the Bäcklund transformation (5.24) is the
supersymmetric sine-Gordon equation for both $\Phi$ and $\widetilde{\Phi}$
separately. The super Bäcklund transformation (5.24) reduces to the classical
Bäcklund transformation of the purely bosonic sine-Gordon equation when
fermions are equal to zero.
6\. Concluding Remarks
In this paper, we have introduced a class of super Bell polynomials which play
an important role in the characterization of bilinear Bäcklund transformation,
Lax pairs and infinite conservation laws of supersymmetric equations. To the
knowledge of the authors, this is the first work on the super Bell polynomials
and their applications to super integrable systems. We believe that there are
still many interesting deep relations between generalized Bell polynomials and
integrable structures, which remain open and worth to be considered. For
instance, (i) How to explore the relations between the super Bell polynomials
with symmetries, Hamiltonian functions, etc. (ii) How to define a class of
discrete Bell polynomials and apply them in discrete equations. We have some
ideas on these questions and will intend to return to them in some future
publications.
Acknowledgment
The work described in this paper was supported by grants from the CityU
(Project No. 7002440), the National Science Foundation of China (No. 10971031)
and Shanghai Shuguang Tracking Project (No. 08GG01).
## References
* [1]
* [2] P. Ramond, Dual theory for free fermions, Phys. Rev. D 3, 2415-2418 (1971).
* [3] A. Neveu and J. H. Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31, 86-112 (1971).
* [4] J. Wess and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys. B 70, 39-50 (1974).
* [5] Yu. I. Manin and A. O. Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Commun. Math. Phys. 98, 65-77 (1985).
* [6] J. M. Rabin, The Geometry of the Super KP Flows, Commun. Math. Phys. 137, 533-552 (1991)
* [7] P. Mathieu, Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys. 29, 2499-2506 (1988).
* [8] B. A. Kupershmidt, A super Korteweg-de Vries equation: An integrable system, Phys. Lett. A, 102, 213-215 ( 1984) Super Korteweg-de Vries equations associated to super extensions of the Virasoro algebra
* [9] B. A. Kupershmidt, This article is not included in your organization’s subscription. However, you may be able to access this article under your organization’s agreement with Elsevier, Phys. Lett. A, 109, 417-423 ( 1985)
* [10] J. Harnad and B. A. Kupershmidt, Super Loop Groups, Hamiltonian Actions and Super Virasoro Algebras, Commun. Math. Phys. 132, 315-347 (1990)
* [11] W. Oevel and Z. Popowicz, The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems, Commun. Math. Phys. 139, 441-460 (1991).
* [12] C. Morosi, L. Pizzocchero, On the BiHamiltonian Structure of the Supersymmetric KdV Hierarchies. A Lie Superalgebraic Approach, Commun. Math. Phys. 158, 267-288 (1993)
* [13] T. Inami and H. Kanno, Lie superalgebraic approach to super Toda lattice and generalized super KdV equations, Commun. Math. Phys. 136, 519-542 (1991)
* [14] E. Ivanov, S. Krivonos, New integrable extensions of $\mathcal{N}=2$ KdV and Boussinesq hierarchies, Physics Letters A 231, 75-81 ( 1997)
* [15] E. Ivanov, S. Krivonos, F. Toppan, $\mathcal{N}=4$ super NLS-mKdV hierarchies, Physics Letters B 405, 85-94 (1997)
* [16] D. Sarma, Constructing a supersymmetric integrable system from the Hirota method in superspace, Nucl. Phys. B 681, 351-358 (2004).
* [17] Q. P. Liu, Darboux transformations for supersymmetric korteweg-de vries equations, Lett Math. Phys. 35, 115-122 (1995).
* [18] Q. P. Liu and X. B. Hu, Bilinearization of $\mathcal{N}=1$ supersymmetric Korteweg de Vries equation revisited, J. Phys. A 38, 371-6378 (2005).
* [19] Q. P. Liu, X. B. Hu and M. X. Zhang, Supersymmetric modified Korteweg-de Vries equation: bilinear approach, Nonlinearity 18, 1597-1603 (2005).
* [20] I. N. McArthur, C. M. Yung, Hirota bilinear form for the super-KdV hierarchy, Mod. Phys. Lett. A 18, 1739-1745 (1993)
* [21] A. S. Carstea, Extension of the bilinear formalism to supersymmetric KdV-type equationsnipne, Nonlinearity 13, 1645-1656 (2000).
* [22] E. G. Fan and Y. C. Hon, Quasi-periodic wave solutions of $\mathcal{N}=2$ supersymmetric KdV equation in superspace, Stud. Appl. Math. in Press
* [23] E. T. Bell, Exponential polynomials, Ann. Math. 35, 258-277 (1934)
* [24] M. Abramowitz and J. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1972.
* [25] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974
* [26] J. Riordan, Combinatorial Identities, Wiley, New York, 1966.
* [27] F. T. Howard, A special class of Bell polynomials, Math. Comput. 35 977-989 (1980)
* [28] S. Noschese and P.E. Ricci, Differentiation of multivariable composite functions and Bell polynomials, J. Comput. Anal. Appl. 5, 333-340 (2003).
* [29] P. Natalini and P.E. Ricci, An extension of the Bell polynomials, Computers Math. Appl. 47 , 719-725 (2004).
* [30] A. Bernardini, P. Natalini and P. E. Ricci, Multidimensional Bell Polynomials of Higher Order, Comput. Math. Appl. 50 1697-1708 (2005)
* [31] R.B. Paris, The asymptotics of the generalised Hermite-Bell polynomials, J. Comput. Appl. Math. 232 216-226 (2009)
* [32] C. Gilson, F. Lambert, J. Nimmo and R. Willox, On the combinatorics of the Hirota D-operators, Proc. R. Soc. Lond. A 452, 223-234 (1996)
* [33] F. Lambert, I. Loris and J.Springael, Classical Darboux transformations and the KP hierarchy, Inverse Probl. 17, 1067-1074 (2001)
* [34] F. Lambert and J. Springael, Soliton equations and simple combinatorics, Acta Appl. Math. 102, 147-178 (2008)
* [35] V. S. Vladimirov, Superanalysis. I. Differential calculus, Theor. Math. Phys. 59, 317-335 (1984).
* [36] V. S. Vladimirov, Superanalysis. II. Integral calculus, Theor. Math. Phys. 60, 743-765 (1984)
* [37] F. A. Berezin: Introduction to Superanalysis, Reidel Publishing Company, Dordrecht, 1987.
* [38] A. Khrennikov, Superanalysis, Kluwer Academic Publishers, Dordrecht, 1999
* [39] G L Lamb, Elements of Soliton Theory, New York: Wiley, 1980
* [40] R. K. Bullough, P. J. Caudrey and H. M. Gibbs, The double sine-Gordon equations: a physically applicable system of equations Solitons, eds. R. K. Bullough and P. J. Caudrey, New York, Springer, 1980
* [41] C. Rogers and W. K. Schief, Backlund and Darboux Transformations, Cambridge, Cambridge University Press, 2002
* [42] S. Sciuto, Exterior Calculus and two-dimensional supersymmetric models, Phys. Lett. B 90 , 75-80 ( 1980)
* [43] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge: Cambridge University Press, 1991
* [44] P. Di Vecchia and S. Ferrara, Classical solutions in two-dimensional supersymmetric field theories, Nucl. Phys. B 130, 93-104 (1977)
* [45] D. Gepner, Space-time supersymmetry in compactified string theory and superconformal models, Nucl. Phys. B 296, 757-778 (1988)
* [46] Z Bajnok, C. Dunning, L Palla, G. Takacs and F. Wagner, SUSY sine-Gordon theory as a perturbed conformal field theory and finite size effects, Nucl. Phys. B 679, 521-544 (2004)
* [47] M. Grigoriev and A. A. Tseytlin, Pohlmeyer reduction of AdS(5) x S(5) superstring sigma model, Nucl. Phys. B 800, 4450-501 (2008)
* [48] B. Grammaticos, A. Ramani and A. S. Carstea, Bilinearization and soliton solutions of the $\mathcal{N}=1$ supersymmetric sine-ordon equation, J. Phys. A, 34, 4881-4886 (2001)
* [49] M. Siddiq and M. Hassan, On the linearization of the super sine-Gordon equation, Europhys. Lett. 70, 149-154 (2005)
* [50] M. Siddiq, M. Hassan and U Saleem, On Darboux transformation of the supersymmetric sine-Gordon equation, J. Phys. A, 39, 7313-7318 (2006)
* [51] A. M. Grundland, A. J. Hariton and L. Snobl, Invariant solutions of the supersymmetric sine-Gordon equation, J. Phys. A, 42 335203 (2009)
* [52] M. Omote, Prolongation structures of the supersymmetric Sine-Gordon equation and infinite-dimensional superalgebras, J. Phys. A, 20, 1941-1950 (1987)
|
arxiv-papers
| 2010-08-25T03:45:23 |
2024-09-04T02:49:12.445811
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Engui Fan, Y. C. Hon",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1008.4198"
}
|
1008.4208
|
Astronomy Letters, 2010 Vol. 36, No. 9, pp. 634–644.
Parameters of the Local Warp of the Stellar-Gaseous
Galactic Disk from the Kinematics of Tycho-2
Nearby Red Giant Clump Stars
V.V. Bobylev
Pulkovo Astronomical Observatory, Russian Academy of Sciences, St-Petersburg
Abstract–We analyze the three-dimensional kinematics of about 82000 Tycho-2
stars belonging to the red giant clump (RGC). First, based on all of the
currently available data, we have determined new, most probable components of
the residual rotation vector of the optical realization of the ICRS/HIPPARCOS
system relative to an inertial frame of reference,
$(\omega_{x},\omega_{y},\omega_{z})=(-0.11,0.24,-0.52)\pm(0.14,0.10,0.16)$ mas
yr-1. The stellar proper motions in the form $\mu_{\alpha}cos\delta$ have then
be corrected by applying the correction $\omega_{z}=-0.52$ mas yr-1. We show
that, apart from their involvement in the general Galactic rotation described
by the Oort constants $A=15.82\pm 0.21$ km s-1 kpc-1 and $B=-10.87\pm 0.15$ km
s-1 kpc-1, the RGC stars have kinematic peculiarities in the Galactic $yz$
plane related to the kinematics of the warped stellar-gaseous Galactic disk.
We show that the parameters of the linear OgorodnikovMilne model that describe
the kinematics of RGC stars in the $zx$ plane do not differ significantly from
zero. The situation in the $yz$ plane is different. For example, the component
of the solid-body rotation vector of the local solar neighborhood around the
Galactic $x$ axis is $M_{\scriptscriptstyle 32}^{\scriptscriptstyle-}=-2.6\pm
0.2$ km s-1 kpc-1. Two parameters of the deformation tensor in this plane,
namely $M_{\scriptscriptstyle 23}^{\scriptscriptstyle+}=1.0\pm 0.2$ km s-1
kpc-1 and $M_{\scriptscriptstyle 33}-M_{\scriptscriptstyle 22}=-1.3\pm 0.4$ km
s-1 kpc-1, also differ significantly from zero. On the whole, the kinematics
of the warped stellar-gaseous Galactic disk in the local solar neighborhood
can be described as a rotation around the Galactic $x$ axis (close to the line
of nodes of this structure) with an angular velocity $(-3.1\pm
0.5)\leq\Omega_{W}\leq(-4.4\pm 0.5)$ km s-1 kpc-1.
## INTRODUCTION
Analysis of the large-scale structure of neutral hydrogen revealed a warp of
the gaseous disk in the Galaxy (Westerhout 1957; Burton 1988). The results of
studying this structure based on currently available data on the HI and HII
distributions are presented in Kalberla and Dedes (2008) and Cersosimo et al.
(2009), respectively. This structure is revealed by the spatial distribution
of stars and dust (Drimmel and Spergel 2001), by the distribution of pulsars
in the Galaxy (Yusifov 2004), by HIPPARCOS OB stars (Miyamoto and Zhu 1998),
and by the 2MASS red giant clump (Momany et al. 2006).
Several models were suggested to explain the nature of the Galactic disk warp:
(1) the interaction between the disk and a nonspherical dark matter halo
(Sparke and Casertano 1988); (2) the gravitational influence of the Galaxys
nearest satellites (Bailin 2003); (3) the interaction of the disk with a near-
Galaxy flow formed by high-velocity hydrogen clouds that resulted from mass
exchange between the Galaxy and the Magellanic Clouds (Olano 2004); (4) an
intergalactic flow (López-Corredoira et al. 2002); and (5) the interaction
with the intergalactic magnetic field (Battaner et al. 1990).
By analyzing nearby stars from HIPPARCOS (1997), Dehnen (1998) showed that the
distribution of their residual velocities in the $V_{y}-V_{z}$ plane agreed
satisfactorily with various rotation models of the warped disk. Miyamoto et
al. (1993) and Miyamoto and Zhu (1998) determined the rotation parameters of
the warped stellargaseous disk by analyzing giant stars of various spectral
types and a sample of HIPPARCOS OB5 stars. Thus, there is positive experience
in solving this problem using data on stars relatively close to the Sun.
Studying the three-dimensional kinematics of stars requires that the
observational data be free from the systematic errors related to the
referencing of the optical realization of the ICRS/HIPPARCOS system to the
inertial frame of reference specified by extragalactic sources. The modern
standard system of astronomical coordinates, ICRS (International Celestial
Reference System), is realized by the catalog of positions for 212 compact
extragalactic radio sources uniformly distributed over the entire sky observed
by the radio interferometry technique (Ma et al. 1998). In the optical range,
the first realization of the ICRS was the HIPPARCOS catalog. The application
of various methods of analysis shows that there is a small residual rotation
of the ICRS/HIPPARCOS system relative to the inertial frame of reference with
$\omega_{z}\approx-0.4\pm 0.1$ mas yr-1 (Bobylev 2004a, 2004b). At present,
there are several new results of comparing individual programs with the
catalogs of the ICRS/HIPPARCOS system. One of our goals is to determine the
most probable components of the residual rotation vector of the optical
realization of the ICRS/HIPPARCOS system relative to the inertial frame of
reference.
The main goal of this paper is to study the local kinematics of the warped
stellargaseous Galactic disk by analyzing the motions of Tycho-2 stars that
belong to the red giant clump (RGC). Occupying a compact region on the
HertzsprungRussell diagram, these stars are a kind of “standard candles”.
Their estimated photometric distances are known with a mean accuracy of at
least 20–30%. RGC stars are distributed uniformly in the spatial region and
over the celestial sphere, which is an important property when the three-
dimensional spatial motions of stars are analyzed.
Fig. 1. Colorabsolute magnitude diagram for 82324 RGC stars from the range of
distances 0.3–1 kpc (a) and the sample of nearby RGC stars with reliable
($e_{\pi}/\pi<10$%) HIPPARCOS parallaxes (b). The isochrones (Yi et al. 2003)
for ages of 1, 5, and 10 Gyr with a nearly solar metallicity, $Z=0.02$ (dashed
lines), and a higher metallicity, $Z=0.04$ (solid lines), are plotted.
## DATA
The characteristic clump, commonly called the red giant clump, on the
HertzsprungRussell diagram is formed mostly by giants with masses from
3$M_{\odot}$ to 9$M_{\odot}$. These stars spend the bulk of their lifetime on
the main sequence as B-type stars. At the core helium burning stage, they
evolve toward the RGC almost without changing their luminosity. At the RGC
stage, despite the slow change in color, the luminosity of such a star remains
almost constant. Therefore, RGC stars are convenient as “standard candles” for
distance determinations. The RGC also incorporates other giants at different
evolutionary stages. In addition, owing to probabilistic selection methods
using reduced proper motions, the RGC sample can be diluted by various stars
from adjacent regions of the HertzsprungRussell diagram, both supergiants and
dwarfs. According to the estimates by various authors, the admixture is small,
may be 10–15% (Rybka 2006; Gontcharov 2008).
We used the list of 97000 RGC stars selected by Gontcharov (2008) by the
dereddened color index and the reduced proper motion based on Tycho-2 (Hog et
al. 2000) and 2MASS (Skrutskie et al. 2006) data. These stars occupy the
region with $0.^{m}5<J-Ks<0^{m}.9$ and $-1^{m}<M_{K_{s}}<-2^{m}$ on the
corresponding diagram. For their selection using the reduced proper motion,
calibration based on HIPPARCOS stars with the most reliable data was
performed. Details of the procedure are described in Gontcharov (2008). An
estimate of the photometric distance and interstellar extinction is available
for each star.
As was shown by Bobylev et al. (2009), the kinematic parameters of the
OgorodnikovMilne model for relatively close ($r<0.2-0.3$ kpc) and distant
($r>1$ kpc) stars are determined with large errors when using the photometric
distances of RGC stars. Therefore, here we use 82 324 stars from the range of
distances 0.3 kpc $<r<1$ kpc, with the mean distance being $r=0.57\pm 0.17$
kpc. To eliminate the halo stars from the sample, the following constraint on
the proper motions was applied:
$\sqrt{(\mu_{\alpha}\cos\delta)^{2}+(\mu_{\delta})^{2}}<300$ mas yr-1.
As was shown by Bobylev et al. (2009), from a statistical analysis of the
velocity dispersions, we found that about 20% of the RGC stars are very young,
while an overwhelming majority of the remaining stars are characterized by the
kinematics of thin-disk stars. We see from Fig. 1, where the color–absolute
magnitude diagram is presented with a grid of isochrones with various ages and
metallicities, that the RGC stars (Fig. 1a) are fairly young. Figure 1b
presents about 650 stars from Gontcharovs catalog selected as RGC ones with
parallaxes from the HIPPARCOS catalog (with a parallax error of less than
10%). Although the segment of the isochrones near the RGC is very sensitive to
metallicity, we clearly see from Fig. 1b that, first, the RGC stars proper
have ages younger than 1 Gyr and, second, the admixture of dwarfs is
insignificant.
## THE OGORODNIKOVMILNE MODEL
We use a rectangular Galactic coordinate system with the axes directed away
from the observer toward the Galactic center $(l$=$0^{\circ}$,
$b$=$0^{\circ},$ the $X$ axis or axis 1), along the Galactic rotation
$(l$=$90^{\circ}$, $b$=$0^{\circ},$ the $Y$ axis or axis 2), and toward the
North Galactic Pole $(b$=$90^{\circ},$ the $Z$ axis or axis 3).
In the linear Ogorodnikov-Milne model (Ogorodnikov, 1965), we use the notation
introduced by Clube (1972, 1973) and used, for example, by Vityazev and
Tsvetkov (2009).
The observed velocity ${\bf V}(r)$ of a star with a heliocentric radius vector
${\bf r}$, is described, to the terms of the first order of smallness
$r/R_{0}\ll 1$, by the equation in vector form
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill{\bf V}(r)={\bf V}_{\odot}+M{\bf r}+{\bf
V^{\prime}},\hfill\hbox to0.0pt{\hss(}1)\cr}}$
where ${\bf V}_{\odot}(X_{\odot},Y_{\odot},Z_{\odot})$ is the peculiar
velocity of the Sun relative to the centroid of the stars under consideration;
$\bf V^{\prime}$ is the residual velocity of the star (here, the residual
stellar velocities are assumed to have a random distribution); $M$ is the
displacement matrix (tensor) whose components are the partial derivatives of
the velocity ${\bf u}(u_{1},u_{2},u_{3})$ with respect to the distance ${\bf
r}(r_{1},r_{2},r_{3})$, where ${\bf u}={\bf V}(R)-{\bf V}(R_{0})$, and $R$ and
$R_{0}$ are the Galactocentric distances of the star and the Sun,
respectively. Then,
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill M_{pq}={\left(\frac{\partial u_{p}}{\partial
r_{q}}\right)}_{\circ},\quad(p,q=1,2,3).\hfill\hbox to0.0pt{\hss(}2)\cr}}$
All nine elements of the matrix $M$ can be determined using the three
components of the observed velocities–the stellar radial velocities and proper
motions. Having only the proper motions, we can write the conditional
equations
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill 4.74r\mu_{l}\cos b=X_{\odot}\sin
l-Y_{\odot}\cos l+\hfill\hbox to0.0pt{\hss(}3)\cr
0.0pt{\hfil$\displaystyle\hfill+r[-\cos b\cos l\sin lM_{11}-\cos
b\sin^{2}lM_{12}-\sin b\sin lM_{13}+\cos b\cos^{2}lM_{21}+\hfill\cr
0.0pt{\hfil$\displaystyle\hfill+\cos b\sin l\cos lM_{22}+\sin b\cos
lM_{23}],\hfill\cr 0.0pt{\hfil$\displaystyle\hfill 4.74r\mu_{b}=X_{\odot}\cos
l\sin b+Y_{\odot}\sin l\sin b-Z_{\odot}\cos b+\hfill\hbox to0.0pt{\hss(4)}\cr
0.0pt{\hfil$\displaystyle\hfill+r[-\sin b\cos b\cos^{2}lM_{11}-\sin b\cos
b\sin l\cos lM_{12}-\hfill\cr 0.0pt{\hfil$\displaystyle\hfill-\sin^{2}b\cos
lM_{13}-\sin b\cos b\sin l\cos lM_{21}-\sin b\cos
b\sin^{2}lM_{22}-\sin^{2}b\sin lM_{23}+\hfill\cr
0.0pt{\hfil$\displaystyle\hfill+\cos^{2}b\cos lM_{31}+\cos^{2}b\sin
lM_{32}+\sin b\cos bM_{33}],\hfill\crcr}}}}}}}}$
from which it follows that the terms should be grouped in several cases. It is
useful to divide the matrix $M$ into symmetric, $M^{\scriptscriptstyle+}$
(local deformation tensor), and antisymmetric, $M^{\scriptscriptstyle-}$
(rotation tensor), parts:
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill M_{\scriptstyle
pq}^{\scriptscriptstyle+}={1\over 2}\left(\frac{\partial u_{p}}{\partial
r_{q}}+\frac{\partial u_{q}}{\partial r_{p}}\right)_{\circ},\quad
M_{\scriptstyle pq}^{\scriptscriptstyle-}={1\over 2}\left(\frac{\partial
u_{p}}{\partial r_{q}}-\frac{\partial u_{q}}{\partial
r_{p}}\right)_{\circ},\quad(p,q=1,2,3).\hfill\hbox to0.0pt{\hss(}5)\crcr}}$
This allows the conditional equations to be written as
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill 4.74r\mu_{l}\cos b=X_{\odot}\sin
l-Y_{\odot}\cos l+\hfill\hbox to0.0pt{\hss(}6)\cr
0.0pt{\hfil$\displaystyle\hfill+r[-M_{\scriptscriptstyle
32}^{\scriptscriptstyle-}\cos l\sin b-M_{\scriptscriptstyle
13}^{\scriptscriptstyle-}\sin l\sin b+M_{\scriptscriptstyle
21}^{\scriptscriptstyle-}\cos b+\hfill\cr
0.0pt{\hfil$\displaystyle\hfill+M_{\scriptscriptstyle
12}^{\scriptscriptstyle+}\cos 2l\cos b-M_{\scriptscriptstyle
13}^{\scriptscriptstyle+}\sin l\sin b+M_{\scriptscriptstyle
23}^{\scriptscriptstyle+}\cos l\sin b-0.5(M_{\scriptscriptstyle
11}^{\scriptscriptstyle+}-M_{\scriptscriptstyle 22}^{\scriptscriptstyle+})\sin
2l\cos b],\hfill\cr 0.0pt{\hfil$\displaystyle\hfill 4.74r\mu_{b}=X_{\odot}\cos
l\sin b+Y_{\odot}\sin l\sin b-Z_{\odot}\cos b+\hfill\hbox to0.0pt{\hss(}7)\cr
0.0pt{\hfil$\displaystyle\hfill+r[M_{\scriptscriptstyle
32}^{\scriptscriptstyle-}\sin l-M_{\scriptscriptstyle
13}^{\scriptscriptstyle-}\cos l-0.5M_{\scriptscriptstyle
12}^{\scriptscriptstyle+}\sin 2l\sin 2b+M_{\scriptscriptstyle
13}^{\scriptscriptstyle+}\cos l\cos 2b+\hfill\cr
0.0pt{\hfil$\displaystyle\hfill+M_{\scriptscriptstyle
23}^{\scriptscriptstyle+}\sin l\cos 2b-0.5(M_{\scriptscriptstyle
11}^{\scriptscriptstyle+}-M_{\scriptscriptstyle
22}^{\scriptscriptstyle+})\cos^{2}l\sin 2b+0.5(M_{\scriptscriptstyle
33}^{\scriptscriptstyle+}-M_{\scriptscriptstyle 22}^{\scriptscriptstyle+})\sin
2b].\hfill\crcr}}}}}}}$
Equation (6) was obtained from Eq. (3) by adding two terms to its right-hand
side, $0.5M_{\scriptscriptstyle 31}$ and $0.5M_{\scriptscriptstyle 32}$, and
subtracting them.
The system of equations (6) and (7) then becomes very convenient for its
simultaneous solution. Indeed, as can be seen from Eq. (6), the two pairs of
unknowns to be determined, $M_{\scriptscriptstyle 13}^{\scriptscriptstyle-}$
and $M_{\scriptscriptstyle 13}^{\scriptscriptstyle+}$, along with
$M_{\scriptscriptstyle 32}^{\scriptscriptstyle-}$ and $M_{\scriptscriptstyle
23}^{\scriptscriptstyle+}$, have identical coefficients, $\sin l\sin b$ and
$\cos l\sin b,$ respectively. In this case, the variables cannot be separated
and can be found only from the simultaneous solution of the system of
equations (6) and (7).
In addition, we see that one of the diagonal terms of the local deformation
tensor remains uncertain. Therefore, we determine differences of the form
$(M_{\scriptscriptstyle 11}^{\scriptscriptstyle+}-M_{\scriptscriptstyle
22}^{\scriptscriptstyle+})$ and $(M_{\scriptscriptstyle
33}^{\scriptscriptstyle+}-M_{\scriptscriptstyle 22}^{\scriptscriptstyle+})$.
The quantities $M_{\scriptscriptstyle
32}^{\scriptscriptstyle-},M_{\scriptscriptstyle
13}^{\scriptscriptstyle-},M_{\scriptscriptstyle 12}^{\scriptscriptstyle-},$
are the components of the solid-body rotation vector of a small solar
neighborhood around the $x,y,z$ axes, respectively. According to our chosen
rectangular coordinate system, the rotations from axis 1 to 2, from axis 2 to
3, and from axis 3 to 1 are positive. $M_{\scriptscriptstyle
21}^{\scriptscriptstyle-}$ is equivalent to the Oort constant $B$. Each of the
quantities $M_{\scriptscriptstyle
12}^{\scriptscriptstyle+},M_{\scriptscriptstyle
13}^{\scriptscriptstyle+},M_{\scriptscriptstyle 23}^{\scriptscriptstyle+}$
describes the deformation in the corresponding plane. $M_{\scriptscriptstyle
12}^{\scriptscriptstyle+}$ is equivalent to the Oort constant $A$.
The diagonal components of the local deformation tensor $M_{\scriptscriptstyle
11}^{\scriptscriptstyle+},M_{\scriptscriptstyle
22}^{\scriptscriptstyle+},M_{\scriptscriptstyle 33}^{\scriptscriptstyle+}$
describe the general local contraction or expansion of the entire stellar
system (divergence). The system of conditional equations (6) and (7) contain
eleven unknowns to be determined by the least-squares method. In addition to
the simultaneous solution of the system of equations (6) and (7), here we
analyze the results of their separate solutions. In this case, Eq. (7) remains
unchanged, while Eq. (6) is reduced to the form
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill 4.74r\mu_{l}\cos b=X_{\odot}\sin
l-Y_{\odot}\cos l+\hfill\hbox to0.0pt{\hss(}8)\cr
0.0pt{\hfil$\displaystyle\hfill+r[M_{\scriptscriptstyle 23}\cos l\sin
b-M_{\scriptscriptstyle 13}\sin l\sin b+M_{\scriptscriptstyle
21}^{\scriptscriptstyle-}\cos b+M_{\scriptscriptstyle
12}^{\scriptscriptstyle+}\cos 2l\cos b-\hfill\cr
0.0pt{\hfil$\displaystyle\hfill-0.5(M_{\scriptscriptstyle
11}^{\scriptscriptstyle+}-M_{\scriptscriptstyle 22}^{\scriptscriptstyle+})\sin
2l\cos b],\hfill\cr}}}}$
where there are only seven independent variables.
## INERTIALITY OF THE OPTICAL ICRS/HIPPARCOS SYSTEM
Table 1 presents all of the currently known results of comparing individual
programs with the catalogs of the ICRS/HIPPARCOS system.
First, we will briefly describe the results, most of which were used by
Kovalevsky et al. (1997) to calibrate the HIPPARCOS catalog and to estimate
its residual rotation relative to the system of extragalactic sources and by
Bobylev (2004b) to solve this problem.
(1) The NPM1 solution. We used the result of comparing the stellar proper
motions from the NPM1 (Klemola et al. 1994) and HIPPARCOS catalogs performed
by the Heidelberg group (Kovalevsky et al. 1997). This solution was obtained
in the range of magnitudes $10^{m}.511^{m}.5,$ where (Fig. 1 in Platais et al.
1998a) the HIPPARCOSNPM1 stellar proper motion differences have a “horizontal”
pattern near zero. In our opinion, the NPM1 proper motions in this magnitude
range are free from the influence of the magnitude equation, which is
significant in this catalog, to the greatest extent.
Table 1: Components of the residual rotation vector of the optical realization
of the ICRS/HIPPARCOS system relative to the inertial frame of reference
Method | $P_{x}/P_{y}/P_{z}$ | $N_{\star}$ | $N_{\hbox{\tiny area}}$ | $\omega_{x},$ mas yr-1 | $\omega_{y},$ mas yr-1 | $\omega_{z},$ mas yr-1
---|---|---|---|---|---|---
NPM1 | 10/5/16 | 2616 | 899 | $-0.76\pm 0.25$ | $+0.17\pm 0.20$ | $-0.85\pm 0.20$
NPM2 | (*) | 3519 | 347 | $-0.11\pm 0.20$ | $-0.19\pm 0.20$ | $-0.75\pm 0.28$
SPM2 | 22/9/28 | 9356 | 156 | $+0.10\pm 0.17$ | $+0.48\pm 0.14$ | $-0.17\pm 0.15$
Kiev | 1/1/1 | 415 | 154 | $-0.27\pm 0.80$ | $+0.15\pm 0.60$ | $-1.07\pm 0.80$
Potsdam | 2/1/3 | 256 | 24 | $+0.22\pm 0.52$ | $+0.43\pm 0.50$ | $+0.13\pm 0.48$
Bonn | 5/3/6 | 88 | 13 | $+0.16\pm 0.34$ | $-0.32\pm 0.25$ | $+0.17\pm 0.33$
HST | 0.1/0.1/0.05 | 78 | | $-1.60\pm 2.87$ | $-1.92\pm 1.54$ | $+2.26\pm 3.42$
EOP | 8/2/— | | | $-0.93\pm 0.28$ | $-0.32\pm 0.28$ | —
PUL2 | 3/1/4 | 1004 | 147 | $-0.98\pm 0.47$ | $-0.03\pm 0.38$ | $-1.66\pm 0.42$
XPM | 28/9/33 | $1\times 10^{6}$ | 1431 | $-0.06\pm 0.15$ | $+0.17\pm 0.14$ | $-0.84\pm 0.14$
VLBI-07 | 6/1/5 | 46 | | $-0.55\pm 0.34$ | $-0.02\pm 0.36$ | $+0.41\pm 0.37$
Minor Pl | 25/5/6 | 116 | | $+0.12\pm 0.08$ | $+0.66\pm 0.09$ | $-0.56\pm 0.16$
Mean 1 | | | | $-0.22\pm 0.19$ | $+0.14\pm 0.10$ | $-0.49\pm 0.23$
Mean 2 | | | | $-0.11\pm 0.14$ | $+0.24\pm 0.10$ | $-0.52\pm 0.16$
Note. (*)the NPM2 solution is not used, $N_{\star}$ is the number of
stars/asteroids, $N_{area}$ is the number of areas on the celestial sphere,
mean 1 is a simple mean (without HST), mean 2 is a weighted mean.
(2) The NPM2 solution. The stellar proper motions from the NPM2 and HIPPARCOS
catalogs were compared by Zhu (2003). However, there are no images of galaxies
on NPM2 photographic plates and, hence, these proper motions are relative.
Therefore, we do not use this solution to derive the mean values of
$\omega_{x},\omega_{y},\omega_{z}$. It is given in Table 1 to emphasize its
similarity to the NPM1 solution.
(3) The SPM2 solution. The stellar proper motions from the SPM2 (Platais et
al. 1998b) and HIPPARCOS catalogs were compared by Zhu (2001).
(4) The PUL2 solution. The parameters $\omega_{x},\omega_{y},\omega_{z}$ were
found by comparing the Pulkovo PUL2 photographic catalog (Bobylev et al. 2004)
and HIPPARCOS.
(5) The KIEV solution. The stellar proper motions from the GPM1 (Rybka and
Yatsenko 1997) and HIPPARCOS catalogs were compared by Kislyuk et al. (1997).
(6) The POTSDAM solution. The parameters $\omega_{x},\omega_{y},\omega_{z}$ of
the Potsdam program were taken from Hirte et al. (1996).
(7) The BONN solution. The results of the Bonn program are presented in
Geffert et al. (1997) and Tucholke et al. (1997).
(8) The EOP solution. The results of the analysis of Earth orientation
parameters (EOP) were taken from Vondrák et al. (1997). Only two rotation
parameters, $\omega_{x},$ and $\omega_{y}$ are determined in this method.
(9) The HST solution. The results of stellar observations with the Hubble
Space Telescope (HST) were taken from Hemenway et al. (1997).
Now, we will point out several new results that have not been used previously
to solve this problem.
(10) The XPM solution. The XPM catalog (Fedorov et al., 2009) contains
absolute proper motions for about 275 million stars fainter than 12m derived
by comparing their positions in the 2MASS and USNO-A2.0 (Monet 1998) catalogs.
The absolutization was made using about 1.5 million galaxies from the 2MASS
catalog of extended sources. Thus, the XPM catalog is an independent
realization of the inertial frame of reference. The stellar proper motions
from the XPM and UCAC2 (Zacharias et al. 2004) catalogs were compared by
Bobylev et al. (2010). The parameters $\omega_{x},\omega_{y},\omega_{z}$ were
calculated using about 1 million stars. Among all of the programs listed in
Table 1, the XPM solution is unique in that it was obtained from the
differences of stars covering the entire sky almost completely, except for the
zone $\delta>54^{\circ},$ where there are no UCAC2 stars.
(11) The VLBI-07 solution. Boboltz et al. (2007) analyzed the positions and
proper motions of 46 radio stars and obtained new parameters of the mutual
orientation of the optical realization (HIPPARCOS) and the radio system.
(12) The “MINOR PLANETS” solution. Chernetenko (2008) estimated the rotation
parameters of the HIPPARCOS system relative to the DE403 and DE405 coordinate
systems of ephemerides by analyzing a long-term series of asteroid
observations. This result suggests that either the dynamical DE403 and DE405
theories need to be improved or the HIPPARCOS system needs to be corrected. We
reduced the weight of this solution by half because of the possible
contribution from the inaccuracy of the dynamical DE403 and DE405 theories.
The weight of each of the comparison catalogs was taken to be inversely
proportional to the square of the random error $e_{\omega}$ in the
corresponding quantities $\omega_{x},\omega_{y},\omega_{z}$ and was calculated
from the formula
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill P_{i}={e_{kiev}}^{2}/{{e_{i}}^{2}},\quad
i=1,...,11.\hfill\hbox to0.0pt{\hss(}9)\crcr}}$
Not all of the authors use the equations to determine
$\omega_{x},\omega_{y},\omega_{z}$ in the form in which they were suggested by
Lindegren and Kovalevsky (1995):
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill\Delta\mu_{\alpha}\cos\delta=\omega_{x}\cos\alpha\sin\delta+\omega_{y}\sin\alpha\sin\delta-\omega_{z}\cos\delta,\hfill\hbox
to0.0pt{\hss(}10)\cr
0.0pt{\hfil$\displaystyle\hfill\Delta\mu_{\delta}=-\omega_{x}\sin\alpha+\omega_{y}\cos\alpha,\hfill\hbox
to0.0pt{\hss(}11)\crcr}}}$
where the catalog–HIPPARCOS differences are on the left-hand sides of the
equations. Therefore, in several cases, the signs of the quoted quantities
were reduced to the necessary uniform form (Zhu 2001, 2003; Boboltz et al.
2007).
The last rows of Table 1 give Mean 1 calculated as a simple mean and Mean 2
that was calculated as a weighted mean and is the main result of our analysis.
Denote the components of the rotation vector around the rectangular equatorial
axes by $\omega_{x},\omega_{y},\omega_{z}$; then,
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill\pmatrix{\Omega_{x}\cr\Omega_{y}\cr\Omega_{z}\cr}={\bf
G}\pmatrix{M_{\scriptscriptstyle 32}^{\scriptscriptstyle-}\cr
M_{\scriptscriptstyle 13}^{\scriptscriptstyle-}\cr M_{\scriptscriptstyle
21}^{\scriptscriptstyle-}\cr}+4.74\pmatrix{\omega_{x}\cr\omega_{y}\cr-\omega_{z}\cr},\hfill\hbox
to0.0pt{\hss(}12)\cr}}$
where
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill{\bf
G}=\pmatrix{-0.0548&+0.4941&-0.8677\cr-0.8734&-0.4448&-0.1981\cr-0.4838&+0.7470&+0.4560\cr}\hfill\hbox
to0.0pt{\hss(}13)\cr}}$
is the well-known transformation matrix between the unit vectors of the
Galactic and equatorial coordinate systems. From Eqs. (9) and (10), it is easy
to see the relationship between $M_{\scriptscriptstyle
13}^{\scriptscriptstyle-}$ and $\omega_{z}$. Assuming the components in the
mean-2 solution to be zero,
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill\pmatrix{\Omega_{x}\cr\Omega_{y}\cr\Omega_{z}\cr}={\bf
G}\pmatrix{M_{\scriptscriptstyle 32}^{\scriptscriptstyle-}\cr
M_{\scriptscriptstyle 13}^{\scriptscriptstyle-}\cr M_{\scriptscriptstyle
21}^{\scriptscriptstyle-}\cr}+4.74\pmatrix{0\cr 0\cr 0.52\cr},\hfill\hbox
to0.0pt{\hss(}14)\cr}}$
for the case where the left- and right-hand sides are expressed in km s-1
kpc-1.
## KINEMATICS OF THE DISK WARP
The results of determining the kinematic parameters of the OgorodnikovMilne
model using a sample of RGC stars derived by simultaneously solving the system
of equations (6) and (7) are presented in Table 2. The second column gives the
solution obtained without applying any corrections to the input data; the
third column gives the solution for the case where the stellar proper motions
in the form $\mu_{\alpha}\cos\delta$ were corrected by applying the correction
$\omega_{z}=-0.52$ mas yr-1 (14) using Eq. (10).
The parameters derived by separately solving Eqs. (8) and (7) are presented in
Table 3.
As can be seen from Table 2, applying the correction affected only three
components of the rotation tensor: insignificantly $M_{\scriptscriptstyle
21}^{\scriptscriptstyle-}$, noticeably $M_{\scriptscriptstyle
32}^{\scriptscriptstyle-}$, and most strongly $M_{\scriptscriptstyle
13}^{\scriptscriptstyle-}$, which is explained by the structure of the matrix
G (13). Since the components $M_{\scriptscriptstyle 13}^{\scriptscriptstyle-}$
and $M_{\scriptscriptstyle 13}^{\scriptscriptstyle+}$ found (the third column
of Table 2) do not differ significantly from zero, they may be set equal to
zero.
Suppose that the values of $M_{\scriptscriptstyle 21}^{\scriptscriptstyle-}$
and $M_{\scriptscriptstyle 21}^{\scriptscriptstyle+}$ (the Oort constants)
found describe only the rotation around the Galactic $z$ axis, while the
motion in the $yz$ plane is independent. Let us now consider the displacement
tensor $M_{W}$ that describes the kinematics in the $yz$ plane:
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill M_{W}=\pmatrix{M_{\scriptscriptstyle
22}&M_{\scriptscriptstyle 23}\cr M_{\scriptscriptstyle
32}&M_{\scriptscriptstyle 33}\cr}=\pmatrix{{\displaystyle\partial
u_{2}}\over{\displaystyle\partial r_{2}}&{\displaystyle\partial
u_{2}}\over{\displaystyle\partial r_{3}}\cr{\displaystyle\partial
u_{3}}\over{\displaystyle\partial r_{2}}&{\displaystyle\partial
u_{3}}\over{\displaystyle\partial r_{3}}\cr}.\hfill\hbox
to0.0pt{\hss(}15)\crcr}}$
As has already been noted, when using only the stellar proper motions, we can
determine only the difference $(M_{\scriptscriptstyle
33}^{\scriptscriptstyle+}-M_{\scriptscriptstyle 22}^{\scriptscriptstyle+})$.
The identity $(M_{\scriptscriptstyle
33}^{\scriptscriptstyle+}-M_{\scriptscriptstyle
22}^{\scriptscriptstyle+})\equiv(M_{\scriptscriptstyle
33}-M_{\scriptscriptstyle 22})$ is valid for the diagonal elements. As can be
seen from Table 2, the value of this quantity differs significantly from zero.
Three cases are possible when analyzing tensor (15):
1) $M_{\scriptscriptstyle 22}\neq 0$, $M_{\scriptscriptstyle 33}=0$;
2) $M_{\scriptscriptstyle 22}=0$, $M_{\scriptscriptstyle 33}\neq 0$;
3) $M_{\scriptscriptstyle 22}=0$, $M_{\scriptscriptstyle 33}=0$.
For the completeness of the picture, note that case 4 is also possible:
$M_{\scriptscriptstyle 22}\neq 0$, $M_{\scriptscriptstyle 33}\neq 0$; since
this requires data on the stellar radial velocities, this case is not
considered here.
Consider case 1, $M_{\scriptscriptstyle 33}=0,$, using the data from the third
column of Table 2, $M_{\scriptscriptstyle 22}=1.3\pm 0.4$ km s-1 kpc-1. The
components of the displacement tensor $M_{W}$, the symmetric deformation
tensor $M_{W}^{\scriptscriptstyle+}$, and the antisymmetric rotation tensor
$M_{W}^{\scriptscriptstyle-}$ are (km s-1 kpc-1)
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill
M_{W}=\pmatrix{1.3_{(0.4)}&3.6_{(0.3)}\cr-1.6_{(0.3)}&0\cr},\hfill\hbox
to0.0pt{\hss(}16)\cr
0.0pt{\hfil$\displaystyle\hfill~{}~{}M_{W}^{\scriptscriptstyle+}=\pmatrix{1.3_{(0.4)}&1.0_{(0.2)}\cr
1.0_{(0.2)}&0\cr},\hfill\hbox to0.0pt{\hss(}17)\cr
0.0pt{\hfil$\displaystyle\hfill
M_{W}^{\scriptscriptstyle-}=\pmatrix{0&-2.6_{(0.2)}\cr-2.6_{(0.2)}&0\cr}.\hfill\hbox
to0.0pt{\hss(}18)\crcr}}}}$
The deformation tensor $M_{W}^{\scriptscriptstyle+}$ in the principal axes is
(km s-1 kpc-1):
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill
M_{W}^{\scriptscriptstyle+}=\pmatrix{\lambda_{1}&0\cr
0&\lambda_{2}\cr}=\hfill\cr 0.0pt{\hfil$\displaystyle\hfill=\pmatrix{1.8&0\cr
0&-0.5\cr},\hfill\crcr}}}$
and the angle between the positive direction of the $0y$ axis and the first
principal axis of this ellipse is $29\pm 6^{\circ}$.
Table 2: Kinematic parameters of the OgorodnikovMilne model found by
simultaneously solving the system of equations (6) and (7)
Parameter | Without correction | With correction
---|---|---
$X_{\odot}$ | $7.99\pm 0.10$ | $7.99\pm 0.10$
$Y_{\odot}$ | $16.40\pm 0.10$ | $16.40\pm 0.10$
$Z_{\odot}$ | $6.72\pm 0.09$ | $6.72\pm 0.09$
$A=M_{21}^{+}$ | $15.82\pm 0.21$ | $15.82\pm 0.21$
$M_{32}^{-}$ | $-1.40\pm 0.18$ | $-2.60\pm 0.18$
$M_{13}^{-}$ | $-1.99\pm 0.18$ | $-0.15\pm 0.18$
$B=M_{21}^{-}$ | $-11.99\pm 0.15$ | $-10.87\pm 0.15$
$M_{11-22}^{+}$ | $-7.75\pm 0.39$ | $-7.75\pm 0.39$
$M_{13}^{+}$ | $-0.47\pm 0.23$ | $-0.47\pm 0.23$
$M_{23}^{+}$ | $1.00\pm 0.22$ | $1.00\pm 0.22$
$M_{33-22}^{+}$ | $-1.26\pm 0.44$ | $-1.26\pm 0.44$
Note: $X_{\odot},Y_{\odot},Z_{\odot}$ in km s-1, other parameters in km s-1
kpc-1.
Table 3: Kinematic parameters of the OgorodnikovMilne model found by
separately solving Eqs. (8) and (7)
Parameter | Without correctiona | With correctiona | Without correctionb | With correctionb
---|---|---|---|---
$X_{\odot}$ | $7.89\pm 0.12$ | $7.89\pm 0.12$ | $8.51\pm 0.21$ | $8.51\pm 0.21$
$Y_{\odot}$ | $16.13\pm 0.13$ | $16.13\pm 0.13$ | $17.40\pm 0.21$ | $17.40\pm 0.21$
$Z_{\odot}$ | — | — | $6.73\pm 0.08$ | $6.73\pm 0.08$
$A=M_{21}^{+}$ | $15.76\pm 0.25$ | $15.76\pm 0.25$ | $15.99\pm 0.48$ | $15.99\pm 0.48$
$M_{32}^{-}$ | — | — | $-1.47\pm 0.25$ | $-2.66\pm 0.25$
$M_{13}^{-}$ | — | — | $-1.46\pm 0.25$ | $0.38\pm 0.25$
$B=M_{21}^{-}$ | $-12.02\pm 0.17$ | $-10.91\pm 0.17$ | — | —
$M_{11-22}^{+}$ | $-8.21\pm 0.46$ | $-8.21\pm 0.46$ | $-4.31\pm 1.03$ | $-4.31\pm 1.03$
$M_{13}^{+}$ | — | — | $0.17\pm 0.34$ | $0.17\pm 0.34$
$M_{23}^{+}$ | — | — | $1.11\pm 0.32$ | $1.11\pm 0.32$
$M_{33-22}^{+}$ | — | — | $0.29\pm 0.59$ | $0.29\pm 0.59$
$M_{23}$ | $2.24\pm 0.46$ | $3.43\pm 0.46$ | — | —
$M_{13}$ | $-2.93\pm 0.43$ | $-1.09\pm 0.43$ | — | —
aThe parameters were found by solving Eq. (8).
bThe parameters were found by solving Eq. (7).
Note: $X_{\odot},Y_{\odot},Z_{\odot}$ in km s-1, other parameters in km s-1
kpc-1.
Equation (1) can be written as
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill{\bf V}={\bf V}_{\circ}+{\rm
grad}~{}F+({\hbox{\boldmath$\omega$}}\times{\bf r}).\hfill\crcr}}$
Analysis of grad $F$ shows (Sedov 1970) that an infinitesimal sphere of radius
$r$ composed of points of the medium at time $t$,
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill x^{2}+y^{2}+z^{2}=r^{2}\hfill\crcr}}$
transforms into a deformation ellipsoid after time $\Delta t$,
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill{x^{*2}\over{(1+\lambda_{1}\Delta
t)^{2}}}+{y^{*2}\over{(1+\lambda_{2}\Delta
t)^{2}}}+{z^{*2}\over{(1+\lambda_{3}\Delta t)^{2}}}=r^{2},\hfill\crcr}}$
which is shown in Fig. 2b for our two-dimensional case.
For case 2, we have $M_{\scriptscriptstyle 33}=-1.3\pm 0.4$ km s-1 kpc-1 and
$M_{\scriptscriptstyle 22}=0$. The deformation tensor
$M_{W}^{\scriptscriptstyle+}$ in the principal axes is (km s-1 kpc-1)
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle M_{W}^{\scriptscriptstyle+}=\pmatrix{0.5&0\cr
0&-1.8\cr},\crcr}}$
the angle between the positive direction of the $0y$ axis and the first
principal axis of this ellipse is $29\pm 6^{\circ}$.
For case 3, where $M_{\scriptscriptstyle 33}=0$ and $M_{\scriptscriptstyle
22}=0$, the deformation tensor $M_{W}^{\scriptscriptstyle+}$ has two roots:
$\lambda_{1}=1.0$ km s-1 kpc-1 and $\lambda_{2}=-1.0$ km s-1 kpc-1; therefore,
the first principal axis of this ellipse is oriented at an angle of
$45^{\circ}$ to the $0y$ axis.
In all three cases, the divergence $0.5(M_{\scriptscriptstyle
22}+M_{\scriptscriptstyle 33})$ is insignificant. It is $+0.6\pm 0.4$, and 0
km s-1 kpc-1 for the first, second, and third cases, respectively.
As a result, we can conclude that the kinematics in the $yz$ plane can be
described as a rotation around the Galactic $x$ axis with an angular velocity
$\Omega_{W}=M_{\scriptscriptstyle 32}^{\scriptscriptstyle-}-\lambda_{1}$. This
angular velocity is $-4.4\pm 0.5$, $-3.1\pm 0.5$, and $-3.6\pm 0.3$ km s-1
kpc-1 for the first, second, and third cases, respectively.
Fig. 2. Distribution of rotation velocity vectors (a) and deformation velocity
vectors (b) in the $yz$ plane.
Let us estimate the linear velocities in the $yz$ plane. The mean photometric
distance for our sample of RGC stars is ${\overline{r}}=0.57\pm 0.17$ kpc. The
linear solidbody rotation velocity is then $M_{\scriptscriptstyle
32}^{\scriptscriptstyle-}\cdot{\overline{r}}=-1.5\pm 0.1$ km s-1. The maximum
deformation velocity is $\lambda_{1}\cdot{\overline{r}}=1.0\pm 0.3$ km s-1 and
the highest (among the cases considered) linear velocity is
$\Omega_{W}\cdot{\overline{r}}=-2.5\pm 0.3$ km s-1.
Figure 2 gives a qualitative picture that reflects the pattern of our results.
The distribution of rotation velocity vectors in the $yz$ plane for the case
of rotation we found is shown in Fig. 2a. Figure 2b shows how a circumference
of unit radius turns into a deformation ellipse after a unit time interval.
According to the results of our separate solution of Eqs. (8) and (7) (the
third and fifth columns of Table 3), for example, for case 1, we have
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hfill
M_{W}=\pmatrix{-0.3_{(0.6)}&3.8_{(0.5)}\cr-1.6_{(0.4)}&0\cr},\hfill\hbox
to0.0pt{\hss(}19)\cr
0.0pt{\hfil$\displaystyle\hfill~{}~{}M_{W}^{\scriptscriptstyle+}=\pmatrix{-0.3_{(0.6)}&1.1_{(0.3)}\cr
1.1_{(0.3)}&0\cr},\hfill\hbox to0.0pt{\hss(}20)\cr
0.0pt{\hfil$\displaystyle\hfill
M_{W}^{\scriptscriptstyle-}=\pmatrix{0&-2.7_{(0.3)}\cr-2.7_{(0.3)}&0\cr}.\hfill\hbox
to0.0pt{\hss(}21)\crcr}}}}$
We can see from our comparison of matrixes (19) and (16), (20) and (17), (21)
and (18) that there are differences only in the diagonal element. This leads
us to conclude that both approaches used (the simultaneous and separate
solution of the equations) are equivalent, because they yield coincident
results.
When writing Eq. (14), we set the components equal to zero, $\omega_{x}=0$ and
$\omega_{y}=0$. As we see from the Mean 2 solution in Table 1, the value of
the component $\omega_{y}$ was found to be nonzero outside the $2\sigma$
range. However, the kinematic parameters calculated by taking this into
account, i.e., $\omega_{y}\neq 0$ and $\omega_{z}\neq 0$, have no significant
differences between the solutions reflected in Tables 2 and 3 and, hence, we
do not give them.
## DISCUSSION
Drimmel et al. (2000) provide arguments for the model of precession of the
warped disk in the $zx$ plane (rotation around the y axis) with an angular
velocity of $-25$ km s-1 kpc-1. However, as can be seen from Table 2, as a
result of applying the correction $\omega_{z}=-0.52$ mas yr-1, the component
of the rotation tensor around the $y$ axis ($M_{\scriptscriptstyle
13}^{\scriptscriptstyle-}$) has an almost zero value and the component of the
deformation tensor in this plane ($M_{\scriptscriptstyle
13}^{\scriptscriptstyle+}$) does not differ significantly from zero.
The angular velocity $\Omega_{W}$ estimated here is inconsistent with the
analysis of the motion of giants of various spectral types (O–M) performed by
Miyamoto et al. (1993), who found a positive direction of rotation around the
$x$ axis (directed toward the Galactic center). Note that the analyzed stellar
proper motions were determined in the FK5 (Fundamental Catalog 5) system,
which is noticeably distorted by the uncertainty in the precession constant.
On the other hand, having analyzed the proper motions of O–B5 stars in the
HIPPARCOS system,Miyamoto and Zhu (1998) also reached the conclusion about a
positive rotation around the $x$ axis.
At present, it is impossible to choose between the three forms of the
deformation tensor considered in the previous section due to the absence of
data on stellar radial velocities and distances. Note that based on a sample
of 3632 relatively close RGC stars with known HIPPARCOS trigonometric
parallaxes and radial velocities, we estimated the parameters
$M_{\scriptscriptstyle 22}=-1.3\pm 3.2$ km s-1 kpc-1 and
$M_{\scriptscriptstyle 33}=-1.1\pm 2.6$ km s-1 kpc-1 (Bobylev et al. 2009).
Consequently, the difference $(M_{\scriptscriptstyle
33}^{\scriptscriptstyle+}-M_{\scriptscriptstyle 22}^{\scriptscriptstyle+})$ is
very close to zero and, hence, the orientation of the deformation ellipse is
close to $45^{\circ}$, as in case 3 considered.
In the Introduction, we listed various hypotheses about the nature of the
Galactic disk warp. In our opinion, the hypothesis about an outflow of gas
from the Magellanic Clouds (Olano 2004) has been worked out in greatest
detail. Assuming the distance to the Large Magellanic Cloud to be $r_{LMC}=50$
kpc, we find that the linear velocity of the rotation we revealed at this
distance, $(-3.1\pm 0.5)\leq\Omega_{W}\leq(-4.4\pm 0.5)$ km s-1 kpc-1, is
$|\Omega_{W}|\cdot r_{LMC}=(155\div 220)\pm 25$ km s-1. Such velocities are in
agreement with the estimate of the motion of high-velocity hydrogen clouds
relative to the local standard of rest, $\approx 200$ km s-1 (Olano 2004), and
the rotation direction we found is in good agreement with the direction of
motion of the Magellanic Clouds around the Galaxy during the past 570 Myr
(Fig. 6 from Olano 2004).
## CONCLUSIONS
We analyzed the kinematics of about 82000 RGC stars from the Tycho-2 catalog.
These stars are a variety of “standard candles”, because they occupy a compact
region on the Hertzsprung-Russell diagram. Therefore, photometric distance
estimates with an accuracy of at least 30% are available for them. The RGC
stars considered lie in the range of heliocentric distances 0.3 kpc $<d<1$
kpc.
Since these stars are distributed fairly uniformly in space, they are of great
interest not only in studying the Galaxys general rotation described by the
rotation parameters in the Galactic $xy$ plane but also in the other two
planes, namely $yz$ and $zx$.
For a reliable description of their kinematic peculiarities, we should be
confident that the observational data (Tycho-2 proper motions) are free from
the systematic errors related to the referencing of the optical realization of
the ICRS/HIPPARCOS system to the inertial frame of reference specified by
extragalactic sources.
Here, we gave much attention to this problem. Based on all of the currently
available data, we determined new, most probable components of the residual
rotation vector of the optical realization of the ICRS/HIPPARCOS system
relative to the inertial frame of reference,
$(\omega_{x},\omega_{y},\omega_{z})=(-0.11,0.24,-0.52)\pm(0.14,0.10,0.16)$ mas
yr-1. This led us to conclude that a small correction, $\omega_{z}=-0.52$ mas
yr-1, should be applied to the Tycho-2 stellar proper motions of the form
$\mu_{\alpha}\cos\delta$.
By applying this correction, we showed that, apart from their involvement in
the general Galactic rotation described by the Oort constants $A=15.8\pm 0.2$
km s-1 kpc-1 and $B=-10.9\pm 0.2$ km s-1 kpc-1, the RGC stars considered have
peculiarities in the $yz$ plane related, in our opinion, to the kinematics of
the warped Galactic stellar-gaseous disk.
The component of the solid-body rotation of the local solar neighborhood
around the Galactic x axis is $M_{\scriptscriptstyle
32}^{\scriptscriptstyle-}=-2.6\pm 0.2$ km s-1 kpc-1. The parameter of the
deformation tensor in this plane $M_{\scriptscriptstyle
23}^{\scriptscriptstyle+}=1.0\pm 0.2$ km s-1 kpc-1 and the difference
$M_{\scriptscriptstyle 33}-M_{\scriptscriptstyle 22}=-1.3\pm 0.4$ km s-1 kpc-1
differ significantly from zero.
On the whole, the kinematics of the warped Galactic stellargaseous disk in the
local solar neighborhood can be described as a rotation around the Galactic x
axis directed from the Sun toward the Galactic center with an angular velocity
$(-3.1\pm 0.5)\leq\Omega_{W}\leq(-4.4\pm 0.5)$ km s-1 kpc-1. Thus, the
rotation is around an axis close to the line of nodes of this structure. In
the case where the difference $M_{\scriptscriptstyle 33}-M_{\scriptscriptstyle
22}$ may be set equal to zero, the angular velocity $\Omega_{W}$ is $-3.6\pm
0.3$ km s-1 kpc-1. If $M_{\scriptscriptstyle 22}=0$ and $M_{\scriptscriptstyle
33}\neq 0$, then $\Omega_{W}=-3.1\pm 0.5$ km s-1 kpc-1. If
$M_{\scriptscriptstyle 22}\neq 0$ and $M_{\scriptscriptstyle 33}=0$, then
$\Omega_{W}$ reaches $-4.4\pm 0.5$ km s-1 kpc-1.
ACKNOWLEDGMENTS
I am grateful to the referees for helpful remarks that contributed to an
improvement of the paper and to A.T. Bajkova for her help in the work. This
study was supported by the Russian Foundation for Basic Research (project nos.
08–02–00400 and 09–02–90443–Ukr_f) and, in part, by the “Origin and Evolution
of Stars and Galaxies” Program of the Presidium of the Russian Academy of
Sciences.
REFERENCES
1\. J. Bailin, Astrophys. J. 583, L79 (2003).
2\. E. Battaner, E. Florido, and M.L. Sanchez-Saavedra, Astron. Astrophys.
236, 1 (1990).
3\. D.A. Boboltz, A.L. Fey, W.K. Puatua, et al., Astron. J. 133, 906 (2007).
4\. V.V. Bobylev, Pisma Astron. Zh. 30, 289 (2004a) [Astron. Lett. 30, 251
(2004)].
5\. V.V. Bobylev, Pisma Astron. Zh. 30, 930 (2004b) [Astron. Lett. 30, 848
(2004)].
6\. V.V. Bobylev, N.M. Bronnikova, and N.A. Shakht, Pisma Astron. Zh. 30, 519
(2004) [Astron. Lett. 30, 469 (2004)].
7\. V.V. Bobylev, A.S. Stepanishchev, A.T. Bajkova, and G.A. Gontcharov, Pisma
Astron. Zh. 35, 920 (2009) [Astron. Lett. 35, 836 (2009)].
8\. V.V. Bobylev, P.N. Fedorov, A.T. Bajkova, and V.S. Akhmetov, Pisma Astron.
Zh. 36, 535 (2010) [Astron. Lett. 36, 506 (2010)].
9\. W.B. Burton, Galactic and Extragalactic Radio Astronomy, Ed. by G.
Verschuur and K. Kellerman (Springer, New York, 1988), p. 295.
10\. J.C. Cersosimo, S. Mader, N. Santiago Figueroa, et al., Astrophys. J.
699, 469 (2009).
11\. Yu.A. Chernetenko, Pisma Astron. Zh. 34, 296 (2008) [Astron. Lett. 34,
266 (2008)].
12\. S.V.M. Clube, Mon. Not. R. Astron. Soc. 159, 289 (1972).
13\. S.V.M. Clube, Mon. Not. R. Astron. Soc. 161, 445 (1973).
14\. W. Dehnen, Astron. J. 115, 2384 (1998).
15\. R. Drimmel and D.N. Spergel, Astrophys. J. 556, 181 (2001).
16\. R. Drimmel, R.L. Smart, and M.G. Lattanzi, Astron. Astrophys. 354, 67
(2000).
17\. P.N. Fedorov, A.A. Myznikov, and V.S. Akhmetov, Mon. Not. R. Astron. Soc.
393, 133 (2009).
18\. M. Geffert, A.R. Klemola, M. Hiesgen, et al., Astron. Astrophys. 124, 157
(1997).
19\. G.A. Gontcharov, Pisma Astron. Zh. 34, 868 (2008) [Astron. Lett. 34, 785
(2008)].
20\. P.D. Hemenway, R.L. Duncombe, E.P. Bozyan, et al., Astron. J. 114, 2796
(1997).
21\. The HIPPARCOS and Tycho Catalogues, ESA SP-1200 (1997).
22\. S. Hirte, E. Schilbach, and R.-D. Scholz, Astron. Astrophys. Suppl. Ser.
126, 31 (1996).
23\. E. Hog, C. Fabricius, V.V. Makarov, et al., Astron. Astrophys. 355, L27
(2000).
24\. P.M.W. Kalberla and L. Dedes, Astron. Astrophys. 487, 951 (2008).
25\. V.S. Kislyuk, SP. Rybka, A.I. Yatsenko, et al., Astron. Astrophys. 321,
660 (1997).
26\. A.R. Klemola, R.B. Hanson, and B.F. Jones, in Galactic and Solar System
Optical Astrometry, Ed. by L.V. Morrison and G.F. Gilmore (Cambridge Univ.,
Cambridge, 1994), p. 20.
27\. J. Kovalevsky, L. Lindegren, M.A.C. Perryman, et al., Astron. Astrophys.
323, 620 (1997).
28\. L. Lindegren and J. Kovalevsky, Astron. Astrophys. 304, 189 (1995).
29\. M. López-Corredoira, J. Betancort-Rijo, and J. Beckman, Astron.
Astrophys. 386, 169 (2002).
30\. C. Ma, E.F. Arias, T.M. Eubanks, et al., Astron. J. 116, 516 (1998).
31\. M. Miyamoto and Z. Zhu, Astron. J. 115, 1483 (1998).
32\. M. Miyamoto, M. Sôma, and M. Yoshizawa, Astron. J. 105, 2138 (1993).
33\. Y. Momany, S. Zaggia, G. Gilmore, et al., Astron. Astrophys. 451, 515
(2006).
34\. D.G.Monet, Bull. Am. Astron. Soc. 30, 1427 (1998).
35\. K.F. Ogorodnikov, Dynamics of Stellar Systems (Fizmatgiz, Moscow, 1965)
[in Russian].
36\. C.A. Olano, Astron. Astrophys. 423, 895 (2004).
37\. I. Platais, V. Kozhurina-Platais, T.M. Girard, et al., Astron. Astrophys.
331, 1119 (1998a).
38\. I. Platais, T.M. Girard, V. Kozhurina-Platais, et al., Astron. J. 116,
2556 (1998b).
39\. S.P. Rybka, Kinem Fiz. Neb. Tel 22, 225 (2006).
40\. S.P. Rybka and A.I. Yatsenko, Astron. Astrophys. Suppl. Ser. 121, 243
(1997).
41\. L.I. Sedov, Mechanics of Continuous Media (Nauka, Moscow, 1970), Vol. 1
[in Russian].
42\. M.F. Skrutskie, R.M. Cutri, R. Stiening, et al., Astron. J. 131, 1163
(2006).
43\. L. Sparke and S. Casertano, Mon. Not. R. Astron. Soc. 234, 873 (1988).
44\. H.-J. Tucholke, P. Brosche, and M. Odenkirchen, Astron. Astrophys. Suppl.
Ser. 124, 157 (1997).
45\. V.V. Vityazev and A.S. Tsvetkov, Pisma Astron. Zh. 35, 114 (2009)
[Astron. Lett. 35, 100 (2009)].
46\. J. Vondrák, C. Ron, I. Pešek, Astron. Astrophys. 319, 1020 (1997).
47\. G. Westerhout, Bull. Astron. Inst. Netherlands 13, 201 (1957).
48\. S.K. Yi, Y.-C. Kim, and P. Demarque, Astrophys. J. Suppl. Ser. 144, 259
(2003).
49\. I. Yusifov, astro-ph/0405517 (2004).
50\. N. Zacharias, S.E. Urban, M.I. Zacharias, et al., Astron. J. 127, 3043
(2004).
51\. Zi Zhu, Publ. Astron. Soc. Jpn. 53, L33 (2001).
52\. Zi Zhu, in JOURNEÉS-2003, Astrometry, Geodynamics and Solar System
Dynamics: From Milliarcseconds to Microarcseconds, Ed. by N. Capitaine
(Observ. de Paris, Paris, 2003), p. 95.
Translated by N. Samus
|
arxiv-papers
| 2010-08-25T05:37:56 |
2024-09-04T02:49:12.454623
|
{
"license": "Public Domain",
"authors": "Vadim V. Bobylev",
"submitter": "Anisa Bajkova",
"url": "https://arxiv.org/abs/1008.4208"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.